A Combined Approach to Solving Applied Metal Forming Problems

Abstract

This paper presents a combined approach to the solution of boundary value problems. The approach implies the joint application of a macro-phenomenological model and multilevel physical constitutive models for describing the behavior of a polycrystalline material. Within the framework of the proposed approach, the boundary value problem is solved for a certain product (structure) using a macro-phenomenological model of the material and the response in the chosen area is clarified using multilevel modeling techniques. The capabilities of the approach were demonstrated in a series of numerical experiments covering metal forming processes. In order to test and verify the proposed technique, in the first numerical experiment, the process of an upsetting of the sample was considered using only a multilevel physical model of the material and using the formulated combined approach. The response in the chosen domain was compared to the results obtained by different modeling approaches. The comparison showed good agreement in terms of the variables of the stress–strain state at the macro level, as well as in terms of the variables characterizing the internal structure of the material. At the same time, using the combined approach, the required computation time was reduced by more than 12 times. In the second and third numerical experiments, the process of sheet rolling and sample pressing was successfully simulated using a combined approach. The formulated approach is computationally efficient and applicable for solving the relevant boundary value problems of design and the optimization of technological processes.

1. Introduction

Nowadays, the development of new and the improvement of the existing technological processes for metals and alloys under severe plastic deformation (SPD) is generally preceded by a theoretical analysis, which implies the use of the relevant mathematical models [1]. A theoretical study of the processes listed above poses the need for the formulation and solution of physically and geometrically nonlinear initial boundary value problems, the solution to which is obtained by different numerical methods. The most used technique is the finite-element method (FEM) [2].

Constitutive models (CMs) (or constitutive relations (CRs)) that describe the behavior of materials, form the core for computational modeling [3]. In engineering practice, the operation of products and their thermomechanical treatment are usually modeled in terms of CMs based on a macro-phenomenological approach to the construction of constitutive relations [4]. The application of this type of CMs is often justified by their simplicity, computational efficiency, and a small number of model parameters to be identified. In modeling such plastic deformation processes as forging [5], cutting [6], rolling, bending, and some others [7], the evolution of the stress–strain state (SSS) is evaluated by Johnson–Cook [8] and Zerilli–Armstrong [9] CRs, or by their modifications, for instance, Rusinek–Klepaczko CRs [10]. Macro-phenomenological CMs are also commonly used in describing the process of the operation of different parts [11,12,13].

It is worth noting that the physical and mechanical properties of metals and alloys and the performance characteristics of the products made from these materials depend, first of all, on the meso- and micro-structure of the materials evolved during materials manufacturing. Therefore, further development of the technological regimes for the treatment of metals and alloys requires CMs that are able to describe the evolution of the structure of materials. Macro-phenomenological CRs do not have such capabilities. In recent decades, the problems under consideration have been solved using the multilevel and multi-scale CMs of polycrystalline materials, which are based on a physical approach and involve an explicit consideration of the material structure evolution [14,15,16]. Multilevel modeling is widely used for materials such as steel, aluminum, nickel, titanium alloys, and many others [17,18,19,20]. The studies that use such models enable descriptions of the processes of initiation crack and failure, phase transitions [21,22], recrystallization [23,24], and the analyzing of the anisotropy of the properties of product materials [25], etc. The explicit consideration of the structure makes it possible to develop functional materials—products with the required operational characteristics [26].

The main disadvantage of most existing multilevel CMs is that the calculations require extremely large computing resources [27,28]. Nevertheless, owing to rapid computer performance growth and new efficient numerical schemes and algorithms, multilevel crystal plasticity CMs are used, among other things, to solve the problems associated with the study of manufacturing technologies for full-size products [16,29]. However, the computational complexity of the problems remains extremely high and may increase many times when modeling objects with significant local inhomogeneities. Therefore, this class of CMs is rarely used in engineering applications (with the design and optimization of products made of metals and alloys taking into account their operating conditions). The problem of high computational resource intensity is aggravated by the need to formulate and solve optimization problems in order to develop rational technological modes for many practically important production processes [30,31]. In this context, the problem of the computational efficiency of these models is still relevant.

