Effective Constitutive Relations Research Articles

Bounds for the effective constitutive relation of a nonlinear composite

For a nonlinear composite, a bound on its effective energy density does not induce a corresponding bound on its constitutive relation, because differentiating a bound on a function does not automatically bound its derivative. In this work, a method introduced by Milton and Serkov for bounding directly the constitutive relation is refined by employing a linear comparison material, in a way similar to that employed by the present authors to obtain bounds of ‘Hashin–Shtrikma’ type for the effective energy of a nonlinear composite. The original Milton–Serkov approach produces bounds with a close relationship to the classical energy bounds of Voigt and Reuss type. The bounds produced in the present implementation are closely related to bounds of Hashin–Shtrikman type for the composite. Indeed, in the case of a linear composite, the present method delivers the Hashin–Shtrikman bounds in their generalized form, valid for any two–point statistics. It is demonstrated for nonlinear examples that the approximate constitutive relation that is obtained by differentiating the energy bound can be on the boundary of the bounding set for the exact constitutive relation, but a simple counterexample is presented to show that this is not always the case.

On damage theory of a cohesive medium

A micro-mechanics damage model is proposed based on homogenization of penny-shaped cohesive micro-cracks (Barenblatt–Dugdale type) in a three dimensional representative volume element. By assuming that macro-hydrostatic stress state has dominant effect on permanent crack opening, a class of pressure sensitive yielding potentials and corresponding damage evolution laws have been derived. The merits of this class of damage models are: (1) Its ability to model and predict material failure and degradation due to cohesive micro-crack growth; (2) its ability to estimate the influence of Poisson's ratio on material's damage. One of the distinguished features of the new damage model is at macro-level the reversible part of effective constitutive relation is characterized as a nonlinear elasticity, whereas the irreversible part of effective constitutive relation is a form of pressure-sensitive plasticity, both of which are significantly different from material's behaviors at micro-level before homogenization.

繊維複合材料の非線形粘弾性特性とその損傷の影響(OS4-5 強度,OS4 微視構造を有する材料の変形と破壊)

Thermoplastic matrix of the composite has time-temperature dependent properties. Effect of damage between fiber and matrix on viscoelastic properties of fiber composites is considered. The interface between fiber and matrix slips from end of fiber during loading. It may cause nonlinear behavior of composites. The nonlinearity is discussed using a homogenization theory with two-scale asymptotic expansion to evaluate viscoelastic properties of composites. Effective constitutive relation and microscopic disturbed displacements are derived from the homogenization theory.

短繊維複合材料の繊維端からのはく離進展とクリープ特性の解析(OS9d 先端材料の機能性評価と材料設計に関する数理解析)

Creep deformation of the short fiber composite with thermoplastic matrix is considered in this study. The composite is modeled as periodical array of short fibers in the microstructure. The thermoplastic matrix has time-temperature dependent properties. Effect of damage between fiber and matrix on the creep property of fiber composites is studied. The interface between fiber and matrix slips from end of fiber during loading. It may cause nonlinear behavior of composites. The nonlinearity is discussed using a homogenization theory with two-scale asymptotic expansion to evaluate viscoelastic properties of composites. Effective constitutive relation and microscopic disturbed displacements are derived from the homogenization theory.

Mechanics of plain-woven fabrics for inflated structures

Pressurized fabric tubes, pressure-stabilized beams (known as air beams) and air-inflated structures are considered to be valuable technologies for lightweight, rapidly deployable structures. Design optimization of an inflated structure depends on a thorough understanding of woven fabric mechanics. In this paper the bending response of woven pressure-stabilized beams have been experimentally tested and analytically investigated. Additionally, the micromechanical effects of interacting tows have been studied through finite element models containing contact surfaces and nonlinear slip/stick conditions. Local unit cell models consisting of pairs of woven tows were created to characterize the effective constitutive relations. The material properties from the unit cell models were then used for the global continuum model subjected to 4-point flexure. An experimental set-up was designed and manufactured for testing of Vectran and PEN air beams. The air beam mid-span deflections were measured as functions of inflation pressure and bending load. Plots of the elastic and shear moduli with respect to the pressure and coefficient of friction have been generated. It was determined that the effective elastic and shear moduli were functions of inflation pressure, the material used and the geometry of the weave. It was shown that pneumatic or pressurized tube structures differ fundamentally from conventional metal structures.

