Castigliano’s principle of minimum strain-energy

Source abstract

Abstract 1—Two important theorems can be established regarding the equilibrium of a body which obeys Hooke’s law. In the first we assume that specified displacements are imposed (by an application of suitable forces) on particular points of the body, and we compare configurations which all satisfy this condition, but of which all but one would require additional forces (some applied at other points) to maintain them in equilibrium.(In all the strains are “compatible”,— i.e., they are expressible in terms of single-valued displacements u, v, w; so the comparison is between configurations which could be obtained one from another by imposing single-valued displacements at all points except those of which the displacements are specified. Mathematically, we vary u, v, w, independently (except at these points) without imposing the requirement that every part of the body must be in equilibrium.) It can be shown that U, the total elastic strain-energy, has its smallest value in the equilibrium configuration: this is the First Theorem of Minimum Strain-Energy. As a special example of maintained displacements we have the “selfstrained” body. Self-straining may arise from various causes,—e.g. stresses produced by casting, rolling or other manufacturing processes, or by differences in temperature at different parts of the same body; but it can always (in imagination) be relieved by making suitable cuts in the body, and conversely, the original state of self-strain can be restored by applying such forces to the cut body as are required to neutralize the “gaps” produced by cutting. Let A and B be two points, on opposite sides of a gap, which are brought into coincidence when the gap is neutralized: then the forces entailed at A and B may be grouped together as a generalized “force”, and the relative displacement of A and B may be regarded as the “corresponding" displacement. Since the gaps have definite magnitudes, definite displacements are required to close them: so, by a generalization of the notions of force and displacement, we may regard a self-strained body as resulting from the imposition of definite displacements upon a body which initially was in a state of ease. Compatible strains in the “cut” body entail non-compatible strains in the resulting (self-strained) body.