Advanced Mechanics of Materials and Applied Elasticity

Abstract

This text is an outgrowth of the material used by the author for several decades in senior and graduate courses for students of mechanical, aerospace and civil engineering. It deals with the problem of computing the stress and displacement fields in solid bodies at two levels of approximation: the level of the linear theory of elasticity and the level of the theories of mechanics of materials. The linear theory of elasticity is based on very few assumptions and can be applied to bodies of any geometry. The theories of mechanics of materials are based on many assumptions in addition to those of the theory of elasticity and they can be applied only to bodies of certain geometries (beams, bars, shafts, frames, plates shells and thin-walled, tubular members). In this text the formulas of the theories of mechanics of materials are derived in a way that the assumptions on which they are based can be clearly understood. Moreover, wherever possible the results obtained on the basis of the theories of mechanics of materials are compared with those obtained on the basis of the theory of elasticity. In the past, the use of the linear theory of elasticity was limited by the fact that only a few problems could be solved using the available classical methods. Thus, approximate theories like the theories of mechanics of materials were formulated for which exact solutions could be found. With the advent of the electronic computer, many problems involving bodies whose geometry does not justify the use of the theories of mechanics of materials, are formulated on the basis of the linear theory of elasticity and solved approximately with the aid of a computer. Thus, a mechanical, civil or aerospace engineer who works in the area of stress analysis and design often uses software based on the linear theory of elasticity. It is important therefore that master’s level students of mechanical, aerospace and civil engineering who specialize in the area of stress analysis and design, should acquire some knowledge of applied elasticity. The book includes 18 chapters and 7 appendices. In the first chapter a brief review of vector analysis is presented followed by a very elementary, but concise introduction to the algebra of symmetric tensors of the second rank. In the theories of mechanics of materials and elasticity one deals with quantities such as stress, strain and moments and product of inertia which are symmetric tensors of the second rank. It is desirable therefore that the student learns at the very beginning the transformation properties of such quantities as well as how to determine their stationary values. The boundary value problems for computing the displacement and stress fields in solid bodies on the basis of the linear theory of elasticity, are formulated in Chapter 5 and applied to simple examples in Chapters 5, 6 (torsion problem) and 7 (plain stress and plain strain problems). The boundary value problems for computing