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)     

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)     

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)     

Geometric Models for Quasicrystals II. Local Rules Under Isometries

The atomic structures of quasicrystalline materials exhibit long range order under translations. It is believed that such materials have atomic structures which approximately obey local rules restricting the location of nearby atoms. These local constraints are typically invariant under rotations, and it is of interest to establish conditions under which such local rules can nevertheless enforce order under translations in any structure that satisfies them. A set of local rules in is a finite collection of discrete sets {Y _i } containing 0, each of which is contained in the ball of radius ρ around 0 in . A set X satisfies the local rules under isometries if the ρ -neighborhood of each is isometric to an element of . This paper gives sufficient conditions on a set of local rules such that if X satisfies under isometries, then X has a weak long-range order under translations, in the sense that X is a Delone set of finite type. A set X is a Delone set of finite type if it is a Delone set whose interpoint distance set X-X is a discrete closed set. We show for each minimal Delone set of finite type X that there exists a set of local rules such that X satisfies under isometries and all other Y that satisfy under isometries are Delone sets of finite type. A set of perfect local rules (under isometries or under translations, respectively) is a set of local rules such that all structures X that satisfy are in the same local isomorphism class (under isometries or under translations, respectively). If a Delone set of finite type has a set of perfect local rules under translations, then it has a set of perfect local rules under isometries, and conversely. Received February 14, 1997, and in revised form February 14, 1998, February 19, 1998, and March 5, 1998.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.     

Micro-Electro-Elastic Modeling of Electric Field- and Stress-Driven Domain Wall Motion

Recently, increasing interest in functional materials such as piezoceramics has been shown. Such materials are characterized by properties, which can be significantly changed by external stimuli, such as stress, electric or magnetic fields. We outline a micro-electro-elastic model for the evolution of electrically and mechanically poled domains incorporating the surrounding free space. To this end, recently developed incremental variational principles (Miehe & Rosato [1]) for local dissipative response need to be extended to gradient-type phase-field models, including an embedding into the free space. The variational setting serves as a natural starting point for a compact and symmetric finite element implementation, considering the mechanical displacement, the electric polarization treated as an order parameter, and the electric potential induced by the polarization as the primary variables. The latter is defined on both the solid domain as well as the surrounding free space. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An adaptive approach for modeling a fiber-matrix composite with the FE2 method

In materials with a complicated microstructre [1], the macroscopic material behaviour is unknown. In this work a Fiber-Matrix composite is considered with elasto-plastic fibers. A homogenization of the microscale leads to the macroscopic material properties. In the present work, this is realized in the frame of a FE² formulation. It combines two nested finite element simulations. On the macroscale, the boundary value problem is modelled by finite elements, at each integration point a second finite element simulation on the microscale is employed to calculate the stress response and the material tangent modulus. One huge disadvantage of the approach is the high computational effort. Certainly, an accompanying homogenization is not necessary if the material behaves linear elastic. This motivates the present approach to deal with an adaptive scheme. An indicator, which makes use of the different boundary conditions (BC) of the BVP on microscale, is suggested. The homogenization with the Dirichlet BC overestimates the material tangent modulus whereas the Neumann BC underestimates the modulus [2]. The idea for an adaptive modeling is to use both of the BCs during the loading process of the macrostructure. Starting initially with the Neumann BC leads to an overestimation of the displacement response and thus the strain state of the boundary value problem on the macroscale. An accompanying homogenization is performed after the strain reaches a limit strain. Dirichlet BCs are employed for the accompanying homogenization. Some numerical examples demonstrate the capability of the presented method. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Note on Class Field Theory for Two-Dimensional Local Rings

The class field theory for the fraction field of a two-dimensional complete normal local ring with finite residue field is established by S. Saito. In this paper, we investigate the index of the norm group in the K ₂-idele class group for a finite Abelian extension of such fields and deduce that the existence theorem does not hold for almost fields in this case.     

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)     

Miscellany on traces in ℓ-adic cohomology: a survey

We discuss classical questions concerning traces of elements of Galois groups or correspondences in ℓ-adic cohomology, mostly over finite or local fields, such as rationality and independence of ℓ, integrality, congruences modulo powers of ℓ or p. We report on the progress that has been made on this topic during the past ten years.     

Forcing Linearity on Greedy Codes

Described in this paper are two different methods of forcing greedy codesto be linear over arbitrary finite fields. Both methods are generalizationsof the binary B-greedy codes as well as the triangular greedycodes over arbitrary fields. One method generalizes to arbitrary orderingsover arbitrary finite fields while the other method generalizes theB-greedy codes to linear codes over arbitrary finite fields.Examples of the first method are computed for triangular greedy codes. Theseexamples give codes similar to triangular codes from B-orderings.Both methods are shown to be substantially different.     

Witt equivalence of function fields of curves over local fields

Two fields are Witt equivalent if their Witt rings of symmetric bilinear forms are isomorphic. Witt equivalent fields can be understood to be fields having the same quadratic form theory. The behavior of finite fields, local fields, global fields, as well as function fields of curves defined over Archimedean local fields under Witt equivalence is well understood. Numbers of classes of Witt equivalent fields with finite numbers of square classes are also known in some cases. Witt equivalence of general function fields over global fields was studied in the earlier work [13 G?adki, P., Marshall, M. Witt equivalence of function fields over global fields. Trans. Am. Math. Soc., electronically published on April 11, 2017, doi: https://doi.org/10.1090/tran/6898 (to appear in print).[Crossref] , [Google Scholar]] by the authors and applied to study Witt equivalence of function fields of curves over global fields. In this paper, we extend these results to local case, i.e. we discuss Witt equivalence of function fields of curves over local fields. As an application, we show that, modulo some additional assumptions, Witt equivalence of two such function fields implies Witt equivalence of underlying local fields.     

