Twinning in pressurized porous elastomers

Abstract

For some time, it has been known that, under appropriate loading conditions, porous elastomers may develop instabilities in their macroscopic response. More recent theoretical work has shown that these composites may undergo ‘twinning’ or ‘domain formation,’ beyond the onset of such macroscopic instabilities. Here, motivated by interest in the possibility of using such porous elastomers for actuation purposes, we generalize this earlier work to account for pore pressure. For this purpose, we consider a class of two-dimensional (2D) porous elastomers consisting of random, isotropic distributions of aligned cylindrical pores in the elastomer. Exploiting the incompressibility of the elastomeric matrix, estimates are generated for the ‘principal’ or ‘mesoscopic’ and ‘relaxed’ or ‘macroscopic’ stored-energy functions of the porous elastomers with pressurized pores, by means of the earlier estimates for the unpressurized porous elastomers. Two specific cases are considered in detail: closed pores containing an ideal gas and open pores with direct pressure control. In the first case, it is found that, while the pressure generally stiffens the response, the onset of the instabilities and the associated twinned microstructures are insensitive to the pore pressure. In the second case, it is found that negative gauge pressures in the pores can be used to trigger the twinning instabilities, even at zero externally applied tractions. In either case, as previously found for the case of unpressurized pores, the response of the porous composite after twinning is perfectly soft in shear, which could prove extremely useful for actuation purposes. In addition, highly unusual behaviors, such as huge changes in shape, are predicted, even before the onset of twinning, by combining direct pressure control with appropriately selected (fixed) traction boundary conditions.

1. Introduction

Interest in soft materials has burgeoned in recent years, especially for applications in soft robotics [1], often as soft actuators [2]. In these applications, it is often desirable to create a material with some type of structure, for example, by fashioning a continuous elastomeric matrix with inclusions of some other phase: rigid metallic particles in magnetorheological elastomers (MREs) [3], liquid metal for flexible conductors [4], open channels into which gas or another fluid may be pumped [5], and so on. Such structure may be at the length scale of the body itself, but may often be on a much smaller scale, lending these materials to analysis using homogenization techniques [6–11]. The effective, or macroscopic behavior of these materials may then be determined, including the effects of the coupled physics that are often important in these systems: magnetic, electrical, chemical, and so on. Indeed, such analyses have been performed in the literature, in particular, for MREs (e.g., Ponte Castañeda and Galipeau [12], Danas et al. [13], Galipeau and Ponte Castañeda [14] and Lefèvre et al. [15]) and dielectric elastomeric composites (e.g., Tian et al. [16], Ponte Castañeda and Siboni [17] and Siboni and Ponte Castañeda [18]).

While the use of homogenization for these material systems has the usual benefit that their overall properties may be estimated without resorting to either simulations or experiments which may be time-intensive and expensive, and yields parameterized models which may be used to explore the design space very efficiently, there is a particular benefit when the materials are soft. In such cases, the behavior is inherently highly nonlinear due to the possibility of large deformations, and such nonlinearity may give rise to bifurcation instabilities for certain loading scenarios—that is, the solution of the governing elastostatics equations may lose uniqueness. These instabilities are often specifically sought after, proving extremely useful for various applications (see, for example, the review of Kochmann and Bertoldi [19]), but may be difficult to design for or analyze in general. Here, too, homogenization may play an important role: once a continuum model for the material behavior is obtained, use can be made of rigorous methods both to predict the presence of instabilities and to estimate the ‘relaxed’ response after their onset, describing the physical changes in the microstructure which result and the new macroscopic behavior (see the review of Ponte Castañeda [20]). It is these instabilities which are the focus of the present work, in the particular context of porous elastomeric composites.

There are several results in the literature aimed at understanding the effective mechanical response of porous elastomers. For the case of a periodic arrangement of pores, an early investigation was given by Abeyaratne and Triantafyllidis [7] (see also Triantafyllidis et al. [21] and Michel et al. [22]), while for the case of a random, isotropic arrangement of pores, explicit models were given for spherical pores under general loading by Lopez-Pamies and Ponte Castañeda [23] and for cylindrical pores under plane strain by Lopez-Pamies and Ponte Castañeda [24]. While all these models consider the pore phase to be vacuous and therefore to sustain no pressure, Idiart and Lopez-Pamies [25] and Lopez-Pamies et al. [26] made use of the incompressibility of the elastomer phase and of a clever change of variables in the governing elastostatics problem to express the effective behavior of a material with pressurized pores in a simple manner in terms of the corresponding behavior of an unpressurized material with the same microstructure. With this, the ‘principal’ or ‘mesoscopic’ response of pressurized porous elastomers may be understood in the absence of instabilities. However, in each of the above works, the material behavior was found to lose stability for sufficiently severe deformations. Abeyaratne and Triantafyllidis [7] noted in the periodic case that the incremental macroscopic response of the composite lost strong ellipticity (SE) under sufficient compression (even when the solid elastic material could not). In the context of elastomeric laminates, Triantafyllidis and Maker [27] compared the onset of such ‘macroscopic’ instabilities, signaled by loss of strong ellipticity (LOSE) of the macroscopic response, with the onset of ‘microscopic’ instabilities, driven by bifurcations at the microstructural level. The relation between microscopic and macroscopic instabilities was again considered by Geymonat et al. [28], who showed that in the periodic case, microscopic instabilities generally occur before macroscopic, and that LOSE corresponds to the limit as the wavelength of the microscopic instability becomes infinite. These instabilities were explored for porous elastomers with a particular set of microstructures and material properties by Triantafyllidis et al. [21] and Michel et al. [22], who gave the eigenmodes of the microscopic instabilities in the periodic case. The post-bifurcation response was considered later by Bertoldi et al. [29].

For random (aperiodic) microstructures such as those under consideration in this work, only macroscopic instabilities are possible. After the onset of such instabilities, it has been found by Avazmohammadi and Ponte Castañeda [30] and Furer and Ponte Castañeda [31,32], in the context of fiber-reinforced and laminated elastomer composites, respectively, that the initially macroscopically uniform deformation of the material may become non-uniform at a scale much larger than that of the microstructure, though still much smaller than that of the specimen. Thus, the ‘macroscopic’ or ‘relaxed’ response (after the onset of such macroscopic instabilities) of the composite is associated with the formation of ‘domains’ or ‘twinning,’ and can lead to ‘soft modes’ of deformation. Motivated by the fact that, in each of the aforementioned works for porous elastomers with random distributions of vacuous pores [7,11,23–25], LOSE was found to occur for sufficiently severe deformations, Caulfield and Ponte Castañeda [33] demonstrated recently that twinning is also predicted to take place in these porous elastomers and determined the relaxed response for a special class of porous elastomers with random distributions of aligned, cylindrical, vacuous pores. In general, however, LOSE will be affected by pore pressure (as observed by Lopez-Pamies et al. [26] in the context of porous elastomers with spherical pores), as will the subsequent relaxed response. Hence, the difficult work of computing the relaxed solution would in general need to be performed independently for any pore pressure and evolution law.

In this paper, we generalize the work of Caulfield and Ponte Castañeda [33], who characterized the macroscopic response of porous elastomers with vacuous pores beyond the onset of instabilities, to include materials with arbitrary pore pressure. We show that, for the same class of porous elastomers undergoing plane strain deformations, the relaxation is in fact independent of pore pressure, such that the relaxed behavior of the pressurized composites may be written in terms of that of the unpressurized composite in a manner completely analogous to the pre-relaxation behavior. This argument holds for isotropic porous elastomers with incompressible matrix phases being loaded in plane strain, provided the energy potential governing the pore pressure is convex (in the determinant of the deformation gradient). Thus, once the pre- and post-instability behavior under plane strain of a porous elastomer with vacuous pores is known, these results may be extended trivially to arbitrary pore pressurization protocols.

The paper is organized as follows. In Section 2, we summarize the general framework by which the principal and relaxed solutions for the behavior of porous elastomers are defined and show how to give the principal solution of the pressurized system in terms of that of the unpressurized. Then, in Section 3, we recall a particular estimate for the effective behavior of a material consisting of an incompressible Neo-Hookean matrix containing aligned, cylindrical pores and undergoing plane strain deformations, for which both the principal and relaxed solutions are already known, and use the results of Section 2 to give the pre-bifurcation behavior of pressurized composites with this microstructure using two representative pressurization schemes. In Section 4, we show that the onset of LOSE is unaffected by the presence of pressure in these materials, and therefore Section 5 argues that the relaxed microstructure is also unaffected. We conclude in Section 6 by giving a sampling of results for these pressurized composites including the effects of relaxation, and remark on notable features pointing toward promising future work in these systems.

2. Theoretical framework for effective behavior of pressurized porous elastomers

In this section, we outline the framework used to define the effective response of porous elastomers and give a simple method, adapted from prior work, to incorporate the effect of pore pressure into the homogenized response. Thus, we consider a general elastomeric composite body occupying reference volume ${\mathrm{\Omega }}_0$ which consists of a solid matrix phase in domain ${\mathrm{\Omega }}_0^{(1)}$ interspersed with gas-filled pores in domain ${\mathrm{\Omega }}_0^{(2)}$ of the reference configuration. The body undergoes a deformation whereby position vectors X in the reference configuration are mapped to position vectors x in the deformed; the deformation is characterized by the deformation gradient tensor, defined by its components ${\mathit{F}}_{\mathrm{ij}}=\partial x_i∕\partial X_j$ . Defining the Jacobian of the deformation by $J=\det \mathit{F}$ , we require $J>0$ to satisfy the material impenetrability condition.

