Theory of Superplasticity and Fatigue of Polycrystalline Materials Based on Nanomechanics of Fracturing and Failure

Fracture nanomechanics is the study of the interconnected process of the growth and birth of cracks and dislocations in the nanoscale. In this paper, it is applied to superplasticity and fatigue of metals and other polycrystalline materials in order to derive the basic equations describing some main features of these phenomena, namely, the fatigue threshold and the enormous neck-free superplastic elongation. It is shown that in most metals and alloys the fatigue threshold is greater than one per cent of the value of fracture toughness. Using the concepts of fracture nanomechanics, we study the superplastic deformation and fracturing of polycrystalline materials under uniaxial extension and calculate the neck-free elongation to failure in terms of strain rate, stress and temperature. Then, we determine the optimum strain rate of the maximum superplastic elongation in terms of temperature, creep index and other material constants. Further, we estimate the critical size o f ultrafine grains necessary to stop the growth of microcracks and open way to the superplastic flow, and find the superplastic deformation of grains, their maximum-possible elongation and the activation energy of superplastic state. Also, we introduce the dimensionless A-number in order to characterize the capability of different materials in yielding the superplastic flow. A t a very high elongation the alloying boundary of grains proves to be broken by a periodical system o f dead fractures of some definite period. It is shown that experimental results of the testing of the Pb-62% Sn eutectic alloy and Zn-22% A l eutectoid alloy at T = 473 K have substantially supported the theory of superplasticity advanced herewith.


INTRODUCTION
.atigue is a property of a material to deteriorate and fail under the action of cyclic, random or variable loads which magnitude is less than ultimate or yield strength. .atigue is caused by the slow, subcritical growth of cracks and microcracks and, hence, is a subject of fracture mechanics [17] and nanomechanics [815], the latter being specifically drawn up for the scales from one nanometer to some hundred micrometers.
Superplasticity is a property of polycrystalline materials with ultrafine grains to enormous neck-free elongations to failure at sufficiently high temperatures. So far, it has been considered as an exotic phenomenon and not as a common state at some conditions. It is of a special interest for fracture nanomechanics [815] because of the nano-and microscale of grains characteristic for the superplastic behavior. It should be noted that superplasticity means, in fact, a creep flow of specific ultrafine-granular materials.
.racture mechanics deals with plastic deformations near the crack tip using the phenomenological theories of plasticity describing, in average, the effect of very large number of dislocations; therefore, it is valid only for a sufficiently large growth of macrocracks [1 7]. .racture nanomechanics takes into account every individual dislocation, including its birth, emission and motion; therefore, it does not need in the theories of plasticity and can describe the however small growth of any cracks or microcracks even for very low loads [815].
The analytical theory of fatigue macrocrack growth used an analysis of plastic deformations near the crack tip [16]. According to this theory, the fatigue crack growth rate for cyclic loadings is equal to [16] This is the empirical Paris law [7]. Extensive tests showed that it works well in most metals and alloys when the number of cycles to failure is not too large.
Both Eq. (1) and Eq. (2) are not valid for extremely low loads when cracks do not propagate. This property is controlled by the threshold stress intensity factor .
K such that, if the stress intensity factor I K is less than . , K the tensile crack does not grow. Evidently, in this case it is necessary to use the methods of fracture nanomechanics.
The value of . K is of paramount practical importance because the fatigue strength of structures, especially made from metals or alloys, is determined by their resistance to the subcritical crack growth. It is difficult to find this value experimentally because of enormous number of cycles necessary for adequate testing. The problem of .
K is addressed below in Sect. 2. Superplasticity discovered just about half a century ago is poorly studied, and all basic features of this phenomenon have been so far unexplained, although well documented in experiments. The following problems are in urgent need to get their analysis and solution: (i) What ultrafine grain size is necessary for superplasticity, and why?
(ii)What is the maximum elongation to failure and why is it so enormous?
(iii) How to characterize the superplastic property of a material in terms of its physical microstructure and temperature?
These and other relevant problems are studied in Sects. 3 to 9 below from the position of the nanomechanics of fracturing.

