16.5.2. Finite Element Nonlinear Discretization

The main task of geometrically nonlinear discretization is to formulate the stiffness matrix. The overall discretization of the solution equation for increments in nodal displacement is based on the tangent element equation:

(16–23)

In the previous equation, the total stiffness matrix is made from 2 components: current and stress stiffness matrices. The current stiffness matrix is defined as:

(16–24)

The stress (geometric) stiffness matrix is defined as:

(16–25)

where

 
 

= the nodal load vector

 

= the strain-displacement matrix

= the deformation gradient matrix

= the constitutive matrix

 

= the second Piola-Kirchhoff stress tensor

 

= the element volume