Hyperelastic composites and efiective behavior
Consider a material made up of N difierent (homogeneous) phases that are distributed, either ran-domly or periodically, in a specimen occupying a volume ›0, with boundary @›0, in the reference conflguration, in such a way that the characteristic length of the inhomogeneities (e.g., voids, par-ticles, etc.) is assumed to be much smaller than the size of the specimen and the scale of variation of the applied loading.
Material points in the solid are identifled by their initial position vector X in the reference conflguration ›0, while the current position vector of the same point in the deformed conflguration › is given by x = ´(X): (2.1)
The deformation gradient tensor F at X, a quantity that measures the deformation in the neighbor-hood of X, is deflned as: F(X) = @´ (X): (2.2)
Note that F is not necessarily continuous across interphase boundaries in the composite, but in this work we assume that the various phases are perfectly bonded so that ´ is everywhere continuous.
Furthermore, in order to satisfy global material impenetrability, the mapping ´ is required to be one-to-one on ›0. Thus, for all points X and X0 2 ›0, ´(X0) = ´(X) if and only if X0 = X: (2.3)
The local form of (2.3) is det F(X) 6= 0 8 X 2 ›0: (2.4)
However, in this work we are interested in physically plausible deformation paths with starting point
F(X) = I 8 X 2 ›0, where I denotes the identity operator in the space of second-order tensors.
Thus, by continuity, if follows from (2.4) that det F(X) > 0 8 X 2 ›0: (2.5)
Note that this condition would be automatically satisfled for incompressible materials, where det F is required to be identically 1.
The constitutive behavior of the phases is characterized by stored-energy functions W (r) (r = 1; :::; N), which are taken to be non-convex functions of the deformation gradient tensor F. Thus, the local stored-energy function of the hyperelastic composite is expressible as: W (X; F) = ´(r)(X) W (r)(F); (2.6) where the characteristic functions ´(r), equal to 1 if the position vector X is inside phase r (i.e., X 2 ›(0r)) and zero otherwise, describe the distribution of the phases (i.e., the microstructure) in the reference conflguration. Note that in the case of periodic distributions, the dependence of ´(r) on X is completely determined once a unit cell D0 has been specifled. In contrast, for random distributions, the dependence of ´(r) on X is not known precisely, and the microstructure is only partially deflned in terms of n¡point statistics. The stored-energy functions of the phases are, of course, taken to be objective, in the sense that W (r)(QF) = W (r)(F) (2.7) for all proper orthogonal Q and all deformation gradients F. Making use of the right polar de-composition F = RU, where R is the macroscopic rotation tensor and U denotes the right stretch tensor, it follows, in particular, that W (r)(F) = W (r)(U). Moreover, to try to ensure material impenetrability, the domain of W (r) is taken to be the set of all second-order tensors with positive determinant: fFj det F > 0g. Further, W (r) are assumed to satisfy the condition: W (r)(F) ! 1 if det F ! 0 + : (2.8)
It is thus seen that W (r) are indeed non-convex functions of F since their domain, fFj det F > 0g, is not convex.1
1 This is easy to check by constructing an example where the sum of two distinct second-order tensors F and F0, with det F > 0 and det F0 > 0, does not have positive determinant (see, e.g., Chapter 31 in Milton, 2002).
Assuming su–cient smoothness for W on F, it is now useful to deflne the local constitutive functions
@W S(r)(F) = @W (r) S(X; F) = (X; F) and (F); (2.9) @F @F as well as @2W @2W (r)
L(r)(F) = L(X; F) = (X; F) and (F): (2.10) @F@F @F@F
It then follows that the local or microscopic constitutive relation for the composite is given by: S(X) = S(X; F); (2.11)
where S denotes the flrst Piola-Kirchhofi stress tensor2. Furthermore, note that the local elasticity, or tangent modulus tensor of the material is given by (2.10)1.
