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20 pages, 337 KiB

Open AccessArticle

Differential Geometry Approach to Continuous Model of Micro-Structural Defects in Finite Elasto-Plasticity

by Sanda Cleja-Ţigoiu

Symmetry 2021, 13(12), 2340; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122340 - 06 Dec 2021

Cited by 2 | Viewed by 1476

This paper concerns finite elasto-plasticity of crystalline materials with micro-structural defects. We revisit the basic concepts: plastic distortion and decomposition of the plastic connection. The body is endowed with a structure of differential manifold. The plastic distortion is an incompatible diffeomorphism. The metric [...] Read more.

This paper concerns finite elasto-plasticity of crystalline materials with micro-structural defects. We revisit the basic concepts: plastic distortion and decomposition of the plastic connection. The body is endowed with a structure of differential manifold. The plastic distortion is an incompatible diffeomorphism. The metric induced by the plastic distortion on the intermediate configuration (considered to be a differential manifold) is a key point in the theory, in defining the defects related to point defects, or extra-matter. The so-called plastic connection is metric, with plastic metric tensor expressed in terms of the plastic distortion and its adjoint. We prove an appropriate decomposition of the plastic connection, without any supposition concerning the non-metricity of plastic connection. All types of the lattice defects, dislocations, disclinations, and point defects are described in terms of the densities related to the elements that characterize the decomposition theorem for plastic connection. As a novelty, the measure of the interplay of the possible lattice defects is introduced via the Cartan torsion tensor. To justify the given definitions, the proposed measures of defects are compared to their counterparts corresponding to a classical framework of continuum mechanics. Thus, their physical meanings can be emphasized at once. Full article

(This article belongs to the Special Issue Applications of Differential Geometry to Continuum Mechanics)

33 pages, 4447 KiB

Open AccessArticle

Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches

by Sergey Lychev, Konstantin Koifman and Nikolay Djuzhev

Symmetry 2021, 13(12), 2331; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122331 - 05 Dec 2021

Cited by 7 | Viewed by 1332

Abstract

The present paper is intended to show the close interrelationship between non-linear models of solids, produced with additive manufacturing, and models of solids with distributed defects. The common feature of these models is the incompatibility of local deformations. Meanwhile, in contrast with the [...] Read more.

The present paper is intended to show the close interrelationship between non-linear models of solids, produced with additive manufacturing, and models of solids with distributed defects. The common feature of these models is the incompatibility of local deformations. Meanwhile, in contrast with the conventional statement of the problems for solids with defects, the distribution for incompatible local deformations in additively created deformable body is not known a priori, and can be found from the solution of the specific evolutionary problem. The statement of the problem is related to the mechanical and physical peculiarities of the additive process. The specific character of incompatible deformations, evolved in additive manufactured solids, could be completely characterized within a differential-geometric approach by specific affine connection. This approach results in a global definition of the unstressed reference shape in non-Euclidean space. The paper is focused on such a formalism. One more common factor is the dataset which yields a full description of the response of a hyperelastic solid with distributed defects and a similar dataset for the additively manufactured one. In both cases, one can define a triple: elastic potential, gauged at stress-free state, and reference shape, and some specific field of incompatible relaxing distortion, related to the given stressed shape. Optionally, the last element of the triple may be replaced by some geometrical characteristics of the non-Euclidean reference shape, such as torsion, curvature, or, equivalently, as the density of defects. All the mentioned conformities are illustrated in the paper with a non-linear problem for a hyperelastic hollow ball. Full article

(This article belongs to the Special Issue Applications of Differential Geometry to Continuum Mechanics)

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19 pages, 329 KiB

Open AccessArticle

Material Geometry of Binary Composites

by Marcelo Epstein

Symmetry 2021, 13(5), 892; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13050892 - 18 May 2021

Cited by 2 | Viewed by 1668

Abstract

The constitutive characterization of the uniformity and homogeneity of binary elastic composites is presented in terms of a combination of the material groupoids of the individual constituents. The incorporation of these two groupoids within a single double groupoid is proposed as a viable [...] Read more.

The constitutive characterization of the uniformity and homogeneity of binary elastic composites is presented in terms of a combination of the material groupoids of the individual constituents. The incorporation of these two groupoids within a single double groupoid is proposed as a viable mathematical framework for a unified formulation of this and similar kinds of problems in continuum mechanics. Full article

(This article belongs to the Special Issue Applications of Differential Geometry to Continuum Mechanics)

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