diff --git "a/19E1T4oBgHgl3EQflQTp/content/tmp_files/load_file.txt" "b/19E1T4oBgHgl3EQflQTp/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/19E1T4oBgHgl3EQflQTp/content/tmp_files/load_file.txt" @@ -0,0 +1,567 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf,len=566 +page_content='On the accuracy of one-way approximate models for nonlinear waves in soft solids Harold Berjamin a aSchool of Mathematical and Statistical Sciences, University of Galway, University Road, Galway, Republic of Ireland Abstract A simple strain-rate viscoelasticity model of isotropic soft solid is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The constitutive equations account for finite strain, incompressibility, material frame-indifference, nonlinear elasticity, and viscous dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' A nonlinear viscous wave equation for the shear strain is obtained exactly, and a corresponding one-way Burgers-type equation is derived by making standard approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Analysis of the travelling wave solutions shows that the two partial differential equations produce distinct solutions, and that deviations are exacerbated when wave amplitudes are not arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In the elastic limit, the one-way approximate wave equation can be linked to simple wave theory, thus allowing direct error measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 1 Introduction In nonlinear acoustics, the Burgers equation is often viewed as the simplest model equation that includes nonlinear wave propagation and diffusion effects (Witham, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' This partial differential equation in space and time can be derived directly from the one-dimensional Navier–Stokes equation by dropping the pressure term, or as a special case of the Westerwelt equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Besides Burgers’ equation, other one-way wave equations have been derived to describe wave propagation in fluids and solids at large amplitudes (Hamilton and Blackstock, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Naugolnykh and Ostro- vsky, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Based on an appropriate scaling of the wave amplitude, such approximate partial differential equations describe unidirectional wave motion for slowly-varying wave profiles of moderate amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' One-way approximate wave equations have found applications in various areas of nonlinear acoustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' For in- stance, works by Radostin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2013) and Nazarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2017) describe compression wave propagation in solids with bimodular elastic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Another example is the Zabolotskaya equation that describes unidirectional plane shear wave propagation in soft solids such as gels and brain tissue (Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=', 2004), see also Cormack and Hamil- ton (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In these latter cases, the underlying three-dimensional constitutive theories were revisited by Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2013) as well as Saccomandi and Vianello (2021) to enforce objectivity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=', invariance by change of observer), leading to slight modifications of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' For these partial differential equations, not many analytical solutions are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Nevertheless, it is sometimes possible to derive exact stationary wave solutions that keep an invariant wave profile throughout the motion, which occurs at a suitable constant speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Those permanent waveforms result from the interaction between nonlinearity and dispersion (here of dissipative nature), a common feature that they share with solitary waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' One might wonder whether it is preferable to seek closed-form travelling wave solutions by using directly the full equations of motion, or by using their one-way approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' As a matter of fact, both approaches have been considered separately in the above literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The present study aims to provide evidence to advocate for a derivation of travelling waves based on the complete equations of motion, thus supporting a remark by Jordan and Puri (2005) in relation with the study by Catheline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2003) — this remark led to the publication of an erratum that briefly discusses the validity of a particular one-way wave equation (Catheline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' For this purpose, we consider the case of shear wave propagation in soft viscoelastic solids of strain rate type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' We derive the simplest three-dimensional constitutive theory that accounts for finite strain, incompressibility, material frame-indifference, and viscous dissipation (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Then, this theory is applied to simple shear deformations, aka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' transverse plane waves (Section 3), including the reduction to a one-way model described by a Burgers-type equation with cubic nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Finally, we investigate the travelling wave solutions deduced from the full equations of motion as well as from the reduced wave equation (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Results show non-negligible discrepancies introduced by the reduction to unidirectional motion as soon as wave amplitudes are no longer infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' These comparisons are reconsidered in the lossless elastic limit where connections between the one-way model and simple wave theory are established (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='03284v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='soft] 9 Jan 2023 2 Strain-rate model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='1 Basic equations In what follows, we present the basic equations of Lagrangian dynamics for incompressible solids (Holzapfel, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' We consider a homogeneous and isotropic solid continuum on which no external body force is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Its motion in the Euclidean space is described by using an orthonormal Cartesian coordinate system (O,x, y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Thus, a particle initially located at some position X of the reference configuration moves to a position x of the current configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The deformation gradient is the second-order tensor defined as F = ∂x/∂X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Introducing the displacement field u = x − X and the identity tensor I = [δi j ] whose components are represented by Kronecker’s delta, we therefore have F = I +Gradu where Grad denotes the gradient operator with respect to the material coordinates X = (x, y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In incompressible solids, isochoricity J = detF ≡ 1 (1) is prescribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Thus, the mass density ρ is constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' It follows also that ˙J = JF −⊺ : ˙F ≡ 0, where the dot de- notes the material time derivative ∂/∂t and the colon indicates double contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Introducing the Eulerian velocity gradient L = ˙FF −1, this condition can be rewritten as trL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Various strain tensors are defined as functions of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Here, constitutive laws are expressed in terms of the Green– Lagrange strain tensor E = 1 2(F ⊺F − I), which is often a preferred choice in physical acoustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' We introduce also its rate ˙E = F ⊺DF obtained by differentiation with respect to time, where D = 1 2(L +L⊺) is the strain rate tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' We note that D is trace-free due to incompressibility (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The motion is governed by the conservation of linear momentum equation ρ ˙v = DivP, where v = ˙x is the velocity field and ρ is the mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The equation of motion involves the Lagrangian divergence of the first Piola–Kirchhoff stress tensor P = FS where S is the second Piola–Kirchhoff stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Those stress tensors are specified later on by the provision of a constitutive law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The present definitions are consistent with notations and conventions used in the monograph by Holzapfel (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In particular, the divergence of the tensor P reads [DivP]i = Pi j,j componentwise, where indices after the coma de- note spatial differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In some other texts, a transposed definition of the divergence is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Then, the equation of motion involves the material divergence of the nominal stress tensor P⊺ instead of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='2 Generalities In the present study, we consider deformable solids whose constitutive behaviour is described by the state vari- ables S = {s,E}, where s is the specific entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The choice of variables S is coherent with the postulate of frame- indifference of the internal energy (Holzapfel, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In fact, a change of observer specified by a superimposed rigid- body motion leaves S invariant, as well as the internal energy U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Note that the internal energy does not depend on rates of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The internal energy per unit volume U is a function of state to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The thermodynamic temperature is defined as the conjugate variable of s in the partial Legendre transform of U/ρ with respect to s (Berjamin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' However, the explicit dependence of U with respect to s is usually omitted in the definition of a strain energy density function W e such that U = W e(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The strain energy W e is regarded as a scalar-valued isotropic function of its arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Thus, its dependence with respect to E can be reduced to a dependence with respect to three scalar invariants I1 = tr(E), I2 = tr(E2), I3 = tr(E3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2) They can be used directly, or other physically meaningful scalar quantities might be defined from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The first and second principles of thermodynamics yield the Clausius–Duhem inequality D = (S −Se) : ˙E = Sv : ˙E ≥ 0, (3) where D is the dissipation, S = Se +Sv is the total second Piola–Kirchhoff stress, Se = −pC −1 + ∂W e ∂E (4) denotes the elastic part, and Sv is a viscous contribution to be specified subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The scalar p is an arbitrary Lagrange multiplier for the incompressibility constraint (1), see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='3 of Holzapfel (2000), and C = I +2E is the right Cauchy–Green strain tensor F ⊺F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Therefore, no dissipation occurs in the elastic case S = Se where the viscous stress tensor Sv is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' According to the dissipation inequality (3), the viscous stress Sv is a function of state and evolution variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' the set S ∪ { ˙E} which is a consistent choice to enforce frame-invariance (Antman, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Ball, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' We introduce a dissipation potential W v(E, ˙E) such that Sv = ∂W v ∂ ˙E (5) 2 defines the viscous stress (Maugin, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In general, the dissipation potential is described by additional invariants (Pioletti and Rakotomanana, 2000) I4 = tr( ˙E), I5 = tr( ˙E2), I6 = tr( ˙E3), I7 = tr( ˙EE), I8 = tr( ˙EE2), I9 = tr( ˙E2E), I10 = tr( ˙E2E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (6) In the present study, we consider Newtonian-type viscosity models whose dissipation potential