Following Hill (1972) and Hill and Rice (1973), under the hypothesis of statistical uniformity and f the above-mentioned separation of length scales, the efiective stored-energy function W of the hy-perelastic composite is deflned by:
min W(X; F) = min (r)(r) (2.12)
W(F) = h i c0 hW (F)i ;
F ( F ) F ( F ) =1
f 2K 2K Xr
where K denotes the set of kinematically admissible deformation gradients:
K( ) = fF j9 x = ´(X) with F = Grad ´(X) in ›0; x = (2.13) F FX on @›0g:
In the above expressions, the brackets h¢i and h¢i(r) denote volume averages|in the undeformed conflguration|over the composite (›0) and over the phase r (›(0r)), respectively, so that the scalars (r) (r) f c0 = h´ i represent the initial volume fractions of the given phases. Note that W physically represents the average elastic energy stored in the composite when subjected to an a–ne displacement boundary condition that is consistent with hFi = F. Note further that, from the deflnition (2.12) and the objectivity of W (r), it follows that W is objective, and hence that W (F) = W (U). (For given in Appendix I.) Here, U represents the macroscopic completeness, the proof of this result is right stretch tensor associated with the macroscopic polar decomposition F = R U, with R denoting the macroscopic rotation tensor (of course, hUi =6 U and hRi 6= R).
In analogy with the local expressions (2.9) and (2.10), and assuming su–cient smoothness for W
on , it is convenient to deflne the following efiective quantities: f
2Recall that S is related to the Cauchy stress tensor T by S = det(F) T F¡T .
It then follows that the global or macroscopic constitutive relation for the composite|that is, the re-lation between the macroscopic flrst Piola-Kirchhofi stress and the macroscopic deformation gradient tensor|is given by (see Appendix II):
S = S(F);
where S = hSi is the average flrst Piola-Kirchhofi stress in the composite. Furthermore, the efiective tangent modulus tensor is given by (2.15).
Having deflned the local and efiective behavior of hyperelastic composites, it is now in order to make pertinent remarks regarding the existence and uniqueness of minimizers for W (X; F) in the deflnition (2.12) for the efiective stored-energy function Wf.
As is well known, imposing the constitutive requirement that W (X; F) be strictly convex in F for all X 2 ›0, namely, W (X; t F + (1 ¡ t)F0) < t W (X; F) + (1 ¡ t)W (X; F0) (2.17) for all t 2 [0; 1] and all pairs F and F0, together with suitable smoothness and growth conditions, ensures that the solution of the Euler-Lagrange equations associated with the variational problem (2.12) exists, is unique, and gives the minimum energy (see, e.g., Hill, 1957; Beju, 1971). However, as explicitly stated above, W (X; F) has been taken to be non-convex with respect to F and cannot satisfy (2.17). This is because|motivated by material impenetrability requirements|the domain fFj det F > 0g of W (X; F) is not convex, and further, W (X; F) is required to satisfy the condition (2.8). Moreover, motivated by experimental evidence, it is also recognized that W (X; F) needs to be non-convex in F in order not to rule out bifurcation phenomena such as buckling. In short, some other constitutive condition on W |less restrictive than convexity|is required to guarantee the existence of minimizers in (2.12) without necessarily guaranteeing the uniqueness of the associated Euler-Lagrange equations.
Ball showed in his celebrated paper in 1977 that if the stored-energy function W (X; F) is (strictly) polyconvex, namely, W (X; F) = f(X; F; Fadj; det F) (2.18)
with f(X; ¢; ¢; ¢) (strictly) convex for each X and Fadj = det(F)F¡T , and if certain growth hypotheses are satisfled, then there exist minimizers for (2.12). Ball’s remarkable existence theorem applies to compressible materials that satisfy the physical condition (2.8), as well as to incompressible materials, which require the constraint det F = 1 (which also poses technical di–culties). It should be noted that even though the constitutive restriction of polyconvexity (2.18) has yet to be given strict physical meaning, is general enough as to include many of the more commonly used hyperelastic stored-energy functions such as the Neo-Hookean, Gent, Arruda-Boyce, Mooney-Rivlin, and Ogden materials.
A less restrictive constitutive condition than polyconvexity is that of quasiconvexity, which was introduced by Morrey in 1952. In this celebrated article, Morrey provided (see also Acerbi and Fusco, 1984) a theorem for the existence of minimizers in problems of the type (2.12) by making use of the constitutive hypothesis of quasiconvexity together with certain growth conditions. Unfortunately, the growth hypotheses are too stringent and prohibit the condition (2.8). Thus, as they stand, the existence theorems for minimizers of integrals of general quasiconvex functions do no apply to flnite elasticity (Ball, 2002). However, it has been suggested (see, e.g., Ball and Murat, 1984 and Ball, 2002) that quasiconvexity might be the more appropriate constitutive requirement|less restrictive than polyconvexity|for existence of energy minimizers in flnite elasticity. That such suggestion is actually correct remains a fundamental open problem in flnite elasticity. A key di–culty in proving this result is that there is no known useful characterization of quasiconvexity, other than its deflnition, which is nonlocal.
