*2.2. Relations in Terms of the Actual Configuration with a Taylor Spin (T-Model)*

The models are often formulated using the Jaumann type rate of the Cauchy stress. The law of elasticity is written in this case as [56,57]:

$$
\sigma^{\dagger} = \dot{\sigma} - \overline{\omega}\_T \cdot \sigma + \sigma \cdot \overline{\omega}\_T = \pi : (d - d^p) \; , \tag{7}
$$

where *π* is the elastic tensor in the actual configuration (defined for the current crystallographic orientation), *d* is the stretching tensor (presented on the basis of the additive decomposition *d* = *d<sup>p</sup>* + *d<sup>e</sup>* ), the plastic part of which is determined as:

$$\mathbf{d}^{p} = \sum\_{k=1}^{K} \frac{1}{2} \dot{\boldsymbol{\gamma}}^{(k)} \left( \mathbf{b}^{(k)} \odot \mathbf{n}^{(k)} + \mathbf{n}^{(k)} \oslash \mathbf{b}^{(k)} \right). \tag{8}$$

The spin *ω<sup>T</sup>* is calculated in the framework of the Taylor's rotation model [58]:

$$\overline{\boldsymbol{\omega}}\_{T} = \frac{1}{2} (\boldsymbol{l} - \boldsymbol{l}^{T}) + \sum\_{\mathbf{x}=1}^{K} \frac{1}{2} \dot{\boldsymbol{\gamma}}^{(k)} (\boldsymbol{\pi}^{(k)} \odot \boldsymbol{b}^{(k)} - \boldsymbol{b}^{(k)} \odot \boldsymbol{\pi}^{(k)}),\tag{9}$$

where *l* is the velocity gradient.

Some recent studies (e.g., [59,60]) have used the law of elasticity written as:

$$
\sigma^{\dagger} = \dot{\sigma} - \overline{\omega}\_{\overline{\Gamma}} \cdot \sigma + \sigma \cdot \overline{\omega}\_{\overline{\Gamma}} = \pi : \begin{pmatrix} \dot{\varepsilon} - \dot{\varepsilon}^p \end{pmatrix} - \sigma tr(\dot{\varepsilon}) \tag{10}
$$

Apparently, these studies suggest that any strain rate can be defined as . *<sup>ε</sup>*, yet . *ε* = *d* remains the most common option. The . *ε <sup>p</sup>* is determined by making use of (8).

It is easy to see that, for . *ε* = *d*, Relation (10) can be written in the form:

$$k^{\bar{l}} = \dot{k} - \overline{\omega}\_T \cdot k + k \cdot \overline{\omega}\_T = \widetilde{\pi} : (d - d^p) \; , \tag{11}$$

where *π* is the elastic tensor for establishing the relation between the stress rate and elastic strain rate shown above, *<sup>π</sup>* <sup>=</sup> *<sup>J</sup>π*.

Relation (11) and equivalent Relation (10) are supposed to be used instead of (7) since the internal energy density (per unit mass) is defined as:

$$\mu = \int\_0^{\frac{\pi}{\ddots}} \left( \frac{1}{\not\!\!\! } \sigma : d \right) dt = \frac{1}{\not\!\!\! } \int\_0^{\frac{\pi}{\ddots}} (I\sigma : d) dt = \frac{1}{\not\!\!\! } \int\_0^{\frac{\pi}{\ddots}} (\kappa : d) dt,\tag{12}$$

According to (12), at small deformations (at *ω<sup>T</sup>* = **0** and for in case of change in volume) when (11) is fulfilled, there will be no energy dissipation during the purely elastic deformation, which does not provide (7). A detailed analysis of the fulfillment of the thermodynamic constraints is given in Section 2.3. Note that, due to the smallness of volume changes in metals and alloys, (11) and (7) give similar results.

The model that involves the kinematic relations and constitutive laws (8), (9), and (11) will be further called the "T-model" (Taylor spin model). The relations of such conceptual structure have been used in many papers, in particular, in [59–61]. Some variants of selfconsistent models [7] are based on the mesolevel equations of the same structure but with correction of spin (9) via subtraction of the antisymmetric part of the Eshelby tensor for considered grain.

#### *2.3. Analysis of the Formulations with an Emphasis on Describing Geometric Nonlinearity and Fulfillment of Thermodynamic Constraints*

In many works cited above, it was shown that, due to the structure of the model relations, the dissipation of energy under inelastic deformation is positive within the framework of both U- and T-models.

The U-model elastic law (3) involves the energetic conjugate stress-strain measures, and this guarantees the absence of energy dissipation during purely elastic deformation. Besides, the finite form of the relations ensures that, after the purely elastic deformation of the material (being in the unloaded configuration) along any closed deformation trajectories, the stresses will be zero. Thus, the conservation conditions for purely elastic deformation [62] are fulfilled within the framework of the U-model.

