On a Matrix Representation Lemma Useful in Determining Maximal Invariance Groups

AbstractBanken (1986, J. Multivariate Anal.19, 156–161) proposed a useful method for determining the group of all affine transformations leaving a multivariate normal testing problem invariant. His main result concerning the derivation of the maximal invariance group is heavily based on a matrix representation lemma which can be considered interesting in its own right. Unfortunately, the proof of this lemma is erroneous and there seems to be no trivial way to correct it. The aim of this note is to show the validity of the assertion

Quantification of 3D spatial correlations between state variables and distances to the grain boundary network in full-field crystal plasticity spectral method simulations

Deformation microstructure heterogeneities play a pivotal role during dislocation patterning and interface network restructuring. Thus, they affect indirectly how an alloy recrystallizes if at all. Given this relevance, it has become common practice to study the evolution of deformation microstructure heterogeneities with 3D experiments and full-field crystal plasticity computer simulations including tools such as the spectral method. Quantifying material point to grain or phase boundary distances, though, is a practical challenge with spectral method crystal plasticity models because these discretize the material volume rather than mesh explicitly the grain and phase boundary interface network. This limitation calls for the development of interface reconstruction algorithms which enable us to develop specific data post-processing protocols to quantify spatial correlations between state variable values at each material point and the points' corresponding distance to the closest grain or phase boundary. This work contributes to advance such post-processing routines. Specifically, two grain reconstruction and three distancing methods are developed to solve above challenge. The individual strengths and limitations of these methods surplus the efficiency of their parallel implementation is assessed with an exemplary DAMASK large scale crystal plasticity study. We apply the new tool to assess the evolution of subtle stress and disorientation gradients towards grain boundaries.Comment: Manuscript submitted to Modelling and Simulation in Materials Science and Engineerin

Monte Carlo Dynamics of driven Flux Lines in Disordered Media

We show that the common local Monte Carlo rules used to simulate the motion of driven flux lines in disordered media cannot capture the interplay between elasticity and disorder which lies at the heart of these systems. We therefore discuss a class of generalized Monte Carlo algorithms where an arbitrary number of line elements may move at the same time. We prove that all these dynamical rules have the same value of the critical force and possess phase spaces made up of a single ergodic component. A variant Monte Carlo algorithm allows to compute the critical force of a sample in a single pass through the system. We establish dynamical scaling properties and obtain precise values for the critical force, which is finite even for an unbounded distribution of the disorder. Extensions to higher dimensions are outlined.Comment: 4 pages, 3 figure

Creep motion in a random-field Ising model

We analyze numerically a moving interface in the random-field Ising model which is driven by a magnetic field. Without thermal fluctuations the system displays a depinning phase transition, i.e., the interface is pinned below a certain critical value of the driving field. For finite temperatures the interface moves even for driving fields below the critical value. In this so-called creep regime the dependence of the interface velocity on the temperature is expected to obey an Arrhenius law. We investigate the details of this Arrhenius behavior in two and three dimensions and compare our results with predictions obtained from renormalization group approaches.Comment: 6 pages, 11 figures, accepted for publication in Phys. Rev.

The depinning transition of a driven interface in the random-field Ising model around the upper critical dimension

We investigate the depinning transition for driven interfaces in the random-field Ising model for various dimensions. We consider the order parameter as a function of the control parameter (driving field) and examine the effect of thermal fluctuations. Although thermal fluctuations drive the system away from criticality the order parameter obeys a certain scaling law for sufficiently low temperatures and the corresponding exponents are determined. Our results suggest that the so-called upper critical dimension of the depinning transition is five and that the systems belongs to the universality class of the quenched Edward-Wilkinson equation.Comment: accepted for publication in Phys. Rev.

Micromechanical and macromechanical effects in grain scale polycrystal plasticity experimentation and simulation

A polycrystalline aluminum sample with a quasi-2D single layer of coarse grains is plastically deformed in a channel die plane strain set-up at ambient temperature and low strain rate. The microtexture of the specimen is determined by analysis of electron back scattering patterns obtained in a scanning electron microscope. The spatial distribution of the plastic microstrains at the sample surface is determined by measurement of the 3D plastic displacement field using a photogrametric pixel-based pattern recognition algorithm. The initial microtexture is mapped onto a finite element mesh. Continuum and crystal plasticity finite element simulations are conducted using boundary conditions which approximate those of the channel die experiments. The experimental and simulation data are analyzed with respect to macromechanical and micromechanical effects on grain-scale plastic heterogeneity. The most important contributions among these are the macroscopic strain profile (friction), the kinematic hardness of the crystals (individual orientation factors), the interaction with neighbor grain, and grain boundary effects, Crystallographic analysis of the data reveals two important points. First, the macroscopic plastic strain path is not completely altered by the crystallographic texture, but modulated following soft crystals and avoiding hard crystals. Second, grain-scale mechanisms are strongly superimposed by effects arising from the macroscopic profile of strain, The identification of genuine interaction mechanisms at this scale therefore requires procedures to filter out macroscopically induced strain gradients. As an analysis tool, the paper introduces a micromechanical Taylor factor, which differs from the macromechanical Taylor factor by the fact that crystal shear is normalized by the local rather than the global von Mises strain. (C) 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved

DAMASK: The Düsseldorf Advanced MAterial Simulation Kit for studying crystal plasticity using an FE based or a spectral numerical solver

AbstractThe solution of a continuum mechanical boundary value problem requires a constitutive response that connects deformation and stress at each material point. Such connection can be regarded as three separate hierarchical problems. At the top-most level, partitioning of the (mean) boundary values of the material point among its microstructural constituents and the associated homogenization of their response is required, provided there is more than one constituent present. Second, based on an elastoplastic decomposition of (finite strain) deformation, these responses follow from explicit or implicit time integration of the plastic deformation rate per constituent. Third, to establish the latter, a state variable-based constitutive law needs to be interrogated and its state updated.The Düsseldorf Advanced MAterial Simulation Kit (DAMASK) reflects this hierarchy as it is built in a strictly modular way. This modular structure makes it easy to add additional constitutive models as well as homogenization schemes. Moreover it interfaces with a number of FE solvers as well as a spectral solver using an FFT.We demonstrate the versatility of such a modular framework by considering three scenarios: Selective refinement of the constitutive material description within a single geometry, component-scale forming simulations comparing di_erent homogenization schemes, and comparison of representative volume element simulations based on the FEM and the spectral solver