is as simple as possi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='3 Consequences of incompressibility First, let us investigate the consequences of the incompressibility constraint (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' As noted in Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2007), the invariants (2) of E are linked through I1 = I2 − 4 3 I3 − I 2 1 +2I1I2 − 2 3 I 3 1 , (7) by virtue of incompressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' This identity follows from the expression of the principal invariants of the unimodular tensor C = I +2E in terms of the invariants Ik, see the Appendix of Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Using the differential version of the incompressibility constraint, the invariants (2)-(6) of E, ˙E satisfy the particular relationship 1 2 I4 = I7 −2I8 +2I1I7 − � I1 − I2 + I 2 1 � I4 (8) deduced from the identity trD = 0, see Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The relationship (7) means that the invariant I1 = tr(E) is no longer linear with respect to the components of the strain tensor E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' instead, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (7) shows that it has terms of polynomial order two and three with respect to the strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Furthermore, due to the relationship (8), the invariant I4 = tr( ˙E) is still linear with respect to the components of the strain-rate tensor ˙E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' However, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (8) shows that I4 is no longer invariant on the strain tensor E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' instead, it has terms of polynomial order one, two and three with respect to the Green–Lagrange strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='4 Constitutive assumptions In weakly nonlinear elasticity, the strain energy density function is sought in the form of a polynomial of the invariants Ik with constant coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Similarly to Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2004), we assume that the internal energyU has a fourth- order polynomial expression with respect to the components of the strain tensor E of the form W e = µI2 + 1 3 AI3 +DI 2 2, (9) where µ ≥ 0 is the shear modulus (in Pa), and the coefficients A, D are higher-order elastic constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Now, let us propose an expression for the dissipation potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' To end up with a linear viscosity model similar to that by Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2013), we assume that the dissipation potential is a second-order polynomial expansion of the strain rate tensor ˙E, and a zeroth-order polynomial of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' This assumption amounts to selecting W v of second order in (E, ˙E), and to ignore the terms proportional to ˙E that produce elastic stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Due to the relationships (7)-(8), we therefore keep W v = ηI5, (10) where η ≥ 0 is the shear viscosity (in Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In the above expression, the absence of bulk viscosity “ζ” is due to the assumption on polynomial orders for the viscous part, and to the incompressibility property (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Setting the bulk viscosity ζ = 2 3η in Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2013) yields the same expressions as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Computation of the tensor derivatives of the potentials (9)-(10) by means of the chain rule for W •(Ik,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=') yields the following elastic (4) and viscous stress contributions (5) Se = −pC −1 +2(µ+2DI2)E + AE2, Sv = 2η ˙E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (11) Thermodynamic consistency (3) is ensured provided that the dissipation D = 2W v is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In fact, the present dissipation potential W v is a homogeneous function of degree two with respect to ˙E (Maugin, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' A sufficient condition for the restriction D ≥ 0 to be always satisfied is that the viscosity η is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 3 3 Plane shear waves 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='1 Nonlinear viscous wave equation Similarly to Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2013), we consider simple shear deformations described by the displacement field u = [u,0,0]⊺ where u = u(z,t) denotes the particle displacement along the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Thus, the deformation gradient tensor reads F = � � 1 0 γ 0 1 0 0 0 1 � �, (12) where γ = ∂u/∂z is the shear strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The velocity field takes the form v = [v,0,0]⊺ where v = ∂u/∂t is the shear velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In the equation of motion ρ ˙v = DivP, the relevant first Piola–Kirchhoff stress component P13 is deduced from the expression of the elastic and viscous parts Pe 13 = µγ+Γγ3 and Pv 13 = η(1+2γ2) ˙γ, where only terms up to order γ3 have been kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The non-negative constant Γ = µ+ A/2+D is a parameter of nonlinearity (Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Hence, upon division by the shear modulus µ, the x-component of the equation of motion produces the nonlinear wave equation 1 c2 ∂2u ∂t2 = ∂2u ∂z2 + 2 3β ∂ ∂z �∂u ∂z �3 +τ ∂ ∂z �� 1+2 �∂u ∂z �2� ∂2u ∂z∂t � , (13) describing transverse wave propagation along the z-direction, where we have introduced the notations c = � µ ρ , β = 3 2 Γ µ, τ = η µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (14) Spatial differentiation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (13) allows to write a similar wave equation for the strain 1 c2 ∂2γ ∂t2 = ∂2γ ∂z2 + 2 3β ∂2 ∂z2 γ3 +τ ∂2 ∂z2 �� 1+2γ2� ∂γ ∂t � , (15) which will be used later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' According to the wave equations (13)-(15), shear waves of infinitesimal amplitude propagate at the shear wave speed c = � µ/ρ in the absence of nonlinearity and viscosity (β = 0, τ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Typically, this sound velocity equals c ≈ 2 m/s in gels (Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=', 