The response of the matrix phase is given by the stored-energy density function $W^{(1)}(\mathit{F})$ , which it is recalled must be such that $W^{(1)}(\mathit{F})=W^{(1)}(\mathit{U})$ to satisfy frame indifference, where $\mathit{F}=\mathit{RU}$ is the polar decomposition yielding the rotation tensor R and the right stretch tensor U . Consistent with the assumption of gaseous inclusions, the response of the pore phase is governed by the potential $W^{(2)}(\mathit{F})=\psi (J)$ , such that the pore pressure is given by $p(J)=-{\psi }^{\prime }(J)$ as a function only of the volumetric deformation. We take $\psi (J)$ to be a convex function, though perhaps not strictly so to allow for the possibility of vacuous pores (for which $\psi (J)=0$ ). The local (heterogeneous) energy density is then defined via

$$W(\mathit{F},\mathbf{X})=\{\begin{array}{cc}{W}^{(1)}(\mathit{F}),\hfill & \mathbf{X}\in {\mathrm{\Omega }}_0^{(1)},\hfill \\ W^{(2)}(\mathit{F}),\hfill & \mathbf{X}\in {\mathrm{\Omega }}_0^{(2)},\hfill \end{array}$$

(1)

with the local first Piola–Kirchhoff (PK) stress given by

$$\mathit{S}(\mathbf{X})=\mathcal{S}(\mathit{F},\mathbf{X})=\frac{\mathrm{\partial W}}{\partial \mathit{F}}(\mathit{F},\mathbf{X})$$

(2)

and the local tangent modulus tensor by

$$\mathbb{L}(\mathbf{X})=\mathcal{L}(\mathit{F},\mathbf{X})=\frac{{\partial }^{2}W}{\partial \mathit{F}\partial \mathit{F}}(\mathit{F},\mathbf{X}).$$

(3)

Then, following Hill [6], the homogenized or mesoscopically averaged response of the composite under an affine deformation boundary condition $\mathbf{x}=\overline{\mathit{F}}\mathbf{X}$ is defined to be

$$\hat{W}(\overline{\mathit{F}})=\underset{\mathit{F}\in \mathcal{K}(\overline{\mathit{F}})}{\min }\frac{1}{|{\mathrm{\Omega }}_0|}{\int }_{{\mathrm{\Omega }}_0}W(\mathit{F},\mathbf{X})d{V}_{\mathbf{X}},$$

(4)

where

$$\mathcal{K}(\overline{\mathit{F}})=\{\mathit{F}:\mathit{F}(\mathbf{X})=\frac{\partial \mathbf{x}}{\partial \mathbf{X}}(\mathbf{X})\mathrm{for}\mathbf{X}\in {\mathrm{\Omega }}_0,\mathbf{x}=\overline{\mathit{F}}\mathbf{X}\mathrm{on}\partial {\mathrm{\Omega }}_0\}$$

(5)

are the kinematically admissible deformations satisfying the imposed boundary condition. The average PK stress $\overline{\mathit{S}}=⟨\mathit{S}⟩$ and the effective modulus tensor $\hat{\mathbb{L}}$ are then given by the appropriate derivatives of $\hat{W}$ via

$$\overline{\mathit{S}}=\frac{\partial \hat{W}}{\partial \overline{\mathit{F}}}(\overline{\mathit{F}})$$

(6)

and

$$\hat{\mathbb{L}}=\frac{{\partial }^2\hat{W}}{\partial \overline{\mathit{F}}\partial \overline{\mathit{F}}}(\overline{\mathit{F}}).$$

(7)

Herein, we term (4) the ‘principal’ or ‘mesoscopic’ solution for the effective stored-energy density.

For many composites, it has been found that, even when the constituent phases have polyconvex energy densities, the resulting homogenized energy $\hat{W}(\overline{\mathit{F}})$ , while initially strongly elliptic, may eventually lose SE. This was first shown for porous elastomers by Abeyaratne and Triantafyllidis [7] (see also Triantafyllidis et al. [21] and Michel et al. [22]) in the context of periodic, 2D microstructures, and later for random porous microstructures in both 2D [24] and 3D [23]. This LOSE signals the presence of instabilities which may lead to a relaxation of the above principal solution. In these cases, the stationary conditions of equation (4) admit multiple solutions, and the ‘relaxed’ or ‘macroscopic’ response is defined [8] by generalizing (4) to

$$\tilde{W}(\overline{\mathit{F}})=\underset{\mathit{F}\in \mathcal{K}(\overline{\mathit{F}})}{\inf }\frac{1}{|{\mathrm{\Omega }}_0|}{\int }_{{\mathrm{\Omega }}_0}W(\mathit{F},\mathbf{X})d{V}_{\mathbf{X}},$$

(8)

such that the presence of minimizing sequences must also be considered; indeed, a twinned microstructure corresponds to just such a solution (see Ball and James [34]). While the solution of the above is in general difficult, a method for at least generating an estimate has been developed based on computation of the quasiconvex envelope $Q\hat{W}$ of the principal solution $\hat{W}(\overline{\mathit{F}})$ by first computing its rank-one convexification $R\hat{W}$ and then, if possible, showing the result to be polyconvex. If this latter step is completed, the estimate is exactly the quasiconvexification of the principal solution according to the set of inequalities (see Ball [35])

$$P\hat{W}\le Q\hat{W}\le R\hat{W},$$

(9)

and is postulated [30] to correspond to the solution of equation (8).

To this end, we will make use of an algorithm developed by Kohn and Strang [36] for determining the rank-one convexification by a sequential lamination procedure, which proceeds by computing

$$R_{k+1}\hat{W}(\overline{\mathit{F}})=\underset{c,\mathbf{N},\mathbf{w}}{\min }\{(1-c)R_k\hat{W}(\overline{\mathit{F}}-c\mathbf{w}\otimes \mathbf{N})+c{R}_k\hat{W}(\overline{\mathit{F}}+(1-c)\mathbf{w}\otimes \mathbf{N})\},$$

(10)

where $R_0\hat{W}(\overline{\mathit{F}})=\hat{W}(\overline{\mathit{F}})$ . If this procedure converges after a finite number of iterations, it yields the rank-one convexification $R\hat{W}(\overline{\mathit{F}})$ .

Such a procedure has been used for several types of elastomeric composites to demonstrate that they can relax via domain formation after the onset of a macroscopic instability as signaled by LOSE. This was first done by Avazmohammadi and Ponte Castañeda [30] for fiber-reinforced elastomers and subsequently for elastomeric laminates [31,32], magnetoelastic laminates [37], and most recently for porous elastomers [33].

2.1. Internal pore pressure

According to the aforementioned framework, the homogenized response of a porous material with pore pressures evolving according to some arbitrary constitutive model may be estimated using the various techniques available in the literature; see Ponte Castañeda [20] for a recent review of these methods for soft materials. However, such estimates will depend on the specific model chosen for the pore phase and therefore the difficult homogenization process would need to be performed for each possible case. To alleviate this difficulty, Idiart and Lopez-Pamies [25], building on work from Vincent et al. [38] and Julien et al. [39], have given a method whereby, if the matrix phase is taken to be incompressible (often a reasonable assumption for elastomers), the effective response of the pressurized composite may be given in terms of the response of the unpressurized composite, i.e., a porous material with vacuous pores. This allows the complex homogenization to be performed only for the vacuous case, while the effects of various types of pore pressure may be subsequently investigated with relative facility. As here we desire the result in a somewhat different form than that given by the above authors, their procedure, which makes use of a change of variables in the relevant boundary value problem, is suitably adapted in Appendix 1; for generality, we also include the effect of a constant external pressure (such as atmospheric pressure) on the composite. The resulting expression for the stored-energy density of the composite with initial pore pressure $P_i=-{\psi }^{\prime }(1)$ and constant external pressure $P_{\mathrm{ext}}$ is

$$\hat{W}(\overline{\mathit{F}})={\hat{W}}_0(\overline{\mathit{F}})+\omega (\overline{J}),$$

(11)

where

$$\omega (\overline{J})=f_o\psi (\frac{\overline{J}-1+f_o}{f_o})+P_{\mathrm{ext}}(\overline{J}-1+f_o)+{\omega }_0.$$

(12)

Here, ${\hat{W}}_0$ is the homogenized energy of the unpressurized composite, and ψ is the potential governing the pressure response of the porous phase, such that the pore pressure $\mathcal{P}(\overline{J})$ is given by its derivative:

$$\mathcal{P}(\overline{J})=-{\psi }^{\prime }({\overline{J}}^{(2)}(\overline{J})),$$

(13)

where the average pore deformation ${\overline{J}}^{(2)}$ may be considered to be a direct function of the macroscopic deformation, due to the incompressibility of the matrix, via

$${\overline{J}}^{(2)}(\overline{J})=\frac{\overline{J}-1+f_o}{f_o},$$

(14)

with $f_o=|{\mathrm{\Omega }}_0^{(2)}|∕|{\mathrm{\Omega }}_0|$ being the initial porosity. Note then that the initial pressure is defined by $P_i=\mathcal{P}(1)$ . The constant ${\omega }_0$ , dependent on both the unpressurized energy ${\hat{W}}_0$ and the pore properties, is defined in equation (63) in Appendix 1, such that the traction-free and zero-energy configurations of the composite coincide. Finally, note that $\omega (\overline{J})$ may be understood as a potential for the pore gauge pressure ${\mathcal{P}}_g(\overline{J})$ at any deformation, via

$${\mathcal{P}}_g(\overline{J})=-{\omega }^{\prime }(\overline{J})=\mathcal{P}(\overline{J})-P_{\mathrm{ext}}.$$

(15)

In what follows, we will make use of both the absolute pressure $\mathcal{P}(\overline{J})$ and the gauge pressure ${\mathcal{P}}_g(\overline{J})$ , with corresponding initial values $P_i$ and $P_i^g$ at $\overline{J}=1$ .

A few comments are in order to ensure proper interpretation of the result (11). First, in all the above, we measure the deformation $\overline{\mathit{F}}$ from the initial (natural) configuration of the unpressurized material, despite the fact that this configuration is in general not the traction-free configuration of the pressurized material; this is in contrast to the authors Idiart and Lopez-Pamies [25], who instead measure deformations from the stress-free configuration. We make this choice to facilitate comparisons between composites with different pressure responses, which otherwise would have no direct correspondence between their deformations (due to their traction-free configurations being generally different).

