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.     

The effect of particle shape and distribution on the macroscopic behavior of magnetoelastic composites

The magnetoelastic homogenization framework and the partial decoupling approximation proposed by Ponte Castañeda and Galipeau (2011) are used to estimate material properties for a class of magnetically susceptible elastomers. Specifically, we consider composites consisting of aligned, ellipsoidal magnetic particles distributed randomly with “ellipsoidal” symmetry under combined magnetic and mechanical loading. The model captures the coupling between the magnetic and mechanical fields, including the effects of magnetic saturation. The results help elucidate the effects of particle shape, distribution, and concentration on properties such as the magnetostriction, actuation stress, magnetic modulus, and magnetization behavior of a magnetorheological composite.     

A Magnetoelastic Field in a Ferromagnetic with an Elliptic Inclusion

A stress–strain problem is solved for an infinite isotropic magnetically soft body containing an elliptic inclusion. It is assumed that the body is in an external magnetic field. The basic characteristics of the stress–strain state and the induced magnetic field are determined and their features at the inclusion are analyzed. Graphs are drawn for the total magnetoelastic and Maxwell stresses versus the ratio of the ellipse axes and the angle of dip, and tabular maximum stresses versus the magnetic induction and the magnetic properties of the material.     

用Eshelby理论研究复合材料线粘弹性本构关系

本文用Eshelby微力学理论分析,得到短纤维增强材料(SFRC)和基体材料两者的粘弹性本构方程之间存在简单的正比关系。发现体积含量为f的短纤维无序取向SFRC一维力学行为,等效于体积含量为F的短纤维单轴取向SFRC在取向轴上的力学行为。     

Modeling the viscoplastic micromechanical response of two-phase materials using Fast Fourier Transforms

A viscoplastic approach using the Fast Fourier Transform (FFT) method for obtaining local mechanical response is utilized to study microstructure-property relationships in composite materials. Specifically, three-dimensional, two-phase digital materials containing isotropically coarsened particles surrounded by a matrix phase, generated through a Kinetic Monte Carlo Potts model for Ostwald ripening, are used as instantiations in order to calculate the stress and strain-rate fields under uniaxial tension. The effects of the morphology of the matrix phase, the volume fraction and the contiguity of particles, and the polycrystallinity of matrix phase, on the stress and strain-rate fields under uniaxial tension are examined. It is found that the first moments of the stress and strain-rate fields have a different dependence on the particle volume fraction and the particle contiguity from their second moments. The average stresses and average strain-rates of both phases and of the overall composite have rather simple relationships with the particle volume fraction whereas their standard deviations vary strongly, especially when the particle volume fraction is high, and the contiguity of particles has a noticeable effect on the mechanical response. It is also found that the shape of stress distribution in the BCC hard particle phase evolves as the volume fraction of particles in the composite varies, such that it agrees with the stress field in the BCC polycrystal as the volume of particles approaches unity. Finally, it is observed that the stress and strain-rate fields in the microstructures with a polycrystalline matrix are less sensitive to changes in volume fraction and contiguity of particles.     

The Stress State of a Ferromagnetic with a Spheroidal Inclusion in a Homogeneous Magnetic Field

The stress–strain state of an infinite isotropic magnetically soft ferromagnetic body with a spheroidal inclusion is analyzed. It is assumed that the body is in an external magnetic field. The basic stress–strain characteristics and the induced magnetic field near and inside the inclusion are analyzed. The plots and the table presented show how the total magnetoelastic and Maxwell stresses near and inside the inclusion depend on the ratio of the spheroid axes, the latitude angle, and the magnetic induction when the medium and the inclusion are dissimilar materials.     

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.     

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.     

Magnetoactive elastomers with periodic and random microstructures

We investigate the behavior of magnetoactive elastomers (MAEs) with periodic and random distributions of circular and elliptical fibers. For the MAEs with periodic microstructures, we develop finite element models and determine the local fields as well as the effective properties of MAEs with rectangular and quasi-hexagonal unit cells. For the MAEs with random microstructures, we derive a closed-form expression for the effective response making use of a recently developed theory (Ponte Castañeda and Galipeau, 2011). In particular, we determine the responses to pure shear loading in the presence of a magnetic field, both of which are aligned with the geometric axes of the fibers, and examine the roles of the deformation, concentration, particle shape, and distribution on the magnetostriction, actuation stress, and the magnetically induced stiffness of the composite. We show that the coupling effects are of second order in the concentration. This is consistent with the fact that these effects are primarily the result of the interaction between inclusions. We also demonstrate explicitly that the magnetomechanical coupling of these MAEs, when subjected to aligned loading conditions, depends not only on the magnetic susceptibility, but also, crucially, on its derivative with respect to the deformation. As a consequence, we find that the magnetoelastic effects may be quite different, even for composites with similar effective susceptibilities.     

