Non-Euclidean Geometry and Defected Structure for Bodies with Variable Material Composition

Lychev, S. A.; Kostin, Georgy; Lycheva, T N; Koifman, K. G.

doi:10.1088/1742-6596/1250/1/012035

Cited by 5 publications

(8 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We assume that the index set I ⊂ R is compact and has either finite or continuum cardinality (In present paper we do not consider the cases for countable cardinality for the index set I. Some relevant observations can be found in [19,24]). With respect to additive manufacturing process, that is under modelling, one can associate the elements from I with time labels that characterize the process evolution.…”

Section: Solid With Variable Materials Compositionmentioning

confidence: 99%

“…The discussion on various dyadic representations of deformation gradient F n, k = Dγ n, k can be found in [24].…”

Section: Strain Measuresmentioning

confidence: 99%

“…Due to the smoothing, the multilayered body B N is replaced by the laminated body B N with shape S N and distortion tensor field H defined on it. This field is inverse to (24) and provides the local relaxation of stressed state.…”

Section: Laminated Structurementioning

confidence: 99%

“…Special attention is paid to non-linear theory of surface growth [22]. One can find the statement and model solutions for the evolutionary problem that relates the specificity of internal geometry with scenario of body fabrication in [23,24].…”

Section: Introductionmentioning

confidence: 99%

“…When the family of locally stress-free reference shapes is given, it is fairly easy to calculate stress-strain state of the shape under given external forces. Meanwhile, in the modeling of additive manufactured solids such a family is unknown and should be defined from the solution of a special evolutionary problem [23,24]. In general, such a problem for the continual family can be classified as a non-linear boundary value problem with delay [49,50], and its solution stands as a high mathematical challenge.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches

2021

View full text Add to dashboard Cite

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.

show abstract

Section: Solid With Variable Materials Compositionmentioning

confidence: 99%

“…The discussion on various dyadic representations of deformation gradient F n, k = Dγ n, k can be found in [24].…”

Section: Strain Measuresmentioning

confidence: 99%

Section: Laminated Structurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches

2021

View full text Add to dashboard Cite

show abstract

Material Affine Connections for Growing Solids

Lychev

Koifman

2020

Lobachevskii J Math

View full text Add to dashboard Cite

The Nonlinear evolutionary problem for self-stressed multilayered hyperelastic spherical bodies

Lychev

Lycheva

Лычёва³

et al. 2020

PNRPU Mechanics Bulletin

View full text Add to dashboard Cite

The present paper studies the evolutionary problem for self-stressed multilayered spherical shells. Their stress-strain state is characterized by incompatible local finite deformations that arise due to the geometric incompatibility of the stress-free shapes of the individual layers with each other. In the considered problem, these shapes are thin-walled hollow balls that cannot be assembled into a single solid without gaps or overlaps. Such an assembly is possible only with the preliminary deformations of individual layers, which cause self-balanced stresses in them. For multilayered structures with a large number of layers, a smoothing procedure is proposed, as a result of which the piecewise continuous functions defining the preliminary deformation of the layers are replaced by continuous distributions. The reference stress-free shape of a body constructed in this way is defined within the framework of geometric continuum mechanics as a manifold with a non-Euclidean (material) connection. For the problem in question, this connection is determined by the metric tensor and its deviation from the Euclidean connection is characterized by the scalar curvature. Generalized representations for Cauchy and Piola stresses are also obtained by the methods of geometric continuum mechanics. Computations, provided for the discrete structure and body with a non-Euclidean reference shape defined by the approximation of deformation parameters, numerically illustrate the convergency of the solution for the discrete model to corresponded solution for the continuous problem if the number of layers is increasing while their total thickness is constant. In modelling it is assumed that the material of the layers is compressible, homogeneous, hyperelastic, and determined by the first-order Mooney - Rivlin elastic potential. Individual layerwise finite deformations are supposed to be centrally symmetric.

show abstract

Non-Euclidean Geometry and Defected Structure for Bodies with Variable Material Composition

Cited by 5 publications

References 24 publications

Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches

Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches

Material Affine Connections for Growing Solids

The Nonlinear evolutionary problem for self-stressed multilayered hyperelastic spherical bodies

Contact Info

Product

Resources

About