In particular, the reference configuration from which $\overline{\mathit{F}}$ is measured and with respect to which the energy density $\hat{W}$ is normalized is that in which an external traction, equal in magnitude and opposite in sign to the internal gauge pressure $P_i^g={\mathcal{P}}_g(\overline{J}=1)$ , is applied to the body. This can be seen by differentiating (11) with respect to $\overline{\mathit{F}}$ at $\overline{\mathit{F}}=\mathit{I}$ , leading to

$$\frac{\partial \hat{W}}{\partial \overline{\mathit{F}}}(\mathit{I})=\overline{\mathit{S}}(\mathit{I})=-P_i^g\mathit{I},$$

(16)

and is shown schematically in Figure 1. In particular, note that when $P_i^g=0$ , the applied traction is given by $\overline{\mathit{S}}(\mathit{I})=\mathbf{0}$ , such that $\overline{\mathit{F}}=\mathit{I}$ indeed corresponds to the initial configuration of the unpressurized composite. This scenario may also arise naturally in other circumstances, as for instance when initially open pores are flooded with air at atmospheric pressure and subsequently closed.

Figure 1.

Schematic of reference configuration $\overline{\mathit{F}}=\mathit{I}$ for the pressurized porous elastomer. (a) Shows the initial configuration of the unpressurized material, with $P_i=P_{\mathrm{ext}}=0$ and $\overline{\mathit{S}}=0$ . (b) Shows the same configuration in the pressurized material, with an initial gauge pressure $P_i^g=P_i-P_{\mathrm{ext}}$ within the pores, $P_{\mathrm{ext}}\ne 0$ , and an (additional) ‘extra’ traction given by $\overline{\mathit{S}}=-P_i^{g}I$ on the external boundary of the porous elastomer.

Second, it must be emphasized that the stress corresponding to the energy (11) via $\overline{\mathit{S}}=\partial \hat{W}∕\partial \overline{\mathit{F}}$ is the ‘extra’ stress which is applied to the material beyond the external ‘ambient’ pressure $P_{\mathrm{ext}}$ , and does not include this pressure. In particular, $\overline{\mathit{S}}=0$ implies that nothing except the ambient pressure acts on the body. It is this state, with $\overline{\mathit{S}}=0$ , which we term the ‘traction-free’ state.

Additionally, the relation (11) should be understood to give only the mesoscopic, pre-relaxation response of the pressurized material, and therefore, in it, the unpressurized energy ${\hat{W}}_0$ should likewise correspond to the principal solution for the stored energy of the vacuous composite. This is because the extra term corresponding to the energy associated with the deformation of the pressurized pores alters the elastic moduli in non-trivial ways which may a priori affect LOSE in the pressurized elastomer as compared to the unpressurized elastomer. Hence, in theory, any relaxation which has been computed for the unpressurized composite must be recomputed for the pressurized scenario and could yield a different relaxed configuration in general. This will be considered in detail in Section 5.

Finally, observe that the additional term in equation (11) is a function only of $\overline{J}$ , so that if ${\hat{W}}_0(\overline{\mathit{F}})$ is isotropic, so is $\hat{W}(\overline{\mathit{F}})$ . This is in accord with our intuition on the effect of pore pressure in an isotropic distribution of pores, and is critical to the subsequent arguments concerning LOSE and twinning in these pressurized composites.

3. Estimates for the mesoscopic response of 2D porous elastomers under plane strain loading

Here, we first summarize previous results providing an estimate for the mesoscopic response of a class of 2D porous elastomers with vacuous pores. Then, using the results of Section 2.1, we extend the results to pressurized pores, considering two possible pressurization schemes: an ideal gas model, corresponding to closed, gas-filled pores, and prescribed pore pressure, corresponding to open pores with direct pressure control. These estimates describe elastomers containing an isotropic distribution of aligned cylindrical pores of circular cross-section which are subjected to plane strain conditions in the transverse plane. An estimate for the mesoscopic stored-energy of such a material with vacuous pores, corresponding to the principal solution of the associated variational problem (4), was given by Lopez-Pamies and Ponte Castañeda [24]. These authors specialized their results for the simple case where the elastomeric matrix is a generalized neo-Hookean material, giving explicit predictions for the pre-bifurcation response. They noted, however, the eventual LOSE of these estimates, signaling the possible development of instabilities, which will be considered in the following section.

Thus, considering the mesoscopic stored-energy of such 2D porous elastomers with an incompressible neo-Hookean matrix, Lopez-Pamies and Ponte Castañeda [24] provided the expression

$${\hat{W}}_0(\overline{\mathit{F}})={\hat{\mathrm{\Phi }}}_0({\overline{\lambda }}_1,{\overline{\lambda }}_2)=\frac{(1-f_o)\mu }{2}(\frac{p_4{v}^4+p_3{v}^3+p_2{v}^2+p_{1}v+p_0}{{(q_2{v}^2+q_{1}v+q_0)}^2}-2)$$

(17)

in terms of the initial porosity $f_o$ and the principal stretches ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$ , where $v=v({\overline{\lambda }}_1,{\overline{\lambda }}_2)$ is a solution of the quartic polynomial equation

$$r_4{v}^4+r_3{v}^3+r_2{v}^2+r_{1}v+r_0=0.$$

(18)

The expressions for the coefficients $p_0,p_1,p_2,p_3,p_4,q_0,q_1,q_2,r_0,r_1,r_2,r_3$ and $r_4$ , being somewhat lengthy, are provided explicitly in Appendix 3, but depend on $f_o$ , ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$ . Consistent with conservation of mass and the incompressibility of the matrix phase, these authors further noted that the evolution of the porosity is given simply by

$$f=1-\frac{1-f_o}{{\overline{\lambda }}_1{\overline{\lambda }}_2},$$

(19)

such that pore closure will occur at $\det \overline{\mathit{F}}={\overline{\lambda }}_1{\overline{\lambda }}_2=1-f_o$ ; thus the energy must be infinite for all deformations ${\overline{\lambda }}_1{\overline{\lambda }}_2<1-f_o$ . Following Caulfield and Ponte Castañeda [33], we denote this region by ${\mathcal{S}}_{\infty }$ .

3.1. Gas-filled pores

A simple experiment to incorporate the effects of pore pressure would be to allow gas to flood the pores at some pressure (say, e.g., air at atmospheric pressure), then close the pores and deform the material. Now, for an ideal gas, the constitutive relation (assuming isothermal conditions) is $\mathrm{PJ}=P_i{J}_i$ for some reference state with deformation $J_i$ and corresponding pressure $P_i$ . A convenient reference state is $J_i=1$ , which yields ${\psi }_{\mathrm{IG}}(J^{(2)})=-P_i\ln (J^{(2)})$ , and thus the contribution to the energy due to the internal and external pressures, using expression (12), is

$${\omega }_{\mathrm{IG}}(\overline{J})=f_o{P}_i\ln (\frac{f_o}{\overline{J}-1+f_o})+P_{\mathrm{ext}}(\overline{J}-1+f_o)+{\omega }_{\mathrm{IG}}^0.$$

(20)

However, we are particularly interested in cases for which the initial internal pressure is equal to the external, $P_i=P_{\mathrm{ext}}$ , so that the initial gauge pressure is $P_i^g=0$ , the traction-free configuration is ${\overline{\mathit{F}}}_0=\mathit{I}$ , ${\omega }_{\mathrm{IG}}^0=-f_o{P}_i$ , and we may rewrite the above more explicitly as

$${\omega }_{\mathrm{IG}}(\overline{J})=f_o{P}_i\left[\frac{\overline{J}-1+f_o}{f_o}-\ln (\frac{\overline{J}-1+f_o}{f_o})-1\right].$$

(21)

Thus the mesoscopic energy for the composite with closed pores containing an ideal gas is

$${\hat{W}}_{\mathrm{IG}}(\overline{\mathit{F}})={\hat{W}}_0(\overline{\mathit{F}})+f_o{P}_i\left[\frac{\overline{J}-1+f_o}{f_o}-\ln (\frac{\overline{J}-1+f_o}{f_o})-1\right].$$

(22)

3.2. Prescribed pore pressure

Our interest in the behavior of pressurized porous elastomers is driven by interest in possible applications in actuation. A large class of these applications would involve direct control of the internal pore pressure by, e.g., maintaining open pores connected to a pump or pressure vessel. Because of this control, the internal pressure is independent of $\overline{J}$ , and may be varied only by the external control mechanism. Thus, for a given setting of the controller, we have ${\psi }_{\mathrm{CP}}(J^{(2)})=-P_i{J}^{(2)}$ , with $P_i$ constant, and

$$\begin{array}{ccc}\hfill {\omega }_{\mathrm{CP}}(\overline{J})& =-(P_i-P_{\mathrm{ext}})(\overline{J}-1+f_o)+{\omega }_{\mathrm{CP}}^0\hfill & \hfill \\ \hfill & =-P_i^g(\overline{J}-1+f_o)+{\omega }_{\mathrm{CP}}^0,\hfill \end{array}$$

(23)

such that the mesoscopic energy of the composite is

$${\hat{W}}_{\mathrm{CP}}(\overline{\mathit{F}})={\hat{W}}_0(\overline{\mathit{F}})-P_i^g(\overline{J}-1+f_o)+{\omega }_{\mathrm{CP}}^0.$$

(24)

Thus, the energy is dependent only on the gauge pressure $P_i^g$ , the difference between the internal and external pressures. The value of the constant ${\omega }_{\mathrm{CP}}^0$ is chosen according to equation (63) such that the traction-free and zero-energy configurations align. Thus

$${\omega }_{\mathrm{CP}}^0=P_i^g({\overline{J}}_0-1+f_o)-{\hat{W}}_0({\overline{\mathit{F}}}_0),$$

(25)

where ${\overline{\mathit{F}}}_0$ is the zero-traction configuration defined by

$$\frac{\partial {\hat{W}}_0}{\partial \overline{\mathit{F}}}({\overline{\mathit{F}}}_0)=P_i^g{\overline{\mathit{F}}}_0^{\mathrm{\ast }}.$$

(26)

Here, ${\overline{\mathit{F}}}_0^{\mathrm{\ast }}={\overline{J}}_0{\overline{\mathit{F}}}_0^{-T}$ is the adjugate of ${\overline{\mathit{F}}}_0$ .

