New integral formulation and self-consistent modeling of elastic–viscoplastic heterogeneous materials

Predicting the overall behavior of heterogeneous materials, from their local properties at the scale of heterogeneities, represents a critical step in the design and modeling of new materials. Within this framework, an internal variables approach for scale transition problem in elastic–viscoplastic case is introduced. The proposed micromechanical model is based on establishing a new system of field equations from which two Navier’s equations are obtained. Combining these equations leads to a single integral equation which contains, on the one hand, modified Green operators associated with elastic and viscoplastic reference homogeneous media, and secondly, elastic and viscoplastic fluctuations. This new integral equation is thus adapted to self-consistent scale transition methods. By using the self-consistent approximation we obtain the concentration law and the overall elastic–viscoplastic behavior of the material. The model is first applied to the case of two-phase materials with isotropic, linear and compressible viscoelastic properties. Results for elastic–viscoplastic two-phase materials are also presented and compared with exact results and variational methods.     

A new class of micro–macro models for elastic–viscoplastic heterogeneous materials

The determination of the effective behavior of heterogeneous materials from the properties of the components and the microstructure constitutes a major task in the design of new materials and the modeling of their mechanical behavior. In real heterogeneous materials, the simultaneous presence of instantaneous mechanisms (elasticity) and time dependent ones (non-linear viscoplasticity) leads to a complex space–time coupling between the mechanical fields, difficult to represent in a simple and efficient way. In this work, a new self-consistent model is proposed, starting from the integral equation for a translated strain rate field. The chosen translated field is the (compatible) viscoplastic strain rate of the (fictitious) viscoplastic heterogeneous medium submitted to a uniform (unknown) boundary condition. The self-consistency condition allows to define these boundary conditions so that a relative simple and compact strain rate concentration equation is obtained. This equation is explained in terms of interactions between an inclusion and a matrix, which lead to interesting conclusions. The model is first applied to the case of two-phase composites with isotropic, linear and incompressible viscoelastic properties. In that case, an exact self-consistent solution using the Laplace–Carson transform is available. The agreement between both approaches appears quite good. Results for elastic–viscoplastic BCC polycrystals are also presented and compared with results obtained from Kröner–Weng's and Paquin et al. (Arch. Appl. Mech. 69 (1999) 14)'s model.     

A micromechanical model of elastoplastic and damage behavior of a cohesive geomaterial

The present study is devoted to the development and validation of a nonlinear homogenization approach of the mechanical behavior of Callovo-Oxfordian argillites. The material is modeled as an heterogeneous composite composed of an elastoplastic clay matrix and of linear elastic or elastic damage inclusions. The macroscopic constitutive law is obtained by adapting the incremental method proposed by Hill [Hill, R., 1965. Continuum micro-mechanics of elastoplastic polycrystals. J. Mech. Phys. Solids 13, 89–101]. The approach consists in formulating the macroscopic tangent operator of the material by considering the nonlinear local behavior of each phase. Due to the matrix/inclusion morphology of the microstructure of the argillite, a Mori–Tanaka scheme is considered for the localization step. The developed model is first compared to Finite Element calculations and then validated and applied for the prediction of the macroscopic stress–strain responses of argillites.     

Sequential linearization method for viscous/elastic heterogeneous materials

The paper addresses the problem of suitable approximation of the interaction between phases in heterogeneous materials that exhibit both viscous and elastic properties. A novel approach is proposed in which linearized subproblems for an inhomogeneity–matrix system with viscous or elastic interaction rules are solved sequentially within one incremental step. It is demonstrated that in the case of a self-consistent averaging scheme, an additional accommodation subproblem, besides purely viscous and elastic subproblems, is to be solved in order to estimate the material response satisfactorily. By examples of an isotropic two-phase material it is shown that the proposed approach provides acceptable predictions in comparison with the existing models.     

Time domain modeling of damping using anelastic displacement fields and fractional calculus

A fractional derivative model of linear viscoelasticity based on the decomposition of the displacement field into an anelastic part and elastic part is developed. The evolution equation for the anelastic part is then a differential equation of fractional order in time. By using a fractional order evolution equation for the anelastic strain the present model becomes very flexible for describing the weak frequency dependence of damping characteristics. To illustrate the modeling capability, the model parameters are fit to available frequency domain data for a high damping polymer. By studying the relaxation modulus and the relaxation spectrum the material parameters of the present viscoelastic model are given physical meaning. The use of this viscoelastic model in structural modeling is discussed and the corresponding finite element equations are outlined, including the treatment of boundary conditions. The anelastic displacement field is mathematically coupled to the total displacement field through a convolution integral with a kernel of Mittag–Leffler function type. Finally a time step algorithm for solving the finite element equations are developed and some numerical examples are presented.     

