The stress wave physics in incompressible, linear elastic, isotropic and homogeneous solids is perhaps the simplest possible physics. In these solids, the incident stress wave maintains its amplitude and base (support) during propagation, thus preserving its mechanical energy. In incompressible, linear elastic, isotropic and homogeneous solids with description, i.e. in linear viscoelastic solids, the fixed energy content (mechanical energy) of a wave such as a stress or velocity pulse continuously results in entropy generation during propagation causing amplitude decay and base elongation of the wave or the pulse, but maintaining the wave shape and wave speed due to incompressibility of the medium. In such physics upon continued evolution, the entire energy of the pulse is converted into entropy resulting in temperature rise of the medium, leading to a constant temperature equilibrium as stationary state of the evolution of wave propagation.

In compressible, isotropic, homogeneous, inviscid solid medium, the compressibility causes change in density, thus for fixed modulus of elasticity, the wave speed v = E / ρ changes in the medium as the density changes, compression resulting in higher density reduces wave speed while tension reducing density causes increase in the wave speed. We consider a simple illustrative example to demonstrate the consequences of changing wave speed. Consider an axial rod consisting of isotropic, homogeneous elastic matter fixed at the left and subjected to a compressive stress pulse of duration 2 Δ t at the right end. Let the dimensionless wave speed be one. At the end of t = 2 Δ t , the wave is completely in the solid rod (Figure 1(a)). As the compressive stress increases from A to B (Figure 1(a)), the density increases from ρ 0 at point A (initial density) to maximum value ρ > ρ 0 at point B resulting in progressively reducing wave speed from A to B. Along B to C, the progressively reducing stress causes progressively reduced value of density from its peak value ρ at point B and eventually at point C, we have density ρ 0 (Figure 1(b)). Thus, from A to B (or a to b), we have progressively reducing wave speed, the minimum value being at point B (or b) and from B to C (or b to c), we have progressively increasing wave speed and at C, we have reference wave speed (same as at A). Thus in the support AB of the pulse, the stress wave at A 2 is moving faster than the wave at B and the wave at A 1 is moving faster than the wave at A 2 resulting in piling up of the stress waves from A to B upon continued evolution causing steepening of wavefront AB as shown in Figure 1(a) at a later value of time ( t n ). This is often referred to as shock front of the stress wave. On the other hand, from BC, the progressively increasing wave speed results in the waves ahead of a wave moving at a faster speed, i.e. wave at B 1 , is moving faster than the wave at B and the wave at B 2 is moving faster than the

Figure 1. Evolution of d σ 11 [ 0 ] and ρ for compressive d σ 11 [ 0 ] . (a) Evolution of compressive d σ 11 [ 0 ] ; (b) Evolution of ρ for compressive d σ 11 [ 0 ] .

wave at B 1 resulting in stretching or shallowing of the wave front BC shown in Figure 1(a) at a time t n > t 1 . Thus, we see that in case of a compressive stress wave propagating in compressive solid medium the shock formation in the wave occurs behind the peak of the wave and the swallowing or rarefaction occurs ahead of the wave. Figure 1(b) shows initial density wave at time t 1 = 2 Δ t and at time t n > t 1 the density wave with shock front and rarefaction occurring in the same manner as in case of stress wave.

Wave propagation studies in linear elastic solids have been a subject of investigation for a long time. Wave physics in compressible solids with dissipation, rheology and isothermal as well as nonisothermal physics have not been investigated much, hence published works in this area are very limited. Recently, Surana et al. [1]-[3] investigated shock physics in compressible solids with dissipation and rheology using simple equation of state under isothermal assumption, i.e. in these studies conversion of mechanical energy into entropy and associated thermal effects existed but were not monitored by not considering energy equation in the mathematical model. In these studies, the wave propagation domain was considered insulated, the entropy generation and thermal effects are only due to dissipation that are small, thus had virtually no effect in the material coefficients, i.e. deformation field and thermal physics were decoupled in these studies. Authors in references [1]-[3] have reported pertinent literature review related to wave propagation, mostly linear wave propagation with some work related to weak shock waves. The mathematical models in almost all cases are compromised descriptions compared to the conservation and balance laws of classical continuum mechanics and constitutive theories are almost never based on representation theorem [4]-[15]. The literature review presented by Surana et al. [1]-[3] and Abboud [16] is quite comprehensive but not repeated here for the sake of brevity. Interested readers can see references [1]-[3][16].

