Variational Based Stability Analysis in Coupled Electro-Mechanics at Finite Strains

Smart or functional materials are in focus in broad fields of recent research. In particular, electroactive polymers (EAPs) with their large actuator strains are used for innovative design in robotics. These materials become practical as soon as the material is robust, reliable and long lived. However, EAPs suffer from different types of instabilities, such as unstable thinning of actuators, local buckling induced by coexistent states and electrical breakdown. One distinguishes between structural instability, i.e. instabilities in the global structural response like buckling or wrinkling and local or material instability, e.g. the formation of microstructures and phase decompositions. These effects are well known and analyzed for purely mechanical phenomena, but not fully developed in the context of electro-mechanically coupled problems. To this end, we develop new variational-based stability criteria in finite electro-elastostatics, by postulating an electro-mechanical quasi-convexity condition for weakly convex energy and weakly convex-concave energy-enthalpy formulations. Starting from distinct variational principles, we develop criteria for local and global stability and construct stability checks, which accompany equilibrium paths for the detection of critical points in finite element models. Numerical examples validate the developed concepts for boundary value problems. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

高体积百分比颗粒增强聚合物材料的有效粘弹性性质

聚合物材料通常表现为粘弹性性质．为了改进聚合物材料的力学性能,通常将某种无机材料以颗粒或纤维的形式填充到聚合物中,从而得到增强、增韧的聚合物基复合材料．提出了一个新的细观力学模型,用于预测颗粒增强聚合物复合材料的有效粘弹性性质,尤其针对高体积百分比的颗粒夹杂复合材料,该方法基于Laplace变换和双夹杂相互作用的弹性模型．计算了玻璃微珠/ED-6复合材料的有效松弛模量以及恒应变率下的应力应变关系．计算结果表明在高体积百分比下该文方法比基于Mori Tanaka方法预测的粘弹性效应明显减弱．     

广义Eady模型的非线性稳定性

对于广义Eady模型,分别讨论了密度函数是常数函数与指数函数两种情形,利用变分原理,考虑到动量守恒的约束条件,得到了优化的Poincare不等式,从而得到了新的非线性稳定性定理,并且得到了在径向长度分别不大于纬向长度的0．84402倍及0．86068倍时（这对于地球的实际情况是成立的）,非线性稳定性判据与线性稳定性判据是一致的．     

广义 Eady 模型的非线性稳定性

对于广义 Eady 模型, 分别讨论了密度函数是常数函数与指数函数两种情形, 利用变分原理, 考虑到动量守恒的约束条件, 得到了优化的 Poincaré不等式, 从而得到了新的非线性稳定性定理, 并且得到了在径向长度分别不大于纬向长度的0.84402倍 及 0.86068 倍时(这对于地球的实际情况是成立的), 非线性稳定性判据与线性稳定性判据是一致的.     

具周期输入的有限连续分布时滞神经网络周期解的全局指数稳定性

在去掉对激励函数有界、连续可导、平均时滞有界的条件下,仅要求激励函数满足Lip-schitz条件和连接权矩阵之间的关系是一个M-矩阵的情形下,利用重合度理论、Dini导数等知识得出了具周期输入的有限连续分布的一类细胞神经网络周期解的存在性.利用Dini导数和不等式分析技术在同样条件下得出了周期解的指数稳定性.推广和改进了前人的结论.并举例说明了所得定理的有效性.     

A Phase-Field Model of Ductile Fracture at Finite Strains

This work outlines a variational-based framework for the phase field modeling of ductile fracture in elastic-plastic solids at large strains. The phase field approach regularizes sharp crack discontinuities within a pure continuum setting by a specific gradient damage model with geometric features rooted in fracture mechanics. Based on the recent works [1, 2], the phase field model of ductile fracture is linked to a formulation of gradient plasticity at finite strains in order to ensure the crack to evolve inside the plastic zones. The thermodynamic formulation is based on the definition of a constitutive work density function including the stored elastic energy and the dissipated work due to plasticity and fracture. The proposed canonical theory is shown to be governed by a rate-type minimization principle, which determines the coupled multi-field evolution problem. Another aspect is the regularization towards a micromorphic gradient plasticity-damage setting which enhances the robustness of the finite element formulation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computational Homogenisation of Polycrystalline Elastoplastic Microstructures at Finite Deformation

During sheet bulk metal forming processes both, flat geometries and three-dimensional structures change their shape significantly while undergoing large plastic deformations. As for forming processes, FE-simulations are often done before in situ experiments, a very accurate material model is required, performing well for a huge variety of different geometrical characteristics. Because of the crystalline nature of metals, anisotropies have to be taken into account. Macroscopically observable plastic deformation is traced back to dislocations within considered slip systems in the crystals causing plastic anisotropy on the microscopic and the macroscopic level. A finite crystal plasticity model is used to model the behaviour of polycrystalline materials in representative volume elements (RVEs) of the microstructure. A multiplicative decomposition of the deformation gradient into elastic and plastic parts is performed, as well as a volumetric-deviatoric split of the elastic contribution. In order to circumvent singularities stemming from the linear dependency of the slip system vectors, a viscoplastic power-law is introduced providing the evolution of the plastic slips and slip resistances. The model is validated with experimental microstructural data under deformation. Through homogenisation and optimisation techniques, effective stress-strain curves are determined and can be compared to results from real forming processes leading to a suitable effective material model. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lotka-Volterra模型全局指数稳定性和相应方程解的分岔

研究具有变系数的时滞Lottka-Volterra模型,证明该模型在适当的条件下存在正的平衡解,并给出了正平衡解指数稳定的充分条件,进一步讨论了模型(1)的相对退化形式的解产生Hopf分岔现象.     

