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Plasticity integration: satisfaction of the consistency requirement
Dear all,
I am trying to formulate the return mapping algorithm with Hill48 yield condition, isotropic hardening law and flow rule referenced to Simo & Hughes (Computational Inelasticity).
In section 2.2.2.1, equation (2.2.9) expresses the Kuhn-Tucker complementarity conditions:
In addition to conditions (2.2.9), satisfies the consistency requirement:
However, on the implementation of Return-Mapping algorithm, the N-R iteration is applied to solve the value of delta plastic multiplier, referenced to “An analysis of a new class of integration algorithms for elastoplastic constitutive relations”, Ortiz & Simo, 1986.
Then, how to make sure the plastic multiplier satisfies the consistency requirement?
It seems that there is no limitation to the consistency requirement during the iteration, as it iterates to force the yield conditions to achieve 0 on the (n+1)th time step.
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what's the relationship between yield function and consistency
Well, I mean what has been done to obey the consistency requirements in the renturn mapping algorithm?
When the return mapping
When the return mapping algorithm converged, will the consistency requirement be automatically satisfied?
Kuhn-Tucker Condition
When the return mapping algorithm comes into play it means that d(gamma)/dt is not zero. To maintian consistency (also implied by Kuhn-Tacker conditions) it is requred to enforce df/dt=0. f is a function of corrected stress which in turn is a function of d(gamma)/dt. Enforcing this condition leads to determination of d(gamma)/dt. Therefore it is evident that determination of d(gamma)/dt is equivalent to the satisfaction of consistency condition.
Mohsen
Jahanshahi,
Jahanshahi,
In return mapping algorithm, only f=0 is enforced by applying Taylor's linear approximation of yield function (f= f_n + (df/dsigma)*delta sigma + (df/dq)*delta q = 0).
Then why the gamma calculated in return mapping algorithm will satisfy consistency condition?
Iterative Solution for gamma
When f is a nonlinear function of gamma you have to use the Taylor linear expansion in an iterative fashion so that finally f = 0. Consider the following
f(gamma_i + delta_gamma_i) = 0 -> f(gamma_i) + df(gamma)/dgamma x delta_gamma_i = 0 -> delta_gamma_i = -f(gamma_i)/[df(gamma)/dgamma] (1)
where df(gamma)/dgamma should be evaluated at gamma_i. Computing delta_gamma_i one updates gamma using
gamma_(i+1) = gamma_i + delta_gamma_i (2)
You have to check whether this value of gamma satisfies f i.e. f(gamma_(i+1)) = 0 within a given tolerance. If so the value is the desired value of gamma otherwise equation (1) should be used again with gamma_(i+1) instead of gamma_i so that finally the correct value of gamma is obtained. The procedure looks like this:
gamma = 0;
for (;;) {
delta = - f(gamma) / df_dg(gamma);
gamma = gamma + delta;
if (fabsl(delta) / fabsl(gamma) < tol)
break;
}
The loop iterates until delta is negligible which means that the correct value of gamma has been obtained.
Mohsen