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)     

Generalized distortional hardening in a thermomechanically coupled framework

In many metals like aluminum the evolution of the microstructure leads to an evolution of the macroscopic yield surface. Clearly, isotropic and kinematic hardening models cannot capture this effect realistically. For that purpose, a new generalized distortional hardening model is proposed. This novel approach belongs to the class of so-called generalized standard materials and is thermodynamically consistent. Although the approach is relatively simple, it shows several advantages like yield surface convexity and yield limit saturation. This novel model is embedded into a thermomechanically coupled finite strain framework. Several numerical simulations show the predictive capabilities. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

A rheological model for arbitrary symmetric distortion of the yield surface

A phenomenological model of metal viscoplasticity, which takes combined isotropic, kinematic, and distortional hardening into account, is motivated by a new rheological model. The distinctive advantage of the material model is that any smooth convex saturated form of the yield surface which is symmetric with respect to the recent loading direction can be captured. In particular, an arbitrary sharpening of the saturated yield locus in the loading direction combined with a flattening on the opposite side can be covered. Moreover, the yield locus evolves smoothly and its convexity is guaranteed at each hardening stage. The underlying two-dimensional rheological analogy can be used to provide insight into the main constitutive assumptions. This rheological model is utilized as a guideline for the construction of phenomenological constitutive relations. The distortion of the yield surface is described with the help of a so-called distortional backstress. Thus, 2nd rank tensors are utilized only. The resulting material model is thermodynamically consistent. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Distribution of dislocation density and residual stresses in plastically deformed specimens: numerical studies

A two-scale approach to the simulation of mechanical properties of metallic materials is considered. On the macroscopic level, the material behavior is described by a phenomenological model of finite strain viscoplasticity with nonlinear kinematic hardening. In particular, the process-induced plastic anisotropy is captured by backstresses. On the microstructural level, the so called “load path sensitive two-population dislocation cell model” is implemented. It describes an evolving dislocation cell structure with dislocation populations for dislocation cell walls and the cell interior. Owing to the coupling with the phenomenological plasticity model, it can describe the evolution of the dislocation densities depending on the load path. The applicability of the multiscale approach to the FEM simulation of severe plastic deformation processes such as Equal Channel Angular Pressing is demonstrated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shakedown analysis of periodic composites with kinematic hardening material model

Lower-bound limit and shakedown analysis of periodic composites with the consideration of kinematic hardening are carried out on the representative volume element level. In combination with homogenization theory, the homogenized macroscopic admissible loading domains are determined. Furthermore, the strengths of periodic composites by using elastic perfectly plastic, unlimited and linear limited kinematic hardening material models are calculated and compared. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The theory of non-linear hardening of an elastoplastic composite material filled with ideally elastic inclusions

A version of the statistiad method of averaging the system of equilibrium equations for an elastoplastic two-component composite material in order to predict its macroscopic non-linear hardening is proposed. This version, unlike the averaging method developed previously in [l, 2], enables one to model and estimate the degree of connectedness of the matrix and the inclusions and to take into account the non-uniformity of the distribution and the development of plastic deformations. Macroscopic governing equations are constructed which describe the non-linear hardening of a composite material outside the elasticity limit, and its effective characteristics are calculated.     

Computational mechanics-based modeling of size-dependent hardening in polycrystals

It has been well-known for a number of years that the macroscopic material response of polycrystalline metals is influenced by the size and morphology of grains. Different size effects may occur, one of which is the Hall-Petch effect. The key aim of this contribution is the computational modeling of grain-size-dependent hardening in polycrystals using a microstructural approach. Here, the focus is on the influence of the microscopic grain boundary conditions on the simulation results. In particular, micro-flexible boundary conditions are discussed and compared to micro-hard and micro-free assumptions. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling thermoplastic material behaviour of dual-phase steels on a microscopic length scale

On a microscopic length scale dual-phase steels exhibit a polycrystalline microstructure consisting of ferrite and martensite. In this work a material model for the temperature dependent hardening behaviour of the ferritic phase is presented. As the dislocation structure determines the resistance to dislocation glide, dislocation densities are introduced as state variables to capture the dependence of the material behaviour on the loading history. Motivated by the elementary processes of multiplication by the Frank-Read-mechanism and annihilation by cross-slip, evolution equations for the dislocation densities are introduced. Based on the interaction of dislocations on different slip systems and the Peierls-stress, the resistance to dislocation motion with its temperature dependence is formulated to describe the hardening behaviour. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigations on autogenous shrinkage of cement paste and mortar

Autogenous shrinkage of cement paste and concrete is defined as the macroscopic length change occurring with no moisture transferred to the exterior surrounding environment. It is a result of chemical shrinkage affiliated with the hydration of cement particles and the ongoing process of self-desiccation. The process of self-desiccation can be modeled starting from the formation of the capillary pore space during hydration in the cement paste. In this proposal a working model will be introduced explaining the difficulties to obtain the autogenous shrinkage strains directly from a simulated or measured microstructure of cement paste. In a second step the autogenous shrinkage of a hardening cement mortar was described on a mesoscopic level. It based on measurements on cement paste. The mortar simply consists of cement paste and a defined fraction of spherical aggregates with a known modulus of elasticity. Furthermore the influence of the interfacial transition zone (ITZ) is studied in numerical simulations. The results of these finite-element-calculations are introduced and compared with testing results of the autogenous shrinkage of hardening mortar samples. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Isotropic models for a heat transfer in periodic composites

A macroscopic model for the non – stationary heat transfer processes in a periodic honeycomb – type anisotropic rigid conductor is formulated. The main aim of this contribution is to show that the macroscopic properties of this conductor are isotropic. (© 2004 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)     

Modelling internal erosion of cohesionless soils using a microstructural parameter

Internal erosion processes are of vital importance for the risk management of geotechnical structures as well as for the understanding of the macroscopic mechanical and hydraulic properties of the subsoil in various man-made constructions. Here, a 4-phase continuum model is presented and numerically applied to illustrative applications. The role of interfacial area and related microstructural parameters is addressed. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemo-attractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a drift-diffusion model is proven. The drift-diffusion models derived in this way include the classical Keller-Segel model. Furthermore, sufficient conditions for kinetic models are given such that finite-time-blow-up does not occur. Examples are given satisfying these conditions, whereas the macroscopic limit problem is known to exhibit finite-time-blow-up. The main analytical tools are entropy techniques for the macroscopic limit as well as results from potential theory for the control of the chemo-attractant density.Present address: Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal     

Homogenization for rate-independent systems

This paper is devoted to the homogenization for a class of rate-independent systems described by the energetic formulation. The associated nonlinear partial differential system has periodically oscillating coefficients, but has the form of a standard evolutionary variational inequality. Thus, the model applies to standard linearized elastoplasticity with hardening. Using the recently developed methods of two-scale convergence, periodic unfolding and the new introduced one, periodic folding, we show that the homogenized problem can be represented as a two-scale limit which is again an energetic formulation, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

About the Microstructural Effects of Polycrystalline Materials and their Macroscopic Representation at Finite Deformation

In the sheet bulk metal forming field, the strict geometrical requirements of the workpieces result in a need of a precise prediction of the material behaviour. The simulation of such forming processes requires a valid material model, performing well for a huge variety of different geometrical characteristics and finite deformation. 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 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. The validation on the macroscopic scale is performed through the reproduction of the experimentally calculated initial yield surface. Additionally, homogenised stress-strain curves from the microstructure build the outcome for a suitable effective material model. Through optimisation techniques, effective material parameters can be determined and compared to results from real forming processes. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)