2007), whereas β ≈ 10 and τ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='12 ms at a loading frequency of 100 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Here, we have obtained the same wave equations than those derived in Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2013) for the particular bulk viscosity ζ = 2 3η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Note in passing the presence of a nonlinear viscous term which is absent in Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='2 Slow scale approximations Similarly to Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2004) and Pucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2019), we proceed now to a reduction of the above wave equa- tion (13) for one-way wave propagation with slowly varying profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' We present two approximations based either on a slow space variable or a slow time variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Slow space Let us follow the scaling procedure in Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' For this purpose, we introduce the following scaling defined by the change of variables {˜z = ϵ2z, ˜t = t − z/c,u = ϵ ˜u}, where ϵ is a small parameter and ˜u = ˜u(˜z, ˜t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Furthermore, we assume that τ is of order ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Note that this set of assumptions corresponds to a slowly- varying profile in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' This Ansatz is then substituted in the equation of motion (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' At leading (cubic) order in ϵ, the motion of soft viscous solids is governed by the scalar equation ϵ3c ∂2 ˜u ∂˜z∂˜t = ϵ3 β c2 �∂ ˜u ∂˜t �2 ∂2 ˜u ∂˜t2 +ϵτ 2 ∂3 ˜u ∂˜t3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (16) Transforming back to the initial displacement u and physical coordinates (z,t) leads to a reduced wave equation c ∂v ∂z + � 1−βv2/c2� ∂v ∂t = τ 2 ∂2v ∂t2 , (17) for the velocity v = ∂u/∂t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 1The wave equation proposed by Catheline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2003) and analysed by Jordan and Puri (2005) cannot be obtained rigorously from the equations of motion unless time derivatives are (questionably) replaced by spatial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 4 Up to the choice of time variable used here (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=', the physical time t instead of the retarded time ˜t), the partial differential equation (17) is identical to the cubic Burgers-type equation of Zabolotskaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' However, the underlying modelling assumptions are not equivalent, since the initial wave equation (13) includes the extra nonlinear viscosity term 2τ∂(γ2 ˙γ)/∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' This additional term is lost in the rescaling procedure given that it is of higher order in ϵ than the leading-order viscous term τ∂ ˙γ/∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In the end, while the modelling efforts by Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2013) aimed at enforcing objectivity lead to a slight modification of the wave equation (more precisely, the addition of a nonlinear viscous term), they do not induce any modification of the transport equation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Slow time For later comparisons, let us derive a similar Burgers-type equation governing the evolution of the strain instead of the velocity by following Pucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' To do so, we introduce the slow-time scaling based on the change of variables {˜t = ϵ2t, ˜z = z −ct,u = ϵ ˜u} where ϵ is a small parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Proceeding in a similar fashion to above, we end up with the nonlinear transport equation ∂γ ∂t +c � 1+βγ2� ∂γ ∂z = τc2 2 ∂2γ ∂z2 , (18) where γ = ∂u/∂z is the shear strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Here too, after keeping leading order terms, we have transformed back to the initial physical coordinates (z,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Therefore, the above partial differential equation may be viewed as a one-way ap- proximation of the wave equation (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Their travelling wave solutions are compared in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 4 Travelling wave solutions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='1 Nonlinear viscous wave equation Let us seek travelling wave solutions to the wave equation (15), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' specific smooth waveforms that propagate at a constant velocity with a steady profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In a similar fashion to Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2013), we first introduce the following rescaled dimensionless variables and coordinates g(¯z, ¯t) = � 2 3βγ(z,t), ¯t = t/τ, ¯z = z/(cτ), (19) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (15), such that ∂2g ∂¯t2 = ∂2g ∂¯z2 + ∂2 ∂¯z2 g 3 + ∂2 ∂¯z2 �� 1+ 3 β g 2 � ∂g ∂¯t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (20) Next, we seek travelling wave solutions of the form g = � ν2 −1G(ξ) where ξ = (ν2 − 1)(¯t − ¯z/ν) involves the dimen- sionless wave velocity ν ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Injecting this Ansatz in the above partial differential equation and integrating twice with respect to ξ with vanishing integration constants yields a nonlinear differential equation for the strain: G = G3 + � 1+αG2� d dξG, (21) where α = 3(ν2 −1)/β is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' From the above differential equation, one observes that travelling wave solutions to the wave equation (15) should connect the equilibrium strains G = 0 and G = ±1 by following a smooth transition that depends on the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Solutions read (Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=', 2013) ξ = −ln � 1 2G �4 3(1−G2) � 1+α 2 � (22) in implicit form, where we have enforced G(0) = 1/2 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Illustrations are provided later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='2 Slow time approximation In a similar fashion, let us now seek travelling wave solutions to the reduced wave equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Thus, we first perform