Stability of Classical Shock Fronts for Compressible Hyperelastic Materials of Hadamard Type

Abstract

This paper studies the uniform and weak Lopatinskiĭ conditions associated with classical (Lax) shock fronts of arbitrary amplitude for compressible hyperelastic materials of Hadamard type in several space dimensions. Thanks to the seminal works of Majda (Mem Amer Math Soc 43(281):v+93, 1983, 41(275):iv+95, 1983) and Métivier (Trans Am Math Soc 296:431–479, 1986, Commun Partial Differ Eqs 15(7):983–1028, 1990, Stability of multidimensional shocks, in: Freistühler H, Szepessy A (eds) Advances in the theory of shock waves, vol 47 of progress in nonlinear differential equations and their applications. Birkhäuser, Boston, pp 25–103, 2001), the uniform Lopatinskiĭ condition ensures the local-in-time, multidimensional, nonlinear stability of such fronts. The stability function (also called Lopatinskiĭ determinant) for shocks of arbitrary amplitude in this large class of hyperelastic materials is computed explicitly. This information is used to establish the conditions for uniform and weak shock stability in terms of the parameters of the shock and of the elastic moduli of the material.

Appendices

Appendix A. Multidimensional Stability of Planar Shock Fronts

For convenience of the reader, in this section we gather basic information about the stability conditions for multidimensional shock fronts. The reader is referred to the books by Benzoni-Gavage and Serre [10], Majda [71, 72] and Serre [88] for more information. Consider a hyperbolic system of n conservation laws in $d \ge 2$ space dimensions of the form

$$\begin{aligned} u_t + \sum _{j=1}^d f^j(u)_{x_j} = 0, \end{aligned}$$

(A.1)

where $x \in {{\mathbb {R}}}^d$ and $t \ge 0$ are space and time variables, respectively, and $u \in {\mathcal {U}}\subset {{\mathbb {R}}}^n$ denotes the vector of n conserved quantities (here ${\mathcal {U}}$ denotes an open connected set). The flux functions $f^j \in C^2({\mathcal {U}}; {{\mathbb {R}}}^n)$, $j = 1, \ldots , d$, are supposed to be twice continuously differentiable and to determine the flux of the conserved quantities along the boundary of arbitrary volume elements. System (A.1) is hyperbolic in ${\mathcal {U}}$ if for any $u \in {\mathcal {U}}$ and all $\xi \in {{\mathbb {R}}}^d$, $\xi \ne 0$, the matrix

$$\begin{aligned} A(\xi ,u) := \sum _{j=1}^d \xi _j A^j(u), \end{aligned}$$

(A.2)

where $A^j(u) := Df^j(u) \in {{\mathbb {R}}}^{n \times n}$ for each j, is diagonalizable over ${{\mathbb {R}}}$ with eigenvalues

$$\begin{aligned} a_1(\xi ,u) \le \ldots \le a_n(\xi ,u), \end{aligned}$$

(A.3)

of class at least $C^1({\mathcal {U}}\times {{\mathbb {R}}}^d; {{\mathbb {R}}})$, called the characteristic speeds. Each eigenvalue $a_j(\xi ,u)$ is semi-simple (algebraic and geometric multiplicities coincide), with constant multiplicity for all $(u,\xi ) \in {\mathcal {U}}\times {{\mathbb {R}}}^d \backslash \{0\}$. The matrix $A(\xi ,u)$ has a complete set of right (column) eigenvectors $r_1(\xi ,u), \ldots , r_n(\xi ,u) \in C^1({{\mathbb {R}}}^d \times {\mathcal {U}};{{\mathbb {R}}}^{n \times 1})$, satisfying $A(\xi ,u) r_j(\xi ,u) = a_j(\xi ,u) r_j(\xi ,u)$ for each j, as well as a complete set of left (row) eigenvectors $l_1(\xi ,u), \ldots , l_n(\xi ,u) \in C^1({{\mathbb {R}}}^d \times {\mathcal {U}};{{\mathbb {R}}}^{1 \times n})$, satisfying $l_j(\xi ,u) A(\xi ,u) = a_j(\xi ,u) l_j(\xi ,u)$.

An important class of weak solutions to (A.1) are known as shock fronts, which are configurations of the form

$$\begin{aligned} u(x,t) = {\left\{ \begin{array}{ll} u^+, &{} x \cdot {\hat{\nu }} > st,\\ u^-, &{} x \cdot {\hat{\nu }} < st, \end{array}\right. } \end{aligned}$$

(A.4)

where $u^\pm \in {\mathcal {U}}$ are constant states, $u^+ \ne u^-$, and ${\hat{\nu }} = (\nu _1, \ldots , \nu _d) \in {{\mathbb {R}}}^d$, $|{\hat{\nu }}| = 1$ is a fixed direction of propagation. The shock speed $s \in {{\mathbb {R}}}$ is not arbitrary but determined by the classical Rankine-Hugoniot jump conditions [27, 65],

$$\begin{aligned} -s \llbracket u \rrbracket + \sum _{j=1}^d \llbracket f^j(u) \rrbracket \nu _j = 0, \end{aligned}$$

(A.5)

where the bracket $\llbracket \cdot \rrbracket$ denotes the jump across the interface or, more precisely,

$$\begin{aligned} \llbracket g(u) \rrbracket := g(u^+) - g(u^-), \end{aligned}$$

for any (vector or matrix valued) function $g = g(u)$. Jump conditions (A.5) are necessary conditions for the configuration (A.4) to be a weak solution to (A.1) and express conservation of the state variables u across the interface, $\Sigma = \{x \cdot {\hat{\nu }} - st = 0 \}$.

