diff --git "a/5NE1T4oBgHgl3EQfBAIU/content/tmp_files/load_file.txt" "b/5NE1T4oBgHgl3EQfBAIU/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/5NE1T4oBgHgl3EQfBAIU/content/tmp_files/load_file.txt" @@ -0,0 +1,1255 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf,len=1254 +page_content='A one-dimensional model for axisymmetric deformations of an inflated hyperelastic tube of finite wall thickness Xiang Yua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='˚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Yibin Fub aSchool of Computer Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Dongguan University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Dongguan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' China bSchool of Computing and Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Keele University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Staffordshire ST5 5BG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' UK Abstract We derive a one-dimensional (1d) model for the analysis of bulging or necking in an inflated hyper- elastic tube of finite wall thickness from the three-dimensional finite elasticity theory by applying the dimension reduction methodology proposed by Audoly and Hutchinson (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Solids, 97, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The 1d model makes it much easier to characterize fully nonlinear axisymmetric defor- mations of a thick-walled tube using simple numerical schemes such as the finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The new model recovers the diffuse interface model for analyzing bulging in a membrane tube and the 1d model for investigating necking in a stretched solid cylinder as two limiting cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is con- sistent with, but significantly refines, the exact linear and weakly nonlinear bifurcation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Comparisons with finite element simulations show that for the bulging problem, the 1d model is capable of describing the entire bulging process accurately, from initiation, growth, to propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The 1d model provides a stepping stone from which similar 1d models can be derived and used to study other effects such as anisotropy and electric loading, and other phenomena such as rupture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Keywords: localized bulging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' necking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' reduced models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' tubes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' stability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' nonlinear elasticity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Introduction Hyperelastic tubes are commonly found in various applications ranging from soft robotics (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2015, 2020) to energy harvesting (Lu & Suo, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Bucchi & Hearn, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Smith, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' They are also used to model human arteries in order to understand pathological conditions such as aneurysms (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Alhayani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Demirkoparan & Merodio, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Varatharajan & DasGupta, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Hejazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Inflation of a hyperelastic tube is one of the few boundary value problems in nonlinear elasticity that have closed-form solutions, and it provides the simplest setup to explain bifurcation, localization, loss of convexity, and “two-phase” deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus, understanding this problem is not only important for applications, but may also shed light on other more complicated stability and bifurcation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' ˚Corresponding author Email address: yuxiang@dgut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='cn (Xiang Yu) Preprint submitted to Journal of the Mechanics and Physics of Solids January 10, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='02845v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='soft] 7 Jan 2023 Simple inflation experiments with commercially available latex rubber tubes show that localized bulging is the dominant deformation form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For almost all realistic constitutive models for rubber, the pressure versus volume curve has an N shape under the condition of fixed resultant axial force (Green & Adkins, 1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This led Yin (1977) and Chater & Hutchinson (1984) to analyze the final observable configuration as that corresponding to a “two-phase” deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The subsequent experimental studies carried out by Kyriakides & Chang (1990, 1991), Pamplona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2006) and Goncalves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2008) have provided a clear picture on how a localized bulge initiates, grows and then propagates under fixed axial force or fixed-ends conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' When the membrane assumption is made, the governing equations for tube inflation can be viewed as a finite-dimensional spatial dynamical system that has two conservation laws/integrals (Pipkin, 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This realization enabled Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2008) to demonstrate explicitly how a localized solution initiates as a zero-wave-number mode from the uniform deformation and how it evolves into a “two-phase” state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The stability of bulging solutions and their sensitivity to imperfections have been studied under the same framework (Pearce & Fu, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Fu & Il’ichev, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Fu & Xie, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Fresh analytical insight into the case of fixed ends has also been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is shown that the bifurcation condition for this case corresponds to the axial force reaching a maximum at a fixed pressure (Fu & Il’ichev, 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' in other words, as pressure is increased, the critical pressure is the value of pressure at which the axial force reaches a maximum when viewed as a function of the axial stretch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Also, in contrast with the case of fixed axial force where the measured pressure approaches a constant value (the propagation pressure), the measured pressure in the case of fixed ends has an N shape where the right branch approaches a master curve that is independent of the pre-axial-stretch or the tube length (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In some practical applications, however, the tube wall may be of moderate or even large thickness and the membrane model no longer applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For example, in the context of aneurysm formation, a human artery can be as thick as a quarter of its outer radius (M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2008), and fiber- reinforcement also seems to reduce the range of validity of the membrane assumption (Wang & Fu, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus, recent studies have begun to consider hyperelastic tubes of finite wall thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2016) showed that the associated bifurcation condition for localized bulging corresponds to the vanishing of the Jacobian determinant of the internal pressure and resultant axial force as functions of the azimuthal stretch and the axial stretch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' see also Yu & Fu (2022) for an alternative derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This provides a framework under which additional effects such as rotation (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2017), double-fiber-reinforcement (Wang & Fu, 2018), bi-laying (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2019), torsion (Althobaiti, 2022), and surface tension (Emery & Fu, 2021a,b,c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Emery, 2023) can be assessed in a systematic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020) conducted a weakly non-linear analysis and derived the bulging solution explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The analytic predictions were corroborated by numerical simulations and experiments (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For