At the same time, the U-model implies the necessity to perform numerical integration of (4) so that *f <sup>p</sup>* can be determined. This can violate the condition of isohoricity of plastic deformation and require corrective techniques. Therefore, ref. [54] proposes an alternative formulation for crystal plasticity that is based on the Kröner–Lee decomposition (also) and logarithmic strain measure, whose rate was decomposed additionally by using elastic corrector rates (defined on the basis of the plastic flow characteristics—shear rates on the slip systems). A major advantage is the lack of a necessity to calculate the exponent when determining *f <sup>p</sup>* at the end of each step. Meanwhile, we need to point out here that the interpretation of the physical meaning of the elastic law when considering it as a relation between the introduced complex measures of the stress-strain state (not only in the context of thermodynamics) is rather complicated. In addition, the fulfillment of the additivity condition for the components of the strain rate measure is guaranteed only if one slip system is active; in the general case, this condition is fulfilled approximately [54].

On the other hand, for the study of technological processes of thermomechanical processing, it is more convenient to formulate the boundary value problem in terms of the actual configuration in the rate form. This enables one to use numerical methods: a step-by-step solution with a redefinition of the configuration of the computational domain (including the contacting surfaces) can be realized. Another significant advantage of this formulation is an additive decomposition of the rate of inelastic deformation into contributions from various mechanisms. So, advanced crystal plasticity models taking into account grain boundary sliding were constructed on the basis of this structure [13,30–32]. The T-model, referring to this type of formulation, uses a transparent measure of the strain rate—stretching tensor *d*.

It should be noted that the use of the corotational derivative in Relation (11) suggests that the linear (verified) constitutive relation is assumed to be valid for the observer related to a moving coordinate system rotating with the Taylor spin [63]. This suggests that the motion is decomposed into the rigid motion of the moving coordinate system and the deformation motion with respect to it [64]. Note that, to be fully consistent with the issue, we should write the measure of the rate of elastic deformation on the right-hand side of (11)

as (*l* − *l <sup>p</sup>* − *ωT*), *l <sup>p</sup>* <sup>=</sup> *<sup>K</sup>* ∑ *k*=1 . *<sup>γ</sup>*(*k*) *b*(*k*) ⊗ *n*(*k*) [65]. However, due to the tensor symmetry, the results will be similar to those obtained when using (*d* − *dp*). In some studies, the authors

explicitly say that they use linear constitutive relations written in the local coordinate systems of crystallites [66,67].

It is evident that, for the relations in terms of the actual configuration in the rate form, the key question is how to determine the spin of the moving coordinate system.

The determination of a hypoelastic law of the form (11), for which the above-mentioned conservation conditions are fulfilled under purely elastic deformation, is a well-known problem in the mechanics of solids [62]. The popular Zaremba–Jaumann [68,69] and Green– Naghdi [70] corotational derivatives do not guarantee the fulfillment of the conservation conditions. Therefore, a logarithmic corotational derivative without this drawback was developed [62,71,72]. However, in [63,73], one can find examples that illustrate the difficulties which can be encountered in modeling the multistage deformation processes (including unloading): if the purely elastic cyclic loading is realized upon completion of elastoplastic deformation and unloading, then the stress hysteresis will be observed. In [73,74], the logarithmic spin, called the kinetic logarithmic spin (k-logspin), was improved. The modernized spin makes it possible to meet the conservation conditions in the simulation of multi-stage deformation processes, including unloading, but the consideration is limited to anisotropic material. Thus, the problem of formulating a hypoelastic constitutive relation of the Form (11) for an anisotropic material that ensures the fulfillment of the conservative conditions has not yet been completely solved. Obviously, this is also the case for the crystal plasticity T-model.

So, both the U-model and the T-model have "a certain extent roughness" formulations, which still arouses the interest of researchers in finding a more elegant solution, with the leveling of these "roughness" formulations (see, e.g., [54]).

Note that, when formulating the T-model, the moving coordinate system (MCS) must be related to the crystallographic coordinate system since the components of the elastic tensor are assumed to be constant in the MCS. In other words, the MCS must be associated with the symmetry elements of the crystal.

According to the Taylor constrained rotation model, the displacement gradient at the mesoscale (grain level) is represented in the form [58]:

$$I = \overline{\omega}\_T + \sum\_{\mathbf{x}=1}^K \dot{\gamma}^{(k)} b^{(k)} \odot \mathfrak{n}^{(k)},\tag{13}$$

whence, by taking the antisymmetric part, we obtain the relation for the lattice spin (9).

Traditionally, the Taylor model is used for rigid-plastic models (this follows from the symmetrized Relation (13)—all deformation is associated with the shears on the slip systems). The closeness of the spin (9) to the "material rotation" spin (2) was demonstrated [9,75]. In both cases, one cannot speak of a direct relation between the MCS and the symmetry elements of the lattice. We also note that during deformation, the crystallographic coordinate system (CSC) can be distorted, while the MCS remains rigid, and therefore, strictly speaking, their identification is an approximation of the mathematical model.

In [63,65,76], an approach to the formulation of geometrically nonlinear relations is proposed. The approach is based on a modified multiplicative decomposition of the deformation gradient with an explicit selection of rigid rotation of the MCS, which is associated with the symmetry elements of crystallites. In the next section, the equations of the corresponding mesoscale formulations are given in a brief form (in the finite form in the lattice unloaded configuration and in the rate form in the current configuration). In Section 4, these ratios are compared with the U-model and T-model and are utilized to compare these formulations to each other.
