#  Review on phase transformations, fracture, chemical reactions, and other
structural changes in inelastic materials***Extended version of paper: Levitas
V.I. Phase transformations, fracture, and other structural changes in
inelastic materials. International Journal of Plasticity, 2021, Vol. 140,
102914, 51 pp., invited review. DOI 10.1016/j.ijplas.2020.102914

Valery I. Levitas  Iowa State University, Departments of Aerospace Engineering
and Mechanical Engineering, Ames, Iowa 50011, USA<EMAIL_ADDRESS>Ames
Laboratory, Division of Materials Science and Engineering, Ames, IA, USA

###### Abstract

Review of selected fundamental topics on the interaction between phase
transformations, fracture, and other structural changes in inelastic materials
is presented. It mostly focuses on the concepts developed in the author’s
group over last three decades and numerous papers that affected us. It
includes a general thermodynamic and kinetic theories with sharp interfaces
and within phase field approach. Numerous analytical (even at large strains)
and numerical solutions illustrate the main features of the developed theories
and their application to the real phenomena. Coherent, semicoherent, and
noncoherent interfaces, as well as interfaces with decohesion and with
intermediate liquid (disordered) phase are discussed. Importance of the
surface- and scale-induced phenomena on interaction between phase
transformation with fracture and dislocations as well as inheritance of
dislocations and plastic strains is demonstrated. Some nontrivial phenomena,
like solid-solid phase transformations via intermediate (virtual) melt,
virtual melting as a new mechanism of plastic deformation and stress
relaxation under high strain rate loading, and phase transformations and
chemical reactions induced by plastic shear under high pressure are discussed
and modeled.

Notations

SCs Structural changes
PTs phase transitions
TRIP transformation-induced plasticity
CRs chemical reactions
RIP reaction-induced plasticity
SMA shape memory alloys
PFA phase field approach
DAC diamond anvil cell
RDAC rotational diamond anvil cell
G graphite
D diamond
FEM finite element method
IM intermediate melt(ing)
VM virtual melt(ing)
MD molecular dynamics
HPP high-pressure phase
$\sf A$, $\sf M$, and ${\sf M}_{i}$ austenite, martensite, and martensitic
variant

Direct tensor notations are used throughout this paper. Vectors and tensors
are denoted in boldface type;
$\,{\mbox{\boldsymbol{$A$}}}\,{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$B$}}}\,=\,\left(A_{ij}\,B_{jk}\right)\,$
and
$\,{\mbox{\boldsymbol{$A$}}}\,{\mbox{\boldsymbol{$:$}}}\,{\mbox{\boldsymbol{$B$}}}\,=\,A_{ij}\,B_{ji}\,$
are the contraction of tensors over one and two nearest indices. Superscripts
$\;-1\,$ and $T$ denote inverse operation and transposition, respectively,
$\;:=\;$ means equals per definition, subscript $\,s\,$ designates
symmetrization of the tensors, the indices 1 and 2 indicate the values before
and after the SC. Subscripts $e$, $t$, and $p$ designate elastic,
transformational, and plastic deformations or deformation gradients.

###### Contents

  1. 1 Introduction
  2. 2 Martensitic transformations in inelastic materials: some background information
  3. 3 Some universal relationships for a coherent interface
  4. 4 Phase transformations in elastic materials
  5. 5 Athermal resistance to interface propagation
  6. 6 Theory of structural changes in inelastic materials with an unstable intermediate state
    1. 6.1 Definition of the structural changes without a stable intermediate state and local equations
    2. 6.2 Thermodynamic driving forces for nucleation and interface propagation
    3. 6.3 Three types of kinetic descriptions
    4. 6.4 Athermal kinetics
    5. 6.5 The postulate of realizability and extremum principle for determination of all unknown parameters for a nucleus and interface
    6. 6.6 Global criterion for structural changes based on stability analyses
    7. 6.7 Thermally-activated kinetics
    8. 6.8  ”Macroscale” thermally-activated kinetics for structural changes
    9. 6.9 Comparison with alternative approaches
  7. 7 Finite strain formalism
  8. 8 Spherical elastic nucleus within an elastic - perfectly plastic sphere
    1. 8.1 Pressure variation and athermal kinetics
    2. 8.2  Phase transformation from graphite to diamond
    3. 8.3 Critical nucleus
    4. 8.4  Macroscale nucleation kinetics
  9. 9 Stress- and strain-induced chemical reactions and phase transformations in a thin layer: propagating interface, shear band, and TRIP and RIP phenomena
    1. 9.1 Analytical solution
    2. 9.2  Strain-induced chemical reactions in a shear band
    3. 9.3 Mechanochemical feedbacks and the effect of TRIP/RIP on the strain-induced structural changes
  10. 10 Phase transition in ellipsoidal inclusion
  11. 11 Nucleation and growth of martensite with coherent, semicoherent, and incoherent interfaces, and interface with decohesion
    1. 11.1 Semicoherent interface and interface with a decohesion
    2. 11.2 Propagation of a semicoherent interface
    3. 11.3 Stress-induced PT in a spherical particle within a matrix and its interaction with plasticity, semicoherence and adhesion
    4. 11.4 Semicoherent interface within a phase field approach
    5. 11.5 New approach to incoherent interface
    6. 11.6 Semicoherent interface within a phase field approach with discrete dislocations
  12. 12 Solid-solid phase transformations via intermediate (virtual) melt
  13. 13 Virtual melting as a new mechanism of plastic deformation and stress relaxation under high strain rate loading
  14. 14 Strain-induced nucleation at shear-band intersection. Application of the global criterion for structural changes
  15. 15 Appearance and growth of a martensitic plate in elastoplastic material
    1. 15.1 Macroscale nucleation of a martensitic plate
    2. 15.2  Growth of martensitic lath within the austenite: effect of inheritance of plastic strain
  16. 16 Phase field approach to the interaction between plasticity and phase transformations
    1. 16.1 PFA to martensitic phase transformations
    2. 16.2 Multivariant martensitic phase transformations and transformations in multiphase materials
    3. 16.3 PFA to dislocations
    4. 16.4 PFA to the interaction of phase transformation and dislocations
    5. 16.5 Complete system of the phase-field equations for the interaction between phase transformation and discrete dislocations
  17. 17 Phase transformations and chemical reactions induced by plastic shear under high pressure
    1. 17.1 Main phenomena
    2. 17.2 Atomistic studies
    3. 17.3 Nucleation and evolution of high-pressure phase at dislocation pileups
    4. 17.4  Macroscale theory and FEM modeling of strain-induced transformations
  18. 18 Scale transitions and phenomenological theories for interaction between phase transformation and plasticity
  19. 19 Fracture and interaction between fracture and phase transformation in inelastic materials
    1. 19.1  Crack propagation in elastoplastic material
    2. 19.2 Interaction between fracture and phase transitions
    3. 19.3 Void nucleation in infinite elastoplastic sphere
    4. 19.4 Alternative approach to void nucleation
    5. 19.5 Phase field approach to fracture
    6. 19.6 Phase field approach to interaction between phase transformation and fracture
  20. 20 Concluding remarks

##  1 Introduction

Solid-solid phase transformations (PTs) are broadly studied in physical,
material, and mechanical experiments. They are utilized in modern technologies
(e.g., thermal and thermomechanical treatments of metals, shape memory and
elastocaloric applications, and high-pressure technologies) and broadly spread
in nature, e.g., in geophysical processes. In most cases, PTs are accompanied
by plastic deformations; in some cases, they are caused by plastic
deformations (plastic strain induced PTs). Googling ”phase transformation and
plasticity” returns 13,600,000 results. It is clear that any attempt of review
on this topic will miss many important aspects. We will focus on topics and
papers, which we worked on for several decades, and on works which have
affected us. There are multiple reviews related to the interaction between PT
and plasticity (e.g., [372, 144, 317] on material aspects, [65, 407] on shape
memory alloys, on the continuum mechanical aspects [228, 110, 246],
transformation-induced plasticity (TRIP) [113, 348], PTs and structural
changes (SCs) under high-pressure torsion [41, 42, 474, 476, 88, 89, 28, 434,
477, 435, 436, 87, 336, 417, 252], and mechanochemistry [203, 287]), and we
will avoid overlapping.

One of the broad topics of this review is the development of general
thermodynamic and kinetic approaches to SCs in inelastic materials within a
sharp interface approach. SCs under consideration include various PTs
(martensitic, reconstructive, melting, sublimation, and others), twinning,
chemical reactions (CRs), and fracture (crack and void nucleation and growth).
A well-known formalism was developed for the description of the evolution and
interaction of various defects or singularities in elastic materials [94, 95,
96, 97, 389, 390, 55, 56]. They include phase interfaces, grain and twin
boundaries, crack tips and voids, and point and linear defects, participating
in such structural changes as PTs, grain evolution, damage, and plastic
deformation. For each of these defects, the rate of energy release or
dissipation rate can be written as
$\,D\,=\,{\mbox{\boldsymbol{$X$}}}\,{\mbox{\boldsymbol{$\cdot$}}}\,\dot{\mbox{\boldsymbol{$q$}}}\geq
0\,$, where $\,\dot{\mbox{\boldsymbol{$q$}}}\,$ is the defect velocity
relative to the material and $X$ is the generalized material/driving force
acting on the defect [94, 95, 97]. The formula for
$\,{\mbox{\boldsymbol{$X$}}}\,=\,\int_{\scriptscriptstyle\Sigma}{\mbox{\boldsymbol{$Q$}}}\,{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$n$}}}\;d\,\Sigma\,$
is derived, where $Q$ is the Eshelby energy-momentum tensor, $\,\Sigma\,$ is
an arbitrary surface with the unit normal $n$ surrounding the defect and
separating it from other defects or the actual surface defect (e.g., phase
interface or grain boundary). This concept was extended for or rediscovered in
various specific fields. For example, in fracture mechanics, a driving force
$X$ was introduced as a path-independent $\,J\,$-integral [389, 390] or
$\,\Gamma\,$-integral [55, 56]; see also [129, 130, 193] for PTs and [334,
335] for Eshelbian mechanics. Since, for elastic materials, dissipation occurs
due to the motion of defects only, adding to $X$ the integral over the volume
without evolving defects does not change the dissipation rate and $X$ –this
explains the independence of $X$ of the surface $\,\Sigma\,$ (or integration
path).

For inelastic materials, the dissipation due to plastic deformation and change
in internal variables contributes to the total dissipation rate
$\,{\mbox{\boldsymbol{$X$}}}\,{\mbox{\boldsymbol{$\cdot$}}}\,\dot{\mbox{\boldsymbol{$x$}}}\,$
in the volume $\,v\,$ surrounded by surface $\,\Sigma\,$ and $X$ depends on
the choice of surface $\,\Sigma\,$. Even for the volume $\,v\,$ tending to
zero, i.e., when it includes the defect only, inelastic dissipation in the
singular point or surface still takes place (e.g., at the crack tip or moving
interface). It is not easy to split the dissipation due to the defect
evolution itself and plastic deformation. That is why it was accepted in works
on fracture mechanics [389, 205, 11, 204] that the strict thermodynamic
criterion for ductile fracture is not developed and other approaches like
energy flow in an infinitesimal or finite-sized process zone [389, 56, 166,
11], total dissipation rate [430, 205], plastic work [210], critical plastic
strain [337, 389], critical crack tip opening displacements [150],
$\,{J}\,$-integral (or $\,\Gamma\,$\- integral) without separation of plastic
dissipation [55, 56], and others are used. A similar situation took place in
PT theory, see Section 6.

For elastic materials, the conditions for both the appearance of a defect
(”nucleation”) and its equilibrium are described by the principle of the
minimum of Gibbs free energy. That is why they coincide. Thus, from a
thermodynamic (but not a kinetic) viewpoint, it is not necessary to treat a
nucleation process separately; it is sufficient to insert a nucleus and study
its equilibrium using the local condition, e.g., for an interface or crack
tip. However, for inelastic materials, defect nucleation and equilibrium
conditions are different. Thus, a defect nucleation problem within inelastic
materials has to be formulated and solved. Defect nucleation and evolution in
inelastic materials cannot be described by the principle of the minimum of
Gibbs energy or with the help of an energy-momentum tensor only. Thus, the new
driving force and extremum principle for the determination of all unknown
parameters (e.g., position, shape, orientation, and internal structure of the
nucleus of a product phase) are required. In contrast to elastic materials,
since the constitutive behavior of inelastic materials is history dependent,
analysis of the entire transformation-deformation process in the transforming
region is required.

Another broad topic of this review is the development of phase field
approaches (PFAs) to different SCs in inelastic materials, see e.g., [9, 189,
413, 449, 447, 452, 146, 195, 394, 253, 272, 271, 182, 306, 177]. PFA is based
on the concept of the order parameters that describe instabilities of the
crystal lattice during PT, twinning, dislocation nucleation, and fracture, as
well the evolution of phase and dislocational structures in a continuous way
by solving Ginzburg-Landau evolution equations for the order parameters.
Typical solutions for these equations are propagating finite-width phase and
twin interfaces, crack surfaces, and dislocation core regions that describe
the evolution of complex microstructures. Thermodynamic potential has as many
minima in the space of the order parameters as many phases and structural
states system possesses. These minima are separated by energy barriers.
Besides, the thermodynamic potential depends on the gradients of the order
parameters, which are concentrated at the finite-width phase and twin
interfaces, crack surfaces, and dislocation cores; this reproduces the
interface, surface, and dislocation core energies.

The sharp-interface approach gives specific expressions for the thermodynamic
driving forces for the nucleation and evolution of defects. It is convenient
for solving problems with relatively simple geometries of interfaces and
defects, and allows analytical solutions for some problems. The PFA includes
additional information about the stability and instability of phases and
different states. It allows for studying the evolution of arbitrary complex
geometries of interfaces and defects, without any computational cost on
tracking interfaces. Thus, both approaches have their advantages and
disadvantages, and they supplement each other. Both will be reviewed in the
current paper.

This review is organized as follows. In Section 2, short background
information on martensitic PTs in inelastic materials is presented, including
some examples of interactions between PTs and plasticity. In Section 3, some
universal kinematic and balance relationships for a coherent interface in an
arbitrary medium are presented. Thermodynamic and kinetic descriptions of PTs
in elastic materials are discussed in Section 4. Athermal resistance to
interface propagation and its relationship to the yield strength are
introduced in Section 5. General sharp-interface thermodynamic and kinetic
theories for SCs in inelastic materials are presented in Section 6. They
include the driving forces and thermodynamic criteria for nucleation and
interface propagation, and the extremum principle for determination of all
unknown parameters that substitutes the minimum Gibbs energy principle for
elastic materials. Three types of kinetics are described: athermal kinetics,
thermally-activated kinetics, and ”macroscale” thermally-activated kinetics.
In addition, the global SC criterion based on stability analyses is
introduced. Peculiarities of the finite strain approaches are summarized in
Section 7. All equations from last two Sections are summarized in ten boxes.
Sections 8, 9, and 10 present some analytical solutions based on the developed
theory, which illustrate the main points of the theory as well as their
application to some real problems. They include SC in a sphere (with
application to PT from graphite to diamond), PTs and CRs in a shear band (with
application to transformation- and reaction-induced plasticity (TRIP and
RIP)), consideration of a propagating interphase, as well as PT in an
ellipsoidal inclusion. Various aspects of the nucleation and growth of a
martensitic region with coherent, semicoherent, and incoherent interfaces, as
well as interfaces with decohesion are analyzed with the finite element method
(FEM) in Section 11. Both the sharp-interface theory and PFA are utilized.
Sections 12 and 13 are devoted to recently revealed nontraditional mechanisms
of PTs in solids via intermediate (virtual) melt and mechanism of plastic
deformation and stress relaxation under high strain rate loading via virtual
melting, both much below the melting temperature, when melting is not
expected. Strain-induced martensite nucleation at the shear-band intersection
was studied with FEM in Section 14. The importance of the application of the
global criterion for SCs based on stability analysis was demonstrated. The
appearance and growth of a martensitic plate in an elastoplastic material for
temperature-induced PT was analyzed with FEM in Section 15. The effect of the
inheritance of plastic strain on a large change in the plate’s shape and on
the arrest of plate growth (i.e., morphological transition from the plate to
lath martensite) is demonstrated. Nanoscale and microscale PFAs to the
interaction between dislocation plasticity and PTs are described in Section
16. In Section 17, a multiscale theory of strain-induced SCs under high
pressure is presented for the interpretation of various phenomena during
compression and shear of materials in a rotational diamond anvil cell. Scale
transitions and phenomenological theories for the interaction between PT and
plasticity are analyzed in Section 18. Fracture and the interaction between
fracture and PT in inelastic materials are described within both the sharp-
interface theory and PFA in Section 19. Concluding remarks are presented in
Section 20.