In view of the above and with the intent to solve the formulated problem, we have developed and tested an original technique (hereinafter referred to as a “combined” approach), which is based on the joint and conjugate application of mathematical models, where macro-phenomenological and multilevel physically-oriented CMs are used. We apply a computationally effective mathematical model that involves the use of macro-phenomenological CRs to solve the boundary value problem in the evaluation of SSS at each point of the studied product (the “global” boundary value problem). The resulting solution describes the SSS of the product to the accuracy level acceptable in engineering applications, but it does not take into account the internal structure of the product. The engineering approach to production problem solving does not always involve a determination of the internal structure at each point of the product; sometimes it is sufficient to identify the problem in separate areas so that the response may be clarified. The choice of these areas is unique for each manufacturing problem and depends on the specific operating conditions of the products. After an identification of the areas for which the properties should be identified, a new boundary value problem (a local boundary value problem) is formulated for each area. For this problem, the boundary conditions are determined from the solution already obtained for the product as a whole, and the application of a multilevel physical model allows for the obtainment of detailed information on the internal structure in the considered area. Thus, a hierarchy of boundary value problems is formed for the selected critical areas of the product structure. The solution of the global problem is encapsulated inside the local problem by correctly setting the boundary conditions. Material models (macro-phenomenological and multilevel) are linked due to identification using the same experimental data at the macrolevel. It should be noted that it is precisely due to the coupling of boundary value problems (transfer of effects) and material models that the developed approach fundamentally differs from the known method of studying a representative volume (RV) [32,33].

The formulated combined approach does not impose restrictions on the choice of macro-phenomenological or multilevel models of the material; however, in this study, the specific models which are specified in Section 2: Materials and Methods are selected as a test example. This study demonstrates the capabilities of the developed approach to provide a solution for the metal forming problems—upsetting, rolling, and pressing—characterized by large displacement gradients and stress measures in some areas. It is shown that the CM internal variables obtained for the selected areas of the original workpiece using the proposed combined approach (first the “global” and then the “local” boundary value problems are solved in terms of a multilevel model) are close to the variables calculated for the entire body by means of a direct multilevel model. According to the literature review conducted by the authors, the proposed approach is new and can help in matters of applying multilevel modeling to problems of designing and optimizing the products and structures made of metals and alloys.

2. Materials and Methods

To find solutions to practically oriented problems, there is no need to evaluate the internal structure at each point of a product, it is sufficient to determine it only in some parts of products, which are responsible for their operational characteristics; the choice of such areas is unique for each product and for each technological regime.

In order to increase the computational efficiency of the existing mathematical models for studying metal and alloy forming processes and because of the need to explicitly describe the internal structure (at least some areas of the workpieces underwent treatment), we have developed a combined approach to the solution of boundary value problems. The approach is based on the joint application of macro-phenomenological and multilevel CMs of the material. The schematic illustrating the approach in given in Figure 1. The algorithm of the combined approach is implemented as follows.

First, we solve the boundary value problem upon evaluation of the SSS of the product as a whole using a computationally efficient mathematical model for describing technological processes with macro-phenomenological CRs. This study is based on the given boundary conditions, information about the geometry of an object, and experimental data on the mechanical properties of the material.

Next, we identify the product areas to get information about the internal structure of the material. The effect of the rest of the investigated body on these areas is determined by applying the kinematic, force, or mixed surface parameters, which act as the boundary conditions necessary for the solution of a “local” boundary value problem for each chosen area of the body. Displacement fields and displacement velocities on the boundaries of the chosen areas are obtained from the solution of a “global” boundary value problem for the body as whole.

The solution to the described boundary value problem found using a physically oriented multilevel CM of the material will allow the SSS to be clarified within the limits of the chosen areas and will give information about the internal structure and its evolution.

