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Anytime you begin to lose stiffness in a static structural analysis, there is always potential for convergence trouble. Â
The sensitivity depends on the application. How localized is the material yielding? How well is the structure constrained?
Perhaps the structure is beginning to buckle and non-convergence is a reflection of a physical instability.
You could test this theory by post processing your force-deflection curve to determine if you have lost all resistance to further loading.
You could also try adding stabilization damping to make convergence more forgiving to material softening.