Configurational forces in a multiscale approach to piezoelectric materials

Due to the growing interest in determining the macroscopic material response of inhomogeneous materials, computational methods are becoming increasingly concerned with the application of homogenization techniques. In this work, a two-scale classical homogenization of an electro-mechanically coupled material using a FE²-approach is discussed. We explicitly formulated the homogenized coefficients of the elastic, piezoelectric and dielectric tensors for small strain as well as the homogenized remanent strain and remanent polarization. In the homogenization different representative volume elements (RVEs), which capture the micro-structure of the inhomogeneous material, are used to represent the macroscopic material response. Two different schemes are considered. In the first case, domain wall movement is not allowed, but in the second case the movement of the domain walls is taken into account using thermodynamic considerations. Later this technique is used to determine the macroscopic and microscopic configurational forces on defects [2]. These defect situations include the driving force on a crack tip. The effect of the applied electric field on configurational forces at the crack tip is investigated. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Aspects of a direct homogenization procedure for electro-mechanically coupled problems

The aim of this work is to discuss a micro–macro homogenization procedure for electro–mechanically coupled problems. In this context a two–scale homogenization ansatz for ferroelectric ceramics based on an FE²-approach is presented. The microscopic discretization of the heterogeneous structure of the polycrystalline material allows for the incorporation of microscopic effects, which are necessary to determine the corresponding overall macroscopic material response. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An adaptive approach for modeling a fiber-matrix composite with the FE2 method

In materials with a complicated microstructre [1], the macroscopic material behaviour is unknown. In this work a Fiber-Matrix composite is considered with elasto-plastic fibers. A homogenization of the microscale leads to the macroscopic material properties. In the present work, this is realized in the frame of a FE² formulation. It combines two nested finite element simulations. On the macroscale, the boundary value problem is modelled by finite elements, at each integration point a second finite element simulation on the microscale is employed to calculate the stress response and the material tangent modulus. One huge disadvantage of the approach is the high computational effort. Certainly, an accompanying homogenization is not necessary if the material behaves linear elastic. This motivates the present approach to deal with an adaptive scheme. An indicator, which makes use of the different boundary conditions (BC) of the BVP on microscale, is suggested. The homogenization with the Dirichlet BC overestimates the material tangent modulus whereas the Neumann BC underestimates the modulus [2]. The idea for an adaptive modeling is to use both of the BCs during the loading process of the macrostructure. Starting initially with the Neumann BC leads to an overestimation of the displacement response and thus the strain state of the boundary value problem on the macroscale. An accompanying homogenization is performed after the strain reaches a limit strain. Dirichlet BCs are employed for the accompanying homogenization. Some numerical examples demonstrate the capability of the presented method. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second order homogenization method based on higher order finite elements

Modeling materials with lattice-like microstructures like open-cell foams requires an extended continuum mechanical setting on the macroscopic scale, e. g. a micropolar or micromorphic theory. In order to avoid the formulation of constitutive equations a higher order numerical homogenization scheme (FE²) is proposed. Therefore, each integration point possesses its own microstructure which, in the present case, consists of beam-like elements representing the cell walls. In this paper, the microstructures are discretized by continuum-based higher order locking free finite elements with high aspect ratios, leading to a numerically efficient treatment of a local displacement-driven boundary value problem according to the macroscopic strain and curvature. The resulting stress distributions in the microstructures are homogenized to macroscopic stresses and couple stresses. The approach is demonstrated by a numerical example. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Notes on the Computation of Effective Electro-Mechanical Moduli

The present work deals with the computational determination of effective material properties of electro-mechanically coupled materials. In this context, a two-scale homogenization ansatz for piezoelectric ceramics based on an FE²-approach is utilized. In this way we are able to compute the overall macroscopic material moduli for microheterogeneous structures. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Variational Homogenization Approach to Electromechanical Hystereses Caused by Domain Wall Movement

