On Some Problems in Transversely Isotropic Elastic Materials

Chen, W. T.

doi:10.1115/1.3625048

Cited by 45 publications

(7 citation statements)

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“…The method used is based on special differential operators and is similar to that proposed in [119] for isotropic plates. Lekhnitskii's solution [117] was analyzed for completeness in [105] and generalized to arbitrary threedimensional bodies in [115,120]. Other general solutions were obtained in [111,118] and adapted to transtropic plates in [79,105].…”

Section: Static Problems For Isotropic Cylindrical Bodiesmentioning

confidence: 99%

“…Lekhnitskii's solution [117] was analyzed for completeness in [105] and generalized to arbitrary threedimensional bodies in [115,120]. Other general solutions were obtained in [111,118] and adapted to transtropic plates in [79,105]. A biharmonic solution to the tension-compression and bending problems was constructed in [56] using Cartesian coordinates.…”

Section: Static Problems For Isotropic Cylindrical Bodiesmentioning

confidence: 99%

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Some results and problems in the three-dimensional theory of plates (Review)

Shaldyrvan

2007

Int Appl Mech

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The Lurie-Vorovich method of homogeneous solutions is discussed with reference to the reduction problem. A general procedure for studying the three-dimensional stress concentration in multiply connected bodies of finite size is proposed. Model problems are solved to examine the influence of geometrical parameters on the stress state. The Lurie-Vorovich method is generalized to media with complicated properties and demonstrated with problems in composite mechanics and crack theory Keywords: layer, thick plate, short cylinder, cubical cylinder, mixed boundary-value problems, homogeneous solutions, isotropic and transversely isotropic media, compositeIntroduction. Solid mechanics is characterized by a collection of concepts and ideas about the processes in natural objects and engineering structures. Experimental and theoretical studies as well as numerical calculations expand our knowledge about the outworld and lead to more sophisticated mathematical models. They more adequately describe phenomena of interest and produce more reliable data on processes in objects under study. Mathematical models are as a rule formalized by posing boundary-value problems whose solution is a complicated and many-sided procedure. The associated difficulties stimulate the modification of available and development of new methods, both analytic and numerical-and-analytic, including direct numerical schemes. An excellent illustration of the foregoing is the problem of strength (reliability) dealing with processes that cause failure of structures. One of the decisive factors leading to failure is stress concentration (SC)-an abrupt local change of the stress field in a solid body. SC is induced by cavities, foreign inclusions, cracks, abruptly changing geometry (corners, curvature discontinuity), nonsmoothness of external load, and boundary sections with different fixation and loading conditions.In the beginning, the theory and design methods were developed by analyzing SC, induced by different factors, based on two-dimensional models of elasticity theory (ET). General methods for solving the plane problem owe their development to Muskhelishvili's methods of complex-variable theory. A considerable contribution to solving the SC problem was made by Savin when he dealt with the engineering design of ground support for mine workings. Very hard and laborious research was done to understand that the stress-strain state (SSS) of a rock mass, even if with SC, can be described by a system of Lamé equations. The results obtained by Savin and his school and summarized in the fundamental monographs [68, 69] made it possible to gain an insight into the complex stress pattern around holes and to ascertain the effect of various factors (the shape of the boundaries, the distance to defects, etc.) on it. Of considerable interest were problems of SC caused by the mutual effect of stress concentrators. The basis for the approaches to solving plane problems for multiply connected domains was laid by

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Section: Static Problems For Isotropic Cylindrical Bodiesmentioning

confidence: 99%

Section: Static Problems For Isotropic Cylindrical Bodiesmentioning

confidence: 99%

Some results and problems in the three-dimensional theory of plates (Review)

Shaldyrvan

2007

Int Appl Mech

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“…Numerous investigators [1][2][3][15][16][17][18][19][20][21][22] have presented analytical solutions for displacement under a point load in a transversely isotropic full space, whose transversely isotropic planes are parallel to the horizontal loading surface. A summary of the existing solutions is given in Table I.…”

Section: Introductionmentioning

confidence: 99%

“…Chowdhury [15] Methods of images and Hankel transforms Vertical All displacements Pan [3] Vector functions Three dimensional All displacements Willis [16] Fourier transforms Vertical All displacements Elliott [17] Potential functions Vertical All displacements Chen [18] Potential functions Vertical All displacements Horizontal…”

Section: Introductionmentioning

confidence: 99%

Elastic solutions of displacements for a transversely isotropic full space with inclined planes of symmetry subjected to a point load

Wang

Liao

2007

Num Anal Meth Geomechanics

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SUMMARYThis work presents analytical solutions for displacements caused by three-dimensional point loads in a transversely isotropic full space, in which transversely isotropic planes are inclined with respect to the horizontal loading surface. In the derivation, the triple Fourier transforms are employed to yield integral expressions of Green's displacement; then, the triple inverse Fourier transforms and residue calculus are performed to integrate the contours. The solutions herein indicate that the displacements are governed by (1) the rotation of the transversely isotropic planes (), (2) the type and degree of material anisotropy (E/E , / , G/G ), (3) the geometric position (r, , ) and (4) the types of loading (P x , P y , P z ). The solutions are identical to those of Liao and Wang (Int. J. Numer. Anal. Methods Geomechanics 1998; 22(6):425-447) if the full space is homogeneous and linearly elastic and the transversely isotropic planes are parallel to the horizontal surface. Additionally, a series of parametric study is conducted to demonstrate the presented solutions, and to elucidate the effect of the aforementioned factors on the displacements. The results demonstrate that the displacements in the infinite isotropic/transversely isotropic rocks, subjected to three-dimensional point loads could be easily determined using the proposed solutions. Also, these solutions could realistically imitate the actual stratum of loading situations in numerous areas of engineering.

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“…Recently exact solutions have been found for a number of problems involving a spheroidal elastic inclusion (or cavity) [16][17][18][19][20]; and experience with problems involving spheroidal geometry lends itself naturally to the present hyperboloidal notch analysis. Recently exact solutions have been found for a number of problems involving a spheroidal elastic inclusion (or cavity) [16][17][18][19][20]; and experience with problems involving spheroidal geometry lends itself naturally to the present hyperboloidal notch analysis.…”

Section: Introductionmentioning

confidence: 99%