To circumvent the problem of non-uniqueness of weak solutions of the form (A.4) one further imposes an entropy condition. The shock front (A.4) is called an admissible (or classical) p-shock if it satisfies Lax entropy condition (cf. [27, 65]): there exists an index $1 \le p \le n$ such that

$$\begin{aligned} \begin{aligned} a_{p-1}({\hat{\nu }},u^-)&< s< a_p({\hat{\nu }},u^-),\\ a_{p}({\hat{\nu }},u^+)&< s < a_{p+1}({\hat{\nu }},u^+), \end{aligned} \end{aligned}$$

(A.6)

where, by convention, if $p=1$ then $a_{p-1}({\hat{\nu }}, u^-) := - \infty$, and if $p = n$ then $a_{p+1}({\hat{\nu }}, u^+) := +\infty$. In the case where $p=1$ or $p=n$ the shock is called extreme. The eigenvalue $a_p({\hat{\nu }},u)$ is called the principal characteristic speed and $r_p({\hat{\nu }},u)$ is the principal characteristic field. It is said that the former is genuinely nonlinear in the direction ${\hat{\nu }}$ (cf. Majda [72]) if $D_u a_p({\hat{\nu }},u)^\top r_p({\hat{\nu }},u) \ne 0$ (or equivalently, $l_p({\hat{\nu }},u) D_u a_p({\hat{\nu }},u) \ne 0$) for all $u \in {\mathcal {U}}$.

Given a base state $u^+ \in {\mathcal {U}}$, the Hugoniot locus is defined as the set of all states in ${\mathcal {U}}$ that can be connected to $u^+$ with a speed satisfying the jump conditions (A.5). The intersection of the Hugoniot locus with those states for which one can find a shock speed satisfying Lax entropy condition (A.6) for some $1 \le p \le n$ is referred to as the p-shock curve. If, in addition, $u^+ \in {\mathcal {U}}$ is a point of genuine nonlinearity of the p-th characteristic family in direction of ${\hat{\nu }}$, for which $a_p({\hat{\nu }}, u^+)$ is a simple eigenvalue and

$$\begin{aligned} D_u a_p({\hat{\nu }}, u^+)^\top r_p({\hat{\nu }}, u^+) > 0, \qquad \text {(respectively, }< 0), \end{aligned}$$

(A.7)

then the p-shock curve locally behaves like

$$\begin{aligned} \begin{aligned} u^-&= u^+ + \epsilon \, r_p({\hat{\nu }}, u^+) + O(\epsilon ^2),\\ s&= a_p({\hat{\nu }}, u^+) + \tfrac{1}{2} \epsilon \, D_u a_p({\hat{\nu }}, u^+)^\top r_p({\hat{\nu }}, u^+) + O(\epsilon ^2), \end{aligned} \end{aligned}$$

(A.8)

and satisfies Lax entropy condition (A.6) if and only if $\epsilon < 0$ (respectively, $\epsilon > 0$). The parameter $\epsilon$ measures the strength of the shock, $|u^+ - u^-| = O(|\epsilon |)$.

It is well known that the nonlinear stability behavior of shock fronts of the form (A.4) is determined by the so called uniform and weak Lopatinskiĭ conditions (see Benzoni-Gavage and Serre [10], Majda [70, 71], Métivier [74,75,76] and the references therein). The analysis to obtain the former departs from a Fourier-Laplace decomposition of the constant-coefficient linearized problem associated with (A.1) at the configuration (A.4). By considering single normal modes of the form $u \sim e^{\lambda t} e^{\mathrm {i}\xi \cdot x}$ with spatio-temporal frequencies lying on the set

$$\begin{aligned} \Gamma _{{\hat{\nu }}}^+ = \left\{ (\lambda ,\xi ) \in {{\mathbb {C}}}\times {{\mathbb {R}}}^d \, : \, \mathrm{Re}\,\lambda > 0, \, \xi \cdot {\hat{\nu }} = 0, \, |\lambda |^2 + |\xi |^2 = 1 \right\} , \end{aligned}$$

(A.9)

as solutions to the linearized problem around the shock (A.4), one arrives at the Lopatinskiĭ determinant or stability function

$$\begin{aligned} \Delta (\lambda ,\xi )= & {} \det \Big ( {\mathcal {R}}_1^-, \ldots , {\mathcal {R}}_{p-1}^-, \, \lambda \llbracket u \rrbracket + \mathrm {i}\sum _{j=1}^d \xi _j \llbracket f^j(u) \rrbracket , {\mathcal {R}}_{p+1}^+, \ldots , {\mathcal {R}}_n^+ \Big ),\nonumber \\&\qquad (\lambda ,\xi ) \in \Gamma _{{\hat{\nu }}}^+, \end{aligned}$$

(A.10)

where ${\mathcal {R}}_1^-(\lambda ,\xi ), \ldots , {\mathcal {R}}_{p-1}^-(\lambda ,\xi ) \in {{\mathbb {C}}}^{n \times 1}$ denotes a basis of the stable subspace of ${\mathcal {A}}^-(\lambda ,\xi )$, and ${\mathcal {R}}_{p+1}^+(\lambda ,\xi ), \ldots , {\mathcal {R}}_{n}^+(\lambda ,\xi ) \in {{\mathbb {C}}}^{n \times 1}$ denotes a basis of the unstable subspace of ${\mathcal {A}}^+(\lambda ,\xi )$, whereupon we define the matrix fields

$$\begin{aligned} {\mathcal {A}}^\pm (\lambda ,\xi ) := (\lambda {{\mathbb {I}}}_n+ \mathrm {i}A(\xi , u^\pm ))(A({\hat{\nu }}, u^\pm ) - s{{\mathbb {I}}}_n)^{-1} \in {{\mathbb {C}}}^{n \times n}, \qquad (\lambda ,\xi ) \in \Gamma _{{\hat{\nu }}}^+. \end{aligned}$$

Notice that, in view of Lax entropy conditions, the shock is not characteristic with $s \ne a_p^\pm$ and hence the matrices $A({\hat{\nu }}, u^\pm ) - s{{\mathbb {I}}}_n$ are not singular. The fact that the stable subspace of ${\mathcal {A}}^-(\lambda ,\xi )$ and the unstable subspace of ${\mathcal {A}}^+(\lambda ,\xi )$ have exactly dimensions $p-1$ and $n-p$, respectively, follows from the hyperbolicity of the matrix fields ${\mathcal {A}}^\pm (\lambda ,\xi )$ on the set $\Gamma _{{\hat{\nu }}}^+$. This result is known in the literature as Hersh’ lemma [49] (see also [10, 55, 88]).

