Elasticity/Hellinger-Reissner principle

Hellinger-Prange-Reissner Variational Principle

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In this case, we assume that the elasticity field is invertible and   is smooth on  . We also assume that   is the set of all admissible states that satisfy the strain-displacement relations, the traction-stress relations and the balance of angular momentum.

Let   denote the set of all admissible states and let   be a functional on   defined by

 

for every  .

Then,

 

at an admissible state   if and only if   is a solution of the mixed problem.