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8.10.2 Material instabilities
A material instability is a point in the stress-strain response where the slope of the stress-strain curve effectively becomes zero. Since FEBio's Newton based solvers use this slope to advance the solution, a zero slope would mean an infinite step and FEBio will not be able to continue.
The slope (or more accurately the elasticity tangent tensor) does not have to be zero to cause problems. For materials that soften at large deformations, this slope may become so small that it renders the global stiffness matrix ill-conditioned. It is unlikely that FEBio will be able to continue and even if it finds a solution, it may not be useful.