Finally, it is fltting to spell out an even less restrictive constitutive condition than quasiconvexity, namely, rank-one convexity. Thus, the stored-energy function W (X; F) is said to be rank-one convex if it satisfles the Legendre-Hadamard condition, namely, if it satisfles:
B(X; F) • min =1f m N jLijkl (X; F)m N lg ‚ 0; (2.19)
where it is recalled that L is given by (2.10)1 and indicial notation has been used to indicate precisely the products involved. (In the absence to explicit notice to the contrary, Latin indices range from 1 to 3, and the usual summation convention is employed). Note that the strict inequality in (2.19) (i.e., strict rank-one convexity) corresponds to strong ellipticity, whose physical meaning is that the hyperelastic composite never admits solutions with discontinuous deformation gradients within the given phases (see, e.g., Knowles and Sternberg, 1977; Hill, 1979). In this connection, it is important to remark that other types of singular solutions, such as cavitation, are not precluded by strong ellipticity. The interested reader is referred to Ball (1982) for a detailed discussion of such material instabilities. On the practical side, for many of the cases considered in this work, it should be mentioned that void nucleation is not expected to occur (Ball, 1982).
The conditions of convexity, polyconvexity, quasiconvexity, and rank-one convexity introduced above satisfy the following chain of implications (see, e.g., Dacorogna, 1989): Convexity ) Polyconvexity ) Quasiconvexity ) Rank¡One Convexity: (2.20)
As explained above, convexity is not a valid assumption for materials in flnite elasticity. On the other hand, polyconvexity, for which Ball’s existence theorems apply, is valid for many common non-linear elastic materials. In addition, polyconvexity|as opposed to quasiconvexity and rank-one convexity|is a relatively easy assumption to impose in practice (at least for isotropic materials). Hence, in this work we will adopt the constitutive assumption of polyconvexity for the local behavior of hyperelastic materials. More speciflcally, we will insist in local strict polyconvexity. In this connection, it is convenient to record that (see, e.g., Marsden and Hughes, 1983) Strict Convexity ) Strict Polyconvexity ) Strict Rank¡One Convexity: (2.21)
This chain of implications, together with (2.20), entails that strictly polyconvex hyperelastic mate-rials are also quasiconvex and strongly elliptic. In this regard, it is important to make the following remark. In spite of the fact that the local behavior is assumed to be locally strongly elliptic, the f efiective stored-energy function W may lose strong ellipticity. This can be seen by recognizing that f
W , as deflned by (2.12), is quasiconvex and therefore|according to (2.20)|rank-one convex, but not necessarily strictly so (Geymonat et al., 1993). One of the issues of interest in this work is establishing under what conditions the overall behavior of the composite can lose strict rank-one convexity, that is, under what conditions min m N ( )m N > 0 (2.22) ceases to hold true. Recall that in this last expression L is given by (2.15).
Speciflc stored-energy functions for the phases
In subsequent chapters dealing with applications to speciflc material systems, we will restrict at-tention to a special class of stored-energy functions W (r) for the phases of hyperelastic composites. In particular, motivated by experimental evidence indicating that elastomers are normally isotropic with respect to the undistorted state, special attention will be given to isotropic stored-energy functions W (r).
Recall that the restriction of isotropy (together with that of objectivity) implies that the stored-energy functions W (r) of the material constituents can be expressed as functions of the principal invariants of the right Cauchy-Green deformation tensor C = FT F:
I1 = trC = ‚12 + ‚22 + ‚32; + ‚22‚32 + ‚32‚12;
I2 = p2 £(trC)2 ¡ trC2⁄ = ‚12‚22 = = ‚1‚2‚3; (2.23) I3 det C or, equivalently, as symmetric functions of the principal stretches ‚1; ‚2; ‚3 associated with F.
