Effective Constitutive Relations Research Articles

Micromechanics methods for evaluating the effective moduli of soft neo-Hookean composites

Most biological soft tissues are multiphase composite materials, and the determination of their effective constitutive relations is a major concern in medical engineering. In this paper, we consider a class of soft two-phase composites, in which both phases are isotropic and hyperelastic neo-Hookean materials. For such an isotropic composite consisting of inclusions uniformly distributed but randomly oriented in a matrix, two constitutive parameters are required to characterize its hyperelastic constitutive relation. Micromechanics methods, including dilute concentration method, Mori–Tanaka method, self-consistent method, and differential method are extended to predict the effective properties of this kind of composites. Analytical solutions are given for the hyperelastic neo-Hookean composites reinforced by spherical particles, long fibers, and penny-shaped platelets, respectively. Finite element simulations are performed to evaluate the accuracy of these theoretical methods.

Willis elastodynamic homogenization theory revisited for periodic media

The theory of elastodynamic homogenization initiated by J.R. Willis is revisited for periodically inhomogeneous media through a careful scrutiny of the main aspects of that theory in the 3D continuum context and by applying it to the thorough treatment of a simple 1D discrete periodic system. The Bloch theorem appears to be central to appropriately defining and interpreting effective fields. Based on some physical arguments, three necessary conditions are derived for the transition from the microscopic description to the macroscopic description of periodic media. The parameters involved in the Willis effective constitutive relation are expressed in terms of two localization tensors and specified with the help of the corresponding Green function in the spirit of micromechanics. These results are illustrated and discussed for the 1D discrete periodic system considered. In particular, inspired by Brillouin's study, the dependency of the effective constitutive parameters on the frequency is physically interpreted in terms of oscillation modes of the underlying microstructure.

Extreme coupling: A route towards local magnetic metamaterials

Genuinely homogeneous metamaterials, which may be characterized by local effective constitutive relations, are required for many spectacular metamaterial applications. Such metamaterials have to be made of meta-atoms, i.e., subwavelength resonators, which exhibit only electric and or magnetic dipole and negligible higher-order multipolar polarizabilities in the spectral range of interest. Here, we show that these desired meta-atoms can be designed by exploiting the extreme coupling regime. Appropriate meta-atoms are identified by performing a multipole analysis of the field scattered from the respective meta-atom. To design those particular meta-atoms it is important to disclose the frequency and angular-dependent polarizability of both dipole moments. We demonstrate the applicability of a purely analytical model to accurately calculate for a normally incident plane wave reflection and transmission from meta-surfaces made of periodically arranged meta-atoms. With our work we identify a possible route towards the engineering of artificial materials while only considering the response from its constituents. Our approach is generally applicable to all spectral domains and can be used to evaluate and design metamaterials made from different constituting materials, e.g., metals, dielectrics, or semiconductors.

Extension of the Nonuniform Transformation Field Analysis to linear viscoelastic composites in the presence of aging and swelling

This study presents a micromechanical modeling by the Nonuniform Transformation Field Analysis (NTFA) of the viscoelastic properties of heterogeneous materials with aging and swelling constituents. The NTFA proposed by Michel and Suquet (2003, 2004) is a compromise between analytical models and full-field simulations. Analytical models, which are available only for specific microstructures, provide effective constitutive relations which can be used in macroscopic structural computations, but often fail to deliver sufficiently detailed information at small scale. At the other extreme, full-field simulations provide detailed local fields, in addition to the composite effective response, but come at a high cost when used in nested Finite Element Methods. The NTFA method is a technique of model reduction which achieves a compromise between both approaches. It is based on the observation that the transformation strains (viscous strains, eigenstrains) often exhibit specific patterns called NTFA modes. It delivers both effective constitutive relations and localization rules which allow for the reconstruction of local fields upon post-processing of macroscopic quantities.A prototype of the materials of interest here is MOX (mixed oxides), a nuclear fuel which is a three-phase particulate composite material with two inclusion phases dispersed in a contiguous matrix. Under irradiation, its individual constituents, which can be considered as linear viscoelastic, are subject to creep, to aging (time dependent material properties) and to swelling (inhomogeneous eigenstrains). Its overall behavior is therefore the result of the combination of complex and coupled phenomena. The NTFA is applied here in a three-dimensional setting and extended to account for inhomogeneous eigenstrains in the individual phases.In the present context of linear viscoelasticity the modes can be identified following two procedures, either in each individual constituent, as initially proposed in Michel and Suquet (2003, 2004), or globally on the volume element, resulting into two slightly different models.For non aging materials, the predictions of both models are in excellent agreement with full-field simulations for various loading conditions, monotonic as well as non proportional loading, creep and relaxation. The model with global modes turns out to be as predictive as the original one with less internal variables. For aging materials, satisfactory results for the overall as well as for the local response of the composites are obtained by both models at the expense of enriching the set of modes. The prediction of the global model for the local fields is less accurate but remains acceptable. Use of the global NTFA model is therefore recommended for linear viscoelastic composites.

