Introduction to Elasticity/Minimizing a functional

Minimizing a Functional in 1-D

In 1-D, the minimization problem can be stated as

Find $u(x)$ such that

U[u(x)] = \int_{x_0}^{x_1} F(x,u,u^{'}) dx

is a minimum.

We have seen that the minimization problem can be reduced down to the solution of an Euler equation

\frac{\partial F}{\partial u} - \frac{d}{dx}\left(\frac{\partial F }{\partial u^{'}} \right) = 0

with the associated boundary conditions

\eta(x_0) = 0 ~\text{and}~ \eta(x_1) = 0

or,

\left.\frac{\partial F}{\partial u^{'}} \right|_{x_0} = 0 ~\text{and} \left.\frac{\partial F}{\partial u^{'}} \right|_{x_1} = 0

Minimizing a Functional in 3-D

In 3-D, the equivalent minimization problem can be stated as

Find $\mathbf{u}(\mathbf{x})$ such that

U[\mathbf{u}(\mathbf{x})] = \int_{\mathcal{R}} F(\mathbf{x},\mathbf{u},\boldsymbol{\nabla}\mathbf{u})~dV

is a minimum.

We would like to find the Euler equation for this problem and the associated boundary conditions required to minimize $U$ .

Let us define all our quantities with respect to an orthonormal basis $(\widehat{\mathbf{e}}_{i})$ .

Then,

\mathbf{x} = x_i\widehat{\mathbf{e}}_{i} ~~;~~~ \mathbf{u} = u_i\widehat{\mathbf{e}}_{i} ~~;~~~ \boldsymbol{\nabla} \mathbf{u} = u_{i,j} \widehat{\mathbf{e}}_{i}\otimes\widehat{\mathbf{e}}_{j}

and

U[\mathbf{u}(\mathbf{x})] = \int_{\mathcal{R}} \tilde{F}(x_i, u_i, u_{i,j})~dV

Taking the first variation of $U$ , we get

\delta U = \int_{\mathcal{R}} \left(\frac{\partial\tilde{F} }{\partial u_i} \delta u_i + \frac{\partial \tilde{F}}{\partial u_{i,j}} \delta u_{i,j}\right) dV

All the nine components of $\delta u_{i,j}$ are not independent. Why ?

The variation of the functional $U$ needs to be expressed entirely in terms of $\delta u_i$ . We do this using the 3-D equivalent of integration by parts - the divergence theorem.

Thus,

\begin{align} \int_{\mathcal{R}} \frac{\partial \tilde{F}}{\partial u_{i,j}} \delta u_{i,j}~ dV &= \int_{\mathcal{R}} \frac{\partial }{\partial x_j} \left(\frac{\partial \tilde{F}}{\partial u_{i,j}} \delta u_i\right) dV - \int_{\mathcal{R}} \frac{\partial }{\partial x_j} \left(\frac{\partial \tilde{F}}{\partial u_{i,j}} \right) \delta u_i~ dV \\ & = \int_{\partial\mathcal{R}} \frac{\partial \tilde{F}}{\partial u_{i,j}} \delta u_i~n_j~dA - \int_{\mathcal{R}} \frac{\partial }{\partial x_j} {}{}\left(\frac{\partial \tilde{F}}{\partial u_{i,j}} \right) \delta u_i~ dV \end{align}

Substituting in the expression for $\delta U$ , we have,

\begin{align} \delta U &= \int_{\mathcal{R}} \frac{\partial \tilde{F}}{\partial u_i} \delta u_i~dV + \int_{\partial\mathcal{R}} \frac{\partial \tilde{F}}{\partial u_{i,j}} \delta u_i~n_j~dA - \int_{\mathcal{R}} \frac{\partial }{\partial x_j} \left(\frac{\partial \tilde{F}}{\partial u_{i,j}} \right) \delta u_i~ dV \\ &= \int_{\mathcal{R}}\left[\frac{\partial \tilde{F}}{\partial u_i} - \frac{\partial }{\partial x_j}\left(\frac{\partial \tilde{F}}{\partial u_{i,j}} \right)\right] \delta u_i~ dV + \int_{\partial\mathcal{R}} \frac{\partial \tilde{F} }{\partial u_{i,j}} \delta u_i~n_j~dA \end{align}

For $U$ to be minimum, a necessary condition is that $\delta U = 0$ for all variations $\delta\mathbf{u}$ .

Therefore, the Euler equation for this problem is

\frac{\partial \tilde{F}}{\partial u_i} - \frac{\partial }{\partial x_j} \left(\frac{\partial \tilde{F}}{\partial u_{i,j}} \right) = 0 ~~~~\forall~~\mathbf{x} \in \mathcal{R}

and the associated boundary conditions are

\frac{\partial \tilde{F}}{\partial u_{i,j}} = 0 ~~~\text{or,}~~~ \delta u_i = 0 ~~~~\forall~~\mathbf{x} \in \partial\mathcal{R}

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