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ANALYTICAL AND NUMERICAL TESTS FOR LOSS OF MATERIAL STABILITY
Author(s) -
SCHREYER H. L.,
NEILSEN M. K.
Publication year - 1996
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19960530)39:10<1721::aid-nme924>3.0.co;2-x
Subject(s) - mathematics , bifurcation , numerical analysis , constitutive equation , finite element method , instability , plane stress , convergence (economics) , mathematical analysis , computer simulation , work (physics) , mechanics , nonlinear system , structural engineering , physics , engineering , statistics , quantum mechanics , economics , thermodynamics , economic growth
Material instability occurs when ellipticity is lost for symmetric constitutive equations. Prior to loss of ellipticity it is possible that the second‐order work of Hill or Drucker becomes negative. There are implications in the literature that numerical solutions cease to be meaningful when a material strain softens and the second‐order work is not positive. The instant that the second‐order work is zero or negative simultaneously with the additional restriction that the strain increments satisfy compatibility is equivalent to the loss of the ellipticity criterion for symmetric constitutive relations. The loss of ellipticity criterion is the appropriate one for identifying when numerical solutions cease to show convergence and may also be a suitable criterion for identifying the instant at which material failure is initiated. An analytical development is provided for loss of ellipticity together with an explicit expression for the normal to the bifurcation plane. Numerical solutions are given for several sample problems. For all cases, 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.