Namely, W (r) may be written as: W (r)(F) = ’(r)(I1; I2; I3) = ‘(r)(‚1; ‚2; ‚3); (2.24) where ‘(r) are symmetric. A fairly general (and relatively simple) class of stored-energy functions (2.24), which has been found to provide good agreement with experimental data for rubberlike materials, is given by:
•(r) W (r)(F) = g(r)(I) + h(r)(J) + 2 (J ¡ 1)2; (2.25)
where I • I1 and J • I3 have been introduced for convenience. The parameter •(r) corresponds to the three-dimensional3 bulk modulus of phase r at zero strain, and g(r) and h(r) are twice-difierentiable, material functions that satisfy the conditions: g(r)(3) = h(r)(1) = 0, gI(r)(3) = „(r)=2; h(Jr)(1) = ¡„(r), and 4gII(r)(3) + h(JJr)(1) = „(r)=3. Here, „(r) denotes the shear modulus of phase r at zero strain, and the subscripts I and J indicate difierentiation with respect to these invariants. Note that when these conditions are satisfled W (r)(F) = (1=2)(•(r) ¡ 2=3„(r))(tr »)2 + 3In terms of the Lam¶e moduli, „0(r) and „(r), •(r) = „0(r) + 2=3„(r). „(r)tr »2 + o(« 3), where » is the inflnitesimal strain tensor, as F ! I, so that the stored-energy func-tion (2.25) linearizes properly. Furthermore, note that to recover incompressible behavior in (2.25), it su–ces to make the parameter •(r) tend to inflnity (in which case W (r)(F) = g(r)(I) together with the incompressibility constraint J = 1).
Experience suggests that \neat » elastomers normally do not admit localized deformations. Within the context of the material model (2.25), this property can be easily enforced by simply insisting that g(I) and h(J) + •2 (J ¡ 1)2 be strictly convex functions of their arguments, which renders the stored-energy function (2.25) strictly polyconvex, and in turn|according to (2.21)|strongly elliptic. Note also that the stored-energy function (2.25) is an extension of the so-called generalized Neo-Hookean (or I1-based) materials to account for compressibility. It includes constitutive models widely used in the literature such as the Neo-Hookean, Arruda-Boyce 8-chain (Arruda and Boyce, 1993), Yeoh (Yeoh, 1993), and Gent (Gent, 1996) models.
In the sequel, we will consider a number of applications in the context of plane-strain deforma-tions. For this type of loading conditions, the problems at hand will be essentially two-dimensional (2D). In this regard, for such problems, it will prove more helpful to work with the 2D form of (2.25) rather than with (2.25) itself. Thus, by flxing|without loss generality|‚3 = 1 and deflning the in-plane principal invariants of C = FT F as:
• 2 2 ; • ‚2; (2.26)
I=‚1 + ‚2 and J = ‚1
the stored-energy function (2.25) under plane-strain conditions can be conveniently rewritten as:
W (r)(F) = g•(r)(I•) + h•(r)(J•) + •(r) ¡ „(r) (J• ¡ 1)2 ; (2.27)
where now the parameter •(r) corresponds to the two-dimensional4 bulk modulus of phase r at zero
(r) •(r) (r) •(r) (r) (r) •(r) (1) = ¡„ (r) , and strain, and g• and h are such that: g• (2) = h (1) = 0, g•I• (2)=„ =2, hJ• (r) •(r) (r) • • 4•g•• (2) + h • •(1) = „ . Here, similar to (2.25), the subscripts I and J indicate difierentiation with II JJ respect to these invariants. Further, the above conditions make the stored-energy function (2.27) linearized correctly in the limit of small deformations. In the sequel, the above-utilized check mark \• » to denote 2D quantities will be dropped if there is no potential for confusion.
