In this study, we presented a numerically robust procedure to evaluate 4th order tangent moduli which are vital for acheiving quadratic convergence of global Newton-Raphson scheme.
In this study, the proposed method is verified for hyperelastic models alone. However, the same can be extended to other constitutive models.
The paper can be found at
Numerically approximated Cauchy integral (NACI) for implementation of constitutive models
10.1016/j.finel.2014.05.016
http://www.sciencedirect.com/science/article/pii/S0168874X14001061
In order to obtain numerical solution of problemsthat involves a hyperelastic material model, we use what is known as incremental/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