In recent years, increasing interest in so-called smart materials such as ferroelectric polymers and ceramics has been shown. Those materials are used in various actuators, sensors, and also in medical devices. In this paper, we outline a micro-macro approach to the modeling of macroscopic hystereses which directly takes into account the microstructural evolution of electrically poled domains. To this end, an incremental variational formulation for a gradient-type phase field model is developed and exploited for the simulation of electromechanically coupled problems. The formulation determines the hysteretic response of the material in terms of an energy-enthalpy and a dissipation function which both depend on the microscopic remanent polarization treated as an order parameter. The gradient-type balance law for the phase field can be considered as a generalization of Biot's equation for standard dissipative materials and may be related to the classical Ginzburg-Landau equation. Furthermore, the variational formulation serves as natural starting point for a compact and symmetric finite element implementation of the coupled micromechanical problem covering the displacement, the electric potential, and the microscopic polarization vector. For this three-field scenario we develop a variational-based homogenization method which determines the overall macroscopic hysteretic properties of a polycrystalline aggregate. The proposed computational method can be used as a numerical laboratory for the improvement of microstructural properties. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational micro-macro transitions at large strains for curvilinear physical directions

Biological tissues such as those involved in the eye, heart, veins or arteries are heterogeneous on one or another spatial scale and can undergo very large elastic strains. Frequently, these tissues are characterized by shell-like structures at the macroscopic scale and the physical material directions follow curvilinear paths. We consider a homogenized macro-continuum formulated in curvilinear convective coordinates with locally attached representative micro-structures. Micro-structures attached to different macroscopic points are assumed to be rotated counterparts according to the curvilinear path of the physical material directions at the macro-scale. The solution of such multi-scale problems according to the computational homogenization scheme [1, 2, 3] would need a different RVE at each macroscopic point. The goal of this paper is to use the same initial RVE at each macroscopic point by generalizing the computational homogenization scheme to a formulation considering different physical spaces at the micro- and macro-scale. The deformation and the reference frame of the micro-structure are assumed to be coupled with the local deformation and the local reference frame at the corresponding point of the macrocontinuum. For a consistent formulation of micro-macro transitions physical reference directions are defined on both scales, where the macroscopic one follows a curvilinear path. To formulate the generalized micro-macro transitions in absolute tensor notation the operations scale-up and scale-down are introduced. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational homogenization of materials with small deformation to determine configurational forces

In this work the mechanical boundary condition for the micro problem in a two-scaled homogenization using a FE² approach is discussed. The strain tensor is often used in the literature for small deformation problem to determine the boundary conditions for the boundary value problem on the micro level. This strain tensor based boundary condition gives consistent homogenized mechanical quantities, e.g. stress tensor and elasticity tensor, but the present work points out that it leads to unphysical homogenized configurational forces. Instead, we propose a displacement gradient based boundary condition for the micro problem. Results show that the displacement gradient based boundary condition can give not only the consistent homogenized mechanical quantities but also the appropriate homogenized configurational forces. The interpretation of the displacement gradient based boundary condition is discussed. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shakedown analysis of periodic composites with kinematic hardening material model

Lower-bound limit and shakedown analysis of periodic composites with the consideration of kinematic hardening are carried out on the representative volume element level. In combination with homogenization theory, the homogenized macroscopic admissible loading domains are determined. Furthermore, the strengths of periodic composites by using elastic perfectly plastic, unlimited and linear limited kinematic hardening material models are calculated and compared. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two-scale simulation of damage in unidirectionally fibre reinforced composite materials

The finite element analysis of engineering structures usually assumes a homogeneous as well as a continuous medium. The heterogeneity of matter, which is always found on a sufficiently small length scale is neglected by replacing the inhomogeneous medium through a model of a mathematically homogenized material. The macroscopic constitutive behaviour is derived from volume averaging procedures that smear the microscopic heterogeneities. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational homogenization of non-stationary transport processes in masonry structures

