On the simulation of nonlinear hardening in metallic materials using the concept of representative directions

We generalize a uniaxial model of finite strain viscoplasticity using the concept of representative directions. As a result, a new phenomenological material model is obtained, which can describe the mechanical behavior under arbitrary loading conditions. The original uniaxial model takes the nonlinear isotropic and kinematic hardening into account, but it does not cover the distortional hardening. We show that the isotropic and kinematic hardening is completely retained during the process of generalization. Moreover, the distortional hardening effects are naturally induced by the concept. The resulting material model is validated by a comparison with real experimental data. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of a viscoplastic material model on a combined quasi-static-electromagnetic forming process

The prediction and simulation of material behavior by finite element methods has become indispensable. Furthermore, various phenomena in forming processes lead to highly differing results. In this work, we have investigated the process chain on a cross-shaped cup in cooperation between the Institute of Applied Mechanics (IFAM) of the RWTH Aachen and the Institute of Forming Technology and Lightweight Construction (IUL) of the TU Dortmund. A viscoplastic material model based on the multiplicative decomposition of the deformation gradient in the context of hyperelasticity has been used [1,2]. The finite strain constitutive model combines nonlinear kinematic and isotropic hardening and is derived in a thermodynamically consistent setting. This anisotropic viscoplastic model is based on the multiplicative decomposition of the deformation gradient in the context of hyperelasticity. The kinematic hardening component represents a continuum extension of the classical rheological model of Armstrong-Frederick kinematic hardening. The constitutive equations of the material model are integrated in an explicit manner and implemented as a user material subroutine in the commercial finite element package LS-DYNA with the electromagnetical module. The aim of the work is to show the increasing formability of the sheet by combining quasi-static deep drawing processes with high speed electromagnetic forming. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

各向异性强化时薄板的塑性分析

本文讨论了当薄板受到反复加载时由于存在鲍兴(Bauschinger)效应必须运用各向异性强化模型。文中导出了在线性随动强化时如何用广义力及广义塑性变形来表示薄板的加载条件,并讨论了可推广应用于非线性随动强化及混合强化(Mixed Hardening)情况。最后还以圆板为例进行了数值计算给出了计算结果。     

Comparison of different distortional hardening models suitable for the analysis of magnesium

HCP metals such as magnesium are characterized by a strong interplay between dislocation slip and deformation-induced twinning. These micromechanical processes result in a complex macroscopic behavior. More precisely, in addition to classical isotropic and kinematic hardening, the shape of the macroscopic yield function changes during deformation as well. This effect which is frequently referred to as distortional hardening is particularly pronounced in case of non-radial loading paths typical for most forming processes. Consequently, a physically sound distortional hardening is of utmost importance for several technically relevant applications. In the present contribution, three different of such enhanced hardening models are critically analyzed. Focus is on the modeling of magnesium. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermoviscoplasticity deduced from enhanced rheological models

A new ideal body is proposed for representing isotropic hardening. Hence, a rheological model of thermoviscoplasticity may be assembled with linear isotropic and kinematic hardening and nonlinear strain rate sensitivity. The related constitutive equations including the yield function and the flow rule are directly deduced from the kinematics and the stress equilibrium of the rheological network and result in a well-known model. Based on the free energy of the rheological network, the equation of heat conduction is obtained with the dissipative heat source term, driven by plastic deformations. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gradient Plasticity for Single Crystals

The kinematic relationship between classical single crystal kinematics and geometrically necessary dislocations will be clarified by a demonstrative example. The starting point for the dynamics is the contribution of geometrically necessary dislocations to the free energy, which leads to a kinematic hardening law. A simple two-dimensional shear example will be discussed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micro‐Macro Modelling of Piezoelectric Fiber Composites

The present work deals with the numerical modeling of 1‐3 periodic composites made of piezoceramic (PZT) fibers embedded in a soft non‐piezoelectric matrix. We especially focus on predicting the effective co‐efficients of the periodic transversely isotropic piezoelectric fiber composites using representative volume element method (unit cell method). The results which are obtained from the FEM technique are compared with analytical homogenization method for different volume fractions. The effective co‐efficients are obtained for rectangular and hexagonal arrangement of unidirectional piezoelectric fiber composites. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling the Springback of Sheet Metals at Large Deformations

