A condensed approach to model the constitutive behavior of multiphase ferroelectric and multiferroic materials

Ferroelectric as well as ferromagnetic materials are widely used in smart structures and devices as actuators, sensors etc. Most of the developed models, describing the nonlinear behavior, are implemented within the framework of the Finite Element Method. Most investigations, however, are restricted to simple boundary value problems under uniaxial or biaxial loading and their goal is the calculation of hysteresis loops or to determine e.g. electromechanical coupling coefficients. Regarding these circumstances, the so-called condensed method (CM) is introduced to investigate the macroscopic polycrystalline ferroelectric material behavior at a macroscopic material point without any kind of discretization scheme. In the presented paper, the CM is extended towards multiphase ferroelectric material behavior. Moreover, first numerical results of a multiphase ferroelectric material at the morphotropic phase boundary are presented. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear numerical simulation of ferroelectric-ferromagnetic multifunctional composites

The theoretical background of nonlinear constitutive multifield behavior is presented. Nonlinear material models describing the ferroelectric or ferromagnetic behaviors are presented. Both physically and phenomenologically motivated constitutive models have been developed for the numerical calculation of the nonlinear magnetostrictive and ferroelectric behaviors. On this basis, the polarization in the ferroelectric and magnetization in the ferromagnetic respectively magnetostrictive phases are simulated and the resulting effects analyzed. The developed tools enable the prediction of the electromagnetomechanical properties of smart multiferroic composites and supply useful means for their optimization. Goals are to improve the efficiency of ME coupling and to reduce damage associated with the poling process. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational characterization of ferroelectric solids with a 3D Preisach model based on an orientation distribution function

Ferroelectric or ferromagnetic materials show an interaction between mechanical deformations and polarization or magnetization. A few multiferroic materials possess both ferroic properties and exhibit a magneto-electric (ME) coupling. These ME properties can be achieved in two-phase composites, which combine ferroelectric and ferromagnetic characteristics. To predict a realistic material behavior and a more precise ME coefficient, the application of suitable material models which describe the nonlinear hysteretic behavior is of particular importance. In the present contribution we focus on the characterization of a nonlinear ferroelectric material behavior, in terms of a 3D Preisach model based on an orientation distribution function. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

High-resolution simulation of microstructures in dual-phase steel

Modern dual-phase (DP) steels are widely used in industry due to their favorable material behavior based on the complex microstructure. In the presented work, a microstructure based FEM model for the elasto-plastic deformation of DP steels is developed in order to accomplish a deeper understanding of the structure-property correlation. The underlying microstructure is obtained by serial section tomography. The following contribution deals with a comparison of different approaches to build a FE mesh, which is based on such microstructures. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constitutive modeling of ferromagnetic and ferroelectric behaviors and application to multiferroic composites

In this paper, the constitutive modeling of nonlinear multifield behavior as well as the finite element implementation are presented. Nonlinear material models describing the magneto-ferroelectric or electro-ferromagnetic behaviors are presented. Both physically and phenomenologically motivated constitutive models have been developed for the numerical calculation of principally different nonlinear magnetostrictive behaviors. Further, the nonlinear ferroelectric behavior is based on a physically motivated constitutive model. On this basis, the polarization in the ferroelectric and magnetization in the ferromagnetic and magnetostrictive phases, respectively, are simulated and the resulting effects analyzed. Numerical simulations focus on the calculation of magnetoelectric coupling and on the prediction of local domain orientations going along with the poling process, thus supplying information on favorable electric-magnetic loading sequences. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Magneto-Mechanically Coupled Phase-Field Model at Large Deformation

The aggregate magneto-mechanical behavior of magneto-rheological elastomers (MREs) stems from the magnetic properties of the ferromagnetic inclusion and the mechanical properties of the matrix material. We propose a large deformation micro-magnetic theory, to predict the behavior and interaction of ferromagnetic particles inside an elastomeric matrix. A rate-type variational principle, with the magnetization as the order parameter is proposed. A large deformation Landau-Lifshitz-Gilbert equation for the time evolution of the magnetization, is obtained directly from the proposed variational principle. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Large-Deformation Phase-Field Approach for the Modeling of Magneto-Sensitive Elastomers