Micromechanical modeling of linear viscoelastic behavior of heterogeneous materials

Study of effective behavior of heterogeneous materials, starting from the properties of the microstructure, represents a critical step in the design and modeling of new materials. Within this framework, the aim of this work is to introduce a general internal variables approach for scale transition problem in linear viscoelastic case. A new integral formulation is established, based on the complete taking into account of field equations and differential constitutive laws of the heterogeneous problem, in which the effects of elasticity and viscosity interact in a representative volume element. Thanks to Green’s techniques applied to space convolution’s term, a new concentration relation is obtained. The step of homogenization is then carried out according to the self-consistent approximation. The results of the present model are illustrated and compared with those provided by Hashin’s and Rougier’s ones, considered as references, and by internal variables models such as those of Weng and translated fields.     

Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements

In this work, a modeling of electroelastic composite materials is proposed. The extension of the heterogeneous inclusion problem of Eshelby for elastic to electroelastic behavior is formulated in terms of four interaction tensors related to Eshelby’s electroelastic tensors. Analytical formulations of interaction tensors are presented for ellipsoidal inclusions. These tensors are basically used to derive the self-consistent model, Mori–Tanaka and dilute approaches. Numerical solutions are based on numerical computations of these tensors for various types of inclusions. Using the obtained results, effective electroelastic moduli of piezoelectric multiphase composites are investigated by an iterative procedure in the context of self-consistent scheme. Generalised Mori–Tanaka’s model and dilute approach are re-formulated and the three models are deeply analysed. Concentration tensors corresponding to each model are presented and relationships of effective coefficients are given. Numerical results of effective electroelastic moduli are presented for various types of piezoelectric inclusions and for various orientations and compared to existing experimental and theoretical ones.     

Residual stresses in random elastic composites: nonlocal micromechanics-based models and first estimates of the representative volume element size

Random elastic composites with residual stresses are examined in this paper with the aim of understanding how the prestress may influence the overall mechanical properties of the composite. A fully non-local effective response is found in perfect analogy with the un-prestressed case examined in (Drugan and Willis, J. Mech. Phys. Solids 44(4):497–524, 1996). The second gradient approximation is considered and the impact of the residual stresses on the estimate of the RVE size is studied whenever the local response is used to describe the mechanical properties of the heterogeneous medium. To this aim, total and incremental formulations are worked out in this paper and the influence of both uniform and spatially varying prestresses are studied. Among other results, it is shown how rapid oscillations of relatively “small” residual stresses in most cases may result in the impossibility of describing the overall behavior of the composite with a local constitutive equation. On the other hand, prestresses with relatively high amplitudes and slow spatial oscillations may even reduce the RVE size required for approximating the mechanical properties of un-prestressed heterogeneous media with a local constitutive equation.     

Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic–plastic bar

We propose a fundamentally new concept to the treatment of material instabilities and localization phenomena based on energy minimization principles in a strain-softening elastic–plastic bar. The basis is a recently developed incremental variational formulation of the local constitutive response for generalized standard media. It provides a quasi-hyperelastic stress potential that is obtained from a local minimization of the incremental energy density with respect to the internal variables. The existence of this variational formulation induces the definition of the material stability of inelastic solids based on convexity properties in analogy to treatments in elasticity. Furthermore, localization phenomena are understood as micro-structure development associated with a non-convex incremental stress potential in analogy to phase decomposition problems in elasticity. For the one-dimensional bar considered the two-phase micro-structure can analytically be resolved by the construction of a sequentially weakly lower semicontinuous energy functional that envelops the not well-posed original problem. This relaxation procedure requires the solution of a local energy minimization problem with two variables which define the one-dimensional micro-structure developing: the volume fraction and the intensity of the micro-bifurcation. The relaxation analysis yields a well-posed boundary-value problem for an objective post-critical localization analysis. The performance of the proposed method is demonstrated for different discretizations of the elastic–plastic bar which document on the mesh-independence of the results.     