tubes of finite wall thickness, the equations that govern their axisymmetric deformations are coupled nonlinear partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Although analytic solutions can be obtained in 2 the near-critical regime using asymptotic methods (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2020), the complexity of the governing equations forbids any further analytic attempts to understand the bulging evolution further away from the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The post-bifurcation behavior in the fully nonlinear regime has so far only been investigated by resorting to Abaqus simulations (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is not satisfactory since the insight provided by full-scale simulations tends to be limited and there are situations where repeated calculations of the bulging profile are required (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' in the assessment of the rupture potential (Hejazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A recent series of studies by Audoly and coworkers has opened the possibility that a 1d reduced model can be derived to describe the fully nonlinear evolution of bulging or necking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In the first of this series, Audoly & Hutchinson (2016), the authors derived a 1d model for tensile necking localization in a 3d prismatic solid of arbitrary cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The key idea of their derivation is a dimension reduction assuming slow variation in the axial direction that respects self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In terms of the language of perturbation analysis, the leading-order solution is almost correct and higher-order terms are only added to restore self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The method was later applied by Lestrigant and Audoly to obtain a diffuse interface model for the characterization of propagating bulges in membrane tubes (Lestringant & Audoly, 2018) and a 1d model for predicting surface tension-driven necking in soft elastic cylinders (Lestringant & Audoly, 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It has also been used recently to derive a 1d model for elastic ribbons (Audoly & Neukirch, 2021) and for tape springs (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The systematic reduction method for deriving 1d strain-gradient models for nonlinear slender structures was further generalized by Lestringant & Audoly (2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is worth pointing out that although the 1d models are built on the assumption that localized solutions vary slowly in the longitudinal direction, it is surprisingly accurate, even in the region where the localization is well developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is illustrated by the numeric examples in the aforementioned work and in the comparative studies by Wang & Fu (2021) and Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This work aims to extend the diffuse interface model of Lestringant & Audoly (2018) for mem- brane tubes to tubes of finite wall thickness, in a similar spirit as the previous work Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2016) and Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020) that extend the bifurcation condition and the weakly nonlinear analysis from membrane tubes to thick-walled tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In contrast with the case under the membrane assumption where the original governing equations are already one-dimensional, the governing equations for the current case are two-dimensional, and the uniformly inflated deformation is no longer homogeneous since the solution depends on the radial variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It will be shown that a 1d reduced model can still be derived and simplified to the form E1dras “ ż L ´L ´ Gpa, λpaqq ` 1 2Dpaqa1pZq2¯ dZ ` Cpaqa1pZq|L ´L, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) where L is the initial half length, Z is the axial coordinate, apZq is the azimuthal stretch on the inner surface (a constant multiple of the deformed inner radius ) and the expressions for Gpa, λpaqq, Dpaq and Cpaq are given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The first term G in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) corresponds to the energy of the uniform deformation, which determines the amplitudes of the two phases in 3 the bulge propagation stage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' the second term accounts for the contribution of the strain gradient to the total energy, which describes how these two phases are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The Euler-Lagrange equation associated with the energy functional (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) is a second-order nonlinear ode for apZq, which is a drastic simplification from the original nonlinear partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This 1d model is validated by comparison with finite element simulations, showing excellent agreement with numerical results even for the propagation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The outline of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 2, we formulate the 3d axisymmetric finite- strain model for a tube of finite wall thickness under inflation and axial stretching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 3, we summarize solutions corresponding to uniform inflation of the tube, making preparation for the subsequent dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4, we carry out the dimension reduction and derive the above-mentioned 1d strain-gradient model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The connection of the 1d model with prior work is given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 6, we validate the 1d model by making comparisons with finite element simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Finally, concluding remarks are given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Three-dimensional finite-strain model We consider a circular cylindrical tube that has a length 2L, inner radius A and outer radius B in its reference configuration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The ratio of the outer radius to the length ε “ B{2L is assumed to be small, thus ε !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The tube deforms axisymmetrically under the combined action of an internal pressure P and a resultant axial force N, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In terms of cylindrical coordinates, the current position vector of a representative point is given by x “ zpZ, Rqez ` rpZ, Rqer, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) where pR, Θ, Zq and pr, θ, zq are the coordinates of a representative point before and after defor- mation, and per, eθ, ezq are the standard basis vectors associated with both pR, Θ, Zq and pr, θ, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The deformation gradient related to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) is given by F “ r Reθ b eθ ` zZez b ez ` zRez b er ` rZer b ez ` rRer b er, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) where zZ :“ Bz{BZ, zR :“ Bz{BR, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We assume that the tube is made of an incompressible isotropic hyperelastic material, associated with the strain energy function Wpλ1, λ2, λ3q, where λ1, λ2, λ3 denote the three principal stretches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Throughout this paper, we identify the indices 1, 2, 3 such that in uniform inflation they coincide with the θ-, z- and r-directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The total potential energy of the tube is composed of the elastic energy and the load potential, which reads E “ ż L ´L ´ ż B A ` wpλ1, λ2q ´ N˚zZ ˘ 2πR dR ´ Pπr2zZ ˇˇ R“A ¯ dZ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) 4 m B G (a) m2 G2 H2 (b) Figure 1: A hyperelastic cylindrical tube of finite thickness in (a) reference (undeformed) configuration and (b) current configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' where wpλ1, λ2q “ Wpλ1, λ2, λ´1 1 λ´1 2 q is the reduced strain energy function and N˚ “ N{pπpB2 ´ A2qq is the resultant axial force per unit cross-sectional area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The elastic model governed by the energy functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) will be used as a starting point for the subsequent dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The governing equations for the two unknown functions rpZ, Rq and zpZ, Rq can be derived by setting the first variation of E to zero, but these equations are not required in the approach that we follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Uniform inflation We now summarise the solution that corresponds to uniform inflation and extension