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.     

Determination of the effective elastic–plastic response of metal–ceramic composites

The mechanical response of metal–ceramic composites is analysed through a homogenization model accounting for the mechanical behaviour of the constituent materials. In order to achieve this purpose a nonlinear homogenization method based on the phase field approach has been suitably implemented into a numerical code. A prescribed homogenized strain state is applied to a unit volume element of a metal–ceramic composite with proportional loading in which all components of the strain tensor are proportional to one scalar parameter. The mechanical response of the material has been modeled by considering a von Mises plasticity model for the metal phase and a Drucker–Prager associative elastic–plastic material model for the ceramic phase. A two stages plasticity has been obtained in which inelastic strain develops in the metal phase followed by a fully plastic response. A comparison with a finite element model of the stress–strain response of an axisymmetric unit cell has been carried out with the purpose to validate the homogenization based modeling presented in the paper. Plastic parameters of a Drucker–Prager yield surface for the homogenized composite have been calculated at different materials compositions. Associative Drucker–Prager plasticity has been found to be accurate for high ceramic content.     

Micromechanical analysis of polymer composites reinforced by unidirectional fibres: Part II – Micromechanical analyses

This paper presents the application of a new constitutive damage model for an epoxy matrix on micromechanical analyses of polymer composite materials. Different representative volume elements (RVEs) are developed with a random distribution of fibres. Upon application of periodic boundary conditions (PBCs) on the RVEs, different loading scenarios are applied and the mechanical response of the composite studied. Focus is given to the influence of the interface between fibre and matrix, as well as to the influence of the epoxy matrix, on the strength properties of the composite, damage initiation and propagation under different loading conditions.     

A combined viscoelastic–viscoplastic behavior of particle reinforced composites

This study formulates a micromechanical model for predicting effective viscoelastic–viscoplastic responses of composites. The studied composites consist of solid spherical particle reinforcements dispersed in a homogeneous matrix. The particle constituent is assumed linear elastic, while the matrix exhibits combined viscoelastic–viscoplastic responses. The Schapery integral model is used for the 3D isotropic non-linear viscoelastic responses. Two viscoplastic models are considered: the Perzyna model, having a rate-independent yield surface and an overstress function, and the Valanis endochronic model based on an irreversible thermodynamics without a yield surface. The Valanis model is suitable for materials when viscoplastic responses occur at early loadings (small stress levels). A unit-cell model with four particle and polymer sub-cells is generated to obtain homogenized responses of the particle-reinforced composites. Available micromechanical models and experimental data in the literature are used to verify the proposed micromechanical model in predicting effective time-dependent and inelastic responses of composites. Field variables in the homogenized composites are compared to the ones in heterogeneous composites. The heterogeneous composites, having detailed particle geometries, are modeled using finite element (FE) method.     

Homogenization of elastic–viscoplastic heterogeneous materials: Self-consistent and Mori-Tanaka schemes

This paper deals with the prediction of the macroscopic behavior of a multiphase elastic–viscoplastic material. The proposed homogenization schemes are based on an interaction law postulated by Molinari et al. [Molinari, A., Ahzi, S., Kouddane, R. 1997. On the self-consistent modelling of elastic–plastic behavior of polycrystals. Mech. Mater., 26, 43–62]. Self-consistent schemes are developed to describe the behavior of disordered aggregates. The Mori-Tanaka approach is used to capture the behavior of composite materials, where one phase can be clearly identified as the matrix. The proposed schemes are developed within a general framework where compressible elasticity and anisotropy of the materials are taken into account. Inclusions can have various shapes and orientations. Illustrations of the homogenization procedure are given for a two-phase composite materials. Comparisons between results of the literature and predictions based on the interaction law are performed and have demonstrated the efficiency of the proposed homogenization schemes.     