A fully coupled transient heat and moisture transport in a masonry structure is examined in this paper. Supported by several successful applications in civil engineering the nonlinear diffusion model proposed by Künzel (1997) [16] is adopted in the present study. A strong material heterogeneity together with a significant dependence of the model parameters on initial conditions as well as the gradients of heat and moisture fields vindicates the use of a hierarchical modeling strategy to solve the problem of this kind. Attention is limited to the classical first order homogenization in a spatial domain developed here in the framework of a two step (meso–macro) multi-scale computational scheme (FE² problem). Several illustrative examples are presented to investigate the influence of transient flow at the level of constituents (meso-scale) on the macroscopic response including the effect of macro-scale boundary conditions. A two-dimensional section of Charles Bridge subjected to actual climatic conditions is analyzed next to confirm the suitability of algorithmic format of FE² scheme for the parallel computing.     

Thermo-chemical modelling of the formation of a composite material

We consider a composite material composed of carbon or glass fibres included in a resin which becomes solid when it is heated up (the reaction of reticulation).

A mathematical model of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period ? >0, thus we get a coupled system of nonlinear partial differential equations.

First we prove the existence and uniqueness of a solution by using a fixed point theorem and we obtain a priori estimates. Then we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero as well as the estimates for the difference of the exact and the approximate solutions.     

Variational Structural and Material Stability Analysis in Finite Electro-Magneto-Mechanics of Active Materials

Soft matter electro-elastic, magneto-elastic and magneto-electro-elastic composites exhibit coupled material behavior at large strains. Examples are electro-active polymers and magneto-rheological elastomers, which respond by a deformation to applied electric or magnetic fields, and are used in advanced industrial environments as sensors and actuators. Polymer-based magneto-electro-elastic composites are a new class of tailor-made materials with promising future applications. Here, a magneto-electric coupling effect is achieved as a homogenized macro-response of the composite with electro-active and magneto-active constituents. These soft composite materials show different types of instability phenomena, which even might be exploited for future enhancement of their performance. This covers micro-structural instabilities, such as buckling of micro-fibers or particles, as well as material instabilities in the form of limit-points in the local constitutive response. Here, the homogenization-based scale bridging links long wavelength micro-structural instabilities to material instabilities at the macro-scale. This work outlines a comprehensive framework of an energy-based homogenization in electro-magneto-mechanics, which allows a tracking of postcritical solution paths such as those related to pull-in instabilities. Representative simulations demonstrate a tracking of inhomogenous composites, showing the development of postcritical zones in the microstructure and a possible instable homogenized material response. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element thermo-viscoelastic creep approach for heterogeneous structures with dissipative correctors

A finite element approach is presented for three-dimensional thermo-viscoelastic macro analysis of polymer-matrix composite structures containing micro-level heterogeneities, a two-scale approach. Due to its ability to account for microstructural details, the asymptotic expansion homogenization approach is employed to first, obtain the homogenized properties for use in the macroscale problem, and second, to study the local micro-level stress distributions influenced by macro effects. The theoretical formulations are described and developed for a thermoviscoelastic solid whose time-dependent stress–strain relationship can be homogenized. Arising from homogenization of the constitutive equation in the time domain is a hereditary dissipative corrector term. The dissipative corrector is time-dependent and accounts for heterogeneous behavior across the junction of dissimilar materials at the microstructural level. The additional term is necessary for the governing constitutive equations to satisfy equilibrium at both length scales. The objectives of this paper are three-fold: (1) develop the micro and macro constitutive equations for a thermoviscoelastic Kelvin–Voight material; (2) develop a computational approach for the constitutive equations; and (3) demonstrate and verify illustrative applications using results from the theoretical developments in the literature wherever available for a viscoelastic homogeneous/heterogeneous material.     

