Introduction to Elasticity/More variational principles

More Variational Principles

Recall that the admissible states appropriate to the minimum principles are required to meet certain field equations and the appropriate boundary conditions.

In some cases, we would like to use variational principles in which the admissible states satisfy as few constraints as possible. Such variational principles are useful for symmetric elastic fields.

Let $I$ be a scalar-valued functional. Let $\mathcal{A}$ be the set of all admissible states. Let $s$ and $\tilde{s}$ be two admissible states $\in\mathcal{A}$. Let $\lambda$ be a Lagrange multiplier such that $s+\lambda\tilde{s}$ is also an admissible state $\in\mathcal{A}$.

Let

$\delta_{\tilde{s}} I[s] = \left.\frac{d}{d\lambda} I[s+\lambda\tilde{s}]\right|_{\lambda=0}$

Then,

$\delta I[s] = 0$

only if $\delta_{\tilde{s}} I[s]$ exists and equals zero for all $\tilde{s}$ that satisfy the above requirements.

There is an infinite number of possible functional $I[s]$ that satisfy these requirements. Two examples are:

• The Hellinger-Reissner functional
• The Hu-Washizu functional