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Structural-analytical mesomechanics of a solid body with reversible martensitic transformations

In the given article the structurally-analytical mesomechanics for environments with martensitic mechanism of mass transfer is developed. Wave equations are derived to describe the deformation process initiated by martensitic transformations at meso and macro-scale levels.


Introduction
This article presents the analysis of structural and mechanical aspects of the problem of structural and analytical mesomechanics of multilevel media with reversible phase transformations of martensitic type and physical justification of the need to use the effective field method as a tool to take into account the interaction of deformation processes, the evolution of damage and reversible martensitic transformations at various scale and structural levels. Particular attention is paid to the factors of inheritance of defective structures in reversible transformations of austenite to martensite, as well as the development of a hierarchy scenario of martensitic transformations and plastic deformation. This scenario allows us to provide a holistic perception of the methodology of structural and analytical mesomechanics.

Analysis of structural and mechanical aspects of the problem
The task of creating the theory of deformation of materials with shape memory effect (SME) is to build a model of a deformable solid body, which should take into account its complex and multi-level structural organization, various deformation mechanisms, including dislocation slip, diffusion, martensitic transformations and evolution of the material damage on the one hand and generating their thermal effects and mechanical stresses on the other.
The task of solid state physics in this regard is to analyze the micro-scale level and build models of mechanical behavior of crystals, which take into account specific physical mechanisms and their influence on the corresponding parameters of the governing equations, taking into account the structural organization of crystals. The physics of plasticity and fracture allows to analyze crystallography and thermodynamics of phase transformations, to describe the laws of motion of structural imperfections in a loaded solid body, using the methodology of the theory of defects, in particular, the apparatus of the theory of dislocations. Significant progress has been made in the study of dislocation plasticity of crystals. A large array of experimental data obtained using a well-developed experimental technique gives a broad idea of the mechanism of formation of elementary acts and laws of plasticity. Here not only the structural and physical mechanisms of inelastic deformation processes are understood but also influence and interaction between structures of one type. It is obvious that macroscopic properties are formed at all stages of the mass transfer process and cannot be reduced to only one of them [1 -4].
It is necessary to formulate such a theory of deformation that would be based on strict physical postulates, it means taking into account real physical processes and at the same time would allow to solve practical problems of engineering nature. Numerous attempts to build such a theory have been made long time ago but the hope to move directly from the micro-scale to the macro-scale area with the help of various methods of orientation and statistical averaging have not been successful [2].
As it is noted in [2,4], the current situation is determined by two fundamental circumstances: first, a consistent and correct description of the evolution of the complex stochastic distribution of martensitic lamellae and their ensembles faces with insurmountable mathematical difficulties and secondly, selforganization of dislocation ensembles and martensitic lamellae leads to a new quality -in a continuous environment there is a move of larger than the dislocation of meso defects [2]. For the transition from the microstructural level to the macroscopic level, it is necessary to take into account the contribution of the evolution of the intermediate meso structural level where the move of the corresponding meso defects dominates, providing the formation of translational-rotational deformation modes.
The formulation of a general algorithm for constructing a model of physical mesomechanics to describe the deformation of a solid body with any internal structure, taking into account the possibility of transformation of the structure during the phase transformations and for arbitrary modes of its loading is an important problem. This article is devoted to one of the possible solutions to this problem based on the development of methods of structural and analytical mesomechanics [4] for materials with shape memory effect.