##  2 Martensitic transformations in inelastic materials: some background
information

We will focus on displacive PTs, which are dominated by the deformation of a
unit crystal cell of the parent phase into a unit cell of the product phase
that is described by transformation deformation gradient
$\mbox{\boldsymbol{$F$}}_{t}=\mbox{\boldsymbol{$R$}}_{t}\cdot\mbox{\boldsymbol{$U$}}_{t}$,
where $\mbox{\boldsymbol{$R$}}_{t}$ is the orthogonal rotation tensor,
$\mbox{\boldsymbol{$U$}}_{t}=\mbox{\boldsymbol{$I$}}+\mbox{\boldsymbol{$\varepsilon$}}_{t}$
is the transformational right stretch (Bain) tensor, and
$\,\mbox{\boldsymbol{$\varepsilon$}}_{t}\,$ is the transformation strain
tensor. Note that $\mbox{\boldsymbol{$U$}}_{t}$ produces mapping of the
stress-free crystal lattice of the parent phase into that for the product
phase at a fixed temperature. Usually, the higher symmetry and higher
temperature phase is called the austenite, and the lower symmetry and lower
temperature phase is called the martensite.

Displacive PTs include martensitic PTs during which atoms do not change their
neighbors and reconstructive PTs in the opposite case. In addition to
$\mbox{\boldsymbol{$U$}}_{t}$, displacive PT involves intra-cell displacements
or shuffles. The diffusion of species does not occur during martensitic PTs.

Figure 1:  Schematics of the transformation between austenite and martensitic
variants as well as between martensitic variants.

Due to the symmetry of crystal lattice, there is a finite number (e.g., 3 for
the cubic to tetragonal PT and 12 for the cubic to monoclinic PT) of
crystallographically equivalent martensitic variants $M_{i}$ (Fig. 1). Lists
of the transformational right stretch tensors for various PTs and for all
martensitic variants are presented, e.g., in [26, 382]. The transformation
strain can be quite large. For example, for cubic to tetragonal PT from Si I
to Si II [464] and cubic to monoclinic PT from rhombohedral graphite to
hexagonal diamond [44] tensors $\mbox{\boldsymbol{$\varepsilon$}}_{t}$ are

$\displaystyle{\mbox{\boldsymbol{$\varepsilon$}}}^{Si\,I-II}_{tj}=\left(\begin{array}[]{ccc}0.243&0&0\\\
0&0.243&0\\\
0&0&-0.514\end{array}\right);\qquad{\mbox{\boldsymbol{$\varepsilon$}}}^{G-D}_{tj}\;=\;\left(\begin{array}[]{ccc}0.024&0&0.105\\\
0&-0.034&0\\\ 0.105&0&-0.35\end{array}\right).$ (7)

The components of $\mbox{\boldsymbol{$\varepsilon$}}_{t}$, $\\{a,a,c\\}$, for
cubic to tetragonal PT from phase I to II in Ge and GaSb, as well as for
$\alpha\rightarrow\beta$ PT in Sn, are in the range $a=0.254-1.281$ and
$-c=0.502-0.510$ [327]. For the layer-puckering mechanism [43] for PT from
cubic diamond and boron nitride to rhombohedral graphite and BN, $a=0.98$ and
$c=1.596$. Transformation strain is generally much larger than the maximum
elastic strain, which varies from $10^{-3}$ for steels to $10^{-2}$ for NiTi
and CuZnAl shape memory alloys.

Most of the martensitic variants are in twin relationship to each other.
Twinning is a simple shear of one part of the crystal lattice with respect to
another up to a position in which it represents a mirror reflection of the
initial lattice (Fig. 2). Thus, twinning is described by the transformation
deformation gradient
${\mbox{\boldsymbol{$F$}}}_{t}={\mbox{\boldsymbol{$I$}}}+\gamma_{t}\,{\mbox{\boldsymbol{$m$}}}{\mbox{\boldsymbol{$n$}}}$,
where $\gamma_{t}$ is the twinning shear strain that occurs in the direction
$m$ in the plane with the normal $n$, which is called the twinning plane.
Generally, twinning is a mechanism of plastic deformation in crystalline
materials whereby jump-like shear deformation of the crystal lattice occurs.
It both competes with and supplements dislocation plasticity. For the body-
centered cubic (bcc) lattices (for example, in Mo, Na, and Cr) and the face-
centered cubic (fcc) metals (for example, in Al, Cu, and Co), the magnitude of
the transformation shear $\,\gamma\,=\,0.707\,$ is very large. For the
hexagonal close-packed (hcp) lattice in Mg $\,\gamma_{t}\,=\,0.137\,$ and in
Zr $\,\gamma_{t}\,=\,0.225\,$. For the body-centered tetragonal lattice in
NiAl $\,\gamma_{t}\,=\,0.150\,$. Various aspects of twinning can be found in
the books [453, 26, 202, 382]. An example of the constitutive equations for
twinning can be found in [339] and references herein. As a mechanism of
plastic deformation, in which twinning competes with dislocations, twinning
usually occurs at a lower temperature, higher strain rates, and smaller grain
size than the dislocation plasticity. Such competition also takes place during
PTs. Since a single martensitic variant is generally not compatible with
austenite and generates large internal stresses, these stresses relax either
by twinning (for example, in shape memory alloys and some steels) or by
plastic slip (for example, in steels), or both, producing an invariant plane
strain variant with averaged transformation deformation gradient
${\mbox{\boldsymbol{$F$}}}_{t}={\mbox{\boldsymbol{$I$}}}+\gamma_{in}\,{\mbox{\boldsymbol{$m$}}}{\mbox{\boldsymbol{$n$}}}+\varepsilon_{in}\,{\mbox{\boldsymbol{$n$}}}{\mbox{\boldsymbol{$n$}}}$,
where $\gamma_{in}$ is the invariant plane shear strain along the direction
$m$ in the invariant (or habit) plane with the normal $n$, and
$\varepsilon_{in}$ is the normal to habit plane strain. Typical values of
$\varepsilon_{in}$ are around zero for NiTi and CuZnAl shape memory alloys,
0.02 to 0.05 for steels, and 0.2 for $\;\;\delta\;\rightarrow\;\alpha$ PT in
plutonium; typical values of $\gamma_{in}$ are 0.1-0.2 for shape memory
alloys, 0.2 for steels, and 0.27 for $\;\;\delta\;\rightarrow\;\alpha$ PT in
plutonium.

Figure 2:  Schematics of the twinning.

Usually, three types of PTs are distinguished: temperature-induced, stress-
induced, and strain-induced. Temperature-induced PTs occur without external
stresses, and nucleation starts at pre-existing defects (e.g., dislocations,
point defects, grain and subgrain boundaries, stacking faults, and twins).
Stress-induced (or assisted) PTs take place under external stresses below the
macroscopic yield strength $\,\sigma_{y}\,$ by nucleation at the same pre-
existing defects as temperature-induced PTs. Strain-induced PTs occur during
plastic deformation by nucleation at new defects generated by plastic flow
[367, 369, 370, 372].

Formally, we define any type of the SCs enumerated above (PT, CRs, twinning,
and fracture) as a thermomechanical deformation process of the growth of
transformation strain $\,\mbox{\boldsymbol{$\varepsilon$}}_{t}\,$ from its
initial value $\,\mbox{\boldsymbol{$\varepsilon$}}_{t1}\,$ in the parent phase
to the final value $\,\mbox{\boldsymbol{$\varepsilon$}}_{t2}\,$ in the product
phase, which is accompanied by a change in all the thermomechanical properties
(elastic moduli of an arbitrary rank, thermal expansion tensor, specific heat,
etc.).

Below we outline some examples of interaction between SCs and inelasticity,
which are interrelated.

1\. PTs and other SCs, possessing a transformation strain, are processes of an
inelastic deformation in materials. For some materials and applications, for
which traditional (dislocation or twinning) plasticity is not desirable, like
shape memory alloys (SMA), strong ceramics, and semiconductors (like Si and
Ge), PT is the main mechanism of inelasticity. For all other cases,
transformation strain supplements dislocation plasticity and twinning.

2\. Microplastic straining, which occurs in SMA during cyclic loading,
accumulates and leads to defect generation, damage, and degradation of
transformational properties (e.g., reduces recoverable strain and increases
stress hysteresis and energy losses).

3\. Slip and twinning in martensite (a combination of two martensitic
variants) are mechanisms for lattice-invariant shear. Along with the
transformation (Bain) strain and crystal lattice rotation, they produce an
invariant plane strain variant [453] between austenite and martensite. This
process is driven by the reduction of the energy of internal stresses; it
promotes nucleation and growth.

4\. Transformation strain in the transforming regions generates internal
stresses, which in most cases exceed the yield strength and produce
accommodational plastic strains within and outside of the transforming
regions. Reduction of the internal stresses increases the thermodynamic
driving force for nucleation. However, stress redistribution caused by plastic
deformation near the growing product phase reduces the thermodynamic driving
force for interface propagation. In elastic materials, the martensitic region
is arrested by a strong obstacle (e.g., grain, subgrain, or twin boundaries,
or another martensitic region), producing plate martensite morphology.
Plasticity stops the growth of martensitic units inside the grain before
reaching an obstacle, leading to a morphological transition to the lath
martensite. Numerous analytical and numerical results illustrating the above
statements are presented in the following Sections.

5\. Plastic deformation generates a group of defects (e.g., dislocation walls
or pileups), and corresponding stress concentrators produce nucleation sites
for PT. At the same time, chaotic defect structures like dislocation forests
resist the interface propagation. Preliminary plastic deformation generally
suppresses martensitic PTs.

6\. Plastic flow that occurs during PT causes strain-induced PT, which
proceeds by nucleation at defects generated during plastic flow, e.g., at a
dislocation pileup or slip-band intersections. While for stress-induced PTs,
the PT stress grows linearly with the temperature increase, for strain-induced
PTs, the PT stress reduces with the temperature rise due to the reducing yield
strength but grows with increasing plastic strain due to strain hardening.

7\. Internal stresses caused by transformation strain in superposition with
external stresses, which may be well below the yield strength, cause plastic
flow. This phenomenon is called TRIP for PTs [113, 348, 367, 374, 463] or RIP
for CRs [278, 279]. They serve as a relaxation mechanism for internal stresses
and as an additional mechanism of inelastic deformation. For cyclic direct-
reverse PT under the external stress, below the yield strength, TRIP is
accumulated in each cycle and may exceed hundreds of percent.

8\. The phase interface, similar to a twin interface, can be presented as an
array of partial dislocations [372]. In such a representation, PT consists of
the nucleation and motion of these dislocations, like dislocation plasticity.

9\. Some PTs, in addition to transformation strain, involve shuffles
(intracell atomic motion) produced by the motion of partial dislocations,
e.g., for bcc-fcc and fcc-hcp PTs [383].

10\. The strong promoting effect of the plastic shear under high pressure on
PTs and CRs will be discussed in Section 17.

Knowledge of the influence of plastic strain and applied and local stress
fields on SCs is very important for the understanding, simulation, and
improvement of the technical processes, as well as for the development of new
technologies and materials. Examples include heat and thermomechanical
treatment of materials; increasing toughness utilizing TRIP; severe plastic
deformation technologies, including high pressure torsion and ball milling,
friction, wear, surface treatment (polishing and cutting); as well as the
interpretation of earthquakes.

##  3 Some universal relationships for a coherent interface

Let the motion of the homogeneously deformed small vicinity of a material
point be described by the function
$\,{\mbox{\boldsymbol{$r$}}}\,=\,{\mbox{\boldsymbol{$r$}}}\left({\mbox{\boldsymbol{$r$}}}_{0},\;t\right)$,
where $r$ and $\;{\mbox{\boldsymbol{$r$}}}_{0}\,$ are the positions vectors in
the actual $\,\Omega\,$ and reference $\,\Omega_{0}\,$ configurations and $t$
is time. The deformation gradient is
$\,{\mbox{\boldsymbol{$F$}}}\,=\,\frac{\displaystyle\partial\,{\mbox{\boldsymbol{$r$}}}}{\displaystyle\partial\,{\mbox{\boldsymbol{$r$}}}_{0}}\;$.
We will not focus on the multivariant structure of martensite here, and will
instead consider a two-phase material.

For a coherent interface, when a jump in displacement across an interface is
absent, but the particle velocity vector $v$ and the deformation gradient $F$
have a jump, the Hadamard compatibility condition is valid in the reference
configuration $\Omega_{0}$ [426]

$\displaystyle\left[{\mbox{\boldsymbol{$F$}}}\right]\,=\,-\left[{\mbox{\boldsymbol{$v$}}}\right]\,{\mbox{\boldsymbol{$n$}}}_{0}/v_{n}\,,\qquad{\rm
whence}\qquad\left[{\mbox{\boldsymbol{$v$}}}\right]\,=\,-\left[{\mbox{\boldsymbol{$F$}}}\right]\,{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$n$}}}_{0}\,v_{n}\qquad{\rm
and}\qquad\left[{\mbox{\boldsymbol{$F$}}}\right]\,=\left[{\mbox{\boldsymbol{$F$}}}\right]{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$n$}}}_{0}\,{\mbox{\boldsymbol{$n$}}}_{0}\;,$
(8)

where $\,{\mbox{\boldsymbol{$n$}}}_{0}$ is the unit normal to the interface in
$\Omega_{0}$, $v_{n}$ is the interface velocity, and
$[\mbox{\boldsymbol{$A$}}]=\mbox{\boldsymbol{$A$}}_{2}-\mbox{\boldsymbol{$A$}}_{2}$
is the jump of parameters across the interface. The conservation of mass at
the interface is expressed as

$\displaystyle\left[\rho_{0}c\right]\,=0,$ (9)

where $\rho_{0}$ is the mass density. Neglecting inertia, the traction
continuity condition at the interface is

$\displaystyle\left[\mbox{\boldsymbol{$P$}}\right]\cdot{\mbox{\boldsymbol{$n$}}}_{0}\,=0,$
(10)

where $P$ is the non-symmetric first Piola-Kirchhoff stress tensor.

A list of all universal equations for the moving interface, including energy
balance, entropy inequality, and inertia, can be found in [206, 1, 386].
Surface stresses are included in [143, 142, 141, 116].

##  4 Phase transformations in elastic materials

As an initial step and for comparison, we will describe an approach to phase
transformations in elastic materials in the reference configuration. Let us
consider a volume $\,V_{0}\,$ of a two-phase material with the prescribed
traction $\mbox{\boldsymbol{$p$}}_{0}$ at the part of the boundary $S_{0p}$
and displacement $u$ at the rest of the boundary $\,S_{0u}\,$;
$S_{0}=S_{0p}\cup S_{0u}$. The dissipation at the interface $\Sigma_{0}$
between phases is neglected. The problem is to find a two-phase configuration
in thermodynamic equilibrium. For isothermal processes the total dissipation
increment due to variation of the position of the interface $\Sigma_{0}$ is

$\displaystyle{\cal D}\delta
t\,=\int\limits_{S_{0p}}\mbox{\boldsymbol{$p$}}_{0}\cdot\delta\mbox{\boldsymbol{$u$}}dS_{0p}-\delta\int\limits_{V_{0}}\rho_{0}\psi\left(\mbox{\boldsymbol{$F$}},\,\theta\right)d\,V_{0}-\delta\int\limits_{\Sigma_{0}}\Gamma_{0}d\,\Sigma_{0}=0,$
(11)

where $\theta$ is the temperature, $\psi$ the Helmholtz free energy per unit
mass, and $\,\Gamma_{0}\,$ is the interface energy per unit reference area.
Note that if $\Gamma_{0}=const$, it does not produce surface stresses and does
not change the traction continuity condition (10) (in contrast to the case
when the surface energy per unit deformed area $\Gamma=const$ or when
$\Gamma_{0}$ depends on strain). If $\mbox{\boldsymbol{$p$}}_{0}$ is fixed,
then Eq. (11) can be presented in the form

$\displaystyle{\cal D}\delta t\,=-\delta G=0;\qquad
G:=-\int\limits_{S_{0p}}\mbox{\boldsymbol{$p$}}_{0}\cdot\mbox{\boldsymbol{$u$}}dS_{0}+\int\limits_{V_{0}}\rho_{0}\psi\left(\mbox{\boldsymbol{$F$}},\,\theta\right)d\,V_{0}+\int\limits_{\Sigma_{0}}\Gamma_{0}d\,\Sigma_{0},$
(12)

where $G$ is the Gibbs energy of the system body+loading. Thus, phase
equilibrium for an elastic material, i.e., the geometry of the interface, is
determined by the stationary value of the Gibbs energy. If the Gibbs energy
has a local minimum, phase equilibrium is stable; otherwise, it is unstable.
If a stable interface does not exist under the prescribed boundary conditions,
then only the single-phase solution is stable.

