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Focused on Russian mechanics contributions for Structural Integrity Multi-level model describing phase transformations of polycrystalline materials under thermo-mechanical impacts

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The problem of constructing the multilevel physical model of inelastic deformation in steels allowing to take into consideration diffusionless solid-state phase (martensitic) transitions is considered. The model structure includes three scale levels
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    P. Trusov et alii, Frattura ed Integrità Strutturale, 49 (2019) 125-139; DOI: 10.3221/IGF-ESIS.49.14   125 Focused on Russian mechanics contributions for Structural Integrity Multi-level model describing phase transformations of polycrystalline materials under thermo-mechanical impacts Peter Trusov, Elena Makarevich, Nikita Kondratev Perm National Research Polytechnic University, Russia tpv@matmod.pstu.ac.ru, http://orcid.org/0000-0001-8997-5493 makareviches@inbox.ru  kondratevns@gmail.com, http://orcid.org/0000-0002-0261-3017  A  BSTRACT .  The problem of constructing the multilevel physical model of inelastic deformation in steels allowing to take into consideration diffusionless solid-state phase (martensitic) transitions is considered. The model structure includes three scale levels with the closed system of equations offered for them. Explicit internal variables reflecting the evolution of the material structure (both the defect structure and the grain one) are introduced at the lower scale levels of the model. The distinctive feature of the developed model is consideration of the lower scale level in such a way that a homogeneous element of this level completely turns into a new phase at a high speed (relative to the kinematic quasi-static loading), that is close to the speed of sound in the crystal medium. Based on the principles of classical thermodynamics the phase transformation criterion is written. According to this criterion, the choice of a transformational system under the martensitic transition is made. The algorithm of the model is developed and its realization features are described in connection with the high-rate restructuring of the face-centered cubic lattice to the body-centered tetragonal one. The result of this restructuring is a severe change in the physic-mechanical properties of the material. K  EYWORDS .  Crystal plasticity; multiscale modeling; phase transformations; microstructure; defect structure; improved steels. Citation:  Trusov, P., Makarevich, E., Kondratev, N., Multi-level model describing phase transformations of polycrystalline materials under thermo-mechanical impacts, Frattura ed Integrità Strutturale, 49 (2019) 125-139. Received: 31.03.2019  Accepted: 16.05.2019 Published:  01.07.2019 Copyright:  © 2019 This is an open access article under the terms of the CC-BY 4.0,  which permits unrestricted use, distribution, and reproduction in any medium, provided the srcinal author and source are credited. I NTRODUCTION dvanced high-strength steels have a wide range of applications in industry and technology, which is constantly increasing due to the excellent combination of their plastic and durability properties [1–2]. Enhanced physics and mechanical properties of steels are achieved by the state of grain and defect structure formed as a result of previous thermo-mechanical processing, mechanisms of its formation and evolution [3]. Numerous experimental studies indicate that phase transformations may be the main reason for the set of physical and mechanical properties in steels, ensuring their   P. Trusov et alii, Frattura ed Integrità Strutturale, 49 (2019) 125-139; DOI: 10.3221/IGF-ESIS.49.14 126  wide usage [4–5]. All known phase transitions for solid state are observed in steels and their alloys, such as polymorphic transformations with a wide range of morphological and kinetic features, eutectoidal (pearlite) transformations, decay of solid interstitial and substitutional solutions, ordering with a change in the local and long-range order in austenite and martensite. The possibility of realization for certain phase transformations and kinetics of transformations depends on the steel composition and the parameters of thermo-mechanical impacts, such as temperature, heating or cooling conditions, holding time, mechanical loading parameters, etc. The important feature of such systems is that the diffusion mobility of metal atoms and carbon is sharply different. Therefore, the crystal lattice restructuring during transformations can occur together with the diffusion redistribution of carbon and alloying elements. Another feature of steels is that during phase transitions of supercooled austenite, the transformation of a face-centered cubic crystal lattice into a body-centered tetragonal lattice can occur simultaneously with the diffusion redistribution of carbon and alloying elements. The experimental study of this issue is quite resource-expensive. Therefore, in solid mechanics, the problem of constructing the models describing the state and evolution of the structure for a material taking into consideration solid-state phase transformations becomes actual. It is widely known, that the physic-mechanical properties of polycrystalline materials and