According to the considered combined approach to modelling technological processes, the problem of choosing the appropriate areas and determining their geometry (shape and position in the body) arises. The choice of particular areas depends on the purpose of the research and the problem formulated; in the context of strength characteristics (or, possibly, any other parameters or internal variables), the areas of stress and/or strain concentrators are of the greatest interest.

Such sub-areas can be identified for the known solution of the “global” boundary value problem using the gradient of, for example, stress or strain tensor. An increase (by a specified number of times) in the stress and strain tensor gradients averaged over the body serves as a criterion for choosing an area in the product under study. In this study, to choose the areas for detailed analysis, we use for certainty the magnitude of a stress tensor. For the convenience of calculation, we assume that the area in the reference configuration is cubic, and the orientation of the cube edges coincides with the axes of a specified laboratory coordinate system.

Consider the statement of the boundary value problem of elastic–plastic deformation research for some products. Let the product occupy, at time t, the spatial domain ${\mathrm{V}}^{\mathrm{t}}$ bounded by the closed surface ${\mathrm{S}}^{\mathrm{t}}$. Generally, the boundary value problem for describing the elastoplastic deformation of a workpiece involves the following system of equations [14,34,35]:

$$\begin{array}{l}\nabla \cdot \dot{\Sigma }-\nabla \cdot \left(\mathrm{v}\cdot \nabla \Sigma \right)=0,R_0\in {\mathrm{V}}^0\\ \dot{\Sigma }+\Sigma \cdot \Omega -\Omega \cdot \Sigma =\Pi :Z^{\mathrm{e}},\\ Z^{\mathrm{e}}=\nabla {\mathrm{v}}^{\mathrm{T}}-\Omega -Z^{\mathrm{v}\mathrm{p}},\\ {\dot{r}}^{\mathrm{t}}=\mathrm{v},\end{array}$$

where $R_0$ is the radius vector of a material point in the reference configuration, $r^{\mathrm{t}}$ is the radius vector of a material point in the actual configuration, $\Sigma$ is the chosen stress measure, $\nabla$ is the gradient operator in the current configuration, $\Omega$ is the spin tensor associated with the chosen corotational derivative, and $\Pi$ is the elastic tensor. In order to use the model for describing a specific material, the problem is supplemented with equations that represent the relationship between $\Omega ,\Pi$ and $Z^{\mathrm{v}\mathrm{p}}$ and the deformation history. To close the problem, the following initial and boundary conditions are introduced:

$${\left.\Sigma \right|}_{t=0}={\Sigma }_0,{\left.u\right|}_{t=0}=u_0\forall R_0\in {\mathrm{V}}^0,$$

$$n\cdot \dot{\Sigma }-\left(\nabla V\cdot n\right)\cdot \dot{\Sigma }+\left(n\cdot \nabla V\cdot n\right)n\cdot \Sigma =\dot{\widehat{\mathrm{T}}},r^{\mathrm{t}}\in S_{\sigma }^{\mathrm{t}},$$

$$V=\widehat{V},r^{\mathrm{t}}\in S_{\mathrm{v}}^{\mathrm{t}},$$

where ${\Sigma }_0,u_0$ are the initial stress and displacement measures, respectively, $n$ is normal to the surface of the area under study at a chosen point, $\widehat{T}$ are the given surface load vectors at the boundary, $\widehat{V}$ is the prescribed velocity field at the boundary, and ${\mathrm{S}}_{\sigma }^{\mathrm{t}}$, ${\mathrm{S}}_v^{\mathrm{t}}$ are the bounding surface areas for which the force and kinematic boundary conditions are specified, respectively.