Let us consider each phase separately, without interfaces. The Gibbs energy
can then be introduced for the parent phase (subscript 1) and product phase
(subscript 2):

$\displaystyle
G_{i}:=-\int\limits_{S_{0pi}}\mbox{\boldsymbol{$p$}}_{0}\cdot\mbox{\boldsymbol{$u$}}dS_{0i}+\int\limits_{V_{0i}}\rho_{0}\psi_{0i}\left(\mbox{\boldsymbol{$F$}},\,\theta\right)d\,V_{0i}=\int\limits_{V_{0i}}\rho_{0i}g_{i}\left(\mbox{\boldsymbol{$P$}},\,\theta\right)d\,V_{0i};\qquad$
(13)
$\displaystyle\rho_{0i}g_{i}\left(\mbox{\boldsymbol{$P$}},\,\theta\right)=\rho_{0i}\psi\left(\mbox{\boldsymbol{$F$}},\,\theta\right)-\mbox{\boldsymbol{$P$}}^{T}\mbox{\boldsymbol{$:$}}\mbox{\boldsymbol{$F$}}.$
(14)

Here, $g_{i}$ is the local Gibbs energy per unit mass of each phase, expressed
in terms of $P$ with the help of the elasticity rule; the divergence theorem
was used to transform surface integral into a volume integral, and the
equilibrium equation has been utilized. Since analytical inversion of the
elasticity rule is in most cases impossible in practice, one can keep the
argument $F$ in $g_{i}$, and $F$ should numerically correspond to the given
$P$.

For a given traction $\mbox{\boldsymbol{$p$}}_{0}$ (or, for homogeneous
stress, a given stress tensor $P$) and temperature, the phase with smaller
Gibbs energy is called the stable phase, and the other is called the
metastable phase. The transition from a metastable to stable phase is
accompanied by the reduction in Gibbs energy and a positive dissipation
increment (see Eq. (12)), i.e., it is thermodynamically possible. However, it
does not mean that this transition will occur because there is usually an
energy barrier between phases. When an energy barrier disappears due to change
in traction or temperature, the metastable phase becomes unstable and
barrierless PT occurs. PT from the phase with the lower Gibbs energy to the
higher energy is thermodynamically impossible. Phases with equal Gibbs energy
are considered to be in thermodynamic equilibrium. This definition has a
physical sense for hydrostatic media, liquids, and gases under prescribed
pressure, because (with neglected surface tension (stresses)) the pressure is
the same in both phases. For solids, phase equilibrium should be considered
across an interface and not the entire stress tensor; only traction remains
the same across an interface (Eq. (12)). Of course, phase equilibrium
conditions are different for different interface orientations. Still, in high-
pressure research, phase equilibrium is defined by equality of the Gibbs
energies of phases under prescribed pressure.

It is also clear that the definition of the Gibbs energy and related
definitions depend on the chosen stress measure and boundary conditions, i.e.,
which components of stresses are fixed at the boundary. Integrating Eq. (12)
over the time $t_{s}$ for the appearance of a nucleus of phase 2 in a finite
volume $V_{n}$, we obtain

$\displaystyle X_{v}=\int\limits^{t_{s}}_{0}{\cal D}dt=-(G_{2}-G_{1})=-\Delta
G=0,$ (15)

i.e., the total dissipation increment is equal to the negative difference
between the Gibbs energy of the final and initial states, and represents the
global thermodynamic driving force for the PT. Eq. (15) will be used as the
main hint and limiting case for checking when we will develop SC theory for
inelastic materials.

Let us transform Eq. (11) for the neglected interface energy using the
divergence theorem and the rule of differentiation for the volume integral,
both for a volume with moving surfaces with discontinuous velocity or a
deformation gradient:

$\displaystyle\int\limits_{S_{0p}}\mbox{\boldsymbol{$p$}}_{0}\cdot\delta\mbox{\boldsymbol{$u$}}dS_{0p}=\int\limits_{V_{0}}\mbox{\boldsymbol{$P$}}^{T}\mbox{\boldsymbol{$:$}}\delta\mbox{\boldsymbol{$F$}}d\,V_{0}-\int\limits_{\Sigma_{0}}\left[{\mbox{\boldsymbol{$v$}}}\right]{\mbox{\boldsymbol{$n$}}}_{0}\mbox{\boldsymbol{$:$}}\mbox{\boldsymbol{$P$}}^{T}\delta
d\,\Sigma_{0}=\int\limits_{V_{0}}\mbox{\boldsymbol{$P$}}^{T}\mbox{\boldsymbol{$:$}}\delta\mbox{\boldsymbol{$F$}}d\,V_{0}+\int\limits_{\Sigma_{0}}\left[{\mbox{\boldsymbol{$F$}}}\right]\mbox{\boldsymbol{$:$}}\mbox{\boldsymbol{$P$}}^{T}v_{n}d\,\Sigma_{0}\delta
t;$ (16) $\displaystyle\delta\int\limits_{V_{0}}\rho_{0}\psi
d\,V_{0}=\int\limits_{V_{0}}\rho_{0}\delta\psi
d\,V_{0}+\int\limits_{\Sigma_{0}}\rho_{0}\left[\psi\right]v_{n}d\,\Sigma_{0}\delta
t,$ (17)

where Eqs. (8) and (10) were used. Substituting Eqs. (16) and (17) into Eq.
(11), we obtain

$\displaystyle{\cal D}\delta
t\,=\int\limits_{V_{0}}\left(\mbox{\boldsymbol{$P$}}^{T}\mbox{\boldsymbol{$:$}}\delta\mbox{\boldsymbol{$F$}}-\rho_{0}\delta\psi\right)d\,V_{0}+\int\limits_{\Sigma_{0}}\left(\mbox{\boldsymbol{$P$}}^{T}\mbox{\boldsymbol{$:$}}\left[{\mbox{\boldsymbol{$F$}}}\right]-\rho_{0}\left[\psi\right]\right)v_{n}d\,\Sigma_{0}\delta
t=0.$ (18)

Due to the independence of both integrals and arbitrariness of $V_{0}$ and
$\Sigma_{0}$, both integrands in Eq. (18) are equal to zero. The first
integrand results in the nonlinear elasticity rule for points of the volume.
The second results in the phase equilibrium condition at the interface

$\displaystyle
X_{\Sigma}:=\mbox{\boldsymbol{$P$}}^{T}\mbox{\boldsymbol{$:$}}\left[{\mbox{\boldsymbol{$F$}}}\right]-\rho_{0}\left[\psi\right]={\mbox{\boldsymbol{$n$}}}_{0}\,{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$P$}}}^{T}\,{\mbox{\boldsymbol{$\cdot$}}}\left[{\mbox{\boldsymbol{$F$}}}\right]{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$n$}}}_{0}-\rho_{0}\left[\psi\right]=\,{\mbox{\boldsymbol{$n$}}}_{0}\,{\mbox{\boldsymbol{$\cdot$}}}\left[{\mbox{\boldsymbol{$H$}}}_{0}\right]{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$n$}}}_{0}=0;\qquad{\mbox{\boldsymbol{$H$}}}_{0}\,:=\,{\mbox{\boldsymbol{$P$}}}^{T}\,{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$F$}}}-\rho_{0}\,\psi\,{\mbox{\boldsymbol{$I$}}},$
(19)

where $X_{\Sigma}$ is the thermodynamic driving force for the interface
motion. Eq. (8) was utilized in the derivations. The expression for
$X_{\Sigma}$ in Eq. (19) is also called the Eshelby driving force for the
interface propagation per the celebrated work [93], where it was derived. The
chemical potential tensor in the reference configuration
$\,{\mbox{\boldsymbol{$H$}}}_{0}\,$ was introduced in [129, 130], where
various aspects of phase equilibrium and different chemical potentials are
discussed (see also [206, 193] and references). For geometrically linear
approximation,
${\mbox{\boldsymbol{$F$}}}=\mbox{\boldsymbol{$I$}}+\mbox{\boldsymbol{$\varepsilon$}}+\mbox{\boldsymbol{$\omega$}}$,
where $\varepsilon$ and $\omega$ are the small symmetric strain and
antisymmetric spin tensors, respectively,
$\mbox{\boldsymbol{$P$}}=\mbox{\boldsymbol{$\sigma$}}$ is the symmetric stress
tensor and Eq. (19) simplifies

$\displaystyle
X_{\Sigma}:=\mbox{\boldsymbol{$\sigma$}}\mbox{\boldsymbol{$:$}}\left[{\mbox{\boldsymbol{$\varepsilon$}}}\right]-\rho\left[\psi\right]={\mbox{\boldsymbol{$n$}}}\,{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$\sigma$}}}\,{\mbox{\boldsymbol{$\cdot$}}}\left[{\mbox{\boldsymbol{$\varepsilon$}}}\right]{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$n$}}}-\rho\left[\psi\right]=\,{\mbox{\boldsymbol{$n$}}}\,{\mbox{\boldsymbol{$\cdot$}}}\left[{\mbox{\boldsymbol{$H$}}}\right]{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$n$}}}=0;\qquad{\mbox{\boldsymbol{$H$}}}\,:=\,{\mbox{\boldsymbol{$\sigma$}}}\,{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$\varepsilon$}}}-\rho\,\psi\,{\mbox{\boldsymbol{$I$}}}.$
(20)

When isotropic surface energy is taken into account, Eq. (19) generalizes to

$\displaystyle{\mbox{\boldsymbol{$n$}}}_{0}\,{\mbox{\boldsymbol{$\cdot$}}}\left[{\mbox{\boldsymbol{$H$}}}_{0}\right]{\mbox{\boldsymbol{$\cdot$}}}\,{\mbox{\boldsymbol{$n$}}}_{0}=2\Gamma_{0}\kappa_{av},$
(21)

where $\kappa_{av}$ is the mean interface curvature.

The following static problem formulations are usual for PTs in elastic
materials:

1\. For the prescribed boundary conditions, find a two-phase solution within a
body. Solutions (possibly multiple) correspond to the stationary value of the
Gibbs energy (Eq. (12)) or local thermodynamic equilibrium condition Eq. (21)
for each point of a phase interface.

2\. For the neglected interface energy for the problem in item 1, one can find
external tractions and displacements at which the two-phase equilibrium
becomes possible for the first time. This is the initiation of the PT.
Examples include the minimum pressure for PT from a low-pressure to high-
pressure phase or the maximum pressure for PT from a high-pressure to low-
pressure phase. The critical conditions for PT initiation can be found from
the solution for a small inclusion of the product phase in item 1 when the
characteristic size of the inclusion tends to be zero while keeping a shape
that minimizes the Gibbs energy. This zero size explains why surface energy
should be neglected.

3\. When surface energy is included, an unstable stationary solution called
the critical nucleus plays an important role. It corresponds to the minimax of
the Gibbs energy, the minimum with respect to the shape (and all internal
parameters, like a twinned structure, if included) and the maximum with
respect to the size or volume of the nucleus $V_{n}$ (Fig. 3). The energy
increment for the critical nucleus, $\Delta G_{n}=G_{n}-G_{par}$ is the
difference between the energy of the critical nucleus and the energy of the
parent phase, which can be any phase the material is initially in. The energy
of the critical nucleus is higher than the energy of the parent phase and its
appearance causes a negative dissipation increment. Since the appearance of
the critical nucleus is not thermodynamically favorable and requires
fluctuations, it cannot be described by traditional continuum thermodynamics
based on two thermodynamics laws. While it was well-known in material
literature that the energy grows during the appearance of a critical nucleus
(i.e., dissipation increment is negative), this statement caused a
psychological problem for the continuum mechanics community, which required
some tutorial explanations in [250]. Earlier, in the treatment of nucleation
in elastic materials in [319], it was specified that nucleation produces,
rather than dissipates, energy and that the subcritical nuclei cannot grow due
to the restriction produced by dissipation inequality. In [243, 244], after
stating that the appearance of the critical nucleus is accompanied by a
negative dissipation increment, thereby requiring thermal fluctuations, the
concept of the thermodynamically admissible nucleus was suggested. In
particular, the nucleus of the radius $r_{t}$ in Fig. 3 corresponds to
$X_{v}=-\Delta G=0$, i.e., it is thermodynamically admissible. However, such a
treatment is not consistent with the existing textbook approaches on
nucleation, and it does not explicitly determine the activation energy. This
concept appears naturally within a phenomenological theory of the appearance
of a finite region of a product phase without going into detail about complex
nucleation-growth processes. The approach in [243, 244] illustrates that, in
general, there is nothing unusual in the nucleation and that the second law is
satisfied for the event, averaged over the size $r>r_{t}$ or the corresponding
time interval. The problem arises because we are interested in the event that
occurs during a shorter period of time, and it is not surprising that the
second law of thermodynamics is not applicable for such a scale.

Figure 3:  Gibbs free energy of the nucleus vs. nucleus’ size $r$ and
definition of the size $r_{c}$ of the critical nucleus, which corresponds to
the maximum (generally, minimax) of the energy (and negative thermodynamic
driving force $-X_{v}$) and the size $r_{t}$ of the thermodynamically
admissible nucleus, corresponding to the zero driving force $X_{v}$.

Indeed, continuum thermodynamics operates with parameters that are averaged
over some time and volume, and any fluctuations are filtered out. As a result,
the nucleation process is generally described by statistical theories (as seen
in [197]), which we will not discuss here.

General ideas of the thermally activated kinetics can be found in classical
nucleation theory [108, 429, 383]. The Arrhenius-type equation for the
nucleation time in a sample with volume $V$ is

$\displaystyle t_{s}\,=\,t_{0}\,\exp\left(\;-\;\frac{Q}{k\,\theta}\right).$
(22)

Based on the probability consideration, the pre-exponential factor $t_{0}$ is
usually considered to be inversely proportional to the volume of the entire
sample $V$. The activation energy $Q$ for thermally activated nucleation is
equal to the energy of the critical nucleus $\Delta G_{n}$. We write

$\displaystyle
Q=\max_{{V_{0n}}}\min_{{shape}}\min_{{structure}}\min_{{position}}\Delta G.$
(23)

The first maximum in Eq. (23) belongs to the definition of the critical
nucleus. All minima are motivated as follows: the smaller the activation
energy, the smaller the nucleation time. Consequently, a nucleus with the
smallest activation energy should appear first. That is why in Eq. (23)
activation energy is minimized with respect to all relevant parameters.

The traditional criterion for thermally activated nucleation is

$\displaystyle Q=\beta k\theta,\qquad\beta=40-80,$ (24)

which is determined from the condition that, for larger $Q$, the nucleation
time exceeds any realistic time of observation for any choice of pre-
exponential parameters in the Arrhenius-type kinetic equation [383, 313, 247,
246]. Here $\,k\,=1.380\cdot 10^{-23}J/K\,$ is the Boltzmann constant. While
criterion (24) is from a material science textbook on PTs [383] and broadly
used in the treatment of PTs by material scientists [383, 313], its
application was limited in the continuum mechanics community [247, 246, 255,
256, 250, 257, 284]. Note that definition (23) includes an extremum principle
for the determination of all unknown parameters of the nucleus.