4. LOSE under plane strain

The above estimates characterize the behavior of the pressurized composite in the absence of instabilities. However, Lopez-Pamies and Ponte Castañeda [24] noted that the stored energy given by expression (17) loses SE for certain loading conditions. Therefore, in this section, we recall the conditions for SE, then specialize them for plane strain of isotropic materials and summarize the results for LOSE in the unpressurized case. (It should be noted that void collapse instabilities [40] are not expected for porous neo-Hookean elastomers.) Finally, we consider the effects of pressurization on LOSE, in particular for the cases described in Section 3.

In general, the homogenized behavior of a heterogenous material is said to be strongly elliptic [28] at a deformation $\overline{\mathit{F}}$ with corresponding incremental modulus tensor $\hat{\mathbb{L}}$ if the inequality

$$\underset{|\mathbf{M}|=|\mathbf{N}|=1}{\min }Q(\hat{\mathbb{L}},\mathbf{M},\mathbf{N})>0$$

(27)

is satisfied, where

$$Q(\hat{\mathbb{L}},\mathbf{M},\mathbf{N})=M_i{N}_j{\hat{L}}_{\mathrm{ijkl}}{M}_k{N}_l.$$

(28)

For plane strain of an isotropic material, this general condition reduces [41,42] to five scalar conditions on the components of $\hat{\mathbb{L}}$ :

$$\begin{array}{ccc}\hfill & {\hat{L}}_{1111}>0,{\hat{L}}_{2222}>0,{\hat{L}}_{1212}>0,{\hat{L}}_{2121}>0,\hfill & \hfill \\ \hfill & {\hat{L}}_{1111}{\hat{L}}_{2222}+{\hat{L}}_{1212}{\hat{L}}_{2121}-{({\hat{L}}_{1122}+{\hat{L}}_{1221})}^2>-2\sqrt{{\hat{L}}_{1111}{\hat{L}}_{2222}{\hat{L}}_{1212}{\hat{L}}_{2121}},\hfill \end{array}$$

(29)

where the expressions for the ${\hat{L}}_{\mathrm{ijkl}}$ are given explicitly by equation (64) in Appendix 2 in terms of the derivatives of $\hat{\mathrm{\Phi }}({\overline{\lambda }}_1,{\overline{\lambda }}_2)$ .

4.1. LOSE of the unpressurized elastomer

Lopez-Pamies and Ponte Castañeda [24] found that the conditions (29) fail to be met in the unpressurized case for certain overall compressive loadings. Caulfield and Ponte Castañeda [33] noted that LOSE first occurs when ${\hat{L}}_{1212}={\hat{L}}_{2121}$ goes to zero, and partitioned the domain of $\hat{\mathrm{\Phi }}({\overline{\lambda }}_1,{\overline{\lambda }}_2)$ into the regions

$$\begin{array}{ccc}\hfill {\mathcal{S}}_{\mathrm{se}}& =\{({\overline{\lambda }}_1,{\overline{\lambda }}_2):{\overline{\lambda }}_2>{\overline{\lambda }}_{\mathrm{se}}({\overline{\lambda }}_1),{\overline{\lambda }}_1{\overline{\lambda }}_2\ge 1-f_o\},\hfill & \hfill \\ \hfill {\mathcal{S}}_{\mathrm{ne}}& ={({\mathcal{S}}_{\mathrm{se}}\cup {\mathcal{S}}_{\infty })}^c,\hfill \end{array}$$

(30)

where ${\overline{\lambda }}_{\mathrm{se}}({\overline{\lambda }}_1)$ defines the curve ${\hat{L}}_{1212}={\hat{L}}_{2121}=0$ , and where it is recalled that ${\mathcal{S}}_{\infty }$ corresponds to ${\overline{\lambda }}_1{\overline{\lambda }}_2<1-f_o$ .

Making use of the invariants $Ī=\overline{\mathit{F}}\cdot \overline{\mathit{F}}$ and $\overline{J}=\det \overline{\mathit{F}}$ , we may alternatively define the energy via ${\hat{W}}_0(\overline{\mathit{F}})={ĝ}_0(Ī,\overline{J})$ . The curve in Ī – $\overline{J}$ space corresponding to ${\overline{\lambda }}_{\mathrm{se}}({\overline{\lambda }}_1)$ is denoted by ${Ī}_{\mathrm{se}}(\overline{J})$ ; using the identity $L_{1212}=2g_{,I}$ this is equivalent to the curve ${ĝ}_{0,I}=0$ . The curves ${\overline{\lambda }}_{\mathrm{se}}({\overline{\lambda }}_1)$ and ${Ī}_{\mathrm{se}}(\overline{J})$ are shown as dashed orange lines in Figure 2 for this material with initial porosity $f_o=0.7$ , along with pore closure as dotted red lines.

Figure 2.

(a) LOSE and pore closure in the porous elastomer in terms of the macroscopic principal stretches ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$ , for an initial porosity $f_o=0.7$ . Initial LOSE is marked by the curve ${\overline{\lambda }}_{\mathrm{se}}({\overline{\lambda }}_1)$ in dashed orange, which it is recalled is equivalent to the curve ${\hat{L}}_{1212}=0$ . (b) Corresponding plot in terms of the invariants Ī and $\overline{J}$ , with the orange dashed curve now defining ${Ī}_{\mathrm{se}}(\overline{J})$ . (Note deformations with $\overline{J}>\frac{1}{2}Ī$ are physically impossible; the grey dash-dotted line therefore bounds the domain of possible deformations, and corresponds to hydrostatic deformation). In both plots pore closure is marked by a dotted red curve.

4.2. LOSE under general pore pressure

We now consider LOSE of the pressurized composite and compare it to the unpressurized. It is recalled that Idiart and Lopez-Pamies [25] found that pressurizing the pores generally tends to stabilize the porous elastomer. More precisely, if the behavior is strongly elliptic for a certain deformation in the absence of pore pressure, it must also be strongly elliptic for that deformation when the pores are pressurized. This is because, using the expression (11) giving the pressurized stored energy in terms of the unpressurized energy ${\hat{W}}_0(\overline{\mathit{F}})$ and the function $\omega (\overline{J})$ , we have

$$Q(\hat{\mathbb{L}},\mathbf{M},\mathbf{N})=Q({\hat{\mathbb{L}}}_0,\mathbf{M},\mathbf{N})+{\omega }^{″}{\overline{J}}^2{(\mathbf{N}\cdot {\overline{\mathit{F}}}^{-1}\mathbf{M})}^2.$$

(31)

Under our assumption of convex $\psi (J)$ , which is equivalent to the convexity of $\omega (\overline{J})$ , the second term is always non-negative, such that $Q(\hat{\mathbb{L}},\mathbf{M},\mathbf{N})\ge Q({\hat{\mathbb{L}}}_0,\mathbf{M},\mathbf{N})\mathrm{\forall }\mathbf{M},\mathbf{N}$ . Therefore, wherever equation (27) is satisfied by the unpressurized modulus ${\hat{\mathbb{L}}}_0$ , it must also be satisfied by the pressurized modulus $\hat{\mathbb{L}}$ . Hence, the addition of pore pressure cannot cause the LOSE domain to grow.

Here, we go beyond this result to examine the particular ‘mode’ which drives LOSE, making use of the specialized conditions (29) for plane strain. We are interested in whether the addition of pore pressure could affect which condition of equation (29) first fails on the LOSE boundary. To this end, using expression (11) for the mesoscopic energy of the pressurized composite, and the expressions (64) for the components of $\hat{\mathbb{L}}$ , the moduli of the pressurized composite are

$$\begin{array}{ccc}\hfill {\hat{L}}_{1111}& ={\hat{L}}_{1111}^0+{\omega }^{″}{\overline{\lambda }}_2^2,\hfill & \hfill \\ \hfill {\hat{L}}_{2222}& ={\hat{L}}_{2222}^0+{\omega }^{″}{\overline{\lambda }}_1^2,\hfill \\ \hfill {\hat{L}}_{1122}& ={\hat{L}}_{1122}^0+{\omega }^{″}{\overline{\lambda }}_1{\overline{\lambda }}_2+{\omega }^{\prime },\hfill \\ \hfill {\hat{L}}_{1212}& ={\hat{L}}_{1212}^0,\hfill \\ \hfill {\hat{L}}_{1221}& ={\hat{L}}_{1221}^0-{\omega }^{\prime },\hfill \end{array}$$

(32)

in terms of the components ${\hat{L}}_{\mathrm{ijkl}}^0$ of the unpressurized modulus ${\hat{\mathbb{L}}}_0$ and the function $\omega (\overline{J})$ characterizing the pore pressure evolution. Because $\omega (\overline{J})$ is convex, ${\omega }^{″}\ge 0$ everywhere, and therefore, it is clear that wherever the first two conditions of equation (29) for SE are satisfied in the absence of pore pressure, they are also satisfied in the presence of arbitrary pore pressure. The shear modulus ${\hat{L}}_{1212}={\hat{L}}_{2121}$ is completely independent of the pore pressure, and hence equations (29)₃ and (29)₄, which are equivalent, are unaffected regardless of choice of $\omega (\overline{J})$ . The fifth condition is more opaque, and in the pressurized case becomes

$$\begin{array}{ccc}\hfill ({\hat{L}}_{1111}^0+{\omega }^{″}{\overline{\lambda }}_2^2)& ({\hat{L}}_{2222}^0+{\omega }^{″}{\overline{\lambda }}_1^2)+{\hat{L}}_{1212}^0{\hat{L}}_{2121}^0-{({\hat{L}}_{1122}^0+{\omega }^{″}{\overline{\lambda }}_1{\overline{\lambda }}_2+{\hat{L}}_{1221}^0)}^2\hfill & \hfill \\ \hfill & +2\sqrt{{\hat{L}}_{1212}^0{\hat{L}}_{2121}^0({\hat{L}}_{1111}^0+{\omega }^{″}{\overline{\lambda }}_2^2)({\hat{L}}_{2222}^0+{\omega }^{″}{\overline{\lambda }}_1^2)}>0.\hfill \end{array}$$

(33)

Note this depends on $\omega (\overline{J})$ only through its second derivative.