The solution of the boundary value problem of a product in terms of macro-phenomenological CRs makes it possible to determine the fields, which characterize all the parameters of the SSS, including the displacement velocity field $\mathbf{V}(R_0,\mathrm{t})$ and the displacement field $u(R_0,\mathrm{t})$ at each point of the product at any instant of time. To study the internal structure and its evolution, it is necessary to apply a multilevel CM-based model, where the structure is considered explicitly, and which is implemented in the framework of the formulated combined approach. The area selection criterion does not depend on the used multilevel CM and is determined in this paper to meet the following requirement: in the chosen area, the values of a stress tensor should be higher than the average values for the body by 15–20%.

For an example, to clarify the response, we consider the two-level elastoviscoplastic statistical model, the main provisions of which are given in [35]. For the area which occupies, at the initial instant of time, the volume ${\mathrm{V}}^{\prime 0}\in {\mathrm{V}}^0$ and is bounded by the closed surface ${\mathrm{S}}^{\prime 0}$, the boundary value problem on the SSS evaluation is solved in terms of the physical oriented multilevel elastoviscoplastic CM. The initial and boundary conditions for the chosen area are found from the solution already obtained for the “global” boundary value problem. Since the area lies within the limits of the body, the velocity gradients $V^{\prime }$ are known for each of their points at any instant of time. Thus, the initial and boundary conditions for the chosen area can be written as follows:

$${\left.\Sigma \right|}_{t=0}={\Sigma }_0,{\left.u\right|}_{t=0}=u_0\forall R_0\in {\mathrm{V}}^{\prime 0},$$

$$V^{\prime }=\widehat{V},\forall R_0\in {\mathrm{S}}^{\prime 0}.$$

The proposed combined approach for solving applied boundary value problems allows one to significantly reduce the computational costs when compared to using a multilevel model for solving a “global” boundary value problem. Finding the solution to these problems will determine the fields characterizing all the parameters of the SSS in the chosen area, including the refined complete fields of displacement and displacement velocities, stresses, strains, as well as the parameters representing the internal structure at each point of the chosen area at any instant of time.

In order to demonstrate the capabilities of the developed approach and to evaluate its adequacy, we performed three numerical experiments in which the behavior of polycrystalline workpieces during an upsetting of the sample, rolling, and pressing was investigated. As a material of products, we used an stainless steel 304L, which was taken due to its wide applicability in different industrial operations, including the processes mentioned above [36,37,38,39].

The calculations were carried out using mathematical models that are based on macro-phenomenological CRs and a multilevel, physically oriented CM. The classical theory of plasticity, implemented in ABAQUS and representing the associated plastic flow rule with the von Mises yield condition and the isotropic hardening rule, was accepted for macro-phenomenological CRs.

As a multilevel CM, a two-level elastoviscoplastic statistical model intended to describe the deformation of polycrystalline materials with consideration of geometric nonlinearity was used [35]. In the material model, two structural-scale levels are distinguished: a macrolevel, the structural element of which is a representative volume that involves a few hundred grains, and a mesolevel—the level of individual grains. With this model, shear along the slip systems in each grain and the grain rotations are explicitly described, which provides a more accurate determination of the representative volume of the material stress–strain state and gives information about the formed texture. Within the framework of the assumptions of the multilevel model used, it was considered that this material is single-phase and has a face-centered crystal lattice; the only physical mechanism of plastic deformation is the movement of the edge dislocations along the main slip systems in the {111} planes in the <110> directions.

3. Results

The selected constitutive models are identified via analyzing the same experimental data on the uniaxial loading of the 304L stainless steel samples. The parameters of the multilevel and phenomenological models are given in

Table 1. Parameters of the CM for describing the elastioviscoplastic deformation of the material.