Since the critical nucleus cannot appear thermodynamically, it should be
introduced ”by hand” into the problem for continuum treatment. This allows the
nucleus slightly larger than the critical one (supercritical nucleus) to grow
in a thermodynamically consistent way. The nucleus slightly smaller than the
critical one (subcritical nucleus) shrinks and disappears. In the phase-field
approach, nucleation is modeled by including a stochastic term in the
Ginzburg-Landau or Cahn-Hilliard equations, which satisfies the dissipation-
fluctuation theorem (see, e.g., [188, 315, 446, 9]).

##  5 Athermal resistance to interface propagation

Athermal resistance for interface propagation is analogous to dry friction in
classical mechanics: interface propagation occurs only if the driving force
$X_{\Sigma}$ exceeds a rate-independent threshold $K_{\Sigma}$. The athermal
resistance is responsible for deviation of the actual SC stress or/and
temperature from their thermodynamical equilibrium values, and consequently,
for stress or/and temperature hysteresis during forward-reverse SC, and for
energy dissipation, when interface velocity is small and viscous friction is
negligible. It is observed for SCs both in elastic (e.g., in shape memory
alloys) and elastoplastic materials. Inclusion of the athermal dissipation in
the description of PT in elastic materials significantly and conceptually
complicates the description of the transformation process, similar to the
mechanics of the system with dry friction. Thus, the entire behavior becomes
loading-path dependent and should be treated incrementally. The thermodynamic
equilibrium does not correspond to the minimum of the Gibbs energy, and the
principle of stationary Gibbs energy cannot be applied for searching a two-
phase equilibrium microstructure. Such features are closer to those for PTs in
elastoplastic materials. Extremum principle substituting the principle of the
stationary Gibbs energy for SCs in elastic materials with athermal friction
were obtained in [240] as a particular case of the corresponding principle for
PTs in elastoplastic materials. Mathematic treatment of the problem for SCs in
elastic materials with athermal interfacial resistance was initiated in [347].

There are several sources of athermal interfacial friction $\,K_{\Sigma}\,$
[125]:

  1. 1.

Peierls barrier, which appears due to the discrete periodic structure of the
crystal lattice, similar to that for dislocations.

  2. 2.

Interaction of a moving interface with a long-range stress field of various
defects, e.g., point defects (solute and impurity atoms, vacancies),
dislocation forest, stacking faults, grain, subgrain, and twin boundaries, and
precipitates.

  3. 3.

Emission of acoustic waves.

When SC is considered in a finite volume, an additional contribution to the
athermal dissipation and SC hysteresis is related to a nucleation barrier. An
athermal threshold $K_{v}$ is determined as a minimum value of the driving
force at which SC occurs, i.e.,

$\displaystyle{X}_{v}\;=\;K_{v}=\;\int_{V_{n}}\rho\,{K}\;d\,V_{n}=K^{0}m_{n}.$
(25)

where $\,K\,$ is a locally determined athermal threshold per unit mass $m_{n}$
and $K^{0}$ is an athermal threshold averaged over a nucleus per unit mass. If
the volume is transformed by continuous interface propagation, then
$X_{\Sigma}=\rho\,{K}$.

By definition, the athermal threshold cannot be overcome by thermal
fluctuations. While the magnitude of $\,K\,$ can be different for direct and
reverse SC and based on a variety of the mechanisms, $K$ is expected to be
dependent on the entire deformation-SC process and the evolving material
microstructure. However, a simple relationship

$\displaystyle
K\,=\,\,L\,\sigma_{y}\left(\theta,\,\mbox{\boldsymbol{$\varepsilon$}}_{p},\,{\mbox{\boldsymbol{$g$}}}\right)\,\varepsilon_{o}$
(26)

was suggested in [238, 240] based on the comparison of some high-pressure
experiments summarized in [28] with the solution of the corresponding
boundary-value problems. Here $\,\sigma_{y}\,$ is the yield strength, which is
the function of the temperature $\theta$, plastic strain
$\mbox{\boldsymbol{$\varepsilon$}}_{p}$, and set of internal variables $g$,
$\,L\,$ is the proportionality factor, and $\varepsilon_{o}$ is the volumetric
transformation strain, which can be included in $L$. The values $L$ for some
PTs are determined in [238, 240]. Physically, the proportionality between
$\,\sigma_{y}\,$ and $\,K\,$ is caused by the fact that $\,\sigma_{y}\,$ and
$\,K\,$ characterize resistance to the dislocation and interface motion,
respectively, through the same material microstructure consisting of various
point, linear, planar, and bulk defects. For large plastic deformations, based
on the regularity revealed in [236], $\,\sigma_{y}\,$ and consequently $\,K\,$
reach their maximum and are independent of plastic strain and strain-history.
Since experiments in [28] were performed in 1980-1983, utilization of modern
diagnostic, like x-ray diffraction with synchrotron radiation, may change our
understanding of the effect of plastic strain on the athermal threshold.

For SMAs, the stress hysteresis and dissipation are proportional to $\,K\,$.
Since both the stress hysteresis and $\,\sigma_{y}\,$ linear depend on plastic
strain, as seen in experiments in [157, 411, 412, 425], this also supports Eq.
(26). A slightly different but close relationship between $K$ and the yield
strength of an austenite $\sigma_{y}^{A}$ for steel is suggested in [125, 126]
and presented in Eq. (64).

##  6 Theory of structural changes in inelastic materials with an unstable
intermediate state

For the description of SC in an elastic solid, the explicit equation for the
thermodynamic driving force for the interface propagation and corresponding
local thermodynamic equilibrium condition (Eqs. (19) and (21)) for each point
of a phase interface are known. They correspond to the stationary value of the
Gibbs energy (Eq. (12)). The activation energy for a critical nucleus Eq. (23)
is also determined in terms of the Gibbs energy. Furthermore, each solution of
the elastic problem (including cases where there are multiple solutions for
given boundary conditions) is independent of the process and depends on the
final boundary conditions.

For inelastic materials, such results are lacking. All processes in plasticity
are loading-history dependent and accompanied by energy dissipation, even for
infinitesimally slow processes. For this reason, a change in the Gibbs energy
alone does not define the driving force, and the stationarity of the Gibbs
energy does not determine the phase equilibrium or critical nucleus. A
conceptually different thermodynamic approach was required to determine the
thermodynamic driving force for nucleation and interface propagation, the
definition of the critical nucleus, the expression for the activation energy
for the critical nucleus, and the extremum principle for the determination of
all unknown parameters of the nucleus. Eq. (15) will be utilized as the main
hint for the formulation of the driving force for SC in inelastic materials:
it should have a sense of the dissipation increment due to SC only for the
appearance of a complete nucleus.

We will present our theory developed in [240, 243, 244] and [250, 257].
Initially [229, 232, 233, 238, 239], the nucleation condition was postulated
in the form of the dissipation increment due to PT only (excluding plastic and
other types of dissipation) reaching its experimentally determined value
related to the athermal threshold. Later in [234, 235, 240, 243, 244], a local
description of PTs was developed for better justification, which we will use
here.

###  6.1 Definition of the structural changes without a stable intermediate
state and local equations

For simplicity and transparency of the main ideas, we will start with the
geometrically linear formulation. The main local equations describing SC are
collected in Box 1.

Traditional additive kinematics (27) is utilized. It is convenient to
introduce a scalar parameter $\xi$ describing SC: SC starts at $\,\xi=0\,$ and
completes at $\,\xi=1\,$. The parameter $\,\xi\,$ is an internal variable
similar to a volume fraction of the product phase in the mixture theory or
order parameter in the PFA. The parameter $\xi$ can be determined by Eq. (28)
or a similar equation based on any other material property of the phases, such
as elastic moduli or entropy. For the Helmholtz free energy, we assume Eq.
(29), where $g$ is a set of internal variables describing plasticity, e.g.,
back stress or dislocation density. Traditional thermodynamic treatment
results in the constitutive equations (30) for stress and entropy, the yield
condition (33), and the evolution equations for plastic strain and internal
variables (34). It is assumed that $\xi$-dependence of all material properties
is known, e.g., based on linear interpolation between properties of two
phases, similar to the simplest mixture theory.

Note that in contrast to the PFA or mixture theory, we do not prescribe the
local evolution equation for $\xi$ describing the SC. We assume that the SC
process at each point of the transforming volume cannot be stopped in an
intermediate stage. Consequently, the material in each material point can only
be in either phase 1 or phase 2. Such SC is coined in [243, 244] as the SC
without a stable local intermediate state, and the following definition is
used.

The SC will be considered as a process of variation of the transformation
deformation gradient and some or all thermomechanical properties in an
infinitesimal or finite transforming volume from the initial to the final
value. This process cannot be stopped at an intermediate state in any
transforming point. Thermodynamic equilibrium for an intermediate value of the
transformation deformation gradient or material properties is impossible.

Such a definition excludes from consideration the Landau-Ginzburg model (see
Section 16.1), in which a smooth transition from phase 1 to phase 2 occurs due
to a nonlocal term, and any intermediate state can be stable inside a diffuse
interface of finite thickness. We will focus on the case with a sharp
interface and local constitutive equations describing the deformation in each
material point.

Box 1. Local equations describing phase transitions and plasticity [234, 235,
240, 243, 244]

1\. Additive decomposition of strain into elastic
$\mbox{\boldsymbol{$\varepsilon$}}_{e}$, plastic
$\mbox{\boldsymbol{$\varepsilon$}}_{p}$, and transformational
$\mbox{\boldsymbol{$\varepsilon$}}_{p}$ parts

$\displaystyle\mbox{\boldsymbol{$\varepsilon$}}\;:=(\mbox{\boldsymbol{$\nabla$}}\mbox{\boldsymbol{$u$}})_{s}=\;\mbox{\boldsymbol{$\varepsilon$}}_{e}\,+\,\mbox{\boldsymbol{$\varepsilon$}}_{p}\,+\,\mbox{\boldsymbol{$\varepsilon$}}_{t}\;.$
(27)

2\. Internal time $\xi$

$\displaystyle\xi\;:=\;\frac{\mid\mbox{\boldsymbol{$\varepsilon$}}_{t}-\mbox{\boldsymbol{$\varepsilon$}}_{t1}\mid}{\mid\mbox{\boldsymbol{$\varepsilon$}}_{t2}-\mbox{\boldsymbol{$\varepsilon$}}_{t1}\mid}\qquad
0\,\leq\,\xi\,\leq\,1.$ (28)

3\. Local constitutive equations

3.1. Helmholtz free energy per unit mass $\psi$, elastic $\psi^{e}$ and
thermal $\psi^{\theta}$ energies

$\displaystyle\psi=\psi\left(\mbox{\boldsymbol{$\varepsilon$}}_{e},\,\theta,\,{\mbox{\boldsymbol{$g$}}},\,\xi\right)=\psi^{e}\left(\mbox{\boldsymbol{$\varepsilon$}}_{e},\,\theta,\,\mbox{\boldsymbol{$\varepsilon$}}_{p},\,{\mbox{\boldsymbol{$g$}}},\,\xi\right)+\psi^{\theta}\left(\theta,\,\mbox{\boldsymbol{$\varepsilon$}}_{p},\,{\mbox{\boldsymbol{$g$}}},\,\xi\right).$
(29)

3.2. Elasticity rule and entropy-temperature relationship

$\displaystyle{\mbox{\boldsymbol{$\sigma$}}}\;=\rho\;\frac{\partial\,\psi\left(\mbox{\boldsymbol{$\varepsilon$}}_{e},\,\theta,\,\mbox{\boldsymbol{$\varepsilon$}}_{p},\,{\mbox{\boldsymbol{$g$}}},\,\xi\right)}{\partial\,\mbox{\boldsymbol{$\varepsilon$}}_{e}}\;;\qquad
s\;=\;-\,\frac{\partial\,\psi\left(\mbox{\boldsymbol{$\varepsilon$}}_{e},\,\theta,\,\mbox{\boldsymbol{$\varepsilon$}}_{p},\,{\mbox{\boldsymbol{$g$}}},\,\xi\right)}{\partial\,\theta}\;.$
(30)

3.3. Dissipation rate per unit mass due to plastic flow $D_{p}$ and variation
of the internal variable $D_{g}$

$\displaystyle
D_{p}:={\mbox{\boldsymbol{$X$}}}_{p}\;:\dot{\mbox{\boldsymbol{$\varepsilon$}}}_{p};\qquad
D_{g}:={\mbox{\boldsymbol{$X$}}}_{g}\;\cdot\dot{\mbox{\boldsymbol{$g$}}}^{T}.$
(31)

3.4. Dissipative forces for plastic flow ${\mbox{\boldsymbol{$X$}}}_{p}$ and
variation of the internal variable ${\mbox{\boldsymbol{$X$}}}_{g}$

$\displaystyle{\mbox{\boldsymbol{$X$}}}_{p}\;:=\;\frac{1}{\rho}{\mbox{\boldsymbol{$\sigma$}}}\,-\;\frac{\partial\,\psi}{\partial\,\mbox{\boldsymbol{$\varepsilon$}}_{p}};\qquad{\mbox{\boldsymbol{$X$}}}_{g}\;:=\;-\,\frac{\partial\,\psi}{\partial\,{\mbox{\boldsymbol{$g$}}}^{T}}\;.$
(32)

3.5. Yield condition

$\displaystyle
f\left({\mbox{\boldsymbol{$X$}}}_{p}\,,\,\theta,\,\mbox{\boldsymbol{$\varepsilon$}}_{p},\,{\mbox{\boldsymbol{$g$}}},\,\xi\right)=\,0\;.$
(33)

3.6. Evolution equations for plastic strain and internal variables

$\displaystyle\dot{\mbox{\boldsymbol{$\varepsilon$}}}_{p}\,=\,{\mbox{\boldsymbol{$f$}}}_{p}\left({\mbox{\boldsymbol{$X$}}}\,,\,\theta,\,\mbox{\boldsymbol{$\varepsilon$}}_{p},\,{\mbox{\boldsymbol{$g$}}},\,\xi\right)\,\quad{\rm
if}\quad f=\,0;\qquad{\rm
and}\quad\dot{f}>0;\qquad\dot{\mbox{\boldsymbol{$\varepsilon$}}}_{p}\,=0\quad{\rm
otherwise}.$
$\displaystyle\dot{\mbox{\boldsymbol{$g$}}}\;=\;{\mbox{\boldsymbol{$f$}}}_{g}\left({\mbox{\boldsymbol{$X$}}}_{g}\,,\,\theta,\,\mbox{\boldsymbol{$\varepsilon$}}_{p},\,{\mbox{\boldsymbol{$g$}}},\,\xi\right)\,.$
(34)

###  6.2 Thermodynamic driving forces for nucleation and interface propagation

Figure 4:  Schematics of the region $\,V\,$ of a material undergoing SCs with
a boundary $\,S\,$. The region $\,V_{n}\,$ with the boundary $\,\Sigma_{n}\,$
represents a new nucleus. SC within a volume covered by a phase interface
$\,\Sigma_{t}\,$ propagating with velocity $v_{n}$ during the time $\,\Delta
t\,$ is shown as well.

Our goal here is to generalize the thermodynamic driving forces for nucleation
and interface propagation derived for elastic materials (Eqs.(15) and (19))
for those in inelastic materials.