Quasi-micromechanical damage model for brittle solids with interacting microcracks

The combination of phenomenological and micromechanical damage mechanics is a promising path to construct an applicable damage model with firm physical foundation. In this paper, a quasi-micromechanical model is presented for microcrack-weakened brittle or quasi-brittle materials. The microcracking damage is characterized in terms of the orientation domain of microcrack growth (DMG) as well as a scalar microcrack density parameter. The DMG describes the complex damage and its evolution associated with microcrack growth, while the scalar microcrack density factor defining the isotropic part of damage yields an easy calculation of the effects of microcrack interaction on effective elastic moduli. The conventional methods for estimating the effective moduli of microcracked solids, based on the concept of effective medium or effective field, are formulated in a universal framework. A novel and efficient scheme is suggested to calculate the microcrack interaction effects on the effective elastic moduli in a very convenient manner. Thereby, a quasi-micromechanical model is established to simulate the constitutive response of microcrack-weakened brittle or quasi-brittle materials under complex loading. Considering various micromechanisms of microcracking damage, the overall effective constitutive relation for different stages including linear elasticity, pre-peak non-linear hardening, stress drop and strain softening are expressed in a unified form.

Effective response of a non-linear composite in external AC electric field

A general effective response is proposed for nonlinear composite media, which obey a current–field relation of the form J= σE+ χ| E| 2 E when an external alternating current (AC) electrical field is applied. For a sinusoidal applied field with finite frequency ω, the effective constitutive relation between the current density and electric field can be defined as, 〈 J( x, ω, t)〉= σ e 〈 E( x, ω, t)〉+ χ e 〈| E( x, ω, t)| 2 E( x, ω, t)〉+⋯, where σ e and χ e are the general effective linear and nonlinear conductive responses, respectively. The angled brackets 〈⋯〉 denotes the ensemble average. As two examples, we have investigated the cylindrical and spherical inclusions embedded in a host and also derived the formulae of the general effective linear and nonlinear conductive responses in dilute limit. For higher volume fraction of inclusions, we have proposed a nonlinear effective medium approximation (EMA) method to estimate the general effective response of nonlinear composites in external AC field. Furthermore, the effective nonlinear responses at harmonics are predicted by using the general effective response.

Micromechanics-based hyperelastic constitutive modeling of magnetostrictive particle-filled elastomers

An effective hyperelastic constitutive model is developed for particle-filled elastomer composites based on the microstructural deformation and physical mechanism of the magnetostrictive particles embedded in the hyperelastic elastomer matrix. Two types of loading conditions are considered––magnetic field and mechanical load. Magnetic eigenstrains are prescribed on the particles due to effect of magnetostriction. The effective constitutive relation of the composites during infinitesimal deformation can be established based on Eshelby's micromechanics approach. Since the elastomers normally exhibit finite hyperelastic deformation, the corresponding hyperelastic constitutive law of the composites is constructed in terms of the strain energy densities of the constituents.

On the construction of effective constitutive relations for porous elastic materials subjected to finite deformations including the case of their superposition

This study deals with a new approach to the construction of effective constitutive relations for porous nonlinearly elastic materials subjected to finite deformations. The approach allows for superimposing the deformations when pores are formed after a preliminary loading. The effective constitutive relations based on this approach are shown to remain unchanged in the case of a superimposed rigid motion. The construction of the effective constitutive relations is performed on the basis of well-known general principles [1, 2]. We isolate in a material a certain representative volume (for the two-dimensional case, this is a certain area) and, studying its mechanical behavior under loading, we can determine properties of the material as a whole. For this volume (area), the static problem of nonlinear elasticity is solved for a given stress at the boundary. Furthermore, deformations and stresses are averaged over the representative volume (area), and the effective constitutive relations are constructed as a dependence between averaged deformations and averaged stresses. When solving problems on effective properties of inhomogeneous materials, a question arises: how to define an effective material in order that this definition would have both a clear physical meaning (for use in experiments) and a sufficiently simple mathematical representation. In addition, in the process of seeking average deformations of porous media, a problem arises associated with the fact that the deformations are not defined in the interior of pores (this is the difference between porous materials and those with elastic inclusions). In the case of small deformations, these issues are solved in a rather simple manner [1, 3, 4]. For example, the deformation tensor of an effective material is