the functional characteristics of finished products are determined by the current state of the structure at various scale levels [6–7]. The latter significantly changes in thermo-mechanical processing of metals. The correct description of the internal structure for a material provides a fundamental opportunity to optimize existing and develop new methods for obtaining materials and products made of them with increased strength and performance characteristics. As a result, in recent decades, models based on explicit considering the mechanisms and the carriers of inelastic deformation (crystal plasticity based multilevel models of inelastic deformation) are of great interest in solid mechanics [8]. As a rule, in such models, the correct description of the existing mechanisms for inelastic deformation requires the introduction of several (two or more) scale levels. The multilevel approach from the point of view of physical description of the occurring processes is rather universal and can be applied to designing structures made of new, not yet existing materials and creating technologies for their manufacturing. The theoretical basis of this approach to the study of inelastic deformation is the methods of mathematical modeling with the introduction of internal variables, supplemented with the model identification and verification procedures. Internal variables make it possible to explicitly include a description of the physical mechanisms, their carriers, and processes accompanying inelastic deformation at various scale levels of a material. Also, the internal variables of a model reflect structural interactions and restructuring the meso- and microstructure of a material. The scale levels involved into consideration are determined by the objectives of the study and the most important mechanisms of inelastic deformation.  At the lower scale levels there is a principal possibility for correct accounting the physical mechanisms of inelastic deformation. Thermo-mechanical effects are transmitted from the macro level to the lower scales and cause changes in the internal structure. In turn, the latter determines the effective characteristics of the material at the macro level. In the framework of the multilevel approach to describe inelastic deformation of metals under thermo-mechanical processing a material point with a necessary set of homogeneous (averaged) characteristics is allocated at the macro-level. A set of homogeneous areas corresponds to this material point at one or several lower scale levels. The multilevel approach allows to describe the response of the material with the constitutive relations of a same type at various scale levels. In the framework of this work, Hooke's law in the rate relaxation form, written in terms of asymmetric measures of strain rates, is used. At the lower scale level, crystallite (a homogeneous part of a polycrystalline material) is considered. Each crystallite has a set of properties: anisotropic elastic modules, lattice orientation, a set of slip systems, transformation systems, the thermal conductivity coefficients. In the models of this type, an important aspect is the correct description of the internal variables evolution being responsible for the properties of both a crystallite and a polycrystal [6]. Constitutive and kinematic equations describing the irreversible deformation at the meso-level due to the slip of dislocations, phase transitions, the evolution equations for critical shear stresses (by different mechanisms), description of rotation for the crystallites, the influence of the temperature changing and the attached stresses on the evolution of the defect structure are included into consideration.  The problem of taking into account the geometric nonlinearity at the upper scale level and the connection for the similar characteristics of the scale levels remains important [9]. M ODELING THE PROCESSES OF INELASTIC DEFORMATION TAKING INTO CONSIDERATION PHASE  TRANSFORMATIONS  t present, in scientific literature there are various models of different types to describe the behavior of steels taking into consideration martensite transformations. The reviews of the works devoted to this problem can be found in the articles [10–11]. Two main approaches to constructing the models of polymorphic transformations can been distinguished. The first one is based on the models with explicit account of the phase boundaries taking into consideration    P. Trusov et alii, Frattura ed Integrità Strutturale, 49 (2019) 125-139; DOI: 10.3221/IGF-ESIS.49.14   127 the conditions at the phase boundary of the deformable material and the kinetics of a new phase evolution. The second one is based on the models with introduction of additional state parameters or model variables characterizing certain features of the material structure “on average” (for example, the concentration of a new phase), and the formulation of relations for them. In the models of the first type, with the explicit introduction of interphase boundaries into consideration [see 12–13], there is an opportunity to describe phase transformations from the point of solid mechanics using the ideas of the classical phase transitions theory by J. Gibbs. Phase boundaries appear in solids as a result of phase transitions. They can be considered as surfaces where deformations have a