A pathwise solution for nonlinear parabolic equations with stochastic perturbations

We analyse here a semilinear stochastic partial differential equation of parabolic type where the diffusion vector fields are depending on both the unknown function and its gradient ∂ _xu with respect to the state variable, ∈ ℝⁿ. A local solution is constructed by reducing the original equation to a nonlinear parabolic one without stochastic perturbations and it is based on a finite dimensional Lie algebra generated by the given diffusion vector fields.     

Embedding Problems of Division Algebras

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent work by Harbater, Hartmann, and Krashen, all admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue field of characteristic zero have been characterized. We show that also certain embedding problems of division algebras over such a field can be solved for admissible groups.     

Variational Structural and Material Stability Analysis in Finite Electro-Magneto-Mechanics of Active Materials

Soft matter electro-elastic, magneto-elastic and magneto-electro-elastic composites exhibit coupled material behavior at large strains. Examples are electro-active polymers and magneto-rheological elastomers, which respond by a deformation to applied electric or magnetic fields, and are used in advanced industrial environments as sensors and actuators. Polymer-based magneto-electro-elastic composites are a new class of tailor-made materials with promising future applications. Here, a magneto-electric coupling effect is achieved as a homogenized macro-response of the composite with electro-active and magneto-active constituents. These soft composite materials show different types of instability phenomena, which even might be exploited for future enhancement of their performance. This covers micro-structural instabilities, such as buckling of micro-fibers or particles, as well as material instabilities in the form of limit-points in the local constitutive response. Here, the homogenization-based scale bridging links long wavelength micro-structural instabilities to material instabilities at the macro-scale. This work outlines a comprehensive framework of an energy-based homogenization in electro-magneto-mechanics, which allows a tracking of postcritical solution paths such as those related to pull-in instabilities. Representative simulations demonstrate a tracking of inhomogenous composites, showing the development of postcritical zones in the microstructure and a possible instable homogenized material response. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rigidity of H–type groups

On H–type groups N the left invariant horizontal vector fields span a subbundle of the tangent bundle, called the horizontal bundle HN. Generalized contact mappings f on N are smooth mappings which preserve HN. The question is: how many such mappings exist? In the case of the Heisenberg group these are the contact mappings in the classical sense and they exist in abundance. In this paper it is shown that if the dimension of the center of N is at least three, then the generalized contact mappings are in the automorphism group of a finite dimensional Lie algebra g. The elements in are the infinitesimal generators of local one parameter subgroups of generalized contact transformations. Rigidity is defined as the property that is finite dimensional. For the case of the complexified Heisenberg group, i.e. the case when the dimension of the center of N is two, it has been shown [RR] that g is infinite dimensional. Received January 4, 2000; in final form March 20, 2000 / Published online April 12, 2001     

Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties

In this paper, the second order statistics of post buckling response of functionally graded materials plate (FGM) subjected to mechanical and thermal loading with nonuniform temperature changes subjected to temperature independent (TID) and dependent (TD) material properties is examined. Material properties such as material properties of each constituent’s materials, volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear kinematic using modified C⁰ continuity. A direct iterative based C⁰ nonlinear finite element method (FEM) combined with mean centered first order perturbation technique (FOPT) proposed by last two authors for the composite plate is extended for Functionally Graded Materials (FGMs) plate with reasonable accuracy to compute the second order statistics (mean and coefficient of variation) of the post buckling load response of the FGM plates. The effect of random material properties with amplitude ratios, volume fraction index, plate thickness ratios, aspect ratios, boundary conditions and types of loadings subjected to TID and TD material properties are presented through numerical examples. The performance of outlined present approach is validated with the results available in literatures and independent Monte Carlo simulation (MCS).     

Local and non‐local ductile damage and failure modelling at large deformation with applications to engineering

The numerical analysis of ductile damage and failure in engineering materials is often based on the micromechanical model of Gurson [1]. Numerical studies in the context of the finite‐element method demonstrate that, as with other such types of local damage models, the numerical simulation of the initiation and propagation of damage zones is strongly mesh‐dependent and thus unreliable. The numerical problems concern the global load‐displacement response as well as the onset, size and orientation of damage zones. From a mathematical point of view, this problem is caused by the loss of ellipticity of the set of partial di.erential equations determining the (rate of) deformation field. One possible way to overcome these problems with and shortcomings of the local modelling is the application of so‐called non‐local damage models. In particular, these are based on the introduction of a gradient type evolution equation of the damage variable regarding the spatial distribution of damage. In this work, we investigate the (material) stability behaviour of local Gurson‐based damage modelling and a gradient‐extension of this modelling at large deformation in order to be able to model the width and other physical aspects of the localization of the damage and failure process in metallic materials.     

Discreteness criterion for subgroups of products of SL(2)

Let G be a finite product of SL(2, K _i )’s for local fields K _i of characteristic zero. We present a discreteness criterion for nonsolvable subgroups of G containing an irreducible lattice of a maximal unipotent subgroup of G. In particular, such a subgroup has to be arithmetic. This extends a previous result of A. Selberg when G is a product of SL₂( \mathbbR \mathbb{R} )’s.