Kink and kink-like waves in pre-stretched Mooney-Rivlin viscoelastic rods

The present paper theoretically investigates kink and kink-like waves propagating in pre-stretched Mooney-Rivlin viscoelastic rods. In the constitutive modeling, the Cauchy stress tensor is assumed to consist of an elastic part and a dissipative part. The asymptotic method is adopted to simplify the nonlinear dynamic equations in the limit of finite-small amplitude and long wavelength. Using the reductive perturbation method, we further derive the well-known far-field equation (i.e. the KdV-Burgers equation), to which two kinds of explicit traveling wave solutions are presented. Examples are given to show the influences of pre-stretch and viscosity on the wave shape and wave velocity. It is shown that pre-stretch could be an effective method for modulating the two types of waves. In addition, such waves may be utilized to measure the viscosity coefficient of the material. The competition between the effects of pre-stretch and viscosity on the kink and kink-like waves is also revealed.


I. INTRODUCTION
Recently, the investigation of nonlinear elastic waves has attracted much research attention due to a wide range of technical and industrial applications, such as geophysical exploration, 1 soft tissue acoustics, 2 dynamics of elastomer, 3 nondestructive evaluation, 4,5 and some biomedical applications. 6Actually, just as in fluid and gas dynamics, which have long been the kernel of the traditional nonlinear science, nonlinear effects are becoming increasingly critical in solid mechanics research. 7Compared with linear waves, nonlinear waves have some distinct and attracting properties.For example, the nonlinear acoustic properties are much more sensitive for detection and characterization of the microstructure in a material than the linear ones. 8s a special kind of nonlinear waves, solitary waves can propagate over a long distance without distortion.The solitary waves were first observed in the field of fluid mechanics in 1834 by John Scott Russell.In 1895, this fascinating phenomenon was successfully interpreted by Korteweg and de Vries who developed the well-known KdV equation.Since then, problems related to solitary wave propagation have been widely investigated.Due to their remarkable characters, solitary waves are a subject of considerable interest in many fields including solid mechanics.With the development of high-resolution optical methods for wave detection, solitary waves have also been observed experimentally in elastic solids. 9,10There are many excellent theoretical works on solitary waves in solids.2][13][14][15][16] Besides, Yong and LeVeque investigated the longitudinal elastic strain solitary waves in a one-dimensional periodically layered medium. 17Maugin studied the possibilities of existence of solitary surface waves travelling over a substrate. 18Dai et al. studied analytically the interaction of two solitary waves in a circular cylindrical rod. 19A detailed review of the study of solitary wave in solids is referred to the paper of Maugin. 20he above mentioned solitary waves are usually generated due to the balance between nonlinearity and dispersion.However, dissipation is always present in a realistic situation.There are also many works in the field of nonlinear waves in solids considering the effect of dissipation.Destrade et al. studied the nonlinear shear waves propagating in viscoelastic materials whose generation is directly linked to the nonlinear viscosity term. 21Hayes and Saccomandi studied the propagation of finite amplitude shear waves in Mooney-Rivlin viscoelastic materials maintained in the static state of a pure homogenous deformation. 22,23Destrade and Saccomandi then extended to the case of inhomogeneous plane waves. 24They also studied the interaction of a longitudinal wave with a transverse wave in viscoelastic materials. 25Zabolotskaya et al. developed an evolution equation for nonlinear shear waves in soft isotropic solids with viscous dissipation. 26It should be noted that the nonlinear elastic and dissipative behavior of rocks has been recently observed in many experiments. 27As is well-known, the combination of nonlinearity, dispersion and dissipation may lead to the generation of kink-shaped solitary waves or simply kink waves. 28or soft or hyperelastic materials, applying a mechanical biasing field can conveniently modulate their effective properties.Chen and Dai studied the influences of pre-stretch and biasing electric field on wave dispersion in a soft cylinder. 29Zhang et al. found that the incremental constitutive relations of a particularly pre-stretched soft electroactive half-space are similar to the ones of a transversely isotropic piezoelectric body. 30Huang et al. studied the effect of mechanical biasing field on the propagation of longitudinal waves in soft periodic structures. 31Stress pulses in metamaterials, which can be modulated by pre-compression, were explored numerically and experimentally by Xu and Nesterenko. 32By using the method of coupled series-asymptotic expansions, Dai and Peng investigated waves in a pre-stretched Blatz-Ko cylinder and concluded that a variety of waves can arise, including solitary waves and kink waves. 33Recently, we theoretically investigated solitary waves propagating in electroactive rods, and found that the biasing electric field can be used to adjust the waves. 34However, to the authors' knowledge, there are few works on nonlinear waves in pre-stretched structures composed of viscoelastic materials, which motivates the present work.
