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Abstract:
Mean-field modeling based on the Eshelby inclusion problem poses some difficulties when the non-linear Maxwell-type constitutive law is used for elasto–viscoplasticity. One difficulty is that this behavior involves different orders of time differentiation, which leads a long-term memory effect. One of the possible solutions to this problem is the additive interaction law. Generally, mean field models solely use the mean values of stress and strain fields per phase, while variational approaches consider the second moments of stresses and strains. It is seen that the latter approach improves model predictions allowing to account for stress fluctuation within the phases. However, the complexity of the variational formulations still makes them difficult to apply in the large scale finite element calculations and for non-proportional loadings. Thus, there is a need to include the second moments within homogenization models based on the additive interaction law. In the present study, the incorporation of the second moments of stresses into the formulation of the additive Mori–Tanaka model of two-phase elastic–viscoplastic material is discussed. A modified tangent linearization of the viscoplastic law is proposed, while the Hill–Mandel’s lemma is used to track the evolution of second moments of stresses. To study the model performance and efficiency, the results are compared to the full-field numerical calculations and predictions of other models available in the literature. Very good performance of the modified tangent linearization is demonstrated from these benchmarks for both monotonic and non monotonic loading responses.

Keywords:
Homogenization , Elasto-viscoplasticity, Non linear composites, Modified tangent linearization, Additive interaction law

Abstract:
A cluster interaction model has been proposed to account for the spatial distribution and morphology of particles when estimating the effective properties of elastic and thermoelastic composites (Molinari and El Mouden, 1996). In the present paper this approach is extended to elastic-viscoplastic composites. To this end the tangent linearization of the non-linear viscoplastic law and the concept of additive interaction equation are used. Although the extension is formulated for the non-linear case, first applications are considered for linear viscoelastic composites, a situation rich enough to evaluate the interest of the cluster interaction approach. Results of the model are compared to numerical homogenization for periodic
unit cells with two cubic configurations.

Keywords:
homogenization, the cluster interaction model, elastic-viscoplastic composite, spatial configuration of inclusions, interaction between inclusions

Abstract:
The goal of the present work is to identify and model the elastic-viscoplastic behavior of electrodeposited copper films under tension-compression loadings. From the experimental point of view, as proposed in the literature, a film of copper is electrodeposited on both sides of an elastic compliant substrate. The overall specimen is next subjected to tensile loading-unloadings. As the substrate remains elastic, the elastic–plastic response of copper under cyclic loading is experimentally determined. A clear kinematic hardening behavior is captured. To model the mechanical response, a new elastic-viscoplastic self-consistent scheme for polycrystalline materials is proposed. The core of the model is the tangent additive interaction law proposed in Molinari (2002). The behavior of the single grain is rate dependent where kinematic hardening is accounted for in the model at the level of the slip system. The model parameters are optimized via an evolutionary algorithm by comparing the predictions to the experimental cyclic response. As a result, the overall response is predicted. In addition, the heterogeneity in plastic strain activity is estimated by the model during cyclic loading.

Keywords:
electrodeposited copper, self-consistent scheme, elasto-viscoplasticity, kinematic hardening, experiments

Abstract:
The goal of the present work is to propose a multi-scale approach for composite materials which accounts for kinematic hardening in the phases. For that purpose, the additive/sequential interaction rule and tangent linearization of viscoplastic response proposed for elastic–viscoplastic material can be extended in a straightforward manner. A two phase composite where each phase is elastic–viscoplastic is considered. The viscoplastic flow is governed by a J2 flow theory with an overstress. To find the overall behavior of the composite, a Mori–Tanaka model is applied. Numerical validation of the proposition is carried out by considering a representative volume element with 30 inclusions. Various configurations have been tested: hard or soft inclusion cases with or without isotropic hardening. It is shown that the quality of the model predictions is not affected by the introduction of the kinematic hardening component in the local constitutive behavior. Namely, in most cases considered in the paper the overall stress–strain response as well as the average stress–strain response per phase is accurately estimated. It has been also verified that the obtained backstress components are consistent with the ones predicted by Finite element calculations with ABAQUS Software.

Keywords:
elastic-viscoplasticity, homogenization, finite element, metal matrix composite, Mori–Tanaka scheme, kinematic hardening

Abstract:
This work deals with the prediction of the macroscopic behavior of two-phase composites, based on the Mori–Tanaka scheme combined with an additive/sequential interaction rule and tangent linearization of viscoplastic response. Cyclic tension compression loadings are considered to further validate the approach. The composite is made of spherical inclusions dispersed in a matrix. Both materials have an elastic–viscoplastic behavior. In a second part, finite element calculations are performed using ABAQUS/STANDARD software in order to validate the proposed homogenization technique. A representative volume element is analyzed with 30 randomly distributed inclusions. Comparisons between the additive tangent Mori–Tanaka scheme and finite element calculations are made for different volume fractions of inclusions, different contrasts in elastic and viscous properties and different strain rates and strain amplitudes. These comparisons demonstrate the efficiency of the proposed homogenization scheme. The effect of isotropization of the viscoplastic tangent stiffness is also investigated. It is concluded that quality of predictions does not benefit from such simplification, contrary to the known result for elastic–plastic case.

Keywords:
Elasto-viscoplasticity, Homogenization, Finite element, Composite, Mori–Tanaka scheme

Abstract:
This work deals with the prediction of the macroscopic behavior of two-phase composites, based on the Mori-Tanaka scheme combined with an additive/sequential interaction rule and tangent linearization of viscoplastic response. Cyclic tension compression loadings are considered to further evaluate the approach. The composite is made of spherical inclusions dispersed in a matrix. Both materials have an elastic-visco-plastic behavior. In the second part, finite element calculations are performed using ABAQUS/STANDARD software in order to validate the proposed homogenization technique. A representative volume element is analyzed with 30 randomly distributed inclusions.

Keywords:
Multiscale Modeling, Elasto-viscoplasticity, Mori Tanaka Scheme, Composite