of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This solution will be referred to as the uniform solution and is indicated by a superposed bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For a more detailed derivation, see Haughton & Ogden (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' First, incompressibility implies that a uniform solution must necessarily be of the form ¯z “ λZ, ¯r “ a a2A2 ` λ´1pR2 ´ A2q, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) where λ and a denote the constant axial stretch and azimuthal stretch on the inner surface, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The three principal stretches are simply ¯λ1 “ ¯r R, ¯λ2 “ λ, ¯λ3 “ d¯r dR “ ¯λ´1 1 ¯λ´1 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) and the azimuthal stretch on the outer surface, denoted by b, is given by b “ ¯λ1|R“B “ a a2A2 ` λ´1pB2 ´ A2q B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) The three associated principal Cauchy stresses ¯σ11, ¯σ22 and ¯σ33 satisfy the relations ¯σ11 ´ ¯σ33 “ ¯λ1w1, ¯σ22 ´ ¯σ33 “ λw2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) 5 where w1 “ Bwp¯λ1, ¯λ2q{B¯λ1 and w2 “ Bwp¯λ1, ¯λ2q{B¯λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The only equilibrium equation that is not satisfied automatically is d¯σ33 d¯r “ ¯σ11 ´ ¯σ33 ¯r “ ¯λ1w1 ¯r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) On integrating this equation from R “ A to R “ B and making use of the boundary conditions that ¯σ33|R“A “ ´P and ¯σ33|R“B “ 0, we obtain P “ Qpa, λq :“ ż a b w1p¯λ1, λq ¯λ2 1λ ´ 1 d¯λ1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) where the second equation defines the function Qpa, λq and we have made use of the identity d¯r ¯r “ ´ d¯λ1 ¯λ1p¯λ2 1λ ´ 1q, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) which can be deduced from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The overall equilibrium in the axial direction implies Mpa, λq ´ 1 2a2P ´ N 2πA2 “ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) where Mpa, λq is given by Mpa, λq :“ 1 A2 ż B A λ´1¯σ22R dR “ ż a b p¯λ2 1 ´ a2qw1p¯λ1, λq ` 2¯λ1λpa2λ ´ 1qw2p¯λ1, λq 2p¯λ2 1λ ´ 1q2 d¯λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) In view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3), the total potential energy of the uniform deformation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) per unit reference length, after scaling by 2π, is Gpa, λq “ ż B A wp¯λ1, λq R dR ´ 1 2PA2a2λ ´ N 2πλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10) The equilibrium equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) can also be obtained from BG{Ba “ 0 and BG{Bλ “ 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Once the loads P and N are specified, the deformation parameters a and λ can be found by solving the equilibrium equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' On differentiating the left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) with respect to λ, we find that it takes the form Hw22pa, λq{A ` OpH2q, where H “ B ´ A is the thickness of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We assume that the strong ellipticity condition is satisfied pointwise which guarantees that w22pa, λq is positive (Knowles & Sternberg, 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This, combined with the implicit function theorem, implies that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) can be inverted to express λ in terms of a uniquely at least when H is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We assume that this remains true for arbitrary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This enables us to view (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) as an implicit equation for λ “ λpaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We remark that λ is also dependent on P, but this dependence is not indicated explicitly for notational brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus, by definition, λpaq is the solution to the implicit equation Mpa, λpaqq ´ 1 2a2P ´ N 2πA2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11) 6 Since λ has been viewed as a function of a, all quantities (except ¯z which also depends on Z) related to the uniform solution are functions of a and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For instance, ¯r denotes the function ¯rpa, Rq “ a a2A2 ` λpaq´1pR2 ´ A2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12) We denote ¯σ33 by ´qpa, Rq so that qpa, Rq :“ ´¯σ33 “ ż ¯λ1 b w1p˜λ1, λq ˜λ2 1λ ´ 1 d˜λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='13) We also define another function mpa, Rq through mpa, Rq :“ 1 R2 ż B R λ´1¯σ22RdR “ ż ¯λ1 b p˜λ2 1 ´ ¯λ2 1qw1p˜λ1, λq ` 2˜λ1λp¯λ2 1λ ´ 1qw2p˜λ1, λq 2p˜λ2 1λ ´ 1q2 d˜λ1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) and record the connections qpa, Aq “ Qpa, λpaqq, mpa, Aq “ Mpa, λpaqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15) The 1d reduced model to be derived in the next section will be expressed in terms of the two functions qpa, Rq and mpa, Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The integrals in these two functions can be evaluated explicitly for some commonly used strain energy functions, including the Gent material model that will be used in our illustrative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Derivation of the one-dimensional model In this section, we apply the dimension reduction methodology proposed by Audoly & Hutchin- son (2016) to derive a one-dimensional model from the full three-dimensional theory formulated in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Optimal correction We start our dimension reduction by assuming that all dependent variables related to the axisymmetric configuration vary slowly in the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' More precisely, it is assumed that all variables depend on Z through the “far distance” variable S “ εZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) Recall that ε is the ratio of the outer radius to the length, which is assumed to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In particular, we now allow a and λ to depend on S and write a “ apSq, λ “ λpapSqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Our aim is to derive a reduced model, an ordinary differential equation, that is satisfied by apSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We recall that apSq is the deformed inner radius divided by a constant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A naive approach would be to use a “ apSq and λ “ λpapSqq to compute the two principal stretches and then derive the equation satisfied by a “ apSq by minimizing the energy functional 7 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This would yield an equation for apSq that is not self-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The correct way is to allow for higher-order correction terms by looking for an asymptotic solution of the form zpZ, Rq “ 1 ε ż S 0 λpapTqq dT ` εv˚pS, Rq ` Opε3q, rpZ, Rq “ ¯rpapSq, Rq ` ε2u˚pS, Rq ` Opε4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) We note that the correction terms in zpZ, Rq and rpZ, Rq are of order ε and ε2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is because the Op1q-term in zpZ, Rq and the Opεq-term in rpZ, Rq correspond to a uniform perturbation and can thus be absorbed into the leading terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' On substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) and truncating at order ε2, we obtain the deformation gradient F “ ¨ ˚ ˝ ¯r{R ` ε2u˚{R 0 0 0 λpapSqq ` ε2v˚ S εv˚ R 0 ε¯raa1pSq ¯rR ` ε2u˚ R ˛ ‹‚, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) where the subscripts represent partial differentiation with respect to the indicated variables (in particular ¯ra “ B¯r{Ba).