Magneto-sensitive materials show magneto-mechanical coupled response and are thus of increasing interest in the recent age of smart functional materials. Ferromagnetic particles suspended in an elastomeric matrix show realignment under the influence of an external applied field, in turn causing large deformations of the substrate material. The magneto-mechanical coupling in this case is governed by the magnetic properties of the inclusion and the mechancial properties of the matrix. The magnetic phenomenon in ferromagnetic materials is governed by the formation and evolution of domains on the micro scale. A better understanding of the behavior of these particles under the influence of an external applied field is required to accurately predict the behavior of such materials. In this context it is of particular importance to model the macro scopic magneto-mechanically coupled behavior based on the micro-magnetic domain evolution. The key aspect of this work is to develop a large-deformation micro-magnetic model that can accurately capture the microscopic response of such materials. Rigorous exploitation of appropriate rate-type variational principles and consequent incremental variational principles directly give us field equations including the time evolution equation of the magnetization, which acts as the order parameter in our formulation. The theory presented here is the continuation of the work presented in [1, 7] for small deformations. A summary of magneto-mechanical theories spanning over multiple scales has been presented in [4]. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A phenomenological constitutive model for magnetostrictive materials and ferroelectric ceramics

The contribution is concerned with a thermodynamic consistent constitutive model for magnetostrictive materials and ferroelectric ceramics. It captures the nonlinear phenomenological behavior which is described by hysteresis effects. Magnetostrictive alloys and ferroelectric ceramics belong to the multifunctional materials. In recent years these materials have become widely–used in actor and sensor applications. They characterize an inherent coupling between deformation and magnetic or electric field. Due to the similarities of the coupled differential equations a uniform approach is applied for both phenomena. The presented three–dimensional material model is thermodynamically motivated. It is based on the definition of a specific free energy function and a switching criterion. Furthermore an additive split of strain and the magnetic or electric field in a reversible and an irreversible part is suggested. The irreversible quantities serve as internal variables, which is analog to plasticity theory. A one–to–one–relation between the two internal variables provides conservation of volume for the irreversible strains. The presented material model can approximate the ferromagnetic or ferroelectric hysteresis curve and the related butterfly hysteresis. Furthermore an extended approach for ferrimagnetic behavior, which occurs in magnetostrictive materials, is presented. Some numerical simulations demonstrate the capability of the presented model. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Material libraries for texturized thin metal sheets in elastic range

An approach for the simulation of the elastic material responses using stochastic generation of material samples and a subdomain-based FEM simulation is presented. Both the single-grain geometries and the lattice orientations are stochastically simulated. This simulations can be used to build a Material Library of responses which must be computed only once for a given material. Furthermore, the Library can be used in a fast two-scale simulation avoiding any detailed computation in the microscale. In this contribution, the basic ideas of the computational algorithm are presented and some results of the material library for Steel DC01 are given. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermomechanical modeling and simulation of aluminum alloys during extrusion process

The purpose of this work is to simulate the microstructure development of aluminum alloys during hot metal forming processes such as extrusion with the help of the Finite Element Method (FEM). To model the thermomechanical coupled behavior of the material during the extrusion process an appropriate material model is required. In the current work a Johnson–Cook like thermoelastic viscoplastic material model is used. To overcome the numerical difficulties during simulation of extrusion such as contact problem and element distortion an adaptive meshing system is developed and applied. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A global damping factor for a non-invasive form finding algorithm

Inverse form finding based on the finite element method (FEM) aims in determining the optimal material (undeformed) configuration when knowing the target spatial (deformed) configuration in a discretized setting. The strategy is to iteratively update the material coordinates and recompute the spatial configuration by a FEM simulation until the computed spatial nodal positions are close enough to a priori given spatial nodal positions. A form finding algorithm is utilized, which is purely based on geometrical considerations and can be coupled with arbitrary external FEM software via subroutines in a non-invasive fashion. At large deformations degenerated elements can occur when updating the material coordinates. Evaluating the mesh quality of the updated material configuration and adjusting a global damping factor before recomputing the next spatial configuration helps to avoid mesh distortions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of material and structural inhomogeneities in timber structures

The numerical simulation of timber structures by means of FEM has been an object of recent research. Most of the material models developed so far are based on idealized assumptions by disregarding inhomogeneities. Here, models to capture structural inhomogeneities in terms of branches and knots and the resulting deviation in grain course in a three-dimensional FE analysis are presented. Besides, naturally varying material properties referred to as material inhomogeneities have to be considered in the structural analysis. Due to the insufficient experimental data, the uncertainty model fuzziness is applied. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Particle Finite Element Simulations of Cutting Processes in Manufacturing