THE RULE O. THUMB: .ATIGUE THRESHOLD
To study the start of the crack growth it is necessary to take into account the atomic structure of the material and emission of individual dislocations from the front of the crack. This account has been provided by the nanomechanics of fracturing, see [815]. According to its basic concept, the minimum value of the crack tip growth corresponding to the stable settlement of a first single pair of elementary edge dislocations emitted from the crack tip, is equal to the interatomic spacing.
This concept allows us to also evaluate the threshold stress intensity factor using Eq. (1) or (2). According to these equations, the crack growth for one cycle of loading, ∆L, is equal to 4 max Evidently, when the crack growth is equal to the interatomic spacing, the maximum stress intensity factor in any loading cycle is equal to the threshold stress intensity factor because the crack can not grow for an amount less than the interatomic spacing. .rom here and from Eq. (3), it follows that 1 4 .
Here, a is the interatomic spacing. This equation requires some significant experimental work with measurements of the controlled fatigue crack growth in order to determine the fitting constant β in Eqs. (1) or (2). Then Eq. (4) allows one to evaluate the value of the threshold stress intensity factor . K for any specific material.
The analysis of several hundreds of experimental works on the growth of fatigue cracks in various metals and alloys has shown that the value of β varies in the range from 0.001 to 1 mm [17]. Since a ≈ 10 7 mm, PHYSICAL < is more complicated for study. In this range, beyond of Eq. (4) a more accurate estimate can be given by using the model of two rows of edge dislocation pile-ups on each side of the tensile crack [11].

CHARACTERIZATION O. POLYCRYSTALLINE MATERIALS
Both local plastic deformations near a crack tip and a crack growth are some interconnected effects taking place in polycrystalline materials simultaneously during the loading part of a loadingunloading cycle. Both of them proceed in primary acts such that the stable settlement of a pair of elementary edge dislocations emitted from the tip of a tensile crack is always accompanied by the crack growth on one interatomic spacing. Plastic deformation is a continuum description of the strain produced by a very large number of dislocations. Everywhere below, we assume that the polycrystalline materials under study are homogenous and isotropic, except for some designated cases. .or the modeling purpose, we use the case of the cubic lattice of atoms.
According to fracture nanomechanics, at the beginning of the process of loading the crack does not grow and no dislocations emanate from the crack tip until the stress intensity factor achieves some critical value of 1 k when the crack grows one interatomic spacing and simultaneously the first pair of elementary edge dislocations emanates from the crack tip and settles down at a certain distance whereby [815] Here µ is the shear modulus of the polycrystalline material, 0 τ is Schmids constant of friction on gliding planes of the crystal lattice, ν is Poissons ratio, a is the the interatomic spacing, and 1 k is the specific value of the open-mode stress intensity factor I K for the cubic lattice of atoms.
This first pair of settled dislocations finds a stable position at distance 1 ρ from the crack tip on the glid-ing planes issuing from the crack-front under angles ±45° to the crack plane in the cubic lattice of atoms so that [815] Here e b is the absolute value of the Burgers vector of the elementary edge dislocation depending on the parameters of the crystal lattice e ( 2 b a = for the cubic lattice).
If the stress intensity factor is less than 1 , k there are no stable set dislocations and no crack growth so that Eq. (6) describes the lowest threshold of the dislocation emission from the crack tip and of the crack growth. When I 1 , K k < no dislocations settle down and the crack does not grow because, despite the infinite stress at the crack tip, all dislocations emitted from the crack tip by thermal fluctuations are unstable and come back to the crack tip.
.or example, using Eqs. (6) and (7) for aluminum crystals [16,17] To compare: fracture toughness of structural aluminum Ic K is of the order of 1GPa mm, while Ic k is of the order of 10 MPa mm, where Ic k is the stress intensity factor of the aluminum lattice for the brittle crack in the cleavage plane corresponding to the dislocation-free growth of the crack.