Sheet metal forming simulations are increasingly replacing traditional build-and-break prototyping because they provide a less expensive and more rapid way to speed up product design time while improving quality and performance. Sheet metal is subjected to stretching, bending and unbending during forming, and an accurate prediction of the formability and springback of the sheet necessitates the use of an appropriate constitutive model. Use of isotropic hardening alone has been identified as a source of inaccurate springback prediction. In order to be able to describe the cyclic hardening behaviour and the Bauschinger effect, a large-deformation elasto-plastic material model with non-linear kinematic hardening is discussed in the present work. The model is derived from a thermodynamical framework and is based on the multiplicative split of the deformation gradient. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermodynamically and variationally consistent modeling of distortional hardening: application to magnesium

To capture the complex elastoplastic response of many materials, classical isotropic and kinematic hardening alone are often not sufficient. Typical phenomena which cannot be predicted by the aforementioned hardening models include, among others, cross hardening or more generally, the distortion of the yield function. However, such phenomena do play an important role in several applications in particular, for non-radial loading paths. Thus, they usually cannot be ignored. In the present contribution, a novel macroscopic model capturing all such effects is proposed. In contrast to most of the existing models in the literature, it is strictly derived from thermodynamical arguments. Furthermore, it is the first macroscopic model including distortional hardening which is also variationally consistent. More explicitly, all state variables follow naturally from energy minimization within advocated framework. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singular approximation of the method of periodic components in statistical mechanics of composite materials

A quasi-periodic model is developed for random structures of composites, when the locations of inclusions are given in terms of random deviations from nodes of an ideal periodic lattice. Solution of the stochastic boundary problem of the theory of elasticity is examined for a quasi-periodic component by the method of periodic components, which is reduced to determination of the field of deviations from the known solution for a corresponding periodic composite. The solution is presented for the tensor of effective elastic properties of a quasi-periodic composite in singular approximation of the method of periodic components in terms of familiar solutions for tensors of the effective elastic properties of composites with periodic and chaotic structures and the parameters of the quasi-periodic structure: the coefficient of periodicity and the tensor of the anisotropy of inclusion disorder. A numerical calculation is performed for the effective transversally isotropic elastic properties of unidirectional fibrous composites with different degrees of fiber disorder.Perm' State Technical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 460–473, July–August, 1997.     

Multiple bifurcation trees of period-1 motions to chaos in a periodically forced,time-delayed,hardening Duffing oscillator

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.     

Anisotropic multiplicative elastoplasticity for the prediction of earings in sheet forming

In this work, the simulation of earings in cup drawing by means of a recently developed anisotropic combined hardening material model is discussed. The model represents a multiplicative formulation of anisotropic elastoplasticity in the finite strain regime with nonlinear kinematic and isotropic hardening. Plastic anisotropy is described by the use of second-order structure tensors as additional arguments in the representation of the yield function and the plastic flow rule. The evolution equations are integrated by a form of the exponential map that preserves the plastic volume and the symmetry of the internal variables. Finite element simulations of cylindrical cup drawing processes are performed by means of ABAQUS/Standard where the discussed material model has been implemented into a user-defined reduced integration solid-shell element. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermo-mechanically coupled modelling of cellular MgO-C refractories under thermal shock

Refractory materials such as magnesium oxide carbon (MgO-C) composites are used in the steel-making industry for furnaces, ladles or oxygen converters. A new class of composites are cellular MgO-C materials, consisting of carbon foams filled with magnesium oxide and inclusions of gas filled pores. Cellular MgO-C composites have the advantage of significantly improving the thermo-mechanical properties [1]. This contribution focuses on the FEM implementation of a fully coupled thermo-mechanical continuum model. It is based on the theory of porous media (TPM) restricted by a kinematic coupling of the displacement field of all constituents. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

