On the instabilities resulting from a non-associated flow rule

Instability resulting from a non-associated flow rule is explored. For a Mohr-Coulomb material model with non-normality, or in the plane strain case for any material model with non-normality, the model is unstable in any equilibrium configuration as soon as it enters the plastic range, provided the elastic response along the wedge path is too small to stabilize the plastic response. This genuine instability of configuration in the form of a shear band with a rotating boundary is conceptually different from the classical shear band in many aspects. It is the first counter-example to the commonly held idea that instability of path always occurs earlier than instability of configuration in the plastic range. For an elastic-plastic extended Mises model, or any elastic plastic material model with a smooth yield surface, combined with non-normality, both the instability of configuration and instability of path are examined. A new type of instability of path has been discovered that is entirely different from the instability of path of a Shanley column in the plastic range. The strong tacit, but not always valid, assumption of continuing equilibrium is responsible for confusion over uniqueness and bifurcation with a non-associated flow rule. Techniques customary for an associated flow rule give results for a non-associated flow rule that are of limited value at best. The possibility is indicated that macroscopic plastic deformation may not develop continuously, but instead develops in a discontinuous manner reminiscent of breakaway of dislocation pile-ups in metals on the microscale and frictional response in geomaterials. However, the jumps may be so small that they are undetectable with the usual external measuring devices. A reassessment of the reliability and validity of commercial codes which employ non-associated flow rules is suggested, and alternative choices are given which are stable in the small in the forward sense.