Finite strain FEM using an existing code that solves small strain elasticity

Question

I have an existing FEM code that solves the linear elasticity problem. I would like to use the same code for large strain rates, still using a simple material law (Saint Venant–Kirchhoff model). [The problem I want to solve is a contact problem in which strain is larger than 1 very locally, close to the contact point]

I believe I have two steps to take: take into account the nonlinear part of strain $\nabla u \cdot \nabla u^T$, and solve the mechanical balance in the deformed configuration.

Can this be done by successive iterations of the linear elasticity problem, e.g. using iterative mesh deformation and a fixed-point algorithm for the nonlinear strain term? Does this converge? References which would highlight the steps to be taken from small-strain elasticity are welcome.

EDIT

I would for instance like to know whether such an algorithm would work (there may be a couple mistakes in the details, for now it's rather the spirit of it I'm enquiring about):

I want to solve $K u + \nabla u^T\cdot \nabla u= Mf$.

Let $(f^i)_{i\leq n}$ be a sequence of functions with $f_n = f$, with each increment small enough.

Let $i=0$, $U^0=0$, $K^0=K$.

• Solve linear elasticity problem $K^i \delta u^i = M f^i$.

• Construct configuration $\Omega^{i+1}$, and a change of variable $X^{i+1}$ from $\Omega^{i}$ to $\Omega^{i+1}$

• let $U^{i+1} = U^i \circ X^{i+1} + \delta u^i$

• Calculate $K^{i+1} = K_{\Omega^{i+1}} + \nabla {U^{i+1}}^T\cdot \nabla$ (there's probably something to do here as we develop $u=U+\delta u$ in the double product, it doesn't write out nicely in tensor notation but should in the linear problem)

Again, the above is not to be taken as something thought over but rather the sort of thing I'd like to find in some report/textbook/publication. But maybe there's a crucial problem I don't see which would be the reason why finite strain codes are not presented as an extension of small strain ones?

Answer 1

We only know how to solve general linear problems efficiently, so every algorithm for general nonlinear problems needs to reduce them to a sequence of linear problems. This is what we do in fixed point methods, Newton's method, etc. In your context, all methods to solve such problems reduce the problem to a sequence of (linear) elasticity problems.

Nonlinear mechanics is not a field I know really well, so I don't know of many examples. I could suggest looking at the step-44 tutorial program of deal.II, though, which solves what I think comes closest to what you want to do. (Disclaimer: I'm one of the principal authors of deal.II.)