Filled elastomers: A theory of filler reinforcement based on hydrodynamic and interphasial effects

Experimental evidence has by now established that (i) the hydrodynamic effect and (ii) the presence of stiff interphases (commonly referred to as bound rubber) “bonding” the underlying elastomer to the fillers are the dominant microscopic mechanisms typically responsible for the enhanced macroscopic stiffness of filled elastomers. Yet, because of the technical difficulties of dealing with these fine-scale effects within the realm of finite deformations, the theoretical reproduction of the macroscopic mechanical response of filled elastomers has remained an open problem.The object of this paper is to put forward a microscopic field theory with the capability to describe, explain, and predict the macroscopic response of filled elastomers under arbitrarily large nonlinear elastic deformations directly in terms of: (i) the nonlinear elastic properties of the elastomeric matrix, (ii) the concentration of filler particles, and (iii) the thickness and stiffness of the surrounding interphases. Attention is restricted to the prominent case of isotropic incompressible elastomers filled with a random and isotropic distribution of comparatively rigid fillers. The central idea of the theory rests on the construction of a homogenization solution for the fundamental problem of a Gaussian elastomer filled with a dilute concentration of rigid spherical particles bonded through Gaussian interphases of constant thickness, and on the extension of this solution to non-Gaussian elastomers filled with finite concentrations of particles and interphases by means of a combination of iterative and variational techniques.For demonstration purposes, the theory is compared with full 3D finite-element simulations of the large-deformation response of Gaussian and non-Gaussian elastomers reinforced by isotropic distributions of rigid spherical particles bonded through interphases of various finite sizes and stiffnesses, as well as with experimental data available from the literature. Good agreement is found in all of these comparisons. The implications of this agreement are discussed.     

A finite-strain constitutive theory for electro-active polymer composites via homogenization

This paper presents a homogenization framework for electro-elastic composite materials at finite strains. The framework is used to develop constitutive models for electro-active composites consisting of initially aligned, rigid dielectric particles distributed periodically in a dielectric elastomeric matrix. For this purpose, a novel strategy is proposed to partially decouple the mechanical and electrostatic effects in the composite. Thus, the effective electro-elastic energy of the composite is written in terms of a purely mechanical component together with a purely electrostatic component, this last one dependent on the macroscopic deformation via appropriate kinematic variables, such as the particle displacements and rotations, and the change in size and shape of the appropriate unit cell. The results show that the macroscopic stress includes contributions due to the changes in the effective dielectric permittivity of the composite with the deformation. For the special case of a periodic distribution of electrically isotropic, spherical particles, the extra stresses are due to changes with the deformation in the unit cell shape and size, 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.     

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.     

Homogenization estimates for fiber-reinforced elastomers with periodic microstructures

This work presents a homogenization-based constitutive model for the mechanical behavior of elastomers reinforced with aligned cylindrical fibers subjected to finite deformations. The proposed model is derived by making use of the second-order homogenization method [Lopez-Pamies, O., Ponte Castañeda, P., 2006a. On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—theory. J. Mech. Phys. Solids 54, 807–830], which is based on suitably designed variational principles utilizing the idea of a “linear comparison composite.” Specific results are generated for the case when the matrix and fiber materials are characterized by generalized Neo-Hookean solids, and the distribution of fibers is periodic. In particular, model predictions are provided and analyzed for fiber-reinforced elastomers with Gent phases and square and hexagonal fiber distributions, subjected to a wide variety of three-dimensional loading conditions. It is found that for compressive loadings in the fiber direction, the derived constitutive model may lose strong ellipticity, indicating the possible development of macroscopic instabilities that may lead to kink band formation. The onset of shear band-type instabilities is also detected for certain in-plane modes of deformation. Furthermore, the subtle influence of the distribution, volume fraction, and stiffness of the fibers on the effective behavior and onset of macroscopic instabilities in these materials is investigated thoroughly.     

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.     