Brittle Versus Ductile Behavior of Crystals
Equations (6) and (7) allow us to also characterize the brittle vs ductile properties of crystals. The crack tip can advance not only by the ductile mechanism with emission of dislocations, but also by a brittle mechanism. If the brittle mechanism is realized, the atomic bond ahead of the crack tip is ruptured without emission of dislocations. It occurs when the superfine scale stress intensity factor I k at the crack tip exceeds the critical value Ic k corresponding to the reversible surface energy of the dislocation-free crack growth.
The superfine scale stress intensity factor is the total stress intensity factor at the crack tip, which is equal to the sum of the stress intensity factors due to the external load, I , K and to the elastic field of dislocations Id .
k The dislocations generated by the crack tip during the loading process create a compression in the superfine scale of the crack tip and, hence, cause the local relaxation of the stresses induced by the external load so that In other words, the generated dislocations play the role of a screen or shield protecting the crack tip from external loads.
A crystal is ideally brittle, if the crack can grow along a cleavage plane without to settle down any dislocations emitted from the crack front, i.e. if the following inequality is met 1 Ic Ic Here γ is the true surface energy of the crystal along the cleavage plane. .or metal crystals, the value of γ has an order of 1 Pa m and, hence, for aluminum the value of Ic k has an order of 10 MPa mm so that due to Eq. (8) the aluminum crystal is very ductile.
To characterize the brittle versus ductile behavior of crystals, it is useful to introduce the dimensionless brittleness number as follows A crystal is absolutely brittle if η > 1 so that the crack grows without the dislocation generation. A crystal is ductile if η < 1 so that the dislocation emission starts before the brittle crack propagation becomes possible. Using Eqs. (6) and (10) The smaller η, the more ductile is the crystal. .or crystal lattices other than the cubic ones, the coefficient in Eq. (12) can be different. The value of η was calculated for several common crystals. .or example, in diamonds it is equal to about 1.5, and for aluminum 0.030.04 depending on the values of the surface energy and Schmids constant. Typically, for most metals the brittleness number varies in the range of 0.01 to 0.1.

SUPERPLASTIC MATERIALS: ULTRA.INE GRAIN SIZE
The discovery of superplastic materials, metglas and graphen marked the greatest advances of material science for the last half a century. On the scale of technological impact, each one of them is comparable with the invention of bronze and iron several thousand years ago noted down by historians as the Bronze and Iron Ages of humankind. Superplasticity is characterized by a very large neckfree elongation to failure at elevated temperatures greater than about a half of the melting temperature m T in Kelvins. .or example, the superplastic Zn22% Al alloy specimen can extend 23.3 times its original length at temperature 473 K and at strain rate 10 2 s 1 while the superplastic Pb62% Sn alloy specimen can extend 76.5 times its original length at temperature 473 K and at strain rate 2.12 × 10 4 s 1 [16]. The melting temperature of zinc and tin which are the base metals of these alloys is equal to 693 and 505 K, correspondingly, while the melting temperature of their alloying components, aluminum and lead, is equal to 933 and 601 K. In the special literature, the base metal component is called also the host or parent metal [1619].
The theory of superplasticity is given in Sect. 6 below. Usually, a material behavior is called superplastic if the elongation to failure exceeds the original length about three times. However, as shown below, the superplastic material can extend to a much greater maximum when the temperature tends to the melting point.
To make a compound liquid metal superplastic, it should be cooled down very fast from the liquid state to the solid one so that the base-metal grains could be able to grow up only to the size less that some critical value, below which no stable dislocations can be born inside the solid grains and no microcracks can grow. The base-metal grain boundary made of other special components of the alloy forms during the crystallization process as a result of pushing them out by the growing base-metal grain. Thus, they create an intergranular boundary layer which prevents from the settledown of individual dislocations within the grains and from the growth of any microcracks inside the grains. There should be, at least, two components in any superplastic metal alloy. Besides, the intergranular layer protects the base metal from corrosion.

Grain Size of Superplastic Materials
According to fracture nanomechanics, the size of grains should be less than the value of 1 ρ characterizing the distance between the first pair of stable dislocations and a crack tip or another stress concentrator like the vertex of a two-sided angle of the grain. The grain size s should satisfy the following requirement 0 .
(1 ) on the nature of the dislocation generator inside the grain and its lattice type. .or the sharpest concentrator, a crack tip, and the cubic lattice α = 0.11 according to Eq. (7). Let us estimate number α in Eq. (13) using a more realistic stress-singularity for the vertex of a two-sided angle between two dissimilar materials. In this case, it can be shown that the balance equation of a pair of edge dislocations near a stress concentrator is given by the following equation [8] e 0 , 0 Here r is the distance between the vertex of the stressconcentrator and the position of dislocations, n is the proper number characteristic for the angular stress singularity [5,6], and K is the intensity of the stress concentration proportional to the external load, with the dimension being 2 N m .

