Computation of Material Forces in Thermoplasticity Based on Alternative Smoothing Algorithms

Different approaches to the computation of material forces in inelastic structures are investigated. Dissipative effects in inelastic materials are described by internal variables. The formulation of balance equations in the material space requires the computation of gradients of these internal variables. The computational evaluation of these gradients in the context of finite element simulations needs a global representation of the internal variable fields. On the one side, this request can be carried out by a global formulation that discretizes the internal variable fields in terms of nodal degrees additional to the displacements. A numerically more effective approach applies smoothing algorithms which project the internal variables of a typical local formulation from the integration points onto the nodal points. In detail, the implementation of two smoothing algorithms for the computation of material forces is dicussed. The L2–projection necessiates the solution of a system of equations on the global level. A patch recovery yields a smoothed solution from an element patch surrounding the nodal point of interest. The performance of both algorithms is compared for the material force computation in finite thermoplasticity. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Configurational-Force-Driven Crack Propagation and Adaptive Mesh Refinement in Finite Inelasticity

We introduce a consistent variational framework for inelasticity at finite strains, yielding dual balances in physical and material space as the Euler equations. The formulation is employed for the simultaneous usage of configurational forces as both driving forces for crack propagation as well as h-adaptive mesh refinement. The theoretical basis builds upon a global balance of internal and external power, where the mechanical response is exclusively governed by two scalar functions, the free energy function and a dissipation potential. The resulting variational structure is exploited in the context of fracture mechanics and yields evolution equations for internal variables. In the discrete setting, we present a geometry model fully separated from the finite element mesh structure that represents structural changes of the material configuration due to crack propagation. Advanced meshing algorithms provide an optimal discretization at the crack tip. Local and global criteria are obtained via error estimators based on configurational forces being interpreted as indicators of an energetic misfit due to an insufficient discretization. The numerical handling is decomposed into a staggered algorithm scheme for the dual set of equilibrium equations in material and physical space and efficient mesh generation tools. Exemplary numerical examples are considered to illustrate the method and to underline the effects of inelastic material behaviour in the presented context. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermodynamics in material space-time

The concept of material forces is well established in the continuum theory of defects. These forces are quantities in a three-dimensional material manifold. The manifold can be augmented to a four-dimensional material space-time by a time-like coordinate. The inelastic phenomena caused by moving defects are highly dissipative. So a thermodynamic theory in material space-time seems to be indispensable. Some first steps towards such a theory are developed here. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Material-Force-Based Refinement Indicators in Adaptive Strategies

A material-force-based refinement indicator for adaptive finite element strategies for finite elasto-plasticity is proposed. Starting from the local format of the spatial balance of linear momentum, a dual material counterpart in terms of Eshelby's energy-momentum tensor is derived. For inelastic problems, this material balance law depends on the material gradient of the internal variables. In a global format the material balance equation coincides with an equilibrium condition of material forces. For a homogeneous body, this condition corresponds to vanishing discrete material nodal forces. However, due to insufficient discretization, spurious material forces occur at the interior nodes of the finite element mesh. These nodal forces are used as an indicator for mesh refinement. Assigning the ideas of elasticity, where material forces have a clear energetic meaning, the magnitude of the discrete nodal forces is used to define a relative global criterion governing the decision on mesh refinement. Following the same reasoning, in a second step a criterion on the element level is computed which governs the local h-adaptive refinement procedure. The mesh refinement is documented for a representative numerical example of finite elasto-plasticity. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exploitation of Configurational Forces in Extended Nonlocal Continua with Microstructure

We investigate aspects of the application of configurational forces in extended nonlocal continua with microstructure. Focussing on multifield approaches to gradient–type inelastic solids, the coupled problem is governed by the macroscopic deformation field, while nonlocal inelastic effects on the microstructure are described by a family of order parameter fields. The dual macro– and micro–field equations are derived within an incremental variational framework. Using an incremental principle, due to the variation with respect to the material position, an additional balance in the material space appears with the dual macro–micro–balances in the physical space. In view of the numerical implementation of this coupled problem by a finite element method, the incremental variational framework is recast into a discrete format in terms of discrete macro– and micro–physical nodal forces and configurational nodal forces. Applying a staggered solution scheme, the configurational branch is used as a postprocessing procedure with all the ingredients known from the solution of the coupled macro–micro–problem. The procedure is implemented for a nonlocal, viscous damage model. The consequences with regard to the configurational nodal forces are assessed by means of a numerical example. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类指定应力问题的变分原理与应用

针对有限元分析中对应力或内力有指定条件的问题,引入非弹性应变作为实现指定应力条件的附加未知量,在小变形条件下描述了指定应力条件应当满足的弹性力学控制方程;以位移和未知非弹性应变作为独立变量,建立了具有指定应力条件问题的势能变分原理和虚功方程;以位移、弹性应变、未知非弹性应变和应力为独立变量,建立了一个含四类变量的广义变...     