Parameters of the Multilevel CM
Parameter	Value	Source
${\Pi }_{1111}$	204.6 GPa	[40]
${\Pi }_{1122}$	137.7 GPa
${\Pi }_{1212}$	126.2 GPa
${\dot{\mathsf{\gamma }}}_0^{(k)}$	0.001 s−1	[41]
$m$	50	(identification)
${\mathsf{\tau }}_{c0}^{(k)}$	0.08 GPa	(identification)
${\mathsf{\tau }}_{sat}^{(k)}$	0.19 GPa	(identification)
$q_{lat}$	1.4	[41]
Parameters of the phenomenological model
Parameter	Value	Source
Density	7900 kg/m3	[42]
Young’s modulus	195 GPa	[43]
Poisson’s ratio	0.29	[42]
Loading diagram	–	[43]
Friction coefficient	0.8	–

.

The model parameters were taken from the relevant literature sources or determined during the identification procedure based on the results of the uniaxial compression tests conducted on a AISI 304L polycrystal [43]. The multilevel model parameter identification was carried out using the multicriterial multidimensional optimization program IOSO [44] and involved the minimization of the standard deviation between the obtained intensity of stress tensor values and the experimental data from [43]. Figure 2 presents the stress–strain curve obtained from the identification.

In the first numerical experiment, the process of the sample upsetting was simulated. For its study, a cube-shaped volume was chosen as the computational domain (Figure 3a). The dimensions of the full workpiece along the designated axes $\mathrm{X},\mathrm{Y},\mathrm{Z}$ of the laboratory coordinate system (LCS) were set, respectively, as 0.5, 0.5, 0.5 (m) ($\mathrm{Y}$ is the axis in the upsetting direction, $\mathrm{X}$, and Z is perpendicular to the compression direction). The image of the workpiece with the constructed finite element mesh (the elements of the C3D8R type) is shown in Figure 3a. The surfaces of the machining tool are not shown; their influence was taken into account in the boundary conditions. The problem was calculating the stress–strain state fields of the cube during the upsetting. The problem statement was supplemented by setting kinematic boundary conditions on the surfaces perpendicular to the compression axis. The movement of the upper boundary was determined by the velocity component of the displacement vector: ${\mathrm{v}}_1=\left\{0,-0.025,0\right\}$ (m/s), at the lower boundary, and a kinematic ban on the displacement speed was implemented, ${\mathrm{v}}_2=\left\{0,0,0\right\}$ (m/s). The remaining borders were considered free.

In the second numerical experiment, the rolling process was modeled. As a computational domain, half of an initially flat plate was considered (Figure 3b). The dimensions of the full workpiece along the $\mathrm{X},\mathrm{Y},\mathrm{Z}$ axes of the laboratory coordinate system (LCS) were set as, respectively, 0.75, 0.240, 0.5 (m) ($\mathrm{X}$—in the direction of rolling, $\mathrm{Y}$—perpendicular to the workpiece plane, and $\mathrm{Z}$—perpendicular to the rolling plane). The workpiece modeled in finite elements (the elements of type C3D8R) is presented in Figure 3b. The forming rolls are assumed to be absolutely rigid bodies, which have the shape of circular solid cylinders with a radius of 0.707 m and with the axes of rotation parallel to the $\mathrm{Z}$ axis; the finite-element mesh (the elements of type R3D4) on the surface of one of the rolls is depicted in Figure 3c. The axes of the cylinders are formed by the intersection of the planes $\mathrm{X}=-0.2$ (m), $\mathrm{Y}=1.101$ (m) and $\mathrm{X}=0.2$ (m), $\mathrm{Y}=1.101$ (m) for the upper and lower cylinders, respectively. The relative position of the workpiece and the rolls are demonstrated in Figure 3d. The task is to calculate the SSS fields of the plate taken through the rolls’ rotation.

The problem statement is supplemented with mixed boundary (including contact) conditions by setting the force of friction between the surfaces in contact, and the boundaries outside the contact zone are considered free. The motion of the rolls is determined by the vectors of the angular velocity of the rotation of the rolls about their axes ${\omega }_{\mathrm{r}1}=\left\{0,0,-2\right\}$, ${\omega }_{\mathrm{r}2}=\{0,0,2\}$ (rad/s), respectively. In addition, the translational motion of the rear edge of the workpiece with the velocity ${\mathrm{v}}_1=0.15$ (m/s) is set as an analogue of feeding for the sheet between the rolls during the process of rolling. To speed up the calculations, the XOY plane is assumed to be a plane of symmetry.