Theory of equilibria of elastic 2-braids with interstrand interaction

Motivated by continuum models for DNA supercoiling we formulate a theory for equilibria of 2-braids, i.e., structures formed by two elastic rods winding around each other in continuous contact and subject to a local interstrand interaction. No assumption is made on the shape of the contact curve. The theory is developed in terms of a moving frame of directors attached to one of the strands. The other strand is tracked by including in this frame the normalised closest-approach chord connecting the two strands. The kinematic constant-distance constraint is formulated at strain level through the introduction of what we call braid strains. As a result the total potential energy involves arclength derivatives of these strains, thus giving rise to a second-order variational problem. The Euler–Lagrange equations for this problem give balance equations for the overall braid force and moment referred to the moving frame as well as differential equations that can be interpreted as effective constitutive relations encoding the effect that the second strand has on the first as the braid deforms under the action of end loads. Hard contact models are used to obtain the normal contact pressure between strands that has to be non-negative for a physically realisable solution without the need for external devices such as clamps or glue to keep the strands together. The theory is first illustrated by a number of problems that can be solved analytically and then applied to several new problems that have not hitherto been treated.

Effective constitutive relations for inelastic composites

The first special boundary value problem in the mechanics of deformable solids is considered to derive the effective constitutive relations for a heterogeneous inelastic body. The problem is reduced to a number of auxiliary boundary value problems for functions dependent on the shape of the body and on the form of constitutive relations. In the case of a layer of nonuniform thickness, the problem of finding the effective constitutive relations is reduced to an operator equation whose solution is sought by an iterative method of successive approximations. An approximate analytical formula is proposed to find the effective constitutive relations for a laminated composite on the basis of known inelastic constitutive relations for its components. This approximate formula takes into account the character of structural anisotropy in a laminated composite and, in the elastic case, yields the exact values of the effective elastic modulus.

Effective Dynamic Constitutive Relations for 3-D Periodic Elastic Composites

ABSTRACTCentral to the idea of metamaterials is the concept of dynamic homogenization which seeks to define frequency dependent effective properties for Bloch wave propagation. Recent advances in the theory of dynamic homogenization have established the coupled form of the constitutive relation (Willis constitutive relation). This coupled form of the constitutive relation naturally emerges from ensemble averaging of the dynamic fields and automatically satisfies the dispersion relation in the case of periodic composites. Its importance is also notable due to its invariance under transformational acoustics. Here we discuss the explicit form of the effective dynamic constitutive equations. We elaborate upon the existence and emergence of coupling in the dynamic constitutive relation and further symmetries of the effective tensors.

Four exact relations for the effective relaxation function of linear viscoelastic composites

This study is devoted to viscoelastic composites composed of individual Maxwell constituents. The effective constitutive relations of such composites exhibit a long memory effect which manifests itself through an integral kernel (the effective relaxation function of the composite). Four asymptotic relations for this integral kernel are derived which require only the resolution of linear elastic (or purely viscous) problems. These four relations can be used in an approximate model with two relaxation times (for incompressible, isotropic composites). The model is exact for specific microstructures but is an approximation in general. Its accuracy is discussed by comparison with full-field simulations.

