The mathematical theory of growing bodies. Finite deformations

Lychev, S. A.; Manzhirov, Alexander V.

doi:10.1016/j.jappmathmech.2013.11.011

Cited by 38 publications

(15 citation statements)

References 10 publications

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“…Using convected coordinates, he constructed for each point a deformation gradient as a composition of the prehistory of deformation and the deformation between the time of addition and the current time. There are many other works on the mechanics of surface growth (some of which use a geometric framework) by Russian researchers [Arutyunyan et al, 1990;Manzhirov, 1995;Lychev and Manzhirov, 2013a;Manzhirov, 2014;Lychev and Manzhirov, 2013a,b;Lychev, 2011]. Most of the recent works are formal.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies

Sozio

Yavari

2017

Journal of the Mechanics and Physics of Solids

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In this paper we formulate the initial-boundary value problems of accreting cylindrical and spherical nonlinear elastic solids in a geometric framework. It is assumed that the body grows as a result of addition of new (stress-free or pre-stressed) material on part of its boundary. We construct Riemannian material manifolds for a growing body with metrics explicitly depending on the history of applied external loads and deformation during accretion and the growth velocity. We numerically solve the governing equilibrium equations in the case of neo-Hookean solids and compare the accretion and residual stresses with those calculated using the linear mechanics of surface growth.

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Section: Introductionmentioning

confidence: 99%

Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies

Sozio

Yavari

2017

Journal of the Mechanics and Physics of Solids

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“…In general this family can be associated with a smooth bundle. The dimension of a base of this bundle defines the class of a growing body [5]. In present paper we will considered the simplest class that corresponds to the three-dimensional bundle over one-dimensional base.…”

Section: Common Definitionsmentioning

confidence: 99%

“…Note that manifolds k represent the preimage of growth boundary. Relations (17.2) introduce on the manifold B the structure of a smooth bundle [5,18]. The interval I represents the base of the bundle while the manifolds˝k represent the fibers.…”

Section: Common Definitionsmentioning

confidence: 99%

“…This concept refers to a new branch of continuum mechanics [4][5][6][7][8], therefore it seems appropriate here to clarify the definition of a growing solid. In a broad sense growing process defines the alteration of the body composition occurring in the course of deformation.…”

Section: Introductionmentioning

confidence: 99%

“…Article [11] is devoted to the development of geometric methods adopted for the mechanics of incompatible strains arising as the result of the growing process. In the works [4][5][6][7]12] growth is investigated as the continuous process of deposition of strained material surfaces to a deformable 3D body.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Discrete and Continuous Growth of Deformable Cylinder

Lychev

Manzhirov

Bychkov

2015

Transactions on Engineering Technologies

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The finite deformations of the growing cylinder of incompressible elastic material are under consideration. We assume that the deformations are axisymmetric and do not change along the axis of cylinder. The discrete and continuous types of growing are studied. The analytical solutions of the corresponding boundaryvalue problems are derived. The results of numerical simulation of both discrete and continuous growth are given.

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Thermal stresses in growing thermoviscoelastic cylinder and their evolution in the course of selective laser melting processing

Fekry

2022

Z Angew Math Mech

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The evolution in time of stresses and strains in a solid during its manufacturing through selective laser melting (SLM) process is studied. The technological technique, which allows to reduce the intensity of internal and residual stresses is proposed and theoretically grounded. The basic concept of the technique is that the additional in-homogeneous inductive heating in the vicinity of the growing boundary is applied in the processing. The term "growing boundary" refers to the part of the boundary, on which the additional melted material is added. The evolutionary problem for a growing thermoelastic body is considered in approach of thermal stresses (for slow additive processes) and in the approach of full coupled thermoelasticity (for fast processing). The time depend distribution of internal stresses and their residual counterparts has been evaluated in analytical form in the case of model cylindrical problem. The analysis of the influence of frequency and amplitude of electromagnetic induction as well as viscoelastic properties of heated material being added has been carried out. The results are analyzed numerically for Titanium and Copper materials.

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The mathematical theory of growing bodies. Finite deformations

Cited by 38 publications

References 10 publications

Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies

Nonlinear mechanics of surface growth for cylindrical and spherical elastic bodies

Discrete and Continuous Growth of Deformable Cylinder

Thermal stresses in growing thermoviscoelastic cylinder and their evolution in the course of selective laser melting processing

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