Computational Homogenization in Micro-Magneto-Elasticity

The overall macroscopic response of magneto-mechanically coupled materials stems from complex magnetization evolution and corresponding domain wall motion occurring on a lower length scale. In order to account for such effects we propose a computational homogenization approach that incorporates a ferromagnetic phase-field formulation into a macroscopic Boltzmann continuum. This scale-bridging is obtained by rigorous definition of rate-type and incremental variational principles. An extended version of the classical Hill-Mandel macro-homogeneity condition is obtained as a consequence. In order to satisfy the unity constraint of the magnetization on the micro-scale, an efficient operator-split method is proposed. (© 2015 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)     

Variational Homogenization of Micro-Electro-Elasticity

Understanding of micromechanical mechanisms in functional materials with electro-mechanical coupling is a highly demanding area of simulation technology and increasing interest has been shown in the last decades. Smart materials are characterized by microstructural properties, which can be changed by external stress and electric field stimuli, and hence find use as the active components in sensors and actuators. In this context, a key challenge is to combine models for microscopic electric domain evolution with variational principles of homogenization. We outline a variational-based micro-electro-elastic model for the micro-structural evolution of electric domains in ferroelectric ceramics. The micro-to-macro transition is performed on the basis of variational principles, extending purely mechanical formulations to coupled electro-mechanics. We focus on an electro-mechanical Boltzmann continuum on the macro-scale with mechanical displacement and electric potential as primary variables. The material model on the micro-scale is described by a gradient-extended continuum formulation taking into account the polarization vector field and its gradient, see Landis [1] and Schrade et al. [2] for conceptually similar approaches. A crucial aspect of the proposed homogenization analysis is the derivation of appropriate boundary conditions on the surface of the representative volume element. In this work we derive stiff Dirichlet-type, soft Neumann-type, and periodic boundary constraints starting from a generalized Hill-Mandel macrohomogeneity condition. Furthermore, we propose techniques to incorporate these boundary conditions in the variational principles of homogenization by means of Lagrange multiplier methods. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Treatment and Stability Analysis of Coupled Electro-Mechanics

Dielectric materials such as electro-active polymers (EAPs) belong to the class of functional materials which are used in advanced industrial environments as sensors or actuators and in other innovative fields of research. Driven by Coulomb-type electrostatic forces EAPs are theoretically able to withstand deformations of several hundred percents. However, large actuation fields and different types of instabilities prohibit the ascend of these materials. One distinguishes between global structural instabilities such as buckling and wrinkling of EAP devices, and local material instabilities such as limit- and bifurcation-points in the constitutive response. We outline variational-based stability criteria in finite electro-elastostatics and design algorithms for accompanying stability checks in typical finite element computations. These accompanying stability checks are embedded into a computational homogenization framework to predict the macroscopic overall response and onset of local material instability of particle filled composite materials. Application and validation of the suggested method is demonstrated by a representative model problem. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of Micro- and Macro-Instability Phenomena in Computational Homogenization of Finite Electro-Statics

Dielectric materials such as electro-active polymers (EAPs) belong to the class of functional materials which are used in advanced industrial environments as sensors or actuators and in other innovative fields of research. Driven by Coulomb-type electrostatic forces EAPs are theoretically able to withstand deformations of several hundred percents. However, large actuation fields and different types of instabilities prohibit the ascend of these materials. One distinguishes between global structural instabilities such as buckling and wrinkling of EAP devices, and local material instabilities such as limit- and bifurcation-points in the constitutive response. We outline variational-based stability criteria in finite electro-elastostatics and design algorithms for accompanying stability checks in typical finite element computations. These accompanying stability checks are embedded into a computational homogenization framework to predict the macroscopic overall response and onset of local material instability of particle filled composite materials. Application and validation of the suggested method is demonstrated by representative model problems. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Regularized Sharp Interface Model for Martensitic Phase Transformations at Large Strains

The macroscopic mechanical behavior of many functional materials crucially depends on the formation and evolution of their microstructure. When considering martensitic shape memory alloys, this microstructure typically consists of laminates with coherent twin boundaries. We suggest a variational-based phase field model for the dissipative evolution of microstructure with coherence-dependent interface energy and construct a suitable gradient-extended incremental variational framework for the proposed dissipative material. We use our model to predict laminate microstructure in martensitic CuAlNi. (© 2011 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)     

Variational Methods of Micro–Macro Transitions in Nonlinear Elasticity

Continuum micromechanics deals with idealized materials where the macroscopic material response is modelled in an averaged or homogenized sense based on the information of the heterogeneous microstructure. In general, an efficient treatment of multiscale systems requires the application of equivalent structural problems where the constituents are governed by overall properties. The key contribution of this paper is the computational exploitation of variational methods for a numerical upper and lower bound estimation of the effective material response. We present aspects for the formulation of an appropriate minimizing principle yielding the displacement fluctuations on the microstructure and the local effective constitutive variables of the macrostructure depending on the choice whether we apply linear displacement, traction or periodic boundary conditions to the displacement fluctuations on the boundary of the microstructure. The proposed concept will be demonstrated in the scope of some representative model problems.     