The function $\Delta$ is jointly analytic in $(\lambda ,\xi ) \in \Gamma _{{\hat{\nu }}}^+$ and homogeneous of degree one. Also, by continuity of the eigenprojections, the Lopatinskiĭ determinant can be defined for all frequencies within the set

$$\begin{aligned} \Gamma _{{\hat{\nu }}} := \left\{ (\lambda ,\xi ) \in {{\mathbb {C}}}\times {{\mathbb {R}}}^d \, : \, \mathrm{Re}\,\lambda \ge 0, \, \xi \cdot {\hat{\nu }} = 0, \, |\lambda |^2 + |\xi |^2 = 1 \right\} , \end{aligned}$$

(see [62, 70, 71, 74] for further information). The stability function $\Delta$ determines the solvability of the linearized equations by wave solutions that violate an $L^2$ well-posedness estimate. Whenever a zero of $\Delta$ occurs then there exist spatially decaying solutions with time growth rate $\exp (t \, \mathrm{Re}\,\lambda )$. Thus, a necessary condition for well-posedness of the linearized problem is that $\Delta$ does not vanish in the open set $\Gamma _{{\hat{\nu }}}^+$. A stronger condition requires $\Delta$ not to vanish in the whole frequency set $\Gamma _{{\hat{\nu }}}$ (allowing time frequencies with $\mathrm{Re}\,\lambda = 0$) and it is sufficient for the well-posedness of the nonlinear system, as the analyses of Majda [70, 71] and Métivier [74, 75] show. To sum up, we have the following

Definition A.1

Consider a planar shock wave of the form (A.4) and its corresponding Lopatinskiĭ determinant defined in (A.10). If $\Delta$ has no zeroes $(\lambda ,\xi )$ in $\Gamma _{{\hat{\nu }}}$ the shock is called uniformly stable (uniform Lopatinskiĭ condition). If $\Delta$ has a zero $(\lambda ,\xi )$ in $\Gamma _{{\hat{\nu }}}^+$ (with $\mathrm{Re}\,\lambda > 0$) the shock is referred to as strongly unstable. In the intermediate case where $\Delta$ has some zero $(\lambda ,\xi )$ with $\mathrm{Re}\,\lambda = 0$ but no zero in $\Gamma _{{\hat{\nu }}}^+$ the shock is said to be weakly stable (weak Lopatinskiĭ condition).

Remark A.2

When a shock is extreme with $p=1$ then there is no stable subspace of $\mathcal {A}^-(\lambda ,{\tilde{\xi }})$ for $(\lambda ,{\tilde{\xi }}) \in \Gamma _{{{\hat{\nu }}}}^+$ and the unstable subspace of $\mathcal {A}^+(\lambda ,{\tilde{\xi }})$ has dimension $n-1$. Therefore, the left stable subspace of $\mathcal {A}^+(\lambda ,{\tilde{\xi }})$ is generated by a single (row) vector $l_+^s(\lambda ,{\tilde{\xi }})$ associated to a unique stable eigenvalue $\beta (\lambda ,{\tilde{\xi }})$ with $\mathrm{Re}\,\beta < 0$. In such a case the expression for the Lopatinskiĭ determinant simplifies to

$$\begin{aligned} {\overline{\Delta }}(\lambda ,{\tilde{\xi }}) = l_+^s(\lambda ,{\tilde{\xi }}) \Big ( \lambda \llbracket u \rrbracket + \mathrm {i}\sum _{j=1}^d \xi _j \llbracket f^j(u) \rrbracket \Big ), \qquad (\lambda ,{\tilde{\xi }}) \in \Gamma _{{{\hat{\nu }}}}^+, \end{aligned}$$

(A.11)

in the sense that $\Delta = 0$ in $\Gamma _{{{\hat{\nu }}}}^+$ if and only if ${\overline{\Delta }} = 0$ in $\Gamma _{{{\hat{\nu }}}}^+$ (see [10, 55, 88]).

When a shock is strongly unstable, the instability is of Hadamard type [45, 88] and it is so violent that we practically never observe the shock evolve in time. In contrast, any small initial perturbation around a strongly stable shock (that is, a small wave impinging on the interface), compatible with the conservation laws and the jump conditions, produces a (local-in-time) solution to the nonlinear system with the same wave structure, that is, made of smooth regions separated by a (modified or curved) shock front. As shown by Majda [71], the strong stability condition ensures the well-posedness of a non-standard constant coefficient initial boundary value problem. The intermediate case of a weakly stable shock for which there exist zeroes of the Lopatinskiĭ determinant on the imaginary axis ($\Delta (\mathrm {i}\tau ,{\tilde{\xi }}) = 0$, for frequencies $(\mathrm {i}\tau , \xi ) \in \partial \Gamma _{{\hat{\nu }}}^+$, $\tau \in {{\mathbb {R}}}$) refers to the existence of surface wave solutions localized near the shock, having the form $\Phi (|x\cdot {\hat{\nu }}|) e^{\mathrm {i}(\tau t + x\cdot \xi )}$ and with amplitude $\Phi$ decaying exponentially as we move away from the interface, $|x\cdot {\hat{\nu }}| \rightarrow \infty$.

Appendix B. Compressible Hyperelastic Materials of Hadamard Type

An hyperelastic material of Hadamard type (cf. [48, 57]) is defined as an elastic material whose energy density function has the general form (1.1), where $\mu > 0$ is a constant and $h : (0,\infty ) \rightarrow {{\mathbb {R}}}$ is a function of class $C^3$. According to custom, let us denote that

$$\begin{aligned} I^{(1)} = \mathrm{tr}\,(U^\top U), \quad I^{(d)} = \det (U^\top U), \quad J = \sqrt{I^{(d)}} = \det U. \end{aligned}$$

$I^{(1)}$ and $I^{(d)}$ are well-known principal invariants of the right Cauchy-Green tensor, $C = U^\top U$, for any given deformation gradient $U \in {{\mathbb {M}}}^d_+$. Hence, energy densities for compressible Hadamard materials have the (Rivlin-Ericksen) form

$$\begin{aligned} W(U) = {\overline{W}}(I^{(1)}, J) = \frac{\mu }{2} I^{(1)} + h(J). \end{aligned}$$

(B.1)

The constant $\mu > 0$ is the classical shear modulus in the reference configuration, describing an object’s tendency to deform its shape at constant volume when acted upon opposing forces. The energy density (B.1) consists of two contributions: the first term is the isochoric part of the energy, quantifying energy changes at constant volume and depending only on $I^{(1)}$, whereas the second one, the volumetric function $h = h(J)$, quantifies energy changes due to changes in volume, and depends only on $J = \det U \in (0,\infty )$. In this paper, it is assumed that the function h satisfies the regularity assumption ($\hbox {H}_1$) ($h \in C^3$), the convexity condition for the energy ($\hbox {H}_2$) ($h'' > 0$) and the material convexity condition ($\hbox {H}_3$) ($h''' < 0$).

