av Teodor Heggelund
Abstract:
A finite element method implementation of lower-order strain gradient plasticity is developed. Its validity is checked against known analytical solutions. The implementation gives expected trends when applied to particle strengthening and void growth.
During implementation, stability issues are encountered. The instability is denoted the tower/canyon defect, and traced back to nodal averaging as a basis for strain gradient computation. A conservative stability criterion for stability is developed, and within the stability limit given by the criterion, no tested models are unstable. Instabilities are shown to appear under a combination of (a) large plastic strains, (b) small length scale and (c) fine element mesh. Nodal averaging underestimates the strain gradient at boundaries.
An alternative to nodal averaging is developed, denoted nodal contributions. Nodal contributions has not been implemented for use in finite element simulations, but analytical verification indicates that nodal contributions is resilient to previously encountered stability issues. Nodal contributions is shown to represent strain gradients exactly for linear strain fields, even at boundaries.
The exact, mathematical solution to the applied lower-order strain gradient plasticity theory is shown to be singular given a prescribed stress field. Iterative solutions based on load incrementation choose one of the possible resulting strain distributions. Nodal averaging is biased towards small strain gradients at boundaries.