Problem Formulation. Let us consider some region $\,V\,$ of a material
undergoing SCs under the prescribed boundary conditions in tractions and
displacements at a boundary $\,S\,$ (Fig. 4). Nucleation of phase 2 will be
considered as the SC that starts and completes in some region $\,V_{n}\,$ with
the boundary $\,\Sigma_{n}\,$ during the nucleation time $\,t_{s}\,$. In
practice, we change the field of the parameter
$\,\xi\left({\mbox{\boldsymbol{$r$}}}\right)\,$ from 0 to 1 in the nucleus,
homogeneously or heterogeneously, which introduces into the nucleus the
transformation strain field
$\,\mbox{\boldsymbol{$\varepsilon$}}_{t}\,\left({\mbox{\boldsymbol{$r$}}}\,,\;\xi\right)\,$
and variation of all material properties from properties of phase 1 to
properties of phase 2— all without changing the boundary conditions. This
process represents the prescribed deformation process, for which the inelastic
boundary value problem and heat conduction equation should be solved
incrementally, producing fields $\sigma$ ,
$\;\mbox{\boldsymbol{$\varepsilon$}}_{p}\,$, $g$, $\;\theta\,$, and all other
participating fields. Note that depending on the goal, we will use a different
definition of a nucleus. In addition to the traditional critical nucleus, we
will also consider the appearance of a macroscopic region that is formed
during nucleation and growth processes. Having this information, one must
answer the following four questions:

1\. What is the thermodynamic driving force for the nucleation and phase
interface propagation?

2\. Is SC possible for the chosen boundary and initial conditions, i.e., what
is the SC criterion?

3\. How do we determine all the unknown parameters of a nucleus and the
transformation process in it, i.e. the position, volume, shape, and
orientation of a nucleus, its internal structure (i.e., actual field
$\,\mbox{\boldsymbol{$\varepsilon$}}_{t}\;({\mbox{\boldsymbol{$r$}}}\,,\;\xi)\,$?
All unknown parameters will be collectively designated as $b$ .

4\. If time-dependent kinetics is considered, what is the nucleation time
$t_{s}$?

The traditional thermodynamic procedure based on two thermodynamic laws has
been applied to find the thermodynamic driving forces presented in Box 2.
Similar to the local dissipation rates due to plastic flow $D_{p}$ and the
change in the internal variable $D_{g}$ (Eqs.(31) and (32)), a similar local
dissipation rate $\,{\cal D}_{\xi}\,$ and dissipative force $X_{\xi}$ for
structural changes Eq. (35) have been derived. One of the main assumptions is
that irreversible processes of plastic flow, a variation of internal variables
and SC are thermodynamically independent; they only interact through stress
fields. However, since we do not want to describe the kinetics of SC in terms
of $\dot{\xi}$, rather complete SC in the point (such as for the point of an
elastic nucleus), we introduce the local thermodynamics driving force for a
complete structural change $X$ by integrating the dissipation rate over the
entire SC in the point, see Eq. (36). This expression can be transformed into
Eq. (37), which is physically clear: the local dissipation increment $X$ due
to SC alone is equal to the total dissipation increment (the first three terms
in Eq. (37)) minus the dissipation increment due to all other dissipative
processes except SC, namely plastic flow and evolution of the internal
variable, which are the two last terms in Eq. (37). At $\,0<\xi<1\,$ a
nonequilibrium process takes place, which requires energy and stress
fluctuations. It is necessary to average the thermodynamic parameters, related
to SC, over the SC duration $\,t_{s}\,$ in order to filter off these
fluctuations, which results in consideration of the dissipation increment.

Since we want to find the thermodynamic driving force for nucleation in a
volume $V_{n}$, we integrate $X$ over the volume of a nucleus $V_{n}$ and add
a change in the surface energy, see Eq. (38). If the initial surface energy is
zero, then the change in the surface energy is just the surface energy of a
nucleus. Finally, the global dissipation rate for nucleation Eq. (40) is the
dissipation increment due to SC only, divided by nucleation time $t_{s}$;
thus, the generalized rate is $\dot{\chi}\,:=\;1\,/\,t_{s}\,$.

It is easy to show that an integral in Eq.(38) can be evaluated over an
arbitrary volume $\,v\,$ containing a single transforming region $\,V_{n}\,$
with SC, see Eq.(39). Indeed, integration over the region $\,v-V_{n}\,$
without SC gives zero contribution to the driving force $\,X_{v}\,$, which is
the dissipation rate due to SC only. The change in surface energy should be
evaluated over the nucleus surface $\,\Sigma_{n}\,$. Thus, instead of a
surface-independent Eshelby integral in the theory of defects and path-
independent J-integral in fracture mechanics, we introduced a region-
independent integral for arbitrary inelastic materials. In contrast to the
$\,\Gamma\,$-integral in [55, 56] and a region-dependent $\,T\,$-integral in
[11], the region-independent integral (39) separates the dissipation increment
due to SC only from other dissipation contributions and is the thermodynamic
driving force for SC. It also includes the temperature variation in the
process of SC.

It is shown in [240] that Eq. (38) for the thermodynamic driving force for
nucleation in a volume $V_{n}$ reduces for SC in elastic materials to the
negative increment of the Gibbs free energy Eq. (15), i.e., our approach is
noncontradictory.

Note that in [243, 244] a concept of SC without a stable intermediate state
for $\,0<\xi<1\,$ was introduced and used for justification that we should not
prescribe an evolution equation for
$\,\dot{\xi}\,=\,f_{\xi}\left(X_{\xi},\;\xi\right)\,\,$, like for other
internal variables. Since for any $\,0<\xi<1\,$ one can chose $X_{\xi}$ to
enforce $\,\dot{\xi}\,=\,0$ and arrest any $\,0<\xi<1\,$ (which contradicts
the definition that only states $\xi=0\,$ and $\xi=1\,$ should be in
thermodynamic equilibrium), this led to the conclusion that an evolution
equation for $\,\dot{\xi}\,$ should not be prescribed. However, thermodynamic
equilibrium for $\,0<\xi<1\,$ may be unstable (e.g., like in Landau theory,
see Section 16.1), and contradiction is seeming. The real reason for avoiding
an evolution equation for $\,\dot{\xi}\,$ is the desire to describe the
complete SC in some region $V_{n}$ and to generalize the approach, as
developed for elastic materials in Eq. (15), for an inelastic material.

Box 2. Dissipation rate due to SC and thermodynamic driving forces [263, 243,
244]

1\. Local dissipation rate $\,{\cal D}_{\xi}\,$ per unit mass and dissipative
force $X_{\xi}$ for structural changes

$\displaystyle{\cal D}_{\xi}\,:=X_{\xi}\dot{\xi};\qquad
X_{\xi}\,=\,\frac{1}{\rho}\,\mbox{\boldsymbol{$\sigma$}}\,{\mbox{\boldsymbol{$:$}}}\,\frac{\partial\mbox{\boldsymbol{$\varepsilon$}}_{t}}{\partial\,\xi}\,-\,\frac{\partial\,\psi}{\partial\,\xi}\,.$
(35)

2\. Local thermodynamics driving force per unit mass for a complete structural
change $X$

$\displaystyle X\,:=\int\limits^{t_{s}}_{0}{\cal
D}_{\xi}dt\;=\int\limits^{t_{s}}_{0}X_{\xi}\,\dot{\xi}dt\;=\int\limits^{1}_{0}X_{\xi}d\xi\,$
(36)

or

$\displaystyle\rho
X\,:=\int\limits^{\mbox{\boldsymbol{$\varepsilon$}}_{2}}_{\mbox{\boldsymbol{$\varepsilon$}}_{1}}{\mbox{\boldsymbol{$\sigma$}}}\,{\mbox{\boldsymbol{$:$}}}\,d\,\mbox{\boldsymbol{$\varepsilon$}}\,-\,\rho\left(\psi_{2}-\psi_{1}\right)-\int\limits^{t_{s}}_{0}\rho\left(s\,\dot{\theta}\,+\,{\mbox{\boldsymbol{$X$}}}_{p}\,{\mbox{\boldsymbol{$:$}}}\,\dot{\mbox{\boldsymbol{$\varepsilon$}}}_{p}+{\mbox{\boldsymbol{$X$}}}_{g}\,{\mbox{\boldsymbol{$:$}}}\,\dot{\mbox{\boldsymbol{$g$}}}\right)\,d\,t\;.$
(37)

3\. Global thermodynamic driving force $X_{v}$ for nucleation in a volume
$V_{n}$, i.e., the total dissipation increment due to SC only during the
complete SC in the transforming region

$\displaystyle X_{v}:=\bar{X}m_{n}\,=\int\limits_{V_{n}}\rho
Xd\,V_{n}-\Delta\,\int\limits_{\Sigma_{n}}\Gamma\;d\,\Sigma_{n}\,.$ (38)

4\. Global thermodynamic driving force for nucleation $X_{v}$ in terms of a
region-independent integral

$\displaystyle X_{v}:=\bar{X}m_{n}\,=\int\limits_{v}\rho
Xd\,v-\Delta\,\int\limits_{\Sigma_{n}}\Gamma\;d\,\Sigma_{n}\,.$ (39)

5\. Global dissipation rate $\,{\cal D}_{v}\,$ for nucleation in a volume
$V_{n}$

$\displaystyle{\cal D}_{v}\,=\;X_{v}\,\dot{\chi}\;\geq
0;\qquad\qquad\dot{\chi}\,:=\;1\,/\,t_{s}\;.$ (40)

6\. The thermodynamic driving force for a phase interface propagation

$\displaystyle
X_{\Sigma}:=\,\mbox{\boldsymbol{$\sigma$}}\,{\mbox{\boldsymbol{$:$}}}\,\left[\mbox{\boldsymbol{$\varepsilon$}}\right]\,-\rho\left[\psi\right]\,-\int\limits^{\theta_{2}}_{\theta_{1}}\,\rho\,s\;d\,\theta\,-\,\int\limits^{t+\Delta\,t}_{t}\rho\,\left({\mbox{\boldsymbol{$X$}}}_{p}\,{\mbox{\boldsymbol{$:$}}}\,\dot{\mbox{\boldsymbol{$\varepsilon$}}}_{p}\,+\,{\mbox{\boldsymbol{$X$}}}_{g}\,{\mbox{\boldsymbol{$:$}}}\,\dot{\mbox{\boldsymbol{$g$}}}\right)\,d\,t\,.$
(41)

7\. Local dissipation rate $\,{\cal D}_{\Sigma}\,$ per unit area for a phase
interface propagation

$\displaystyle{\cal D}_{\Sigma}\,:=\;X_{\Sigma}\,v_{n}\;\geq 0.$ (42)

If $\,\psi\,$ depends only on $\,\mbox{\boldsymbol{$\varepsilon$}}_{e}\,$ ,
$\;\theta\;$, and $\;\xi\,$, and if the elastic properties of phases are the
same (i.e.,
$\,\psi^{e}\,=\psi^{e}(\mbox{\boldsymbol{$\varepsilon$}}_{e}\,,\,\theta)$),
$\psi^{\theta}=\psi^{\theta}\left(\theta\right)$, surface energy is negligible
for isothermal approximation and homogeneous $\,\theta\,$ in the nucleus, then
Eqs. (37) and (38) result in the following very simplified expression for the
thermodynamic driving force for nucleation

$\displaystyle\bar{X}\,=\,\varphi\,-\,\;\Delta\,\psi^{\theta}\left(\theta\right)\qquad\qquad\varphi\,:=\;\frac{1}{m_{n}}\,\int_{V_{n}}\int^{\mbox{\boldsymbol{$\varepsilon$}}_{2}^{t}}_{\mbox{\boldsymbol{$\varepsilon$}}^{t}_{1}}{\mbox{\boldsymbol{$\sigma$}}}\,{\mbox{\boldsymbol{$:$}}}\,d\,\mbox{\boldsymbol{$\varepsilon$}}^{t}\;d\,V_{n}\;.$
(43)

It looks strange that Eq. (43) does not explicitly contain any plastic strain
and looks similar to that for an elastic material. However, stress variation
within the nucleus during the SC depends on the evolution of the plastic
strain field, which essentially affects the transformation work $\varphi$ and
$\bar{X}$.

Let us consider SC within a volume covered by a phase interface
$\,\Sigma_{t}\,$ propagating with velocity $v_{n}$ during the time $\,\Delta
t\,$ (Fig. 4). After some transformations that include universal conditions
(8)-(10) for a coherent interface, one arrives at Eqs.(41) and (42) that
determine the thermodynamic driving force for a phase interface propagation
and the corresponding dissipation rate. Note that for evaluation of the
integrals in Eq.(41) one has to perform the same procedure as for SC in a
nucleus $V_{n}$: produce $V_{n}$ by infinitesimally advancing an interface,
change $\xi$ incrementally from 0 to 1 in an infinitesimal layer $V_{n}$, and
solve the thermomechanical boundary-value problem for $V_{n}$. For such a
volume $V_{n}$, we obtain $\rho\bar{X}=X_{\Sigma}$. Examples for such
analytical solutions can be found in [238, 240, 243, 244] for small strain, in
[278, 279, 242] for large strains (Section 9), and for FEM solutions in [170,
263, 172, 266] for FEM solutions (Sections 11.2 and 15.2). The important point
is that for the isothermal process and neglected plastic dissipation, and
dissipation due to internal variable, Eq. (41) transforms into the Eshelby
driving force (20) for interface propagation in an elastic material.

###  6.3 Three types of kinetic descriptions

The following kinetic descriptions will be considered.

  * 1.

Athermal or rate-independent kinetics, for which real time and rate are
irrelevant. SC occurs in the chosen region instantaneously when SC criterion
is satisfied. To some extent, this idealization is similar to the rate-
independent plasticity.

  * 2.

Thermally-activated kinetics of the appearance of the critical nucleus,
similar to that in classical homogeneous nucleation theory. Like a classical
theory, for which the typical size of a critical nucleus is a few to tens of
nanometers, this theory is applicable at the nanoscale.

  * 3.

”Macroscale” thermally-activated kinetics, for which time for the appearance
of a macroscale region is postulated. The main assumptions here are quite
different than in classical nucleation theory.

All kinetic approaches should include some extremum principle to determine all
unknown parameters of a nucleus, like its position, shape, internal structure,
etc. They should substitute the principle of the stationary Gibbs energy (12)
and the extremum principle incorporated in the definition of the activated
energy (23).

###  6.4 Athermal kinetics

We consider the appearance of an arbitrary macroscopic region of phase 2 by
some nucleation and growth process. It is accepted in SC criterion (44) in Box
3 that SC in the chosen volume occurs when the thermodynamic driving force per
unit mass $\bar{X}$ is equal to the athermal threshold $\;K^{0}\,$. Note that
the same criterion is applied for SC in elastic materials.

SC condition (44) does not require fulfillment of any condition for each point
of the nucleus, i.e., it has nonlocal nature caused by surface energy.
However, for large volumes, the effect of surface energy is negligible, and an
integral SC condition without surface energy was applied in many papers,
including [393, 194, 110, 229, 232, 233, 238, 239]. This, in particular, means
that for $\,K=0\,$ the dissipation increment due to SC may be negative in some
points, both in elastic and inelastic materials.

Similar criteria (45) and (46) have to be satisfied for points of the
interface, both at time $t$ and $t+\Delta t$, where the subscript $\Delta$
denotes that a parameter is determined at time $\,t+\Delta\,t\,$. Two
equations for the propagating interface are a consequence of the lack of a
kinetic equation for interface propagation and the fact that SC occurs in the
infinitesimal volume covered by a moving interface during the time interval
$[t;t+\Delta\,t]$. Similarly, for time-independent plasticity, the yield
condition should be satisfied at time $t$ and $t+\Delta t$, which results in
the consistency condition in addition to the yield condition.

However, the SC criterion (44) is just one scalar equation that is not
sufficient for the determination of all unknown parameters $b$ (e.g., position
and shape of the nucleus, internal structure, and transformation path in the
nucleus or at the interface, etc.) among all possible parameters
${\mbox{\boldsymbol{$b$}}}^{*}$. To resolve this problem, the extremum
principles (47) and (48) were derived for the nucleus and propagating
interface. This was done using the postulate of realizability, see below. For
$K=0$ and elastic materials, principle (47) reduces to the principle of the
minimum of Gibbs energy. The extremum principle (48) is considered for time
$\,t+\Delta\,t\,$ only because for time $t$ it was met at the previous time
step.

###  6.5 The postulate of realizability and extremum principle for
determination of all unknown parameters for a nucleus and interface

In order to derive the extremum principle (47), the plausible assumption that
we called the postulate of realizability [230, 231, 240, 243, 244] was
formulated. For time-independent kinetics, it consists of two steps:

  * 1.

