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Incremental stress-strain relations

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We will assume that strain increments include elastic components and plastic components and they are small. An incremental stress-strain relation, analogous to the relation of elasticity can be formulated.

During an increment of plastic straining dF=0 thus:

 (141)

By substitution:

 (142)

The resulting equation to solve for the plastic multiplier dλ is:

 (143)

where Pλ is the row matrix in which both work hardening and strain hardening are included in this expression:

 (144)

Rearranging the terms yield:

 (145)

where I is a unit matrix and Eep is the generalized tangent constitutive matrix. If Q=F (associated flow rule) this matrix Eep is symmetric. For F<0 (yield has not occur) or F=0 and dF<0 (unloading from plastic state) then Eep=E.

The tangent stiffness matrix kt is now given by:

 (146)

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