Recall that in the material here under consideration, with unpressurized stored energy (17), LOSE is driven by ${\hat{L}}_{1212}^0=0$ in the unpressurized case. From equation (32)₄, ${\hat{L}}_{1212}$ for the pressurized material vanishes on precisely the same curve. But as noted above, the domain of LOSE cannot expand by the addition of pore pressure, and therefore the domain of LOSE is unchanged in the pressurized composite as compared to the unpressurized, such that ${\overline{\lambda }}_{\mathrm{se}}({\overline{\lambda }}_1)={\overline{\lambda }}_{\mathrm{se}}^0({\overline{\lambda }}_1)$ and ${Ī}_{\mathrm{se}}({\overline{\lambda }}_1)={Ī}_{\mathrm{se}}^0({\overline{\lambda }}_1)$ . Moreover, in both the pressurized and the unpressurized composites, these curves bounding the domains of LOSE are given by ${\hat{L}}_{1212}={\hat{L}}_{1212}^0=0$ , or equivalently ${ĝ}_{,I}={ĝ}_{0,I}=0$ .

For the particular case of prescribed pore pressure, we may go further to say that all five of the LOSE conditions fail in precisely the same domains regardless of the pressure or lack thereof, because each of the conditions depends only on ${\omega }^{″}$ and ${\omega }_{\mathrm{CP}}^{″}(\overline{J})=0$ . For the case of gas-filled pores, the domains where equations (29)₁, (29)₂, and (29)₅ fail are found to shrink when the pores are pressurized, but in such a way that overall LOSE is unchanged, as a consequence of the fact that LOSE is still driven by the vanishing of the transverse shear modulus (which is not affected by pore pressure).

5. Twinning under plane strain

Once it is known under what conditions stability is lost, the next question becomes in what way the material relaxes from the principal solution so as to recover stability in the ‘macroscopic’ solution. Caulfield and Ponte Castañeda [33] explored the possibility of relaxation via domain formation, using the lamination procedure discussed in Section 2 to demonstrate the existence of a singly twinned microstructure which yields a macroscopically stable response according to the Legendre–Hadamard condition (66) (that is, equation (29) without the strict inequality). As will be demonstrated below, the same arguments hold for the pressurized composites discussed herein, as a consequence of the fact that the behavior of the pressurized composite is still isotropic and that LOSE is still driven by $L_{1212}$ going to zero. Therefore, the exact same twinned microstructure will occur, and for exactly the same deformations (when measured from the reference configuration of the unpressurized material), for these pressurized composites, leading to a simple expression for their macroscopic energy in terms of that of the unpressurized composite.

Hence, we here summarize the analysis of Caulfield and Ponte Castañeda [33], applying it to the pressurized porous elastomer with mesoscopic energy $\hat{W}(\overline{\mathit{F}})=ĝ(Ī,\overline{J})$ , and refer the interested reader to their work for further details. To begin, we make use of the expression (10) for constructing the rank-one convexification $R\hat{W}$ , the first step of which involves computing

$$R_1\hat{W}(\overline{\mathit{F}})=\underset{c,\mathbf{N},\mathbf{w}}{\min }\{(1-c)\hat{W}(\overline{\mathit{F}}-c\mathbf{w}\otimes \mathbf{N})+c\hat{W}(\overline{\mathit{F}}+(1-c)\mathbf{w}\otimes \mathbf{N})\},$$

(34)

where we have used the fact that $R_0\hat{W}(\overline{\mathit{F}})=\hat{W}(\overline{\mathit{F}})$ . This result physically corresponds to a laminated microstructure in which the body breaks into lamellar domains with normals N (in the reference configuration) and deformations

$${\overline{\mathit{F}}}^{(I)}=\overline{\mathit{F}}-c\mathbf{w}\otimes \mathbf{N},{\overline{\mathit{F}}}^{(\mathrm{II})}=\overline{\mathit{F}}+(1-c)\mathbf{w}\otimes \mathbf{N}$$

(35)

in corresponding volume fractions $1-c$ and c, such that the average deformation $\overline{\mathit{F}}=(1-c){\overline{\mathit{F}}}^{(I)}+c{\overline{\mathit{F}}}^{(\mathrm{II})}$ still satisfies the macroscopically prescribed deformation. Thus, the determination of a twinned microstructure amounts to finding the twinning parameters in equation (35) defining ${\overline{\mathit{F}}}^{(I)}$ and ${\overline{\mathit{F}}}^{(\mathrm{II})}$ . This requires solving the optimality conditions of the minimization problem (10), namely,

$$\begin{array}{ccc}\hfill ({\overline{\mathit{S}}}^{(\mathrm{II})}-{\overline{\mathit{S}}}^{(I)})\mathbf{N}& =\mathbf{0},\hfill & \hfill \\ \hfill \mathbf{w}\cdot ({\overline{\mathit{S}}}^{(\mathrm{II})}-{\overline{\mathit{S}}}^{(I)})\mathbf{M}& =0,\hfill \\ \hfill \hat{W}({\overline{\mathit{F}}}^{(\mathrm{II})})-\hat{W}({\overline{\mathit{F}}}^{(I)})& =\mathbf{w}\cdot {\overline{\mathit{S}}}^{(I)}\mathbf{N}=\mathbf{w}\cdot {\overline{\mathit{S}}}^{(\mathrm{II})}\mathbf{N},\hfill \end{array}$$

(36)

where N is the unit vector in the reference configuration giving the normal to the layers, $\mathbf{M}\perp \mathbf{N}$ , and ${\overline{\mathit{S}}}^{(I)}$ and ${\overline{\mathit{S}}}^{(\mathrm{II})}$ are the stresses corresponding to the deformation gradients ${\overline{\mathit{F}}}^{(I)}$ and ${\overline{\mathit{F}}}^{(\mathrm{II})}$ in the layers. Note that these conditions correspond respectively to continuity of the traction vector across the layers, continuity in the w-direction of the traction on faces orthogonal to the layers, and a generalized bi-tangency condition for the stresses along the rank-one direction ${\overline{\mathit{F}}}^{(\mathrm{II})}-{\overline{\mathit{F}}}^{(I)}=\mathbf{w}\otimes \mathbf{N}$ .

Now, to aid in determining a solution to the above, recall [43] that any twinned microstructure must satisfy the condition that the right-stretch tensors in each layer are related to one another simply by a rotation—i.e., denoting the deformation in one phase by ${\overline{\mathit{F}}}^{(I)}={\overline{\mathit{R}}}^{(I)}{\overline{\mathit{U}}}^{(I)}$ and similarly in the other by ${\overline{\mathit{F}}}^{(\mathrm{II})}={\overline{\mathit{R}}}^{(\mathrm{II})}{\overline{\mathit{U}}}^{(\mathrm{II})}$ , we have

$${\overline{\mathit{U}}}^{(\mathrm{II})}=\mathit{Q}{\overline{\mathit{U}}}^{(I)}{\mathit{Q}}^T$$

(37)

for some proper orthogonal Q . This implies both invariants must be equal across the two layers, such that

$$\begin{array}{ccc}\hfill {\overline{J}}^{(I)}& =\det {\overline{\mathit{F}}}^{(I)}=\det {\overline{\mathit{F}}}^{(\mathrm{II})}={\overline{J}}^{(\mathrm{II})}=Ĵ,\hfill & \hfill \\ \hfill {Ī}^{(I)}& ={\overline{\mathit{F}}}^{(I)}\cdot {\overline{\mathit{F}}}^{(I)}={\overline{\mathit{F}}}^{(\mathrm{II})}\cdot {\overline{\mathit{F}}}^{(\mathrm{II})}={Ī}^{(\mathrm{II})}=Î.\hfill \end{array}$$

(38)

Note that this also implies that both twinned deformations have the same principal stretches and, due to isotropy, the same mesoscopic energy.

Using the expression (35) for ${\overline{\mathit{F}}}^{(I)}$ and ${\overline{\mathit{F}}}^{(\mathrm{II})}$ , condition equation (38)₁ requires $\mathbf{w}\cdot {\overline{\mathit{F}}}^{\mathrm{\ast }}\mathbf{N}=0$ , i.e., w has the form

$$\mathbf{w}=\frac{w}{|\overline{\mathit{F}}\mathbf{M}|}\overline{\mathit{F}}\mathbf{M},$$

(39)

with $\mathbf{M}\perp \mathbf{N}$ and where $w=|\mathbf{w}|$ . Moreover, this form for w yields

$${\overline{J}}^{(I)}={\overline{J}}^{(\mathrm{II})}=Ĵ=\overline{J},$$

(40)

such that the volumetric deformation in the twins is equal to the macroscopically applied $\overline{J}$ . On the contrary, using equations (35) and (38)₂, we may directly obtain an expression for the twin volume fractions as

$$c=\frac{1}{2}+\frac{1}{w^2}(\mathbf{w}\cdot \overline{\mathit{F}}\mathbf{N}).$$

(41)

This allows for explicit definition of the first invariant in the twins via

$$Î=Ī+\frac{w^2}{4}-\frac{1}{w^2}{(\mathbf{w}\cdot \overline{\mathit{F}}\mathbf{N})}^2,$$

(42)

with N and w as yet still unknown.

Using these observations, we may write the stresses ${\overline{\mathit{S}}}^{(I)}$ and ${\overline{\mathit{S}}}^{(\mathrm{II})}$ in terms of derivatives of the mesoscopic energy $ĝ(Î,\overline{J})$ and show that the optimality condition equation (36)₁ requires

$$\frac{\mathrm{\partial ĝ}}{\mathrm{\partial I}}(Î,\overline{J})=0.$$

(43)

This condition defining $Î(\overline{J})$ in equation (42), along with equations (39) and (41) for the twinned deformation parameters due to the twinning conditions, satisfies the four scalar optimality conditions (36). Note that the lamination orientation, determined by N, is therefore indeterminate from the analysis. Thus, for each possible N, a w may be chosen in equation (42) such that Î satisfies (43) and in terms of which c is determined by equation (41); the resulting microstructure, however, has precisely the same energy as the corresponding microstructure for a different choice of N, as a result of the twinning conditions (38) and the material isotropy. Note also that the laminated microstructure depends on the material’s stored-energy density only through the condition (43), which is equivalent to ${\hat{L}}_{1212}({\overline{\mathit{F}}}^{(I)})={\hat{L}}_{1212}({\overline{\mathit{F}}}^{(\mathrm{II})})=0$ .

