# Two-Dimensional Problems in Elasticity

## 3.1 Introduction

As has been pointed out in Sec. 1.1, the approaches in widespread use for determining the influence of applied loads on elastic bodies are the mechanics of materials or elementary theory (also known as technical theory) and the theory of elasticity. Both must rely on the conditions of equilibrium and make use of a relationship between stress and strain that is usually considered to be associated with elastic materials. The essential difference between these methods lies in the extent to which the strain is described and in the types of simplifications employed.

The mechanics of materials approach uses an assumed deformation mode or strain distribution in the body as a whole and hence yields the average stress at a section under a given loading. Moreover, it usually treats separately each simple type of complex loading, for example, axial centric, bending, or torsion. Although of practical importance, the formulas of the mechanics of materials are best suited for relatively slender members and are derived on the basis of very restrictive conditions. On the other hand, the method of elasticity does not rely on a prescribed deformation mode and deals with the general equations to be satisfied by a body in equilibrium under any external force system.

The theory of elasticity is preferred when critical design constraints such as minimum weight, minimum cost, or high reliability dictate more exact treatment or when prior experience is limited and intuition does not serve adequately to supply the needed simplifications with any degree of assurance. If properly applied, the theory of elasticity should yield solutions more closely approximating the actual distribution of strain, stress, and displacement.

Thus, elasticity theory provides a check on the limitations of the mechanics of materials solutions. We emphasize, however, that both techniques cited are approximations of nature, each of considerable value and each supplementing the other. The influences of material anisotropy, the extent to which boundary conditions depart from reality, and numerous other factors all contribute to error.

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