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Consequently, the three principal stretches are given by λ1 “ ¯λ1 ` ε2 u˚ R , λ2 “ ¯λ2 ` ε2´ v˚ S ` λp¯r2 aa1pSq2 ` v˚2 R q ` 2¯λ3¯raa1pSqv˚ R 2pλ2 ´ ¯λ3q ¯ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) where ¯λ1 and ¯λ2 are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) but with a and λ replaced by apSq and λpapSqq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) and expanding to second order, we see that E can be written, in terms of the un-scaled variables, as E “ 2π ´ ż L ´L GpapZq, λpapZqqq dZ ` E2 ¯ ` OpLε3q, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) where E2 represents the term of order ε2 and is given by E2 “ ż L ´L ´ ż B A ´ pw2 ´ N˚qvZ ` w2 λp¯r2 aa12 ` v2 Rq ` 2¯λ3¯raa1vR 2pλ2 ´ ¯λ2 3q ¯ R dR ` ż B A w1u dR ´ 1 2PA2a2vZ|R“A ´ PAaλu|R“A ¯ dZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) In the above expression, vpZ, Rq “ εv˚pS, Rq and upZ, Rq “ ε2u˚pS, Rq denote the unscaled dis- placements, and here and hereafter we write apZq for apSq and so a1 now denotes a1pZq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is seen that the only reason for introducing S above is to identify all terms of order ε2 that should be kept in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' With this task accomplished, the scaled variable S will no longer appear in the subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Also, w1 “ w1p¯λ1, λq, w2 “ w2p¯λ1, λq in which λ is a function of a and ¯λ1 is a function of a and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 8 Our formulation in terms of the reduced strain energy function requires the solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) to satisfy the incompressibility condition automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This can be achieved by eliminating u in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) with the use of detpF q “ 1 which takes the form λp¯ruqR ` ¯rp¯rRvZ ´ ¯raa1vRq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) To this end, we make use of the equilibrium equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) and write ż B A w1u dR “ λ ż B A ¯σ33,R¯ru dR “ λ¯σ33¯ru|B A ´ λ ż B A ¯σ33p¯ruqR dR “ λPAau|R“A ´ ż B A q¯rp¯rRvZ ´ ¯raa1vRq dR, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) where we have replaced ¯σ33 by ´qpa, Rq (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='13)) and have used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) to eliminate p¯ruqR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' On eliminating u in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) with the use of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8), we can recast E2 in the form E2 “ ż L ´L ´ ż B A ` pλ´1¯σ22 ´ N˚qvZ ` 1 2ζp¯r2 aa12 ` v2 Rq ` ξ¯raa1vR ˘ R dR ´ 1 2PA2a2vZ|R“A ¯ dZ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) where ζ “ λw2{pλ2 ´ ¯λ2 3q, ξ “ q¯λ1 ` ¯λ3ζ{λ, and we have made use of the connection λw2 ´ q “ ¯σ22 that follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4)2 with ¯σ33 “ ´q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Then upon using integration by parts, we obtain E2 “ ż L ´L ´ ż B A KpR, v, vRq dR ` PA2aa1v|R“A ¯ dZ ` ´ ż B A pλ´1¯σ22 ´ N˚qvR dR ´ 1 2PA2a2v|R“A ¯ˇˇˇ Z“L Z“´L, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10) where KpR, v, vRq is given by KpR, v, vRq “ ´pλ´1¯σ22qaa1Rv ` 1 2Rζp¯r2 aa12 ` v2 Rq ` Rξ¯raa1vR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11) In the last expression, pλ´1¯σ22qa denotes the partial derivative of λ´1¯σ22 with respect to a with R fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' To find the remaining correction field v “ vpZ, Rq, we assume that the leading-order stretch apZq is prescribed and seek the correction v such that the total potential energy is stationary (Audoly & Hutchinson, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' As a result, the optimal v satisfies the Euler-Lagrange equation and the boundary conditions BK Bv ´ d dR ´ BK BvR ¯ “ 0, A ď R ď B, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12) BK BvR “ PA2aa1, R “ A, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='13) BK BvR “ 0, R “ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) 9 Solution of the above boundary value problem requires satisfaction of the solvability condition ż B A BK Bv dR “ ´PA2aa1, that is ż B A pλ´1¯σ22qaa1R dR “ PA2aa1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is automatically satisfied in view of the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) for Mpa, λq and the equilibrium con- dition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Written out explicitly, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) take the form d dRpRζvRq “ ´ ´ pλ´1¯σ22qaR ` d dRpRξ¯raq ¯ a1, A ď R ď B, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15) RζvR “ ´Rξ¯raa1, R “ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16) Integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15) subject to the boundary condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16) leads to vR “ cpa, Rqa1pZq, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='17) where cpa, Rq is defined by cpa, Rq “ ´ ¯ra ¯λ1λ2 ` 1 Rζ ´ R2 B Bampa, Rq ´ ¯r¯raqpa, Rq ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18) Once vR is found, the optimal correction v can be obtained by integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='17) from B to R, which yields v “ ´ ˆż B R cpa, Tq dT ˙ a1pZq, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='19) where we have neglected the function arising from integration since it can be absorbed into λpapZqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' One-dimensional energy functional Substituting the correction function v found in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='19) back into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10), after some simplification (which is detailed in Appendix A), we obtain the final expression for the energy functional of the 1d model E1dras “ ż L ´L ´ Gpa, λpaqq ` 1 2Dpaqa1pZq2¯ dZ ` Cpaqa1pZq|L ´L, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) where the gradient moduli D and C are given by Dpaq “ ż B A Rζp¯r2 a ´ cpa, Rq2q dR, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='21) Cpaq “ ż B A pλ´1¯σ22 ´ N˚q˜cpa, RqR dR ´ 1 2PA2a2˜cpa, Aq, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22) 10 with ˜cpa, Rq “ ´ şB R cpa, Tq dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The associated equilibrium equation is obtained by extremizing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) with respect to apZq and is found to take the form A2aλpaqpQpa, λpaqq ´ Pq ´ 1 2D1paqa1pZq2 ´ Dpaqa2pZq “ 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) where we have used the fact that BG{Bλ “ 0 as it is used to find the implicit relation between λ and a (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Since Z does not explicitly appear in the integrand of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) due to the translational invariance of the current problem in Z, by the Beltrami identity, the equilibrium equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) admits a first integral of the form Gpa, λpaqq ´ 1 2Dpaqa1pZq2 “ constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='24) We remark that the variational problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) is ill-posed due to the presence of the boundary terms Cpaqa1pZq|L ´L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is because the variational structure of the problem is broken when higher-order terms are dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' There are two possible ways to get around this issue (Lestringant & Audoly, 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The first one is to simply ignore the boundary terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', to set Cpaq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The second one is to add an Opε2q-term to apZq so that the boundary terms go away, which is rigorous but slightly more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It has previously been verified in Lestringant & Audoly (2020a) that the simple and rigorous approaches yield curves that can hardly be distinguished visually in any of the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' To summarize, the second-order nonlinear ordinary differential equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) is our approxi- mate 1d model that governs the variation of the inner radius (which is A times apZq) in the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Once apZq is determined, the 3d deformation is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We note that the func- tion Qpa, λpaqq is explicit for most of the commonly used strain energy functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The only slight complication is that the function Dpaq is given by an integral;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='21), but the functions mpa, Rq, qpa, Rq, and hence cpa, Rq and the integrand in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='21) all have explicit expressions for most of the commonly used strain energy functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus, only one numerical integration is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This can easily be implemented on a symbolic manipulation platform such as Mathematica (Wolfram, 1991) as we shall show later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Connections with previous work We now demonstrate that the 1d model derived in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4 can recover the 1d model of Lestringant & Audoly (2018) for membrane tubes and that of Audoly & Hutchinson (2016) for solid cylinders under appropriate limits, and it can also reproduce the same weakly nonlinear bulging solution as that based on the exact 3d theory (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Membrane limit We first consider the reduction of the 1d model in the membrane limit when the tube thickness H approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The general axisymmetric deformation is now described by r “ rpZq, θ “ Θ, z “ zpZq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) and the three principal stretches are given by λ1 “ r R, λ2 “ a r12 ` z12, λ3 “ 1{pλ1λ2q, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) where R denotes the constant radius of the mid-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The total energy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) reduces to E “ 2π ż L ´L ´ w ´ 1 2P ˚λ2 1z1 ´ N˚z1¯ dZ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) where P ˚ denotes the pressure scaled by H{R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Setting the first variation δE to zero then gives the governing equations w1 ´ R ´w2 λ2 r1¯1 ´ P ˚λ1z1 “ 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) w2 λ2 z1 ´ 1 2P ˚λ2 1 “ N˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) Under the assumption that |r1| !