It is evident, that if strict inequality (47)1 is valid for all variable
parameters $\bf b^{*}$, then SC cannot occur, because the SC criterion (44) is
not met.

  * 2.

Let us vary boundary conditions and check the inequality (47) for all variable
parameters $\bf b^{*}$ for each of them. The main assumption is that if the SC
criterion is satisfied for the first time for some parameter $b$ and SC can
occur, this SC will occur.

The postulate of realizability is quite a natural assumption expressing the
stability concept. If some dissipative process (PT, fracture, plastic flow,
contact sliding, etc.) can occur from an energetic point of view, it must
occur, since various perturbations provoke the initiation of a process.
Recollecting Murphy’s law that ”anything that can go wrong will go wrong,” the
postulate of realizability can be considered its optimistic and thermodynamic
version.

Numerous applications of the postulate of realizability to derive constitutive
equations for plastic flow and plastic spin for anisotropic plasticity,
friction, nonlinear nonequilibrium thermodynamics, PTs, twinning, CRs,
fracture, as well as for stability analysis [263, 171, 230, 231, 240, 241,
243, 244, 286, 278, 279, 264, 265, 266] lead to the impression that this
postulate catches a general essential property of dissipative systems.
Mathematical treatment of the extremum principle for PT in elastic materials
with athermal friction that follows from the postulate of realizability was
performed in [347]. This paper has initiated significant mathematical
literature on the study of the rate-independent and hysteretic systems,
including PT in ferroelastic, ferromagnetic, ferroelectric, and multiferroic
materials (e.g., SMA), elastoplasticity, damage, crack propagations, friction,
delamination, etc., which can be found from citations of the paper [347].

The formulation based on SC condition (44) and extremum principle (47) is
consistent in the limiting case of elastic materials and $K=0$ with the
classical description based on the principle of the stationary or minimum of
the Gibbs free energy (12).

When simplified expression (43) is used for the driving force for nucleation,
extremum principle (47) results in the maximum of transformation work
$\,\varphi\,$. Even in this simplest case, the SC criterion (44) includes the
history of stress variation $\sigma$ in the nucleus during SC, i.e., we cannot
define the SC condition using only the initial stresses before SC. We have to
solve the elastoplastic problem and determine the variation of stresses in the
nucleus during SC in order to calculate the transformation work $\,\varphi\,$
in Eq. (43).

There is a major problem in the application of the time-independent kinetics
for homogeneous materials under prescribed stresses. The minimum of Gibbs free
energy for elastic materials with $K=0$ and the maximum driving force for
elastoplastic materials may be reached when the entire volume transforms
homogenously after meeting the SC criterion $\bar{X}=K$. Indeed, as a
corroborating argument, if the energy of the external surface does not change
during PT, then the negative surface term disappears from $\bar{X}$. Also, the
negative contribution to $\bar{X}$ due to the positive energy of internal
stresses during nucleation disappears as well. Our statement can be easily
proved for the nucleation of a spherical product phase within a parent sphere.

However, homogenous PT in a large volume is unphysical (with some exceptions
[280, 258, 12, 13], some which are discussed in Sections 13, 17.2) because of
local barriers, which we filtered out while integrating $X$ over $\xi$ from 0
to 1. That is why the nucleus of some not-strictly-justified size is
considered in analytical or numerical solutions, in many cases equal to a
single finite element [263], region [264, 266], band [170, 263, 454]. In some
cases, plasticity arrests PTs [266], see also Section 15.2. For heterogeneous
boundary conditions or fields in bulk, or prescribed displacements, which
control the amount or transformed phase via total transformation strain, the
size of the nucleus is determined via SC conditions and extremum principle.

Generally, only the kinetic treatment, due to its accounting for the
activation barrier and energy, results in the finite size of the nucleus. It
will be described in Sections 6.7 and 6.8.

###  6.6 Global criterion for structural changes based on stability analyses

Time-independent problem formulation simplifies a solution, but nothing comes
for free. Indeed, it leads in some cases to the necessity of some additional
principles. In particular, for some problems under the given increment of
boundary conditions the SC criterion and extremum principle (47) allow several
different solutions, e.g., nucleation in different places or propagation of
different interfaces. In most cases, at least two following solutions are
possible: (a) the solution without SC (since all equations of continuum
mechanics can be satisfied without SC as well) and (b) the solution with SC.
That is, it is possible that SC will not occur even when the SC criterion and
extremum principle (47) are met. Such a problem was first revealed in [230,
231]. It was proposed to use the stability consideration to choose the unique
solution among all possible solutions. Remarkably that the postulate of
realizability was utilized again to formulate the stability criterion and
corresponding extremum principle. Since the general extremum principle [230,
231] is too bulky, we will utilize its simplified version (50) either for the
prescribed displacements $u$ or traction vector $p$ at the external boundary
$\,S\,$.

It follows from the principles (50) or (51) that the stable solution minimizes
the work of external stresses for prescribed displacements and maximizes the
work of external stresses at given tractions. Thus, the fulfillment of the SC
criterion and extremum principle (47) is not sufficient for the occurrence of
SC, and only the extremum principle (Eq. (50) or (51))— called the global SC
criterion—offers the final solution. Application of stability analysis to
strain-induced nucleation at a shear-band intersection can be found in [264]
and in Section 14 and in [172] and in Section 19.2 for competition between PT
and fracture.

Box 3. Athermal kinetics of SCs [263, 243, 244]

1\. Thermodynamic SC criterion for a nucleus

$\displaystyle\bar{X}\;=\;K^{0}\;.$ (44)

2\. Thermodynamic SC criterion for interface propagation

2a. At time $t$

$\displaystyle{X}_{\Sigma}\;=\;K_{\Sigma}\;.$ (45)

2b. At time $t+\Delta t$

$\displaystyle{X}_{\Sigma\Delta}\;=\;K_{\Sigma\Delta}\;.$ (46)

3\. Extremum principle for determination of all unknown parameters $b$ among
all possible ${\mbox{\boldsymbol{$b$}}}^{*}$

3a. For a nucleus

$\displaystyle\bar{X}\,\left({\mbox{\boldsymbol{$b$}}}^{*}\right)-\,K^{0}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\,<\;0\;=\;\bar{X}\,\left({\mbox{\boldsymbol{$b$}}}\right)-\,K^{0}\left({\mbox{\boldsymbol{$b$}}}\right)\,.$
(47)

3a. For an interface

$\displaystyle
X_{\Sigma\Delta}\,\left({\mbox{\boldsymbol{$b$}}}^{*}\right)-\,K_{\Sigma\Delta}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\,<\;0\;=\;X_{\Sigma\Delta}\,\left({\mbox{\boldsymbol{$b$}}}\right)-\,K_{\Sigma\Delta}\left({\mbox{\boldsymbol{$b$}}}\right)\,.$
(48)

4\. Athermal threshold $\;K^{0}\,$

$\displaystyle
K^{0}\,=\,\,L\,\sigma_{y}\left(\theta,\,\mbox{\boldsymbol{$\varepsilon$}}_{p},\,{\mbox{\boldsymbol{$g$}}}\right)\,\varepsilon_{o}\;.$
(49)

5\. Extremum principle for choosing the stable solution, i.e., the global SC
criterion

$\displaystyle\int\limits_{S}\int\limits^{{\mbox{\boldsymbol{$u$}}}_{2}}_{{\mbox{\boldsymbol{$u$}}}_{1}}{\mbox{\boldsymbol{$p$}}}\,{\mbox{\boldsymbol{$\cdot$}}}\,d\,{\mbox{\boldsymbol{$u$}}}\;d\,S\,\Rightarrow\;{\sl
min}\quad\mbox{(for prescribed }\;{\mbox{\boldsymbol{$u$}}}\;\mbox{ at the
external surface }\;S\,)\,.$ (50)
$\displaystyle\int\limits_{S}\int\limits^{{\mbox{\boldsymbol{$u$}}}_{2}}_{{\mbox{\boldsymbol{$u$}}}_{1}}{\mbox{\boldsymbol{$p$}}}\,{\mbox{\boldsymbol{$\cdot$}}}\,d\,{\mbox{\boldsymbol{$u$}}}\;d\,S\,\Rightarrow\;{\sl
max}\quad\mbox{(for prescribed }\;{\mbox{\boldsymbol{$p$}}}\;\mbox{ at the
external surface }\;S\,)\,.$ (51)

Let us summarize the main steps for the solution of the boundary-value
problems for the athermal kinetics based on equations in Boxes 1-3. The
initial stages of the solution are the same as described in the problem
formulation in Section 6.2. We consider a body under prescribed boundary
conditions that do not change during the nucleation event (Fig. 4). We choose
a potential nucleation region $V_{n}$, changing the field of the parameter
$\,\xi\left({\mbox{\boldsymbol{$r$}}}\right)\,$ from 0 to 1 in it
homogeneously or heterogeneously, which introduces into a nucleus the
transformation strain field
$\,\mbox{\boldsymbol{$\varepsilon$}}_{t}\,\left({\mbox{\boldsymbol{$r$}}}\,,\;\xi\right)\,$
and variation of all material properties from properties of the parent phase 1
to properties of the product phase 2. For such a prescribed deformation
process in $V_{n}$, the inelastic boundary value problem and heat conduction
equation should be solved incrementally, producing fields $\sigma$ ,
$\;\mbox{\boldsymbol{$\varepsilon$}}_{p}\,$, $g$, $\;\theta\,$, and all other
participating fields. Then we calculate the driving force $\,\bar{X}\,$
(Eq.(38)) and athermal resistance $\,K^{0}\,$ (Eq.(49)). Next, we vary the
possible SC region and way of variation of the transformation strain and
properties from initial to final values in it and find such a PT region and
way of varying transformation strain and properties, which maximizes the net
driving force $\,\bar{X}\,-\,K^{0}\,$. It is important for the athermal
kinetics to begin solving for the boundary conditions for which
$\,\bar{X}\,-\,K^{0}<0\,$, i.e., the SC is impossible. Then we change boundary
conditions incrementally to increase the net driving force until we find
boundary conditions for which $\,\bar{X}\,-\,K^{0}=0\,$ for the first time for
the ”optimized” nucleus. Then we compare the obtained solution with the
solution without the nucleus for the same load increment (or with other
solutions with SCs, if they exist) and find out using extremum principle (50),
which solution is more stable. If with the nucleus, we keep this nucleus in
the transformed state with all corresponding fields and use as the initial
conditions for the next loading increment, for which we repeat the same
procedure. If without a nucleus, then we use the solution without the nucleus
as the initial conditions for the next loading increment, for which we repeat
the same procedure. We do the same for all following load increments.
Interface propagation does not require special treatment. If under a given
load increment, the next transformed volume is in touch with the previous one,
then the interface propagation occurs; otherwise, this is the appearance of a
new nucleus. For interface propagation, $K_{\Sigma}$ should be used instead of
$K^{0}$, but in most cases, this difference is neglected.

###  6.7 Thermally-activated kinetics

In the equations presented in Box 4, we follow Eqs. (22)-(24) as much as
possible for SC in elastic materials. The main problem is that since the
change in the Gibbs energy is not the thermodynamic driving force for SC, it
cannot be used for the definition of the activation energy. Since according to
Eqs. (38) and (47) $(\bar{X}-K^{0})m_{n}$ is the net thermodynamic driving
force for SC, it is utilized in the definition (53). Also, Eq. (53) includes
additional minimization with respect to the transformation path along which a
critical nucleus appears. For example, one can consider a homogeneous SC
process in the entire critical nucleus of a fixed radius, or the formation of
a critical nucleus by the motion of a sharp interface from zero to the
critical radius, or a homogenous SC process in the region of the smallest
size, which can be treated as a nucleus within a continuum approach and then
grows to the critical size by interface propagation, or any general scenario
involving heterogeneous fields [250, 257]. For SC in elastic materials, the
transformation path is irrelevant for the definition of $\,Q\,$ because
$\Delta G_{n}$ is path-independent. This is not the case for nucleation in an
inelastic material, for which stress variation depends on the entire
transformation process, and so does $\bar{X}$.

This fact has one more consequence. In elasticity, if a supercritical nucleus
appears, it will grow because the Gibbs potential reduces and the dissipation
rate is positive during growth. However, due to the path-dependence of plastic
solutions, one cannot say whether the supercritical nucleus will grow,
disappear, or remain unchanged after nucleation. Thus, an additional condition
concerning growth after appearance should be checked.

Due to the path-dependence, the driving force for interface propagation for
forward $X_{\Sigma}^{f}$ and reverse $X_{\Sigma}^{r}$ PT can be different. The
driving force for reverse SC is defined in the same way as for forward SC, but
when $\xi$ varies from 1 to 0. The kinetic equation for the interface
propagation may be based on experiments, atomistic simulations, or some
dislocation models [134, 133, 372, 144, 125, 126]. The following scenarios are
presented in item 5 of Box 4.

The cases 5.1 and 5.2 are similar to PT in elastic materials, i.e., the
supercritical nucleus will grow, and the subcritical nucleus will shrink.
Other cases do not have the counterparts for PTs in elastic materials. For
case 5.3, the evolution of the nucleus is determined by the sign of the
resultant interface velocity, where $f^{f}(X_{\Sigma}^{f})$ and
$f^{r}(X_{\Sigma}^{r})$ are the kinetic equations for forward and reverse PTs,
respectively. Case 5.4 describes the nucleus that cannot evolve, i.e., it is
stable rather than the critical nucleus. Finally, in 5.5, a gaseous nucleus
may grow independent of the interfacial driving force due to loss of
mechanical stability if the pressure in the gas exceeds the resistance of the
material $-\sigma_{n}$ to plastic expansion and surface stress. For gaseous or
any other hydrostatic media, athermal resistance is zero.

The main steps for the solution of the boundary-value problems for the
thermally activated kinetics are similar to those for athermal kinetics with
the following difference. The actual nucleus is chosen by the minimization of
activation energy instead of the net driving force, and the nucleation time is
defined. In addition, growth condition are checked to choose the next
transforming region. The growth conditions are based on parameters at time
$t+\Delta t$, i.e., again, the next transformed volume is checked like for the
athermal kinetics.

The examples of the application of equation in Box 4 for SC in an inelastic
material are given in [250, 257] for sublimation (and melting and chemical
reaction), i.e. when a nucleus is a hydrostatic medium, and in [247, 246] for
nucleation of a high-pressure solid phase in a low-pressure solid phase.

Note that the typical size of a critical nucleus determined by the nucleation
criterion (54) is in the range of a few to few tens of nanometers. The effect
of surface energy is the leading one for such sizes.

