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.     

A general hyperelastic model for incompressible fiber-reinforced elastomers

This work presents a new constitutive model for the effective response of fiber-reinforced elastomers at finite strains. The matrix and fiber phases are assumed to be incompressible, isotropic, hyperelastic solids. Furthermore, the fibers are taken to be perfectly aligned and distributed randomly and isotropically in the transverse plane, leading to overall transversely isotropic behavior for the composite. The model is derived by means of the “second-order” homogenization theory, which makes use of suitably designed variational principles utilizing the idea of a “linear comparison composite.” Compared to other constitutive models that have been proposed thus far for this class of materials, the present model has the distinguishing feature that it allows consideration of behaviors for the constituent phases that are more general than Neo-Hookean, while still being able to account directly for the shape, orientation, and distribution of the fibers. In addition, the proposed model has the merit that it recovers a known exact solution for the special case of incompressible Neo-Hookean phases, as well as some other known exact solutions for more general constituents under special loading conditions.     

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.     

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.     

Homogenization-based constitutive models for porous elastomers and implications for macroscopic instabilities: II—Results

In Part I of this paper, we developed a homogenization-based constitutive model for the effective behavior of isotropic porous elastomers subjected to finite deformations. In this part, we make use of the proposed model to predict the overall response of porous elastomers with (compressible and incompressible) Gent matrix phases under a wide variety of loading conditions and initial values of porosity. The results indicate that the evolution of the underlying microstructure—which results from the finite changes in geometry that are induced by the applied loading—has a significant effect on the overall behavior of porous elastomers. Further, the model is in very good agreement with the exact and numerical results available from the literature for special loading conditions and generally improves on existing models for more general conditions. More specifically, we find that, in spite of the fact that Gent elastomers are strongly elliptic materials, the constitutive models for the porous elastomers are found to lose strong ellipticity at sufficiently large compressive deformations, corresponding to the possible onset of “macroscopic” (shear band-type) instabilities. This capability of the proposed model appears to be unique among theoretical models to date and is in agreement with numerical simulations and physical experience. The resulting elliptic and non-elliptic domains, which serve to define the macroscopic “failure surfaces” of these materials, are presented and discussed in both strain and stress space.     

Homogenization-based constitutive models for porous elastomers and implications for macroscopic instabilities: I—Analysis

The purpose of this paper is to provide homogenization-based constitutive models for the overall, finite-deformation response of isotropic porous rubbers with random microstructures. The proposed model is generated by means of the “second-order” homogenization method, which makes use of suitably designed variational principles utilizing the idea of a “linear comparison composite.” The constitutive model takes into account the evolution of the size, shape, orientation, and distribution of the underlying pores in the material, resulting from the finite changes in geometry that are induced by the applied loading. This point is key, as the evolution of the microstructure provides geometric softening/stiffening mechanisms that can have a very significant effect on the overall behavior and stability of porous rubbers. In this work, explicit results are generated for porous elastomers with isotropic, (in)compressible, strongly elliptic matrix phases. In spite of the strong ellipticity of the matrix phases, the derived constitutive model may lose strong ellipticity, indicating the possible development of shear/compaction band-type instabilities. The general model developed in this paper will be applied in Part II of this work to a special, but representative, class of isotropic porous elastomers with the objective of exploring the complex interplay between geometric and constitutive softening/stiffening in these materials.     

Cavitation in elastomeric solids: I—A defect-growth theory

It is by now well established that loading conditions with sufficiently large triaxialities can induce the sudden appearance of internal cavities within elastomeric (and other soft) solids. The occurrence of such instabilities, commonly referred to as cavitation, can be attributed to the growth of pre-existing defects into finite sizes. This paper introduces a new theory to study the phenomenon of cavitation in soft solids that: (i) allows to consider general 3D loading conditions with arbitrary triaxiality, (ii) applies to large (including compressible and anisotropic) classes of nonlinear elastic solids, and (iii) incorporates direct information on the initial shape, spatial distribution, and mechanical properties of the underlying defects at which cavitation can initiate. The basic idea is to first cast cavitation in elastomeric solids as a homogenization problem of nonlinear elastic materials containing random distributions of zero-volume cavities, or defects. This problem is then addressed by means of a novel iterated homogenization procedure, which allows to construct solutions for a specific, yet fairly general, class of defects. These include solutions for the change in size of the defects as a function of the applied loading conditions, from which the onset of cavitation — corresponding to the event when the initially infinitesimal defects suddenly grown into finite sizes — can be readily determined. In spite of the generality of the proposed approach, the relevant calculations amount to solving tractable Hamilton-Jacobi equations, in which the initial size of the defects plays the role of “time” and the applied load plays the role of “space”. When specialized to the case of hydrostatic loading conditions, isotropic solids, and defects that are vacuous and isotropically distributed, the proposed theory recovers the classical result of Ball (1982) for radially symmetric cavitation. The nature and implications of this remarkable connection are discussed in detail.     