Remark B.1

Hayes [48] calls restricted Hadamard materials to those which, in addition to ($\hbox {H}_1$) and ($\hbox {H}_2$), satisfy

$$\begin{aligned} h'(J) \le 0, \quad \text {for all } \, J > 0, \end{aligned}$$

(B.2)

a condition which guarantees that the elastic medium fulfills the ordered forces inequality of Coleman and Noll [19]. Even though some of the examples of elastic materials presented in this paper satisfy inequality (B.2), the latter plays no role in the shock stability analysis.

1.1 B.1. Stress fields

We now derive the first Piola-Kirchhoff and Cauchy stress tensors from any energy density function of the form (1.1).

Lemma B.2

For a general compressible elastic model with energy density of the form $W = {\overline{W}}(I^{(1)}, J)$ in any dimension $d \ge 2$, the first Piola-Kirchhoff stress tensor is given by

$$\begin{aligned} \sigma (U) = 2 \frac{\partial {\overline{W}}}{\partial {I^{(1)}}} U + \frac{\partial {\overline{W}}}{\partial J} \big ( \mathrm {Cof}\,U\big ), \qquad U \in {{\mathbb {M}}}^d_+. \end{aligned}$$

(B.3)

Moreover, the Cauchy stress tensor is

$$\begin{aligned} T(U) = \frac{2}{J} \frac{\partial {\overline{W}}}{\partial {I^{(1)}}} UU^\top + \frac{\partial {\overline{W}}}{\partial J} {{\mathbb {I}}}_d, \qquad U \in {{\mathbb {M}}}^d_+. \end{aligned}$$

(B.4)

Proof

Follows from elementary computations: since $I^{(1)} = \sum _{h,k=1}^d U_{hk}^2$ then clearly $\partial _{U_{ij}} I^{(1)} = 2 U_{ij}$, $1 \le i, j \le d$; on the other hand, expression (2.11) above yields

$$\begin{aligned} \frac{\partial W}{\partial U_{ij}} = 2 \frac{\partial {\overline{W}}}{\partial {I^{(1)}}} U_{ij} + \frac{\partial {\overline{W}}}{\partial J} (\mathrm {Cof}\,U)_{ij}, \qquad 1 \le i,j \le d. \end{aligned}$$

This shows (B.3). Now, since the Cauchy stress tensor T is related to $\sigma$ by $\sigma = J T U^{-\top }$ (cf. [4, 17]), apply (1.2) to obtain (B.4), as claimed. $\square$

We immediately have

Corollary B.3

For compressible hyperelastic materials of Hadamard type, the first Piola-Kirchhoff stress tensor is given by

$$\begin{aligned} \sigma (U) = \mu U + h'(J) \, \mathrm {Cof}\,U , \qquad U \in {{\mathbb {M}}}^d_+. \end{aligned}$$

(B.5)

Furthermore, the Cauchy stress tensor is

$$\begin{aligned} T(U) = \frac{\mu }{J} UU^\top + h'(J) {{\mathbb {I}}}_d, , \qquad U \in {{\mathbb {M}}}^d_+. \end{aligned}$$

(B.6)

Proof

Follows directly from (B.1) and Lemma B.2. $\square$

Given any deformation gradient $U \in {{\mathbb {M}}}^d_+$, the principal stretches $\vartheta _j > 0$, $j = 1, \ldots , d$, are the square roots of the eigenvalues of the symmetric right Cauchy-Green tensor. Therefore,

$$\begin{aligned} I^{(1)} = \mathrm{tr}\,(U^\top U) = \sum _{j=1}^d \vartheta _j^2, \quad J = \det U = \prod _{j=1}^d \vartheta _j. \end{aligned}$$

The following observation is a generalization of the result established by Currie [24] in dimension $d = 3$.

Proposition B.4

For any $d \ge 2$ the possible range for $I^{(1)}$ is given by

$$\begin{aligned} \mathcal {D} = \{ (I^{(1)}, J) \in {{\mathbb {R}}}\times (0, \infty ) \, : \, I^{(1)} \ge d J^{2/d}\}. \end{aligned}$$

Proof

It is a straightforward application of the inequality of arithmetic and geometric means on the principal stretches,

$$\begin{aligned} I^{(1)} = \mathrm{tr}\,(U^\top U) = \vartheta _1^2 + \ldots + \vartheta _d^2 \ge d \, \Big ( \vartheta _1^2 \cdots \vartheta _d^2 \Big )^{1/d} = d \, (\det U)^{2/d} = d J^{2/d}. \end{aligned}$$

$\square$

The boundary of the domain $\partial \mathcal {D} = \{ (I^{(1)},J) \, : \, I^{(1)} = d J^{2/d}\}$ is associated to pure pressure deformations, and the value $(I^{(1)},J) = (d,1) \in \partial \mathcal {D}$ corresponds to no deformations, $U = {{\mathbb {I}}}_d$, with a reference configuration in which $\vartheta _j = 1$ for all $1 \le j \le d$.