Homogenized elastic–viscoplastic behavior of anisotropic open-porous bodies with pore pressure

Constitutive modeling is studied for the homogenized elastic–viscoplastic behavior of pore-pressurized anisotropic open-porous bodies made of metallic base solids at small strains and rotations. For this purpose, by describing micro–macro relations relevant to periodic unit cells of anisotropic open-porous bodies subjected to pore pressure, constitutive features are discussed for the viscoplastic macrostrain rate in steady states. On the basis of the constitutive features found, the viscoplastic macrostrain rate is represented as an anisotropic function of Terzaghi’s effective stress, which is shown using Hill’s macrohomogeneity condition. The resulting viscoplastic equation is used to simulate the homogenized elastic–viscoplastic behavior of an ultrafine plate-fin structure subjected to uniaxial/biaxial loading in addition to pore pressure. The corresponding finite element homogenization analysis is also performed for comparison. It is demonstrated that the developed viscoplastic equation simulates well the anisotropic effect of pore pressure in the viscoplastic range in spite of there being no anisotropic factor and no fitting parameter in Terzaghi’s effective stress itself.     

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.     

An anisotropic elastic–viscoplastic model for soft clays

Experimental evidences have shown deficiencies of the existing overstress and creep models for viscous behaviour of natural soft clay. The purpose of this paper is to develop a modelling method for viscous behaviour of soft clays without these deficiencies. A new anisotropic elastic–viscoplastic model is extended from overstress theory of Perzyna. A scaling function based on the experimental results of constant strain-rate oedometer tests is adopted, which allows viscoplastic strain-rate occurring whether the stress state is inside or outside of the yielding surface. The inherent and induced anisotropy is modelled using the formulations of yield surface with kinematic hardening and rotation (S-CLAY1). The parameter determination is straightforward and no additional experimental test is needed, compared to the Modified Cam Clay model. Parameters determined from two types of tests (i.e., the constant strain-rate oedometer test and the 24 h standard oedometer test) are examined. Experimental verifications are carried out using the constant strain-rate and creep tests on St. Herblain clay. All comparisons between predicted and measured results demonstrate that the proposed model can successfully reproduce the anisotropic and viscous behaviours of natural soft clays under different loading conditions.     

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.     

Comparative analysis of the two‐constant generalizations of hooke's law for isotropic elastic materials at finite strains

For ten models of the isothermal behavior of materials, the solutions of boundaryvalue problems are studied for five types of the experimentally reproducible uniform stress–strain state with unchanged directions of the principal axes. It is found that, for three models, the governing equations are similar to the relations of Hooke's law and valid within the same range of the ratio between the shear and bulk moduli. In these models, the specific strain energy can be represented as a sum of the energies due to changes in volume and shape. The ranges where the other three known models exhibit incorrect behavior are determined.     

A new formulation of the effective elastic–plastic response of two-phase particulate composite materials

In this paper the double-inclusion model, originally developed to determine effective linear elastic properties of composite materials, is reformulated and extended to predict the effective nonlinear elastic–plastic response of two-phase particulate composites reinforced with spherical particles. The resulting problem of elastic–plastic deformation of a double-inclusion embedded in an infinite reference medium subjected to an incrementally applied far-field strain is solved by the finite element method. The proposed double-inclusion model is evaluated by comparison of the model predictions to the available exact results obtained by the direct approach using representative volume elements containing many particles. It is found that the double-inclusion formulation is capable of providing accurate prediction of the effective elastic–plastic response of two-phase particulate composites at moderate particle volume fractions.     

Micromechanical analysis of polymer composites reinforced by unidirectional fibres: Part I – Constitutive modelling

Micromechanical analyses of unidirectional continuous-fibre reinforced composite materials were performed to study the mechanisms of deformation and fracture of the constituents, and their influence on the mechanical properties of the composite. Special focus was given to the matrix material behaviour as well as to the interface between constituents. The matrix was modelled using a pressure dependent, elasto-plastic thermodynamically consistent damage model. Cohesive elements were used to model the interface between matrix and fibres. Part I of this paper details the continuum model developed for a typical epoxy matrix. Part II will focus on micromechanical analyses of composite materials and the estimation of its elastic and strength properties.     

Evolution of plastic zones in dynamically loaded plates using different elastic–viscoplastic laws

In the framework of viscoplastic theory many different laws were developed, accounting for material behaviors like creep, relaxation or evolution of overstresses. Though each model is able to predict in uni-axial material tests the values of stresses depending on plastic strains and plastic strain rates the question is if solutions of simulations are still realistic if the viscoplastic law is applied on structural deformations. In the present study strain rate sensitive metal plates are subjected to shock waves. The purpose is to compare simulation results obtained with different elastic–viscoplastic laws to experiments in order to determine the most appropriate material model. By subjecting circular metal plates experimentally to shock wave loadings realistic deformation histories are measured. The measurements are compared to simulation results obtained with different viscoplastic laws. The aim is to find out the accuracy of each model concerning the predictions of displacements, shape formings, spread of plastic zones and evolutions of inner bending moments.     