Macroscopic and microscopic instabilities
Next, it is important to recall that more mathematically precise deflnitions of the efiective energy Wf, other than (2.12), have been given by Braides (1985) and M˜uller (1987) for periodic microstructures. Such deflnitions generalize the classical deflnition of the efiective energy for periodic media with convex energies (Marcellini, 1978), by accounting for the fact that, in the non-convex case, it is not su–cient to consider one-cell periodic solutions, as solutions involving interactions between several unit cells may lead to lower overall energies. Physically, this corresponds to the possible development of \microscopic » instabilities in the composite at su–ciently large deformation (see Appendix III for a more precise deflnition of microscopic instabilities). In this connection, it is important to 4 In terms of the Lam¶e moduli, „0(r) and „(r), •(r) = „0(r) + „(r). remark that Geymonat et al. (1993), following earlier work by Triantafyllidis and Maker (1985) for laminated materials, have shown rigorously that loss of strong ellipticity in the homogenized behavior of the composite corresponds to the development of long-wavelength (i.e., \macroscopic ») instabilities in the form of localized shear/compaction bands. Furthermore, the \failure surfaces » deflned by the loss of strong ellipticity condition of this homogenized behavior provide upper bounds for the onset of other types of instabilities.
In view of the di–culties associated with the computation of the microscopic instabilities men-tioned in the previous paragraph, especially for composites with random microstructures, a more pragmatic approach will be followed here. By assuming|for consistency with the classical theory of linear elasticity|that W (r) = 12 » ¢ L(linr) » + o(« 3) as F ! I, where » denotes the inflnitesimal strain tensor and L(linr) are positive-deflnite5, constant, fourth-order tensors, it is expected (except for very special cases) that, at least in a neighborhood of F = I, the solution of the Euler-Lagrange equations associated with the variational problem (2.12) is unique, and gives the minimum energy. As the deformation progresses into the nonlinear range, the composite material may reach a point at which this \principal » solution bifurcates into lower-energy solutions. This point corresponds to the onset of a microscopic instability beyond which the applicability of the \principal » solution becomes questionable. However, it is still possible to extract useful information from the principal solution by computing the associated macroscopic instabilities from the loss of strong ellipticity of the homogenized behavior. This means that, in practice, we will estimate the efiective stored-energy function (2.12) by means of the stationary variational statement: W ( ) = stat c0(r) hW (r)(F)i(r); where it is emphasized that the energy is evaluated at the above-described \principal » solution of the relevant Euler-Lagrange equations. From its deflnition, it is clear that W (F) = W (F) up to the onset of the flrst the onset of the flrst microscopic instability. Beyond this point, and up to macroscopic instability, W ( )•W( ). The point is that while the microscopic instabilities are macroscopic instabilities are easy to estimate from W (F). Furthermore, it is di–cult to compute, the f c flrst instability is indeed often the case (Geymonat et al., 1993; Triantafyllidis et al., 2006) that the c a long-wavelength instability, f c in which case W (F) = W (F) all the way up to the development of a macroscopic instability, as characterized by the loss of strong ellipticity of the homogenized moduli associated with W (F). More generally, the flrst instability is of flnite wavelength (i.e., small compared to the size of the specimen), but even in this case, it so happens, as we have already mentioned, that the loss of strong ellipticity of the homogenized energy W (F) provides an upper bound for the development of microscopic instabilities. In other words, the composite material will become unstable before reaching the \failure surface » deflned by the macroscopic instabilities. Furthermore, recent work (Michel, 2006) suggests that the macroscopic instabilities may be the more relevant ones for random systems, since many of the microscopic instabilities in periodic systems tend to disappear as the periodicity of the microstructure is broken down.