Finally, noting that for all the material models considered above, ${\hat{L}}_{1212}=0$ corresponds precisely to the LOSE boundary, we have $Î(\overline{J})={Ī}_{\mathrm{se}}(\overline{J})$ , and the energy of the rank-one laminate for the pressurized material is of the form

$$R_1\hat{W}(\overline{\mathit{F}})=r_1ĝ(Ī,\overline{J})=\{\begin{array}{cc}ĝ(Ī,\overline{J}),\hfill & \mathrm{if}\overline{\mathit{F}}\in {\mathcal{S}}_{\mathrm{se}},\hfill \\ ĝ({Ī}_{\mathrm{se}}(\overline{J}),\overline{J}),\hfill & \mathrm{if}\overline{\mathit{F}}\in {\mathcal{S}}_{\mathrm{ne}},\hfill \\ \infty ,\hfill & \mathrm{if}\overline{\mathit{F}}\in {\mathcal{S}}_{\infty }.\hfill \end{array}$$

(44)

At this point, it should be recalled that the corresponding result for the rank-one laminate in the unpressurized case takes the analogous form [33]

$$R_1{\hat{W}}_0(\overline{\mathit{F}})=r_1{ĝ}_0(Ī,\overline{J})=\{\begin{array}{cc}{ĝ}_0(Ī,\overline{J}),\hfill & \mathrm{if}\overline{\mathit{F}}\in {\mathcal{S}}_{\mathrm{se}},\hfill \\ {ĝ}_0({Ī}_{\mathrm{se}}(\overline{J}),\overline{J}),\hfill & \mathrm{if}\overline{\mathit{F}}\in {\mathcal{S}}_{\mathrm{ne}},\hfill \\ \infty ,\hfill & \mathrm{if}\overline{\mathit{F}}\in {\mathcal{S}}_{\infty }.\hfill \end{array}$$

(45)

In particular, note that the function ${Ī}_{\mathrm{se}}(\overline{J})$ defining the twin deformations in the pressurized case is precisely the same as the one arising in the unpressurized case, as a result of the fact that ${\hat{L}}_{1212}={\hat{L}}_{1212}^0=0$ gives the boundary of LOSE in both cases. An important consequence of this fact is that the twinned configurations of the pressurized and unpressurized materials are exactly the same.

It then remains to determine whether the stored energy (44) is rank-one convex and polyconvex. To this end, note that equations (11), (44), and (45) allow us to write

$$r_1ĝ(Ī,\overline{J})=r_1{ĝ}_0(Ī,\overline{J})+\omega (\overline{J}),$$

(46)

as a result of the fact that the volumetric deformation within the layers is equal to the macroscopic $\overline{J}$ . Now, Caulfield and Ponte Castañeda [33] showed that the function $r_1{ĝ}_0(Ī,\overline{J})$ is convex in both its arguments and non-decreasing in the first, and that therefore the associated energy $R_1{\hat{W}}_0(\overline{\mathit{F}})$ is polyconvex. But it is clear from equation (46) that if these conditions are satisfied for $r_1{ĝ}_0(Ī,\overline{J})$ they must also be satisfied for $r_1ĝ(Ī,\overline{J})$ , as a result of the convexity of $\omega (\overline{J})$ and its independence of Ī. Therefore the pressurized energy $R_1\hat{W}(\overline{\mathit{F}})$ is polyconvex as a result of the polyconvexity of $R_1{\hat{W}}_0(\overline{\mathit{F}})$ .

Recalling the set of inequalities (9), $R_1\hat{W}(\overline{\mathit{F}})$ is therefore also rank-one convex, such that the sequential lamination procedure (10) yields the rank-one convexification after this one lamination. Finally, $R_1\hat{W}(\overline{\mathit{F}})$ is also quasiconvex by equation (9), and we have $\tilde{W}(\overline{\mathit{F}})=R_1\hat{W}(\overline{\mathit{F}})$ . The relaxed energy of the pressurized porous elastomer, by equations (46) and (45), is therefore

$$\tilde{W}(\overline{\mathit{F}})=\tilde{g}(Ī,\overline{J})={\tilde{g}}_0(Ī,\overline{J})+\omega (\overline{J})=\{\begin{array}{cc}{ĝ}_0(Ī,\overline{J})+\omega (\overline{J}),\hfill & \mathrm{if}\overline{\mathit{F}}\in {\mathcal{S}}_{\mathrm{se}},\hfill \\ {ĝ}_0({Ī}_{\mathrm{se}}(\overline{J}),\overline{J})+\omega (\overline{J}),\hfill & \mathrm{if}\overline{\mathit{F}}\in {\mathcal{S}}_{\mathrm{ne}},\hfill \\ \infty ,\hfill & \mathrm{if}\overline{\mathit{F}}\in {\mathcal{S}}_{\infty },\hfill \end{array}$$

(47)

or, equivalently,

$$\tilde{W}(\overline{\mathit{F}})={\tilde{W}}_0(\overline{\mathit{F}})+\omega (\overline{J}).$$

(48)

Thus, the macroscopic energy $\tilde{W}$ of the pressurized composite may be written simply in terms of the macroscopic energy ${\tilde{W}}_0$ of the unpressurized composite, in a manner exactly analogous to the mesoscopic energy (11). Moreover, the twinned configuration of both the pressurized and unpressurized elastomers are identical, and are given by equation (35) with equations (39) and (41)–(43). This argument holds for any pressurized, isotropic porous elastomer undergoing plane strain deformations as long as LOSE is driven by failure of the SE condition (29)₃.

6. Results and discussion

In this section, we present representative results for the two types of pressurization protocols described above: first closed pores containing an ideal gas in Section 6.1, then open pores with prescribed pressure in Section 6.2. All pressures and stresses are normalized by the ground shear modulus μ of the matrix elastomer. The initial porosity of each material is chosen to be $f_o=0.7$ at $\overline{\mathit{F}}=\mathit{I}$ . While we primarily consider the response of these materials under applied tractions or displacements, note that in the prescribed pressure scenario the pressure itself becomes a controlled loading parameter. Thus, we conclude with simple results exploring direct pressure control.

6.1. Gas-filled pores

We begin by considering porous elastomers with closed pores which are filled with an ideal gas. The material is simultaneously subjected to a constant exterior pressure $P_{\mathrm{ext}}$ , modeling atmospheric pressure. The initial pore pressure $P_i$ is equal to $P_{\mathrm{ext}}$ in the undeformed configuration $\overline{\mathit{F}}=\mathit{I}$ , then subsequently evolves with the deformation according to the ideal gas law. Below we choose several values for $P_i$ and present the corresponding stored energy, followed by the material’s response to applied deformations and tractions.

Thus, Figure 3 shows the level curves of the stored-energy density when $P_i=0.1$ and $P_i=1$ . Figure 3(a) and (b) give the mesoscopic energies $\hat{W}$ , while Figure 3(c) and (d) give the macroscopic $\tilde{W}$ , incorporating the effects of relaxation and twinning. The energy is shown in terms of the principal stretches ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$ , making use of the form for the energy $W(\overline{\mathit{F}})=\mathrm{\Phi }({\overline{\lambda }}_1,{\overline{\lambda }}_2)$ . In each figure, the strongly elliptic regime is highlighted in green, the infinite-energy in white, and the not strongly elliptic or twinned in blue. LOSE and the onset of twinning are marked by a dashed orange line, and pore closure at $\det \overline{\mathit{F}}=1-f_o=0.3$ is shown by the dotted red line. Finally, the initial and zero-traction configurations are marked by a white diamond and a red circle, respectively; observe from equation (16) that these coincide in this case as a result of our choice $P_i=P_{\mathrm{ext}}$ .

Figure 3.

Level curves of the principal $\hat{W}=\hat{\mathrm{\Phi }}$ and relaxed $\tilde{W}=\tilde{\mathrm{\Phi }}$ energies of elastomers with pressurized, closed pores having initial pressures $P_i=0.1$ ((a) and (c)) and $P_i=1$ ((b) and (d)), as functions of ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$ , for an initial porosity of $f_o=0.7$ . The orange dashed line denotes the boundary of LOSE and onset of twinning, ${\overline{\lambda }}_{\mathrm{se}}({\overline{\lambda }}_1)$ , while the red dotted line denotes pore closure. The reference configuration $\overline{\mathit{F}}=\mathit{I}$ corresponds to the white diamond, and the zero-traction configuration ${\overline{\mathit{F}}}_0$ is shown by the red circle; note that these align in this case. The strongly elliptic, not strongly elliptic, and infinite energy regions are highlighted in green, blue, and white, respectively.

It is clear in Figure 3 that, the values of ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$ at which LOSE and the onset of twinning occur are completely unaffected by the change in internal pressure, just as predicted. However, the energies are affected, and in particular convexity in ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$ : note in Figure 3(a) that joint convexity in these variables is lost sufficiently within ${\mathit{S}}_{\mathrm{ne}}$ , while in Figure 3(b) convexity is lost only further into ${\mathit{S}}_{\mathrm{ne}}$ . Hence, it is seen that the pore pressure affects loss of convexity of the energy in these variables, while leaving unchanged both LOSE and loss of rank-one convexity. This emphasizes the fact that convexity of the energy in any particular choice of loading parameters is in general a different condition from SE.

Figure 4 gives the nonvanishing components of the average PK stress $\overline{\mathit{S}}$ as functions of the deformation, as measured by the logarithmic strain $\ln (\overline{\lambda })$ . Hydrostatic loading is shown in Figure 4(a), and uniaxial is shown in Figure 4(b). Pore closure is marked by a vertical red dotted line, while LOSE and the onset of twinning are shown by an orange dashed line. Three initial pore pressures are included: $P_i=0.01$ , $0.1$ , and 1. In Figure 4(a), the principal solutions for the stress are included as dashed curves along with the relaxed values as solid curves, while in Figure 4(b), the two components of the relaxed stress are given, with ${\overline{S}}_{11}$ as a solid line and ${\overline{S}}_{22}$ as a dotted.