Effective 3D Constitutive Relations of Composites with Evolving Damage

In this study it is shown that a computational procedure, termed “discrete damage space homogenization method” (DDSHM), can predict the constitutive response of layered composite materials containing growing cracks. The fully 3D effective constitutive law for a specific layered composite architecture, as defined by the DDSHM, was integrated into the ABAQUS commercial finite-element program using the user-defined material feature. Calculations were carried out for flat laminates with penny-shaped microcracks to illustrate the computational efficiency of the method for general analysis of composite materials with growing cracks. Results show that, given basic information about the fracture toughness of the material, the DDSHM is able to predict important material parameters including the load at initiation of cracking, damage growth rate, and resulting effect on the macroscopic stiffness.

Effective elastoplastic behavior of metal matrix composites containing randomly located aligned spheroidal Inhomogeneities. Part II: applications

Based on the proposed formulation in Part I of this sequel (Ju, J.W., Sun, L.Z., Int. J. Solids Struct. 38, 183–201), effective elastoplastic constitutive relations are implemented in this article for metal matrix composites (MMCs) with randomly located and unidirectionally aligned spheroidal particles. First, we investigate the uniaxial elastoplastic stress–strain behavior of MMCs. In particular, we perform comparisons among the theoretical uniaxial stress–strain predictions, existing finite element results and experimental data for MMCs to illustrate the capability of the proposed method. Furthermore, the effect of stress triaxiality is discussed under either the purely hydrostatic or axisymmetric loading on the overall elastoplastic behavior of composites. The proposed initial effective yield surfaces for composites are demonstrated and compared with those of the experimental data. As a special case of the incompressible ductile material containing aligned spheroidal voids, the initial effective yield criterion is studied and compared with that of mathematical upper bound. Finally, viscoplastic extension is briefly presented.

On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media

Higher order (so-called strain gradient) homogenised equations are rigorously derived for an infinitely extended periodic elastic medium with the periodicity cell of a small size ε, in the presence of a fixed body force f , via a combination of variational and asymptotic techniques. The coefficients of these equations are explicitly related to solutions of higher order unit cell problems. The related higher order homogenised solutions are shown to be best possible in a certain variational sense, and it is shown that these solutions are close to the actual solutions up to higher orders in ε. We derive a rigorous full asymptotic expansion for the energy I ε, f and also show that its higher order terms are determined by the higher order homogenised solutions. The resulting variational construction generates higher order effective constitutive relations which are in agreement with those proposed by phenomenological strain gradient theories.

On the constitutive relations of materials with evolving microstructure due to microcracking

This paper discusses the practical implementation of a thermodynamically based procedure for the derivation of the effective nonlinear constitutive relations for composites with evolving microstructure. Examples of the procedure for the case of an elastic periodic composite with several growing cracks are presented. These examples are intended to show how the use of this procedure differs from the traditional global-local analysis approach and how well suited it is for use with general purpose structural analysis packages.

Dual finite element methods in homogenization for elastic–plastic fibrous composite material

Abstract The problem of homogenization for a periodic, elastic–perfectly plastic, fiber reinforced, composite material is considered. The overall mechanical behavior of the material is described using the anisotropic model of elastic–plastic body with kinematic hardening. The appropriate initial–boundary value problem, set for one repeatable cell of the composite, is solved in order to find the effective constitutive relations. The cell problem is solved using the finite element method formulated in two dual forms: in displacements and in stresses. Stress functions are used in the latter formulation.

Phase separation of rigid-rod suspensions in shear flow.

We analyze the behavior of a suspension of rigid-rod-like particles in shear flow using a modified version of the Doi model, and construct diagrams for phase coexistence under conditions of constant imposed stress and constant imposed strain rate, among paranematic, flow-aligning nematic, and log-rolling nematic states. We calculate the effective constitutive relations that would be measured through the regime of phase separation into shear bands. We calculate phase coexistence by examining the stability of interfacial steady states and find a wide range of possible "phase" behaviors.