Conditional Lagrange derivative and its application

Our paper aims to show what to do in material modeling for an inelastic body. The method we use is based on the virtual work σ ′δε and on the variation of some function W with respect to the strain tensor. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stress L∞-estimates and the uniqueness problem for the quasistatic equations to the model of Bodner-Partom

This paper proves the uniqueness result for global in time large solutions of quasistatic equations to an inelastic model of material behavior of metals, provided that an a priori L^∞-estimation for the Cauchy stress tensor holds. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Investigation of elastoplastic effects of cables under large spatial deformation

Cables are complex components consisting of a multi-layer structure and various materials. The structural setup includes for example conducting wires, isolating shields and protecting sheaths. This leads to several inelastic effects under large deformations like pull-out of wires, delamination of layers or friction between the constituents. The materials used in cables belong to different material classes and consequently show different behavior under load. Elastoplastic behavior has to be expected for metallic wires, whereas polymer layers behave viscoelastically. The combination of these inelastic effects caused by the structure and constituents of cables motivates the inclusion of inelasticity in the material model on a phenomenological level. Since cables are flexible, slender structures, they can be described physically correctly by the theory of Cosserat rods. In this context, the constitutive equations are formulated in terms of the sectional quantities. The related model parameters have to be determined in suitable experiments. As cables undergo large multiaxial deformations in applications, uniaxial experiments are not sufficient for their characterization. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stress L∞-estimates and the uniqueness problem for the equations to the model of Bodner–Partom in the two-dimensional case

This paper is a continuation of the work [9]. We prove the uniqueness result for global in time large solutions of dynamic equations to an inelastic model of material behaviour of metals in the two-dimensional case, provided a higher regularity of the solutions. Moreover, the 𝕃^p-stability for p<2 of the solutions in the case of homogeneous boundary data is established. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Determination of material parameters of heterogeneous viscoelastic curing polymers

Two goals characterize the present contribution: First, the development of a numerical approach for determining the properties of the material microstructure, and second, the shift of the focus of the inverse analysis from investigating a purely elastic material toward the parameter identification related to heterogeneous inelastic materials. As a rule, the constitutive laws in this case involve a greater number of material parameters the determination of which requires different kinds of tests. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A geometrically nonlinear elongation of a plane with an elliptic inclusion

We deduce relations in geometrically nonlinear formulation for the stresses and displacements on the boundary of an elliptic cut containing an inclusion with elastic characteristics different from the elastic characteristics of the base material. We find the dependence of the stress concentration on the size of the tensile force. We show the necessity of taking account of the geometric nonlinearity for low-modulus materials with nonuniformities. We give the dependence of the stress concentration on the size of the tensile forces for various ratios of elastic constants for the inclusion and the base material.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 89–95.     

Influence of the nonlinear matrix material behaviour on the effective properties of textile-reinforced thermoplastics

For a consistent lightweight design the consideration of the nonlinear macroscopic material behaviour of composites, which is amongst others driven by damage and strain-rate effects on the mesoscale, is required. Therefore, a modelling approach using numerical homogenization techniques is applied to predict the effective nonlinear material behaviour of the composite based on the finite element simulation of a representative volume element (RVE). In this RVE suitable constitutive relations account for the material behaviour of each constituents. While the reinforcing glass fibres are assumed to remain linear elastic, a viscoplastic constitutive law is applied to represent the strain-rate dependent, inelastic deformation of the matrix material. In order to analyse the influence of the nonlinear matrix material behaviour on the global mechanical response of the composite, effective stress-strain-curves are computed for different load cases and compared to experimental observations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inelastic behavior of the single domain of metal-ceramic composites with lamellar microstructure

Metal-ceramic composites are widely applied in the different brunches of industry. The composites are produced by squeeze-casting of the ceramic preform by molten aluminum alloy. The lamellar microstructure is obtained during freezing of ceramic suspension. The internal structure of the domains can be controlled via freeze-casting parameters. The material has high anisotropy level and its effective properties depend on lamella orientation. The aim of this study is numerical simulation of the inelastic behavior of the material and its verification by experiment. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimisation of Structures with Inelastic Deformations

In context of design optimisation, the treatment of inelastic, path-dependent materials is a topic of interest. As opposed to purely elastic materials, it is necessary to store and analyse the deformation history in order to appropriately describe inelastic material behaviour. For design optimisation of structures sensitivities of all quantities of influence have to be computed so as to use gradient based optimisation algorithms. Considering path-dependent materials the sensitivities of internal variables that represent the deformation history have to be additionally calculated. A numerical effective way of determining sensitivities is the variational sensitivity analysis. This approach is applied to develop a powerful and effective algorithm to compute sensitivities of the structural response with respect to their geometric design. (© 2016 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)