This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1892 Excerpt: ...becomes of importance. Kirchhoff's theorem of the uniqueness of solutions holds for the cases we shall investigate in this volume, viz. those of infinitesimal strain in a body all whose dimensions are finite. 67. Possibility of Solution. We add here a theorem relating to the possibility of solving the general equations ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1892 Excerpt: ...becomes of importance. Kirchhoff's theorem of the uniqueness of solutions holds for the cases we shall investigate in this volume, viz. those of infinitesimal strain in a body all whose dimensions are finite. 67. Possibility of Solution. We add here a theorem relating to the possibility of solving the general equations. Suppose the surface-displacements given, and that there is no bodily force, we have to shew that there exist functions u, v, w, satisfying the differential equations of equilibrium and the boundary-conditions. Let u, v, w and u', v', w' be two systems of functions of x, y, z which are finite, continuous, and one-valued within the surface S, limiting the body, and at that surface become equal to the given displacements. It is clear that there is an infinite number of such systems of functions. Form with u, v, w the energy-function W, which is a definite quadratic function of the six strains, expressed by certain differential coefficients of u, v, w; and form in like manner with u', v', w' the energy-function W, and let Denote the six strains corresponding to u, v, w by the letters elt e2, ...e6, and the corresponding six stresses by P1, P2, ...P6, and let similar quantities derived from the u, v', w system be denoted since f is a homogeneous quadratic function of its six arguments. Then it is easy to shew that the right hand side of (53) can be transformed into dy--(similar terms in v' v and w'--w) 4-///f(e--e) dxdydz...(54), and therefore, since the function f(e'--e) is necessarily positive, it follows from (53) that V is always greater than V, if the equation and the two similar equations hold good, and therefore these equations have a solution if V have a minimum value. Now W is a sum of positive terms, and, at the surface, W must be ...
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