discontinuity whereas a field of displacement is continuous. The microstructure of a material changing in the process of the phase transition generates its own transformation deformations and modification in the elastic modules. So that, at the phase boundary some components of the strain tensor can break, and it leads to limitations on the constitutive relations. Appearance of an equilibrium discontinuous deformation field in an elastic body requires some regions in the deformation space where the Hadamard's inequality, being a necessary condition for stability with respect to infinitely small deformations, is failed. To maintain an equilibrium at the phase boundary, the following conditions must be satisfied: continuity keeping, force continuity, and a thermodynamic condition being an analogous of the chemical potentials' equality in the Gibbs theory. The latter condition imposes restrictions on determining the shape of the phase boundary and the corresponding deformations at the boundary. Since, even if the defining relations allow the existence of two-phased states, not all deformations can be realized at the phase boundary. This leads to the concept of a phase transitions' zone, which boundary determines the limit surface of transformation in the space of deformations. For the second type models, the phase field method is often applied, which is used to describe both diffusion transformations and diffusionless (martensite) ones at the meso-level (the simulated area consists of several grains) in many cases [see, for example, 14]. This approach assumes the presence of a “blurred” (“diffusion”) boundary between the phases in contrast to the classical methods using the concept of “sharp boundary”, when the multiphase structure is described by the position of the boundary and the set of differential equations is solved together with the flow equations and the constitutive equations at the boundary for each of the areas. In the “diffusion boundary” approach, the form and mutual arrangement of the regions occupied by individual phases are described by a set of parameters determining their fraction φ i  .  The value of the parameter can vary from 0 to 1; φ i  =0 corresponds to the area where there is no phase i  , φ i  =1 corresponds to the single-phase region. Thus, the microstructure (with the exception of grain boundaries, defects, etc.) can be described by a set of single-phased regions separated by boundaries where more than one value φ i    is different from zero. In the “diffusion boundary” approach, the change in the shape of the regions (and hence, the position of the boundary) over time is implicitly determined by the change in the fractions of phases. The time change of the phase fractions is described by the kinetic equation obtained in terms of thermodynamics of irreversible processes, i.e. the linear relationship between the rate of change for the phase fractions and the derivative of the thermodynamic potential for this parameter is used. Phase transformations occurring in isothermal conditions are most often investigated, and free energy is taken as a thermodynamic potential, but there are works studying non-isothermal processes where entropy is chosen as a thermodynamic potential [15]. In most studies devoted to description of thermo-mechanical processes, the so-called direct models of the first type are used [16] when a set of finite elements is matched to each grain and a model is used to describe the phase transitions for each of the elements. The usage of such models for modeling the real processes in three-dimensional formulation requires significant computational resources. Therefore, the statistic type models [17] are actual, where the set of homogeneous elements of the lower scale level constitutes a representative volume with homogeneous properties at a higher scale. Within this paper, a multilevel model of the hybrid type to describe the behavior of steels under thermo-mechanical loading, taking into consideration the phase transformations, is proposed by the authors. Within the framework of this model at the macro level, a direct type model is used. To determine the response of each material point at this scale level, a statistic model is applied, comprising the elements from a lower scale. The structure of this model includes internal variables being divided into two groups: explicit ones and implicit ones. Explicit internal variables are included into the constitutive relations at the considered scale level, and the implicit ones are the parameters of evolution equations. To connect the internal variables of two above mentioned groups, the closing equations are applied. When using the approach with explicit introduction of internal variables, the following hypothesis is accepted. The reaction of the material at any moment is determined by the current values of thermo-mechanical characteristics, internal variables and parameters of external influence. The considered hypothesis allows to give up rather complicated constitutive relations in the operator form. At the same time, the material memory about the prehistory of influences is preserved due to the evolving internal variables being the impact history carriers in this case. Within this article, the structure of the proposed model is described, its scale levels are introduced, the constitutive relations and evolutionary equations are given, as well as the algorithm for the implementation of the model.   