Concluding Remarks

In this chapter, we have developed new constitutive models for the macroscopic response of compos- ites with hyperelastic phases and particulate microstructures, subjected to general, three-dimensional, finite deformations. For this purpose, we have made use of a suitable extension of the tangent second- order (TSO) homogenization theory of Ponte Casta˜neda and Tiberio (2000), which is capable of ac- counting for the strongly nonlinear overall incompressibility constraint (for incompressible behavior of the phases), as well as for the reorientation of the particles with the deformation. Thus, for in- compressible elastomers reinforced with random distributions of aligned, ellipsoidal, rigid inclusions, the expression (1.77) was derived for the macroscopic stored-energy functionWc( ¯F) in terms of equa- tion (1.79) for the evolution of the particle orientation ¯R(2)_{, the stored-energy functionW}(1)

µ of the

elastomeric phase (with ground-state shear modulusµ(1)_{), the initial concentration of the particles}

0 0.2 0.4 0.6 0.8 1 1 5 10 15 20 25

c

λ

lock

J

=50

J

=100

J

=500

Figure 1.10: TSO estimates for the macroscopic stretch, ¯λlock

1 , at which an particle-reinforced incom-

pressible Gent elastomer locks up under three different loadings: Pure shear (PS), Uniaxial tension (UT) and Equibiaxial tension (ET). The results are shown as a function of particle concentrationc

for different values of the matrix lock-up parameterJm.

by expression (1.106). In particular, closed-form, analytical results were obtained for neo-Hookean rubbers reinforced by isotropic distributions of spherical particles under general loading conditions (see expressions (1.134) to (1.141)). For this case, it was found that the macroscopic stored-energy function exhibits dependence on the second invariant of the right Cauchy–Green deformation tensor (even when the matrix response is assumed to depend only on the first invariant), in agreement with theoretical expectations. In addition, it was also found that the macroscopic response of Gent-type elastomers reinforced with isotropic distributions of spherical particles is strongly elliptic, and there- fore shear-band localization instabilities of the type found by Lopez-Pamies and Ponte Casta˜neda (2006b) and Agoras et al. (2009b) for fiber-reinforced composites loaded in compression along the long axis of the fibers were not found in this case.

The TSO theory was also tested for a 2-D problem consisting of transverse shear loading of elastomers reinforced with cylindrical fibers of elliptical cross-section, where it was found to recover exactly the generalized second-order (GSO) results of Lopez-Pamies and Ponte Casta˜neda (2006a) for dilute concentration of elliptical fibers in a neo-Hookean elastomeric matrix. For more general material behavior (e.g., Gent) and non-dilute conditions, the new TSO theory is still in relatively good agreement with the GSO predictions, although it can lead to much stiffer predictions for neo- Hookean matrix behavior, when the TSO theory predicts “geometric” lock up, at sufficiently large deformations. However, for more realistic situations, when lock up due to the matrix behavior is present, the differences are relatively minor. In any case, comparisons with FEM simulations for

5 10 15 20 0 2 4 6 8 10

I

1

W

I

2

W

(b)

(a)

c = 0.1

Matrix

Matrix

Matrix

c = 0.1

Figure 1.11: NewTSO estimates for the effective response of a rigidly particle-reinforced elastomer with an incompressible neo-Hookean matrix. (a) The effective energy Wc versus the macroscopic invariant ¯I1under three different loadings: Pure shear (PS), Uniaxial tension (UT) and Equibiaxial

tension (ET) (b) The effective energy Wc versus the invariant ¯I2 for two different values of the

invariant ¯I1.

realistic values of the matrix locking strain and macroscopic stretches, are in excellent agreement even for relatively high concentrations (i.e., up to 30 %).

It should be emphasized that while there are presently other homogenization theories for hy- perelastic composites (e.g., the GSO (Lopez-Pamies and Ponte Casta˜neda, 2006a) and sequentially laminated (deBotton, 2005) homogenization methods), the new TSO method developed in this work offers a good balance of generality and accuracy. Indeed, to the best of our knowledge, the TSO estimates developed in this work are the first homogenization estimates for reinforced elastomers withgeneralparticle shape. While only the case of spherical inclusions has been developed in detail here, results are also available for the response of elastomers reinforced with ellipsoidal inclusions under general (non-aligned) loading conditions. Due to the anisotropy of these material systems and the important effects of particle reorientation, which can lead to loss of ellipticity of the macroscopic response, the analysis of these results is quite a bit more involved and will be considered in detail in a future publication.