The construction of effective relations for waves in a composite

Implications of a recent general formulation for the effective dynamic response of a composite, in which “effective displacement” is defined as a weighted average, which could, for instance, be an average over the matrix material, are developed. A general formula is already known [J.R. Willis, Effective constitutive relations for waves in composites and metamaterials, Proc. R. Soc. A 467 (2011) 1865–1879], but it is expressed in terms of the Greenʼs function of the actual composite. A corresponding formula, expressed relative to a comparison medium, is developed here. The property of self-adjointness of the problem for the actual medium is transmitted to the corresponding problem for the “effective medium”. This permits, in the case of self-adjointness, variational characterizations of the effective response, both directly and in a formulation of “Hashin–Shtrikman” type relative to a comparison medium. The exposition is for waves in a viscoelastic composite but it applies equally to other physical examples, including electromagnetic waves.

A comparison of two formulations for effective relations for waves in a composite

Effective constitutive relations for waves in composites with random microstructure were proposed by Willis (2009) as relations between ensemble averages of stress and momentum, and “effective” strain and velocity which were related to a weighted ensemble average of displacement, and results of an example one-dimensional calculation were presented, explicitly demonstrating the possibility of coupling between mean stress and effective velocity, and mean momentum density and effective strain, even in the long-wavelength (or homogenization) limit. Relations of this type have recently been recognized to be inevitably non-unique, and a quite general prescription for defining unique relations has been advanced (Willis, 2011). The present work compares and contrasts the effective relations obtained by either formulation, for the example considered in 2009. The work of 2009 is generalized to the extent that the constituent materials are taken to have some dissipation. It emerges explicitly that an “effective elastic constant” obtained by the method of 2009 can display an apparent energy gain rather than loss. This is not the only term that contributes, however, and it is shown that the effective material remains dissipative, as it should. It is also confirmed, both theoretically and in the computation, that either formulation leads to exactly the same mean stress and momentum density, and to the same dissipation.

Estimation of material properties for linear elastic strain gradient effective media

In this paper materials with microstructures, composed of linear elastic constituents are considered. Spherical volume elements with spherical inclusions are used to obtain Voigt and Reuss bounds for the material properties of a corresponding strain gradient effective continuum. The bounds are derived following the general line of argumentation well established for simple effective continua presenting in detail the major differences arising in the case of a strain gradient continuum. Furthermore, an alternative method for the approximation of the material properties is developed, where the volume element is exposed entirely to a kinematically admissible quadratic displacement field. Since the resulting stresses do not fulfill the equilibrium conditions, a second strain field is superimposed which is approximated by a third order polynomial. The coefficients of this polynomial are determined from an energy principle together with a constraint equation in order to fulfill the equilibrium conditions in an integral sense and to assure that the resulting effective constitutive relations are insensitive to rigid body motions. Based on this approximate solution, the components of the sixth-order material tensor which relates the gradients of the macroscopic strain with the corresponding higher order stresses can be approximated analytically. In spite of the fact that these estimations are no bounds in general, useful results for the effective material parameters can be derived. This is shown by comparing the obtained approximations with the corresponding numerical results from finite element solutions of the boundary value problem for a spherical volume element considering different void volume fractions.

Fluctuations and symmetries in two-dimensional active gels

Motivated by the unique physical properties of biological active matter, e.g., cytoskeletal dynamics in eukaryotic cells, we set up effective two-dimensional (2d) coarse-grained hydrodynamic equations for the dynamics of thin active gels with polar or nematic symmetries. We use the well-known three-dimensional (3d) descriptions (K. Kruse et al., Eur. Phys. J. E 16, 5 (2005); A. Basu et al., Eur. Phys. J. E 27, 149 (2008)) for thin active-gel samples confined between parallel plates with appropriate boundary conditions to derive the effective 2d constitutive relations between appropriate thermodynamic fluxes and generalised forces for small deviations from equilibrium. We consider three distinct cases, characterised by spatial symmetries and boundary conditions, and show how such considerations dictate the structure of the constitutive relations. We use these to study the linear instabilities, calculate the correlation functions and the diffusion constant of a small tagged particle, and elucidate their dependences on the activity or nonequilibrium drive.