The mathematical model used in the present work includes energy equation as part of the group of PDEs constituting the complete mathematical model. This aspect of the mathematical model is not considered in references [1]-[3]. The energy equation is essential to determine the conversion of mechanical work (some) into heat energy through entropy generation. Tracking of entropy generation through various phases of shock physics is helpful and essential in understanding the extend to which mechanical work is converted into thermal energy especially during shock formation and shock reflections. Higher entropy values indicate more mechanical energy being converted into thermal energy. The solutions of the mathematical model consisting of nonlinear PDEs in space and time are obtained by using space-time residual functional based space-time finite element formulation established and used extensively by Surana et al. [17] in which the space-time local approximation in hpk scalar product spaces permits higher order global differentiability in space and time, a novel feature of the computational framework.

The research presented here is extension of the works of Surana et al. [1]-[3] for nonisothermal shock physics. In the work presented here, the energy equation is integral part of the mathematical model, hence entropy generation due to mechanical work and due to thermal physics is monitored and the resulting thermal physics associated with the shock waves is considered in addition to mechanical deformation. The three distinct aspects of this work presented here are described in the following.

The first aspect is the derivation of the details of the mathematical model consisting of: conservation and balance laws in Lagrangian description based on classical continuum mechanics: conservation of mass, balance of linear momenta, balance of angular momenta, energy equation, entropy inequality, constitutive theories and equation of state. In the derivation of the conservation and balance laws compressibility physics requires consideration of finite deformation, finite strain physics, hence we consider contravariant second Piola-Kirchhoff stress tensor and covariant Green’s strain tensor as measures of stress and strain. Dissipation physics is described by rates of Green’s strain tensor up to order n, hence ordered rate dissipation mechanism, yielding a spectrum of dissipation coefficients corresponding to the strain rates. Rheology or memory mechanism is incorporated using rates of contravariant second Piola-Kirchhoff stress tensor up to orders m, hence yielding ordered rate rheology mechanism with a spectrum of relaxation times. Energy equation accounts for the conversion of mechanical energy into entropy as well as other aspects of entropy generation. Entropy inequality cast in Helmholtz free energy density provides mechanism for determining constitutive tensors and the initial determination of their argument tensors. These argument tensors are augmented due to dissipation physics and rheology and some constitutive tensors are redefined due to consideration of rheology. Constitutive theories are derived using representation theorem [4]-[15][18][19]. Simple equation of state is considered in which the thermodynamic pressure only depends upon density. Equation of state is used to define equilibrium contravariant second Piola-Kirchhoff stress tensor. This mathematical model consists of nonlinear partial differential equations in dependent variable, space coordinates and time, hence constitutes IVP. The mathematical model in R 3 has closure. This mathematical model in R 3 is reduced to R 1 for pure one dimension wave physics studies in compressible nonlinear elastic medium with dissipation and rheology.

