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# Consistent linearized tangent moduli for hyperelastic materials... is it always a positive definite matrix?

In order to obtain numerical solution of problemsthat involves a hyperelastic material model, we use what is known as i**ncremental/iterative **solution techniques of Newthon's type. The basic idea is to contruct a discrete system of nonlinear equation, **KU=F, **and solving it using a Newton's method or a modified version of it. As we know, its lead to a systematic linearization of the internal force vector and by the chain rule to the linearization of the material model.

Now, getting back to my question, I would like to discuss if we can proof that the ``stiffness matrix'' or jacobian matrix obtained by a consistent linearization process of a hyperelastic material model is always positive definite. In concrete, if we have a hyperelastic material model whose scalar-valued energy function W is given by the Flory-Rehner free-energy function, could we argue that the ``stiffness matrix'' is positive definite?

Thanks,

Mario Juha

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## Comments

## As usual, it depends on the function

Dear Mario,

There are some strain energy functions for which a consistent tangent tensor will not be positive definite. For instance, the Ogden strain energy model may have parameters chosen which provide a great fit to test data but which result in unstable behavior based on Drucker's postulate regarding second order work. It is guaranteed that at least at the point of material instability for these cases the tangent tensor will not be positive definite.

Off the top of my head, I do not know the Flory-Rehner form of the strain energy function. If it is a relatively simple, non-damaging one based on invariants, one can PROBABLY show that the consistent tangent tensor is positive definite provided all coefficients are positive.

Cheers,

Matt

Matt Lewis

Los Alamos, New Mexico

## The stiffness matrix is not always positive-definite

Dear Mario, The tangential stiffness matrix is not always positive-definite. The positive-definiteness depends on the physics of the model. If the material has instability, the load-displacement curve would be discontinuous, and the stiffness matrix will have negative eigenvalues. In the Flory-Rehner model of polymeric gels, in some parameter ranges, the gel is unstable at some point, and would phase separate spontaneously. Tanaka used the same model to explain the temperature-sensitiveness of some gels, which is discontinuous. Wei

## Re:Consistent linearized tangent moduli for hyperelastic materia

Thanks, Dear Wei and Matt.

Now, it explains why for some parameters and boundary conditions I got negatives eigenvalues in my jacobian matrix. Now, I am running several hyperelastic problem that used the Flory-Rehner free energy function as presented in http://imechanica.org/node/3163, but instead of using abaqus I am using my own code. I can reproduce most of the results, but for some other configurations my code fail, but that is another story.

I was concerned about efficient numerical algorithms to solve the system of equations. I am using LAPACK routines to solve the system of equations that results from the linearization process. I was tempted to use the following subroutine dposv_, which is intended for symmetric positive definite matrix, instead of dgesv_. The former uses the Cholesky decomposition algorithm to factorize the jacobian and the latter use an LU decomposition with partial pivoting. The solution time is significantly reduced if we can use algorithms based on symmetric positive definite matrices. Do you have suggestions?

On the other hand, is the banded structure of the stiffness matrix unaffected by the material model? Or does it just depends on the nodal connectivity?. That is something I do not understand fully. I know that in linear cases when we have incompressible materials models and we use the segregeted d,p form (T.J.R. Huges. Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover, 2000) the banded structure is destroyed. Does happen the same with hyperelastic materials?

This is a nice topic and I am very interested on it. Dr. Hong thanks for your excellent paper. It have helped me during my research.

Mario

## Re: tangent moduli

Mario,

In our paper (http://imechanica.org/node/6594), we derived an explicit formula for the tangent modulus (Eq. 53) of a hydrogel based on the same model of Wei Hong's works. We coded it as a user subroutine (UMAT) in ABAQUS and was able to solve several problems, including one with instability.

RH

## Re:Consistent linearized tangent moduli for hyperelastic materia

Dear Mario,

To complement other people's comments, l would like to point out that irrespective of the constitutive model, the tangent stiffness matrix can be non-definite due to geometric effects. This is the case in the vicinity of a buckling point, or geometric instability.

Best regards,

Marino

Marino Arroyo

http://www-lacan.upc.es/arroyo/

Universitat Politecnica de Catalunya

Campus Nord, C2-204

Jordi Girona 1-3

Barcelona, 08034

Tel: +34 93 4054653

Tel: +34 93 4011805

Fax: +34 93 4011825

## geometric instability

Dear Marino,

Thank you for pointing out the non-positive-definite tangent stiffnes due to geometric instability. I agree completely. It is numreically challenging to solve such instability problems though. For example, in ABAQUS, we ran into numerical stability and convergence issues at the point of buckling or creasing. What would be the best numerical method you suggest to tackle such problmes?

Best regards,

RH

## Re: geometric instability

Dear Rui,

Indeed, Newton's method will have a hard time to converge to an equilibrium solution. If you seek for stable equilibria, those likely to be observed, a good idea is combining Newton's method with a line search method that guarantees that the total potential energy decreases in every iteration. This is very robust.

Best regards,

marino

Marino Arroyo

http://www-lacan.upc.es/arroyo/

Universitat Politecnica de Catalunya

Campus Nord, C2-204

Jordi Girona 1-3

Barcelona, 08034

Tel: +34 93 4054653

Tel: +34 93 4011805

Fax: +34 93 4011825

## Re:geometric instability

There are several methods to do it. One of them is the arc-length method or use descent and gradient minimization methods as presented in "J.T. Oden. Finite Elements of Nonlinear Continua. Dover.2000". I am planning work on it right now. Unfortunately, I am working alone and I do not move as fast as I want.

Mario

## alternative control schemes

If your problem does not have snap back (but can have peaks or snap through) , a fairly simple displacement control scheme will work (I used this technique in a corotational truss program see outline of algorithm starting on page 7 http://people.wallawalla.edu/~louie.yaw/Co-rotational_docs/2Dcorot_truss...). Although the algorithm is for the truss program the key ideas would still be applicable to your situation possibly.

If your problem has snap back, snap through, peak points and possibly winding loops, an alternative to arc length control is generalized displacement control which works really well. I have programmed it in matlab for my corotational truss program and also for a corotational 2D beam program.

A really good paper describing generalized displacement control is

Y. B. Yang, L. J. Leu, and Judy P. Yang, Key Considerations in Tracing the Postbuckling

Response of Structures with Multi Winding Loops, Mechanics of Advanced Materials and

Structures, 14:175-189, 2007.

regards,

Louie

## The method in the blog

The method in the blog http://imechanica.org/node/4124 can also solve this non-positive definite case.