Computation of consistent tangent moduli for rate-dependent, anisotropic single crystals

Hi,

I am working on implementing a rate-dependent crystal plasticity (RDCP) constitutive model into a UMAT (Abaqus 6.9-2). The correct derivation of the CTM is required to ensure quadratic convergence of the global iterations. However I have some problems achieving quadratic convergence over all the iterations. Usually from the first to the second the error is the square root of the first one, but then convergence is linear, typically err(i+1) = err(i)/10 or so. In addition, the convergence speed depends on the time increment used, and I do not know whether it should be so or to which extent.

I am wondering several things, could not find any answer in the literature so any reference or comment is welcome.

• The CTM depends on the time increment because the constitutive model is rate-dependent, and the increment of stress delta_sig created by a strain increment delta_str ultimately depends on delta_t (e.g. because delta_sig is a function of the internal variables that are integrated over delta_t using a midpoint rule). Therefore the CTM will depend on the global time increment used to solve the BVP. What are the conditions required on the increment length to ensure quadratic convergence of the iterations? Is there any study dealing with the effect on the global time increment onto the convergence properties of RDCP models?
• Which delta_str should be applied to the system for derivation of the CTM? Since the material is rate-dependent, the twin question is: over which time length should the CTM be derived? Should it be the same delta_t & delta_str as applied during the iteration?
• The material is in addition anisotropic, meaning that deriving the CTM by applying at the end of the increment delta_str =deps (1,1,1,1,1,1) in vector form does not a priori give the same result as if a perturbation delta_str = (deps1, deps2,... deps6) with different components is applied. Which perturbation is to be applied to derive a correct CTM?

Thank you so much.