The present article focuses on the investigation of kink and kink-like longitudinal waves in pre-stretched Mooney-Rivlin viscoelastic rods.The Cauchy stress is split into an elastic part, which is derived from the classical Mooney-Rivlin elastic material, and a dissipative part, which is identical to the one in fluid.In the limit of finite-small amplitude and long wavelength, we simplify the three-dimensional (3D) nonlinear governing equations to one-dimensional (1D) ones by making use of the asymptotic expansions of variables as in Dai and Huo. 16Then, using the reductive perturbation method gives rise to the far-field equation (the KdV-Burgers equation).Finally, two kinds of explicit wave solutions are presented, namely the kink and kink-like waves, which correspond to the saddle-node heteroclinic orbit and the saddle-focus heteroclinic orbit of the equation, respectively.Examples are given to show the influences of pre-stretch and viscosity on the wave shape and wave velocity.The potential application of such waves is to measure the viscosity coefficient of the material.The competition between the effects of pre-stretch and viscosity on kink and kink-like waves is also uncovered.

A. Basic formulations
Let a material point, in an undeformed body which occupies a region Γ 0 with the outward normal N in the reference configuration, be identified by its position vector X.After a time T, the material point is at the position vector x, which occupies a region Γ with the outward normal n in the current configuration.Thus, the motion of the body can be described by For an incompressible Mooney-Rivlin material with energy density function Ω, the (elastic) Cauchy stress tensor can be described by the constitutive relations 29 where Ω m = ∂Ω/∂I m (m = 1, 2), I is the unit tensor, b = FF T is the left Cauchy-Green strain tensor with F = ∂x/∂X being the deformation gradient tensor, P is the undetermined pressure related to the constraint of incompressibility detF = 1, and I m are the scalar invariants: where "tr" is the trace operator.
To describe the effects of dissipation, we adopt the following viscous stress tensor for the incompressible viscoelastic materials 21,25 where D = 1/2(L + L T ) is the rate of deformation tensor with L = (∂F/∂T)F −1 , η is the viscosity coefficient which should be positive, and T is the time.A detailed discussion of proper formation of the viscous stress tensor can be found in Ref. 35.Through simple combination, the nonlinear constitutive equation of a viscoelastic material may be expressed by where with µ being the shear modulus and β a material constant.The equations of motion, in the absence of body forces in Γ 0 , are given by where Σ = F −1 τ is the nominal stress tensor and "Div" is the divergence operator.The boundary conditions in Γ 0 are given by where t A is defined by t A dA = t a da.Here dA and da are the unit areas in Γ 0 and Γ, respectively, and t a is the applied mechanical traction vector per unit area in Γ.

B. Longitudinal waves with small but finite amplitude
To study the axisymmetric wave motion in a circular rod, we prefer adopting cylinder coordinates, with (R, Θ, Z) and (r, θ, z) corresponding to the reference and current configurations, respectively.Considering the axisymmetric motion superposed on a finite static axisymmetric deformation: where λ 1 and λ 2 are the pre-stretches, U and W are the displacements along the r and z directions, respectively, p 0 is the pressure in the deformed state, and p is the incremental pressure.From Eq. ( 8), we obtain Furthermore, we can arrive and To obtain Eq. ( 11), we have made use of the constraint of incompressibility detF = 1 and neglected the terms which are higher than the first order.As a convention, here and hereafter, the subscript letter denotes partial differentiation, while the subscript letter inside the brackets denotes coordinate direction.
With Eqs. ( 9) and ( 11) substituted into Eq.( 5), we get the expressions of the Cauchy stress tensor including the effect of viscosity.The nominal stress tensor can be derived as well.It is noted that for the finite but small disturbance, it is reasonable to neglect the higher order terms.Furthermore, for weakly viscoelastic materials, it is reasonable to assume that the viscous stresses are much smaller than the elastic stresses. 36Similar assumption was adopted by Zabolotskaya et al. 26 and Catheline et al. 37 Thus, for the elastic stresses, we neglect the terms which are higher than the second order, while for the viscous stresses, we only retain the first order terms.With the approximate expressions for nominal stresses thus obtained (see Appendix A), the equations of motion ( 6) can be rewritten as We assume that the lateral surface is free from tractions.From Eq. ( 7), we get the boundary conditions as which reduce to at R = a, where p 0 = 2λ 2 1 Ω 1 + 2λ 4 1 Ω 2 + 2λ 2 Ω 2 which can be obtained from the boundary condition Σ (Rr ) = 0 of the rod in the deformed state.