混凝土材料的粘塑性损伤本构模型

基于不可逆热力学,引入运动硬化、等向硬化和损伤内变量,构造了相应的自由能函数和流动势函数,推导出了混凝土材料的粘塑性损伤本构模型．数值模拟的结果表明,该模型能够避开屈服面和破坏准则的基本假设来描述混凝土材料的以下特性:压缩载荷作用下的体积膨胀现象;应变率敏感性;峰值后由损伤和破坏引起的应力软化和刚度退化现象A·D2由于此模型避开了根据各种变形阶段选择与其相应的本构模型的繁琐计算,因此更便于纳入复杂工况下应力分析有限元程序中．     

正交各向异性材料的混合硬化弹塑性本构方程

建立了混合硬化正交各向异性材料的屈服准则,进而推导了与之相关的塑性流动法则.根据简单应力状态的实验曲线,可得到广义等效应力-应变关系.初始屈服曲面与材料的弹性常数有关,材料退化为各向同性且只考虑各向同性硬化时,屈服函数退化为Huber-Mises屈服函数,相关的本构方程退化为Prandtl-Reuss方程.     

Micromechanics of Electro- and Magneto-active Soft Composites

We study the coupled behavior in soft active microstructured materials undergoing large deformations in the presence of an external electric or magnetic field. We focus on the role of the microstructures on the coupled behavior, and examine the phenomenon in the composites with (a) periodic composites with rectangular and hexagonal periodic unit cells, and (b) in composites with the random distributions of active particles embedded in a soft matrix. We show that for these similar microstructures exhibit very different responses in terms of the actuation, and the coupling phenomenon. Next, we consider the macroscopic and microscopic instabilities in the active composites. We show that the external field has a significant influence of the instability phenomena, and can stabilize or destabilize the composites depending on the direction relative to composite geometry. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical treatment of a finite strain viscoplasticity model based on a double multiplicative split

A viscoplastic material model of overstress type is analyzed. The model is based on the kinematic assumption of a double multiplicative split in order to simulate a nonlinear kinematic hardening. We investigate theoretically and numerically two implicit time–stepping methods, which were used lately in engineering literature to integrate the evolution equations in the context of multiplicative plasticity/viscoplasticity. A common feature of both methods is that the plastic incompressibility constraint is exactly satisfied. Moreover, we show that both methods preserve the symmetry of internal variables, which is inherent to the problem. According to testing results, both methods are equivalent regarding robustness, accuracy, and computational efficiency. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A non-conforming finite element for limit analysis of periodic composites

In this work, a non-conforming three-dimensional finite element coupled with direct methods and homogenization technique is presented for the limit analysis of periodic metal-matrix composites. Using this element, which is constructed from bilinear shape functions and enriched by internal second-order polynomials, limit analysis of composite material can be efficiently carried out. Accuracy and overall performance are illustrated through comparison with different structural solid elements in the context of direct as well as incremental methods. It is shown that the limit domain of periodic composites for different fiber distributions and volume fractions provides a foundation for the structural design. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Basis Reduction for the Shakedown Problem for Bounded Kinematic Hardening Material

Limit and shakedown analysis are effective methods for assessing the load carrying capacity of a given structure. The elasto–plastic behavior of the structure subjected to loads varying in a given load domain is characterized by the shakedown load factor, defined as the maximum factor which satisfies the sufficient conditions stated in the corresponding static shakedown theorem. The finite element dicretization of the problem may lead to very large convex optimization. For the effective solution a basis reduction method has been developed that makes use of the special problem structure for perfectly plastic material. The paper proposes a modified basis reduction method for direct application to the two-surface plasticity model of bounded kinematic hardening material. The considered numerical examples show an enlargement of the load carrying capacity due to bounded hardening.     

Homogenization of magneto-electro-elastic multilaminated materials

In this work, based on the periodic unfolding homogenizationtechnique, the limiting equations modelling the behaviour ofthree-dimensional magneto-electro-elastic periodic structuresare rigorously established. The local problems and the correspondinghomogenized coefficients of the elastic, dielectric, magneticpermittivity, piezoelectric, piezomagnetic and magneto-electric(ME) tensors are explicitly described. The homogenization modelis exemplified for laminated composites and a unified generalformula for all effective properties of periodic multilaminatedmagneto-electro-elastic composites is obtained. This formulais applied to investigate the global behaviour for the importantcase of transversely isotropic constituents and any finite numberof layers in each periodic cell. Examples that provide theoreticalevidence of the presence of both a product property and theME effect are given.