The cutting of metals is an important process in manufacturing and challenges established methods in the field of computational mechanics. The particle finite element method (PFEM) combines the benefits of particle based methods and the standard finite element method (FEM) to account for large deformations and separation of material. In cutting simulations the workpiece is realised as a set of particles, whose boundary is detected by the α-shape method. After the boundary detection, the particles are meshed with finite elements. Since metals show a plastic behavior under large deformations, a suitable material model needs to be considered. Numerical examples show the effect of the choice of the parameter α on the cutting force. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust semi-coarsening multilevel preconditioning of biquadratic FEM systems

While a large amount of papers are dealing with robust multilevel methods and algorithms for linear FEM elliptic systems, the related higher order FEM problems are much less studied. Moreover, we know that the standard hierarchical basis two-level splittings deteriorate for strongly anisotropic problems. A first robust multilevel preconditioner for higher order FEM systems obtained after discretizations of elliptic problems with an anisotropic diffusion tensor is presented in this paper. We study the behavior of the constant in the strengthened CBS inequality for semi-coarsening mesh refinement which is a quality measure for hierarchical two-level splittings of the considered biquadratic FEM stiffness matrices. The presented new theoretical estimates are confirmed by numerically computed CBS constants for a rich set of parameters (coarsening factor and anisotropy ratio). In the paper we consider also the problem of solving efficiently systems with the pivot block matrices arising in the hierarchical basis two-level splittings. Combining the proven uniform estimates with the theory of the Algebraic MultiLevel Iteration (AMLI) methods we obtain an optimal order multilevel algorithm whose total computational cost is proportional to the size of the discrete problem with a proportionality constant independent of the anisotropy ratio.     

On the adaptive coupling of FEM and BEM in 2–d–elasticity

Summary. This paper concerns the combination of the finite element method (FEM) and the boundary element method (BEM) using the symmetric coupling. As a model problem in two dimensions we consider the Hencky material (a certain nonlinear elastic material) in a bounded domain with Navier–Lamé differential equation in the unbounded complementary domain. Using some boundary integral operators the problem is rewritten such that the Galerkin procedure leads to a FEM/BEM coupling and quasi–optimally convergent discrete solutions. Beside this a priori information we derive an a posteriori error estimate which allows (up to a constant factor) the error control in the energy norm. Since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh–refinement we state adaptive algorithms for the –version of the FEM/BEM–coupling. Illustrating numerical results are included. Received April 15, 1994 / Revised version received January 8, 1996     

Wave Propagation in a beam with random material properties

Modern composite materials, e.g., carbon fibre reinforced plastics (CFRP), exhibit a complex micro structure due to their fabrication process. The latter, being usually omitted in mechanical models through the homogenization of elastic properties, has a strong influence on the propagation of ultrasonic guided waves [1, 2]. Though it is possible to model the wave phenomena deterministically, taking into account a realistic distribution of fibres and polymer matrix, it is desirable to develop an improved model for the finite element analysis (FEM), which consider the stochastic properties in a more general way. In the current work, an approach for the simulation of waves in a isotropic beam with random material properties is presented. For the numerical computations with the FEM the Young's modulus was discretized by the Karhunen-Loève Expansion (KLE). Numerical investigations on the excited and propagating guided waves are presented. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of regenerative tissue for replacing the Bruch's membrane

In the present study the anisotropic mechanical properties such as elasticity and diffusion of bovine Bruch's membrane (BM) and a collagen foils (CF) are compared with each other. For this reason, a constitutive material law is developed and implemented into a FEM software. Based on tensile tests, the material parameters of both materials are identified. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modified Landweber tterations in a multilevel setting applied to the determination of nonlinear material parameters in piezoelectricity

An efficient solution of the inverse problem of identifying nonlinear dependencies in hyperbolic systems of PDEs, here piezoelectric material parameter curves, is the aim of this work. The dominant material tensor entries in the coupled field equations which describe the electromechanical interplay are approximated by functions depending on the physical field quantities electric field or mechanical stress. In order to solve this nonlinear and ill-posed problem of parameter curve identification efficiently, modified Landweber iterations (steepest descent and minimal error) will be studied. A multilevel approach is expedient due to the discretization of the unknown parameter curves and high computational efforts solving the forward problem (transient, nonlinear FEM computations). Theoretical investigations concerning convergence and regularization properties of the methods in a multilevel scenario will be presented, along with numerical results from an example in piezoelectricity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)