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.     

Homogenization-based constitutive models for magnetorheological elastomers at finite strain

This paper proposes a new homogenization framework for magnetoelastic composites accounting for the effect of magnetic dipole interactions, as well as finite strains. In addition, it provides an application for magnetorheological elastomers via a “partial decoupling” approximation splitting the magnetoelastic energy into a purely mechanical component, together with a magnetostatic component evaluated in the deformed configuration of the composite, as estimated by means of the purely mechanical solution of the problem. It is argued that the resulting constitutive model for the material, which can account for the initial volume fraction, average shape, orientation and distribution of the magnetically anisotropic, non-spherical particles, should be quite accurate at least for perfectly aligned magnetic and mechanical loadings. The theory predicts the existence of certain “extra” stresses—arising in the composite beyond the purely mechanical and magnetic (Maxwell) stresses—which can be directly linked to deformation-induced changes in the microstructure. For the special case of isotropic distributions of magnetically isotropic, spherical particles, the extra stresses are due to changes in the particle two-point distribution function with the deformation, and are of order volume fraction squared, while the corresponding extra stresses for the case of aligned, ellipsoidal particles can be of order volume fraction, when changes are induced by the deformation in the orientation of the particles. The theory is capable of handling the strongly nonlinear effects associated with finite strains and magnetic saturation of the particles at sufficiently high deformations and magnetic fields, respectively.     

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.     

Cavitation in elastomeric solids: II—Onset-of-cavitation surfaces for Neo-Hookean materials

In Part I of this work we derived a fairly general theory of cavitation in elastomeric solids based on the sudden growth of pre-existing defects. In this article, the theory is used to determine onset-of-cavitation surfaces for Neo-Hookean solids where the defects are isotropically distributed and vacuous. These surfaces correspond to the set of all critical Cauchy stress states at which cavitation ensues; general three-dimensional loadings are considered. Their computation requires the numerical solution of a nonlinear first-order partial differential equation in two variables. The theoretical results indicate that cavitation occurs only for stress states where the three principal Cauchy stresses are tensile, and that the required hydrostatic tensile component increases with increasing shear components. These results are confronted to finite-element simulations for the growth of a small spherical cavity in a Neo-Hookean block under multi-axial loading. Good agreement is found for a wide range of loading conditions. Comparisons with earlier results available in the literature are also provided and discussed. We conclude this work by devising a closed-form approximation to the theoretical surface, which is of remarkable accuracy and mathematical simplicity.     

Neo-Hookean fiber-reinforced composites in finite elasticity

The response of a transversely isotropic fiber-reinforced composite made out of two incompressible neo-Hookean phases undergoing finite deformations is considered. An expression for the effective energy-density function of the composite in terms of the properties of the phases and their spatial distribution is developed. For the out-of-plane shear and extension modes this expression is based on an exact solution for the class of composite cylinder assemblages. To account for the in-plane shear mode we incorporate an exact result that was recently obtained for a special class of transversely isotropic composites. In the limit of small deformation elasticity the expression for the effective behavior agrees with the well-known Hashin-Shtrikman bounds. The predictions of the proposed constitutive model are compared with corresponding numerical simulation of a composite with a hexagonal unit cell. It is demonstrated that the proposed model accurately captures the overall response of the periodic composite under any general loading modes.     