From Eq. ( 10), the constraint of material incompressibility can be reduced to where the terms which are higher than the second order have been omitted, and the following relation has been noticed Eqs. ( 12), ( 13) and ( 15)-( 17) can be used to completely describe the nonlinear dynamics of viscoelastic rods in the limit of finite but small amplitude.
For such axisymmetric problems, S = R 2 will be a more natural radial variable than R. 16 Also, as in Ref. 16, we introduce the transformation U = RV (Z, S,T).These two changes of variables are now used to simplify the governing equations ( 12), ( 13) and ( 15)- (17).For convenience, we will further adopt the following scales to non-dimensionalize the governing equations where µ is the shear modulus of the material, h is a characteristic axial displacement, and l is a characteristic wavelength, c is a characteristic speed to be determined later (see Eq. ( 41)).For long waves with finite but small amplitudes, ε = h/l is a small dimensionless parameter.The following dimensionless material constants will also be needed: Thus, in view of Eqs. ( 19) and ( 20), we get the dimensionless governing equations as follows: The corresponding boundary conditions are at s = δ, where δ = a 2 /l 2 is also a small parameter for long waves.As can be seen, the variable R doesn't appear explicitly in the resulting system of governing equations, and s seems to be a more natural radial variable as compared with R in the original system.Eqs. ( 21)-( 25) are complex two-dimensional (2D) nonlinear partial differential equations, which are still too difficult to get an analytical solution.For a slender rod, to further simplify the equations, we adopt the asymptotical method introduced by Dai and Huo 16 to tackle such a complicated system.First, the unknowns (w, v and p) can be expanded in the neighborhood of s = 0 as follows: Substituting Eqs. ( 26)-( 28) into Eqs.( 21)-( 25) and setting the coefficient of each power of s to be zero, we can transform the 2D problem to the 1D problem involving only two variables (x and t).
The governing equations are The corresponding boundary conditions are where δ/ε = O(1) is assumed.The above seven equations give a set of 1D nonlinear equations for seven unknowns w i v j and p j (i = 0, 1, 2 and j = 0, 1), in which we have neglected O(δε, ε 2 ) terms.

A. Derivation of the KdV-Burgers equation
To derive the far-field equation, we follow the procedure of the reductive perturbation method and introduce the following transformation: and w i , v j and p j (i = 0, 1, 2 and j = 0, 1) have the following perturbation expansions Inserting Eqs. ( 36) and (37) into Eqs.( 29)- (35), we obtain at O(ε 0 ) where H 0 = w 00ξ , w 10 , w 20 , v 00 , v 10 , p00 , p10 T , and where In order to obtain the nontrivial solutions, we set which gives It determines the characteristic speed c, which depends on the pre-stretch λ 1 .This result coincides with the one obtained in Ref. 16 if we set λ 1 = λ 2 = 1.
Substituting Eq. ( 41) into Eq.( 38), we obtain The left eigenvector L e of the coefficient matrix M 0 is where )∂/∂ξ.Similar to Eq. ( 38), another seven equations at O(ε 1 ) can be easily obtained.Making use of Eq. ( 42), we can simplify these equations to where Q 1 ∼ Q 8 are coefficients (in vector form) given in Appendix B. In order to suppress the secular term, we multiply the left-hand side of Eq. ( 44) with the left eigenvector L e to get the following nonlinear evaluation equation where , where η0 = µ −1 c T η/h is also a dimensionless viscosity coefficient, which is independent of the pre-stretch, and c T =  µ/ρ is the shear wave velocity.Eq. ( 45) is the KdV-Burgers type equation with the nonlinear coefficient C 1 , the dissipative coefficient C 2 , and the dispersive coefficient C 3 .In this paper, we only consider the case of λ 1 ≥ 1 (i.e. the rod is subjected to a pre-stretch, not pre-compression).Thus, we have C 3 > 0. Due to the balance of nonlinearity, dissipation and dispersion, there exists a steady kink (or kink-like) wave propagating in the rod.If the dissipation is neglected, we can reduce the KdV-Burgers equation into the KdV equation.Furthermore, when λ 1 = 1, the KdV equation thus obtained is identical to the one in our previous paper 34 if the electroelastic coupling is neglected there.

