# In an incremental nonlinear FE procedure does the material stiffness matrix need to be updated?

A typical nonlinear FE problem can be given by:

$$\mathbf{R} = \mathbf{K}\rho + \mathbf{F}_{NL}(\rho),$$

where R is the vector of applied forces, K is the material or linear stiffness matrix, $F_{NL}$ is the nonlinear force function and $\rho$ is the vector of displacements.

The tangent stiffness matrix is:

$$K_T = K + \dfrac{\partial F_{NL}}{\partial \rho}(\rho)$$

If an incremental procedure is applied, $\dfrac{\partial F_{NL}}{\partial \rho}$ will need to be updated at load application step, since it is a function of the deformed configuration.

But why does the material stiffness matrix $K$ not need to be updated? $K$ is formed by rotating local stiffness matrices and assembling them into a global frame of reference. Surely when the geometry becomes deformed $K$ will need updating also, even if it is the linear stiffness matrix?

• That is for a 2D beam such that $\varepsilon_{xx} = \dfrac{\partial u}{\partial x} + \dfrac{1}{2} \left( \dfrac{\partial v}{\partial x} \right)^2$. – simplytheaverage Dec 9 '17 at 7:18