During the third numerical experiment, the process of pressing was considered. The modeled object was a workpiece for a sample, which, in its initial configuration, had the shape of a cylindrical rod of the radius $R=0.243$ m and the length $W_{\mathrm{r}}=0.3$ m; the finite-element mesh of its computational domain (the elements of type C3D8R) is given in Figure 3e. The upper and lower stamps have the same shape, and their geometry is associated with the required final shape of the sample; the finite-element model of the stamps (the elements of type R3D4) is presented in Figure 3f. The centers of the upper and lower stamps in the reference configuration are determined by the coordinates $\left\{0,0.16,0\right\}$ and $\left\{0,-0.3,0\right\}$, for the upper and lower stamps, respectively, in the reference frame, the origin of which coincides with the cylinder center. The stamps are assumed to be absolutely rigid. The relative position of the workpiece and the stamps are shown in Figure 3g.

Kinematic boundary conditions were used in modeling the pressing process. It was assumed that the lower stamp is stationary (rigid fixation conditions), and the upper stamp moves with the velocity ${\mathrm{V}}_{\mathrm{Y}}=-0.03$ m/s toward the lower stamp. When the points of the workpiece and stamp surfaces came into contact, the conditions of friction were imposed.

In the context of the first problem, the calculation results of the upsetting process parameters obtained from the solution of the global boundary value problem in terms of a multilevel model were compared with the modeling results determined using the proposed combined approach. The stress intensity fields for the workpiece as a whole are shown in Figure 4a,b. A detailed comparison of the results was performed for the area identified for the combined approach (Figure 4c). Figure 4d presents the stress–strain curve, “stress intensity—accumulated strain intensity” (obtained in two different ways), for the variables averaged over the considered domain; the results show good agreement. One of the important characteristics reflecting the evolution of the grain structure is the orientation of crystallites, and it is for this reason that the orientations of grains in the chosen area of the deformed body were used for comparison. The corresponding pole figures, drawn using the results of the application of each model are given in Figure 4e; the results are in good qualitative agreement. It is possible to see a discrepancy between the results of the multilevel and macro-phenomenological modeling. This is due to two main factors. Firstly, the macro-phenomenological model is based on isotropy and the hypothesis of a single curve, which does not take into account the initial or formed in the process of intense plastic deformations anisotropy of elastic-plastic properties. Multilevel models, on the contrary, describe with a high degree of accuracy the formed texture, and anisotropic properties of alloys [14,15,16]. Secondly, during the considered processes of forming blanks or products in local areas, deformation along the complex trajectories is realized. If multilevel models with a high degree of accuracy describe the mechanical effects of complex loading [45], then simplified macro-phenomenological models do not have this advantage. The test calculations carried out by the authors showed good agreement between the results of the multilevel and macro-phenomenological models under monotonic loading along the ray trajectories at small and moderate values of deformation (about 30–40%). The full conjugation of the macro-phenomenological and multilevel models is a complex problem that goes beyond the scope of the current study. The calculations were performed on a machine with an Intel Core i9 14900KF clocked at 3.20 GHz. The computation time was 201 min and 31 s for the multi-scale modeling and 15 min and 57 s for the combined approach. Using the combined approach, it was possible to achieve an acceleration of more than 12 times.

Another value that characterizes the internal structure in the product area can be the averaged values of the critical stresses in the main slip systems, that is, in the {111} planes in the <110> direction. The dependence of this value on the deformation of the chosen area of the object is given in Figure 4f for each approach considered here. As can be seen, the results show good qualitative and quantitative agreement.

