Investigating a problem from the theory of elasticity for a half-space with cylindrical cavities for which boundary conditions of contact type are assigned

Проценко, В. С.; Miroshnikov, Vitaly

doi:10.15587/1729-4061.2018.139567

Cited by 9 publications

(8 citation statements)

References 8 publications

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“…Such justification was taken into account in the problem of elasticity theory for a half-space with cylindrical cavities in displacements in [28] and for a mixed type in [29]. But in [28,29], only a half-space is considered and there is no connection with another boundary surface to create a layer. This limit is taken into account in [30], where the problem is solved for a layer with one cylindrical cavity in displacements.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

Solution of the problem of the theory of elasticity and analysis of the stress state of a fibrous composite layer under the action of transverse compressive forces

Miroshnikov

Savin

Younis

et al. 2022

EEJET

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An analytical-numerical solution of the spatial problem of elasticity theory for a composite in the form of a layer with two longitudinal endless continuous cylindrical inclusions is proposed. Homogeneous, isotropic materials of the layer and inclusions differ from each other in the modulus of elasticity and Poisson's ratio. Normal stresses are set on the upper and lower boundaries of the layer. The object of study is the stress state of such a composite. The problem is the lack of a high-precision method for calculating multiply connected bodies of this type. The solution of the problem is based on the generalized Fourier method for the Lame equations in various coordinate systems. The problem is reduced to an infinite system of linear algebraic equations, which is solved by the reduction method. In a numerical study, the stress state was obtained inside the composite bodies, within their conjugations, and on the isthmus between inclusions. It has been established that extreme stresses sρ=–0.9306 MPa, sf=–0.5595 MPa, tρf=–0.315 MPa occur on the mating face. Analysis of the stress state indicates the need to take into account the normal stresses on the mating surface. This is due to the presence of a binder, which may differ in physical characteristics from the main components of the composite. The results have logical physical correctness and, in simplified versions, are fully consistent with the results of similar problems from other approved sources. In the work, the transition formulas in the basic solutions between different coordinate systems, the conjugation conditions for different bodies, and the strict fulfillment of the equilibrium conditions for given boundary functions are simultaneously applied. This made it possible to obtain a high-precision solution of a new problem in the theory of elasticity for a layer with cylindrical inclusions and given only stresses on the boundary surfaces. The proposed method of calculation can be applied in the design of structures made of fibrous composites in the aircraft industry and construction.

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Section: Literature Review and Problem Statementmentioning

confidence: 99%

Solution of the problem of the theory of elasticity and analysis of the stress state of a fibrous composite layer under the action of transverse compressive forces

Miroshnikov

Savin

Younis

et al. 2022

EEJET

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“…The authors of work [19] solved a boundary problem for the half-space with two cylindrical cavities, on the boundary surfaces of which the contact type conditions are assigned. The stress-strain of the layer with a cylindrical cavity on a hard base is investigated in [20].…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…The stress-strain of the layer with a cylindrical cavity on a hard base is investigated in [20]. he addition theorems of the solutions to the Lamé equation for the half-space and cylinder were also used to solve the problems in [19,20].…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…Hence, it follows that different methods have solved the boundary problems related to the theory of elasticity for unlimited elastic bodies with cavities of finite size. Works [16][17][18][19][20] report solving boundary problems for the half-space and layer with an infinite cylindrical cavity or cavities in different statements. However, the mixed problem in elasticity theory for half-space with an infinite cylindrical cavity in the case where displacements are assigned at the boundary of the half-space, and stresses on the surface of the cylinder, is not solved.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

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Solving a one mixed problem in elasticity theory for half-space with a cylindrical cavity by the generalized Fourier method

Ukrayinets¹,

Murahovska²,

Prokhorova³

2021

EEJET

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Designing and constructing underground structures for various purposes, such as tunnels, mines, mine workings, necessitate the development of procedures for calculating their strength and reliability. The physical model of such objects worth considering is a homogeneous isotropic half-space that contains an infinitely long hollow cylinder, located parallel to its border. One can explore problems related to the mechanics of deformable solids for such a multiply connected body. This paper reports the proofs of addition theorems for the basic solutions to the Lamé equation regarding the half-space and cylinder recorded, respectively, in the Cartesian and cylindrical coordinate systems. This result is important from a theoretical point of view in order to substantiate a numerical-analytical method ‒ the generalized Fourier method. This method makes it possible to solve spatial boundary problems from the theory of elasticity and thermo-elasticity for isotropic and transversal-isotropic multiply connected bodies. Similar to the classical Fourier method, the general solutions to equilibrium equations have been used here but in several coordinate systems rather than one. From a practical point of view, this method has made it possible to investigate the combined problem of elasticity theory regarding the multiply-connected body described above. The analysis of the stressed-strained state of this elastic body has made it possible to draw conclusions on determining those regions that are most vulnerable to destruction. The highest values are accepted by normal stresses in the region between the boundaries of the half-space and the cylinder. Changing the σy component along the Ox axis corresponds to the displacements assigned on the half-space. The τxy component contributes less to the distribution of stresses than σx and σy. The applied aspect of using the reported results is the possibility to apply them when designing underground structures

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“…The generalized Fourier method [12] is used for problems with several boundary surfaces. Based on this method, the problems for a half-space in displacements [10], in stresses [11,14,20], and a mixed problem with conditions of contact type [21] are solved.…”

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Stress State of an Elastic Layer with a Cylindrical Cavity on a Rigid Foundation

Miroshnikov

2020

Int Appl Mech

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The third basic problem of elasticity for a layer with a cylindrical cavity parallel to its surfaces is solved. Stresses on the cavity and the upper boundary of the layer are specified. On the lower boundary, the displacements are given. The solution to the spatial problem of elasticity is obtained by the generalized Fourier method for the Lame system of equations. The infinite systems of linear algebraic equations obtained by satisfying the boundary conditions are solved by the reduction method. As a result, the displacements and stresses are obtained at different points of the elastic body. We carried out a comparative analysis for the stress-strain state of the layer with a cylindrical cavity and without a cavity as well as for different distances to the lower boundary of the layer.

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Investigating a problem from the theory of elasticity for a half-space with cylindrical cavities for which boundary conditions of contact type are assigned

Cited by 9 publications

References 8 publications

Solution of the problem of the theory of elasticity and analysis of the stress state of a fibrous composite layer under the action of transverse compressive forces

Solution of the problem of the theory of elasticity and analysis of the stress state of a fibrous composite layer under the action of transverse compressive forces

Solving a one mixed problem in elasticity theory for half-space with a cylindrical cavity by the generalized Fourier method

Stress State of an Elastic Layer with a Cylindrical Cavity on a Rigid Foundation

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