The second aspect of the work presented here is the unconditionally stable computational infrastructure used to obtain accurate solutions of IVPs that has built in measure of error in the computed solution (without the knowledge of theoretical solution) and built in adaptivity mechanism to improve the accuracy of the computed solutions. Based on Surana et al. [17], the space-time coupled method in which the integral forms are constructed using space-time residual functional and calculus of variation and the space-time local approximations are in higher-order scalar product spaces permitting higher degree of polynomials as well as higher order global differentiability is highly meritorious. In this approach, when the space-time integrals over the space-time discretization are Riemann and when the integrated sum of squares of space-time residual functional (I) approaches zero i.e. of the order of O ( 10 − 8 ) or lower, the PDEs in the mathematical model are satisfied accurately in the point-wise sense. Thus, proximity of residual functional (I) to zero is an absolute measure of error. This approach obviously does not require theoretical solutions. Adaptively to improve the solution to achieve lower values of I based on h,p,k is inherent in this computational method. Converged solution is obtained for the first space-time strip (or slab). This is followed by computation of the solution for the second space-time strip (or slab) using ICs from the first time strip. This is continued till the desired time is reached. This approach is efficient compared to space-time mesh. In this approach only after obtaining converged solution for the current space-time strip, the solution is advanced to the next space-time strip, hence ensuring converged solution for the all space-time strips, hence the entire space-time domain.

The third aspect of this work is model problem studies using 1D wave physics in compressible and incompressible solid medium with dissipation and rheology. In each study wave propagation, shock formation, propagation of waves with shocks, reflection of waves with shocks, interaction of waves, entropy generation during evolution resulting in propagating temperature waves with and without shocks, their reflections and interactions one considered.

The mathematical model in Lagrangian description consisting of conservation and balance laws of classical continuum mechanics, constitutive theories, and their dimensionless forms for nonlinear and nonisothermal 3D deformation physics of thermoelastic and thermoviscoelastic solid mediums with and without rheology is presented. This mathematical model is specialized for 1D wave propagation. Details of the space-time coupled finite element method based om space-time residual functional and calculus of variations and the solution procedure for nonlinear algebraic equations for a space-time strip with time marching are also presented in the paper. Extensive model problem studies are presented to clearly illustrate accurate simulation of complex shock physics. Summary and conclusion are given in the last section of the paper.

Thus we observe that rate of entropy density generation is quite complex in polymeric solids. We note the following:

1) In the absence of rheology, η w 2 = 0 .

2) In the absence of dissipation and rheology, η w 2 = 0 and η w 4 = 0 .

3) If the matter is incompressible, then η w 1 = 0 . In this case, the stress, strain, and strain rate measures become d σ 11 , ε 11 , and ε ˙ 11 , in which d σ 11 is Cauchy stress, ε 11 , ε ˙ 11 are linear strain and strain rate.

4) We note that rate of entropy density generation η w 3 is never zero in elastic solids as it is due to strain and strain rate. This aspect is absent in fluent media due to absence of strain.

5) When the medium is inviscid and incompressible only η w 3 is nonzero.

6) Rate of entropy density η and total entropy Ψ.

All conservation and balance laws are independent of the volume of matter, hence can be viewed as being valid for unit volume. η , hence η q and η w represents rate of specific entropy i.e. entropy per unit volume. We integrate η over each space-time finite element and then sum them over the elements to obtain Ψ, total entropy as referred and used in the paper.

In the model problem studies we integrate η , hence η g and η w i ; i = 1 , 2 , 3 , 4 over the discretization of each space-time strip ( Ω ¯ x t ( i ) ) ; i = 1 , 2 , ⋯ , n obtain entropy Ψ and entropies Ψ q , Ψ w i ; i = 1 , 2 , 3 , 4 . For ( Ω ¯ x t ( i ) ) T we can write

or

Substitution of η w from (58) in (60) follows. In the model problem studies Ψ is reported.

The mathematical model considered for the model problem studies consists of dimensionless partial differential Equations (38)-(43) in dependent variable u 1 ,   d σ 11 [ 0 ] , q 1 , θ and v 1 that are functions of space coordinates x 1 and time t . The solutions of the initial value problems defined by these equations is obtained by using space-time coupled finite element method based of space-time residual functional [17] in which the local approximation are ( u 1 ) h e , ( d σ 11 [ 0 ] ) h e , ( q 1 ) h e , ( θ ) h e and ( v 1 ) h e for the dependent variables u 1 ,   d σ 11 [ 0 ] , q 1 , θ and v 1 are over a space-time element Ω ¯ x t e . The local approximations are p-version hierarchical with higher order global differentiability in space as well as time, hence are in H ( Ω ¯ x t e ) ( p , k ) scalar product spaces.