Chapter 2

Application of the TSO theory to

short fiber-reinforced composites:

I–Analytical results

In this chapter, we present a homogenization-based constitutive model for the mechanical behav- ior of non-spherical particle-reinforced elastomers with random microstructures subjected to finite deformations. The model is based on the improved version of the tangent second-order (TSO) method, developed in Chapter 1, for two-phase, hyperelastic composites, and is able to directly account for the shape, orientation, and concentration of the particles. After a brief summary of the TSO homogenization method, we describe its application to composites consisting of an incom- pressible rubber reinforced by aligned, spheroidal, rigid particles, undergoing generally non-aligned, three-dimensional loadings. While the results are valid for finite particle concentrations, in the dilute limit they can be viewed as providing a generalization of Eshelby’s results in linear elasticity. In particular, we provide analytical estimates for the overall response and microstructure evolution of the particle-reinforced composites with generalized neo-Hookean matrix phases under non-aligned loadings. For the special case of aligned pure shear and axisymmetric shear loadings, we give closed- form expressions for the effective stored-energy function of the composites with neo-Hookean matrix behavior. Moreover, we investigate the possible development of “macroscopic” (shear band-type) instabilities in the homogenized behavior of the composite at sufficiently large deformations. These instabilities whose wavelengths are much larger than the typical size of the microstructure are de- tected by making use of the loss of strong ellipticity condition for the effective stored-energy function of the composites. The analytical results presented in this chapter will be complemented in the next chapter by specific applications for several representative microstructures and loading configurations.

2.1

Introduction

For the purposes of the present chapter, the most relevant work was carried out by Lopez-Pamies and Ponte Casta˜neda (2006b) for random distributions of rigid elliptical fibers in an elastomeric phase, or more precisely for plane strain loading of continuous fiber-reinforced elastomers in the transverse plane where the fibers exhibit elliptical cross-section. These estimates demonstrated for the first time the strong effect of particle rotations, which, under certain conditions, could induce strong geometric softening leading to the possible development of macroscopic instabilities through loss of ellipticity. These homogenization estimates were also compared with full-field numerical simulations by Moraleda et al. (2009) and found to be in fairly good quantitative agreement at least for neo- Hookean matrix phases. The existence of long wave length instabilities (Geymonat et al., 1993), as well as other types of “microscopic” instabilities, in the context of two-dimensional fiber-reinforced composites with periodic microstructures has also been documented recently (Michel et al., 2010).

As already mentioned, in this chapter, we will make use of the improved version of the TSO method, developed in Chapter 1, to investigate the effect of particle shape on the macroscopic response, microstructure evolution and macroscopic instabilities in short-fiber-reinforced elastomers subjected to general finite-strain loadings. As mentioned in Chapter 1, the resulting estimates of this method for the macroscopic response of the reinforced elastomers are consistent with the overall incompressibility constraint, expected on physical grounds. Also, the TSO method is able to provide analytical estimates for the evolution of the relevant microstructural variables, including most notably the rotation of the ellipsoidal particles under general loading conditions. In particular,

we verified in Chapter 1 that the improved version of the TSO method leads to predictions that are very similar—and in some cases identical—to the predictions of the more sophisticated GSO method (Lopez-Pamies and Ponte Casta˜neda, 2006a), at least for the case of two-dimensional elliptical particles (Lopez-Pamies and Ponte Casta˜neda, 2006b). In this context, it should be noted that the GSO method requires the use of the field fluctuations in the linear comparison composite and is therefore more difficult to implement, especially for the complex three-dimensional microstructures of interest in this work.

The structure of this chapter is as follows. For convenience and clarity, Sections 2.2 and 2.3 summarize the basic elements of the nonlinear homogenization methods and, in particular, the tan- gent second-order theory developed in Chapter 1. Section 2.4 deals with the specific application of the TSO theory for elastomers reinforced with aligned, rigid, spheroidal particles. This section in- cludes closed-form, analytical expressions for the homogenized stored-energy function of transversely isotropic, reinforced elastomers with neo-Hookean matrix phases under aligned, triaxial loading con- ditions (see expressions (2.35), (2.39), and (2.42)). Section 2.5 spells out the general conditions of strong ellipticity used to determine the “macroscopic” instabilities for incompressible, transversely isotropic, hyperelastic composites under aligned and non-aligned loading conditions. These condi- tions are provided in terms of appropriate traces of the associated effective incremental modulus tensor, which, in turn, can be written in terms of the derivatives of the associated effective stored- energy function with respect to the macroscopic kinematical variables. Then, these conditions are specialized for the class of (rigid) particle-reinforced elastomers undergoingaxisymmetric andpure

shear loading conditions. Finally, some conclusions are drawn in Section 2.6. In the next chapter, use will be made of the analytical results presented in Sections 2.4 and 2.5 of this chapter to investigate in more detail the influence of the microgeometry, matrix properties, and loading conditions on the effective constitutive behavior of the reinforced elastomers, including the associated microstructure evolution and the possible development of macroscopic instabilities.

2.2

Preliminaries on hyperelastic composites and their effec-