Consider the second order stress tensor representation ij = ,kk ij , ij ,

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Consider the second order stress tensor representation σij = ∅,kkδij –∅,ij, where∅is a proposed stress function. First show that this scheme gives a divergence-less stress; i.e. σij,j = 0. Next using this representation in the general compatibility equations (5.3.3) with no body forces gives the result:

ij + 1 - v 1+v ,kkij = 0

Finally show that for the two-dimensional plane stress case, this representation reduces to the ordinary Airy form from Section 7.5.

Equation 5.3.3

Tij, kk + 1 1+v - kk, ij V 1-v -dijFk,k - Fij - Fj,i

Data from section 7.5

Numerous solutions to plane strain and plane stress problems can be determined through the use of a

This assumption is not very restrictive because many body forces found in applications (e.g., gravity

With equilibrium now satisfied, we focus attention on the remaining field equations in the stress

This relation is called the biharmonic equation, and its solutions are known as biharmonic functions. Thus,expect that all parts of the solution field be identical. Problems with multiply connected regions or

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