Deformation Driven Homogenization of Fracturing Solids

The paper discusses numerical formulations of the homogenization for solids with discrete crack development. We focus on multi–phase microstructures of heterogeneous materials, where fracture occurs in the form of debonding mechanisms as well as matrix cracking. The definition of overall properties critically depends on the developing discontinuities. To this end, we extend continuous formulations [1] to microstructures with discontinuities [2]. The basic underlying structure is a canonical variational formulation in the fully nonlinear range based on incremental energy minimization. We develop algorithms for numerical homogenization of fracturing solids in a deformation–driven context with non–trivial formulations of boundary conditions for (i) linear deformation and (ii) uniform tractions. The overall response of composite materials with fracturing microstructures are investigated. As a key result, we show the significance of the proposed non–trivial formulation of a traction–type boundary condition in the deformation–driven context. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational Homogenization of Piezoelectric Materials using FE2

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, two-scale classical (first-order) homogenization of electro-mechanically coupled problems 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. The homogenization of the coupled problem is done using different representative volume elements (RVEs), which capture the microstructure of the inhomogeneous material, to represent the macro material response. Later this technique is used to determine the macroscopic and microscopic configurational forces on certain defects. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive Finite–Element Procedures for Homogenization Analysis of Heterogeneous Micro–Structures

The paper investigates adaptive finite element procedures for the treatment of homogenize macro‐continuums with locally attached micro‐structures of nonlinear elastic and elastoplastic constituents. The deformation of the microstructure is assumed to be driven by a macroscopic strain mode. In this context we investigate different constraints for the fluctuation field base on variational formulations. For the computation of the overall response of the microstructures we develop a distinct adaptive finite element procedure. We consider possible a priori and a posteriori error estimators for the homogenization analysis in connection with an adaptive remeshing procedure.     

Micro-Electro-Elastic Modeling of Electric Field- and Stress-Driven Domain Wall Motion

Recently, increasing interest in functional materials such as piezoceramics has been shown. Such materials are characterized by properties, which can be significantly changed by external stimuli, such as stress, electric or magnetic fields. We outline a micro-electro-elastic model for the evolution of electrically and mechanically poled domains incorporating the surrounding free space. To this end, recently developed incremental variational principles (Miehe & Rosato [1]) for local dissipative response need to be extended to gradient-type phase-field models, including an embedding into the free space. The variational setting serves as a natural starting point for a compact and symmetric finite element implementation, considering the mechanical displacement, the electric polarization treated as an order parameter, and the electric potential induced by the polarization as the primary variables. The latter is defined on both the solid domain as well as the surrounding free space. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Obtaining Cosserat material parameters by homogenization of a Cauchy continuum

An increasing importance of composites with sandwich architecture and fibre-reinforced components is recognizable especially in aerospace and light weight industry. Due to the inner structure such materials often exhibit a complex behavior. If the ratio of micro- and macroscopic length scales, l and L, violates the condition l/L ≪ 1, a higher order continuum should be used to describe the macroscopic material behavior correctly. The numerical simulation requires reliable material constants, for which the experimental determination is laborious and sometimes impossible. Alternatively homogenization methods can be used for the numerical identification of overall material parameters. A short introduction to the linear Cosserat theory is followed by an extended homogenization procedure to derive the macroscopic material constants of a linear Cosserat continuum. The parameters obtained with a heterogeneous cell are used to simulate different bending load cases. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Linking macroscopic deformation processes to microstructure evolution using an FE-FFT-based micro-macro transition and non-conserved phase-fields

The purpose of this work is the multiscale FE-FFT-based prediction of macroscopic material behavior, micromechanical fields and bulk microstructure evolution in polycrystalline materials subjected to macroscopic mechanical loading. The macroscopic boundary value problem (BVP) is solved using implicit finite element (FE) methods. In each macroscopic integration point, the microscopic BVP is embedded, the solution of which is found employing fast Fourier transform (FFT), fixed-point and Green's function methods. The mean material response is determined by the stress-strain relation at the micro scale or rather the volume average of the micromechanical fields. The evolution of the microstructure is modeled by means of non-conserved phase-fields. As an example, the proposed methodology is applied to the modeling of stress-induced martensitic phase transformations in metal alloys. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Phase Field Modeling of Laminate Deformation Microstructure in Finite Gradient Crystal Plasticity

The macroscopic mechanical behavior of many materials crucially depends on the formation and evolution of their microstructure. In this work, we consider the formation and evolution of laminate deformation microstructure in plasticity. Inspired by work on the variational modeling of phase transformation [5] and building on related work on multislip gradient crystal plasticity [9], we present a new finite strain model for the formation and evolution of laminate deformation microstructure in double slip gradient crystal plasticity. Basic ingredients of our model are a nonconvex hardening potential and two gradient terms accounting for geometrically necessary dislocations (GNDs) by use of the dislocation density tensor and regularizing the sharp interfaces between different kinematically coherent plastic slip states. The plastic evolution is described by means of a nonsmooth dissipation potential for which we propose a new regularization. We formulate a continuous gradient-extended rate-variational framework and discretize it in time to obtain an incremental-variational formulation. Discretization in space yields a finite element formulation which is used to demonstrate the capability of our model to predict the formation and evolution of laminate deformation microstructure in f.c.c. Copper with two active slip systems in the same slip plane. (© 2012 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)     

Thermodynamically consistent modeling of yield surface distortion

From a macroscopic point of view, the deformation of most metals results in an evolution of the symmetry groups characterizing the isotropy of the considered materials. With respect to plastic deformation for instance, the shape of the macroscopic yield surface evolves during deformation. In the present paper, a novel constitutive framework capturing this evolution is proposed. This framework is based on the fundamentals of thermodynamics. Furthermore, it also shows a variational structure such that all state variables follow jointly from minimizing an incrementally defined energy functional. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)