Figure 2(a) shows space time domain Ω ¯ x t = Ω ¯ x × Ω ¯ t . Figure 2(b) shows discretization Ω ¯ x t T of Ω ¯ x t in space-time strip Ω ¯ x t ( i ) ; i = 1 , 2 , ⋯ , n given by the

Figure 2. Space-time domain and its discretizations using space-time elements and space-time strips. (a) Space-time domain Ω ¯ x t ; (b) Discretization Ω ¯ x t into a space-time strip; (c) Discretization of i th space-time strip Ω ¯ x t using space-time coupled finite elements.

following.

Consider i t h space-time strip discretized using space-time finite elements (Figure 2(c)). Let the local approximations for u 1 ,   d σ 11 [ 0 ] , q 1 , θ and v 1 be given by (equal order equal degree)

Approximations ( u 1 ) h e , ( d σ 11 [ 0 ] ) h e , ( q 1 ) h e , ( θ ) h e and ( v 1 ) h e over ( Ω ¯ x t ( i ) ) T are given by

in which { δ } u 1 , { δ } d σ 11 [ 0 ] , { δ } q 1 , { δ } θ and { δ } v 1 be degrees of freedom for an element Ω ¯ x t e for u 1 ,   d σ 11 [ 0 ] , q 1 , θ and v 1 and the degrees of freedom for u 1 ,   d σ 11 [ 0 ] , q 1 , θ and v 1 for ( Ω ¯ x t ( i ) ) T are given by

The total degrees of freedom for ( Ω ¯ x t ( i ) ) T are given by { δ }

substituting ( u 1 ) h e , ( d σ 11 [ 0 ] ) h e , ( q 1 ) h e , ( θ ) h e and ( v 1 ) h e in Equations (38), (39) and (41)-(43), we obtain residual equations E i e ; i = 1 , 2 , ⋯ , 5 over Ω ¯ x t e .

these hold ∀ x 1 ∈ Ω ¯ x t e , the domain of space-time element e . We construct space-time residual functional I over ( Ω ¯ x t ( i ) ) T given by

In which E i e ; i = 1 , 2 , ⋯ , 5 are residual equations for an element e with domain Ω ¯ x t e given by (67)-(71).

A necessary condition for obtaining an extremum of I is that its first variation must vanish at the extremum provided functional I is differentiable in its arguments. Using (72), we can write

Let

and

In which { g e } is a nonlinear function of { δ e } . Following Surana et al. [17] we find a solution { δ } that satisfies { g } = 0 iteratively using Newton’s linear method with line search. Details are given in the following. Let { δ } 0 be an assumed or starting solution, then

Let { Δ δ } be corrections to { δ } 0 such that

We expand (77) in Taylor Series about { δ } 0 and retain only up to linear term in { Δ δ } (Newton’s linear method or Newton-Raphson method).

in which

which is second variation of I.

Using (73)

Surana etal. [17] have shown that a sufficient condition or an extremum principle is possible if δ 2 E i , δ 2 E i e is neglected in (81). Justification and validity of this approximation are described in reference [17]. Thus, now we have

or

where

We calculate { Δ δ } using

and the improved solution is given by (using line search [17])

In which α * is determined using the smallest value of I.

This is referred to as line search. We check for convergence i.e. when | g i | max ≤ Δ , Δ = O ( 10 − 6 ) or lower the solution { δ } is the solution.

Main steps in computations:

1) Assume { δ } 0 , starting or assumed solution

2) Calculate { g e } = ∑ i = 1 5 ( E i e , δ E i e ) = { f e }

3) Assemble { g e } to obtain { g } = ∑ e { g e }

4) Calculate δ 2 I e = ∑ i = 1 5 ( δ E i e , δ E i e ) Ω ¯ x t e