P. Trusov et alii, Frattura ed Integrità Strutturale, 49 (2019) 125-139; DOI: 10.3221/IGF-ESIS.49.14 128  T HE STRUCTURE OF MULTI - SCALE MODEL WITH PHASE TRANSFORMATIONS   solid phase transformation in a polycrystalline material is understood as a polymorphic transformation leading to changing the physic-mechanical properties of some region in the material at a micro- and/or meso-level as a result of the crystal lattice transformation under external influences (loading, temperature, etc.). A phase is understood as some part of a grain being characterized by a specific type of crystal lattice, a chemical composition, a type of solid solutions, etc., at a fixed moment of a thermo-mechanical loading process. From the point of mathematical modeling and solid mechanics a phase is understood as a certain sub-region inside a material which behavior under deformation is described by the constitutive relations of the fixed type with a specified (determined from the solution of some auxiliary subtasks) set of properties being defined by the parameters and the current value of internal variables.  With a goal of modeling the phase transformations of polycrystalline materials in thermo-mechanical processing the multi-level model of a hybrid type is developed. Within this model three structure-scale levels are taken into consideration inside the material. They are macro-level, meso-level I and meso-level II. Internal variables are added into the structure of the model at each scale level being the carriers of impacts' history. The macro-level is supposed to be the material representative  volume (consisting of some hundreds of grains). To analyze a behavior of this volume, the boundary value problem is offered to be formulated and solved (for the chosen computational domain) with determination of fields for stresses, strains and temperature in the considered material volume. The finite element method procedure is applied for the numerical solution of the problem at the macro-level. Within the framework of the proposed multilevel model the problem of determining the reaction of a material to the applied thermo-mechanical impact is essentially nonlinear. A step-by-step procedure (in time) is used to solve it. Decomposition of the whole problem according to the physical processes is realized.  The subtasks of determining the stress-strain state, the temperature and the problem of determining the phase composition of a material are considered being connected (using a step-by-step procedure). The solution of this problem allows to determine the impacts (velocity gradient, temperature and temperature rate of change) at each point of the considered area (i.e. within each finite element), which are then given down to a deeper scale level within the multilevel model. Thus, within the framework of the finite element scheme at the macro level, the finite elements themselves are precisely the elements of meso-I. As a basic constitutive relation at the meso-I, the generalized Hooke's law in the rate relaxation form and the heat equation are used. The mentioned constitutive relations contain in their structure explicit internal variables. The values of those variables depend on the history of the impacts on the material, are changed in the process of deformation and are determined from the deeper scale levels as a response to the thermo-mechanical effects. For example, the tensor of elastic properties of a material, the inelastic strain rate tensor, the heat capacity coefficient, the thermal conductivity tensor, the power of internal heat sources can be such variables. Herewith, an element of the meso-I is understood as a certain subdomain of a grain, the state of which at each moment of the thermo-mechanical loading process is assumed to be homogeneous in all parameters characterizing the state of the meso-I element, and within which the crystal lattice of the material can be considered as approximately perfect (a grain is supposed to consist of crystallites with a minor misorientation). In turn, a meso-I element is represented as an aggregate, consisting of N elements of meso-II, such as subgrains, fragments, cells, phase components. Wherein, to determine the response of the meso-I element, the modified statistical model (taking into account the relative position of the neighboring meso-II elements) is used [17]. Herewith, a size of a meso-II element is assumed to be so small that its state can be considered as a homogeneous one according to all parameters at each time moment. The velocity gradient, temperature, and the rate of temperature changing are transmitted from meso-I to meso-II as impact factors. A model based on crystal plasticity is applied for the meso-II element. The Voigt hypothesis is used when transmitting exposure from meso-I to meso-II. At any fixed moment of a thermo-mechanical loading process each meso-II element is supposed to be in some specific phase (i.e. it is always single phased), but the phase characterizing it can change as a result of external influence, which leads to a change in all its properties. Belonging to a particular phase determines the basic properties of the meso-II element, including the type of its crystal lattice. Orientation