RESIDUAL DISPLACEMENT DUE TO ARBITRARY PLASTIC STRAINS

The displacement of the boundary of an elastic body in which inelastic strains exist or initiate due to various conditions can affect the intended functioning of the machine element and thus the service life of the assembly. Plastic strains are often initiated when the body is loaded beyond the elasticity limit of its material. The subsequent deformation of the boundary may affect the interaction with the coupling elements, resulting in further perturbations of the contact conditions. This is especially true when load is transmitted through a small contact region, such as in gears or bearings. Assessment of this permanent deformation is thus essential to predict and prolong the service life of various machine elements working under severe contact loads. The mathematical model employed in this work is built around the Betti’s reciprocal theorem, which exposes a property of the Elasticity equations in the presence of inelastic strains. The resulting equations prove that the residual displacement due to plastic strains in the bulk of the body can be expressed by superposition of effects, as a convolution product. The Green’s functions for the latter convolution are integrated to derive the influence coefficients needed in the numerical approach. The model discretisation is rectangular, with the volume of plastic strains approximated with a reunion of adjacent cuboids with uniform strains. The contributions of each elementary cuboid to the displacement components in a specific point from the boundary, i.e., the influence coefficients, are derived and expressed analytically. With these elements, an acceleration technique for the computation of elastic displacement can be extended to residual displacement. Calculations are performed with the aid of a Matlab computer program developed based on the newly derived relations. Five configurations of plastic strains are considered and the program predictions are compared with existing results from the literature. The good agreement gives confidence in the advanced computer tool and encourages its implementation in a more complex program for the simulation of the elastic-plastic contact.