II II II II II I - Waste Isolation Pilot Plant - U.S. Department of Energy

where C is the fourth order tensor of elastic coefficients defhwd by Equation (4.5.12). Combining the strain rate
decomposition defined in Equation (4.5.8) with Equations (4.5.19) and (4.5.20) yields
(4.5.21)
We note that because Q is deviatoric, C:Q = 2P Q and Q:C:Q = 21.L. Then using the normality condition,
Equation (4.5.9), the definition of equivalent plastic strain, Equation (4.5.10), and Equation (4.5.21)
2HIy=Q:6tr–
3
Y2Y
(4.5.22)
and since Q is deviatoric ( ~ btr = 2V Q: d ), we can determine y from Equation (4.5.22) as
y = (1+14’/3~)
Qd . (4.5.23)
The current normal to the yield surface, Q, and the total strain rate, d, are known quantities. Hence, from
Equation (4.5.23), ~ can be determined which can be used in Equation (4.5.9) to determine the plastic part of the
strain rate which, with the additive strain rate decomposition and the elastic stress rate of Equations (4.5.8) and
(4.5. 12), completes the definition of the rate equations.
We still must explain how to integrate the rate equations subject to the constraint that the stress must remain on
the yield surface. We will show how that is accomplished in Section 4.5.4.
4.5.2 Kinematic Hardening
Just as before with the isotropic hardening case, we write a von Mises yield condition but now in terms of the
stress difference
f(~) =+&~=K2 . (4.5.24)
It is important to remember that a and ~ are deviatoric tensors. The consistency condition is written for
kinematic hardening as
f(g) = o (4.5.25)
because the size of the yield surface does not grow with kinematic hardening; therefore, k = O. Using the chain rule
on Equation (4.5.25)
(4.5.26)
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