1.2 B.2. Compressible neo-Hookean materials

The simplest interpretation of an elastic Hadamard material is as a compressible extension of a neo-Hookean incompressible solid. Incompressible hyperelasticity is restricted to isochoric (volume preserving) deformations with $J = \det U =1$, which is a kinematic constraint. The best known incompressible hyperelastic model is the neo-Hookean material [63, 84, 99], whose energy function (in arbitrary space dimensions) is given by

$$\begin{aligned} W_{\mathrm {nH}}(U) = {\overline{W}}_{\mathrm {nH}}(I^{(1)}) = \frac{\mu }{2} (I^{(1)} - d). \end{aligned}$$

(B.7)

This strain-energy function provides a reliable and mathematically simple constitutive model for the nonlinear deformation behavior of isotropic hyperelastic materials, such as vulcanized rubber, similar to Hooke’s law. It predicts typical effects known from nonlinear elasticity within the small strain domain (in contrast to linear elastic materials the stress-strain curve for a neo-Hookean material is not linear). It was first proposed by Rivlin in 1948 [84]. Notably, the energy function (B.7) may also be derived from statistical theory, in which rubber is regarded as a three-dimensional network of long-chain molecules that are connected at a few points (cf. [12, 52]).

The incompressibility hypothesis works well for vulcanized rubber (under very high hydrostatic pressure the material undergoes very small volume changes). There are other materials, however, which are either slightly compressible, or which may undergo considerable volume changes (like foamed rubber). Therefore, compressible models are needed in order to describe these elastic responses. Furthermore, it is known that incompressibility can cause numerical difficulties in the analysis of finite elements, and in such cases nearly incompressible models are often used [54, 66]. As a result, either motivated by numerical or by physical considerations, compressibility is often accounted by the addition of a strain energy describing the purely volumetric elastic response. In the case of the neo-Hookean model, compressible extensions have the form

$$\begin{aligned} W(U) = {\overline{W}}(I^{(1)},J) = {\overline{W}}_{\!\mathrm {nH}}(I^{(1)}) + {\overline{W}}_{\!\mathrm {vol}}(J). \end{aligned}$$

This decoupled representation of the energy as the sum of isochoric and volumetric energies is very common for isothermal deformations. A compressible extension should satisfy ${\overline{W}}(I^{(1)},1) = {\overline{W}}_{\!\mathrm {nH}}(I^{(1)})$, that is, ${\overline{W}}_{\!\mathrm {vol}}(1) = 0$. In the case of energies of the form (1.1) we clearly have an isochoric contribution given by the neo-Hookean energy density (B.7) and a volumetric response given by ${\overline{W}}_{\!\mathrm {vol}}(J) = h(J) +\tfrac{1}{2}\mu d$. Pence and Gou [81] discuss nearly incompressible versions of the neo-Hookean model, as well as the requirements on the material moduli for the models to be compatible with the small-strain regime. In the next section we review such requirements and extrapolate them to arbitrary space dimensions.

1.3 B.3. Compressible theory of infinitesimal strain

Since undeformed configurations are stress free, one requires that $\sigma = 0$ whenever $U = {{\mathbb {I}}}_d$. In the case of a Hadamard material, this requirement leads, upon substitution into formula (B.5), to the following relation between the shear modulus and the function h,

$$\begin{aligned} h'(1) = - \mu . \end{aligned}$$

(B.8)

This relation can be interpreted as a free stress condition for no deformations in the incompressible boundary, precisely at $(I^{(1)},J) = (d,1) \in \partial \mathcal {D}$.

The mean pressure field is defined as (see, e.g., [100], p. 545),

$$\begin{aligned} {\overline{p}} := - \frac{1}{d} \mathrm{tr}\,(T(U)) = - \frac{1}{d} \mathrm{tr}\,\Big ( \frac{2}{J} \frac{\partial {\overline{W}}}{\partial I^{(1)}} UU^\top + \frac{\partial {\overline{W}}}{\partial J} {{\mathbb {I}}}_d\Big ) = - h'(J) - \frac{\mu }{d} \frac{I^{(1)}}{J}. \end{aligned}$$

For symmetric deformation states, $U = J^{1/d} {{\mathbb {I}}}_d$ (or equivalently, $(I^{(1)},J) \in \partial \mathcal {D}$), Pence and Gou [81] define

$$\begin{aligned} - \,{\hat{p}}(J) := - {\overline{p}}(dJ^{2/d},J) = h'(J) + \mu J^{\frac{2}{d}-1} = -p_{\mathrm {hyd}}(J) + \mu J^{\frac{2}{d}-1}, \end{aligned}$$

where

$$\begin{aligned} p_{\mathrm {hyd}} (J) = - \frac{\partial {\overline{W}}_{\!\mathrm {vol}}}{\partial J} = - h'(J), \end{aligned}$$

(B.9)

is the hydrostatic pressure (cf. [52, 99]), or the pressure the material experiences when the shear strain is zero. The appropriate definition of the bulk modulus of infinitesimal strain theory is therefore

$$\begin{aligned} \kappa := - \left. \frac{d {\hat{p}}}{dJ} \right| _{J=1} = {\hat{p}}'(1), \end{aligned}$$

describing volumetric elasticity or how resistant to compression the elastic medium is. Consequently, for a Hadamard material with strain energy of the form (1.1) we have $\partial {\overline{W}} / \partial I^{(1)} = \tfrac{\mu }{2}$ and $\partial {\overline{W}} / \partial J = h'(J)$, yielding

$$\begin{aligned} {- {\hat{p}}'(J) =} \mu \Big ( \frac{2}{d} -1\Big ) J^{\frac{2}{d}-2} + h''(J), \end{aligned}$$

and the following relation between the bulk and shear moduli:

$$\begin{aligned} \kappa = \mu \Big ( \frac{2}{d} -1\Big ) + h''(1). \end{aligned}$$

(B.10)

Since the strain energy must be positive for small strains (linear physical theory for small deformations), on restriction to infinitesimal deformations the shear and bulk moduli must be positive to ensure compatibility with the linear response (cf. [15]). The Poisson ratio can then be defined in arbitrary dimensions as

$$\begin{aligned} {\overline{\nu }} := \frac{d \kappa - 2 \mu }{2 \mu + d(d-1) \kappa }, \end{aligned}$$

measuring the ratio of strain in the direction of load over the strain in orthogonal directions. This definition extends the well known formulae for the Poisson ratio in dimension $d=2$, ${\overline{\nu }} = \frac{\kappa - \mu }{\kappa + \mu }$, and in dimension $d = 3$, ${\overline{\nu }} = \frac{3\kappa - 2\mu }{2(3\kappa + 2\mu )}$ (see [73, 96]). Although the admissible thermodynamic range for the Poisson ratio is $-1 \le {\overline{\nu }} \le 1/2$ in dimension $d = 3$ [81], and $-1 \le {\overline{\nu }} \le 1$ in dimension $d = 2$ [73], the standard range for consideration is ${\overline{\nu }} > 0$ (${\overline{\nu }}$ is usually positive for most materials³ because interatomic bonds realign with deformation). To sum up, in this paper it is assumed that

$$\begin{aligned} \mu> 0, \quad \kappa> \frac{2}{d} \mu > 0. \end{aligned}$$

(B.11)

