The subject of Elasticity is concerned with the determination of the stresses and displacements in a body as a result of applied mechanical or thermal loads, for those cases in which the body reverts to its original state on the removal of the loads. In this book, we shall further restrict attention to the case of linear infinitesimal elasticity, in which the stresses and displacements are linearly proportional to the applied loads and the displacements are small in comparison with the characteristic length dimensions of the body. These restrictions ensure that linear superposition can be used and enable us to employ a wide range of series and transform techniques which are not available for non-linear problems.
Most engineers first encounter problems of this kind in the context of the subject known as Mechanics of Materials, which is an important constituent of most undergraduate engineering curricula. Mechanics of Materials differs from Elasticity in that various plausible but unsubstantiated assumptions are made about the deformation process in the course of the analysis. A typical example is the assumption that plane sections remain plane in the bending of a slender beam. Elasticity makes no such assumptions, but attempts to develop the solution directly and rigorously from its first principles, which are Newton’s laws of motion, Euclidian geometry and Hooke’s law. Approximations are often introduced towards the end of the solution, but these are mathematical approximations used to obtain solutions of the governing equations rather than physical approximations that impose artificial and strictly unjustifiable constraints on the permissible deformation field.
KeywordsPrincipal Stress Stress Component Principal Direction Transformation Rule Maximum Tensile Stress
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