Dynamic analogues of Somigliana's formula for unsteady dynamics of elastic media with an arbitrary degree of anisotropy

Alekseyeva, L.A.; Zakir’yanova, G. K.

doi:10.1016/0021-8928(94)90066-3

Cited by 2 publications

(3 citation statements)

References 3 publications

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“…Initial-boundary value problem II. Find a solution of system (1) that satisfies boundary conditions (33), (34), and (36) and front conditions (5)- (7).…”

Section: Statement Of the Initial Bvpmentioning

confidence: 99%

“…These equations can be numerically solved using the boundary element method. In special cases of nonstationary boundary value problems in elasticity theory (M ¼ N ¼ 2, 3), these equations were solved in [4,[6][7][8].…”

Section: Theorem 53 If a Classical Solution Of The Dirichlet (Neumamentioning

confidence: 99%

“…For systems of hyperbolic equations, the BIE method is developed. Here, the ideas for solving nonstationary BVPs for the wave equations in multidimensional space [3,4] are used and the methods were elaborated for boundary value problems of dynamics of elastic bodies [5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations of Mathematical Physics

Alexeyeva¹,

Zakir’yanova²

2020

Mathematical Theorems - Boundary Value Problems and Approximations

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The method of boundary integral equations is developed for solving the nonstationary boundary value problems (BVP) for strictly hyperbolic systems of second-order equations, which are characteristic for description of anisotropic media dynamics. The generalized functions method is used for the construction of their solutions in spaces of generalized vector functions of different dimensions. The Green tensors of these systems and new fundamental tensors, based on it, are obtained to construct the dynamic analogues of Gauss, Kirchhoff, and Green formulas. The generalized solution of BVP has been constructed, including shock waves. Using the properties of integrals kernels, the singular boundary integral equations are constructed which resolve BVP. The uniqueness of BVP solution has been proved.

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“…Initial-boundary value problem II. Find a solution of system (1) that satisfies boundary conditions (33), (34), and (36) and front conditions (5)- (7).…”

Section: Statement Of the Initial Bvpmentioning

confidence: 99%

Section: Theorem 53 If a Classical Solution Of The Dirichlet (Neumamentioning

confidence: 99%

See 1 more Smart Citation

Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations of Mathematical Physics

Alexeyeva¹,

Zakir’yanova²

2020

Mathematical Theorems - Boundary Value Problems and Approximations

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Generalized Solutions of Boundary Value Problems of Dynamics of Anisotropic Elastic Media

Alexeyeva¹,

Zakir’yanova²

2005

Journal of the Mechanical Behavior of Materials

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The method of boundary integral equations (BIE) for the solution of non-stationary boundary value problems (BVP) of dynamics of anisotropic elastic mediums is elaborated. The central moment of this method is constructing the fundamental solutions of equations system, kernels of BIE. Here the fundamental solutions in two-and three-dimensional cases (N, M=2,3) are considered and their properties are studied. In the space of generalized functions the solutions of initial BVP are obtained and their integral representations, regular inside a range of definition are given. Generalizing the Green and the Gauss formulas for the generalized solutions of these equations, singular integral equations for the solution of non-stationary BVP are constructed. The uniqueness theorem of the solutions, including for the class of shockwaves, is presented. STATEMENT OF NONSTATIONARY BOUNDARY VALUE PROBLEMSLet u(x,t) be the solution of the system of hyperbolic equations which describes dynamics of anisotropic elastic mediums. We consider it in the cases of plane (N=2) and space (N=3) deformation: L.A. Alexeyeva, SUMMARY Lij(S x A)Uj (x,t) + G,(x,t) = 0, (x,t)eR N+] (2.1) L ij (d x ,d t ) = C l ?'d m d l -S iJ d?, i,j =1,M, m,l =1, Ν ij V m u l s^lm _ ,-ml ij ~ ij ~ ^ ji · (2.2) 259 Brought to you by | Western University

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Dynamic analogues of Somigliana's formula for unsteady dynamics of elastic media with an arbitrary degree of anisotropy

Cited by 2 publications

References 3 publications

Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations of Mathematical Physics

Singular Boundary Integral Equations of Boundary Value Problems for Hyperbolic Equations of Mathematical Physics

Generalized Solutions of Boundary Value Problems of Dynamics of Anisotropic Elastic Media

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