The classical Lamé moduli of an elastic material are the shear modulus $\mu > 0$ (second Lamé parameter) and $\Lambda$ (first Lamé parameter)⁴; the former can be related to the bulk and shear moduli by

$$\begin{aligned} \Lambda = \kappa - \frac{2\mu }{d}; \end{aligned}$$

see [17, 99]. Notice that, under assumption (B.11), $\Lambda > 0$.

Remark B.5

In view of (B.9), condition ($\hbox {H}_3$) implies that $p_{\mathrm {hyd}}'' (J) = - h'''(J) > 0$ for all $J \in (0, \infty )$. Hence, hypothesis ($\hbox {H}_3$) can be interpreted as a material convexity condition for zero shear strain.

1.4 B.4. Examples

The following models belong to the class of compressible hyperelastic materials of Hadamard type, whose energy density functions have the form (1.1) and satisfy assumptions ($\hbox {H}_1$) and ($\hbox {H}_2$). They have been proposed in the materials science literature to describe different elastic responses. It is worth mentioning that there exist compressible models with energies of the form (1.1) but which do not satisfy the convexity assumption ($\hbox {H}_2$) for all deformations $J \in (0,\infty )$, such as the original Simo-Pister model [93] (see also [47]), or the Ogden $\beta$-log model [79] (see eq. (6.137), p. 244 in [52]).

1.4.1 $\mathrm {(a)}$ Ciarlet-Geymonat model

As a first example consider the following volumetric strain energy function

$$\begin{aligned} h_{\mathrm {CG}}(J) = - \frac{d}{2} \mu - \mu \log J + \left( \frac{\kappa }{2} - \frac{\mu }{d}\right) (J -1 )^2, \end{aligned}$$

(B.12)

where $\mu$ and $\kappa$ are the shear and bulk moduli, respectively, satisfying (B.11). Notice that $h_{\mathrm {CG}}(1) = -d\mu /2$ and therefore the energy density ${\overline{W}}_{\mathrm {CG}} = \frac{\mu }{2} I^{(1)} + h_{\mathrm {CG}}(J)$ is normalized as ${\overline{W}}_{\mathrm {CG}}(d,1) = 0$. It also satisfies (B.8) and (B.10) as the reader may easily verify. Finally, in view of (B.11) there holds the convexity condition ($\hbox {H}_2$) as

$$\begin{aligned} h_{\mathrm {CG}}''(J) = \frac{\mu }{J^2} + \Big ({\kappa } - \frac{2 \mu }{d}\Big ) > 0, \qquad J \in (0, \infty ). \end{aligned}$$

In addition, it holds that

$$\begin{aligned} h_{\mathrm {CG}}'''(J) = - \frac{2 \mu }{J^3} < 0, \end{aligned}$$

for all $J \in (0,\infty )$. This model is an extension to arbitrary spatial dimensions of the strain energy

$$\begin{aligned} {\overline{W}} = \frac{\mu }{2} (I^{(1)} -3) + \left( \frac{\kappa }{2} - \frac{\mu }{3}\right) (J -1 )^2 - \mu \log J, \end{aligned}$$

proposed by Ciarlet and Geymonat [18] (see also [80]) in dimension $d = 3$. It is a special form of the family of compressible Mooney-Rivlin materials (see Ciarlet [17], section 4.10, p. 189, formula (iii) in the limit $b \rightarrow 0$). $h_{\mathrm {CG}}$ is defined for all deformations $J \in (0,\infty )$ and satisfies $h_{\mathrm {CG}} \rightarrow \infty$ as $J \rightarrow \infty$ and as $J \rightarrow 0^+$.

1.4.2 $\mathrm {(b)}$ Blatz model

The energy function

$$\begin{aligned} h_{\mathrm {B}}(J) = - \frac{d}{2} \mu + \Big (\kappa - \frac{2}{d} \mu \Big ) \big ( J-1\big ) - \Big ( \kappa + \Big (\frac{d-2}{d}\Big ) \mu \Big ) \log J, \end{aligned}$$

(B.13)

where, once again, $\mu$ and $\kappa$ are the shear and bulk moduli, respectively, generalizes to arbitrary dimensions $d \ge 2$ the modified compressible neo-Hookean form of the energy proposed by Blatz [13] (see eq. (48), p. 36), in dimension $d = 3$:

$$\begin{aligned} {\overline{W}} = \frac{\mu }{2} (I^{(1)} - 3) + \Big (\kappa - \frac{2}{3} \mu \Big ) \big ( J-1\big ) - \Big ( \kappa + \frac{\mu }{3} \Big ) \log J. \end{aligned}$$

This function fulfills normalization, $h_{\mathrm {B}}(1) = -d\mu /2$, as well as conditions (B.8) and (B.10), as it is easily verified. Moreover,

$$\begin{aligned} h_{\mathrm {B}}''(J) = \frac{1}{J^2} \Big ( \kappa + \frac{(d-2) \mu }{d} \Big ) > 0, \quad h_{\mathrm {B}}'''(J) = -\frac{2}{J^3} \Big ( \kappa + \frac{(d-2) \mu }{d} \Big ) < 0, \end{aligned}$$

for all $J \in (0,\infty )$. Notice that $h_{\mathrm {B}} \rightarrow \infty$ as $J \rightarrow \infty$ or as $J \rightarrow 0^+$. This energy was selected by Blatz as a candidate strain energy density to describe thermostatic properties of homogeneous isotropic continuous elastomers (elastic polymers).

