Principle of Minimum Potential Energy and Reciprocal Theorem

1 Principle of Minimum Potential Energy and Reciprocal Theorem 

To illustrate the equivalence of the energy minimization principle advanced above and the linear elastic equilibrium equations of section 2.4.2, we resort to evaluating the functional derivative of eqn (2.77) and setting it equal to zero, or in mathematical terms 8Y /8ui = 0. Equivalently, working in indicial notation, we find 

8U = ]- ( Cijki8 ij kidV + f Cijki ij8 kidV - f fi8uidV - f ti8uidA. 

Exploiting the symmetries of the elastic modulus tensor and the strain tensor, the first two terms may be joined to form Oij8 ijdV. Rewriting the integrand crij8eij 

In addition to the theoretical advantages offered by the principle of minimum potential energy, our elastic analyses will also be aided by the reciprocal theorem. This theorem is a special example of a more general class of reciprocal theorems and considers two elastic states (u ), or(i)) and where each 

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