B. Travelling wave solutions
Eq. ( 45) is not a standard KdV-Burgers equation.We take the following transformation Inserting Eq. ( 46) into Eq.( 45), we obtain Eq. ( 47) admits both travelling wave solutions corresponding to saddle-node and saddle-focus heteroclinic orbits, respectively. 39In order to obtain the travelling wave solution, we assume Substituting Eq. ( 48) into Eq.( 47) and integrating once with respect to ζ both sides of the equation, we can get where A is an integral constant, which depends on the initial conditions.When the variable ζ approaches infinity, Λ, Λ ζ and Λ ζζ should gradually become zero.Thus, it's reasonable to set A = 0.The solution corresponding to the saddle-node heteroclinic orbit for the KdV-Burgers equation was first obtained in Jeffrey and Xu 40 through a nonlinear transformation method.Alternatively, the solution may be obtained through the expansion of tangential function, 41 as follows where v = 6C 2 2 /25C 3 has been determined in the process of derivation.For the saddle-focus heteroclinic orbit, the analytical solution can be obtained by following Liu and Liu 39 for C 3 > 0 as where v will be a given parameter.This wave can be divided into two parts: the right part is a solitary wave in which the dissipative term is neglected, and the left is a damped oscillation due to dissipation.Inserting Eqs. ( 50) and (51) into Eq.( 46), we can get the expression of w 00ξ , which in turn gives rise to the following leading order of the travelling wave solutions: and Eqs. ( 52) and ( 53) are the kink and kink-like waves, respectively.It should be pointed out that the kink-like wave in Eq. ( 53), which corresponds to the saddle-focus heteroclinic orbit, is often overlooked in literatures.However, such a wave profile is exactly the same as what has been observed in fluid. 39,42Thus, it should be important both in practice and in science and technology.The kink wave in Eq. ( 52) corresponds to the saddle-node heteroclinic orbit.As expected, the kink (kink-like) waves can be modulated by the pre-stretch λ 1 .Substituting Eqs. ( 52) and (53) into Eq.( 42), we can get the expressions of other physical quantities, including the expression of v 00 , which is very important to the experiment. 9In the next part, we will discuss the influence of pre-stretch and viscosity on the wave shape and wave velocity.

IV. NUMERICAL RESULTS AND DISCUSSIONS
In this part, we shall discuss through numerical examples but based on the analytical solutions obtained in the last section how the pre-stretch and viscosity affect the wave shape and wave velocity.In the calculation, we take β = 1/6 and ε = δ = 0.3 for example.It is noted that the Mooney-Rivlin model can be degenerated into the neo-Hookean model when β = 1/2.In this case, the kink wave solutions are still available, which can be proven from Eq. (45).
For the discussion about the influence of pre-stretch on the wave velocity, it becomes inappropriate to use the dimensionless variable t since it depends on the pre-stretch, see Eq. (19).Thus, we employ the following new dimensionless time variable: where c T =  µ/ρ is defined below Eq. (45).Then, Eqs. ( 52) and (53) can be rewritten as where are the wavelength, wave velocity, and wave amplitude, respectively, and where c p2 = (1 + εv)  4λ 6 1 Ω1 + 2 Ω1 + 6λ 8 1 Ω2 is also the wave velocity.Fig. 1 depicts three different kink waves underlying different pre-stretches at t 0 = 0.It should be noted that these wave solutions have a saddle-node herteroclinic orbit, and result from the balance of dissipation, dispersion, and nonlinearity (see Eq. ( 55)).If the dissipation is ignored, the dissipative coefficient C 2 would be zero, so the kink waves would not be generated.It is seen that the pre-stretch makes the wave lower and wider.Therefore, the pre-stretch has a repressive function on the wave.Fig. 2  viscosity coefficients at t 0 = 0. Comparing these waves with each other, we find that the kink wave with a larger viscosity will have a higher amplitude and narrower wavelength.If the viscosity is small enough (for example, η0 = 0.5 as in Fig. 2), the wave will gradually become flatted.From Figs. 1 and 2, we recognize that the pre-stretch and viscosity of the material have absolutely opposite influence on such kind of kink waves.So it is not surprising that waves propagating in the rod with two different viscosity coefficients may have very similar wave shapes if the underlying pre-stretches are appropriately applied, see Fig. 3. From Eq. ( 42) 1 , we find that the radial displacement is proportional to the longitudinal strain with the same wave velocity.This enables us to determine the nonlinear wave characteristics in rods through measuring the radial displacement which is experimentally more feasible. 