I'm a little bit lost with this. I have two questions.
1) I want to write variational (weak) formulation for a single deformable body, that can undergo large discplacements (rotation), but small strains. I searched lots of places. Finally, I managed to find this:
div(S) + rho*F = rho*a,
where a is acceleration, F is external force, and S (attention!) is a nominal stress, or first Piola-Kirchoff strees, defined as S = J*(F^-1)*sigma. J is a jacobian of F, F itself is a transformation gradient, sigma is Cauchy stress tensor. What already bothers me is that Wikipedia has a different definition for S: S = J*sigma*(F^-T). But whatever, assume I use the first definition. The equation is written with respect to the underformed, initial configuration, and the derivatives in div are taken also in underformed configuration.
So, to write weak form, multiply by vector function u and integrate over the initial configuration V0. The first term turns into:
- integral over V0 (S : grad u) dv0 + integral over boundary of V0 (S * n0) u dA0
where dA0 means elementary surface on the boundary of V0, and n0 is outward unit normal for that boundary. The problem is in the last term, with integration over the boundary.
How do I connect this term with the provided external force on the surface????
I know that for Cauchy stress it's done: (sigma*n)dA = dP, where P is the force that's given. Note n and dA here are given for the current configuration. But what do I get for (S * n0) dA0 ? If I express S in terms of sigma, I end up with (J*(F^-1)*sigma*n0) dA0. As far as I know, there is a relation dA0 * n0 = 1/J * (F^T) * dA * n, so plugging it results in:
(F^-1)*sigma*(F^T)*n*dA. Term F^T is getting in the way between sigma and n, so they cannot be multiplied directly in order to give force P.
I'm sure I missed something somewhere, maybe in the equations or definition of the nominal stress or something. Could you please provide me with the correct variational formulation for the large displacement case and explain it in some depth?
2) My second question relates to the FEM discretization of the above. I plan to simulate mechanics of a very stiff solid (with very large Young's modulus and Poissoin's ratio ) so to a human eye it would look like a rigid body. This implies that J is almost equal to 1 ( since F is almost an orthogonal transformation ). For discretization, do I really need to write EVERY term in the weak form explicitly in terms of FEM variables, or can I simplify by using J = 1? What is the normal way for these kind of applications?
I thank you for all the answers I can get!