The script hierarchy of scale and structural levels
Deformation of a loaded object deformed by initiation of reversible martensitic reactions can be represented as a multilevel relaxation process. We will briefly consider the main stages of the origin and development of mass transfer processes in reversible phase transformations of martensitic nature. For convenience, figure 1 shows the scheme of scale and structural levels, adopted during the construction of the model.
According to the data of physical studies, the first loss of shear stability occurs at the micro level, in the local areas of the crystal, where the nuclei of the martensitic phase occur. Microconcentrators of stresses arising at the structural inhomogeneities cause martensitic transformation of the crystal lattice in certain crystallographic planes and directions. Such local structural transformation appears as the emergence and development of martensitic nuclei in the austenitic matrix. The indicated stage is shown in figure 1A, where the shaded areas are schematically represented by martensite nuclei. Since the influence of microconcentrators is short-range, the resulting distortion transformations with a good approximation can be characterized by translational modes of deformation.
In the process of deformation the density of the forming martensitic crystals increases, the variants of martensitic transformations that do not coincide with the favorably oriented directions in the field of external stresses due to the loading trajectories in the stress space are involved.
The mentioned circumstances cause the need for self-organization of martensitic domains in accordance with the given boundary conditions, that means with the directions of the principal axes of the strain tensor at each moment of the deformation process. As a result, there are self-encoded groups of martensitic crystals forming dissipative structures within the initial structure of the material, in accordance with figure 1b. Such dissipative structures contribute to the appearance of translationalrotational deformation modes according to the shift+rotation scheme and are classified as mesoscale level 1.
At the considered scale and structural level, the hydrodynamic characteristics of the substance, depending on the history of thermomechanical action and the type of stress-strain state, are clearly manifested [3,4].  Figure 1. Scheme of scale and structural levels of a deformable object made of materials with a martensitic mechanism of mass transfer, the shaded areas represent the nuclei of the martensitic crystals: a -micro, b -meso-1, c -meso-2, d -macro-1.
With the development of martensitic reactions, the density of structural imperfections of a larger scale level increases and, at a certain critical value of meso concentrators, when the distortion of the transformation at the meso-1 level reaches the limit value, the shear stability is lost in the local zones of the next mesostructural level, meso-2, figure 1c. Thus, structural changes in areas of considerable length and in arbitrary directions, not necessarily coinciding with crystallographic variants of shifts at martensitic transformations, become possible. Extended band-pass deformation structures appear, they originate on meso stress concentrators and spread over long distances through many structural elements regardless of the crystallographic orientation. At this scale level, the stochastic features of the formation of the translational-rotational field of the deformable material are of particular importance [3,4].
Self-organization of complex stochastic processes in various volumes of the large-scale level meso-2 leads to the formation of a non-trivial tensor properties at the scale level of particulate substances, i.e., at the level of macro-1, figure 1d. Inhomogeneous distribution of macrodeformation in the loaded actual It should be noted that the scheme of structural and scale levels of the deformation process given in figure 1 under the conditions of phase transformations of martensitic type, is very conditional, however, it allows to properly represent the sequence and the main stages of modeling, which in the following sections are formulated in the form of the corresponding connected defining relations. A brief overview of mathematical models of martensitic transformations initiated by phase deformations and structural stresses is given further.

Model of martensitic transformations process at the micro-scale level
Characteristic averaging volume V0 >> a 3 ), where a -is the lattice parameter [2].
Micro-level deformation Ф In the formulae (1) In the formulae (2)  Distortion rate Φ ik   and the intensity of structural strains Φ ik q at the considered scale level we will describe a system of integro-differential equations that takes into account the condition of preserving the continuity of the environment. Within the development of the model the continuity condition of the environment means the need to harmonize the sliding velocity of the considered bands and shifts in the random meso bands of deformation. Using a technique developed in statistical mechanics of continuum, the following analytical relations can be formulated for meso-2: In the formulae ( Here, -macroscopic strain tensors characterizing the contribution of mass transfer processes at micro, meso-1 and meso-2 structural and scale levels, respectively. The correlations (1)-(5) are discussed in more detail in [4].

Mesomechanics model taking into account the wave character of deformation
As it is known [4], the peculiarity of the meso bands motion in the deformable solid is the complex nature of the interaction of effective shear stresses with the subsystem of structural imperfections, which in the given boundary conditions (determined by the direction of the main axes of the stress state in the stress space at "soft" loading trajectories or corresponding directions in the deformation space under rigid loading conditions) leads to the wave nature of the plastic flow spread.
We take as a generalized coordinate the shear component of the distortion tensor Ф 31  at the meso-2 scale level [4]. It is necessary to enter the tensor parameter to construct the wave equation  . This technique is inconsistent for the transfer processes described at the same scale level. In our case of multilevel analysis, this approach is natural and methodically justified, since it allows us to organize additional coherence of the processes of evolution of deformation structures at the micro -, meso -and macro-scale levels. Taking into account the abovementioned, we introduce a tensor parameter ik d in the form: The wave equation for the shear strain component at the meso structural level of meso-2 can be represented by the following equation: Here   -Laplace operator for hydrodynamic orientation space {2}, m0, m1, m2, АВ -the material constants. Availability is dependent on the orientation of coordinates and time, the components of the right side of the equation (7) indicates a strong attenuation of the waves of plastic flow in a structurally heterogeneous environment. The specificity of the initiation of meso structure vibrations in the deformation process allows us to characterize it as an active excitable environment.
The available experimental data [4] convincingly testify about the auto wave nature of shifts of meso bands of deformation at the meso structural level. Equation (7) is a natural generalization of equation (3) and takes into account the wave nature of the deformation process at the meso structural level.
Taking into account the description of auto wave processes at the scale level of macro-1, leads to the appearance in the equation for the calculation of the components of the tensor of inelastic deformation   Having performed the orientation averaging of the relations (7) of the mesoscale level, we will obtain the desired wave equations at the macroscale level in the form: the effective stress. The indicated parameters are functionals depending on the history of loading in time and in the space of stresses (deformations), and are calculated by the method described in [4] in the derivation of the defining relations for the macro-scale level.