9Fig. 4 shows three different radial displacements of kink waves propagating in the viscoelastic rod.As expected, we find that the influences of the pre-stretch and viscosity on the wave shape are the same as on the axial strain.It is noticed that the radial displacements for λ 1 = 1.1, η0 = 0.5 and λ 1 = 1.2, η0 = 0.71 are with different shapes, unlike the situation for the axial strains in Fig. 3.This is simply due to the fact that the proportional factor in Eq. ( 42) 1 depends on the pre-stretch.All the results show that for kink waves, the viscosity coefficient of the material has a dominant effect on the wave shape.Thus, we may use this property to measure the viscosity coefficient of the material.For example, we can use the measured wave amplitude to calculate the viscosity coefficient by where H v is the wave amplitude of the radial displacement.Figs.5-7 show the variations of wavelength, wave velocity and wave amplitude with the pre-stretch.From Fig. 5, we find that, with the increase of viscosity, the wavelength becomes smaller and with the increase of pre-stretch, the wavelength becomes larger, which may be explained intuitively that the dispersion is strengthened due to the decrease of the radius of the rod when it is stretched.It is interesting that the influence of the viscosity on the wavelength becomes significant when the pre-stretch is large enough.Furthermore, the influence of viscosity on the wave velocity become smaller as the pre-stretch increases, see Fig. (for example, η0 = 2.0), the variation of wave velocity becomes no longer monotonous.Thus, there should be a range where the wave velocity will decrease with the increase of pre-stretch.This is somehow against the intuition.On the other hand, just as expected, the influence of viscosity on the wave amplitude will gradually become small with the increase of pre-stretch.When the pre-stretch is big enough, the wave amplitude will approach to zero, see Fig. 7.
Fig. 8 depicts the kink-like waves, which correspond to the saddle-focus heteroclinic orbit of the KdV-Burgers equation.The other parameters are fixed as η0 = 1 and v = 20.At the right-most part of the wave, it behaves like a solitary wave for which the dissipation can be neglected.When the wave amplitude arrives at the maximum point, the field is controlled by damped oscillation due to dissipation, and the amplitude will gradually become smaller and smaller.Finally, the axial strain will approach a constant.This kind of waves actually reflects the cascading down process of energy, which is an important property of turbulence. 39By comparing Figs.8(a), 8(b) and 8(c) with each other, the influence of pre-stretch on such kind of waves can be uncovered.Just like the kink waves, the wave amplitude of kink-like waves will become lower and the wavelength will become wider with the increase of pre-stretch.Furthermore, we also find that the left part of the wave in Fig. 8  decays more rapidly than the corresponding ones in Fig. 8(b) and Fig. 8(c).Therefore, we can get the conclusion that the pre-stretch can weaken the effect of viscosity on of the kink-like waves.

V. CONCLUDING REMARKS
We studied the propagation of kink and kink-like waves in pre-stretched Mooney-Rivlin elastic rods with the consideration of viscous dissipation.The Cauchy stress tensor consists of an elastic part and a dissipative part.Several asymptotic expansions were introduced to simplify the 3D governing equations for a rod to the 1D ones.The boundary conditions on the lateral surface of the rod were satisfied asymptotically.Using the reductive perturbation method, we obtained the KdV-Burgers equation, which admits analytical and explicit wave solutions.
Two kinds of travelling wave solutions for the KdV-Burgers equation are given in the present paper.They correspond to the saddle-node heteroclinic orbit and saddle-focus heteroclinic orbit of travelling wave solutions, respectively.For the discussions, we mainly paid our attention to the influences of pre-stretch and viscosity on the wave shape and wave velocity.We found that the pre-stretch will make the kink waves lower and wider.Moreover, the pre-stretch can also be used to modulate the wave velocity.Furthermore, a larger viscosity coefficient will lead to a higher and narrower wave.Thus, we may use kink waves to measure the viscosity coefficient of the material.
Last but not least, we uncover the competition between the influences of pre-stretch and viscosity on kink (kink-like) waves.For example, for the wave with a saddle-node heteroclinic orbit, as the pre-stretch increases, the effect of viscosity on the wavelength will become more remarkable; while its effect on the wave amplitude and wave velocity becomes smaller with the increase of pre-stretch.Furthermore, if the viscosity coefficient is large enough, the variation of wave velocity will no longer monotonously vary with the pre-stretch.For the wave with a saddle-focus heteroclinic orbit, we uncover that the pre-stretch can weaken the effect of viscosity, which will decrease the wave amplitude.