Experimental investigation of the coupled magneto-mechanical response in magnetorheological elastomers

Magnetorheological elastomers (MREs) are materials made of a soft elastomer matrix filled with magnetizable particles. These flexible composites that deform in response to an externally applied magnetic field are of special interest in advanced engineering applications such as actuators, artificial muscles or shape control. However, no systematic characterization of their coupled response has been undertaken so far, thus limiting the efficient design of MRE-based devices. In this study, we propose a framework—relying on both specially designed samples and a dedicated experimental setup—to characterize experimentally the coupled magneto-mechanical response of MREs since magnetization within the sample is nearly uniform and structural-dependent effects are minimized. The influence of particle content and arrangement within the composite are particularly studied and the corresponding experimental results give some insight into the underlying microstructural mechanisms that are responsible for the macroscopic deformation of MREs under combined magnetic and mechanical loading conditions. Such data is crucial for the design of new MRE composite materials in which the microstructure is optimized (to have the largest coupling effect with minimal energy input).     

Deflections of a rubber membrane

This paper deals with the problem of the transverse deflection of a natural rubber membrane that is fixed along a circular boundary. Uniaxial experiments were performed in order to characterize the constitutive behaviour of the rubber material in terms of several constitutive models available in the literature. These constitutive models were used to develop computational estimates for the quasi-static load-displacement response of a rigid spherical indentor that deflects the rubber membrane in a controlled fashion and to determine the deflected shape of the membrane at specified load levels. Both axisymmetric and asymmetric deflections of the rubber membrane were investigated. The paper provides a comparison of the experimental results for the membrane deflections with results derived from computational simulations.     

Macroscopic constitutive relations for elastomers reinforced with short aligned fibers: Instabilities and post-bifurcation response

This paper is concerned with the characterization of the macroscopic response and possible development of instabilities in a certain class of anisotropic composite materials consisting of random distributions of aligned rigid fibers of elliptical cross section in a soft elastomeric matrix, which are subjected to general plane strain loading conditions. For this purpose, use is made of an estimate for the stored-energy function that was derived by Lopez-Pamies and Ponte Castañeda (2006b) for this class of reinforced elastomers by means of the second-order linear comparison homogenization method. This homogenization estimate has been shown to lose strong ellipticity by the development of shear localization bands, when the composite is loaded in compression along the (in-plane) long axes of the fibers. The instability is produced by the sudden, collective rotation of a band of fibers to partially release the high stresses that develop in the elastomer matrix when the composite is compressed along the stiff, long-fiber direction. Consistent with the mode of the impending instability, a lower-energy, post-bifurcation solution is constructed where “striped domain” microstructures consisting of layers with alternating fiber orientations develop in the composite. The volume fractions of the layers and the fiber orientations within the layers adjust themselves to satisfy equilibrium and compatibility across the layers, while remaining compatible with the imposed overall deformation. Mathematically, this construction is shown to correspond to the rank-one convex envelope of the original estimate for the energy, and is further shown to be polyconvex and therefore quasiconvex. Thus, it corresponds to the “relaxation” of the stored-energy function of the composite, and can in turn be viewed as a stress-driven “phase transition,” where the symmetry of the fiber microstructures changes from nematic to smectic.     

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.     

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.     

The crack tip zone shielding effect for ductile particle reinforced brittle materials

The crack tip zone shielding effect for the ductile particle reinforced brittle materials is analyzed by using a micromechanics constitutive theory. The theory is developed here to determine the elastoplastic constitutive behavior of the composite. The elastoplastic particles, with isotropic or kinematical hardening, are uniformly dispersed in the brittle elastic matrix. The method proposed is based on the Mori-Tanaka's concept of average stress in the composite. The macroscopic yielding condition and the incremental stress strain relation of the composite during plastic deformation are explicity given in terms of the macroscopioc applied stress and the microstructural parameters of the composite such as the volume fraction and yield stress of ductile particles, elastic constants of the two phases, etc. Finally, the contribution of the plastic deformation in the particles near a crack tip to the toughening of the composite is evaluated. The project supported by National Natural Science Foundation of China     

