Theory and Numerics of Rate-Dependent Incremental Variational Formulations in Ferroelectricity

This paper is concerned with macroscopic continuous and discrete variational formulations for domain switching effects at small strains, which occur in ferroelectric ceramics. The developed new three–dimensional model is thermodynamically–consistent and determined by two scalar–valued functions: the energy storage function (Helmholtz free energy) and the dissipation function, which is in particular rate–dependent. The constitutive model successfully reproduces the ferroelastic and the ferroelectric hysteresis as well as the butterfly hysteresis for ferroelectric ceramics. The rate–dependent character of the dissipation function allows us also to reproduce the experimentally observed rate dependency of the above mentioned hysteresis phenomena. An important aspect is the numerical implementation of the coupled problem. The discretization of the two–field problem appears, as a consequence of the proposed incremental variational principle, in a symmetric format. The performance of the proposed methods is demonstrated by means of a benchmark problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

剪切变形下非晶态高聚物的力学行为

基于非平衡态热力学理论，提出了一个适用于不可压材料的新的热粘弹性本构模型．该模型将橡胶弹性理论中的非高斯分子网络模型推广到计及粘性和热效应的情形．通过引入一组二阶张量形式的内变量，建议了一个新的Helmholtz自由能表达式，从而可以用来合理描述内变量的演化规律．根据以上模型，重点研究了热粘弹性材料在简单剪切变形下的力学行为，考察了由于分子链取向分布的变化而产生的“粘性耗散诱导”各向异性，讨论了应变率效应和由于粘性耗散而导致的热软化效应对剪应力的影响．理论预测结果与G’Sell等人的实验数据的定性比较表明了新的本构模型的有效性。     

On dynamic finite element analysis of viscoplastic thin-walled structures with non-local damage: A phase-field approach

[EN] In this work, a nonlocal damage model is proposed for dynamic analysis of viscoplastic shell structures using the phase-field approach. A phase-field variable on the mid surface is introduced to characterize the nonlocal damage as well as the transition between undamaged and damaged phase. The total free energy in [1] is modified as a sum of Helmholtz free-energy and Ginzburg-Landau one. The latter is defined as a function of the phase-field variable and its corresponding gradient. This enhancement gives rise to an introduction of gradient parameters in terms of a substructure-related intrinsic length-scale. The evolution of the phase-field based damage variable can be found from the minimum principle of the dissipation potential [3]. The performance of the proposed model is demonstrated through numerical results of a plate with a circular hole. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Delamination of Grain-Interfaces in Silicon Nitride

Intergranular cracking due to delamination of grain interfaces along with the development of bridging grains is the most important mechanism for the high fracture toughness of silicon nitride. In this line, an interface behavior, which is extending the Coulomb friction concept into the tensile domain has been implemented into a thermodynamical consistent frame work of Helmholtz free energy and dissipation. The model is used to describe the fracture process in a simple model geometry with a β-Si₃N₄ grain embedded into a precracked matrix of oxynitride glass. The material model considers the thermoelastic anisotropy of the grain and the thermal residual stresses, which evolve during the cooling of the model from the glass transition temperature to room temperature. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interaction of phase-transformations and plasticity – a multi-phase micro-sphere approach

We present an efficient model for the simulation of solid to solid phase-transformations in polycrystalline materials. As a basis, we implement a scalar-valued Gibbs-energy-barrier-based phase-transformation model making use of statistical physics. In this work, we particularly adopt the model for the simulation of phase-transformations between an austenitic parent phase and a martensitic tension and compression phase. The incorporation of plasticity phenomena is established by enhancing the Helmholtz free energy functions of the material phases considered, where the plastic driving forces acting in each phase are derived from the overall free energy potential. The coupled model is embedded into a micro-sphere formulation in order to simulate three-dimensional boundary value problems—a technique well-established in the context of computational inelasticity at small strains. It is shown that the model is capable of reflecting experimentally observed behaviour. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multi-Scale and Multi-Component Approach for Solidification Processes

This articel focuses on a thermo-mechanical model for numerical simulation of solidification processes considering two different scales, both in time and space. The macro-scale implies two phases (solid and liquid steel), and is described by use of the theory of porous media (TPM). Moreover, a strong thermal coupling is addressed as well as finite J₂ elastic-plastic solid behavior. The physics of solidification during heat dissipation is covered on the micro-scale in the framework of phase-field modeling. Therefore, a Ginzburg-Landau type free energy function is used. After discussing the main model details, a numerical example will demonstrate the principal performance of the presented model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of phase-transformations based on numerical minimization of intersecting Gibbs energy potentials