n−
In the case of a crack tip, we get n = 0.5 and (14) has one minimum at These equations generalize Eqs. (6) and (7) for the crack tip.
Hence the first pair of stable edge dislocations emitted by this stress concentrator settles at distance 1 r = ρ when the external load measured by K achieves the value of 1 k in Eq. (15). No stable equilibrium dislocation position exists for 1 . K k < According to Eq. (14), if 1 K k > and the first pair of stable dislocations is set down, their position r grows when K increases, until the second pair of stable dislocations is born [815]. Equations (15) support the estimate of coefficient α in Eq. (13) for 0.1 < n ≤ 0.5.
Using computerized calculations, the process of the dislocation generation by a crack tip in large crystals was studied up to many thousands of individual dislocations, and the diagram of the stress intensity factor versus the crack growth was calculated, see [813] for more detail. In the framework of this discrete atomic model, the fracture toughness of the material was calculated as the limit of the stable growth of the crack characterized by the maximum point on this diagram, after which the unstable growth of the crack occurs.
However, a crack tip does not grow at I I c k k < and does not produce any stable dislocations, if the size of grains in a polycrystalline material is less than 1 . ρ Thus, the value of 1 ρ in fracture nanomechanics characterizes the critical ultrafine grain size of the superplastic state of the material if I which is usually the case. This value is estimated by Eq. (13).
In superplastic materials under low temperatures, the ultimate and yield strengths depend on the grain size as follows [46] Here B σ is the ultimate strength, y σ is the yield strength, 0 s is the characteristic grain diameter, I I c k k < is the fracture toughness, and IIc K is the slip toughness of the material.
.or example, the decrease of the grain size from 10 µm to 10 nm causes the 31-fold increase of the yield strength of some superplastic metals [18]. The equations, analogous to Eqs. (16), are called the Petch HallStrawCottrell equations. However, under high temperatures close to the melting point the effect of surface tension dominates so that these equations turn to be invalid, see below. Also, in the nanoscale the effect of surface tension becomes essential.
It is noteworthy that a microcrack inside the grain starts on to grow as an absolutely brittle one only after the superfine stress intensity factor achieves the value of Ic k so that the tensile stress σ and the size of the microcrack c d are connected as follows [14] c I c .
.or example, in the aluminum-base grain under low tensile stress typical for the superplastic state, e.g. σ = 33 MPa, the critical size of the brittle microcrack equals c d = 100 µm which is much greater than the grain size in the superplastic state so that the brittle failure cannot occur because Ic 100 MPa mm. k =

Critical Grain Size in Some Superplastic Metals
According to Eqs. (13) and (15), the aluminum grain size in superplastic state should be less than about 2 µm. Evidently, this result of calculation is valid for any Al-base alloys.
Using Eq. (13) for the crack tip concentrator and Table 6.2 in Ref. [14], let us provide the results of calculation of the critical size of grains in superplastic state for the following base metals: 2.3 µm in Al, 1.8 µm in Cu, 2.6 µm in Au, 0.9 µm in Ni, 2 µm in Pb, 3.2 µm in Ag, 5.8 µm in Zn, 0.2 in α-.e, 4.5 µm in Sn.
All of these metals, except for tin and zinc, have the face-centered-cubic crystal lattice (fcc), in accordance with the theoretical model served to derive Eq. (13). Tin has the tetragonal lattice, and zinc the hexagonalclose-packed lattice (hcp); therefore, the above figures for these metals, probably, need some corrections.
It is worthy of keeping in mind that these figures correspond to the minimum values of Schmids constant observed in experiments [16,17]. This constant which is the friction stress on the gliding planes of a pure crystal, is very sensitive to any impurities so that any obstacles on these planes, beyond of atomic forces of the crystal, significantly influence its value. The interstitial atoms of alloying elements, which diameter is greater than the interatomic spacing of the lattice of the parent metal, can significantly increase Schmids constant and, hence, decrease the critical size of grains for the superplastic state to be realized. Any distortions of the crystal lattice like original dislocations on other planes also increase this constant. On the other hand, vacancies in the crystal lattice of the base metal can decrease Schmids constant and, hence, increase the critical size of grains. Also, according to Eq. (15), a slighter singularity of a two-sided angle can lead to an increase of this critical size as compared to the crack tip concentrator.

