CENTER FOR COMPUTATIONAL MATHEMATICS COLLOQUIUM

                  UNIVERSITY OF COLORADO AT DENVER



TITLE:   LOSS OF ELLIPTICITY AS A MATHEMATICAL INDICATOR OF MATERIAL FAILURE
 

SPEAKER: Howard L. Schreyer, Department of Mechanics, 
         University of New Mexico
         

DATE:    Monday, April 7, 1997


PLACE:   Math Conference Room 626 
         UCD Building, 1250 14th St., Denver


TIME:    noon (Refreshments served at 11:45 am) 



ABSTRACT

        In solid mechanics there are several families of constitutive 
equations of which typical examples are elasticity, plasticity, 
viscoelasticity, continuum damage, and viscoplasticity.  In addition, there are 
several engineering criteria which are meant to indicate when material failure 
may occur.  Examples are the criteria of Rankine, Tresca, and von Mises; 
various fracture criteria based on maximum principal stress, maximum principal 
strain and energy; and in addition there are DruckerUs stability postulate and 
HillUs second-order work criterion.  Needless to say, most mathematicians would 
not recognize any of these conditions which is surprising because the set of 
governing equations are merely partial differential equations with varying 
coefficients.
        When most solid bodies are loaded, the constitutive equation evolves 
from that of linear elasticity to one of the examples given above.  If a 
perturbation is applied at each step of the loading path, the problem remains 
well posed until the tangent tensor relating stress rate and strain rate yields 
an acoustic tensor with a zero eigenvalue.  This single point can be identified 
with any one of the following three conditions: (i) the material is unstable, 
(ii) a discontinuous bifurcation exists, or (iii) the original ellipticity of 
the system is lost.  Here, a fourth interpretation is proposed, namely, 
material failure has been initiated.  The purpose of the last interpretation is 
to unify a wide range of engineering terminology into a single, well-defined, 
mathematical concept.
        The presentation will provide the details of the above interpretation 
and give numerical results which show convergence with mesh refinement even 
though material softening is present and the criteria of Hill and Drucker are 
violated.  A possible approach for maintaining well posedness past the 
initiation of failure will also be discussed.