Onset of macroscopic instabilities in fiber-reinforced elastomers at finite strain

In this paper, we investigate theoretically the possible development of instabilities in fiber-reinforced elastomers (and other soft materials) when they are subjected to finite-strain loading conditions. We focus on the physically relevant class of “macroscopic” instabilities, i.e., instabilities with wavelengths that are much larger than the characteristic size of the underlying microstructure. To this end, we make use of recently developed homogenization estimates, together with a fundamental result of Geymonat, Müller and Triantafyllidis linking the development of these instabilities to the loss of strong ellipticity of the homogenized constitutive relations. For the important class of material systems with very stiff fibers and random microstructures, we derive a closed-form formula for the critical macroscopic deformation at which instabilities may develop under general loading conditions, and we show that this critical deformation is quite sensitive to the loading orientation relative to the fiber direction. The result is also confronted with classical estimates (including those of Rosen) for laminates, which have commonly been used as two-dimensional (2-D) approximations for actual fiber-reinforced composites. We find that while predictions based on laminate models are qualitatively correct for certain loadings, they can be significantly off for other more general 3-D loadings. Finally, we provide a parametric analysis of the effects of the matrix and fiber properties and of the fiber volume fraction on the onset of instabilities for various loading conditions.     

Finite deformation of incompressible fiber-reinforced elastomers: A computational micromechanics approach

The in-plane finite deformation of incompressible fiber-reinforced elastomers was studied using computational micromechanics. Composite microstructure was made up of a random and homogeneous dispersion of aligned rigid fibers within a hyperelastic matrix. Different matrices (Neo-Hookean and Gent), fibers (monodisperse or polydisperse, circular or elliptical section) and reinforcement volume fractions (10–40%) were analyzed through the finite element simulation of a representative volume element of the microstructure. A successive remeshing strategy was employed when necessary to reach the large deformation regime in which the evolution of the microstructure influences the effective properties. The simulations provided for the first time “quasi-exact” results of the in-plane finite deformation for this class of composites, which were used to assess the accuracy of the available homogenization estimates for incompressible hyperelastic composites.     

Microscopic and macroscopic instabilities in finitely strained fiber-reinforced elastomers

The present work is a detailed study of the connections between microstructural instabilities and their macroscopic manifestations — as captured through the effective properties — in finitely strained fiber-reinforced elastomers, subjected to finite, plane-strain deformations normal to the fiber direction. The work, which is a complement to a previous and analogous investigation by the same authors on porous elastomers, (Michel et al., 2007), uses the linear comparison, second-order homogenization (S.O.H.) technique, initially developed for random media, to study the onset of failure in periodic fiber-reinforced elastomers and to compare the results to more accurate finite element method (F.E.M.) calculations. The influence of different fiber distributions (random and periodic), initial fiber volume fraction, matrix constitutive law and fiber cross-section on the microscopic buckling (for periodic microgeometries) and macroscopic loss of ellipticity (for all microgeometries) is investigated in detail. In addition, constraints to the principal solution due to fiber/matrix interface decohesion, matrix cavitation and fiber contact are also addressed. It is found that both microscopic and macroscopic instabilities can occur for periodic microstructures, due to a symmetry breaking in the periodic arrangement of the fibers. On the other hand, no instabilities are found for the case of random microstructures with circular section fibers, while only macroscopic instabilities are found for the case of elliptical section fibers, due to a symmetry breaking in their orientation.     

Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

Many composite materials, including biological tissues, are modeled as non-linear elastic materials reinforced with elastic fibers. In the current paper, the full set of dynamic equations for finite deformations of incompressible hyperelastic solids containing a single fiber family are considered. Finite-amplitude wave propagation ansätze compatible with the incompressibility condition are employed for a generic fiber family orientation. Corresponding non-linear and linear wave equations are derived. It is shown that for a certain class of constitutive relations, the fiber contribution vanishes when the displacement is independent of the fiber direction.Point symmetries of the derived wave models are classified with respect to the material parameters and the angle between the fibers and the wave propagation direction. For planar shear waves in materials with a strong fiber contribution, a special wave propagation direction is found for which the non-linear wave equations admit an additional symmetry group. Examples of exact time-dependent solutions are provided in several physical situations, including the evolution of pre-strained configurations and traveling waves.     