On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—Theory

This work presents an analytical framework for determining the overall constitutive response of elastomers that are reinforced by rigid or compliant fibers, and are subjected to finite deformations. The framework accounts for the evolution of the underlying microstructure, including particle rotation, which results from the finite changes in geometry that are induced by the applied loading. In turn, the evolution of the microstructure can have a significant geometric softening (or hardening) effect on the overall response, leading to the possible development of macroscopic instabilities through loss of strong ellipticity of the homogenized incremental moduli. The theory is based on a recently developed “second-order” homogenization method, which makes use of information on both the first and second moments of the fields in a suitably chosen “linear comparison composite,” and generates fairly explicit estimates—linearizing properly—for the large-deformation effective response of the reinforced elastomers. More specific applications of the results developed in this paper will be presented in Part II.     

Localization of deformation and loss of macroscopic ellipticity in microstructured solids

Localization of deformation, a precursor to failure in solids, is a crucial and hence widely studied problem in solid mechanics. The continuum modeling approach of this phenomenon studies conditions on the constitutive laws leading to the loss of ellipticity in the governing equations, a property that allows for discontinuous equilibrium solutions. Micro-mechanics models and nonlinear homogenization theories help us understand the origins of this behavior and it is thought that a loss of macroscopic (homogenized) ellipticity results in localized deformation patterns. Although this is the case in many engineering applications, it raises an interesting question: is there always a localized deformation pattern appearing in solids losing macroscopic ellipticity when loaded past their critical state?In the interest of relative simplicity and analytical tractability, the present work answers this question in the restrictive framework of a layered, nonlinear (hyperelastic) solid in plane strain and more specifically under axial compression along the lamination direction. The key to the answer is found in the homogenized post-bifurcated solution of the problem, which for certain materials is supercritical (increasing force and displacement), leading to post-bifurcated equilibrium paths in these composites that show no localization of deformation for macroscopic strain well above the one corresponding to loss of ellipticity.     

On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: II—Application to cylindrical fibers

In Part I of this paper, we presented a general homogenization framework for determining the overall behavior, the evolution of the underlying microstructure, and the possible onset of macroscopic instabilities in fiber-reinforced elastomers subjected to finite deformations. In this work, we make use of this framework to generate specific results for general plane-strain loading of elastomers reinforced with aligned, cylindrical fibers. For the special case of rigid fibers and incompressible behavior for the matrix phase, closed-form, analytical results are obtained. The results suggest that the evolution of the microstructure has a dramatic effect on the effective response of the composite. Furthermore, in spite of the fact that both the matrix and the fibers are assumed to be strongly elliptic, the homogenized behavior is found to lose strong ellipticity at sufficiently large deformations, corresponding to the possible development of macroscopic instabilities [Geymonat, G., Müller, S., Triantafyllidis, N., 1993. Homogenization of nonlinearly elastic materials, macroscopic bifurcation and macroscopic loss of rank-one convexity. Arch. Rat. Mech. Anal. 122, 231-290]. The connection between the evolution of the microstructure and these macroscopic instabilities is put into evidence. In particular, when the reinforced elastomers are loaded in compression along the long, in-plane axis of the fibers, a certain type of “flopping” instability is detected, corresponding to the composite becoming infinitesimally soft to rotation of the fibers.     

Molecular simulation guided constitutive modeling on finite strain viscoelasticity of elastomers

Viscoelasticity characterizes the most important mechanical behavior of elastomers. Understanding the viscoelasticity, especially finite strain viscoelasticity, of elastomers is the key for continuation of their dedicated use in industrial applications. In this work, we present a mechanistic and physics-based constitutive model to describe and design the finite strain viscoelastic behavior of elastomers. Mathematically, the viscoelasticity of elastomers has been decomposed into hyperelastic and viscous parts, which are attributed to the nonlinear deformation of the cross-linked polymer network and the diffusion of free chains, respectively. The hyperelastic deformation of a cross-linked polymer network is governed by the cross-linking density, the molecular weight of the polymer strands between cross-linkages, and the amount of entanglements between different chains, which we observe through large scale molecular dynamics (MD) simulations. Moreover, a recently developed non-affine network model (Davidson and Goulbourne, 2013) is confirmed in the current work to be able to capture these key physical mechanisms using MD simulation. The energy dissipation during a loading and unloading process of elastomers is governed by the diffusion of free chains, which can be understood through their reptation dynamics. The viscous stress can be formulated using the classical tube model (Doi and Edwards, 1986); however, it cannot be used to capture the energy dissipation during finite deformation. By considering the tube deformation during this process, as observed from the MD simulations, we propose a modified tube model to account for the finite deformation behavior of free chains. Combing the non-affine network model for hyperelasticity and modified tube model for viscosity, both understood by molecular simulations, we develop a mechanism-based constitutive model for finite strain viscoelasticity of elastomers. All the parameters in the proposed constitutive model have physical meanings, which are signatures of polymer chemistry, physics or dynamics. Therefore, parametric materials design concepts can be easily gleaned from the model, which is also demonstrated in this study. The finite strain viscoelasticity obtained from our simulations agrees qualitatively with experimental data on both un-vulcanized and vulcanized rubbers, which captures the effects of cross-linking density, the molecular weight of the polymer chain and the strain rate.     