the substitutions (19) to get ∂g ∂¯t + ∂ ∂¯z � g + 1 2 g 3 � = 1 2 ∂2g ∂¯z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (23) In order to obtain wave solutions that correspond to the same strain values at infinity as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='1, we introduce a slightly different scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Indeed, let us inject the Ansatz g = � ν2 −1G(χ) with χ = (ν2 − 1)(ϑ¯t − ¯z) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (23), where ϑ = 1+ 1 2(ν2 −1) is the new dimensionless velocity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Thus, we arrive at the differential equation G = G3 + d dχG (24) 5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='5 2 ν ϑ Figure 1: Scaled velocity ϑ = 1+ 1 2(ν2 −1) for the ‘slow-time’ reduced model in terms of the scaled velocity ν for the full wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' of which the strain values 0 and 1 are steady states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Enforcing the initial value G = 1/2 at χ = 0 gives G = 1 � 1+3e−2χ , (25) which does not involve any extra parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' One observes that this expression corresponds to the case α = 0 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (21)-(22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' One might proceed in a similar fashion with the Burgers-type equation (17) corresponding to the slow space approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Similarly to (19), we perform the substitutions r(¯z, ¯t) = � 2β/3v(z,t)/c in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (17) to get ∂r ∂¯z + ∂ ∂¯t � r − 1 2r 3 � = 1 2 ∂2r ∂¯t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (26) Next, we introduce r = � ν2 −1V (ψ) where ψ = (ν2 −1)(¯t − ¯z/κ) involves the dimensionless velocity κ defined by the relationship κ−1 = 1− 1 2(ν2 −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' This way, we obtain the same differential equation V = V 3 + d dψV for the dimension- less velocity V as previously for the strain (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Therefore, within the scope of the present study, the slow time and slow space approximations lead to related travelling wave solutions that describe the evolution of distinct kinematic variables (strain and velocity, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='3 Comparison Let us compare the solutions (22)-(25) obtained for the full wave equation (15) and the one-way model (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' First, one observes that these travelling waves of same amplitude do not propagate at the same speed, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Indeed, given the expression of ϑ, we can express the relative error E = ϑ/ν−1 on the scaled velocity as a function of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' To ensure that the latter remains less than 5% (respectively 1%), we obtain the requirement ν ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' ν ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='1) marked by dotted lines in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Now, let us observe that for a unit kink covering the range 0 ≤ G ≤ 1, the corresponding shear strains satisfy 0 ≤ γ ≤ � α/2 (27) where α = 3(ν2 − 1)/β was introduced earlier on, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In other words, the coefficient α in the differential equation (21) is related to the maximum strain of travelling waves, and these bounds are valid for both models at hand due to application of the rescaling procedure (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Thus, restrictions of the wave speed ν can be expressed in terms of the strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' To ensure that the velocity error E remains less than 5% (respectively 1%), we therefore require γ � β ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' γ � β ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Note that the parameter of nonlinearity can take such values as β ≈ 10 for gels (Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Therefore, the slow scale approximation has a very restricted validity for such a soft viscoelastic material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' This property is further illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 2, where we have represented the evolution of the relative velocity ν−1 (or ϑ−1) in terms of the maximum strain amplitude, both for the full wave equation and its one-way approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' According to the expression of α above, we have the relationship ϑ−1 = 1 3β( � α/2)2 in the case of the one-way approx- imate model, which produce lines of slope two in log-log coordinates (dashed lines in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' However, for the full wave equation, this relationship between the wave speed ν and the strain amplitude is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' For fixed values of the nonlinearity parameter β, differences between the one-way model and the full wave equation become visible at large strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 3, we display the evolution of the waveforms (22)-(25) in terms of the scaled coordinates ξ, χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In the case of the full wave equation (21), the parameter α takes the values {0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='2,3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' It appears that the waveforms so-obtained follow a drastically different evolution when parameters are modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In particular, the wavefront deduced from the full wave equation (solid lines) does not exhibit the same invariance and symmetry properties as the wavefront deduced from the one-way model (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 6 10−1 100 10−3 10−2 10−1 100 β = 3 β = 1 Strain amplitude Relative velocity one-way wave eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Figure 2: For the full wave equation (solid line) and the ‘slow-time’ reduced model (dashed line), we represent the evolution of the relative velocity ν − 1 (respectively, ϑ − 1) in terms of the strain amplitude � α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The axes have a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' −2 0 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='5 1 α ξ, χ G wave eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' one-way Figure 3: Steady waveforms deduced from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (22)-(25) for increasing values of the parameter 0 ≤ α ≤ 3 (arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Evolution of the scaled shear strain G in terms of the related dimensionless coordinate ξ or χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 7 5 Simple waves In the lossless case, exact one-way wave equations can be derived by using the method of Riemann invariants, see for instance the introductory example by John (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Such particular wave solutions called simple waves keep one Riemann invariant constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In other words, the particle velocity v = R−−Q(γ) withQ(γ) = c �γ 0 � 1+2βg 2 dg depends explicitly on the strain γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' The scalar R− is an arbitrary constant, for instance R− ≡ 0 in some specific boundary-value problems (Berjamin and Chockalingam, 2022), which will be assumed satisfied from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Spatial differentiation of the velocity then produces ∂γ ∂t +c � 1+2βγ2 ∂γ ∂z = 0, (28) where we have used the equality of mixed partials ∂v/∂z = ∂γ/∂t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Obviously, the lossless one-way wave equation (18) with τ = 0 is an approximation of (28) for 2βγ2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Let us analyse this requirement in a more quantitative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' To ensure that the relative error on the advection velocity E = 1+a � 1+2a −1 for a = βγ2 remains less than 5% (respectively 1%), we obtain the requirement a ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='44 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' a ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Application of the square root leads to the restriction γ � β ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='66 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' γ � β ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='40) which is slightly more constraining than in the case of viscoelastic travelling waves (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Along a simple wave, computation of the partial derivative of the velocity v = R− −Q(γ) with respect to time pro- duces c ∂v ∂z + � 1+2βγ2�−1/2 ∂v ∂t = 0, (29) where the strain γ = Q−1(−v) can be expressed formally as a function of the velocity, despite no analytical expression of the inverse function Q−1 of Q is known in the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' If |v| is small, then we can use the approximation γ ≃ −v/c of the strain which follows from the asymptotic equivalence of Q ∼ cγ at small strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Next, the (·)−1/2- factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (29) can be approximated by the polynomial expression 1−βγ2 as long as 2βγ2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' This way, we have shown that the one-way wave equation (17) is an approximation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (29) obtained for R− = 0 and 2βv2/c2 ≪ 1 in the elastic limit τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' This observation is consistent with the discussions in Catheline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In summary, the lossless ‘slow-space’ and ‘slow-time’ reductions (17)-(18) with τ = 0 are approximate governing equations for simple waves with small values of βv2/c2 and of βγ2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 6 Conclusion For a specific strain-rate viscoelasticity theory of soft solids, we have shown that one-way approximate wave propaga- tion models produce significantly different travelling wave solutions than the full equations of motion as soon as the wave amplitude is not infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Similar observations are reported in the literature in relation with shear shock formation (Berjamin and Chockalingam, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In the elastic limit, we have examined the validity of one-way approx- imations in relation with simple wave theory, thus leading to dedicated criteria of validity involving small velocity and strain amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' We conclude that these approximations should be used with care given their limited accuracy, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Nevertheless, they might remain useful for the interpretation of experimental results where their validity is not always severely penalised (Catheline et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=', 2003, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Acknowledgments The author is grateful to Michel Destrade (Galway, Ireland) for support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement TBI-WAVES — H2020- MSCA-IF-2020 project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 101023950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' A Consequence of incompressibility This Appendix is devoted to the derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' We start with the Cayley–Hamilton identity for the right Cauchy– Green tensor C = F ⊤F, which reads C 3 −I C 2 +II C −III I = 0, (30) where I, II, III are the principal invariants of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' In the case of volume-preserving motions (1), the tensor C is uni- modular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' we have III = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Next, multiplication of (30) by C −1 ˙E on the right side, substitution of C = I + 2E and computation of the trace entails the relationship (I4 +4I7 +4I8)−(3+2I1)(I4 +2I7) +(3+4I1 +2I 2 1 −2I2)I4 = 0, (31) 8 where we have used the incompressibility property trD = tr(C −1 ˙E) = 0, the definition of the invariants (2)-(6), and the relationship between I, II and the invariants Ik used here (Destrade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' Rearranging terms, we get the desired identity (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' References S.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='1121/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content='1802533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'} +page_content=' 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E1T4oBgHgl3EQflQTp/content/2301.03284v1.pdf'}