Box 4. Thermally-activated kinetics [250]

1\. The Arrhenius-type kinetic equation for nucleation time

$\displaystyle t_{s}\,=\,t_{0}\,\exp\left(\;-\;\frac{Q}{k\,\theta}\right).$
(52)

2\. Activation energy for the appearance of a critical nucleus

$\displaystyle
Q=\max_{{m_{0n}}}\min_{{shape}}\min_{{position}}\min_{{path}}\min_{{structure}}(-(\bar{X}-K^{0})m_{n}).$
(53)

3\. Criterion for thermally activated nucleation

$\displaystyle Q=\beta k\theta,\qquad\beta=40-80.$ (54)

4\. Interface propagation kinetics

$\displaystyle v_{n}^{f}=f^{f}(X_{\Sigma}^{f}-K_{\Sigma}^{f})\qquad{\rm
for}\qquad X_{\Sigma}^{f}>K_{\Sigma}^{f};$ $\displaystyle
v_{n}^{r}=f^{f}(X_{\Sigma}^{r}-K_{\Sigma}^{r})\qquad{\rm for}\qquad
X_{\Sigma}^{r}>K_{\Sigma}^{r}.$ (55)

5\. Growth conditions for a critical nucleus

$\displaystyle{\bf 5.1.}\;X_{\Sigma\Delta}^{f}>K_{\Sigma\Delta}^{f}\qquad{\rm
and}\qquad X_{\Sigma\Delta}^{r}<K_{\Sigma\Delta}^{r}\quad\Rightarrow\quad{\rm
growth\;(supercritical\;nucleus)}$ (56) $\displaystyle{\bf
5.2.}\;X_{\Sigma\Delta}^{f}<K_{\Sigma\Delta}^{f}\qquad{\rm and}\qquad
X_{\Sigma\Delta}^{r}>K_{\Sigma\Delta}^{r}\quad\Rightarrow\quad{\rm
shrinking\;(subcritical\;nucleus)}$ (57) $\displaystyle{\bf 5.3.}$
$\displaystyle X_{\Sigma\Delta}^{f}\geq K_{\Sigma\Delta}^{f}\;\;{\rm
and}\;\;X_{\Sigma\Delta}^{r}\geq K_{\Sigma\Delta}^{r}\;\;\Rightarrow\;\;$
$\displaystyle
v_{res}=f^{f}(X_{\Sigma^{f}\Delta}-K_{\Sigma\Delta}^{f})-f^{r}(X_{\Sigma^{r}\Delta}-K_{\Sigma\Delta}^{r})\quad\Rightarrow\quad{\rm
competing\;kinetics}\ $ $\displaystyle{\bf 5.4.}\;X_{\Sigma\Delta}^{f}\leq
K_{\Sigma\Delta}^{f}\qquad{\rm and}\qquad X_{\Sigma\Delta}^{r}\leq
K_{\Sigma\Delta}^{r}\quad\Rightarrow\quad{\rm
stable\;(rather\;than\;critical)\;nucleus}$ (59) $\displaystyle{\bf
5.5.}\;p_{g}>-\sigma_{n}+2\Gamma_{2}/R\qquad\Rightarrow\quad{\rm
growth\;of\;a\;gaseous\;nucleus\;by\;mechanical\;instability\;}$ (60)

###  6.8  ”Macroscale” thermally-activated kinetics for structural changes

As it is shown in [243, 244] and Section 6.5, the time-independent model can
lead to some contradictions. That is why development of phenomenological time-
dependent kinetics is necessary. It is not related to the appearance of a
critical nucleus, for which size and energy (i.e., activation energy) can be
calculated. We consider plausible phenomenological kinetics for the time of
appearance of an arbitrary region of a product phase of volume $V_{n}$ or mass
$m_{n}$, because in irreversible thermodynamics, the kinetic equation between
rate and force
$\;\dot{\chi}\,=1/t_{s}=\,f\left(X_{v}\,,\,{\mbox{\boldsymbol{$.$}}}{\mbox{\boldsymbol{$.$}}}{\mbox{\boldsymbol{$.$}}}\right)\,$
has to be formulated. We coin this region as a macroscale nucleus. Motivated
by the approach in Section 6.7, we consider in Box 5 a size-dependent
Arrhenius-type kinetics, which includes both thermal activation and an
athermal threshold $\,K^{0}\,$, see Eq.(62). Here, $\,E_{a}\,$ is the
activation energy per unit mass at $\,\bar{X}-K^{0}\,=\,0\,$, which is an
experimentally-determined parameter (in contrast to Eq.(53) for traditional
thermal activation), $\,N\,=\,6.02\cdot 10^{23}\,$ is Avogadro’s number
(number of atoms in 1 mol), $\;R\,=\,8.314\,$ ${\sl J}/\left({\sl
K\;mol}\right)$ is the gas constant, and $\,n\,$ is the number of atoms in the
volume which undergoes thermal fluctuations during the entire macroscopic
nucleation process (an experimentally fitting parameter). The actual
activation energy per unit mass
$\;\bar{E}_{a}\,:=\,E_{a}\,-\,\bar{X}\,+\,K^{0}\geq\,0$, otherwise, there is
no need for thermal fluctuations, and barrierless nucleation occurs. The
characteristic time $\,t_{0}\,$ has a meaning of a nucleation time for
$\bar{E}_{a}\,=0$. Since $k=R/N$, without $n$ using the same arguments as for
formulating Eq.(24), we obtain that $\;\bar{E}_{a}m_{n}\leq(40-80)k\theta$,
i.e., the nucleus size would be similar to the size of the critical nucleus,
i.e., few to tens of nanometers. The effective temperature $\,\theta_{ef}\,$
takes into account that the temperature may vary significantly during the SC,
e.g., during CR [278, 279]. As the simplest assumption, we define the
effective temperature either as a temperature averaged over the transformation
process and transforming volume or use a similar definition in terms of
inverse temperature. As a nucleation criterion, we accept that the nucleation
time is shorter or equal to the accepted observation time $t_{ob}$. With the
help of the postulate of realizability [230, 231, 243, 244], the principle of
the minimum of transformation time, Eq.(63), (or the maximum of transformation
rate) is obtained.

We used a specific model for the interface velocity kinetics developed in
[125, 126, 372] and combined it with our continuum thermodynamic treatment in
[266]. It is taken into account that the athermal threshold $K^{0}$ in Eq.
(64) consists of two parts due to solute atoms $\,K_{\mu}\,$ and dislocation
forest hardening $K_{d}=B[\sigma_{y}^{A}(\bar{q})-\sigma_{y}^{A}(0)]$, where
$B$ is the proportionality factor, $\bar{q}$ is the accumulated plastic strain
averaged over the small volume covered by a moving interface during time
$\Delta t$, $\sigma_{y}^{A}(\bar{q})$ is the plastic strain dependence of the
yield strength of the austenite, and $A$ is a parameter in this dependence.
Dependence of $K^{0}$ on $\bar{q}$ is consistent with the relationship (49)
and dependence of the yield strength on $q$ for a specific steel in [125,
126]. In the kinetic equation for the interface propagation (65)
$\,v_{n\,0}\,$ is the characteristic velocity on the order of the shear-wave
velocity, $\,Q_{0}\,$ is the activation energy, $\,W_{0}\,$ is the height of
the barrier above which thermal fluctuations are not required, and $\,p\,$ and
$\,b\,$ are constants. All material parameters estimated for the alloy Fe \-
22.31 Ni \- 2.888 Mn are presented in [125, 126, 266], and the application of
the kinetics for the lath martensite growth is presented in [266] and Section
15.2.

Box 5. ”Macroscale” thermally-activated kinetics for SCs [243, 244, 266]

1\. Thermodynamic SC criterion for a macroscale nucleus

$\displaystyle\bar{X}\,\geq\,K^{0}\,.$ (61)

2\. Arrhenius-type kinetic equation and nucleation criterion

$\displaystyle
t_{s}\,=\,t_{0}\,\exp\left(\;-\;\frac{\left({\bar{X}\,-\,K^{0}}\,-\,E_{a}\right)m_{\,n}}{R\,\theta_{ef}}\;\frac{N}{n}\;\right)\leq
t_{ob}\qquad\mbox{at}\qquad 0\,\leq\,\bar{X}\,-\,K^{0}\,\leq\,E_{a}\,$ (62)
$\displaystyle t_{s}\,=\,\infty\qquad\mbox{at}\qquad\,\bar{X}\,\leq\,K^{0}\,.$

3\. Principle of the minimum of transformation time

$\displaystyle
t_{s}\;=\;t_{0}\,\exp\;-\;\frac{\left(\bar{X}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\,-\,K^{0}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)-\,E_{a}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\right)m_{n}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)}{R\,\theta_{ef}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)}\;\frac{N}{n}\quad\longrightarrow\;\;{\sl
min}\;\,.$ (63)

4\. Interface propagation criterion

$\displaystyle\bar{X}\geq
K^{0}=K_{\mu}+K_{d}=K_{\mu}+B[\sigma_{y}^{A}(\bar{q})-\sigma_{y}^{A}(0)]=K_{\mu}+A\bar{q}^{0.5}.$
(64)

5\. Kinetic equation for an interface propagation

$\displaystyle v_{n}\,=\,v_{n\,0}\,\mbox{\sl
exp}\left[-\,\frac{Q_{0}}{k\,\theta}\left(1-\left(\frac{\bar{X}-K^{0}}{W_{0}}\right)^{p}\;\right)^{b}\right]\quad{\rm
for}\quad 0\leq\bar{X}-K^{0}\leq W_{0}.$ (65)

A straightforward way to reduce the transformation time is to reduce the mass
(i.e., size) of the nucleus as much as possible. However, the increasing
contribution of the surface energy to $\bar{X}$ will then lead to a violation
of the SC criterion (61). In this case, the minimum nucleus size will be
determined by thermodynamic ”static” constraint
$\;\,\bar{X}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)-\,K^{0}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)=\,0\;$,
which should be explicitly imposed. That is why we called this nucleus a
thermodynamically admissible nucleus. Corresponding equations are presented in
Box 6. The expression for the transformation time (67) and for the principle
of the minimum of the transformation time (68) are becoming significantly
simpler. For mutually independent $\,E_{a}\,$, $\,m_{n}\,$ and
$\,\theta_{ef}\,$, the principle of the minimum of the transformation time
results in three principles, namely in the principle of the minimum of
transforming mass, the minimum of activation energy per unit mass, and the
maximum of effective temperature. Since $\,E_{a}\,$ is a fitting constant
independent of ${\mbox{\boldsymbol{$b$}}}^{*}$, and if the temperature
variation is neglected, then the main principle is the principle of the
minimum of transforming mass.

Box 6. ”Macroscale” thermally-activated kinetics for SCs when the minimum in
principle (63) violates the thermodynamic criterion of SC (61), and it is
included as a constraint [243, 244]

1\. Thermodynamic ”static” SC criterion for a macroscale nucleus

$\displaystyle\bar{X}\,=\,K^{0}\,.$ (66)

2\. Arrhenius-type kinetic equation and nucleation criterion for a
thermodynamically admissible nucleus satisfying criterion (66)

$\displaystyle
t_{s}\,=\,t_{0}\,\exp\left(\frac{\,E_{a}m_{n}}{R\,\theta_{ef}}\;\frac{N}{n}\;\right)\leq
t_{ob}.$ (67)

3\. Principle of the minimum of transformation time

$\displaystyle
t_{s}=t_{0}\,\exp\frac{E_{a}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)m_{n}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)}{R\,\theta_{ef}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)}\frac{N}{n}\quad\longrightarrow\;\;{\sl
min}\quad\Rightarrow\quad\frac{E_{a}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\,m_{n}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)}{\theta_{ef}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)}\;\rightarrow\;{\sl
min}.$ (68)

4\. For mutually independent $\,E_{a}\,$, $\,m_{n}\,$ and $\,\theta_{ef}\,$

$\displaystyle
E_{a}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\;\rightarrow\;{\sl
min}\;;\qquad\qquad
m_{n}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\;\rightarrow\;{\sl
min}\;;\qquad\qquad\theta_{ef}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\;\rightarrow\;{\sl
max}\;.$ (69)

For homogenous fields within a macroscale nucleus, the formulated equations
are further elaborated in Box 7. Here, we consider a nucleus of arbitrary
shape with a surface area $\Sigma_{n}$ and characteristic size
$\,\displaystyle{\bar{r}\,=\;\frac{V_{n}}{\Sigma_{n}}}\;$. The thermodynamic
SC criterion for a macroscale nucleus (70) (which is equality, like for time-
independent kinetics) allows us to introduce the new concept of a
thermodynamically admissible nucleus  (see also Fig. 3), for which
characteristic size is determined by Eq.(71). The size of a thermodynamically
admissible nucleus is larger than the size of a critical nucleus, which has an
equal probability of growing and disappearing. It may be qualitatively related
to the concept of the operational nucleus in [372], which reached the size
required for fast growth. However, as it is described in Box 4, due to the
path-dependence plastic flow theory and the driving forces for nucleation and
growth, there is no guarantee that our thermodynamically admissible nucleus
will grow; this should be checked for each specific case.

Eq. (70) allows us to elaborate Eq.(72) for the time of SC. While applying the
principle of the minimum of transformation time (73), we, for simplicity,
assume constant temperature, $K^{0}$, and $E_{a}$. This principle then results
in the principle of minimum volume (or mass) of the nucleus.

Box 7. ”Macroscale” thermally-activated kinetics for SCs in Box 6 for a
thermodynamically admissible nucleus for homogeneous fields [243, 244]

1\. Thermodynamic ”static” SC criterion for a nucleus

$\displaystyle\left(\,X\,-\,K^{0}\,\right)\,\rho\,\Sigma_{n}\,\bar{r}\,=\,\Delta\Gamma\,\Sigma_{n}\quad\Rightarrow\left(\,X\,-\,K^{0}\,\right)\,\rho\,\bar{r}\,=\,\Delta\Gamma\,.$
(70)

2\. The characteristic size of a thermodynamically admissible nucleus

$\displaystyle\bar{r}=\frac{\Delta\Gamma}{\rho\left(\,X\,-\,K^{0}\,\right)}\,.$
(71)

3\. Arrhenius-type kinetic equation and nucleation criterion for a
thermodynamically admissible nucleus satisfying criterion (70)

$\displaystyle t_{s}=t_{o}{\sl
exp}\;\left(\frac{E_{a}\rho\Sigma_{n}\bar{r}}{R\theta_{ef}}\frac{N}{n}\right)=t_{o}{\sl
exp}\;\left(\frac{E_{a}\Delta\Gamma\Sigma_{n}}{\left(\,X\,-\,K^{0}\,\right)R\theta_{ef}}\frac{N}{n}\right)\leq
t_{ob}.$ (72)

4\. Principle of the minimum of transformation time

$\displaystyle t_{s}=t_{o}{\sl
exp}\left(\frac{E_{a}\rho\Sigma_{n}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\bar{r}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)}{R\theta_{ef}}\frac{N}{n}\right)=t_{o}{\sl
exp}\left(\frac{E_{a}\Delta\Gamma\Sigma_{n}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)}{\left(\,X\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\,-\,K^{0}\,\right)R\theta_{ef}}\frac{N}{n}\right)\longrightarrow{\sl
min}\quad\Rightarrow$
$\displaystyle\Sigma_{n}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\bar{r}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\longrightarrow{\sl
min}.$ (73)

When a change in surface energy $\Delta\Gamma$ is very small or even zero,
according to Eq.(71), it is possible that the transforming mass is becoming
smaller than the mass of a single atom or molecule or the mass of $\,n\,$
atoms, which undergo thermal fluctuations. This should be avoided by the
constraint (74), which changes the equations in Box 5 to those in Box 8. It is
taken into account in Eqs. (76) and (77) where $\,k\,=\,\frac{R}{N}$ and the
mass of the nucleus is fixed.

Box 8. ”Macroscale” thermally-activated kinetics for SCs in Box 5 when a mass
of a nucleus is smaller than mass of $n$ atoms $nm_{a}$ [243, 244]

1\. The constraint for a mass of a nucleus

$\displaystyle m_{n}=nm_{a}.$ (74)

2\. Thermodynamic SC criterion for a macroscale nucleus

$\displaystyle\bar{X}\,\geq\,K^{0}\,.$ (75)

3\. Arrhenius-type kinetic equation

$\displaystyle
t_{s}\,=\,t_{0}\,\exp\left(\;-\;\frac{\left({\bar{X}\,-\,K^{0}}\,-\,E_{a}\right)m_{\,a}}{k\,\theta_{ef}}\;\;\right)\qquad\mbox{at}\qquad
0\,\leq\,\bar{X}\,-\,K^{0}\,\leq\,E_{a}\,$ (76) $\displaystyle
t_{s}\,=\,\infty\qquad\mbox{at}\qquad\,\bar{X}\,\leq\,K^{0}\,.$

4\. Principle of the minimum of transformation time

$\displaystyle
t_{s}\;=\;t_{0}\,\exp\;-\;\frac{\left(\bar{X}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\,-\,K^{0}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)-\,E_{a}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)\right)m_{a}}{k\,\theta_{ef}\left({\mbox{\boldsymbol{$b$}}}^{*}\right)}\;\quad\longrightarrow\;\;{\sl
min}\;\,.$ (77)

A general scheme for the application of SC criterion and the principle of the
minimum of transformation time for the macroscale nucleation kinetics is
similar to that for thermally activated kinetics. For the interface
propagation, a small shift of the interface is produced at each step and the
volume covered as an interface is considered as a new transformed region.
After calculating $X$ and $K^{0}$ and the maximization of $X-K^{0}$ with
respect to internal structure (if any), interface velocity is calculated using
Eq.(65).