1.4.3 $\mathrm {(c)}$ Neo-Hookean Ogden compressible foam material

The energy function

$$\begin{aligned} h_{\mathrm {O}}(J) = - \frac{d}{2} \mu + \frac{\mu }{2 c_1} \big ( J^{-2c_1} - 1\big ), \end{aligned}$$

(B.14)

where

$$\begin{aligned} c_1 = \frac{{\overline{\nu }}}{1- (d-1) {\overline{\nu }}} = \frac{d \kappa - 2 \mu }{2 d \mu } > 0, \end{aligned}$$

was proposed by Ogden [79] to model highly compressible rubber-like materials for which significantly volume changes can occur with relatively little stress (such as foams). It belongs to what is known in the literature as the family of Ogden compressible rubber foam materials (see [69], p. 161):

$$\begin{aligned} {\overline{W}} = \sum _{p=1}^N \frac{\mu _p}{\alpha _p} \Big ( \sum _{j=1}^d \vartheta _j^{\alpha _p} - d \Big ) \, + \, \sum _{p=1}^N \frac{\mu _p}{\alpha _p c_p} (J^{- \alpha _p c_p}-1), \end{aligned}$$

specialized here to $N =1$ (neo-Hookean), $\mu _1 = \mu > 0$, $\alpha _1 = 2$ and $c_1$ given above. This neo-Hookean element of the family has been used as a basis for residually stressed extensions for energies that account for elastic responses of blood arteries in medical applications (cf. [41]). Notice that $h_{\mathrm {O}}(1) = -d\mu /2$ (normalization) and relations (B.8) and (B.10) hold. Moreover, the convexity condition holds as

$$\begin{aligned} h_{\mathrm {O}}''(J) = \frac{\mu ( 2c_1 +1)}{J^{2(c_1+1)}} > 0, \quad h_{\mathrm {O}}'''(J) = - \frac{2\mu (c_1+1)(2c_1 +1)}{J^{2c_1+3}} < 0, \end{aligned}$$

for all $J \in (0,\infty )$. Notably $h_{\mathrm {O}} \rightarrow \infty$ as $J \rightarrow 0^+$ but $\lim _{J \rightarrow \infty } h_{\mathrm {O}}(J)$ exists.

1.4.4 $\mathrm {(d)}$ Levinson-Burgess model

Consider the following volumetric function

$$\begin{aligned} h_{\mathrm {LB}}(J) = - \frac{d}{2} \mu + \frac{\mu }{2} \Big ( {\overline{c}} (J^2 -1) + 2({\overline{c}} + 1) (1 - J) \Big ), \end{aligned}$$

(B.15)

where

$$\begin{aligned} {\overline{c}} = \frac{\kappa }{\mu } - \frac{2}{d} + 1 > 0. \end{aligned}$$

This is a generalization to any space dimension $d \ge 2$ of the three dimensional material considered by Kirkinis et al. [59],

$$\begin{aligned} {\overline{W}} = \frac{\mu }{2} \left( I^{(1)} -3 + \left( \frac{\kappa }{\mu } + \frac{1}{3}\right) (J^2 -1) - 2 \left( \frac{\kappa }{\mu } + \frac{1}{3} + 1\right) (J-1) \right) , \end{aligned}$$

which is, in turn, a special case of a compressible polynomial material introduced by Levinson and Burgess [67] to account for weakly compressible elastic media with Poisson ratio close to $\tfrac{1}{2}$ (in dimension $d = 3$). Notice that $h_{\mathrm {LB}}(1) = -d\mu /2$ (normalization), it satisfies (B.8) and (B.10), and

$$\begin{aligned} h_{\mathrm {LB}}''(J) = \mu {\overline{c}} > 0, \quad h_{\mathrm {LB}}'''(J) \equiv 0, \end{aligned}$$

for all $J \in (0,\infty )$.

1.4.5 $\mathrm {(e)}$ Simo-Taylor material

The Simo-Taylor model [94] (see also [47]),

$$\begin{aligned} h_{\mathrm {ST}}(J) = - \frac{d}{2} \mu - \mu \log J + \frac{\Lambda }{2} \Big ( \frac{J^2}{2} - \log J - \frac{1}{2}\Big ), \end{aligned}$$

(B.16)

where $\mu$ is the shear modulus and $\Lambda = \kappa - 2\mu /d > 0$ is the first Lamé parameter, clearly satisfies $h_{\mathrm {ST}}(1) = -d\mu /2$ (normalization) and conditions (B.8) and (B.10). Furthermore, the convexity condition ($\hbox {H}_2$) holds, as

$$\begin{aligned} h_{\mathrm {ST}}''(J) = \frac{\Lambda }{2} + \big ( \mu + \frac{\Lambda }{2} \big ) \frac{1}{J^2} > 0, \end{aligned}$$

for all $J \in (0,\infty )$. Observe also that

$$\begin{aligned} h_{\mathrm {ST}}'''(J) = - (2 \mu + \Lambda ) \frac{1}{J^3} < 0, \qquad J \in (0,+\infty ). \end{aligned}$$

When $J \rightarrow 0^+$ or $J \rightarrow \infty$, $h_{\mathrm {ST}}$ grows unboundedly. This energy form can be derived from (Gaussian) statistical mechanics of long-chain molecules with entropic sources of compressibility modeled thorough the logarithmic terms (cf. Bischoff et al. [12]).

1.4.6 $\mathrm {(f)}$ Special compressible Ogden-Hill material

The volumetric response function

$$\begin{aligned} h_{\mathrm {OH}} (J) = - \frac{d}{2} \mu + \frac{1}{b} \big ( J -1)^2, \end{aligned}$$

