Compatibility conditions: number of independent equations and boundary conditions

Andrianov, Igor V.; Topol, Heiko

doi:10.1016/b978-0-32-390543-5.00011-6

Cited by 9 publications

(8 citation statements)

References 31 publications

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“…The formulation of boundary value problems of thermoelasticity with respect to stresses and deformations is an urgent problem in solid mechanics. Boundary value problems of thermoelasticity with respect to stresses and strains can be formulated within the framework of the Saint-Venant deformation compatibility condition [1]. It is known that the conditions for compatibility of deformations using the Duhamel-Neumann relation and the equation of motion can be written in the form of the Beltrami-Michell equations for stress and temperature [2].…”

Section: Introductionmentioning

confidence: 99%

New Coupled Thermoelasticity Boundary-Value Problems in Strains

Khaldjigitov,

Djumayozov

2024

Preprint

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This work is devoted to the formulation and numerical solution of coupled boundary value problems of thermoelasticity with respect to deformations. Two equivalent coupled boundary value problems of thermoelasticity with respect to strain and temperature are proposed. The first consists of six differential equations of thermoelasticity found within the framework of the compatibility conditions of Saint-Venant deformations and the heat influx equation with the corresponding initial and boundary conditions. In the second case, the first three of the six differential equations of thermoelasticity are replaced with three differentiated equations of motion. The validity of the formulated two boundary value problems of thermoelasticity is justified by comparing their numerical, obtained by the variable direction method and recurrence relations, as well as solving a similar related problem regarding displacements.

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Section: Introductionmentioning

confidence: 99%

New Coupled Thermoelasticity Boundary-Value Problems in Strains

Khaldjigitov,

Djumayozov

2024

Preprint

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“…Model equations of a new type for stresses were proposed in the works of B.E. Pobedri [1,2]. Dynamic boundary value problems with respect to stresses are considered in the works of Konovalov [9].…”

Section: Introductionmentioning

confidence: 99%

Model Equations of the Theory of Elasticity in Strains: Classical and New Formulations

Khaldjigitov,

Djumayozov

2024

E3S Web Conf.

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The article is devoted to the construction of model equations of the theory of elasticity with respect to deformations. Classical and new versions of boundary value problems of the theory of elasticity in strains are considered. In the classical version, model equations in strains are constructed within the framework of the Beltrami-Michell equations. A new version of model equations in strains is based on a new formulation of boundary value problems of the theory of elasticity in stresses. Discrete equations are constructed using the finite-difference method for two-dimensional problems. The well-known problem of tension a rectangular plate with a parabolic load applied on opposite sides has been solved. By comparing the numerical results of boundary value problems in classical and new formulations, as well as the Timoshenko-Goodier solution, the validity of the formulated model equations in strains and the reliability of the obtained numerical results are ensured.

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“…The formulation of the boundary value problems of thermoelasticity with respect to stresses and strains is an urgent problem in solid mechanics. The boundary value problems of thermoelasticity with respect to stresses and strains can be formulated within the framework of the Saint-Venant deformation compatibility condition [1]. With the support of the equation of motion and the Duhamel-Neumann relation, it is known that the requirements for compatibility of deformations can be expressed in terms of the stress tensor as the Beltrami-Michell equations [4,5].…”

Section: Introductionmentioning

confidence: 99%

Coupled Problem on Thermo-Elasticity in Strains for an Isotropic Parallelepiped

Djumayozov,

Eshmanova

2024

E3S Web Conf.

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This work is devoted to mathematical and numerical modeling of the coupled dynamic problem of thermoelasticity in deformations. A numerically related boundary value problem of thermoelasticity in deformations for a parallelepiped with the corresponding initial and boundary conditions is formulated and solved. Grid equations are constructed using the finite-difference method in the form of explicit and implicit schemes. In this case, the solution of the explicit scheme is reduced to recurrent relations with respect to deformations and temperature. In the case of implicit difference schemes, the equations are solved by sequential application of the sweep method. The validity of the formulated boundary value problems is justified by comparing the numerical results obtained with different methods based on two different models.

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Compatibility conditions: number of independent equations and boundary conditions

Cited by 9 publications

References 31 publications

New Coupled Thermoelasticity Boundary-Value Problems in Strains

New Coupled Thermoelasticity Boundary-Value Problems in Strains

Model Equations of the Theory of Elasticity in Strains: Classical and New Formulations

Coupled Problem on Thermo-Elasticity in Strains for an Isotropic Parallelepiped

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