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 1, we have λ2 “ z1 ` r12 2z1 ` ¨ ¨ ¨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) As an algebraic equation for z1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) has an asymptotic solution of the form z1 “ gpλ1q ` k1pλ1qr12 ` ¨ ¨ ¨ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) where the leading-order term gpλ1q obviously satisfies the algebraic equation w2pλ1, gpλ1qq ´ 1 2P ˚λ2 1 ´ N˚ “ 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) and the function k1pλ1q can easily be found but is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) determines gpλ1q uniquely under the assumption w22 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' With the use of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7), we may expand the integrand in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) to order r12 and obtain E “ 2π ż L ´L ´ wpλ1, gpλ1qq ´ 1 2P ˚λ2 1gpλ1q ´ N˚gpλ1q ` 1 2 w2pλ1, gpλ1qq gpλ1q r12¯ dZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) This is the reduced model derived by Lestringant & Audoly (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We now show that our general 1d model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) can recover this 1d model under the limit H Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' To this end, we first note that the uniformly deformed configuration is now described by ¯r “ aR, ¯z “ λZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10) 12 In particular, we have ¯ra “ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Since qpa, Rq and mpa, Rq involve integrals from R to B, they go to zero as H Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Consequently, the cpa, Rq defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18) takes the simple form cpa, Rq “ ´ R aλ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11) Taking the limit H Ñ 0 in ζ “ λw2{pλ2 ´ ¯λ2 3q yields ζ “ a2λ3w2 a2λ4 ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12) Substituting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20), we obtain lim HÑ0 E1dras RH “ ż L ´L ´ wpa, λpaqq ´ 1 2P ˚a2λpaq ´ N˚λpaq ` 1 2R2 w2pa, λpaqq λpaq a1pZq2¯ dZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='13) Note that the modulus Cpaq vanishes in the membrane limit because of the equilibrium in the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The integrand on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='13) is the same as that on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) if we identify λ1, gpλ1q and r1 with apZq, λpaq, and Ra1pZq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Solid cylinder limit Next we consider the other extreme limit corresponding to A Ñ 0 and P Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The uniform solution takes the form ¯z “ λZ, ¯r “ aR (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) with a “ λ´1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The three principal stretches are ¯λ1 “ ¯λ3 “ λ´1{2, ¯λ2 “ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15) In particular, we have w1p¯λ1, ¯λ2q “ 0, w2p¯λ1, ¯λ2q “ ˆw1pλq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16) where ˆwpλq “ Wpλ´1{2, λ, λ´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16)1 that qpa, Rq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Note that the deforma- tion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) is homogeneous, so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) implies that mpa, Rq “ A2pB2 ´ R2q R2pB2 ´ A2qmpa, Aq “ A2pB2 ´ R2q R2pB2 ´ A2qMpa, λpaqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Differentiating this expression with respect to a and noting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11), we obtain Bmpa, Rq{Ba “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus cpa, Rq reduces to cpa, Rq “ ´ R λ3{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='17) The elastic modulus ζ is easily calculated as ζ “ λ2 ˆw1pλq λ3 ´ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18) 13 Substituting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='17) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20), we obtain 2πE1drλs “ ż L ´L ´ πB2 ˆwpλq ` πB4 16 ˆw1pλq λ4 λ1pZq2 ´ Nλ ¯ dZ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='19) where we have made use of the relation a1pZq “ λ1pZq{p2λ3{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This recovers the 1d model of Audoly & Hutchinson (2016) specialized to an incompressible circular cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Comparison with exact weakly nonlinear analysis Finally, we carry out a weakly nonlinear near-critical analysis using our 1d model and compare the resulting amplitude equation with that obtained by Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020) from the exact 3d theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We focus on localized solutions in an infinitely long tube of finite wall thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Denote by a8 the limit of apZq as Z Ñ 8 and λ8 “ λpa8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11) that P “ Qpa8, λ8q, N “ 2πA2Fpa8, λ8q, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) where Fpa8, λ8q is defined by Fpa8, λ8q “ Mpa8, λ8q ´ 1 2a2 8Qpa8, λ8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='21) We look for a localized solution that bifurcates from the uniform solution by writing apZq “ a8 ` ypZq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22) where ypZq is a small perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Substituting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22) into the 1d equilibrium equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) and expanding in terms of ypZq to quadratic order with the use of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20), we obtain Dpa8qy2pZq “ ωpa8, λ8qypZq ` γpa8, λ8qypZq2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) where the two coefficient functions ωpa, λq and γpa, λq are given by ωpa, λq “ A2 2aλ a2Qλ ` 2Fλ Ωpa, λq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='24) γpa, λq “ A2 aλpa2Qa ` 2Faq Fapa2Qλ ` 2Fλq2 Γpa, λq ` A2ψpa, λqΩpa, λq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='25) In the above expressions, Qa “ BQpa, λq{Ba, Qλ “ BQpa, λq{Bλ, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' and Ωpa, λq and Γpa, λq are defined by Ωpa, λq “ BQ Ba BF Bλ ´ BQ Bλ BF Ba , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='26) Γpa, λq “ BΩ Ba BF Bλ ´ BΩ Bλ BF Ba , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='27) and ψpa, λq is not written out as it is not required in the weakly nonlinear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 14 The solution to the linearized equation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) changes character when the sign of ωpa8, λ8q changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus a bifurcation occurs when ωpa8, λ8q “ 0, or equivalently, Ωpa8, λ8q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='28) Note that Qpa8, λ8q and Fpa8, λ8q represent respectively the functional dependence of P and N on a8 and λ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus the above bifurcation condition is simply the vanishing of the Jacobian determinant of P and N as functions of a8 and λ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is consistent with previous work Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2016) and Yu & Fu (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We consider two typical loading scenarios: either the resultant axial force N or the axial stretch at infinity λ8 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The latter case is used to approximate the case of fixed axial length, which can be realized more easily experimentally or in Abaqus simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Let us first assume that the resultant axial force N “ Nc is fixed, where Nc is the prescribed axial force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Denote by pacr, λcrq the root