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Initial strain method

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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:

 (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:

 (164)

and

 (165)

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

 for Von-Misees materials (166)

 for Mohr-Coulomb materials (167)

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

 (168)

where where t represents the second deviatoric stress invariant:

 (169)

and

 (170)

 , etc. (171)

 (179)

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

 (180)

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

The viscoplastic strain rate is evaluated numerically by the expression:

 (181)

where

 (182)

 (183)

 (184)

 (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):

 (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.