Parameter–identification of macroscopic material models based on virtual testing of given material mesostructures

For many heterogeneous materials such as composites and polycrystals, the material modeling for the constituents on a representative mesoscale can be considered as known, including concrete values of their inherent material parameters. Typical examples are isotropic elastic–plastic models for the constituents of composites or anisotropic crystal–plasticity models for the grains of polycrystals. This knowledge can be exploited with regard to the modeling of the homogenized macroscopic response. In particular, parameters in macroscopic models may be identified by virtual experiments provided by a computational deformation–driving of representative mesostructures. This paper outlines the general concept for the parameter–identification of macroscopic materialmodels based on the virtual testing of given material mesostructures. The virtual test data are obtained in the form of multi–dimensional stress–strain paths by applying different deformation gradients to a given mesostructure. After specifying a corresponding macroscopic material model covering the observed effects on the macroscale, the material parameters are identified by a least–square–type optimization procedure that optimizes the macroscopic material parameters. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenized Diffusion Limit of a Vlasov–Poisson–Fokker–Planck Model

The approximation by diffusion and homogenization of the initial-boundary value problem of the Vlasov–Poisson–Fokker–Planck model is studied for a given velocity field with spatial macroscopic and microscopic variations. The L¹-contraction property of the Fokker–Planck operator and a two-scale Hybrid-Hilbert expansion are used to prove the convergence towards a homogenized Drift–Diffusion equation and to exhibit a rate of convergence.     

Computation of the effective nonlinear material behaviour of composites using X-FEM - Analysis of the matrix material behaviour

For a consequent lightweight design the consideration of the nonlinear macroscopic material behaviour of composites, which is amongst others driven by damage and strain–rate effects on the mesoscale, is required. Therefore, the modelling approach using numerical homogenization techniques based on the simulation of representative volume elements which are modelled by the extended finite element method (X–FEM) is currently extended to nonlinear material behaviour. While the glass fibres are assumed to remain linear elastic, a viscoplastic constitutive law accounts for strain–rate dependence and inelastic deformation of the matrix material. This paper describes the process of finding suitable constitutive relations for the polymeric matrix material Polypropylene in the small–strain regime. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational Multi-Scale Stability Analysis of Periodic Electroactive Polymer Composites at Finite Strains

This work is dedicated to multi-scale stability analysis, especially macroscopic and microscopic stability analysis of periodic electroactive polymer (EAP) composites with embedded fibers. Computational homogenization is considered to determine the response of materials at macro-scale depending on the selected representative volume element (RVE) at micro-scale [4, 5]. The quasi-incompressibility condition is considered by implementing a four-field variational formulation on the RVE, see [7]. Based on the works [1–3, 6, 8] the macroscopic instabilities are determined by the loss of strong ellipticity of homogenized moduli. On the other hand, the bifurcation type microscopic instabilities are treated exploiting the Bloch-Floquet wave analysis in context of finite element discretization, which allows to detect the changed critical size of periodicity of the microstructure and critical macroscopic loading points. Finally, representative numerical examples are given which demonstrate the onset of instability surfaces, the stable macroscopic loading ranges, and further a periodic buckling mode at a microscopic instability point is presented. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Étude d'un modèle thermo-chimique de formation d'un matériau composite

We consider a composite material constituted of carbon or glass fibres included in a resin which becomes solid when it is heated up (reaction of reticulation). The mathematical modelling of the cure process is given by a kinetic equation describing the evolution of the reaction of reticulation coupled with the heat equation. The geometry of the composite material is periodic, with a small period $?>0$. First we prove the existence and uniqueness of a solution by using Schauder's fixed point theorem. Then, by using an asymptotic expansion, we derive the homogenized problem which describes the macroscopic behaviour of the material. We prove the convergence of the solution of the problem to the solution of the homogenized problem when ? tends to zero and we obtain an error estimate in a case of weak non-linearity. Finally we solve numerically the homogenized problem. To cite this article: S. Meliani et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).