For a second and third example, the processes of rolling and pressing by means of the formulated combined approach was modeled. The stress intensity field at each point of the product is shown in Figure 5a,b. The “stress intensity—accumulated strain intensity” diagrams and the pole figures for this area, obtained using the combined approach, are given in Figure 5c–e. It should be noted that the stress–strain curves (Figure 5c) were obtained by averaging over the elements of the selected region, which explains their difference from the original identification diagram (Figure 2). In the case of pressing, the finite elements entering the region have a close stress–strain state (see Figure 5b); therefore, the stress–strain curve is close to the specified experimental curve (Figure 2). For rolling, the considered structural elements experience sequential loading in the cases when their location is in the vicinity of the rollers, the next intensive loading when passing between the rollers occurs, and partial un-loading after leaving the rollers occurs. The first inclined section of the red curve (Figure 5c, deformation interval 0–0.12) corresponds to the moment when some of the finite elements of the selected region have already passed into the plastic stage of deformation, while the other part of the elements of this region is still elastically deformed. The subsequent flat section of the curve (Figure 5c, deformation range 0.12–0.22) is caused by the exit of that part of the finite elements, which are in the state of plastic deformation, to the state of hardening saturation, while the remaining elements of the selected region are still in the elastic state. The second inclined section on this curve (Figure 5c, deformation range 0.22–0.4) corresponds to the passage of the considered region of the section between the shafts, where the conditions on the contact surface become close to the adhesion conditions (insignificant slippage). In this case, the stress state is close to uniform compression; therefore, the system remains quite rigid, which explains the observed large value of the tangent modulus. The values of the averaged stresses in the final section of the diagram coincide both with the value of the hardening saturation state of the identification diagram (Figure 2) and with the value of the hardening saturation state for the pressing process (Figure 5c, blue curve). Each of the curves in Figure 5c represent a stress–strain curve for active loading; therefore, the processes of the region leaving the zone between the rollers and the lifting of the die, accompanied by un-loading, are not shown in this diagram.

As can be seen, in general, the results of the multilevel modeling enacted in the two ways described above show good agreement. This confirms the adequacy of the developed combined approach, which is based on the approximate solution of the “global” problem on the evaluation of the inelastic deformation of workpieces. The approach uses mathematical models with simplified macro-phenomenological CRs and implies a determination of the critical areas of the treated workpiece and the finished product. The application of the displacement fields (displacement velocities) and/or stress fields obtained from the solution of the “global” problem will make it possible to improve the accuracy of calculations when using the mathematical models based on multilevel physically oriented CMs, and to determine the internal variables for characterizing the meso- and micro-structure of the chosen polycrystalline areas.

4. Discussion and Conclusions

In this study, we developed a combined approach to solving boundary value problems by means of the mathematical models based on macro-phenomenological CRs. The approach was applied to the problem of the evaluation of the SSS of the workpiece as a whole, with further clarification of the response and a determination of the internal structure in the chosen areas using a physically oriented CM. To demonstrate the formulated approach, its application to the study of metal forming processes was considered using the example of the three technological processes (upsetting, rolling, and pressing) of products made of 304L steel. The computational experiments demonstrated that the approach is able to describe the SSS and the internal structure with satisfactory accuracy. The results for the chosen areas obtained using the combined approach were compared with the results found by direct multilevel modeling for the body as a whole; it is shown that the results agree well.

The combined approach makes it possible to significantly improve the computational efficiency of solving the problems related to studying the processes of manufacturing large-size parts by intensive plastic deformation techniques. These processes require an explicit consideration of the evolution of the internal structure of the material in some areas. The choice of the areas is not limited to the formulated approach and can depend on the features of the modeled processes determined during the preliminary calculations performed using engineering mathematical models based on macro-phenomenological CRs, or it can be set heuristically based on the personal experience of the researcher.

In the future, we plan to use the developed approach for solving the boundary value problems related to the evaluation of the SSS, which arise due to the need to conduct a detailed study of real technological forming processes, and to design and create of functional materials and products.