A new constitutive theory for fiber-reinforced incompressible nonlinearly elastic solids

We consider an incompressible nonlinearly elastic material in which a matrix is reinforced by strong fibers, for example fibers of nylon or carbon aligned in one family of curves in a rubber matrix. Rather than adopting the constraint of fiber inextensibility as has been previously assumed in the literature, here we develop a theory of fiber-reinforced materials based on the less restrictive idea of limiting fiber extensibility. The motivation for such an approach is provided by recent research on limiting chain extensibility models for rubber. Thus the basic idea of the present paper is simple: we adapt the limiting chain extensibility concept to limiting fiber extensibility so that the usual inextensibility constraint traditionally used is replaced by a unilateral constraint. We use a strain-energy density composed with two terms, the first being associated with the isotropic matrix or base material and the second reflecting the transversely isotropic character of the material due to the uniaxial reinforcement introduced by the fibers. We consider a base neo-Hookean model plus a special term that takes into account the limiting extensibility in the fiber direction. Thus our model introduces an additional parameter, namely that associated with limiting extensibility in the fiber direction, over previously investigated models. The aim of this paper is to investigate the mathematical and mechanical feasibility of this new model and to examine the role played by the extensibility parameter. We examine the response of the proposed models in some basic homogeneous deformations and compare this response to those of standard models for fiber reinforced rubber materials. The role of the strain-stiffening of the fibers in the new models is examined. The enhanced stability of the new models is then illustrated by investigation of cavitation instabilities. One of the motivations for the work is to apply the model to the biomechanics of soft tissues and the potential merits of the proposed models for this purpose are briefly discussed.     

The rectilinear shear of fiber-reinforced incompressible non-linearly elastic solids

In this paper we study the problem of rectilinear shear of a slab of transversely isotropic incompressible non-linearly elastic material. In particular, the material under consideration is a base neo-Hookean model augmented with a function that accounts for the existence of a unidirectional reinforcement. The slab is of infinite length in two dimensions and finite thickness in the other one and is clamped to two rigid plates. Closed form analytic solutions are found for this problem. It is shown that, depending on the reinforcement strength and the fiber orientation in the undeformed configuration, weak solutions, i.e. solutions for which the smoothness required by the differential equations is relaxed, are to be expected. These solutions give rise to fiber kinking. It is shown that: (i) both sides of the kink involve fiber contraction; (ii) a suitable intermediate deformation between the two conjoined kink deformation states is non-elliptic.     

CAVITATED BIFURCATION FOR INCOMPRESSIBLE HYPERELASTIC MATERIAL

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Free-energy density functions for nematic elastomers

The recently proposed neo-classical theory for nematic elastomers generalizes standard molecular-statistical Gaussian network theory to allow for anisotropic distributions of polymer chains. The resulting free-energy density models several of the novel properties of nematic elastomers. In particular, it predicts the ability of nematic elastomers to undergo large deformations with exactly zero force and energy cost—so called soft elasticity. Although some nematic elastomers have been shown to undergo deformations with unusually small applied forces, not all do so, and none deform with zero force. Further, as a zero force corresponds to infinitely many possible deformations in the neo-classical theory, this non-uniqueness leads to serious indeterminacies in numerical schemes. Here we suggest that the neo-classical free-energy density is incomplete and propose an alternative derivation that resolves these difficulties. In our approach, we use the molecular-statistical theory to identify appropriate variables. This yields the choice for the microstructural degrees of freedom as well as two independent strain tensors (the overall macroscopic strain plus a relative strain that indicates how the deformation of the elastomeric microstructure deviates from the macroscopic deformation). We then propose expressions for the free-energy density as a function of the three quantities and show how the material parameters can be measured by two simple tests. The neo-classical free-energy density can be viewed as a special case of our expressions in which the free-energy density is independent of the overall macroscopic strain, thus supporting our view that the neo-classical theory is incomplete.     

Microscopic and macroscopic instabilities in finitely strained porous elastomers

The present work is an in-depth study of the connections between microstructural instabilities and their macroscopic manifestations—as captured through the effective properties—in finitely strained porous elastomers. The powerful second-order homogenization (SOH) technique initially developed for random media, is used for the first time here to study the onset of failure in periodic porous elastomers and the results are compared to more accurate finite element method (FEM) calculations. The influence of different microgeometries (random and periodic), initial porosity, matrix constitutive law and macroscopic load orientation on the microscopic buckling (for periodic microgeometries) and macroscopic loss of ellipticity (for all microgeometries) is investigated in detail. In addition to the above-described stability-based onset-of-failure mechanisms, constraints on the principal solution are also addressed, thus giving a complete picture of the different possible failure mechanisms present in finitely strained porous elastomers.     

