*2.2. Structural Problem Formulation*

The general idea of the method is that the stress-strain state depends on some of the parameters vectors. Assuming that the anisotropy of the material is provided by the microarchitecture, we consider the forming material isotropic [13,16]. The parameters vector *λ* describes the material microarchitecture and influences the macro-stiffness tensor. On the other side, we should add an additional vector with initializing parameters, which describe the stress-strain state of the microarchitecture. Let us call it the initializing vector *γ*, which, obviously, depends on the invariants of the stress tensor *f*. The initializing vector can be interpreted as the control function of the microarchitecture changes. So, we propose that the stiffness tensor can be presented as a function of the parameters vector, the initializing vector, and the spatial coordinate:

$$\stackrel{\rightharpoonup}{C} = \tilde{\mathcal{C}} \left( \stackrel{\rightharpoonup}{\lambda} \left( \gamma, \stackrel{\rightharpoonup}{\lambda} \right), \stackrel{\rightharpoonup}{\gamma} \left( f(\tilde{\sigma}), \stackrel{\rightharpoonup}{\lambda} \right), \stackrel{\rightharpoonup}{\tilde{\chi}} \right) \tag{9}$$

Let us consider region *V* as the number of basic cells. For each basic cell we assume:

$$\begin{cases} \stackrel{\rightarrow}{\lambda} \begin{pmatrix} \gamma, \stackrel{\rightarrow}{\mathcal{X}} \end{pmatrix} = \stackrel{\rightarrow}{\lambda}(\gamma) \\ \stackrel{\rightarrow}{\gamma} \begin{pmatrix} f(\tilde{\sigma}), \stackrel{\rightarrow}{\mathcal{X}} \end{pmatrix} = \stackrel{\rightarrow}{\gamma}(f(\tilde{\sigma})) \end{cases} \tag{10}$$

This approach considers a basic cell as a micro-construction with constant macroproperties. The parameters vector *λ* should be changed according to values of the initializing vector *γ*. So, if we introduce the control function *U* the problem can be rewritten:

$$\begin{cases} \begin{array}{l} \stackrel{\rightarrow}{\lambda} \left( \gamma, \stackrel{\rightarrow}{\mathbf{x}} \right) = \stackrel{\rightarrow}{\lambda} \left( \gamma, \mathcal{U} \left( \stackrel{\rightarrow}{\mathbf{x}} \right) \right) \\\ \mathcal{U} \left( \stackrel{\rightarrow}{\mathbf{x}} \right) = f \left( \stackrel{\rightarrow}{\gamma} \left( f(\tilde{\sigma}), \stackrel{\rightarrow}{\mathbf{x}} \right) \right) \\\ \stackrel{\leftarrow}{\mathcal{C}} = \mathcal{C} \left( \stackrel{\rightarrow}{\lambda} \left( \gamma, \mathcal{U} \left( \stackrel{\rightarrow}{\mathbf{x}} \right) \right), \stackrel{\rightarrow}{\mathbf{x}} \right) \end{array} \tag{11}$$

This means that the state of the initializing vector *γ* determines the changes of microarchitecture in terms of the parameters vector *λ*, and the microarchitecture influences the macro-stiffness tensor. Let us consider the investigated region as a composition of basic cells; each one describes the microarchitecture of a material. Each basic cell can be described by the parameters vectors and can be changed according to the initializing vector. To implement such an approach, the basic cell should be determined in order to define the parameters vector and its relationship with the stiffness tensor.