Figure 4.

PK stresses for the composite with gas-filled pores, normalized by the matrix ground shear modulus μ, for (a) hydrostatic deformation and (b) uniaxial compaction, plotted in terms of the logarithmic strain. Sample curves are given for initial internal pressures of $P_i=0.01$ (green), $P_i=0.1$ (blue), and $P_i=1$ (purple). Each composite has a porosity $f_o=0.7$ at $\overline{\mathit{F}}=\mathit{I}$ . The onset of twinning and ultimate pore closure are marked by vertical dashed orange and dotted red lines, respectively.

The most notable feature in Figure 4 is the kink in the relaxed stress at the onset of twinning, resulting in abrupt softening followed by continued stiffening under further compression. In contrast, Figure 4 shows that the principal solution for the stress stiffens smoothly, and hence the relaxed behavior is much softer than the principal in the twinned domain. Observe also that the stress approaches $-\infty$ at pore closure as a result of the volume of the contained gas approaching zero.

Figure 5 shows the response of the material to uniaxial stress. Figure 5(a) plots the PK stress ${\overline{S}}_{11}$ against the resulting principal stretch ${\overline{\lambda }}_1$ , with ${\overline{S}}_{22}$ fixed to be 0. Figure 5(b) shows the resulting deformation paths in ${\overline{\lambda }}_1$ – ${\overline{\lambda }}_2$ space. The results are given for five initial pore pressures, including $P_i=0$ , the unpressurized case. Pore closure is marked in Figure 5(b) by a dotted red curve, and the onset of twinning by a dashed orange curve.

Figure 5.

Uniaxial stress applied to pressurized elastomers with gas-filled pores. (a) shows standard stress-stretch curves, with ${\overline{S}}_{11}$ applied and ${\overline{S}}_{22}=0$ fixed. (b) shows the deformations in ${\overline{\lambda }}_1$ – ${\overline{\lambda }}_2$ space produced by these uniaxial stresses. The onset of twinning is marked by the orange dashed curve, while pore closure is shown by the dotted red curve.

Note in Figure 5(b) that for uniaxial states of stress the material does not undergo twinning for any of the pore pressures shown. Observe also in Figure 5(a) that, particularly in tension, the uniaxial stress required to produce a given stretch is very similar between cases despite the initial pore pressures spanning several orders of magnitude. This is partly explained by recalling an ambient external pressure is also applied to the material equal in magnitude to the initial internal pressure, tending to counteract the effect of the internal pressure. Moreover, note in Figure 5(b) that the lateral stretch ${\overline{\lambda }}_2$ for a given ${\overline{\lambda }}_1$ varies significantly between the pressures shown, even when ${\overline{S}}_{11}$ is similar.

6.2. Prescribed pore pressure

We now present results for the case of prescribed pore pressure. Here an external mechanism controls the pore pressure $P_i$ , such that it is independent of the deformation. A constant external pressure $P_{\mathrm{ext}}$ is also applied to the material, modeling the effect of atmospheric pressure. In the results below, then, we characterize the pore pressure by the gauge pressure $P_i^g=P_i-P_{\mathrm{ext}}$ . We first fix the pore pressure to be constant and give the stored energy along with the material’s response to applied tractions and deformations, then give simple results for direct pressure control, varying the pore pressure and showing the resulting deformations.

Figure 6 shows the level curves of the stored energy of the porous elastomer with prescribed pressure in terms of the principal stretches ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$ . Figure 6(a) and (c) show the principal energy ${\hat{\mathrm{\Phi }}}_{\mathrm{CP}}$ and the relaxed energy ${\tilde{\mathrm{\Phi }}}_{\mathrm{CP}}$ for $P_i^g=-1$ , respectively. Figure 6(b) and (d) show the corresponding results for $P_i^g=0.1$ . The strongly elliptic regime is highlighted in green, the not strongly elliptic in blue, and the infinite-energy regime in white. LOSE is marked by a dashed orange line, and pore closure by a dotted red line. The white diamond marks the reference configuration ${\overline{\lambda }}_1={\overline{\lambda }}_2=1$ , and the red circle shows the zero-traction configuration $\overline{\mathit{S}}=0$ .

Figure 6.

Level curves of the principal $\hat{W}=\hat{\mathrm{\Phi }}$ and relaxed $\tilde{W}=\tilde{\mathrm{\Phi }}$ energies of elastomers with prescribed, fixed pore gauge pressures $P_i^g=-1$ ((a) and (c)) and $P_i^g=0.1$ ((b) and (d)), as functions of ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$ , for an initial porosity of $f_o=0.7$ . The orange dashed line denotes the boundary of LOSE and onset of twinning, ${\overline{\lambda }}_{\mathrm{se}}({\overline{\lambda }}_1)$ , while the red dotted line denotes pore closure. The reference configuration $\overline{\mathit{F}}=\mathit{I}$ corresponds to the white diamond, and the zero-traction configuration ${\overline{\mathit{F}}}_0$ is shown by the red circle; unlike for the ideal gas case, these in general do not align. The strongly elliptic, not strongly elliptic, and infinite energy regions are highlighted in green, blue, and white, respectively.

Observe in Figure 6(a) and (b) that the principal solution for the energy loses joint convexity in ${\overline{\lambda }}_1$ and ${\overline{\lambda }}_2$ sufficiently within ${\mathit{S}}_{\mathrm{ne}}$ , and in Figure 6(c) and (d) that the relaxation causes the energy to become convex in this region. However, note in Figure 6(a) that convexity is also lost in tension, though this does not affect LOSE and is unaffected by the relaxation. As noted in reference to Figure 3, this emphasizes the fact that SE does not generally correspond to convexity in a given set of loading parameters.

The stress-free configuration, corresponding to the minimum energy, is seen in Figure 6 to be different from the reference configuration. It may correspond to either an expansion or contraction compared to the reference configuration (the stress-free state of the unpressurized elastomer), depending on whether the pore gauge pressure is positive or negative, respectively. Because of this dependence of the traction-free configuration on the prescribed pressure, by varying the pore pressure the material could be made to deform, consistent with physical intuition. This possibility will be explored explicitly below. Note also in Figure 6(a) and (c) that the traction-free configuration for $P_i^g=-1$ is almost on the twinning boundary. Recall that twinning begins when the shear modulus $L_{1212}$ goes to zero—thus, we observe that the contours of the energy around the stress-free state are elongated in the direction perpendicular to the hydrostatic axis ${\overline{\lambda }}_1={\overline{\lambda }}_2$ , corresponding to softening in shear. This softening could be useful for applications in actuation, allowing the material to undergo large deformations for a small input.

Next, Figure 7 gives the response of the material to applied deformations for three prescribed internal pressures. Figure 7(a) shows hydrostatic compaction, and plots both the principal and relaxed values for the hydrostatic PK stress against the logarithmic strain $\ln (\overline{\lambda })$ . Figure 7(b) gives results for uniaxial compaction, and plots ${\overline{S}}_{11}$ and ${\overline{S}}_{22}$ as solid and dotted curves, respectively, against the uniaxial logarithmic strain $\ln ({\overline{\lambda }}_1)$ while ${\overline{\lambda }}_2$ is fixed to be 1. Pore closure is shown by a dotted red line, and onset of twinning by an orange dashed line.

Figure 7.

PK stresses for the composite with constant prescribed pore pressure, normalized by the matrix linearized shear modulus μ, for (a) hydrostatic deformation and (b) uniaxial compaction, plotted in terms of the logarithmic strain. Sample curves are given for pore gauge pressures of $P_i^g=-1$ (green), $P_i^g=-0.5$ (blue), and $P_i^g=0.1$ (purple). Each composite has a porosity $f_o=0.7$ at $\overline{\mathit{F}}=\mathit{I}$ . The onset of twinning and ultimate pore closure are marked by vertical dashed orange and dotted red lines, respectively.

Observe in Figure 7 that when $P_i^g\ne 0$ the horizontal intercepts of the stress-strain curves are not at the origin, corresponding to the fact that the reference and zero stress configurations do not coincide. Their vertical intercepts, where ${\overline{\lambda }}_1={\overline{\lambda }}_2=1$ , are at $\overline{\mathit{S}}=-P_i^g\mathit{I}$ . Observe also that, in contrast to the case of gas-filled pores, the stress reaches a finite value at pore closure. This is because the pore pressure remains at a constant, finite value due to the pressure control, rather than approaching infinity as in the closed, gas-filled pore scenario.

In Figure 8, we give results for uniaxial tension and compression. Figure 8(a) shows ${\overline{S}}_{11}$ versus ${\overline{\lambda }}_1$ with ${\overline{S}}_{22}$ fixed to be zero. Figure 8(b) gives the corresponding deformation paths through ${\overline{\lambda }}_1$ – ${\overline{\lambda }}_2$ space. The internal pore pressure is held constant throughout the deformations, and the results are given for six pressures: $P_i^g=-1.5,-1,-0.5,0,0.1,$ and $0.15$ . As usual, pore closure is marked with a dotted red curve while onset of twinning is marked with a dashed orange curve.

Figure 8.

Uniaxial stress applied to pressurized elastomers with prescribed pore pressure. (a) shows standard stress-deformation curves, with ${\overline{S}}_{11}$ applied and ${\overline{S}}_{22}=0$ fixed. (b) shows the deformations in ${\overline{\lambda }}_1$ – ${\overline{\lambda }}_2$ space produced by these uniaxial stresses. The onset of twinning is marked by the orange dashed curve, while pore closure is shown by the dotted red curve.