Multiscale modeling of oriented thermoplastic elastomers with lamellar morphology

Thermoplastic elastomers (TPEs) are block copolymers made up of “hard” (glassy or crystalline) and “soft” (rubbery) blocks that self-organize into “domain” structures at a length scale of a few tens of nanometers. Under typical processing conditions, TPEs also develop a “polydomain” structure at the micron level that is similar to that of metal polycrystals. Therefore, from a continuum point of view, TPEs may be regarded as materials with heterogeneities at two different length scales. In this work, we propose a constitutive model for highly oriented, near-single-crystal TPEs with lamellar domain morphology. Based on small-angle X-ray scattering (SAXS) and transmission electron microscopy (TEM) observations, we consider such materials to have a granular microstructure where the grains are made up of the same, perfect, lamellar structure (single crystal) with slightly different lamination directions (crystal orientations). Having identified the underlying morphology, the overall finite-deformation response of these materials is determined by means of a two-scale homogenization procedure. Interestingly, the model predictions indicate that the evolution of microstructure—especially the rotation of the layers—has a very significant, but subtle effect on the overall properties of near-single-crystal TPEs. In particular, for certain loading conditions—namely, for those with sufficiently large compressive deformations applied in the direction of the lamellae within the individual grains—the model becomes macroscopically unstable (i.e., it loses strong ellipticity). By keeping track of the evolution of the underlying microstructure, we find that such instabilities can be related to the development of “chevron” patterns.     

A new variational estimate for the effective response of hyperelastic composites

A variational method for predicting the effective properties of hyperelastic composites in terms of available estimates for “hyperelastic comparison composites” is proposed. In some cases this estimate can produce a lower bound on the effective energy-density function. This nonlinear-comparison variational procedure is specialized to classes of fiber and statistically isotropic composites with the aid of appropriate choices of comparison composites with neo-Hookean phases. The end results are given in terms of closed-form expressions for the effective strain energy-density functions, from which the stress-strain relations can be extracted analytically. Explicit analytical estimates for the overall responses of composites whose phases behaviors are governed by the Gent model are obtained. The results for the fiber composites are compared with corresponding finite element simulations of periodic fiber composites as well as with other available estimates. A fine agreement between the predictions obtained via the various estimates is revealed even in the limit of infinite contrast between the properties of the phases.     

On the mechanics of mother-of-pearl: A key feature in the material hierarchical structure

Mother-of-pearl, also known as nacre, is the iridescent material which forms the inner layer of seashells from gastropods and bivalves. It is mostly made of microscopic ceramic tablets densely packed and bonded together by a thin layer of biopolymer. The hierarchical microstructure of this biological material is the result of millions of years of evolution, and it is so well organized that its strength and toughness are far superior to the ceramic it is made of. In this work the structure of nacre is described over several length scales. The tablets were found to have wavy surfaces, which were observed and quantified using various experimental techniques. Tensile and shear tests performed on small samples revealed that nacre can withstand relatively large inelastic strains and exhibits strain hardening. In this article we argue that the inelastic mechanism responsible for this behavior is sliding of the tablets on one another accompanied by transverse expansion in the direction perpendicular to the tablet planes. Three dimensional representative volume elements, based on the identified nacre microstructure and incorporating cohesive elements with a constitutive response consistent with the interface material and nanoscale features were numerically analyzed. The simulations revealed that even in the absence of nanoscale hardening mechanism at the interfaces, the microscale waviness of the tablets could generate strain hardening, thereby spreading the inelastic deformation and suppressing damage localization leading to material instability. The formation of large regions of inelastic deformations around cracks and defects in nacre are believed to be an important contribution to its toughness. In addition, it was shown that the tablet junctions (vertical junctions between tablets) strengthen the microstructure but do not contribute to the overall material hardening. Statistical variations within the microstructure were found to be beneficial to hardening and to the overall mechanical stability of nacre. These results provide new insights into the microstructural features that make nacre tough and damage tolerant. Based on these findings, some design guidelines for composites mimicking nacre are proposed.     

Global load-sharing model for unidirectional hybrid fibre-reinforced composites

A promising strategy to increase the tensile failure strain of carbon fibre-reinforced composites is to hybridise carbon fibres with other, higher-elongation fibres. The resulting increase in failure strain is known as the hybrid effect. In the present article, a global load-sharing model for hybrid composites is developed and used to carry out a parametric study for carbon/glass hybrids. Hybrid effects of up to 15% increase in failure strain are predicted, corresponding reasonably well to literature data. Scatter in the carbon fibre strength is shown to be crucial for the hybrid effect, while the scatter in glass fibre strength is much less important. In contrast to reports in earlier literature, the ratio of failure strains of the two fibres has only a small influence on the hybrid effect. The results provide guidelines for designing optimal hybrid composites.     

