3D Simulation of Point Defect Migration in Ferroelectrics

Electric fatigue in functional materials involves a set of phenomena which lead to the degradation of materials with an increasing number of electrical cycles. Ionic and electronic charge carriers, later 0 as point defects, interact with each other and with microstructural elements in the bulk and with interfaces, which can lead to degradation, or finally to mechanical damage and dissociation reactions, see e.g. [1]. With this in mind, efforts are made to calculate the fields caused by point defects to simulate their interaction as well as to verify the used material parameters. Here, a material with linear electro mechanical coupling is used. The applied methods are integral transforms (Radon Transform) and a combination of Difference Methods and a Fast Fourier Transform to obtain solutions in an infinite domain and under periodic boundary conditions, respectively. The point defect interaction is studied within the framework of material or configurational forces. These forces are used in combination with reasonable kinetic laws to simulate defect migration, cf. [2]. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Point defect interaction in tranversally isotropic ferroelectrics

Functional materials, especially ferroelectrics are used in many devices like actuators, sensors and electronic devices. Due to high amounts of mechanical and electrical load cycles, fatigue phenomena may occur. This so called electric fatigue causes a decrease of the electromechanical coupling capability. It is assumed, that the ability to switch polarisation states, which is the reason for the ferroelectric effect, is decreased in the presence of point defects. These defects are ionic and electronic charge carriers, which can interact with each other, with microstructural elements in the bulk and with interfaces. Accumulation of defects can primarily lead to degradation, because of the loss of polarisation switchability. The interaction of defects in the bulk is simulated to get a better understanding of the defect accumulation processes. A model based on configurational forces can be used to obtain thermodynamic consistent kinetic laws. The material used is transversally isotropic and modelled with linear electromechanical coupling. The focus is on the influence of this material anisotropy on the defect interaction. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Qualitative properties of a continuum theory for thin films

We discuss qualitative aspects of a continuum theory for thin films rigorously derived in [B. Schmidt, On the passage from atomic to continuum theory for thin films, preprint 82/2005, Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig]. The stored energy density is examined for convexity properties and limiting behavior under large and small strains. A study of the dependence of the theory on relaxation parameters leads to the result that the scale of convergence used in [B. Schmidt, On the passage from atomic to continuum theory for thin films, preprint 82/2005, Max-Planck Institut für Mathematik in den Naturwissenschaften, Leipzig] is the only scale for which a limiting theory that also accounts for atomic relaxation effects is non-trivial.     

Prediction of microstructure in a Cosserat continuum using relaxed energies

We develop a nonlinear incompressible multiphase material model in a Cosserat continuum with microstructure. The free energy of the material is enriched with an interaction potential taking into account the intergranular kinematics at the continuum scale. As a result the total energy becomes non-convex, thus giving rise to the development of microstructural phases. To guarantee the existence of minimizers an exact quasi-convex envelope of the corresponding energy functional is derived. As a result a three phase material energy appears, among them two of the phases are with microstructure in the translational motion (displacment field) and micromotion (microrotation field), whereas the third phase is without internal structure. The corresponding relaxed energy is then used for finding the minimizers of the two field minimization problem corresponding to a Cosserat continuum. Results from a numerical example predicting the development of microstructure in the material are presented. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The interaction of a system of weakened zones at the interface of elastic media in a tensile stress field

The interaction of a system of crack-like defects with distributed cohesive forces over the whole surface of the edges, located at the interface of two elastic half-planes and which open under the action of forces at infinity, is considered. A dislocation approach is used to describe the model of each defect: the discontinuity in the asymmetric shifts is specified in the form of a basis function with free parameters that satisfies a number of physical constraints. The free parameters of the model are determined when finding an analytical solution of the problem. The key questions are: what is the minimum load at which just one of these weakened zones is converted into the nucleus of a crack or when one of the connecting bridges separating these zones is fractured and, also, under what conditions can the interaction of the defects be neglected ? The model is extended with a relation which enables an explicit opening - bonding force dependence to be obtained.     

Towards Phase Field Modeling of Ductile Fracture in Gradient-Extended Elastic-Plastic Solids

The modeling of failure in ductile metals must account for complex phenomena at a micro-scale as well as the final rupture at the macro-scale. Within a top-down viewpoint, this can be achieved by the combination of a micro-structure-informed elastic-plastic model with a concept for the modeling of macroscopic crack discontinuities. In this context, it is important to account for material length scales and thermo-mechanical coupling effects due to dissipative heating. This can be achieved by the construction of non-standard, gradient-enhanced models of plasticity with a full embedding into continuum thermodynamics [1,2]. The modeling of macroscopic cracks can be achieved in a convenient way by recently developed continuum phase field approaches to fracture based on regularized crack discontinuities. This avoids the use of complex discretization methods for crack discontinuities, and can account for complex crack patterns within a pure continuum formulation. Moreover, the phase field modeling of fracture is related to gradient theories of continuum damage mechanics, and fits nicely the structure of constitutive models for gradient plasticity. The main focus of this work is the extensions to gradient thermoplasticity and phase field formulation of ductile fracture, conceptually in line with the work [3]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A priori error analysis of two force-based atomistic/continuum models of a periodic chain

