I wish to find the functional whose minimisation yields the follwoing equation on the vector function u ( λ + μ ) ∇ ( ∇ ⋅ u ) + μ ∇ 2 u = 0, the Navier equation of linear elasticity. I know that this equation has a vaiational principle from physical reasons, but struggle to find it. The term containing the laplacian is easier to handle, I can not even prove simmetry of the weak form for the first one: how to integrate by parts (grad div u) v to obtain a term symmetric in u e v?

Question

I wish to find the functional whose minimisation yields the follwoing equation on the vector function u   (  λ  +  μ  )  ∇  (  ∇  ⋅  u  )  +  μ      ∇          2        u  =  0, the Navier equation of linear elasticity. I know that this equation has a vaiational principle from physical reasons, but struggle to find it. The term containing the laplacian is easier to handle, I can not even prove simmetry of the weak form for the first one: how to integrate by parts (grad div u) v to obtain a term symmetric in u e v?

podesect · Accepted Answer

By a corollary of the divergence theorem:      ∫    Ω    ∇  (  ∇  ⋅  u  )  ⋅  v  =  −      ∫    Ω    ∇  ⋅  u    ∇  ⋅  v  +      ∫          ∂      Ω        (  ∇  ⋅  g  )  v  ⋅  n  .

I wish to find the functional whose minimisation yields the follwoing equation on the vector functio

Answered question

Answer & Explanation

New Questions in Linear algebra