A thermo-mechanically coupled theory for fluid permeation in elastomeric materials: Application to thermally responsive gels

An elastomeric gel is a cross-linked polymer network swollen with a solvent, and certain gels can undergo large reversible volume changes as they are cycled about a critical temperature. We have developed a continuum-level theory to describe the coupled mechanical deformation, fluid permeation, and heat transfer of such thermally responsive gels. In discussing special constitutive equations we limit our attention to isotropic materials, and consider a model based on a Flory–Huggins model for the free energy change due to mixing of the fluid with the polymer network, coupled with a non-Gaussian statistical–mechanical model for the change in configurational entropy—a model which accounts for the limited extensibility of polymer chains. We have numerically implemented our theory in a finite element program. We show that our theory is capable of simulating swelling, squeezing of fluid by applied mechanical forces, and thermally responsive swelling/de-swelling of such materials.     

Magneto-elastic modeling of composites containing chain-structured magnetostrictive particles

Magneto-elastic behavior is investigated for two-phase composites containing chain-structured magnetostrictive particles under both magnetic and mechanical loading. To derive the local magnetic and elastic fields, three modified Green's functions are derived and explicitly integrated for the infinite domain containing a spherical inclusion with a prescribed magnetization, body force, and eigenstrain. A representative volume element containing a chain of infinite particles is introduced to solve averaged magnetic and elastic fields in the particles and the matrix. Effective magnetostriction of composites is derived by considering the particle's magnetostriction and the magnetic interaction force. It is shown that there exists an optimal choice of the Young's modulus of the matrix and the volume fraction of the particles to achieve the maximum effective magnetostriction. A transversely isotropic effective elasticity is derived at the infinitesimal deformation. Disregarding the interaction term, this model provides the same effective elasticity as Mori-Tanaka's model. Comparisons of model results with the experimental data and other models show the efficacy of the model and suggest that the particle interactions have a considerable effect on the effective magneto-elastic properties of composites even for a low particle volume fraction.     

A novel analytical model and energy analysis of thermal stresses in two-phase composites

This paper deals with analytical modelling of thermal stresses in a multi-particle-matrix system with isotropic spherical particles. These particles are periodically distributed in an isotropic infinite matrix. This model system which is characterized by microstructural parameters (particle volume fraction, particle radius) is applicable to two-phase composites of a precipitate-matrix type with isotropic phases. The thermal stresses originate during a cooling process due to a difference in thermal expansion coefficients. The analytical modelling which is based on fundamental equations of solid continuum mechanics represents a combination of different mathematical procedures applied to equilibrium and compatibility equations. This novel analytical model is compared with that which is based on mathematical procedures applied to the equilibrium equations only. The energy analysis of both analytical models which is applied to the SiC-Al₂O₃ composite is presented.     

A Magnetoelastic Field in a Ferromagnetic Medium with a Spherical Cavity

The problem on the stress–strain state of an infinite isotropic body made of a magnetically soft material and containing a spherical cavity is considered. It is assumed that the body is under an external magnetic field. The basic characteristics of the stress–strain state and the magnetic field induced are determined and their singularities near the cavity are studied. Graphs are presented for the total magnitoelastic and Maxwell stresses as functions of the magnetic induction, the angle of dip, and the mechanical and magnetic properties of the material     

The effect of the irregular interface geometry in deformation and fracture of a steel substrate–boride coating composite

Presented in this paper is a computational analysis of the mechanisms involved in plastic deformation and fracture of a composite with coating under compressive and tensile loading. Using a steel specimen surface-hardened by diffusion borating, a role of the irregular geometry of the interface between the base material and hardened surface layer is investigated. In order to describe the mechanical behavior of the steel substrate and brittle coating, use is made of a plastic flow model including isotropic strain hardening and a fracture model, respectively. Using the Huber fracture criterion, the model takes into account the difference in the critical strength values for different types of local compressive and tensile states. It is shown that the irregular, serrated shape of the substrate–coating interface retards propagation of a longitudinal crack into this coating and prevents it from spalling under external compression of this composite. It is found out that even in the case of a simple uniaxial compression of the mesovolumes of this composite the boride “teeth” are subjected to tensile stresses, whose values are comparable with those of the external compressive load, and the direction of crack propagation and the general fracture behavior largely depend on the external loading conditions.     

