ANALYTICAL AND NUMERICAL TESTS FOR LOSS OF MATERIAL STABILITY

Material instability occurs when ellipticity is lost for symmetric con stitutive equations. Prior to loss-of ellipticity it is possible that the second-order work of Hill or Drucker becomes negative. There are i mplications in the literature that numerical solutions cease to be mea ningful when a material strain softens and the second-order work is no t positive. The instant that the second-order work is zero or negative simultaneously with the additional restriction that the strain increm ents satisfy compatibility is equivalent to the loss of the ellipticit y criterion for symmetric constitutive relations, The loss of elliptic ity criterion is the appropriate one for identifying when numerical so lutions cease to show convergence and may also be a suitable criterion for identifying the instant at which material failure is initiated. A n analytical development is provided for loss of ellipticity together with an explicit expression for the normal to the bifurcation plane. N umerical solutions are given for several sample problems. For all case s, the numerical solutions based on the finite element method conform to the theoretical expectations that unique numerical solutions exist prior to the point at which ellipticity is lost.