As noted regarding Figure 7, in Figure 8(a) none of the curves with $P_i^g\ne 0$ pass through the origin. The horizontal intercept is at ${\overline{\lambda }}_1>1$ for $P_i^g>0$ , and at ${\overline{\lambda }}_1<1$ for $P_i^g<0$ . This is due to the fact that the reference and traction-free configurations do not coincide for $P_i^g\ne 0$ . Observe also in Figure 8(a) that as the internal pressure is decreased from 0.15 the stiffness at ${\overline{S}}_{11}=0$ initially increases, then decreases rapidly until it is perfectly soft at $P_i^g=-1.5$ , the dark green curve. In fact, this zero stiffness is maintained for a wide range of deformation, from ${\overline{\lambda }}_1\approx 0.36$ until ${\overline{\lambda }}_1\approx 1.07$ . Note in Figure 8(b) that this behavior is due to the material entering the twinned regime. Thus, as the prescribed pore pressure is decreased, the paths through ${\overline{\lambda }}_1$ – ${\overline{\lambda }}_2$ space move closer to LOSE and the onset of twinning, resulting in the softening observed in Figure 8(a). When $P_i^g=-1.5$ the curve passes through the twinned region, resulting in perfectly soft behavior.

Finally, observe in Figure 8(b) that the deformation paths with $P_i^g\le 0$ follow the usual behavior for uniaxial tension: as the material is stretched in the ${\overline{\lambda }}_1$ direction, it contracts in the ${\overline{\lambda }}_2$ direction due to the Poisson effect, corresponding to a negative slope. However, in the curves with $P_i^g>0$ , this behavior reverses, and the slope becomes positive in sufficient tension. Thus here as ${\overline{\lambda }}_1$ is increased, ${\overline{\lambda }}_2$ simultaneously increases. This counterintuitive behavior is due to the prescription of constant pore pressure, which drives the expansion in the lateral direction ${\overline{\lambda }}_2$ .

Figure 9 gives results in which the pore pressure is directly varied while the external traction boundary conditions are fixed. The deformation path through ${\overline{\lambda }}_1$ – ${\overline{\lambda }}_2$ space is plotted as the pores are depressurized from $P_i^g=0.125$ down to $P_i^g=-1.75$ . Thus, in Figure 9(a), ${\overline{S}}_{22}=-0.125$ , and each curve corresponds to a different fixed value of ${\overline{S}}_{11}$ , ranging from $-0.5$ to 0. In Figure 9(b), we fix the Cauchy stress $\overline{\mathit{T}}=(1∕\overline{J})\overline{\mathit{S}}{\overline{\mathit{F}}}^T$ . We set ${\overline{T}}_{22}=-0.125$ , and likewise show seven curves for fixed values of ${\overline{T}}_{11}$ ranging from $-0.5$ to 0. Onset of twinning is marked by a dashed orange line and pore closure by a dotted red line.

Figure 9.

Deformation paths caused by direct control of pressure while maintaining various biaxial stress boundary conditions. The curves begin at $P_i^g=0.125$ , the upper right-most point of each curve, and the pressure decreases by 0.125 between each subsequent datapoint, to a minimum value of $P_i^g=-1.75$ . (a) shows the results when the PK stress is fixed on the boundary, with ${\overline{S}}_{22}=-0.125$ for every curve and ${\overline{S}}_{11}$ fixed in each curve at a value ranging between $-0.5$ and 0. (b) shows the results for fixed Cauchy stress on the boundary, with ${\overline{T}}_{22}=-0.125$ for every curve and ${\overline{T}}_{11}$ fixed in each curve at a value ranging between $-0.5$ and 0. The onset of twinning is marked by the orange dashed curve, while pore closure is shown by the dotted red curve.

Each curve in Figure 9(a) begins as physical intuition would suggest, with the material contracting in both directions as the pore pressure is decreased. As the twinned regime is approached, however, the softening of the material bends the curves away from the hydrostatic axis, with only the curve ${\overline{S}}_{11}={\overline{S}}_{22}=-0.125$ continuing straight along the hydrostatic axis ${\overline{\lambda }}_1={\overline{\lambda }}_2$ . Where the material enters the twinned regime, the equilibrium paths turn abruptly back and, interestingly, become exactly linear. This is due to the energy in this region being a function only of $\det (\overline{\mathit{F}})$ and therefore behaving like an elastic fluid, which cannot support normal (Cauchy) stress differences in equilibrium conditions. Hence ${\overline{T}}_{11}={\overline{T}}_{22}$ , corresponding to the condition

$${\overline{\lambda }}_2=\frac{{\overline{S}}_{11}}{{\overline{S}}_{22}}{\overline{\lambda }}_1$$

(49)

in terms of the PK stress for symmetric $\overline{\mathit{F}}$ .

In Figure 9(b), likewise, the material contracts biaxially as the depressurization begins. As in Figure 9(a), the deformation paths bend away from the hydrostatic axis near to the onset of twinning, ultimately expanding in one direction while continuing to contract in the other. Because the Cauchy stress is applied, only the curve with ${\overline{T}}_{11}=-0.125$ can satisfy the necessary condition ${\overline{T}}_{11}={\overline{T}}_{22}$ in the twinned regime, and thus for all other loadings the material does not twin. But after twinning occurs the Cauchy stress is a function only of $\det (\overline{\mathit{F}})={\overline{\lambda }}_1{\overline{\lambda }}_2$ , and hence the deformation, under equilibrium conditions, becomes indeterminate for ${\overline{T}}_{11}={\overline{T}}_{22}=-0.125$ within the twinned region.

Note the sensitivity to the loading conditions which is displayed in Figure 9. First, observe the purple and red curves in Figure 9(b), corresponding to ${\overline{T}}_{11}=-0.13$ and ${\overline{T}}_{11}=-0.12$ , respectively. As these boundary conditions are nearly identical, the deformation paths almost coincide as the depressurization begins. But just before the onset of twinning they sharply bend in opposite directions: because neither corresponds to hydrostatic stress, the material cannot undergo twinning. Thus a minute change in the applied Cauchy stress can result in completely different deformations under sufficient depressurization. Observe the corresponding purple and red curves in Figure 9(a), with ${\overline{S}}_{11}=-0.13$ and ${\overline{S}}_{11}=-0.12$ . Because here the PK stress is applied, the material may enter the twinned region along a deformation path following equation (49), and therefore the two curves remain close throughout the depressurization. Hence, the deformation is highly sensitive to both the magnitude of the applied stress, as seen in Figure 9(b) for the Cauchy stress, and the exact nature of the applied stress, with applied PK and Cauchy stresses resulting in qualitatively different behavior.

7. Concluding remarks

In previous work, Caulfield and Ponte Castañeda [33] demonstrated that a certain class of porous elastomers with vacuous pores which were known [24] to lose SE in plane strain may recover macroscopic stability by twinning, taking on a rank-one laminate microstructure at a length scale large compared to the pores, but still small compared to the specimen. Here, we extend this work to consider the effect of internal pore pressure, motivated by interest in using this pressure as a control parameter for actuation purposes. Interestingly, we have seen that the presence of pore pressure does not eliminate the instabilities which were observed in the vacuous elastomer. Rather, provided that the unpressurized state is used as the reference configuration, neither the deformations at which the instabilities occur nor the mechanism of relaxation are changed by the presence of pore pressure. Thus, the pressurized material also enters a twinned configuration after being compressed sufficiently and moreover takes on the associated perfectly soft modes of deformation noted by Caulfield and Ponte Castañeda [33] in the vacuous case.

We have also seen that, in the case of prescribed pore pressure, the material response depends only on the pore gauge pressure, i.e., the difference between the pressure within the pores and the external pressure. An important implication of this is that, both prior and subsequent to twinning, applying a negative pressure within the pores is exactly equivalent to applying a positive pressure of the same magnitude on the boundary of the material, and vice-versa. Hence, because twinning may be achieved in the vacuous composite by application of sufficient compressive pressure as an external traction, twinning may also be achieved by applying a negative (gauge) pressure within the pores. By combining control of this internal pressure with the external traction or displacement boundary conditions, we expect that corresponding deformations or tractions may be produced which are useful for actuation, in particular because within the twinned regime the composite is perfectly soft in shear. Large changes in shape, therefore, could in theory be achieved by the application of minimal tractions. Moreover, because both the energy and the Cauchy stress within the twinned material are functions only of the volumetric deformation, the mapping from an applied deformation to the corresponding state of stress is not one-to-one—rather, a continuum of deformations exist all yielding the same stress. Thus, the twinning transformation can have significant implications on the macroscopic response of the material which could prove useful for actuation purposes.

It should be emphasized that the simplifications in the computation of the LOSE condition and the associated relaxation of the energy for the pressurized porous elastomers considered in this work are not expected to hold for more general classes of porous elastomers and loading conditions. Indeed, both problems are simplified by our restriction here to a transversely isotropic material undergoing plane strain in the transverse plane. This reduces the complexity of the conditions of SE and the dimensionality of the minimization problem giving the rank-one convexification. The result may, therefore, be different if either the assumption of isotropy or the restriction to plane strain deformations are relaxed. In fact, Lopez-Pamies et al. [26] have considered instabilities in 3D isotropic composites consisting of spherical pores distributed randomly within an incompressible elastomeric matrix. They examined the effect of an ideal gas within the pores and indeed found that LOSE is changed by the presence of pore pressure in comparison to the vacuous case, with the result being that pore pressure delays the onset of instabilities in the pressurized elastomer as compared to the unpressurized elastomer. Finally, note that here LOSE in the unpressurized material is driven by the transverse shear modulus vanishing; the fact that the shear modulus of the composite is independent of the pore pressure is a critical reason for LOSE being unchanged when pore pressure is added. Of course, this pressure does directly affect other components of the tangent modulus, and thus in a material for which LOSE is driven, say, by the vanishing of one of the normal moduli, we would again expect LOSE to be affected by the pore pressure.

More generally, the results of this work provide further evidence that relaxation via twinning may be a prevalent feature of elastomeric composites undergoing finite deformations. Twinning has now been shown to occur in fiber-reinforced elastomers [30], elastomeric laminates [31,32,37], and porous elastomers with both vacuous [33] and, now, pressurized pores. Additionally, while in this work the case of ideal gas inclusions was specifically considered, the arguments given here are general enough to include any elastic fluid within the pores. Thus, across a broad range of microstructures (both particulate and layered) and phase constitutive behaviors (soft, stiff, and even gaseous), domain formation has been found occur beyond the onset of macroscopic instabilities, with important and interesting implications for the macroscopic behavior of such composites and their numerous potential applications.