The effect of moisture-induced swelling on the absorption capacity of transversely isotropic elastic polymer–matrix composites

The interaction between humid air and transversely isotropic fiber-reinforced composites with swelling polymeric matrix is considered. A model is proposed for the water saturation level in a polymer when stresses are applied, that uses directly obtainable material parameters only. In a composite, the reinforcements modify the water uptake of the polymer matrix because of the internal stresses that are induced by its restricted swelling, and this effect is evaluated. As a consequence of the coupling between stresses and absorption capacity, the sorption isotherm of a composite is ruled by the (non-linear) Langmuir equation when the unreinforced matrix obeys the (linear) Henry’s law.     

A thermo-mechanically-coupled theory accounting for hydrogen diffusion and large elastic–viscoplastic deformations of metals

In this paper we develop a thermodynamically-consistent coupled-theory which accounts for diffusion of hydrogen, diffusion of heat, and large elastic–viscoplastic deformations of metals. The theory should be of utility in the analysis of hydrogen diffusion in elastic–plastically-deforming solids, an analysis which is an essential prerequisite for theoretical and numerical efforts aimed at modeling the integrity of structural components used for hydrogen gas storage and distribution.     

A finite element analysis of the morphology of the twinned-to-detwinned interface observed in microstructure of the Cu–Al–Ni shape memory alloy

A detailed morphology of the twinned-to-detwinned interface in microstructure of 2H-martensite phase of the Cu–Al–Ni shape memory alloy is observed by optical methods (optical microscopy, white-light interferometry). Based on these observations, a finite element model of the transition layer is constructed and applied to calculate the elastic stress distribution inside the observed microstructure. The results show that the real micromorphology does not correspond to the minimum of the sum of the elastic and surface energy, and that the maxima of the stress field necessary for the existence of this morphology are comparable to the elasticity limits of the 2H-martensite.     

Micromechanical models to guide the development of synthetic ‘brick and mortar’ composites

This paper describes a micromechanical analysis of the uniaxial response of composites comprising elastic platelets (bricks) bonded together with thin elastic perfectly plastic layers (mortar). The model yields closed-form results for the spatial variation of displacements in the bricks as a function of constituent properties, which can be used to calculate the effective properties of the composite, including elastic modulus, strength and work-to-failure. Regime maps are presented which indicate critical stresses for failure of the bricks and mortar as a function of constituent properties and brick architecture. The solution illustrates trade-offs between elastic modulus, strength and dissipated work that are a result of transitions between various failure mechanisms associated with brick rupture and rupture of the interfaces. Detailed scaling relationships are presented with the goal of providing material developers with a straightforward means to identify synthesis targets that balance competing mechanical behaviors and optimize material response. Ashby maps are presented to compare potential brick and mortar composites with existing materials, and identify future directions for material development.     

Variational methods for the steady state response of elastic–plastic solids subjected to cyclic loads

Solids (or structures) of elastic–plastic internal variable material models and subjected to cyclic loads are considered. A minimum net resistant power theorem, direct consequence of the classical maximum intrinsic dissipation theorem of plasticity theory, is envisioned which describes the material behavior by determining the plastic flow mechanism (if any) corresponding to a given stress/hardening state. A maximum principle is provided which characterizes the optimal initial stress/hardening state of a cyclically loaded structure as the one such that the plastic strain and kinematic internal variable increments produced over a cycle are kinematically admissible. A steady cycle minimum principle, integrated form of the aforementioned minimum net resistant power theorem, is provided, which characterizes the structure’s steady state response (steady cycle) and proves to be an extension to the present context of known principles of perfect plasticity. The optimality equations of this minimum principle are studied and two particular cases are considered: (i) loads not exceeding the shakedown limit (so recovering known results of shakedown theory) and (ii) specimen under uniform cyclic stress (or strain). Criteria to assess the structure’s ratchet limit loads are given. These, together with some insensitivity features of the structure’s alternating plasticity state, provide the basis to the ratchet limit load analysis problem, for which solution procedures are discussed.