The force-based quasicontinuum (QCF) approximation is a non-conservative atomistic/continuum hybrid model for the simulation of defects in crystals. We present an a priori error analysis of the QCF method, applied to a one-dimensional periodic chain, that is valid for an arbitrary interaction range, large deformations, and takes coarse-graining into account. Our main tool in this analysis is a new concept of atomistic stress. Moreover, we formulate a new atomistic/continuum coupling mechanism based on coupling stresses instead of forces and extend the a priori analysis to this new method. We show that the new method has several theoretical advantages over the original QCF method.     

Influence of the type of defect and conditions of its interaction with a matrix on a wave field scattered by it under antiplane strain

We investigate the influence of a thin strip-like defect (crack or inclusion) and conditions of its interaction with a matrix on a wave field scattered by the defect. It is assumed that the matrix is under conditions of antiplane strain and that plane harmonic longitudinal-shear waves propagate in it. To determine the wave field, we formulate and solve boundary-value problems for a body with the corresponding defect by the method of discontinuous solutions. Main attention is given to a characteristic of the scattered field such as the total scattering cross-section. It is established that there exist angles of propagation of waves for which total scattering cross-sections differ significantly for defects of different types and different conditions of interaction between a defect and a matrix. This proves the possibility to determine the type of defect and conditions of its interaction with a matrix using the total scattering cross-section.     

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)     

Modeling and Simulation of Damage Processes based on a Gradient-Enhanced Free Energy Function

Taking into account softening effects in connection with conventional inelastic material models can cause ill-posed boundary value problems. These problems can be established by obtaining no unique solution for the resulting algebraic system or by having a strong mesh dependence of the numerical results. This is the consequence of losing ellipticity of the governing field equations. A possible approach to solve these problems is to introduce a non-local field function in the model which includes an internal material length scale. For this purpose a gradient-enhanced free energy function is used for the current continuum damage model from which two variational equations are resulting. Calculations with less effort can be achieved due to the enhancement of the free energy function in comparison to other approaches. The mentioned model is applied to a material with locally varying damage properties (yield limits). Furthermore, the model is able to describe crack propagation in cases of completely damaged material. Therewith, a matrix material including precipitates, such as carbides, is modeled. This allows to investigate ship screws, which usually exhibit the mentioned composition, with regard to the influence of cavitation. Cavitation describes the implosion of risen vapor bubbles, whereby the impact on screws causes heavy damages which can lead to a complete destruction. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase field modeling of complex crack patterns in multi-physics problems

Recently developed continuum phase field models for brittle fracture show excellent modeling capability in situations with complex crack topologies including branching in the small and large strain applications. This work presents a generalization towards fully coupled multi-physics problems at large strains. A modular concept is outlined for the linking of the diffusive crack modeling with complex multi field material response, where the focus is put on the model problem of finite thermo-elasticity. This concerns a generalization of crack driving forces from the energetic definitions towards stress-based criteria, the constitutive modeling of degradation of non-mechanical fluxes on generated crack faces. Particular assumptions are made on the generation of convective heat exchanges approximating surface load integrals of the sharp crack approach by distinct volume integrals. The coupling effect is also shown in generation of cracks due to thermally induced stress states. We finally demonstrate the performance of the phase field formulation of fracture at large strains by means of representative numerical examples. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of electrodynamic-mechanical coupling phenomena in smart magnetoelectroelastic materials

The coupling of electric, magnetic and mechanical phenomena may have various reasons. The famous Maxwell equations of electrodynamics describe the interaction of transient magnetic and electric fields. On the constitutive level of dielectric materials, coupling mechanisms are manyfold comprising piezoelectric, magnetostrictive or magnetoelectric effects. Electromagnetically induced specific forces acting at the boundary and within the domain of a dielectric body are, within a continuum mechanics framework, commonly denoted as Maxwell stresses. In transient electromagnetic fields, the Poynting vector gives another contribution to mechanical stresses. First, a system of transient partial differential equations is presented. Introducing scalar and vector potentials for the electromagnetic fields and representing the mechanical strain by displacement fields, seven coupled differential equations govern the boundary value problem, accounting for linear constitutive equations of magnetoelectroelasticity. To reduce the effort of numerical solution, the system of equations is partly decoupled applying generalized forms of Coulomb and Lorenz gauge transformations [1,2]. A weak formulation is given to establish a basis for a finite element solution. The influence of constitutive magnetoelectric coupling on electromagnetic wave propagation is finally demonstrated with a simple one-dimensional example. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于遗传算法的岩体参数辨识研究

