Stéphane Berbenni

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

Abstract 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 Kroner–Wengs and Paquin et al. (Arch. Appl. Mech. 69 (1999) 14)s model.

Micromechanical modeling of the elastic-viscoplastic behavior of polycrystalline steels

Abstract 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.

Comparison of different homogenization approaches for elastic–viscoplastic materials

Homogenization of linear viscoelastic and non-linear viscoplastic composite materials is considered in this paper. First, we compare two homogenization schemes based on the Mori–Tanaka method coupled with the additive interaction (AI) law proposed by Molinari et al (1997 Mech. Mater. 26 43–62) or coupled with a concentration law based on translated fields (TF) originally proposed for the self-consistent scheme by Paquin et al (1999 Arch. Appl. Mech. 69 14–35). These methods are also evaluated against (i) full-field calculations of the literature based on the finite element method and on fast Fourier transform, (ii) available analytical exact solutions obtained in linear viscoelasticity and (iii) homogenization methods based on variational approaches. Developments of the AI model are obtained for linear and non-linear material responses while results for the TF method are shown for the linear case. Various configurations are considered: spherical inclusions, aligned fibers, hard and soft inclusions, large material contrasts between phases, volume-preserving versus dilatant anelastic flow, non-monotonic loading. The agreement between the AI and TF methods is excellent and the correlation with full field calculations is in general of quite good quality (with some exceptions for non-linear composites with a large volume fraction of very soft inclusions for which a discrepancy of about 15% was found for macroscopic stress). Description of the material behavior with internal variables can be accounted for with the AI and TF approaches and therefore complex loadings can be easily handled in contrast with most hereditary approaches.

Homogenization of multicoated inclusion-reinforced linear elastic composites with eigenstrains: Application to thermoelastic behavior

A new micromechanical approach for arbitrary multicoated ellipsoidal elastic inclusions with general eigenstrains is developed. We start from the integral equation of the linear elastic medium with eigenstrains adopting the Greens function technique and we apply an ‘(n+1)-phase’ model with a self-consistent condition to determine the homogenized behavior of multicoated inclusion-reinforced composites. The effective elastic moduli and eigenstrains are obtained as well as the residual stresses through the local stress concentration equations. The effective eigenstrains are determined either with the concentration tensors obtained here by the present model, or, more classically, with Levins formula. In order to assess our micromechanical model, some applications to the isotropic thermoelastic behavior of composites with and without interphase are given. In particular, ‘four-phase’ and ‘three-phase’ models are derived for isotropic homothetic spherical inclusion-reinforced materials, and the results are successfully compared to exact analytical solutions regarding the effective elastic moduli and the effective thermal expansion.

A new micromechanics-based scale transition model for the strain-rate sensitive behavior of nanocrystalline materials

An original two-step “three phase” elastic–viscoplastic scale transition model is developed based on the combined self-consistent and Mori–Tanaka schemes. A coated inclusion is embedded within a matrix, wherein the inclusion represents grain interiors and the coating of the inclusion mimics the effects of grain boundaries and triple junctions. The predominant behavior within the grain interiors is captured through dislocation glide, whereas grain boundary (GB) dislocation emission and absorption, as well as thermally assisted GB sliding, describe the deformation processes within the coating describing the GB affected zone. Furthermore, an imperfect interface is assumed between the inclusion and the coating to account for viscoplastic grain boundary sliding along a stick-slip mechanism. Results and discussion focus on the competitive roles of GB sliding, GB dislocation emission/absorption, dislocation sweep in grain cores and collective dislocation plasticity, and the origins of the pronounced strain rate sensitivity of fcc NC materials.

On the Use of EBSD to Study the Heterogeneity of Plastic Deformation

This paper deals with the study of plastic heterogeneity. It aims to study the role of both grain size and orientation distributions in the development of such heterogeneity. The considered material is an IF steel. EBSD maps have been made on the same areas before and after several degrees of extension. Parameters such as GOS (Grain Orientation Spread) or GOS/D (D the diameter of the grain) or GND (Geometrically Necessary Dislocation) densities have been determined for the whole set of grains as well as for subpopulations (smallest grains, largest grains for example). It appears that the character of neighboring grains plays a more important role than any of these parameters alone.

