By J.R. Barber
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Additional info for Elasticity
The name arises, of course, because rigid-body displacement is the only class of displacement that can be experienced by a rigid body. 2 Strains and their relation to displacements Components of strain will be denoted by the symbol, e, with appropriate suffices As in the case of stress, no special symbol is required for shear strain, though we shall see below that the quantity defined in most elementary texts (and usually denoted by differs from that used in the mathematical theory of Elasticity by a factor of 2.
A biharmonic function) whose derivatives satisfy certain boundary conditions on the surfaces. e. by using the variation of tractions along the boundaries to give a clue to the kind of function required, but then exploring the stress fields developed from a wide range of such functions and selecting a combination which can be made to satisfy the required conditions. The disadvantage with this method is that it requires a wide experience of particular solutions and even then is not guaranteed to be successful.
G. It can then be shown that the average stresses so defined satisfy the plane stress equations exactly. This is referred to as the generalized plane stress formulation. In practice, of course, the gain in rigour is illusory unless we can also establish that the stress variation across the section is small, so that the local values are reasonably close to the average. A fully three-dimensional theory of thin plates under in-plane loading shows that the plane stress assumption is a good approximation except in regions whose distance from the boundary is comparable with the plate thickness2.
Elasticity by J.R. Barber