目前,对于岩体流固耦合分析研究已经很多,而耦合分析常常受困于计算参数的取值,因此对两场耦合模型中的计算参数反演分析是非常必要的.根据实测的水头、位移资料,利用遗传算法,建立了等效连续岩体渗流场与应力场耦合计算参数辨识模型.并对某算例在库水位下降情况下,以渗流场与应力场耦合正分析计算结果作为"实测值",进行两场耦合参数辨识分析.从参数辨识的结果来看,验证了所提出的思路、方法以及程序的正确性和可行性.两场耦合计算参数进行反演分.析,对于两场耦合模型的建立和计算结果的可靠性是非常有意义的.     

Modeling of microscopic failure in heterogeneous materials

The proper modeling of state-of-the-art engineering materials requires a profound understanding of the nonlinear macroscopic material behavior. Especially for heterogeneous materials the effective macroscopic response is amongst others driven by damage effects and the inelastic material behavior of the individual constituents [1]. Since the macroscopic length scale of such materials is significantly larger than the fine-scale structure, a direct modeling of the local structure in a component model is not convenient. Multiscale techniques can be used to predict the effective material behavior. To this end, the authors developed a modeling technique based on representative volume elements (RVE) to predict the effective material behavior on different length scales. The extended finite element method (XFEM) is used to model discontinuities within the material structure independent of the underlying FE mesh. A dual enrichment strategy allows for the combined modeling of kinks (material interfaces) and jumps (cracks) within the displacement field [2]. The gradual degradation of the interface is thereby controlled by a cohesive zone model. In addition to interface failure, a non-local strain driven continuum damage model has been formulated to efficiently detect localization zones within the material phases. An integral formulation introduces a characteristic length scale and assures the convergence of the approach upon mesh refinement [3]. The proposed method allows for an efficient modeling of substantial failure mechanisms within a heterogeneous structure without the need of remeshing or element substitution. Due to the generality of the approach it can be used on different length scales. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase field simulations on the interaction of ferroelectric domains with point defects

One of the suspected micro-mechanical mechanisms causing electric fatigue in ferroelectric materials is the hindering and blocking of domain wall movement. These blocking or pinning phenomena are thought to be due to point defects which interact with domain walls and applied external loads. A phase field model employing the spontaneous polarization as an order parameter is used to simulate the inhomogeneous material behavior. The coupled field equations are solved using the Finite Element Method. The influence of a stationary point defect on a domain wall is shown in a numerical simulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of Cosserat material parameters obtained by homogenization of a Cauchy continuum

If a higher order continuum theory is used in a numerical simulation, the material parameters have to be derived. 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 micropolar continuum and the homogenization approach is followed by the discussion of the identified properties. Therefore some parameter studies are presented. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical modeling of atomistic-to-continuum coupling based on bridging domain method

Nano-submodeling is an approach that enables insertion of nano-refined submodel (atomistic) in the global model (continuum). In this work analysis of the spurious effects that may arise in the concurrent atomistic-to-continuum coupling is performed. The coupling is based on the overlapping domain decomposition (ODD) method called bridging domain [1, 2] (similar is Arlequin [3] method) where different models are overlapped and the displacements compatibility is enforced via Lagrange multipliers (LM). Some coupling options such as energy weighting, coupling zone geometry and LM field interpolation are tested. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional cell model analysis of ductile damage under dynamic loading condition

The paper deals with the damage and fracture behavior of ductile metals under dynamic loading conditions. The in [1–3] presented phenomenological continuum damage and fracture model, which takes into account the rate- and temperature-dependence of the material, provides reasonable results of experiments with high strain rates while the identification of the corresponding material parameters results difficult from the available experimental data. This lack of information can be resolved by micro-mechanical numerical simulations of void containing unit-cells. In this context results of dynamic micro-mechanical simulations are presented which can be used to study the damage effects on the micro-scale and to validate the rate-dependent continuum damage model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison of displacement and stress fields upon diffraction of elastic waves at different defects: Cracks and thin rigid intrusions

This paper considers the problem of diffraction of elastic waves at rectilinear defects (cracks or thin rigid intrusions) in an unbounded elastic medium. A discontinuous solution of the Helmholtz equation is used to reduce the problem to a singular integral equation in unknown jumps. There is a detailed analysis and comparison of the stressed state close to the defects and the diffraction field at large distances from the defect for defects of different types. Translated fromDinamicheskie Sistemy. Vol. 12, pp. 14–23, 1993.