Here we will consider equations of linear elasticity; that is, the behavior of materials undergoing a reasonable load corresponding to their operating range. This context also includes so-called brittle materials, for which no plasticity or viscoplasticity takes place prior to the final rupture. For mathematicians, these equations belong to the class of elliptical equations whose solutions are very smooth, provided that the geometry of the domain under consideration and the coefficients involved in the equations are smooth themselves. Unfortunately, industrial structures are usually far from having these properties. Their geometry shows angles where it is impossible to define any tangent; there is clearly a loss of boundary smoothness. Structures often involve various materials whose elastic properties differ; the elastic stiffness coefficients (Young’s modulus, for example) are thus discontinuous when passing through the contact surface between components.

All of this causes a lack of smoothness in the elastic solution, becoming extreme in some situations: strains and stresses become infinite in some locations. Such a situation is questionable; no material can undergo such an overburden and other mechanisms, such as plasticity, damage or rupture, will occur and release these stresses. In the admissible range of loads considered here, these relaxation mechanisms will develop in very small areas that slightly disturb ...

Start Free Trial

No credit card required