See pdfs in
http://ftp.lstc.com/anonymous/outgoing/jday/composites:
crash_composites_paper.pdf
HIGH_VELOCITY_IMPACT_mat_059_Langlie_Cheng.pdf
mat_059_Hallquist_Cheng.pdf
____________________________________________________________________

According to the mat_059 subroutine for solids, the slot labeled as "plastic strain" in LS-Prepost is 
occupied by the variable "check".  There are 22 extra history variables available for output starting with 
the variable fs1.  Given in sequence, the extra history variables are

          ...,fs1(nlq),fs2(nlq),fs3(nlq),
     &fs4(nlq),q11(nlq),q12(nlq),q13(nlq),q31(nlq),
     &q32(nlq),q33(nlq),ff1(nlq),ff2(nlq),ff3(nlq),
     &ff4(nlq),fs5(nlq),fs6(nlq),fs7(nlq),fs8(nlq),
     &         ff5(nlq),ff6(nlq),ff7(nlq),ff8(nlq)

fs1, fs2, ..., fs8 are failure flags that indicate failure modes 1, 2, ..., 8, respectively.  
(These are stored as history variables 1,2,3,4,15,16,17,18 if NEIPH is set to 18 in 
*database_extent_binary).  A value of zero for 
fs* signifies failure in mode *.  For example, if fs3 is zero, then failure has occurred in mode 3.  The 
variable "check" is the average of the eight fs* values.

The eight modes of failure and the condition that triggers each mode are ...

1.  Longitudinal tension:
sig1^2/xxt^2 + sig4^2/sba^2 + sig6^2/sca^2  > 1.0
(sig1 positive)

2.  Transverse tension:
sig2^2/yyt^2 + sig4^2/sba^2 + sig5^2/scb^2  > 1.0
(sig2 positive)

3.  Through-thickness shear (combined with long. tension):
sig1^2/xxt^2 + sig6^2/sca^2  > 1.0
(sig1 positive)

4.  Delamination (through-thickness tension):
sig3^2/zzt^2 + sig5^2/scb^2 + sig6^2/sca^2  > 1.0
(sig3 positive)

5.  Through-thickness shear (combined with transverse tension):
sig2^2/yyt^2 + sig5^2/scb^2   > 1.0
(sig2 positive)

6.  Longitudinal compression:
sig1^2/xxc^2   > 1.0
(sig1 negative)

(***Following two had typos (in mat59_solids, not source code) prior to 3/29/2011 ***)

7.  Transverse compression:
sig2^2/(sba + scb)^2 + (sig2 / yyc)*((yyc^2/(sba + scb)^2) - 1.0) + sig4^2/sba^2 + sig5^2/scb^2  > 1.0

8.  Through-thickness compression:
sig3^2/(sca + scb)^2 + (sig3 / zzc)*((zzc^2/(sca + scb)^2) - 1.0) + sig6^2/sca^2 + sig5^2/scb^2  > 1.0





ff1, ff2, ... ff8 are scaling factors applied to various components of stress 
when failure modes 1, 2, ..., 8, 
repsectively, have been triggered.  Once failure mode * is triggered, 
it takes 100 time steps for ff* to go 
from a starting value of 1.0 to a final value of 0.0.

Local stress components scaled by ff* when failure mode * is triggered:

Mode 1:  sig1 and all 3 shears
Mode 2:  sig2 and all 3 shears
Mode 3:  sig1 and all 3 shears
Mode 4:  sig3 and all 3 shears
Mode 5:  sig2 and all 3 shears
Mode 6:  sig1 and all 3 shears
Mode 7:  sig2 and all 3 shears
Mode 8:  sig3 and all 3 shears

'Failure' in one direction in mat59 solid does not trigger 
element deletion (see http://ftp.lstc.com/anonymous/outgoing/jday/composites/1brick/mat2_22_59.k) 

Here's an excerpt from the mat59 (for solids) source code...

c     compute pressure for failed material
c
     do 70 i=lft,llt
     check(i)=(fs1(i)+fs2(i)+fs3(i)+fs4(i)+fs5(i)+fs6(i)+
    .          fs7(i)+fs8(i))*.125
     if (check(i).lt..99) then
     if (sig1(i)+sig2(i)+sig3(i).eq.0.0) then
     pres =bulk*log(def(i))
     presm  =  min(0.0,pres)
     sig1(i)=presm
     sig2(i)=presm
     sig3(i)=presm
     sig4(i)=1.1101e-16
     sig5(i)=1.1101e-16
     sig6(i)=1.1101e-16
     ifail(i)=1
     failur=.true.
     endif
     endif

I believe that when ifail(i) is set to 1, the element is deleted.   
Run http://ftp.lstc.com/anonymous/outgoing/jday/composites/mat59_sol.k.gz
and monitor the variable check ("plastic strain" in LS-Prepost), 
Also plot sigx, sigy, and sigz in element 37002.  When all are at zero, the element is deleted. 
A simple 1-element model that also illustrates element deletion is 
http://ftp.lstc.com/anonymous/outgoing/jday/composites/mat2_22_59fail.k in which 
I sequentially induce failure modes 2, 1, and 4.   These modes 
and the history variables that flag failure in these modes (extra 
history variables 2, 1, and 4) are described above.
It's not until all three normal stresses sigy, sigx, and sigz are 
set to zero (due to reaching failure in their respective modes) that the 
element is deleted.
  
This also explains why an element in uniax tension but with PR=0 will be deleted when
sig in the loaded direction reaches the failure criterion whereas when PR=1e-6, the element is not
deleted.  IN the latter case, the transverse stresses are very small but nonzero and so the 
element deletion criterion (sig1=sig2=sig3=0) is not met.  See invalid bug 2796.