scholarly journals The mathematical foundations of anelasticity: existence of smooth global intermediate configurations

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200462
Author(s):  
Christian Goodbrake ◽  
Alain Goriely ◽  
Arash Yavari
Keyword(s):  
Euclidean Space ◽  
Deformation Gradient ◽  
Deformation Tensor ◽  
Dimensional Euclidean Space ◽  
Multiplicative Decomposition ◽  
Mathematical Basis ◽  
Isometric Embeddings ◽  
Higher Dimensional ◽  
Mathematical Foundations ◽  
Central Tool

A central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground. Here, we derive a sufficient condition for the existence of global intermediate configurations, starting from a multiplicative decomposition of the deformation gradient. We show that these global configurations are unique up to isometry. We examine the result of isometrically embedding these configurations in higher-dimensional Euclidean space, and construct multiplicative decompositions of the deformation gradient reflecting these embeddings. As an example, for a family of radially symmetric deformations, we construct isometric embeddings of the resulting intermediate configurations, and compute the residual stress fields explicitly.

Inequalities for random flats meeting a convex body

1985 ◽  
Vol 22 (03) ◽  
pp. 710-716 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Finite Number ◽  
Random Point ◽  
Dimensional Euclidean Space ◽  
Random Direction ◽  
Higher Dimensional

We choose a uniform random point in a given convex bodyKinn-dimensional Euclidean space and through that point the secant ofKwith random direction chosen independently and isotropically. Given the volume ofK, the expectation of the length of the resulting random secant ofKwas conjectured by Enns and Ehlers [5] to be maximal ifKis a ball. We prove this, and we also treat higher-dimensional sections defined in an analogous way. Next, we consider a finite number of independent isotropic uniform random flats meetingK, and we prove that certain geometric probabilities connected with these again become maximal whenKis a ball.

Constitutive theories based on the multiplicative decomposition of deformation gradient: Thermoelasticity, elastoplasticity, and biomechanics

2004 ◽  
Vol 57 (2) ◽  
pp. 95-108 ◽  
Author(s):  
Vlado A Lubarda
Keyword(s):  
Elastic Strain ◽  
Soft Tissues ◽  
Deformation Gradient ◽  
Elastic Strain Energy ◽  
Stretch Ratio ◽  
Deformation Tensor ◽  
Multiplicative Decomposition ◽  
Constitutive Theories ◽  
Constitutive Analysis ◽  
Plastic Parts

Some fundamental issues in the formulation of constitutive theories of material response based on the multiplicative decomposition of the deformation gradient are reviewed, with focus on finite deformation thermoelasticity, elastoplasticity, and biomechanics. The constitutive theory of isotropic thermoelasticity is first considered. The stress response and the entropy expression are derived in the case of quadratic dependence of the elastic strain energy on the finite elastic strain. Basic kinematic and kinetic aspects of the phenomenological and single crystal elastoplasticity within the framework of the multiplicative decomposition are presented. Attention is given to additive decompositions of the stress and strain rates into their elastic and plastic parts. The constitutive analysis of the stress-modulated growth of pseudo-elastic soft tissues is then presented. The elastic and growth parts of the deformation gradient and the rate of deformation tensor are defined and used to construct the corresponding rate-type biomechanic theory. The structure of the evolution equation for growth-induced stretch ratio is discussed. There are 112 references cited in this review article.

Matrix Resolving Function in the Nonstationary Linear Group Pursuit Problem Concepting Multiple Capture

2021 ◽  
pp. 2150016
Author(s):  
N. N. Petrov
Keyword(s):  
Euclidean Space ◽  
Linear Group ◽  
Sufficient Conditions ◽  
Dimensional Euclidean Space ◽  
Pursuit Problem ◽  
Mathematical Basis ◽  
Group Pursuit ◽  
Finite Dimensional

In finite-dimensional Euclidean space, an analysis is made of the problem of pursuit of a single evader by a group of pursuers, which is described by a system of the form [Formula: see text] The goal of the group of pursuers is the capture of the evader by no less than [Formula: see text] different pursuers (the instants of capture may or may not coincide). Matrix resolving functions, which are a generalization of scalar resolving functions, are used as a mathematical basis of this study. Sufficient conditions are obtained for multiple capture of a single evader in the class of quasi-strategies. Examples illustrating the results obtained are given.

