Initial strain method

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Initial strain method

 

In this method the material is allowed to sustain stresses outside the failure criterion for finite "periods". Instead of plastic strains, we refer to viscoplastic strains and are generated as a rate that is related to the amount by which yield has been violated by:

 

 

p_63_image010

 (163)

                                                 

where F is the yield function and Q is the plastic potential function.

The increment of viscoplastic strain, which is accumulated from one iteration to the next, is obtained through multiplication the strain rate by a pseudo time step as:

 

 

p_63_image012

(164)

                                             

and

 

 

p_63_image014

(165)

                                 

where the time step for numerical stability depends on the assumed failure criterion as:

 

 

p_63_image016

for Von-Misees materials  

(166)

 

 

p_63_image018

for Mohr-Coulomb materials

(167)

                   

 

 

The derivatives of the plastic potential function Q with respect to stresses are expressed as:

 

 

p_63_image020

(168)

                               

where p_63_image022 where t represents the second deviatoric stress invariant:

 

 

p_64_image002

(169)

             

and

 

 

p_64_image004

(170)

 

 

p_64_image006, etc.    

(171)

 

 

p_64_image008

(179)

                       

                                                     

where the first invariant (mean stress invariant) s is given by the relation:

 

 

p_64_image010

(180)

                                       

It may be noted that in geotechnical applications, plane strain conditions are enforced and in the above equations p_64_image012.

The viscoplastic strain rate is evaluated numerically by the expression:

 

 

p_64_image014

(181)

                           

where

 

 

p_64_image016

 

 

 

 

(182)

 

 

p_64_image018

 

 

 

 

 (183)

                               

 

p_65_image002

 

 

 

(184)

 

 

p_65_image004

(185)

               

 

The self-equilibrating body loads are accumulated at each time step within each load step by summing the following integrals for all yielded elements (F>0 at Gauss points):

 

 

p_65_image006

(186)

                           

This process is repeated at each time step iteration until no integration point stresses violate the failure criterion within a given tolerance. The convergence criterion is based on a dimensionless measure of the amount by which the displacement increment vector Ui changes from one iteration to other.

 

 

 

 

 

 

 

 

 

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