Experiments and modeling of iron-particle-filled magnetorheological elastomers

Magnetorheological elastomers (MREs) are ferromagnetic particle impregnated rubbers whose mechanical properties are altered by the application of external magnetic fields. Due to their coupled magnetoelastic response, MREs are finding an increasing number of engineering applications. In this work, we present a combined experimental and theoretical study of the macroscopic response of a particular MRE consisting of a rubber matrix phase with spherical carbonyl iron particles. The MRE specimens used in this work are cured in the presence of strong magnetic fields leading to the formation of particle chain structures and thus to an overall transversely isotropic composite. The MRE samples are tested experimentally under uniaxial stresses as well as under simple shear in the absence or in the presence of magnetic fields and for different initial orientations of their particle chains with respect to the mechanical and magnetic loading direction.Using the theoretical framework for finitely strained MREs introduced by Kankanala and Triantafyllidis (2004), we propose a transversely isotropic energy density function that is able to reproduce the experimentally measured magnetization, magnetostriction and simple shear curves under different prestresses, initial particle chain orientations and magnetic fields. Microscopic mechanisms are also proposed to explain (i) the counterintuitive effect of dilation under zero or compressive applied mechanical loads for the magnetostriction experiments and (ii) the importance of a finite strain constitutive formulation even at small magnetostrictive strains. The model gives an excellent agreement with experiments for relatively moderate magnetic fields but has also been satisfactorily extended to include magnetic fields near saturation.     

Chemorheological response of elastomers at elevated temperatures: Experiments and simulations

An experimental study and a method for simulating the constitutive response of elastomers at temperatures in the chemorheological range (90-150 °C for natural rubber) are presented. A comprehensive set of uniaxial experiments for a variety of prescribed temperature histories is performed on natural rubber specimens that exhibit finite elasticity, entropic stiffening with temperature, viscoelasticity, scission, and oxygen diffusion/reaction effects. The simulation approach is based on a multi-network framework for finite elasticity, isothermal incompressibility, thermal expansion, and temperature-induced degradation. The model extends previous work to account for kinetics of scission for arbitrary time-varying temperature histories and incorporates the effects of viscoelastic relaxation and diffusion-limited oxidative scission. The model is calibrated to experiments performed on a commercially-available filled natural rubber material, and numerical simulations are compared favorably to experiments for a variety of temperature histories.     

炭黑增强橡胶复合材料力学行为的三维数值模拟

本文基于炭黑填充橡胶复合材料具有周期性细观结构的假设,采用一种新的、改进的随机序列吸附算法建立了三维多球颗粒随机分布式代表性体积单元,并通过细观力学有限元方法对炭黑颗粒填充橡胶复合材料的力学行为进行了模拟仿真。研究结果表明：采用改进的随机序列吸附算法所建立的模型更加便于有限元离散化;模拟中周期性边界条件的约束,使其更加符合实际约束的真实情况;炭黑填充橡胶复合材料的有效模量明显高于未填充橡胶材料,并随着炭黑颗粒所占体积分数的增加而增大;通过比较发现,本文提出的多球颗粒随机分布式三维数值模型对复合材料的应力-应变行为和有效弹性模量的预测结果与实验结果吻合良好,证实了该模型能够用于炭黑颗粒增强橡胶基复合材料有效性能的模拟分析。     

Tangent Second-Order Estimates for the Large-Strain, Macroscopic Response of Particle-Reinforced Elastomers

An approximate homogenization method is proposed and used to obtain estimates for the effective constitutive behavior and associated microstructure evolution in hyperelastic composites undergoing finite-strain deformations. The method is a modified version of the “tangent second-order” procedure (Ponte Castañeda and Tiberio in J. Mech. Phys. Solids 48:1389, 2000), and can be used to provide estimates for the nonlinear elastic composites in terms of corresponding estimates for suitably chosen “linear comparison composites”. The method makes use of the “tangent” moduli of the phases, evaluated at suitable averages of the deformation gradient, and yields a constitutive relation accounting for the evolution of characteristic features of the underlying microstructure in the composites, when subjected to large deformations. Satisfaction of the exact, macroscopic incompressibility constraint is ensured by means of an energy decoupling approximation splitting the elastic energy into a purely “distortional” component, together with a “dilatational” component. The method is applied to elastomers containing random distributions of aligned, rigid, ellipsoidal inclusions, and explicit analytical estimates are obtained for the special case of spherical inclusions distributed isotropically in an incompressible neo-Hookean matrix. In addition, the method is also applied to two-dimensional composites with random distributions of aligned, elliptical fibers, and the results are compared with corresponding results of earlier homogenization estimates and finite element simulations.     

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.     

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.