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

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?

• This book seemed a good reference when I was considering doing this (I never went to implementation though). I believe the approach you describe fits under the "Updated Lagrangian" framework. Oct 13 '15 at 0:43
• @GeoMatt22: Thanks! Sorry I hadn't seen your comment at first. The ToC looks promising, I'll borrow it next time I make the trip to library.
– Joce
Oct 13 '15 at 15:38

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.)

• Thanks for pointing this link, I'll pore over it. I guess there $\delta u$ is the increment compared to a deformation around which one linearizes the problem? What I'd be really looking for is an abstract iterative algorithm which I could adapt to my case.
– Joce
Oct 12 '15 at 20:51
• I believe so. As for abstract algorithm, have you looked at the books by Simo & Hughes, or by Daya Reddy? Oct 13 '15 at 2:45
• Is it respectivelyComputational Inelasticity and Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements ? They're table of contents doesn't mention closely related problems, the latter is very general and probably has only small strains, and the former focuses on nonlinear material laws (plasticity).
– Joce
Oct 13 '15 at 9:07
• Have you looked at Belytcshko's "Nonlinear Finite Elements" that I recommended above? Chapter 4 discusses the Lagrangian Meshes (Updated vs. Total), and Chapter 6 discusses Solution Algorithms (including Newton & Linearization techniques). An old pre-publication set of notes is available here. Oct 13 '15 at 11:07
• You got the book by Simo and Hughes right. The one by Reddy is simply called "Plasticity". Oct 13 '15 at 11:38