Analytical expressions of incompatibility stresses at Σ3⟨111⟩ twin boundaries and consequences on single-slip promotion parallel to twin plane

Strong incompatibility stresses may develop at grain or twin boundaries because of elastic and plastic anisotropies. Their knowledge at twin boundaries may be of interest for a better understanding of the mechanical behaviour of fcc materials that can display lamellar twin structures, such as twinning-induced plasticity (TWIP) steels or general nanotwinned materials. In this paper, incompatibility stresses arising at general twin boundaries are explicitly derived for a given twin volume fraction. They are deduced from the solutions of the general infinite bicrystal, which is equivalent to a periodic layered structure. In the case of pure elasticity and twin boundaries, the result is of remarkable simplicity. The incompatibility stress field reduces to a shear stress acting upon a plane orthogonal to twin plane. Simple analytical expressions of the resolved shear stresses are also determined according to the twin-boundary orientation, the twin volume fraction and the elastic anisotropy factor. Such expressions allow performing a comprehensive study of slip initiation. In particular, there exists a large physical domain, depending on the three above parameters, where simultaneous slip parallel to twin plane in the parent and in the twin is greatly promoted. There is also a restricted domain where simultaneous single slip parallel to twin plane is promoted. The conditions for these promotions are realistic considering the literature data on TWIP steels. The present results, hence, support the high ductility and strong contribution of kinematic hardening observed in TWIP steels and agree with composite hardening models with single- and multi-slip-deforming grains.

Variant selection of twins with low Schmid factors in cross grain boundary twin pairs in a magnesium alloy

Samples of magnesium AZ31 alloys are deformed in compression at room temperature under a strain rate of 1×10−3 s−1. The initial texture with respect to the loading direction is favorable for {10-12} extension twinning during the deformation. At an engineering strain of 2.75%, many extension twins are found to be connected with each other at grain boundaries, forming cross grain boundary twin pairs. Some have low positive or even negative Schmid factors (SFs). The variant selection of them are interpreted in terms of shear accommodations. The observed twin variants require the least or no accommodation through deformation modes with high CRSSs, but the most or more accommodation through those with low CRSSs.

A Time-Incremental Eshelby-Based Homogenization Scheme for Viscoelastic Heterogeneous Materials

A time-incremental Eshelby-based homogenization scheme for Maxwellian heterogeneous materials is proposed and discussed. This is based on the exact solution of the heterogeneous Eshelby ellipsoidal inclusion problem obtained in the time domain. In contrast with hereditary methods, the effective behavior and the evolution laws of the averaged stresses per phase are solved incrementally in the time domain without the need of inverse Laplace or Laplace–Carson transforms. This is made through a time differential equation to exactly solve a volume term in the integral equation that was generally approximated in previous internal variable methods. The present formulation works for any arbitrary anisotropic ellipsoidal Maxwellian inclusion embedded in an isotropic Maxwellian matrix without any other restrictive assumptions. In order to show the interest of the present approach, a Mori–Tanaka homogenization scheme is applied to two-phase composites using the developed strain rate concentration equations. The results are reported and discussed in comparisons with other existing methods, including hereditary approaches and more recent internal variable approaches, in order to show the efficiency of the present time-incremental homogenization scheme.

Micromechanical Modeling of the Elasto-Viscoplastic Behavior and Incompatibility Stresses of β-Ti Alloys

Near β titanium alloys can now compete with quasi-α or α/β titanium alloys for airframe forging applications. The body-centered cubic β-phase can represent up to 40% of the volume. However, the way that its elastic anisotropy impacts the mechanical behavior remains an open question. In the present work, an advanced elasto-viscoplastic self-consistent model is used to investigate the tensile behavior at different applied strain rates of a fully β-phase Ti alloy taken as a model material. The model considers crystalline anisotropic elasticity and plasticity. It is first shown that two sets of elastic constants taken from the literature can be used to well reproduce the experimental elasto-viscoplastic transition, but lead to scattered mechanical behaviors at the grain scale. Incompatibility stresses and strains are found to increase in magnitude with the elastic anisotropy factor. The highest local stresses are obtained toward the end of the elastic regime for grains oriented with their <111> direction parallel to the tensile axis. Finally, as a major result, it is shown that the elastic anisotropy of the β-phase can affect the distribution of slip activities. In contrast with the isotropic elastic case, it is predicted that {112} <111> slip systems become predominant at the onset of plastic deformation when elastic anisotropy is considered in the micromechanical model.