Effective constitutive relations for waves in composites and metamaterials

Definitions of ‘effective fields’ for a randomly inhomogeneous material are offered, which guarantee automatic satisfaction of the equations of motion. The important case of a medium with periodic microstructure is included. In this special case, the definitions are completely explicit and can be applied without reference to random media. The presentation is mostly expressed in terms of electromagnetic waves. The reasoning is applicable also to other types of waves and its realization for elastodynamics is briefly outlined towards the end. Some of the effective fields are defined directly as ensemble averages, ensuring the exact satisfaction of the equations of motion, but the effective ‘kinematic’ fields to which they are related are defined more generally, as weighted averages. The main result of this work is an explicit formula for the tensor of effective properties. The important issue of uniqueness (or not) of the effective properties is explained and resolved. Self-adjointness of the original problem is not assumed. An attractive feature of the formulation is that self-adjointness at the local level implies self-adjointness at the level of the ‘effective medium’.

Damage Micromechanics for Constitutive Relations and Failure of Microcracked Quasi-Brittle Materials

This article addresses several issues in damage micromechanics for microcracked brittle or quasi-brittle solids, with major attention paid on summarizing some recent works of the present authors. First, we discuss the methods for characterizing evolutionary damage of microcracks and introduce the concept of orientation domain of microcrack growth. Second, we describe how to calculate the macroscopically effective constitutive relation of microcracking solids under complex loading. Third, the effective medium or effective field methods for estimating the effective elastic moduli of microcracked solids are formulated into a universal framework. Fourth, a subregion method is suggested to calculate the effective elastic moduli, fracture strength, and failure process of solids containing a large number and/or a high density of microcracks. Additionally, this method is combined with the finite element method to simulate the damage evolution and failure process of engineering structures.

Porous materials with two populations of voids under internal pressure: I. Instantaneous constitutive relations

This study is devoted to the mechanical behavior of uranium dioxide (UO 2) which is a porous material with two populations of voids of very different size subjected to internal pressure. The smallest voids are intragranular and spherical in shape whereas the largest pores located at the grain boundary are ellipsoidal and randomly oriented. In this first part of the study, attention is focused on the effective properties of these materials with fixed microstructure. In a first step, the poro-elastic properties of these doubly voided materials are studied. Then two rigorous upper bounds are derived for the effective poro-plastic constitutive relations of these materials. The first bound, obtained by generalizing the approach of Gologanu et al. (Gologanu, M., Leblond, J., Devaux, J., 1994. Approximate models for ductile metals containing non-spherical voids-case of axisymmetric oblate ellipsoidal cavities. ASME J. Eng. Mater. Technol. 116, 290–297) to compressible materials, is accurate at high stress-triaxiality. The second one, which derives from the variational method of Ponte Castañeda (Ponte Castañeda, P., 1991. The effective mechanical properties of non-linear isotropic composites. J. Mech. Phys. Solids 39, 45–71), is accurate when the stress triaxiality is low. A N-phase model, inspired by Bilger et al. (Bilger, N., Auslender, F., Bornert, M., Masson, R., 2002. New bounds and estimates for porous media with rigid perfectly plastic matrix. C.R. Mécanique 330, 127–132), is proposed which matches the best of the two bounds at low and high triaxiality. The effect of internal pressures is discussed. In particular it is shown that when the two internal pressures coincide, the effective flow surface of the saturated biporous material is obtained from that of the drained material by a shift along the hydrostatic axis. However, when the two pressures are different, the modifications brought to the effective flow surface in the drained case involve not only a shift along the hydrostatic axis but also a change in shape and size of the surface.

Plastic yield and collapse mechanism of planar lattice structures

Lattice structures possess a huge potential for energy absorbing applications, and the postinitial collapse region should be analyzed with respect to design principles in such cases. This paper presents an analytical method to calculate the ultimate yield surfaces of statically indeterminate planar lattice structures, based on the assessment of static equilibrium of the unit cell before and after initial yielding. The material of the unit cell wall is assumed to be elastic, perfectly plastic. Three statically indeterminate planar lattice structures: the diamond cross cell, the statically-indeterminate square cell (SI-square cell), the new Kagome cell (N-Kagome), are analyzed. The parametric studies reveal the roles of various geometrical parameters on the performance of each structure. The SI-square cell is utilized as an example to demonstrate the evolution of structural yielding, thus providing an insight into the collapse mode of lattice structures. Furthermore, the stress-strain relationships of the SI-square and N-Kagome cells are also calculated, and the effective constitutive relations of both lattices are found to be linearly hardening, which is validated by finite element (FE) simulations.