Mixed Variational Principles and Robust Finite Element Design of Gradient Plasticity at Finite Strains

The modeling of size effects in elastic-plastic solids, such as the width of shear bands or the grain size dependence in polycrystals, must be based on non-standard theories which incorporate length-scales. This is achieved by models of strain gradient plasticity, incorporating spatial gradients of selected micro-structural fields which describe the evolving dissipative mechanisms. The key aspect of this work is to provide a rigorous incremental variational formulation and mixed finite element design of additive finite gradient plasticity in the logarithmic strain space. We start from a mixed saddle point principle for metric-type plasticity, which is specified for the important model problem of isochoric plasticity with gradient-extended hardening/softening response. To this end, we propose a novel finite element design of the coupled problem incorporating a local-global solution strategy of short- and long-range fields. This includes several new aspects, such as extended Q1P0-type and MINI-type finite elements for gradient plasticity [4]. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phenomenological and Numerical Simulation of Shape Memory Alloys at Finite Strains

A phenomenological material law for pseudo elastic NiTi shape memory alloys (SMA) is presented. The model was derived from a thermodynamical framework and is well-suited to describe the thermomechanical coupled behaviour of the material. The material law, which was originally derived for small deformations, was extended to finite deformations using the Eulerian frame, in particular Hencky's logarithmic strain and the logarithmic rate. A first emphasis is on the physical interpretation of the material parameters and their identification. A second focus lies in the presentation of a structural example for the implementation of the material law into a commercial Finite Element code. Additionally a comparison of the numerical and experimental data of the presented example is performed. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An asymmetric Least-Squares Mixed Finite Element for Elasto-Plasticity at Small Strains

We present a mixed finite element based on a modified least-squares formulation for rate-independent elasto-plasticity. Due to kink-like points in the least-squares functional, the first variation is not always continuous and a standard Newton method could fail in order to minimize the least-squares functional. In order to keep the availability of the Newton method, we introduce a modified least-squares approach, which guarantees the continuity of the resulting weak form. Finally, a numerical example is presented to show the applicability and performance. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

高体积分数颗粒增强复合材料有效线性与非线性介电性质的研究

基于两球相互作用的近似解,提出了一种解析方法来预测颗粒增强复合材料的线性有效介电性质,该方法可以应用于颗粒体积分数较高(可达50%)的情况．利用割线方法还研究了该类复合材料的非线性有效性质．结果表明当外加电场较弱时该文提出的方法与Stroud和Hui的方法比较接近,当外场较强时与Yu和Hui的方法一致．     

Automatic Implementation of Elasto-plastic Incremental Formulations at Finite Strains using Hyper-Dual Numbers

Numerical simulation of standard dissipative materials undergoing finite strains remains an important and challenging topic in computational mechanics. The incremental variational formulation (IVF), firstly proposed by Ortiz et al. [1], provides a general variational framework which is suitable for the implementation of a broad range of constitutive laws for standard dissipative materials. The IVF recasts the inelasticity theory as an equivalent optimization problem where the incremental stress potential is minimized with respect to the internal variables. However, their implementation often requires more effort than classical formulations due to high-order tensor derivatives. In this contribution, a novel implementation of IVFs is presented to arrive at a fully automatic and robust scheme with computer accuracy using hyper-dual numbers (HDNs). The HDNs, which are originally developed by Fike [2], derive exact and automatic derivative calculations without any cumbersome choice of perturbation values. Its uncomplicated implementation for associative finite strain elasto-plasticity and its performance is illustrated by a representative numerical example. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational Formulation and Numerical Implementation of Diffusion in Hydrogels at Finite Strains

Hydrogels are polymeric materials with a cross-linked network which can absorb water. Due to their bio-compatibility, hydrogels have many applications in biology and medicine. Recently modeling the mechanical behavior of hydrogels has attracted a great deal of attention among researchers, see e.g., [1] and [2]. Following our previous works [3], [4] and [5] we now present a variational framework for swelling phenomenon in hydrogels. The variational formulation of the problem can be done using a saddle-point principle or a minimization principle. Saddle-point principle has to fulfill the Ladyzhenskaya-Babuška-Brezzi (LBB) condition in order to lead to a stable finite element scheme. The key aspect of our proposed minimization principle is its advantage with regard to an unconstrained fem implementation. In this work we aim to compare the numerical performance of these two variational formulations for swelling of hydrogels. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

多重尺度法在平面Couette流稳定性分析中的应用*

本文用文[3]所建议的多重尺度法分析了经过修正的平面Couette流动的线性稳定性性质。在平面Couette流的分析中,用多重尺度法可以找到不稳定的Tollmien-Schlichting波,但却不能找到最不稳定的模态,通过与文[3]相比较,本文对这种方法进行了探讨。     

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)     

Stability of Preconditioned Finite Volume Schemes at Low Mach Numbers

A finite volume method for inviscid unsteady flows at low Mach numbers is studied. The method uses a preconditioning of the dissipation term within the numerical flux function only. It can be observed by numerical experiments that the preconditioned scheme combined with an explicit time integrator is unstable if the time step Δt does not satisfy the requirement to be as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to , M → 0, though producing unphysical results. A comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis is presented, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M^–2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. The theoretical results are afterwards confirmed by numerical experiments. AMS subject classification (2000) 35L65, 35C20, 76G25