Wave propagation in elastic media with cracks. Part II: Transient nonlinear response of a cracked matrix

The transient dynamic response of an elastic medium containing a random array of aligned penny-shaped cracks is studied. Nonlinearity in the response due to contact between crack faces during the motion is taken into account. The mean value of wave fields is considered and an effective constitutive relation is adopted which contains an internal variable defining the mean opening of the cracks. An integral representation of the effective displacement field is given using the Green function of the uncracked body. One-dimensional nonlinear waves are studied using the solution of the single scattering problem proposed by the authors in a previous paper. Numerical examples showing the role of the characteristic crack dimension on the overall nonlinear response of the cracked medium are presented.

Three-dimensional elastoplastic finite element modelling of deformation and fracture behaviour of rubber-modified polycarbonates at different triaxiality

A periodic face-centred cuboidal cell model is provided to account for inter-particle interaction, and a particle-crack tip interaction model is developed to study the interaction between a blunting model I crack tip and the closest array of initially spherical rubber particles in an effective medium. Three-dimensional elastoplastic finite element analysis has been preformed to study the deformation and fracture behaviour of rubber-modified polycarbonates. The effective elastoplastic constitutive relation is derived by the method of homogenisation and local stress and strain distributions are obtained to explore the role of rubber cavitation in the toughening process at different stress triaxiality. 3D elastoplastic finite element results are compatible with experimental observations, that is, rubber particles can act as stress concentrators to initiate crazing or shear yielding in the matrix but they behave differently from voids at high triaxiality. Rubber cavitation plays an important role in the toughening process under high tensile triaxial stresses.

A study of the effective properties of mass and stiffness microstructures—a multiresolution approach

The theory of multiresolution decomposition and wavelets is used to study the effective properties of a thin elastic plate with surface mass density or stiffness heterogeneity, subjected to time-harmonic forcing. The heterogeneity possesses micro- and macro-scale variations, and has a macroscale outer dimension. It is shown that the microscale mass variation has practically no effect on the macroscale plate response, whereas microscale stiffness variation can have a significant effect. We derive an effective constitutive relation pertaining to a microscale stiffness variation. It is shown that it is possible to synthesize classes of different stiffness microstructures that have the same footprint on the macroscale component of the plate response. An effective, smooth, stiffness heterogeneity associated with the classes is developed. The results are first derived analytically and then supported by numerical simulations.

The effects of friction between crack faces on nonlinear wave propagation in microcracked media

The effective behaviour of a composite consisting of an elastic matrix containing a dilute array of aligned, randomly positioned, ‘closing’, penny-shaped cracks is considered. In Smyshlyaev and Willis (1996) and Woolfries (1998), the faces of the cracks were taken to be frictionless. A similar formulation is studied here when the crack faces permit no sliding when closed, i.e. they have infinite friction. These conditions are written down mathematically and their implications explored under “low-frequency” assumptions. The resulting effective constitutive relations, depending on a field of internal variables, contain conditional inequalities. The tangential components contain a convolution term which describes the effect of the cracks' history on the present tangential stress. The Fourier transform of this term was given in Smyshlyaev and Willis (1994). Here the inverse transform is carried out. The mean wave is taken to be at an angle to the common crack normal. The behaviour of tangential components now interacts with the rest of the system, which was not the case in Smyshlyaev and Willis (1996) and Woolfries (1998) when the mean wave was parallel to the crack normal. The resulting equations on effective variables are reduced by evaluating the Green's function for the system. This allows a simple numerical method to solve initial- or boundary-value problems. The case of no friction may easily be compared to the case of infinite friction by relaxing the condition of no slip, and using the variational inequality formulation of Smyshlyaev and Willis (1996). Numerical routines are carried out for this case also.

Artificial nonreciprocal uniaxial magnetoelectric composites

In this communication we explore the possibilities of using magnetostatic resonators as inclusions for artificial magnetoelectric composites (as suggested by Kamenetskii). The appropriate form of effective constitutive relations is identified; the properties of eigenwaves and reflection and transmission coefficients in slabs are analyzed for uniaxial structures. Potential applications are briefly addressed. © 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett 15: 260–263, 1997.