of the system of equations ωpa8, λ8q “ 0, Fpa8, λ8q “ Nc, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='29) at which the bifurcation occurs according to the previous discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In the vicinity of the bifurca- tion point, the amplitude equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) reduces to Dpacrqy2pZq “ ω1pacr, λcrqpa8 ´ acrqypZq ` γpacr, λcrqypZq2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='30) where the prime on ω denotes d{da8 “ B{Ba8 ` pB{Bλ8qpdλ8{da8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The above equation admits a localized solution of the form ypZq “ ´3ω1pacr, λcrq 2γpacr, λcrq pa8 ´ acrq sech2 ´1 2 d ω1pacr, λcrq Dpacrq pa8 ´ acrqZ ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='31) On the other hand, the weakly nonlinear amplitude equation derived from the 3d theory (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2020) takes the form c2 1pZq “ λ2 crk1pa8 ´ acrqc1pZq ` λ2 crk2c1pZq2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='32) where c1pZq and ypZq are related by ypZq “ kc1pZq (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='33) with k “ ´2λpaq{λ1paq|a“acr, and k1 and k2 are constants available in Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' One can see that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='30) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='32) are identical provided k1 “ ω1pacr, λcrq λ2crDpacrq , k2 “ kγpacr, λcrq λ2crDpacrq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='34) We have verified numerically that this is indeed the case, but the current expressions on the right hand sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='34) are more compact and revealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 15 The case of fixed λ8 can be handled similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Let pacr, λcrq be the solution to the system of equations ωpa8, λ8q “ 0, λ8 “ λc, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='35) where λc is a given constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In the vicinity of the bifurcation point, the amplitude equation parallel to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='30) is of the form Dpacrqy2pZq “ ω1pacr, λcrqpa8 ´ acrqypZq ` γpacr, λcrqypZq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='36) where the prime on ω now signifies B{Ba8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Similar to the previous case, one can verify that the above amplitude equation is the same as its counterparts in Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Comparison with Abaqus simulations In this section, we demonstrate the power of the 1d model by applying it to investigate localized bulging in an inflated tube of finite wall thickness in the fully nonlinear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Previous studies on this problem usually treat the tube as a finite length tube, but the problem can be analyzed more easily and very accurately by assuming the tube to be of infinite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This assumption only fails when the tube is very short and when bulging is no longer localized in the axial direction (Wang & Fu, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The reason is that bulging solutions decay exponentially towards the two ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus in the following analysis, we shall assume that the tube is effectively infinite and focus on solutions subject to decaying boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This assumption is validated by comparison with Abaqus simulations based on tubes of finite lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We shall consider the two loading scenarios discussed in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3 and compare the predictions of the 1d model with Abaqus simulations, which allows us to quantify the accuracy of our 1d model and determine its range of validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In all numerical calculations and Abaqus simulations, we use the Gent material model W “ ´µ 2 Jm ln ´ 1 ´ λ2 1 ` λ2 2 ` λ2 3 ´ 3 Jm ¯ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) where µ is the shear modulus and Jm is a material constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We take µ “ 1 which is equivalent to scaling all stress variables by µ and Jm “ 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2 which is typical for rubber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The geometry of the tube is taken to be H{Rm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4 and 2L{Rm “ 40, where Rm “ pA ` Bq{2 is the average radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In the Abaqus simulations, to ensure that localized bulging occurs in the middle of the tube, a small section with length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1L around the middle point of the tube is weakened by taking its shear modulus to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9999 times that of the rest of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The 1d differential equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) subject to appropriate end conditions (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) later) can be solved numerically with the aid of the symbolic computation software Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Although the gradient modulus Dpaq involves an integral that cannot be evaluated analytically, this integral can be defined numerically in Mathematica with the built-in command ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='NumericQ and can be manipulated as elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Numerically solving the 1d equation is significantly faster than Abaqus simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The 1d equation can typically be solved in a few seconds on a personal computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The case of fixed axial force We first consider the loading scenario whereby the resultant axial force N is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' As mentioned earlier, we assume that the tube is infinitely long and focus on the solution that satisfies the decaying boundary condition lim ZÑ8 apZq “ a8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) A linear analysis shows that the solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) satisfying (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) decays exponentially as Z Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus we have limZÑ8 a1pZq “ 0 automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We assume that the bulging solution is symmetric with respect to Z “ 0 so that a1p0q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We write λ8 “ λpa8q, a0 “ ap0q and λ0 “ λpap0qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Since pa8, λ8q satisfy the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8), we have Mpa8, λ8q ´ 1 2a2 8Qpa8, λ8q ´ N 2πA2 “ 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) Qpa8, λ8q ´ P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) From the definition of λ0 and the conservation law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='24), we see that pa0, λ0q satisfies Mpa0, λ0q ´ 1 2a2 0Qpa8, λ8q “ Mpa8, λ8q ´ 1 2a2 8Qpa8, λ8q, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) Gpa0, λ0q “ Gpa8, λ8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) Either a8 or P can be taken to be the load parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' When a8 is specified, one can first obtain λ8 from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The associated P is computed according to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Then by solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5), one obtains the “initial” values a0 and λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The localized solution can be found by solving the initial value problem A2aλpaqpQpa, λpaqq ´ Pq ´ 1 2D1paqa1pZq2 ´ Dpaqa2pZq “ 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) ap0q “ a0, a1p0q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) As a first example, fixing the axial force N to be zero, we find from the bifurcation condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='29) that localized bulging takes place at a8 “ acr “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='86 with a critical pressure Pcr “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' As we trace the bifurcation solution away from the bifurcation point, the pressure drops while the bulge grows until it has almost reached a maximum amplitude after which the bulge will propagate at a constant pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' From Maxwell’s equal-areal rule, the propagation pressure is PM “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 2 shows the dependence of