Hello there,(adsbygoogle = window.adsbygoogle || []).push({});

I am struggling in proving the following.

The principle of Minimum energy for an elastic body (no body forces, no applied tractions) says that the equilibrium state minimizes

$$\int_{\Omega} \nabla^{(s)} u D (\nabla^{(s)}u)$$

among all vectorial functions u satisfying the boundary conditions, where $$\nabla^{(s)} = \frac{1}{2} (u_{i,j}+u_{j,i})$$ and D is a constant tensor $$D_{ijkl}$$

The principle is expressed in terms of displacements, so one would expect that its Euler Lagrange Equation coinciides with the equlibrium equation of elasticity expressed in terms of displacements, the Navier equations, $$A \nabla (\nabla \cdot u) + B \nabla^{2} u = 0$$, A e B constants.

How to prove that? I am quite shaky in dimensions higher than 1.

I tried writing the first variation, after introducing $$u_{var} = U + \epsilon u$$ as

$$\int_{\Omega} \nabla^{(s)} u D (\nabla^{(s)}U)$$

and now by integration by parts I recover a Laplacian, as in Navier's equation (second term), but not the term $$\nabla (\nabla \cdot u)$$, any help would be so appreciated, thanks

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Variational Principle and Vectorial Identities

Loading...

Similar Threads - Variational Principle Vectorial | Date |
---|---|

I Euler’s approach to variational calculus | Feb 18, 2018 |

A Maximization problem using Euler Lagrange | Feb 2, 2018 |

A Maximization Problem | Jan 31, 2018 |

A Derivation of Euler Lagrange, variations | Aug 26, 2017 |

Need guidance on where to start on the study of the variational principle | Mar 22, 2012 |

**Physics Forums - The Fusion of Science and Community**