Computational aspects in finite strain plasticity analysis of geotechnical materials

This paper presents a framework to describe the constitutive behaviour of geotechnical materials in the context of multiplicative finite strain. A suitable energy function is chosen allowing the hyperelastic response to be energy conserving. The corresponding tangent modulus is derived to ensure quadratic rates of convergence of the Newton–Raphson procedure in the finite element solution. Standard element tests are given to demonstrate the performance of the algorithms developed.     

A homogenization theory of strain gradient single crystal plasticity and its finite element discretization

In this study, a homogenization theory based on the Gurtin strain gradient formulation and its finite element discretization are developed for investigating the size effects on macroscopic responses of periodic materials. To derive the homogenization equations consisting of the relation of macroscopic stress, the weak form of stress balance, and the weak form of microforce balance, the Y-periodicity is used as additional, as well as standard, boundary conditions at the boundary of a unit cell. Then, by applying a tangent modulus method, a set of finite element equations is obtained from the homogenization equations. The computational stability and efficiency of this finite element discretization are verified by analyzing a model composite. Furthermore, a model polycrystal is analyzed for investigating the grain size dependence of polycrystal plasticity. In this analysis, the micro-clamped, micro-free, and defect-free conditions are considered as the additional boundary conditions at grain boundaries, and their effects are discussed.     

A finite strain, finite band method for modeling ductile fracture

We present a finite deformation generalization of the finite thickness embedded discontinuity formulation presented in our previous paper [A.E. Huespe, A. Needleman, J. Oliver, P.J. Sánchez, A finite thickness band method for ductile fracture analysis, Int. J. Plasticity 25 (2009) 2349-2365]. In this framework the transition from a weak discontinuity to a strong discontinuity can occur using a single constitutive relation which is of importance in a range of applications, in particular ductile fracture, where localization typically precedes the creation of new free surface. An embedded weak discontinuity is introduced when the loss of ellipticity condition is met. The resulting localized deformation band is given a specified thickness which introduces a length scale thus providing a regularization of the post-localization response. The methodology is illustrated through several example problems emphasizing finite deformation effects including the development of a cup-cone failure in round bar tension.     

Modeling of dielectric viscoelastomers with application to electromechanical instabilities

Soft dielectrics are electrically-insulating elastomeric materials, which are capable of large deformation and electrical polarization, and are used as smart transducers for converting between mechanical and electrical energy. While much theoretical and computational modeling effort has gone into describing the ideal, time-independent behavior of these materials, viscoelasticity is a crucial component of the observed mechanical response and hence has a significant effect on electromechanical actuation. In this paper, we report on a constitutive theory and numerical modeling capability for dielectric viscoelastomers, able to describe electromechanical coupling, large-deformations, large-stretch chain-locking, and a time-dependent mechanical response. Our approach is calibrated to the widely-used soft dielectric VHB 4910, and the finite-element implementation of the model is used to study the role of viscoelasticity in instabilities in soft dielectrics, namely (1) the pull-in instability, (2) electrocreasing, (3) electrocavitation, and (4) wrinkling of a pretensioned three-dimensional diaphragm actuator. Our results show that viscoelastic effects delay the onset of instability under monotonic electrical loading and can even suppress instabilities under cyclic loading. Furthermore, quantitative agreement is obtained between experimentally measured and numerically simulated instability thresholds. Our finite-element implementation will be useful as a modeling platform for further study of electromechanical instabilities and for harnessing them in design and is provided as online supplemental material to aid other researchers in the field.