(B.17)

where $\mu > 0$ is the shear modulus and $b > 0$ is an empirical coefficient, yields an energy density ${\overline{W}}_{\mathrm {OH}} = \frac{\mu }{2} I^{(1)} + h_{\mathrm {OH}} (J)$ that also belongs to the class of compressible Hadamard materials. Notice that $W_{\mathrm {OH}}(d,1) = 0$ (normalization) but $h_{\mathrm {OH}}'(1) = 0$ and, thus, it does not satisfy the free stress condition (B.8). It does satisfy the convexity condition as

$$\begin{aligned} h_{\mathrm {OH}}''(J) = \frac{2}{b} > 0, \quad h_{\mathrm {OH}}'''(J) \equiv 0, \end{aligned}$$

for all $J \in (0,\infty )$. Also, $h_{\mathrm {OH}} \rightarrow \infty$ as $J \rightarrow \infty$, whereas $h_{\mathrm {OH}} (0^+)$ is well-defined. This model is a particular case of the well-known family of compressible Ogden-Hill materials [50, 51, 79]

$$\begin{aligned} {\overline{W}} = \sum _{p=1}^N \frac{\mu _p}{\alpha _p} \Big ( \sum _{j=1}^d \vartheta _j^{\alpha _p} - d \Big ) \, + \, \sum _{p=1}^N \frac{1}{b_p^2} (J-1)^{2N}, \end{aligned}$$

specialized to $N =1$, $\mu _1 = \mu > 0$, $\alpha _1 = 2$ and $b_1 = b > 0$. The family was proposed to model highly compressible materials such as low density polymer foams (cf. [31, 77]). The parameter $b > 0$ is adjusted from experimental data. It is a modulus that measures compressibility: if b is small then the material is highly compressible, whereas if b is large then the material can be considered as nearly incompressible. It is used in the analysis of elastomers, as well as in the design of O-rings, seals and other industrial products [69].

1.4.7 $\mathrm {(g)}$ Simo-Miehe model

The following energy function proposed by Simo and Miehe [92] (see also [52]):

$$\begin{aligned} h_{\mathrm {SM}}(J) = - \frac{d}{2} \mu + \frac{\kappa }{4} \big ( J^2 - 1 - 2 \log J \big ), \end{aligned}$$

(B.18)

was introduced in the context of finite-strain viscoplasticity. Note that this volumetric energy attains a minimum at $J=1$, with $h_{\mathrm {SM}}'(1) = 0$, and therefore it does not satisfy the free stress condition (B.8). It does, however, satisfy the convexity condition as

$$\begin{aligned} h_{\mathrm {SM}}''(J) = \frac{\kappa }{2} \Big (1 + \frac{1}{J^2}\Big ) > 0, \end{aligned}$$

for all deformations. Moreover,

$$\begin{aligned} h_{\mathrm {SM}}'''(J) = - \frac{\kappa }{J^3}< 0, \qquad J \in (0,\infty ). \end{aligned}$$

Also, $h_{\mathrm {SM}}$ increases unboundedly as $J \rightarrow 0^+$ and as $J \rightarrow \infty$.

1.4.8 $\mathrm {(h)}$ Bischoff, Arruda and Grosh model

Bischoff et al. [12] proposed the following volumetric response function:

$$\begin{aligned} h_{\mathrm {BAG}}(J) = - \frac{d}{2} \mu + \frac{{\overline{c}}}{b^2} \big ( \cosh (b(J-1)) - 1\big ), \end{aligned}$$

(B.19)

where the constants ${\overline{c}}, b$ are positive empirical constants which should be calibrated from experimental data. Notice that $h_{\mathrm {BAG}}'(1) = 0$ and $J = 1$ is a minimum; thus, it does not satisfy (B.8). The convexity condition holds as

$$\begin{aligned} h_{\mathrm {BAG}}''(J) = {\overline{c}} \cosh ( b(J-1)) > 0, \end{aligned}$$

for all $J \in (0,\infty )$. However,

$$\begin{aligned} h_{\mathrm {BAG}}'''(J) = {\overline{c}} b \sinh ( b(J-1)), \end{aligned}$$

yielding $h_{\mathrm {BAG}}'''(1) = 0$, as well as $h_{\mathrm {BAG}}'''(J) > 0$ if $J > 1$ and $h_{\mathrm {BAG}}'''(J) < 0$ if $J < 1$. Note also that $h_{\mathrm {BAG}} \rightarrow \infty$ as $J \rightarrow \infty$ but $h_{\mathrm {BAG}}(0^+)$ is well defined. This model was proposed to account for the contributions of entropy and initial energy to volume change. Its derivation follows non-Gaussian statistics of long chain molecules, which is necessary for large deformations. It can be interpreted as a non-Gaussian, higher order representation of the Ogden-Hill model (B.17) in the small volume changes regime, inasmuch as the series expansion around $J = 1$ yields

$$\begin{aligned} h_{\mathrm {BAG}}(J) = - \frac{d}{2} \mu + \frac{{\overline{c}}}{2}(J-1)^2 + O((J-1)^4). \end{aligned}$$

Remark B.6

The energy densities presented above are divided into two categories. Models (a) thru (e) can be interpreted as compressible versions of the neo-Hookean material in the sense described by Pence and Gou [81]: they satisfy the free stress condition (B.8) and the hydrostatic pressure condition (B.10), both at the incompressible limit with no deformation, and represent materials which are nearly incompressible. In contrast, models (f) thru (h) are designed to fit experimental data involving phenomenological observations such as, for example, when foam polymers undergo large changes in volume [54]. In these models, $h'(1) = 0$, so that the volumetric function h provides a direct penalization of volume departing from $J =1$. All models (a) thru (h) provide neo-Hookean behavior in the incompressible limit, namely, ${\overline{W}}(I^{(1)},1) = {\overline{W}}_{\!\mathrm {nH}}(I^{(1)})$, and reduce to the standard linearly elastic material response when deformations are small (that is, when $| \tfrac{1}{2}(U^\top U - {{\mathbb {I}}}_d) | \ll 1$).

Remark B.7

All the model examples presented here are physically motivated energy functions that satisfy assumptions ($\hbox {H}_1$) and ($\hbox {H}_2$) for all possible deformations and, therefore, they belong to the general class of compressible hyperelastic Hadamard materials. (It is to be observed that the family does not include other hyperelastic models found in the literature, such as the compressible versions of the Blatz-Ko, Murnaghan or Varga models, just to mention a few; see [52, 80] and the references therein.) Notably, the convexity of the energy (property ($\hbox {H}_2$)), implies that all energy functions are rank-one convex in the whole domain of U with $\det U > 0$, making the elastodynamics equations hyperbolic in the whole domain of their state variables. The stability results of this paper apply to materials which, in addition, satisfy the material convexity condition ($\hbox {H}_3$).