Minimum weights of pressurized hollow sandwich cylinders with ultralight cellular cores

Long, open-ended, hollow sandwich cylinders with ultralightweight cellular cores are optimized under uniform internal pressure for minimum weight design. Five different core topologies are considered: Kagomé truss, single-layered pyramidal truss, double-layered pyramidal truss, single-layered corrugated core and double-layered corrugated core. The highly porous cellular materials are homogenized to obtain effective constitutive relations. Close-formed solutions are presented for the forces and stresses in individual structural members of the sandwich, which are then validated by finite element calculations. Optimization of the sandwich-walled hollow cylinder is achieved using a quadratic optimizer, subjected to the constraints that none of the following failure modes occurs: facesheet yielding; facesheet punch shearing (active only for truss-cored sandwiches); core member buckling; core member yielding. In comparison with hollow cylinders having solid walls, truss-core sandwich cylinders and single-layer corrugated core sandwich cylinders are found to have superior weight advantages, especially for more heavily loaded cases. With the consideration of both weight efficiency and failure modes, sandwich-walled hollow cylinders having Kagomé truss core with pyramidal sub-geometry have the best overall performance in comparison with other core topologies.

Parallel 3-D Simulation of a Fault Gouge Using the Lattice Solid Model

Despite the insight gained from 2-D particle models, and given that the dynamics of crustal faults occur in 3-D space, the question remains, how do the 3-D fault gouge dynamics differ from those in 2-D? Traditionally, 2-D modeling has been preferred over 3-D simulations because of the computational cost of solving 3-D problems. However, modern high performance computing architectures, combined with a parallel implementation of the Lattice Solid Model (LSM), provide the opportunity to explore 3-D fault micro-mechanics and to advance understanding of effective constitutive relations of fault gouge layers. In this paper, macroscopic friction values from 2-D and 3-D LSM simulations, performed on an SGI Altix 3700 super-cluster, are compared. Two rectangular elastic blocks of bonded particles, with a rough fault plane and separated by a region of randomly sized non-bonded gouge particles, are sheared in opposite directions by normally-loaded driving plates. The results demonstrate that the gouge particles in the 3-D models undergo significant out-of-plane motion during shear. The 3-D models also exhibit a higher mean macroscopic friction than the 2-D models for varying values of interparticle friction. 2-D LSM gouge models have previously been shown to exhibit accelerating energy release in simulated earthquake cycles, supporting the Critical Point hypothesis. The 3-D models are shown to also display accelerating energy release, and good fits of power law time-to-failure functions to the cumulative energy release are obtained.

Broad-bandwidth and low-loss metamaterials: Theory, design and realization

In this paper, we summarize some recent activities in the field of metamaterial research at the National University of Singapore (NUS). Integral equations are applied for electromagnetic modelling of supernatural materials. Some special characteristics of the metamaterials are shown. Moreover, quasi-static Lorentz theory and numerical method (i.e., the method of moments for solving the electric field integral equation) and the transmission line theory are both presented to obtain the effective constitutive relations of metamaterials, respectively. Finally, feasibility of fabricating metamaterials based on analysis of equivalent transmission line model in the microwave spectrum and even higher is also shown and correspondingly some broad-bandwidth and low-loss metamaterial structures are designed and synthesized.

Elasticity of planar fiber networks

A micromechanics model is proposed for the elasticity of planar fiber networks (FNs). The FN is created by random deposition of linearly elastic straight rods within a region. The rods are bonded rigidly at contacts. Under external in-plane loading, the FN deformation consists of fiber bending, elongation, and contraction. An effective constitutive relation for fiber network is developed by averaging the strain energy dissipated by all possible fiber deformations in all directions. Numerical calculations are performed to analyze the effects of fiber aspect ratio and fiber concentration on the effective stiffness of the planar random FN. Finite element analysis (FEA) is performed and compared with the theoretical predictions of the effective FN moduli at several fiber concentrations. FEA results are in good agreement with theoretical predictions. The present model can be used for the prediction of mechanical properties, scaling analysis, and optimization of fiber assemblies.