Higher order conservation integrals in elasticity and application

摘要

Without using Saint Venant’s compatibility equations and any other condition, the satisfaction of Navier’s displacement equations is a sufficient condition, from which each of the displacements, stresses and strains as well as Airy stress function satisfy a bi-harmonic equation, and the first stress and strain invariants satisfy a harmonic equation in elasticity in two- and/or three-dimensional space, respectively. Noether’s theorem is applied to their functionals in order to obtain a class of higher order non-trivial conservation integrals. For each of the displacements, stresses and strains as well as Airy stress function, a directly proportional function is a symmetry-transformation of their functionals, from which their conservation integrals in physical space follow. All conformal transformations of Euclidean space are symmetry-transformations of all the above equation’s functionals, from which their conservation integrals in material space follow. Especially, the conformal transformation in planar Euclidean space can be expressed in terms of the real and imaginary parts of an analytic function, which is a symmetry-transformation of the functionals for the first stress and strain invariants in two-dimensional elasticity. It is found that by adjusting this analytic function in the obtained conservation integral, a finite value can be obtained in calculating it with a path encircling the material point with any order singularity. Stress field near the tip of a crack is used to show the application of this kind of the conservation integral.