ABSTRACT
Whereas a fluid can have unlimited displacements resulting from integration of velocity over time, an elastic, nonplastic solid has finite displacements. A translation and a rotation are displacements that change the position but not the shape of a body. The remaining part of the displacement represents the deformation of the body, specified by a strain tensor (Section 4.1). The deformations are associated with stresses (Section 4.2), that is, the surface forces that together with volume forces balance the inertia force in the momentum equation. The stresses and strains are related by an elastic constitutive relation (Section 4.3) that depends on the properties of a material. Using the preceding relations, it is found that the displacement vector satisfies a second-order differential equation (Section 4.4); in the case of bounded media, the differential equation has a unique solution satisfying boundary conditions such as given (or zero) stresses/forces/displacements at a boundary. There are three main methods to approach problems of steady plane elasticity. Method I is based on the solution of the momentum equation (Section 4.4) for the displacement vector; for example, a radial (azimuthal) displacement such as the velocity of a source/sink (vortex) leads to the stresses due to a line pressure (torque) in a massive or hollow cylinder or cylindrical cavity or shell (Section 4.5). Method II relies on all components of the stress tensor deriving from a scalar stress function that is biharmonic, so that its double Laplacian is zero (Section 4.6); for example, this method can be used to determine the displacements, stresses, and strains in a wedge, with applied or concentrated force or torque or distributed fluid loading as in a dam (Section 4.7). Method III uses the divergence (curl) of the displacement vector, that is, the linearized relative area change (rotation), with which can be formed a complex elastic potential that is an analytic function, whose real and imaginary parts satisfy the Laplace equation; an example is a monopole that corresponds to a line force in an unbounded medium (Section 4.8). Because the equations of elasticity are linear, the principle of superposition applies. It can be combined with method II (method III) to specify the displacements, strains, and stress in a semi-infinite (infinite) medium [Section 4.7 (Section 4.8)] due to an arbitrary load distribution on the boundary (in the interior). An example of multiple loads is the case of the displacements, stresses, and strains on a wheel (Section 4.9) due to the superposition of (1) its own weight; (2) a traction/braking force associated with forward motion/retardation; and (3) a torque applied by a driving engine.