Table of contents :
2.1 Hyperelastic composites and e®ective behavior
2.1.1 Hyperelastic materials
2.1.2 E®ective behavior
2.1.3 Constitutive hypotheses
2.1.4 Macroscopic and microscopic instabilities
2.2 Bounds and estimates
2.3 Second-order homogenization method
2.3.1 On the speci¯c choice of the variables F(r) and L(r) for isotropic phases
2.4 E®ective behavior of two-phase hyperelastic composites with \particulate » microstructures
2.4.1 Classical bounds
2.4.2 The linear comparison composite
2.4.3 Second-order homogenization estimates: compliant particles
2.4.4 Second-order homogenization estimates: porous elastomers
2.4.5 Second-order homogenization estimates: rigid particles
2.5 Microstructure evolution
2.5.1 Range of validity of the HS-type second-order estimates
2.6 Macroscopic stability
2.7 Concluding remarks
2.8 Appendix I. Overall objectivity of fW
2.9 Appendix II. On the relation S = @fW=@F
2.10 Appendix III. Microscopic and macroscopic instabilities in periodic elastomers
2.11 Appendix IV. On the limit of eL as F ! I
2.12 Appendix V. Earlier versions of the second-order homogenization method
2.12.1 Tangent second-order estimates
2.12.2 Second-order estimates with °uctuations: F(r) = F(r)
2.13 Appendix VI. The tensor P for cylindrical ¯bers and laminates
3 Porous elastomers: cylindrical voids, random microstructure
3.1 Plane-strain loading of transversely isotropic, random porous elastomers
3.1.1 Second-order homogenization estimates
3.1.2 Tangent second-order homogenization estimates
3.1.3 Comparisons with exact results
3.1.4 Loss of strong ellipticity
3.2 Results for plane-strain loading: random porous elastomers
3.2.1 Hydrostatic loading
3.2.2 Uniaxial loading
3.2.3 Pure shear loading
3.2.4 Failure surfaces
3.3 Concluding remarks
3.4 Appendix I. In-plane components of the tensor P for cylindrical inclusions with circular cross-section
3.5 Appendix II. Second-order estimates for transversely isotropic porous elastomers with incompressible Neo-Hookean matrix phase
3.6 Appendix III. Coe±cients associated with the incompressible limit for the second-order estimate of Neo-Hookean porous elastomers
3.7 Appendix IV. Tangent second-order estimates for transversely isotropic porous elastomers with incompressible Neo-Hookean matrix phase
4 Porous elastomers: cylindrical voids, periodic microstructure
4.1 Plane-strain loading of periodic porous elastomers
4.1.1 Second-order homogenization estimates
4.1.2 Loss of strong ellipticity
4.2 Results for plane-strain loading: periodic porous elastomers
4.2.1 Hydrostatic loading
4.2.2 Aligned uniaxial loading
4.2.3 Failure surfaces
4.3 Concluding remarks
4.4 Appendix I. Expressions for the microstructural tensor P
4.5 Appendix II. Onset of percolation
5 Porous elastomers: spherical voids
5.1 Overall behavior of isotropic porous elastomers
5.1.1 Earlier estimates
5.1.2 Second-order homogenization estimates
5.1.3 Small-strain elastic moduli
5.1.4 Exact evolution of porosity
5.1.5 Loss of strong ellipticity
5.2 Results and discussion
5.2.1 Axisymmetric loadings
5.2.2 Plane-strain loadings
5.3 Concluding remarks
5.4 Appendix I. Second-order estimates for isotropic porous elastomers with compressible matrix phases
5.5 Appendix II. Second-order estimates for isotropic porous elastomers with incompressible matrix phases
6 Hyperelastic laminates
6.1 E®ective behavior of hyperelastic laminates
6.1.1 Tangent second-order homogenization estimates
6.1.2 Microstructure evolution
6.2 Plane-strain loading of Neo-Hookean laminates
6.3 Results and discussion
6.3.1 Aligned pure shear
6.3.2 Pure shear at an angle
6.4 Concluding remarks
7 Reinforced elastomers: cylindrical ¯bers, random microstructure
7.1 Plane-strain loading of ¯ber-reinforced, random elastomers
7.1.1 Second-order homogenization estimates: compliant ¯bers
7.1.2 Second-order homogenization estimates: rigid ¯bers
7.1.3 Loss of strong ellipticity
7.2 Results for plane-strain loading: random reinforced elastomers
7.2.1 Pure shear: circular rigid ¯bers and incompressible matrix
7.2.2 Aligned pure shear: rigid ¯bers and incompressible matrix
7.2.3 Pure shear at an angle: rigid ¯bers and incompressible matrix
7.2.4 Aligned pure shear: compliant ¯bers and compressible matrix
7.2.5 Simple shear: rigid ¯bers and incompressible matrix
7.3 Concluding remarks
7.4 Appendix I. Incompressibility limit for rigidly reinforced elastomers: cylindrical ¯bers
A Second-order homogenization estimates incorporating ¯eld °uctuations in ¯nite elasticity1
A.1 Hyperelastic composites and e®ective behavior
A.2 The second-order variational procedure
A.3 Application to particle-reinforced elastomers
A.3.1 Lower bounds
A.3.2 Second-order estimates
A.4 Plane strain loading of transversely isotropic, ¯ber-reinforced Neo-Hookean composites
A.5 Concluding remarks