We present a novel approach for the simulation of solid to solid phase-transformations in polycrystalline materials. To facilitate the utilization of a non-affine micro-sphere formulation with volumetric-deviatoric split, we introduce Helmholtz free energy functions depending on volumetric and deviatoric strain measures for the underlying scalar-valued phase-transformation model. As an extension of affine micro-sphere models [5], the non-affine micro-sphere formulation with volumetric-deviatoric split allows to capture different Young's moduli and Poisson's ratios on the macro-scale [1]. As a consequence, the temperature-dependent free energy assigned to each individual phase takes the form of an elliptic paraboloid in volumetric-deviatoric strain space, where the energy landscape of the overall material is obtained from the contributions of the individual constituents. For the evolution of volume fractions, we use an approach based on statistical physics–taking into account actual Gibbs energy barriers and transformation probabilities [2]. The computation of individual energy barriers between the phases considered is enabled by numerical minimization of parametric intersection curves of elliptic Gibbs energy paraboloids. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vacuum and Thermal Energies for Two Oscillators Interacting Through A Field

We consider a simple (1+1)-dimensional model for the Casimir–Polder interaction consisting of two oscillators coupled to a scalar field. We include dissipation in a first-principles approach by allowing the oscillators to interact with heat baths. For this system, we derive an expression for the free energy in terms of real frequencies. From this representation, we derive the Matsubara representation for the case with dissipation. We consider the case of vanishing intrinsic frequencies of the oscillators and show that the contribution from the zeroth Matsubara frequency is modified in this case and no problem with the laws of thermodynamics appears.     

Energy estimates for reaction-diffusion processes of charged species

We consider electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistical relations. We investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. Here the essential idea is an estimate of the free energy by the dissipation rate. The same properties are obtained for a fully implicit time discretized version of the problem. Moreover, we provide a space discretized scheme for the electro-reaction-diffusion system which is dissipative (the free energy decays monotonously). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase-Field Modeling of Hydraulic Fracture

Hydraulically driven fracture has gained more and more research activity in the last few years, especially due to the growing interest of the petroleum industry. Key challenge for a powerful simulation of this scenario is an effective modeling and numerical implementation of the behavior of the solid skeleton and the fluid phase, the mechanical coupling between the two phases as well as the incorporation of the fracture process. Existing models for hydraulic fracturing can be found for example in [1], where the crack path is predetermined, or in [2] who use a phase field fracture model in an elastic framework, however without incorporating the fluid flow. In this work we propose a new compact model structure for the Biot-type fluid transport in porous media at finite strains based on only two constitutive functions, that is the free energy function ψ and a dissipation potential ϕ that includes the incorporation of an additional Poiseuille-type fluid flow in cracks. This formulation is coupled to a phase field approach for fracture and is fully variational in nature, as shown in [3]. In contrast to formulations with a sharp-crack discontinuity, the proposed regularized approach has the main advantage of a straight-forward modeling of complex crack patterns including branching. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An enhanced‐physics‐based scheme for the NS‐α turbulence model

We study a new enhanced‐physics‐based numerical scheme for the NS‐alpha turbulence model that conserves both energy and helicity. Although most turbulence models (in the continuous case) conserve only energy, NS‐alpha is one of only a very few that also conserve helicity. This is one reason why it is becoming accepted as the most physically accurate turbulence model. However, no numerical scheme for NS‐alpha, until now, conserved both energy and helicity, and thus the advantage gained in physical accuracy by modeling with NS‐alpha could be lost in a computation. This report presents a finite element numerical scheme, and gives a rigorous analysis of its conservation properties, stability, solution existence, and convergence. A key feature of the analysis is the identification of the discrete energy and energy dissipation norms, and proofs that these norms are equivalent (provided a careful choice of filtering radius) in the discrete space to the usual energy and energy dissipation norms. Numerical experiments are given to demonstrate the effectiveness of the scheme over usual (helicity‐ignoring) schemes. A generalization of this scheme to a family of high‐order NS‐alpha‐deconvolution models, which combine the attractive physical properties of NS‐alpha with the high accuracy gained by combining α‐filtering with van Cittert approximate deconvolution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On the Description of Growth in Saturated Living Tissues

