Integration of the Fundamental Problem of Static Elasticity

Abstract

In the previous chapter I have established integral properties of the equations of equilibrium of a continuous medium with respect to the sequence of monomials constructed with the Cartesian coordinates, showing that it is possible to derive some inequalities based on such properties. But their scope is more general, for it is possible to set up an integration method based on those integral properties and valid for homogeneous, isotropic or anisotropic, slightly deformable bodies (Grioli [5]). By this method the principal unknowns are just the six components of stress Y _rs , but the fact that these, rather than the displacements, are the principal unknowns does not seem to be disadvantageous: almost always knowledge of the stress and of the forces prescribed by the constraints is more interesting than knowledge of the displacements, while it is well known that any method giving the displacements directly in the form of a power series generally has the disadvantage that the convergence of the derivatives of the power series—by which stress and constraint forces are expressed—is numerically slow, and the number of terms that one may have calculated is not sufficient to approximate well the derivatives of the power series. Instead, this inconvenience is absent if one determines displacement from the stresses by an integration procedure.