Finite strain analysis of crack tip fields in incompressible hyperelastic solids loaded in plane stress

A new method is developed to determine the dominant asymptotic stress and deformation fields near the tip of a Mode-I traction free plane stress crack. The analysis is based on the fully nonlinear equilibrium theory of incompressible hyperelastic solids. We show that the dominant singularity of the near tip stress field is governed by the asymptotic solution of a linear second order ordinary differential equation. Our method is applicable to any hyperelastic material with a smooth work function that depends only on the trace of the Cauchy-Green tensor and is particularly useful for materials that exhibit severe strain hardening. We apply this method to study two types of soft materials: generalized neo-Hookean solids and a solid that hardens exponentially. For the generalized neo-Hookean solids, our method is able to resolve a difficulty in the previous work by Geubelle and Knauss (1994a). Our theoretical results are compared with finite element simulations.     

FE analysis of stress and strain fields in finite random composite bodies

In this paper the mechanical behaviour of finite random heterogeneous bodies is considered. The analysis of non-local interactions between heterogeneities in microscopically heterogeneous materials is necessary when the spatial variation of the load or the dimensions of the body, relative to the scale of the microstructure, cannot be ignored. Microstructures can be periodic but generically they are random. In the first case, an exact calculation can be performed but in the second case recourse has to be made either to simulation or to some scheme of approximation. One such scheme is based on a stochastic variational principle. The novelty of the present work is that a stochastic variational principle is projected directly onto a finite-element basis so that all subsequent analysis is performed within a finite-element framework. The proposed formulation provides expressions for the local stress and strain fields in any realization of the medium, from which expressions for statistically-averaged quantities can be derived. Then an approximation of Hashin-Shtrikman type is developed, which generates a FE-based numerical procedure able to take account of interactions between random inclusions and boundary layer effects in finite composite structures. Finally, two examples are presented, namely a cylinder with square cross-section subjected to mixed boundary conditions of different types on different faces and a rectangular body containing a centre crack. The results show that in the vicinity of the boundary or close to the crack tip, the strain and the stress in the matrix and in the inclusions differ considerably from those obtained by the formal application of conventional homogenization.     

A finite strain elastic-viscoplastic self-consistent model for polycrystalline materials

A large strain elastic-viscoplastic self-consistent (EVPSC) model for polycrystalline materials is developed. At single crystal level, both the rate sensitive slip and twinning are included as the plastic deformation mechanisms, while elastic anisotropy is accounted for in the elastic moduli. The transition from single crystal plasticity to polycrystal plasticity is based on a completely self-consistent approach. It is shown that the differences in the predicted stress-strain curves and texture evolutions based on the EVPSC and the viscoplastic self-consistent (VPSC) model proposed by Lebensohn and Tomé (1993) are negligible at large strains for monotonic loadings. For the deformations involving unloading and strain path changes, the EVPSC predicts a smooth elasto-plastic transition, while the VPSC model gives a discontinuous response due to lack of elastic deformation. It is also demonstrated that the EVPSC model can capture some important experimental features which cannot be simulated by using the VPSC model.     

A coupled damage-plasticity model for the cyclic behavior of shear-loaded interfaces

The present work proposes a novel thermodynamically consistent model for the behavior of interfaces under shear (i.e. mode-II) cyclic loading conditions. The interface behavior is defined coupling damage and plasticity. The admissible states’ domain is formulated restricting the tangential interface stress to non-negative values, which makes the model suitable e.g. for interfaces with thin adherends. Linear softening is assumed so as to reproduce, under monotonic conditions, a bilinear mode-II interface law. Two damage variables govern respectively the loss of strength and of stiffness of the interface. The proposed model needs the evaluation of only four independent parameters, i.e. three defining the monotonic mode-II interface law, and one ruling the fatigue behavior. This limited number of parameters and their clear physical meaning facilitate experimental calibration. Model predictions are compared with experimental results on fiber reinforced polymer sheets externally bonded to concrete involving different load histories, and an excellent agreement is obtained.     

Modelling of photodegradation effect on elastic-viscoplastic behaviour of amorphous polylactic acid films

Polylactic acid (PLA) films were subjected to accelerated ultra-violet (UV) ageing. The UV irradiation leads to the alteration of the chemical structure which influences directly the mechanical response of the polymer. The chemical modification of the polymer was followed by gel permeation chromatography. Uniaxial tension tests were conducted at 50 °C and for different strain rates in order to characterize the large deformation response of PLA. The influence of UV irradiation on the alteration of the large deformation response of PLA was examined. A physically based elastic-viscoplastic model was used to describe the mechanical response of virgin PLA. The photodegradation effect was incorporated into the constitutive model to capture the stress-strain behaviour up to failure of aged PLA. To that end, the measured molecular weight was used as a direct input into the model. The model is shown to be in good agreement with experimental results over a wide range of UV irradiation doses.