the pressure on ap0q and the bulging amplitude on a8 based on Abaqus simulations and use of the 1d model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The bulging solutions given by Abaqus simulations and the 1d model at the four states marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 2(a) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is seen that the 1d solution agrees well with Abaqus simulations in the entire post-bifurcation regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Remarkably, the 1d solution remains highly accurate even in the final propagation stage, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Note also that the Abaqus simulations and 1d calculations are conducted for 2L “ 40Rm and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This verifies our earlier claim that the tube can effectively be viewed to be infinitely long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 17 ● ● ● ● ● ● 1d model Abaqus simulation 1 2 3 4 5 6 7 a(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='30 P a b c d (a) 1d model Abaqus simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9a∞ 0 1 2 3 4 5 6 a(0) - a∞ (b) Figure 2: Dependence of (a) pressure on ap0q and (b) bulging amplitude on a8 based on Abaqus simulations and the 1d model for fixed N “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Abaqus simulation 1d model 0 2 4 6 8 Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4 a(Z) (a) Abaqus simulation 1d model 0 2 4 6 8 Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 a(Z) (b) Abaqus simulation 1d model 0 2 4 6 8 Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 a(Z) (c) Abaqus simulation 1d model 0 2 4 6 8 Z 1 2 3 4 5 6 a(Z) (d) Figure 3: Bulging solutions given by Abaqus simulations and the 1d model at the four states marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 2(a) for fixed N “ 0: (a) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3, (b) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='25, (c) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22, (d) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The case of fixed ends Next, we consider the loading scenario whereby the tube is first stretched to a specified length 2ℓ and then its two ends are fixed to prevent further axial displacement (whether the radial dis- placement is restricted or not at the ends is immaterial since the tube is assumed to be sufficiently long).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In the previous subsection, we have solved the problem for a specified axial force N or equivalently a specified λ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For the current problem with a given ℓ, we define λc “ ℓ{L and we need to find λ8 such that the following condition is satisfied: ż L 0 λpapZqq dZ “ λcL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) This can be achieved by a shooting procedure: for each N, we compute the left-hand side using the procedure outlined in the previous subsection and adjust N such that the left-hand side and the right-hand side are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The procedure may be started by taking λ8 “ λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' However, solving the present problem by the shooting procedure requires a lot of adjustments by hand due to the fact that the bulging solution may start to grow after decaying for a range of Z values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' To find solutions for the current case in a more robust way, we use the finite difference method instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' To implement the finite difference method, we partition the domain r0, Ls using a uniform mesh Z0, Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' , Zn with mesh size h “ L{n and coordinate of the j-th grid point given by Zj “ jh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We use aj to represent the numerical approximation of apZjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Applying the central difference scheme, we discretize the differential equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) as A2ajλpajqpQpaj, λpajqq ´ Pq ´ 1 2D1pajq ´aj`1 ´ aj´1 2h ¯2 ´ Dpajqaj`1 ´ 2aj ` aj´1 h2 “ 0, j “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' , n ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10) The left boundary condition is given by a1p0q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11) We see from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) that the solution to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) subject to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) has the asymptotic behavior apZq „ a8 ` a1e´κZ as Z Ñ 8, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12) where a1 is a constant and κ “ d ωpa8, λ8q Dpa8q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Because of this, we may replace the decaying condition boundary (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) by the “soft” asymptotic condition a1pLq ` κpapLq ´ a8q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='13) 19 To avoid the loss of accuracy at the two endpoints, we introduce two additional unknowns a´1 and an`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Then the left and right boundary conditions yield a1 ´ a´1 2h “ 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) an`1 ´ an´1 2h ` κpan ´ a8q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15) Solving for a´1 and an`1 from the above equations, and substituting them into the difference equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10) at j “ 0 and j “ n, we obtain the discrete boundary conditions with truncation errors of order h2: A2a0λpajqpQpa0, λpa0qq ´ Pq ´ 2Dpa0qa1 ´ a0 h2 “ 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16) A2anλpanqpQpan, λpanqq ´ Pq ´ 1 2D1pajqκ2pan ´ a8q2 ´ 2Dpanqan´1 ´ an ´ hκpan ´ a8q h2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='17) Finally, the fixed-length restriction (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) gives 1 2λpa0q ` n´1 ÿ j“1 λpajq ` 1 2λpanq ´ λcL h “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18) We use the pressure P as the loading parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' When P is given, one can first solve (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) to express a8 as a function of λ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Then N can be viewed as a function of λ8 due to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It follows that λpµq and Dpµq also depend on λ8 through their dependence on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This implicit dependence should be considered when solving the above algebraic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Setting n to be a sufficiently large number, say n “ 100, and solving the system of nonlinear algebraic equations consisting of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='17) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18) for aj’s and λ8 with a suitable initial guess, we obtain the finite-difference solution for the present problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We may use the weakly nonlinear solution with fixed λ8 “ λc “ ℓ{L as an initial guess in the near-critical regime and continue the solution to the fully nonlinear regime by always using the solution at the previous step as the initial guess for the current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' When the total length is fixed to be ℓ “ 2L, then initially λ8 “ 2 and localized bulging takes place at a8 “ acr “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='74 with a critical pressure Pcr “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='198 according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4, we have shown the dependence of the pressure on ap0q and the bulging amplitude on a8 based on Abaqus simulations and use of the 1d model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The bulging solutions determined by Abaqus simulations and the 1d model at the four states indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4(a) are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is observed that the agreement between the 1d model and Abaqus simulations is again excellent in the fully nonlinear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 6 shows the actual variation of P against ap0q predicted by the 1d model when L is varied and the averaged stretch λc is fixed or λc is varied but L is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' These results confirm the theoretical prediction of Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2022) that the right branches of these curves all converge 20 ● ● ● ● ● ● ● ● 1d model Abaqus simulation 0 1 2 3 4 5 6 7 a(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20 P a b c d (a) 1d model Abaqus simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8a∞ 1 2 3 4 5 6 a(0) - a∞ (b) Figure 4: Dependence of (a) pressure on ap0q and (b) bulging amplitude on a8 based on Abaqus simulations and using the 1d model for fixed length ℓ{L “ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Abaqus simulation 1d model 0 2 4 6 8 10 12Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8 a(Z) (a) Abaqus simulation 1d model 0 2 4 6 8 10 12Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 a(Z) (b) Abaqus simulation 1d model 0 2 4 6 8 10 12Z 1 2 3 4 5 a(Z) (c) Abaqus simulation 1d model 0 2 4 6 8 10 12Z 1 2 3 4 5 6 7 a(Z) (d) Figure 5: Bulging solutions based on Abaqus simulations and the 1d model the at the four states indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4(a) for fixed length ℓ{L “ 2: (a) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='19, (b) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18, (c) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='173, (d) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' to a master curve that is independent of L or λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' These curves all terminate at the point where the axial stress near each end of the tube has become compressive enough so that secondary Euler buckling or axisymmetric wrinkling becomes possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 21 L=15 L=20 L=40 0 1 2 3 4 5 6 7 a(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22 P (a) λc=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 λc=2 λc=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8 0 1 2 3 4 5 6 7 a(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='25 P (b) Figure 6: Variation of P against ap0q predicted by the 1d model when (a) the total length is fixed with λc “ 2 and L “ 15, 20 and 40, respectively and (b) L is fixed at 20 and λc “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5, 2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Conclusion We have derived a 1d model for the analysis of axisymmetric deformations of an inflated cylin- drical tube of finite wall thickness, and established its range of validity by comparing its predictions with those of Abaqus simulations for two typical loading scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The comparison shows that the 1d model performs extremely well in both the near-critical and fully nonlinear regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The dimension reduction started from three-dimensional finite elasticity theory and is performed in terms of the energy functional and principal stretches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A key ingredient of the dimension reduc- tion is the assumption of slow variation of the leading-order solution in the axial direction without any restriction on its amplitude, which results in a 1d model that is simple but is still capable of capturing the strain-gradient effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is in contrast with the traditional asymptotic analysis where the leading order solution is assumed to be a small-amplitude perturbation from the primary deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is because of this difference that the 1d model has a much larger range of validity than the expansion methods around the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The nonlinearity of the strain is kept in the 1d model, reflected by the nonlinear potential Gpa, λpaqq and the nonlinear strain gradient modulus Dpaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Our expression for the strain gradient coefficient Dpaq is quite simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For the Gent material model, Dpaq can be calculated by integrating once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We remark that although the derivation presented in this work is variational, the 1d model can also be derived by substituting the asymptotic solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) into the 3d governing equations and solving the resulting equations at successive orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The 1d model is amenable to asymptotic and numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The bifurcation condition and the weakly nonlinear amplitude equation predicted by the model are exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In fact, the expressions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='24) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='25) derived using the 1d model are more compact and more revealing than their counterparts in Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A major advantage of the 1d model is that the entire evolution process of bulging or necking can be determined using the finite difference method which is more accessible and much easier to implement than commercial packages such as Abaqus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This advantage 22 would become even more significant when other fields such as electric loading and residual stresses were also present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Such extra fields and new geometries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' axisymmetric necking of a stretched plate (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2022) ) will be considered in our future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A Mathematica code that produces all the results presented in the paper is available on GitHub (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='com/yfukeele).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No 12072224) and the Engineering and Physical Sciences Research Council, UK (Grant No EP/W007150/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Simplifying the one-dimensional energy functional Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='19) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11), we can write the integral of KpR, v, vRq as ż B A KpR, v, vRq dR “ pI1 ` I2 ` I3qa12, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) where I1 “ ż B A pλ´1¯σ22qaR ż B R cpa, Tq dT dR, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) I2 “ 1 2 ż B A Rζp¯r2 a ` cpa, Rq2q dR, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) I3 “ ż B A Rξ¯racpa, Rq dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) By interchanging the order of integration, we can rewrite I1 as I1 “ ż B A ż B R pλ´1¯σ22qaRcpa, Tq dT dR “ ż B A ż T A pλ´1¯σ22qaRcpa, Tq dR dT “ ż B A cpa, Tq B Ba ´ ż T A λ´1¯σ22R dR ¯ dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14), we have ż T A λ´1¯σ22R dR “ A2mpa, Aq ´ T 2mpa, Tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) Inserting (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) and noting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15)2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11), we can simplify I1 as I1 “ PA2a ż B A cpa, Rq dR ´ ż B A cpa, RqR2 B Bampa, Rq dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) 23 Noting that ξ “ q¯λ1 ` ¯λ3ζ{λ, the integral I3 can be calculated as I3 “ ż B A R ´ q¯λ1 ` ¯λ3 λ ζ ¯ ¯racpa, Rq dR “ ż B A ´ ¯r¯raq ` Rζ ¯ra ¯λ1λ2 ¯ cpa, Rq dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) Adding up the three integrals, we obtain ż B A KpR, v, vRq dR ` PA2aa1v|R“A “pI1 ` I2 ` I3qa12 ´ PA2aa12 ż B A cpa, Rq dR “a12 ż B A ´ ´ cpa, RqR2 B Bampa, Rq ` 1 2Rζp¯r2 a ` cpa, Rq2q ` ´ ¯r¯raqpa, Rq ` Rζ ¯ra λ1λ2 ¯ cpa, Rq ¯ dR “a12 ż B A p1 2Rζp¯r2 a ` cpa, Rq2q ´ Rζcpa, Rq2q dR “1 2a12 ż B A Rζp¯r2 a ´ cpa, Rq2q dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) This gives the expression of the coefficient Dpaq announced in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The expression of Cpaq in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22) follows by a straightforward substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' References Alhayani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', Rodr´ıguez, J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Solids Struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 176, 173–184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Ye, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', & Fu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Weakly nonlinear analysis of localized bulging of an inflated hyperelastic tube of arbitrary wall thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Solids, 135, 103804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Yin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Non-uniform inflation of a cylindrical elastic membrane and direct determination of the strain energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Elast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 7, 265–282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', & Fu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' An analytic derivation of the bifurcation conditions for localization in hyperelastic tubes and sheets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 73, 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'}