of the axes of the moving coordinate system [18–19] is considered as known for each meso-II element. This orientation changes during a deformation process (as a result of rotation for the meso-II element or as a result of the transformation the element lattice after the realized phase transition). The moving coordinate system is associated with the lattice of the element (but, doesn't coincide with it in general case, as the lattice may have distortions in the deformation process). As a result of thermal and mechanical effects (transmitted from the upper scale level) in case of fulfillment the thermodynamic criterion a phase transition can occur in a meso-II element. Herewith, due to the homogeneity of the all parameters' values for the meso-II element, it is assumed that its entire volume undergoes a phase transformation simultaneously. All processes in the meso-II element are considered in its moving coordinate system being oriented definitely with respect to the laboratory coordinate system. Wherein, the following modes are realized in the meso-II element: inelastic deformation by    P. Trusov et alii, Frattura ed Integrità Strutturale, 49 (2019) 125-139; DOI: 10.3221/IGF-ESIS.49.14   129 shears on slip systems, the lattice rotation i.e. quasi-rigid motion with a spin tensor being determined by one or another rotation law (here, the Taylor model is used), temperature deformation, deformations of an element caused by a phase transformation, elastic distortions of a crystal lattice in a crystallite, the changing the lattice type of the crystallite and all its properties (internal variables) as a result of phase transformation, the changing the orientation of the moving coordinate system for the element as a result of a phase transformation, the formation of internal heat sources as a result of inelastic deformation processes and phase transformations at meso-II level. The acting stresses in the element, the inelastic strain rate tensor, the power of internal heat sources (due to latent heat of phase transformations and inelastic deformation), the orientation tensor of a meso-II element, characterizing the current orientation of the moving coordinate system for the element with respect to the fixed laboratory coordinate system, and the information about phase composition are transmitted as a response from meso-II to meso-I. The statistical averaging procedures are carried out at the meso-I level (among all meso-II elements, taking into consideration the current orientations of the moving coordinate systems for the elements) to determine the values of the model internal variables at an integration point for a finite element (for simplex-elements, this matches the data for an entire finite element). Thus, the meso-I element constitutive relations are “formed” in the process of solving the problem depending on transformation of the material structure at a deeper scale level in the thermo-mechanical processing and may change during the process itself. The constitutive relations constructed by this way for the meso-I element are then used to solve the boundary value problem at the macro level at the next time step.  The subject for discussion in this article is a description of the structure and the implementation features of the model at the meso-levels I and II.  About motion decomposition for the meso-II element  As in solving the boundary value problems connected with description of thermo-mechanical processing for polycrystalline materials, as a rule, it is necessary to take into consideration large displacement gradients, the problem under consideration, being essentially nonlinear, is posed and solved in a rate form. Within the framework of the model, description of kinematics for the meso-II element is based on introduction of the multiplicative decomposition for the deformation gradient by the following way: e =            etrptrp ffffffrfff  , (1)  where e  f  , tr  f  ,   f  ,  p f   are elastic, transformation, temperature and plastic components of the deformation gradient; e  f   is the component of the deformation gradient characterizing the elastic distortion of the lattice with respect to the rigid moving coordinate system connected with the lattice of the initial phase; r  is the rotation tensor describing the material rotation together with the moving coordinate system (from the initial position of the moving coordinate system to its current position);  p f   is the plastic component of the deformation gradient that does not change the symmetry properties of the material. Based on the accepted decomposition (1) the transposed velocity gradient is defined as follows:                                                    111111111111 . eeeTeetrtrTe etrtrTeetrpptrTe  lfffffrrffrffrf frffffrffrffffffrf   (2)  When considering metallic polycrystals, the magnitude of the elastic distortions of the lattice can be assumed to be small, therefore  e  fI  and the relation (2) is converted into the following form:                                        11111111 . eeTtrtrTtrtrT trpptrT  lffffrrrffrrffffrrffffffr  (3)  Then, the additive decomposition of the transposed velocity gradient in the actual configuration based on the multiplicative decomposition (1) can be represented as follows        etrp ll  ω lll  , (4)
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