Recent developments in the evaluation of the 3D fundamental solution and its derivatives for transversely isotropic elastic materials

The problem of constructing fundamental solutions to the thermoelasticity problem for a piecewise-homogeneous transversely isotropic space is reduced to the matrix Riemann problem in the space of generalized slow growth functions. As a result of the solution of which, were obtained expressions in explicit form for the components of the stress tensor and the displacement vector in plane of connection of transversely isotropic elastic half-spaces containing concentrated stationary heat sources. The temperature distribution is investigated depending on the thermophysical characteristics of the half-space materials.

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“…Then, considering the conditions (3), ( 4) and the results of [8][9][10][11][12][13][14], with respect to…”

Section: Construction Of the Fundamental Solution Problem Of Thermoel...mentioning

confidence: 99%

Inhomogeneous transversely isotropic space under influence of concentrated power and temperature sources

Kryvyi

Morozov

2022

J. Phys.: Conf. Ser.

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“…Once again, the second-order derivative may be written as a singular part by a modulation function (e,g,, Refs, [17,35])…”

Section: Explicit and Unique Expressions For The Derivative Of Q"mentioning

confidence: 99%

Unique and Explicit Formulas for Green's Function in Three-Dimensional Anisotropic Linear Elasticity

Buroni¹,

Sáez

2013

Journal of Applied Mechanics

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Unique, explicit, and exact expressions for the first-and second-order derivatives of the three-dimensional Green's function for general anisotropic materials are presented in this paper. The derived expressions are based on a mixed complex-variable method and are obtained from the solution proposed by Ting and Lee (Ting and Lee, 1997, "The Three-Dimensional Elastostatic Green's Function for General Anisotropic Linear Elastic Solids," Q. J. Mech. Appl. Math. 50, pp. 407^26) which has three valuable features. First, it is explicit in terms of Stroh's eigenvalues Pa fa = 1,2,3j on the oblique plane with normal coincident with the position vector; second, it remains welt-defined when some Stroh's eigenvalues are equal (mathematical degeneracy) or nearly equal (quasi-mathematical degeneracy); and third, they are exact. Therefore, both new proposed solutions inherit these appealing features, being explicit in terms of Stroh's eigenvalues, simpler, unique, exact and valid independently of the kind of degeneracy involved, as opposed to previous approaches. A study of all possible degenerate cases validate the proposed scheme.

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“…For isotropic and linear elastic materials, three-dimensional, simple and closed-form elastostatic fundamental solutions are available. In contrast, the corresponding three-dimensional fundamental solutions for anisotropic materials are much more complex, and they have closedform analytical expressions only in some special cases such as for transversely isotropic materials [9][10][11][12][13][14][15]. By using either the Fourier or the Radon transform, contour or infinite line integral expressions for the three-dimensional displacement fundamental solutions have been presented by Fredholm [16], Lifshitz & Rozenzweig [17], Barnett [18], Mura [1], Wang [19] and Lee [20], among others.…”

Section: Introductionmentioning

confidence: 99%

Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials

et al. 2016

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Novel unified analytical displacement and stress fundamental solutions as well as the higher order derivatives of the displacement fundamental solutions for three-dimensional, generally anisotropic and linear elastic materials are presented in this paper. Adequate integral expressions for the displacement and stress fundamental solutions as well as the higher order derivatives of the displacement fundamental solutions are evaluated analytically by using the Cauchy residue theorem. The resulting explicit displacement fundamental solutions and their first and second derivatives are recast into convenient analytical forms which are valid for non-degenerate, partially degenerate, fully degenerate and nearly degenerate cases. The correctness and the accuracy of the novel unified and closed-form three-dimensional anisotropic fundamental solutions are verified by using some available analytical expressions for both transversely isotropic (non-degenerate or partially degenerate) and isotropic (fully degenerate) linear elastic materials.

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Recent developments in the evaluation of the 3D fundamental solution and its derivatives for transversely isotropic elastic materials

Cited by 6 publications

References 18 publications

Inhomogeneous transversely isotropic space under influence of concentrated power and temperature sources

Inhomogeneous transversely isotropic space under influence of concentrated power and temperature sources

Unique and Explicit Formulas for Green's Function in Three-Dimensional Anisotropic Linear Elasticity

Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials

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