Mechanical and structural investigation of isotropic and anisotropic thermoplastic magnetorheological elastomer composites based on poly(styrene-b-ethylene-co-butylene-b-styrene) (SEBS)

Novel smart thermoplastic magnetorheological elastomer composites containing micron-sized magnetic carbonyl iron (CI) particles were prepared with a poly(styrene-ethylene-butylene-styrene) (SEBS) triblock copolymer utilized as the thermoplastic matrix rubber, and the structures and properties of the CI-SEBS composites were examined. The CI particles were uniformly dispersed in the composites prepared in the absence of the magnetic field at high temperatures T (>T $_{\rm g}^{\rm S})$ , and this isotropic composite exhibited a larger storage modulus G^?? compared to the SEBS matrix at room temperature (<?<T $_{\rm g}^{\rm S})$ where the EB phase therein was rubbery while the PS phase was in the glassy state. In contrast, the SEBS composite prepared under the magnetic field (with the intensity ???< 2.5?T) at high T (>T $_{\rm g}^{\rm S})$ contained a chain structure of CI particles. This chain structure became longer and better aligned on an increase of ?? up to a saturation of the particle magnetization and on an increase of the time interval of applying the field (that allowed the particles to move and equilibrate their aligned structure). The modulus G^?? of this ??pre-structured?? composite measured for both cases of ?? = 0 and ???> 0 in the direction perpendicular to the chain structure at room temperature was enhanced compared to G^?? of the isotropic composites. This difference of the filler effect (for ???=?0) and the magnetorheological effect (for ???> 0) between the pre-structured and isotropic composites was enhanced when the chain structure of the CI particles in the pre-structured composites became longer and better aligned. A mechanism(s) of this enhancement was discussed in relation to the morphologies (particle distribution) in the composites with the aid of a filler model and a molecular expression of the stress due to magnetically interacting particles.     

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     

Finite element modelling of Al/SiCp metal matrix composites with particles aligned in stripes—a 2D–3D comparison

3D Finite element calculations comparing to axisymmetric calculations have been performed to predict quantitatively the tensile behaviour of composites reinforced with ceramic particles aligned in stripes. The analyses are based on a unit cell model, which assumes the periodic arrangement of reinforcements. The results are presented in such a manner that can be directly compared for all possible aspect ratios and inclusion volume fractions. It is indicated that varying the distance between the stripes when particle volume fraction is kept constant significantly influences the overall mechanical behaviour of composites. Whereas during elastic deformation 3D and axisymmetric formulations predict quantitatively similar results, the mechanical behaviour perpendicular to the stripe direction predicted by 3D and axisymmetric models may differ depending on the inclusion volume fraction. Nevertheless an appreciable strengthening in the stripe direction independent on the model and deformation stage is predicted.     

Changes in the natural frequency of a ferromagnetic rod in a magnetic field due to magnetoelastic interaction

Based on the magnetoelastic generalized variational principle and Hamilton＇s principle, a dynamic theoretical model characterizing the magnetoelastic interaction of a soft ferromagnetic medium in an applied magnetic field is developed in this paper. From the variational manipulation of magnetic scale potential and elastic displacement, all the fundamental equations for the magnetic field and mechanical deformation, as well as the magnetic body force and magnetic traction for describing magnetoelastic interaction are derived. The theoretical model is applied to a ferromagnetic rod vibrating in an applied magnetic field using a perturbation technique and the Galerkin method. The results show that the magnetic field will change the natural frequencies of the ferromagnetic rod by causing a decrease with the bending motion along the applied magnetic field where the magnetoelastic buckling will take place, and by causing an increase when the bending motion of the rod is perpendicular to the field. The prediction by the mode presented in this paper qualitatively agrees with the natural frequency changes of the ferromagnetic rod observed in the experiment.     

Viscoelastic Deformation of a Reinforced Plate with a Crack

The deformation of a viscoelastic reinforced composite is studied. The composite has an axis of elastic symmetry and consists of transversally isotropic fibers and a viscoelastic matrix, which differ by the volume concentration and mechanical characteristics. The material is modeled by a transversely isotropic homogeneous linearly viscoelastic medium with some averaged characteristics. A plate fabricated from the composite in question is weakened by a through mode I crack and is subjected to constant tensile forces. The viscoelastic properties of the matrix material are described by a convolution operator. The Volterra principle is used to derive expressions for the viscoelastic characteristics and crack opening. The irrational function of the integral operator that describes the crack opening is expanded into an operator continued fraction and is represented as the sum of base operators     

Assessing continuum postulates in simulations of granular flow

Continuum mechanics relies on the fundamental notion of a mesoscopic volume “element” in which properties averaged over discrete particles obey deterministic relationships. Recent work on granular materials suggests that a continuum law may be inapplicable, revealing inhomogeneities at the particle level, such as force chains and slow cage breaking. Here, we analyze large-scale three-dimensional discrete-element method (DEM) simulations of different granular flows and show that an approximate “granular element” defined at the scale of observed dynamical correlations (roughly three to five particle diameters) has a reasonable continuum interpretation. By viewing all the simulations as an ensemble of granular elements which deform and move with the flow, we can track material evolution at a local level. Our results confirm some of the hypotheses of classical plasticity theory while contradicting others and suggest a subtle physical picture of granular failure, combining liquid-like dependence on deformation rate and solid-like dependence on strain. Our computational methods and results can be used to guide the development of more realistic continuum models, based on observed local relationships between average variables.