###  6.9 Comparison with alternative approaches

Since we are not aware of any previous work on CRs in plastic materials within
a framework of the materials without an intermediate stable state but papers
by [243, 244, 278, 279], we will focus on the PTs only. However, as it is
demonstrated in [243, 244, 278, 279], formal continuum theory for PTs and CRs
is practically the same.

####  6.9.1 ”Macroscale nucleation”

As it is usual in physical literature, instead of a general theory, analytical
solutions for some tractable model problems were found first. Melting of a
small spherical particle in an elastoplastic space was presented in the first
publication [312] on the topic. The appearance of the spherical nucleus in a
finite-size sphere under external pressure was treated in [393]. The
ellipsoidal nucleus in an infinite space with the stress-free boundary was
analyzed in [194]. The shape of an inclusion corresponding to minimum energy
losses during growth was found, much like the counterpart of a ”critical
macroscopic nucleus,” while surface energy was not included in this and most
of the other works on PTs in elastoplastic materials. A Landau-type theory was
applied in [17] to study the emergence of spherical and plate-like regions of
the product phase in an infinite elastoplastic space without external stresses
and neglecting surface energy. In these works, the deformation theory of
plasticity was utilized, which is thermodynamically equivalent to a nonlinear
elasticity instead of elastoplasticity. In these papers, the PT criterion and
extremum principle for the determination of some unknown parameters are the
same as for PT in elastic materials, i.e., the total Gibbs energy is
minimized. As it is discussed above, this principle is not applicable for
elastoplastic materials. Still, due to specific simple problems, some results
are either correct or give reasonable hints on some new effects. Thus, it was
found in [393] that (in contrast to elastic materials) nucleation and
interface propagation conditions in elastoplastic materials are not
equivalent.

Only in [393], for the appearance of a spherical nucleus, was it hypothesized
that the mechanical work should be smaller than the change in thermal energy,
which for some simplifications coincides with the driving force in Eq. (43).
Unfortunately, in the next paper [194] the principle of the minimum of free
energy was implemented again. Numerous investigations of PT in elastoplastic
materials in [110, 332, 331, 333] were also based on the comparison of Gibbs
free energy before and after PT. A very different approach was developed in
[369, 371, 372] based on the dislocation representation of the martensitic
interfaces and considering heterogeneous nucleation at the dislocation wall
and corresponding overall nucleation kinetics, as well as martensite growth in
an elastic and elastoplastic material. Various physical, material, and
mechanical features of the nucleation theory were critically reviewed in
[373].

In [229, 232, 233, 238, 239] we formulated a nucleation condition in the form
that the dissipation increment due to PT only (excluding plastic and other
types of dissipation) reaches its experimentally determined value (see
Eq.(44)). To justify this condition, a local description of PTs, similar to
the ones presented in Boxes 1-3, was developed in [234, 235, 240]. The
postulate of realizability [230, 231, 238, 238, 240] was formulated and
applied to the derivation of the extremum principle (38) for finding all
unknown parameters of the nucleus, see [229, 232, 233, 234, 235, 238, 240] for
athermal kinetics. The ”macroscale” thermally activated kinetics equations
presented in Boxes 5-8, were suggested in [243, 244] for small strains with
application to some analytical solutions and in [172, 266] for finite strains.
An approach in line with the classical nucleation theory but for elastoplastic
material presented in Box 4 was developed in [247, 246] for nucleation at a
dislocation pileup, in [254, 255, 250, 257] for sublimation, melting, CR, and
sublimation via the virtual melting within elastoplastic material, and in
[256] for void nucleation due to fracture, sublimation, sublimation via the
virtual melting, and melting and evaporation within an elastoplastic space.

####  6.9.2 Interface propagation

In material science literature [134, 133, 372, 144, 125, 126], various
contributions to the driving force for interface propagation were discussed.
The growth of a thin ellipsoidal martensitic region in a viscoplastic material
was approximately modeled in [372, 144]. Despite very approximate stress-
strain fields, some very important features were revealed. In particular, it
was found that plastic deformation near the tip of an inclusion stops its
lengthening. The first FEM treatment of the appearance and thickening of a
martensitic plate with fixed ends was performed in [332, 331]. In [393],
growth of a spherical region in a spherical elastoplastic matrix was solved.
The defect heredity for the same problem was treated for the first time in
[192]. The driving force for PT was not localized to an interface, but it
represented the variation of the Gibbs free energy combined with the
plastically dissipated heat in the whole body. In the next paper by [194], an
alternative approach for the thermodynamic equilibrium for an elastoplastic
ellipsoidal region in the elastic matrix was utilized. This means that the
correct approach to the interface propagation and final equation were not
known in physical literature. All the balance equations for the points of a
propagating phase interface in a viscoplastic material, including PT
criterion, are obtained in [206]. However, the PT is assumed to be much faster
than the plastic relaxation, and the plastic strain increment does not have a
jump at the phase interface. In this case, the PT conditions are the same as
for elastic materials, which does not explain the strong influence of plastic
strain on PT. In this case, the plastic dissipation at the interface is
absent, and our driving force (41) for isothermal processes coincides with the
Eshelby driving force [93], i.e., like for elastic materials.

The Eshelby thermodynamic driving force was suggested to be used for the
interface propagation condition in elastoplastic materials in [228] and
independently in [110, 332, 331]. This was similar to the condition for
elastic materials with an athermal threshold, but different from Eq. (41) even
for the isothermal process. Note that the Eshelby driving force represents the
total dissipation increment at the moving interface, including plastic
dissipation and dissipation due to changes in internal variables not related
to SCs.

An approach in which the thermodynamic driving force for nucleation and
interface propagation represent the dissipation increment due to PT only,
i.e., excluding all other types of dissipation—in particular, plastic
dissipation has been developed in [229, 230, 231, 232, 233, 240]. The result
coincides with the isothermal version of Eq. (41). This expression was
justified in [234, 235, 240], utilizing a local description of PTs.

Thus, two different expressions for the driving force for interface
propagation in plastic materials were used in literature: based on the Eshelby
driving force in [60, 59, 114, 115] and based on the dissipation increment due
to PT only in [171, 263, 240, 243, 244, 266]. It was shown in [245] that using
the Eshelby driving force, along with its maximization, leads to a conceptual
contradiction for plastic materials. In contrast, Eq. (41) does not exhibit
this type of contradiction. Recently, the correctness of Eq. (41) and phase
equilibrium condition (45) was confirmed utilizing the phase field approach to
coupled PTs and discrete dislocations at the nanoscale in [183, 184] and
coupled PTs and slip bands at the microscale in [259, 92].

A combination of the strict equation for the driving force with physically-
based expressions for athermal and thermal parts of the interfacial friction
from [125, 126, 372] was suggested in [266], and it was applied to the
martensite growth problem, see Section 15.2.

####  6.9.3 Extremum principle for PT in plastic materials

Maximizing the mechanical work in order to find the habit-plane (or invariant
plane strain) variant was suggested in [378]. When the invariant plane strain
includes dislocation plastic shear in addition to the transformation (Bain)
strain and lattice rotation, this represents the maximization of the Eshelby
thermodynamic driving force for the interface propagation and total
dissipation increment for nucleation. However, when the lattice invariant
shear includes twinning only, this represents the maximization of the
transformation work, since twinning in martensite represents the appearance of
two twin-related martensitic variants, which is a part of the PT process. This
case is equivalent to the maximization of the total dissipation increment for
nucleation and interface propagation. The Eshelby driving force was also
maximized with respect to the habit-plane variant in [110, 332, 331, 114, 60,
59].

The transformation work uses local stress in place of nucleation before the PT
was maximized in [121, 333]. This generalizes the extremum principle in [378]
for the presence of internal stresses. However, this contradicts for the
limiting case of elastic materials with the principle of the minimum of Gibbs
energy, because the variation of stresses in the course of the PT is not taken
into account. It leads to significant errors because stresses change
drastically during the PT (i.e., growth of transformation strain) and even
change a sign (see, e.g. [238, 243, 244, 393] and Section 10). In [110, 332,
331, 333], the difference in the Gibbs energies after and before the PT
governed the PT. In [454], an alternative potential was suggested to be
maximized. The common point of the above-mentioned works is that the PT
conditions are not directly related to the second law of thermodynamics and
dissipation due to the PT. That is why they were not justified, and it is
problematic to comprehend the physical sense of the proposed criteria and
extremum principles and strictly compare them.

A plausible assumption called the postulate of the realizability was
formulated in [230, 231, 238, 240], which results in the maximization of the
net driving force $\,\bar{X}-K\,$, both for macroscale nucleation and
interface motion (Eq. (47) for athermal kinetics. It involves the justified
expressions (37) and (38) for $\bar{X}$ and (41) for $X_{\Sigma}$, i.e., based
on the dissipation increment due to PT only. At constant $\,K$, this principle
simplifies to the maximization of $\,\bar{X}\,$ or $X_{\Sigma}$, and, in the
simplest case, to the principle of maximum transformation work, which takes
into account the stress variation in the nucleus during the transformation
process. Such transformation work significantly differs from
$\,{\mbox{\boldsymbol{$\sigma$}}}\,{\mbox{\boldsymbol{$:$}}}\,\mbox{\boldsymbol{$\varepsilon$}}_{t}\,$
and for some problems can possess the opposite sign [263]. For thermally
activated kinetics, the principle of the minimum transformation time was
derived in [243, 244] using the postulate of realizability (see Eq. (63)). One
benefit is that such a principle was intuitively formulated and routinely used
in material science [108, 429, 383, 372] for any nucleation events as a part
of a concept of a critical nucleus (see Eqs. (22) and (23)). Indeed, the
critical nucleus corresponds to the minimax of the energy, maximum with
respect to the size and minimum with respect to all other parameters (shape,
position, etc.) From a practical point of view, our main conceptual result for
thermally activated kinetics is the expression (53) for the activation energy
for inelastic materials, which subtracts all types of dissipations, but due to
PT, as well as athermal resistance. At the same time, arriving at an
intuitively known principle of the minimum of SC time using the postulate of
realizability increases its plausibility for other and more general systems.
Furthermore, the postulate of realizability can be used to derive some known
and new extremum principles in various fields, like the plastic flow rule and
the problem of plastic spin in plasticity, nonlinear nonequilibrium
thermodynamics, twinning, ductile fracture, CRs, and stability analysis [263,
171, 230, 231, 240, 241, 243, 244, 286, 278, 279, 264, 265, 266].

The idea that the solution without PT (even if PT criterion could be met) is
one of the possible solutions to be compared with the solution with PT was
formulated in [230, 231, 240]. This led to the formulation of the global SC
criterion based on stability analysis and its specific expression (50),
derived again with the help of the postulate of realizability in [230, 231,
240]. The nontrivial choice of the stable solution among several possible ones
was demonstrated in [171, 230, 231, 264] for PT and in [172] for fracture.

##  7 Finite strain formalism

As discussed in the Introduction, for various PTs the transformation strains
are quite large. High pressure can cause large elastic strains. For perfect
crystal deviatoric strains are often finite. For example, based on the first
principle simulations, under uniaxial compression elastic strain reaches 0.16
for Si I [464] and 0.38 for graphite [122] before lattice instability and
initiation transformation to Si II and diamond, respectively. For
nanotechnology applications, large elastic strains can be reached in defect-
free volumes (e.g., nanofilms, nanotubes, and nanowires) and near strong
stress concentrators (like dislocation pileups) and lattice misfits, when
plastic relaxation is suppressed. TRIP strain can be very large, theoretically
infinite, in a thin layer (see Eq. (122). One of the great examples of large
plastic deformations during thermally-induced lath martensite nucleation in
steel, for which invariant plane shear strain is 0.2 and volumetric
transformation strain is just 0.02, is presented in [266] (see Section 15.2).
Also, for plastic strain-induced PTs, applied plastic shear during high-
pressure torsion can be $1-$10 and larger [28, 302, 87, 41, 42, 29, 474, 476].
Additionally, lattice and material rotations can be finite even for small
transformation strains. The crystallographic theory of the martensite [453,
16, 26] includes finite transformation strains and rotations, as well as
lattice-invariant shears due to twinning (or slip) but neglects elastic
strain. Continuum thermodynamic theory of PTs in elastic materials that also
includes interfacial instabilities but does not discuss crystallography of the
martensite is presented by [130].

Thermodynamic theory for large elastic, transformational, and plastic
deformations and rotations was drafted in [235] and developed in [240], and is
used for various analytical [240, 278, 279, 242, 254, 250, 257, 245] and
numerical [266, 264, 171] solutions. A summary of this theory is contained in
Boxes 9 and 10. A few equations in Box 9 are similar to those in Box 11 in
Section 16.5 for the PFA to PTs and discrete dislocations.

Multiplicative decomposition of the deformation gradient $F$ (78) is justified
in [271] as a noncontradictory and most economical way to describe discrete
dislocation plasticity in A and Mi and inheritance of dislocation during
cyclic direct-reverse PTs. Specific sequence of the terms in Eq. (78) is
important. The simplest way to justify the sequence in [240, 271] was that
${\mbox{\boldsymbol{$F$}}}_{e}{\mbox{\boldsymbol{$\cdot$}}}{\mbox{\boldsymbol{$U$}}}_{t}$
represents deformation of a crystal lattice and in some theories [315, 396,
98, 388, 99, 18, 176, 439] it is not decomposed into elastic and
transformational parts. Thermoelastic deformation gradient
${\mbox{\boldsymbol{$F$}}}_{e}$ can be multiplicatively decomposed into
elastic and thermal parts [240, 236]. Theory in [240] includes additional
terms in the multiplicative decomposition, namely, plastic deformation
gradients in A, Mi, and during the transformation process. However, they never
were used for the solution of a specific problem. As analyzed in [271],
splitting plastic deformation gradients into several components makes this
theory unnecessararily sophisticated and introduces undesired features. Some
general approaches for justification of kinematic decomposition based on
physical principles are presented in [240, 236].

In addition to the reference stress-free configuration $\Omega_{0}$
corresponding with $\mbox{\boldsymbol{$F$}}=\mbox{\boldsymbol{$I$}}$ and the
current deformed configuration $\Omega$, additional stress-free configuration
$\Omega_{t}$, obtained after elastic unloading without reverse PT from the
current configuration $\Omega$, and the stress-free configuration
$\Omega_{p}$, obtained after elastic unloading with reverse PT from $\Omega$
are introduced (Fig. 5). Parameters defined in these configurations will be
marked with corresponding subscripts. Elastic and plastic Lagrangian strain
tensors used in the constitutive equations are introduced in Eq. (79). Eq.
(80) introduces Jacobian determinants describing ratios of elemental volumes
(and mass densities) in different configurations. Internal time $\xi$ is
introduced in Eq. (81) the same way as for small strains, but in terms of the
finite-strain measure of the transformation strain. Multiplicative
decomposition (78) of the deformation gradient leads to the additive
decomposition of the deformations rate Eq. (82). The Helmholtz free energy
(83) consists of elastic and thermal parts. Note the elasticity rule is
determined experimentally or with atomistic simulations in configuration
$\Omega_{t}$, which is the reference state for
${\mbox{\boldsymbol{$F$}}}_{e}$. To transform it to the energy per unit mass,
the multiplier $J_{t}/\rho_{0}$ is used: $J_{t}$ transforms it in the energy
per unit reference volume and $1\rho_{0}$ in the energy per unit mass. Such a
finite-strain correction was introduced in [288, 297] for large anisotropic
compositional expansion during lithiation-delithiation of silicon and then in
[300] for PFA to PTs. The simplest expression for elastic energy (82) is
assumed, while higher-order terms can be added [240, 272]. Application of
thermodynamic laws results, in particular, to the expressions (85) and (86)
for the first Piola-Kirchhoff $P$ and Cauchy $\sigma$ stress tensor, which are
presented in the general form and for elastic energy (82). Expressions
(31)-(90) for dissipation rate due to plastic flow $D_{p}$ and variation of
the internal variable $D_{g}$, corresponding dissipative forces, the yield
conditions and evolution equations for plastic strain and internal variables
appear similar to those at small strain but using corresponding finite-strain
measures. Specification for single crystals will be presented in Box 11.

Figure 5:  Multiplicative decomposition of the deformation gradient into