A comprehensive model for biological tissues must include the anisotropic tissue structure, the interstitial liquid wich saturated the tissue and the growth mechanism of the tissue. In the present contribution this is done by use of a three phasic model with a solid, liquid and nutrient phase in the framework of the porous media theory (TPM). In order to characterize the transversal isotropic skeleton behavior, an invariant formulation of the Helmholtz free energy function and the permeability tensor is suggested. The growth mechanism is characterizes by a mass transfer between the nutrient and solid phase. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigation of the deformation characteristics and heat generation in pneumatic aircraft tires Part II. Thermal modeling

A numerical procedure to determine the temperature rise in aircraft tires under free rolling conditions is presented in this article. Energy dissipation from cyclic inelastic deformation is considered the main heat generation source. This modeling considers the deformation process of the tire to be a steady-state problem, where all concurrent cycles are assumed to be the same as the first. The inelastic energy is determined by imposing a phase lag between the strain and the stress fields. The phase lag is assumed to be frequency independent in the range of interest, in keeping with the experimental observations in aircraft tire materials. It is further assumed that the inelastic energy is completely converted into volumetric heat input for a transient thermal conduction analysis. A conduction model is described and results are compared against thermocouple data recorded by Clark and Dodge [1].     

On the Navier-Stokes Equations with Energy-Dependent Nonlocal Viscosities

We discuss the mathematical modeling of incompressible viscous flows for which the viscosity depends on the total dissipation energy. In the two-dimensional periodic case, we begin with the case of temperature-dependent viscosities with very large thermal conductivity in the heat convective equation, in which we obtain the Navier-Stokes system coupled with an ordinary differential equation involving the dissipation energy as the asymptotic limit. Letting further the latent heat to vanish, we derive the Navier-Stokes equations with a nonlocal viscosity depending on the total dissipation of energy. Bibliography: 7 titles.Dedicated to V. A. Solonnikov on the occasion of his 70th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 71–91.     

Existence of frequency modes coupling seismic waves and vibrating tall buildings

We prove in this paper an existence result for frequency modes coupling seismic waves and vibrating tall buildings. The derivation from physical principles of a set of equations modeling this phenomenon was done in previous studies. In this model all vibrations are assumed to be anti-plane and time harmonic so the two dimensional Helmholtz equation can be used. A coupling frequency mode is obtained once we can determine a wavenumber such that the solution of the corresponding Helmholtz equation in the lower half plane with relevant Neumann and Dirichlet conditions at the interface satisfies a specific integral equation at the base of an idealized tall building. Although numerical simulations suggest that such wavenumbers should exist, as far as we know, to date, there is no theoretical proof of their existence. This is what this present study offers to provide.     

Stable long-term simulation of dynamically loaded elastomers

A well–known problem in long–term simulations of flexible solid bodies is the restriction to small time steps of standard time integrators in order to obtain a stable simulation. One approach is to achieve exact energy conservation while simulating a nonlinear elastic body (see Reference [1] and the references therein). Additionally, total linear and total angular momentum is conserved for a free motion. This approach leads to a qualitatively improved solution, because the approximated time evolution exactly fulfills the same physical laws as the exact time evolution. Incorporating energy dissipation, the energy conserving time integration is extended to an energy consistent time integration. It turned out that such an energy consistent time integration is also of great advantage when computing finite motions of flexible solid bodies with material dissipation (see References [2,3]). This paper points out that an energy consistent time discretisation is also advantageous for dynamic finite deformation thermoviscoelasticity under dynamic loads. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hard versus soft impacts in oscillatory systems modeling

The applicability of the soft and hard impact models in modeling of vibro-impact systems is discussed in the paper. We derive the conditions which allow the same rate of energy dissipation in dynamical systems which use both impact models. The advantages and disadvantages of both models in modeling are discussed. We show that in the case of the stiff base both methods give the same results but the elastic base application of the hard impact model leads to wrong results.     

Geometrical Aspects of the Incorporation of Free Space in Magnetomechanics at Finite Strains

This paper addresses the modeling of finite magnetoelasticity with a particular focus on the incorporation of free space energy storage and magnetic body forces. First, a coordinate-invariant formulation of an extended Neo-Hookean model with magnetostrictive coupling is presented. Then, a finite element model for finite magnetomechanics based on a variational stationarity principle is formulated. The influence of magnetic body forces is studied in an exemplary boundary value problem. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)