Homogenization of the viscoelastic heterogeneous materials with multi-coated reinforcements: an internal variables formulation

In this work, a new homogenization method to estimate the effective behavior of viscoelastic heterogeneous materials with multi-coated reinforcements is presented. Unlike classical methods that are based on the Laplace transform, the present internal variables formulation operates directly in the time domain. Using the Green’s function techniques, the micromechanical approach is based on establishing a new integral equation adapted to scale transition methods. Using this integral equation, we apply a generalized self-consistent scheme to determine the local stress concentration equations and the effective behavior of multi-coated inclusion-reinforced materials. To assess the reliability of our model, some applications to the isotropic viscoelastic heterogeneous materials with homothetic spherical inclusions are given. The model is applied to the case of two-phase and three-phase materials, and the results are compared to exact solutions. Results for three-phase materials are presented regarding the influence of soft and stiff viscoelastic interphase on the effective behavior of heterogeneous materials.     

A continuum model for the mechanical behavior of nanowires including surface and surface-induced initial stresses

The continuum modeling of the mechanical behavior of nanowires has recently attracted much attention due to its simplicity and efficiency. However, there are still some critical issues to be solved. In this paper, we demonstrate the importance of accounting for the effects of initial stresses in the nanowires that are caused by deformation due to surface stresses; we note that such initial stresses have previously been neglected in most existing continuum models. By considering the local geometrical nonlinearity of strains during the incremental flexural motion, a new formulation of the Euler–Bernoulli beam model for nanowires is developed through the incremental deformation theory, in which effects of the surface stress, the surface-induced initial stress and surface elasticity are naturally incorporated. It is found through comparisons to existing experimental and computational results for both fcc metal and ceramic nanowires that the surface-induced initial stresses, which are neglected in the Young–Laplace model, can significantly influence the overall mechanical properties of nanowires. We additionally demonstrate and quantify the errors induced by using the Young–Laplace model due to its approximation of surface stresses acting on only the top and bottom surfaces of nanowires.     

Interacting elliptic inclusions by the method of complex potentials

An accurate series solution has been obtained for a piece-homogeneous elastic plane containing a finite array of non-overlapping elliptic inclusions of arbitrary size, aspect ratio, location and elastic properties. The method combines standard Muskhelishvili’s representation of general solution in terms of complex potentials with the superposition principle and newly derived re-expansion formulae to obtain a complete solution of the many-inclusion problem. By exact satisfaction of all the interface conditions, a primary boundary-value problem stated on a complicated heterogeneous domain has been reduced to an ordinary well-posed set of linear algebraic equations. A properly chosen form of potentials provides a remarkably simple form of solution and thus an efficient computational algorithm. The theory developed is rather general and can be applied to solve a variety of composite mechanics problems. The advanced models of composite involving up to several hundred inclusions and providing an accurate account for the microstructure statistics and fiber–fiber interactions can be considered in this way. The numerical examples are given showing high accuracy and numerical efficiency of the method developed and disclosing the way and extent to which the selected structural parameters influence the stress concentration at the matrix–inclusion interface.     

Identification of elastic property and loading fields from full-field displacement measurements

A method is introduced to identify simultaneously elastic properties and loading fields from a measured displacement field. Since the mechanical behavior of micro-electro-mechanical systems (MEMS) is governed by surface effects, this type of identification tool is thought to be of major interest. However, increasing the number of parameters to retrieve affects the redundancy necessary for an accurate identification. A finite-element formulation of a distance between measured and statically admissible (SA) displacement fields is shown to be equivalent to a standard least-squares distance to kinematically admissible (KA) fields if the used modeling is suitable. Any deviation from this equivalence is then the signature of a modeling error. Balancing the distance to KA and SA displacement fields allows one to retrieve unknown modeling parameters. This method is detailed on heterogeneous Euler–Bernoulli beams submitted to an unknown loading field and applied to experimental displacement fields of micro-cantilevers obtained with an electrostatic set-up. An elastic property field and a parameterized loading field are then identified, and the quality of the identification is assessed.     

A variational formulation for the incremental homogenization of elasto-plastic composites

This work addresses the micro–macro modeling of composites having elasto-plastic constituents. A new model is proposed to compute the effective stress–strain relation along arbitrary loading paths. The proposed model is based on an incremental variational principle (Ortiz, M., Stainier, L., 1999. The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Eng. 171, 419–444) according to which the local stress–strain relation derives from a single incremental potential at each time step. The effective incremental potential of the composite is then estimated based on a linear comparison composite (LCC) with an effective behavior computed using available schemes in linear elasticity. Algorithmic elegance of the time-integration of J₂ elasto-plasticity is exploited in order to define the LCC. In particular, the elastic predictor strain is used explicitly. The method yields a homogenized yield criterion and radial return equation for each phase, as well as a homogenized plastic flow rule. The predictive capabilities of the proposed method are assessed against reference full-field finite element results for several particle-reinforced composites.     