Comparison of Hyper- and Hypoelastic-Based Formulations of Elastoplasticity with Applications to Metal Forming

2013 ◽  
Vol 554-557 ◽  
pp. 2321-2329 ◽  
Author(s):  
Tim Brepols ◽  
Ivaylo N. Vladimirov ◽  
Stefanie Reese
Keyword(s):  
Metal Forming ◽  
Finite Strain ◽  
Fe Simulation ◽  
Kinematic Hardening ◽  
Deformation Gradient ◽  
Deformation Tensor ◽  
Drawing Process ◽  
Metallic Materials ◽  
Multiplicative Decomposition ◽  
Plastic Parts

The aim of this work is to examine two specific finite strain elastoplasticity models in terms of their applicability in metal forming processes, namely a hyperelastic-based model which relies upon a multiplicative decomposition of the deformation gradient into elastic and plastic parts and a hypoelastic-based model which makes use of an additive elastic-plastic split of the rate of deformation tensor. Both models allow for nonlinear isotropic and kinematic hardening and were implemented as user material subroutines (UMAT) into ABAQUS/Standard. Various sample calculations were performed to assess the respective properties and capabilities of the models. The FE simulation of a deep drawing process produced nearly congruent results for both models which suggests that they are equally well-suited for modeling metallic materials in metal forming processes.

scholarly journals Higher horospherical limit sets for G-modules over CAT(0)-spaces

2021 ◽  
pp. 1-31
Author(s):  
ROBERT BIERI ◽  
ROSS GEOGHEGAN
Keyword(s):  
Group Theory ◽  
Euclidean Space ◽  
Discrete Group ◽  
Tropical Geometry ◽  
Dimensional Euclidean Space ◽  
Limit Sets ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Suitable Space

Abstract The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.

Solutions of equivariance for a polynomial-like iterative equation

2000 ◽  
Vol 130 (5) ◽  
pp. 1153-1163 ◽  
Author(s):  
Weinian Zhang
Keyword(s):  
Fixed Point ◽  
Euclidean Space ◽  
Closed Subgroup ◽  
General Linear Group ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Dimensional Euclidean Space ◽  
Continuous Solutions ◽  
Iterative Equation ◽  
Higher Dimensional

Using fixed point theorems we discuss continuous solutions of Γ-equivariance for a polynomial-like iterative equation on the real line, where Γ is a closed subgroup of the general linear group GL(R). Our main results guarantee the existence of solutions with certain kinds of symmetry. We show that, under restrictive hypotheses, similar results can be proved in a higher-dimensional case, where the symmetry group is a topologically finitely generated subgroup of the group generated by rotations and dilations in N-dimensional Euclidean space.

Basic Principles of Data Mining

2010 ◽  
pp. 266-289
Author(s):  
Karl-Ernst Erich Biebler
Keyword(s):  
Data Mining ◽  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Topological Order ◽  
Data Types ◽  
Basic Principles ◽  
Mathematical Structures ◽  
Related Data ◽  
Higher Dimensional ◽  
Discriminant Analyses

This chapter gives a summary of data types, mathematical structures, and associated methods of data mining. Topological, order theoretical, algebraic, and probability theoretical mathematical structures are introduced. The n-dimensional Euclidean space, the model used most for data, is defined. It is executed briefly that the treatment of higher dimensional random variables and related data is problematic. Since topological concepts are less well known than statistical concepts, many examples of metrics are given. Related classification concepts are defined and explained. Possibilities of their quality identification are discussed. One example each is given for topological cluster and for topological discriminant analyses.

An algorithm for estimating the vector fields` curl of the earth’s surface motion from geodetic data

2019 ◽  
Vol 949 (7) ◽  
pp. 51-56
Author(s):  
B.T. Mazurov
Keyword(s):  
Vector Fields ◽  
Displacement Vector ◽  
Research Work ◽  
Strain Tensor ◽  
Deformation Tensor ◽  
Mathematical Basis ◽  
Displacement Vectors ◽  
Geodynamic Processes ◽  
Multiple Measurements ◽  
Mathematical Foundations

According to the changes of geodetic elements (coordinates, heights, directions) after multiple measurements, it is possible to represent the field of geodetic points’ displacement vector. When studying the stress-strain state of the earth’s surface, the vectors obtained can be used not only to calculate the earth’s deformation tensor in the area under study, but also the differential characteristics of the vector field, called divergence and rotor (vortex, curl). The author proposes the way to determinethe rotor according to discrete geodesic observations of displacement vectors on the surface of the surveyed area. The most important continuation of this research work is the method of geodynamic systemsmathematical modeling for predictive purposes. In order to study the complex (nonlinear) geodynamic processes, an appropriate mathematical basis should be chosen. Here the attention is drawn to the attraction of the mathematical foundations of field theory. The variants of the curl definition are proposed – one of the vector fields’ differential characteristics. To assess the characteristics of vector fields when using multiple geodetic measurements, the finite element method can be used. The division of the surveyed area into triangles enables you to determine the characteristics of the deformation after calculating the elements of the strain tensor.

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