Hi,
I want to know the differences between those models in formulations and the pros and cons of each model.
In addition,
I want to know the representative constitutive model of both models.
please, help me.
In general, flow stress behavior of a metal involves consideration of effects of strain, strain rate, and temperature. Due to the strain rate sensitivity, the strength of the material under dynamic loading is different to the corresponding static value. in most metals, increase in strain rate would lead to increase of the yield stress. for example the Johnson-cook constitute model takes into account quasi- static yielding, strain hardening, strain-rate hardening and thermal softening of material. the Cowper-Symonds equation determines the relation between yield or ultimate dynamic stresses and the corresponding static values in different strain rates.
in general, if the material is rate dependent (i.e steel), and the problem involves dynamic loading, rate dependent plasticity model should be used.
In case of a rate-independent crystal plasticity model you have the problem of choosing the set of active slip (or twin) systems, because the yield surface has corners. However, this problem can be overtaken by using the regularized yield surface, cf. e. g. [Gambin, 1992: Refined analysis of elastic-plastic crystals; Kowalczyk, Gambin, 2004: Model of plastic anisotropy evolution with texture-dependent yield surface]. If you choose large enough regularization exponent, you are close to yield surface stemming from sum of Schmid laws for every system.
Otherwise, you can use rate-dependent model in which you have the power law. Then, there is no problem with the choice of active systems. The classical formulation: works by Asaro and Needleman, 1985, Hutchinson, 1976. The rate-dependent formulation is used e.g. in the popular VPSC code [Lebensohn, Tome, 1993].
When you use regularization or rate-dependent model it is relatively easy to get results, but the disadvantage is loosing the possibility to observe bifurcation in the yield surface corner, from which division of the grain can result (when different parts start to deform on different systems).
The two previous answers are nice illustrations of the fact that one must be careful with the terms rate-dependency and strain rate-dependency in a crystal plasticity modelling context. When you talk about rate-dependency of a crystal plasticity model, it is indeed about the numerical approach that is used to 'round off' the corners of the yield surface in order to have a unique, unambiguous (yet approximate) solution for every applied strain or stress. When you talk about strain rate dependency of a material, it is about the physical response of the material to changes in loading rate (i.e. applied strain rate). Although the rate-dependency of a crystal plasticity model can be used to introduce strain rate effects, it is often not a straight-forward approach, as the mathematical formulation might at first sight seem very similar, but it is used a very different levels (i.e. at the slip system level for the rate dependency of the crystal plasticity model, and on the macroscopic level for the strain rate dependent models, e.g. Cowper-Symonds or Johnson-Cook strength models). The higher strain rates you are considering, the further also you will be 'rounding off' the yield surface, and hence the further you are stepping away from the Schmid law concept. Next to this, many crystal plasticity codes normalize the applied deformation rate; the influence of the globally applied strain rate is then reduced to a small shape difference in yield surface but without any significant influence on the yield strength of the material. This ultimately means that you can have any combination of a strain-rate (in)dependent material model with a rate (in)dependent crystal plasticity model. I have myself been working with the Los Alamos VPSC code in the past to try to model dynamic deformation of Ti6Al4V:
Article Importance of twinning in static and dynamic compression of ...
A rate independent material, at least in the concept of hypo-elasticity is one which is a homogeneous of degree one function in the rate of deformation.
See Walter Noll the Great (1955).
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