The LS-DYNA Theory Manual, in the description of mat 3, defines effective plastic strain as follows,

(equation)

Characteristics of effective plastic strain are:

-  Calculated incrementally using the equivalent plastic strain rate.

-  It's non-negative and monotonically increasing.

-  Increases whenever the material is actively yielding, i.e.,whenever the state of stress is on the yield surface.

https://www.dynasupport.com/tutorial/computational-plasticity/the-equations-for-isotropic-von-mises-plasticity  
uses the same equation as above in defining  "equivalent plastic strain".

[Thus] effective plastic strain and equivalent plastic strain are interchangeable terms, that is, they mean the same thing.

jd
Ticket#2018110510000052

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Effective plastic strain is a 
monotonically increasing scalar value which is calculated incrementally as a function of  
(Dp)ij, the plastic component of the rate of deformation tensor.  In tensorial notation,
this is expressed as...

epspl  =  integral over time of (depspl) = integral [ sqrt( 2/3  (Dp)ij   (Dp)ij ) ] dt

This particular definition applies to J2 plasticity (Ref: Ticket#2017110110000132).

(See the equation for effective plastic strain in the description of Material Model 3 in 
the LS-DYNA Theory Manual.  To understand the notation, look under the heading of "Summation Convention" in this page...
http://www.brown.edu/Departments/Engineering/Courses/En221/Notes/Index_notation/Index_notation.htm or 
see http://ftp.lstc.com/anonymous/outgoing/support/FAQ_docs/tensor_notation.png for a screenshot.)

Effective plastic strain grows whenever the material is actively yielding, i.e.,
whenever the state of stress is on the yield surface. 

In contrast, the tensorial strain values, written by LS-DYNA when
STRFLG is set to 1 in *database_extent_binary, are not necessarily 
monotonically increasing as they reflect the current, total (elastic + plastic) 
state of deformation.  To fringe the tensorial strains in LS-PrePost, click
Fcomp > Strain.

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RE: Effective strain

Effective strain is NOT the same thing as effective plastic strain.

Effective strain, expressed in tensorial notation, is 
   epseff = sqrt[ 2/3  (epsdev)ij  (epsdev)ij ]  
where epsdev are deviatoric strains.
(See the description of mat 19 in the Theory Manual.)

In terms of tensorial strains, 
   epseff = sqrt(2.0*(sx^2+sy^2+sz^2)/3.0 + (sxy^2+syz^2+szx^2)/3.0)

In terms of principal strains, 
   epseff = sqrt(2.0*(s1^2+s2^2+s3^2)/3.0)

The effective strain calculation in LS-PrePost is as follows:
  tensor strains sx,sy,sz,sxy,syz,szx
  mean strain p=(sx+sy+sz)/3
  deviatoric strains dx = sx - p, dy = sy - p, dz = sz - p
  aa = sxy^2 + syz^2 + sxz^2 - dx * dy - dy * dz - dx * dz
  effective strain es = sqrt(4*abs(aa)/3) 


See http://ftp.lstc.com/anonymous/outgoing/support/FAQ/effstrain.tar for a test case for 
shells which includes a spreadsheet.
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For a simple, 1-element illustration of various strain measures, 
run http://ftp.lstc.com/anonymous/outgoing/support/FAQ_kw/mat24.cycle.k and then overlay time histories of 
z-strain, effective strain, and effective plastic strain.


Other measures of strain can be fringed in LS-PrePost but these are calculated by
LS-PrePost from nodal coordinates, e.g., 
Fcomp > Infin  (infinitesimal or engineering strain)
Fcomp > Green
Fcomp > Almansi
Be aware the infinitesimal strain is affected by rotation of the element as illustrated by
ftp://ftp.lstc.com/outgoing/support/FAQ_kw/spinup.bar.expl_dr.k.
Strain measures are addressed in 
http://en.wikipedia.org/wiki/Deformation_(mechanics)
http://en.wikipedia.org/wiki/Finite_strain_theory#Finite_strain_tensors
http://en.wikipedia.org/wiki/Infinitesimal_strain_theory

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RE: von Mises stress
 
For definition of effective stress, aka von Mises stress, try the following links:
http://en.wikipedia.org/wiki/Von_Mises_yield_criterion
http://www.continuummechanics.org/cm/vonmisesstress.html
http://en.wikipedia.org/wiki/T-criterion

In tensorial notation, effective stress is sqrt[ 3/2 (sij)(sij) ]
where sij are deviatoric stresses.

sigvm = 1/sqrt(2) * sqrt[ (sigx-sigy)^2 + (sigy-sigz)^2 + (sigz-sigx)^2 + 6*sigxy^2 + 6*sigyz^2 + 6*sigzx^2 ]
 
or in terms of principal stresses,  

sigvm = 1/sqrt(2) * sqrt((sig1-sig2)^2 + (sig2-sig3)^2 + (sig3-sig1)^2)