Abstract
A NEW method for solving two-dimensional elasticity problems has recently been suggested by N. Louat1. It is perhaps worth pointing out that his method falls within the scope of a very general approach to elasticity, which can be briefly described as follows. Regard the body D (assumed homogeneous) as embedded within an infinite elastic medium of the same material, and introduce a distribution of point-forces over its boundary S; since these are acting in an infinite medium, they produce a known displacement field throughout the medium, including D and S; if so, any prescribed boundary displacements (or, more indirectly, prescribed boundary tractions) can be achieved by a suitable boundary distribution of point-forces, which hence qualify to generate the relevant elastic fields within D. Mathematically expressed, we formulate the system of three linear functional relations : where q,p are vector variables defining points on S, and where dq denotes the surface differential at q; the σi(q) are the three independent point-force components per unit area at q, and the Uj(p) are the three independent displacement components at P; Kij(q–p) stands for the ,jth displacement component at p produced by σi = 1 at q. Given Uj on S, relations (1) become three coupled Fred-holm integral equations for the σi(q), which can then by utilized to generate Uj at any point P of D simply by writing P for p in Kij. By appropriate differentiations and linear combinations of these equations, the problem of given boundary tractions can also be formulated. These considerations hold for bodies of any shape or form, isotropic and anisotropic. Generally speaking, however, not even approximate analytic solutions would be available. Digital computing techniques offer the prospect of achieving effective numerical solutions to problems of technological interest, though the difficulties associated with singular kernels such as Kij and its derivatives should not be underrated.
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References
Louat, N., Nature, 196, 1081 (1962).
Jaswon, M. A., Report on Integral Equation Methods in Potential Theory and Elasticity (to be published).
Eshelby, J. D., Proc. Roy. Soc., A, 241, 376 (1957).
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JASWON, M. Integral Equation Methods in Elasticity. Nature 198, 572 (1963). https://doi.org/10.1038/198572a0
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DOI: https://doi.org/10.1038/198572a0
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