Geometric Nonlinearities
When deformations in a material are significant, the influence of the changing geometry on the equilibrium equations can no longer be neglected. These effects are termed geometric nonlinearities. While often associated with strains exceeding a few percent, geometric nonlinearities also arise from large rigid-body rotations even if the strains remain small (e.g., in slender structures).
Analyses capturing these effects are referred to as large strain or finite strain analyses. In numgeo, this is activated by flagging large deformations (adding nlgeom to the step definition). In such cases, the system stiffness becomes a function of the displacements, changing as the body changes shape or rotates.
Deformation gradient
The theory of large strain computations is established by defining basic physical quantities (motion and deformation) and their corresponding mathematical relationships. The applied loads acting on a body cause it to move from one position to another. This motion is defined by tracking position vectors in the “deformed” (current) and “undeformed” (reference) configurations.
Let the position vectors in the “deformed” and “undeformed” states be represented by \(\boldsymbol{x}\) and \(\boldsymbol{X}\) respectively. The motion (displacement) vector \(\boldsymbol{u}\) is defined as:
The deformation gradient \(\boldsymbol{F}\) maps infinitesimal vectors from the reference configuration to the current configuration:
The information contained in the deformation gradient \(\boldsymbol{F}\) includes the volume change, the rotation, and the shape change of the deforming body. The local volume change is calculated from the determinant of \(\boldsymbol{F}\):
The rotation and shape change contributions of \(\boldsymbol{F}\) are decoupled using the right polar decomposition theorem:
where \(\boldsymbol{R}\) is the orthogonal rotation matrix and \(\boldsymbol{U}\) is the symmetric right stretch (shape change) matrix.
Stress update
The nlgeom procedure in numgeo differs depending on the constitutive formulation. Specifically, distinct approaches are used for general materials (rate/incremental form) versus hyperelastic materials (total form):