Stress concentrations in the particulate composite with transversely isotropic phases

The accurate series solution have been obtained of the elasticity theory problem for a transversely isotropic solid containing a finite or infinite periodic array of anisotropic spherical inclusions. The method of solution has been developed based on the multipole expansion technique. The basic idea of method consists in expansion the displacement vector into a series over the set of vectorial functions satisfying the governing equations of elastic equilibrium. The re-expansion formulae derived for these functions provide exact satisfaction of the interfacial boundary conditions. As a result, the primary spatial boundary-value problem is reduced to an infinite set of linear algebraic equations. The method has been applied systematically to solve for three models of composite, namely a single inclusion, a finite array of inclusions and an infinite periodic array of inclusions, respectively, embedded in a transversely isotropic solid. The numerical results are presented demonstrating that elastic properties mismatch, anisotropy degree, orientation of the anisotropy axes and interactions between the inclusions can produce significant local stress concentration and, thus, affect greatly the overall elastic behavior of composite.     

Multi-scale modeling of grouted sand behavior

The mechanical properties of sand: stiffness, cohesion and, to a less extent, friction angle can be increased through the process of grouting. A constitutive model adapted for cohesive-frictional materials from a homogenization technique which allowed us to integrate constitutive relations at the grain level has been developed to obtain constitutive equations for the equivalent continuous granular medium. A representative volume was obtained by mobilizing particle contacts in all orientations. Thus, the stress–strain relationship could be derived as an average of the behavior of these local contact planes. The local behavior was assumed to obey a stress-dependent elastic law and Mohr–Coulomb’s plastic law. The influence of the cement grout was modeled by means of adhesive forces between grains in contact, which were added to the contact forces created by an external load. The intensity of these adhesive forces is a function of nature and amount of grout present inside the material and can be reduced due to a damage mechanism at the grain contact during loading. In this paper, we present several examples of simulation which show that the model can reproduce with sufficient accuracy the mechanical improvement induced by grouting as well as the damage of the grain cementation during loading.     

Numerical calculation of the elastic and plasticbehavior of saturated porous media

In this investigation the field equations governing the mechanical behavior of a fluid-saturated porous media are analyzed and built up for the study of elastic dynamical problems and quasi-static problems in case of elastic–plastic material behavior. The investigations are limited to small deformation in order to apply a geometrical linear approach. The two constituents are assumed to be microscopically incompressible. A numerical solution is derived by means of the standard Galerkin procedure and the finite element method.     

Micromechanical modeling of the elastic–viscoplastic behavior of polycrystalline steels

A self-consistent model developed to describe the elastic–viscoplastic behavior of heterogeneous materials is applied to low carbon steels to simulate tensile tests at various strain rates in the low temperature range. The choice of crystalline laws implemented in the model is discussed through the viscoplastic flow rule and several strain-hardening laws. Comparisons between three work-hardening models show that the account of dislocation annihilation improves the results on simulations at large strains. The evolution of the Lankford coefficients and texture development are also successfully simulated. Some microstructural aspects of deformation such as the stored energy and the evolution of the flow rates are discussed. By including the dislocation density on each slip system as internal variable, intragranular heterogeneities are underscored.     

Micro–macro analysis of shape memory alloy composites

The aim of the paper is to develop a micro–macro approach for the analysis of the mechanical behavior of composites obtained embedding long fibers of Shape Memory Alloys (SMA) into an elastic matrix. In order to determine the overall constitutive response of the SMA composites, two homogenization techniques are proposed: one is based on the self-consistent method while the other on the analysis of a periodic composite. The overall response of the SMA composites is strongly influenced by the pseudo-elastic and shape memory effects occurring in the SMA material. In particular, it is assumed that the phase transformations in the SMA are governed by the wire temperature and by the average stress tensor acting in the fiber. A possible prestrain of the fibers is taken into account in the model.Numerical applications are developed in order to analyze the thermo-mechanical behavior of the SMA composite. The results obtained by the proposed procedures are compared with the ones determined through a micromechanical analysis of a periodic composite performed using suitable finite elements.Then, in order to study the macromechanical response of structural elements made of SMA composites, a three-dimensional finite element is developed implementing at each Gauss point the overall constitutive laws of the SMA composite obtained by the proposed homogenization procedures. Some numerical applications are developed in order to assess the efficiency of the proposed micro–macro model.