The Theory of Singularities of the Elastic .ield
Let us provide a short resume of this theory given in [5,6,14]. There are point and linear singularities. The first ones are point inclusions like interstitial or foreign atoms in the lattice of a parent metal, or point holes like vacancies or small bubbles in this lattice. The second ones are the fronts of cracks and dislocations, and the vertex of the two-sided angle between dissimilar materials different from π. It is necessary to distinguish the S-singularities from the N-singularities.
The S-singularities possess the self-controlled fields independent of applied loads. These are point singularities and dislocations. They obey Saint-Venants principle and can drift in the stress field produced by applied loads without to change their own fields like some invariable particles.
The N-singularities do not possess their own independent fieldstheir fields are produced by applied loads and determined by them. These are crack fronts and vertices of two-sided angles. They do not obey Saint-Venants principle. The crack front can move in the stress field, usually with changing its own field in the process of motion.
The drift of any singularity is governed by the corresponding driving force; most of these forces were discovered during last century. The first discoveries for massive elastic solids were: Irwins law for opening mode cracks, Peach-Koehlers law for edge dislocations, and Eshelbys law for point inclusions. Using invariant integrals, dozens of other laws were found out for other important field singularities including point holes, cracks and dislocations of arbitrary modes in anisotropic solids, interface cracks, cavities, cracks and bubbles between shells, plates, membranes and solids, solid-liquid contact fronts and many others [4 6, 14, 15].
The drift of point inclusions and point holes in the field of cracks and dislocations was studied earlier in [14] under some simplifying assumptions, e.g. of zero Schmids stress. In particular, all point inclusions and holes proved to drift into the crack-front or dislocation-core. It is easy to predict some effects of nonezero Schmids stress, for example, appearance of some critical distances from the crack-fronts and dislocation-cores such that outside of these distances point inclusions and holes cannot drift, at all.
As to the original point singularities, like vacancies or foreign atoms in the identically stretched grains of base superplastic metals, its evident that they do not drift in the process of flow because their driving forces equal zero for this case of the zero-gradient field of tensile stress. And so, vacancies and foreign atoms can substantially effect only upon the bulk properties of the base metal grains, for example, upon µ, 0 τ and Ic . k In real distorted grains, some drift of vacancies and foreign atoms towards the grain boundary occurs; however, this effect is small and ignored in the present model, see the next section. Certainly, the diffusion and migration of vacancies and foreign atoms governed by the temperature gradient plays some role most essential at high temperature.

Critical Temperatures of the Superplastic State
In ultrafine grains of subcritical size, the activity of any stable individual dislocations is suppressed so that the grains can deform only by the uniform flow along gliding planes [ Let us study the extension and flow of a bar of a superplastic material at sufficiently high temperatures and low tensile loads. We assume that all grains of the material have one and same volume and the material is incompressible. The first assumption is justified by the simplicity of the following analysis, while the latter one is close to reality for large deformations.
In polycrystalline materials, the large deformation and flow can, in principle, be caused by a relative movement of neighboring grains (the sliding mechanism). However, it forms both open and sliding interface cracks, which size in the superplastic flow would be much greater than the grain size; these cracks would grow in the process of flow so that their healing by diffusion would require an enormous concentration and activity of foreign atoms. Thats why we take into account only the most probable mechanism of the superplastic flow of each grain, with no material discontinuities arising between the neighboring grains.

Grain Shape
.or very large deformations, as a result of stretching by tensile load every grain acquires a shape of a prolonged prism. In order to densely pack the space, the prism cross-section can be either equilateral triangle, or square, or regular hexagon. All prisms have length l and cross-section area g A so that 0 lA V = where 0 V is the constant volume of any grain in this model. In the process of flow, length l increases and area g A decreases. The thin intergranular layer of alloying components cohering grains plays the part of surface tension for the flowing base material. Out of possible three forms of the prism cross-section, square has a minimum perimeter for the prisms of the same volume and length and, hence, provides for a minimum of surface energy of prisms at any moment of flow. Therefore, according to the principle of minimum surface energy, the cross-section of all prisms is the square with side s so that 2 g A s = where . l s >> To be densely packed in the space, both front and rear faces of the prisms should be flat. This packing order leaves no empty spacings between the prisms. All succeeding calculations are assumed for the packing of this kind.
And so, in the process of superplastic deformation the grains of arbitrary shape become long identical prisms of square cross-section, with flat front and rear faces.

Superplastic Zero-Dilatancy .low
In the process of flow, a bar of initial length 0 L and initial cross-section area b0 A acquires length L and cross-section area b A so that b 0 b0 . LA L A = Besides, the number of grains in each cross-section of the bar remains constant during this process so that we have Here l is the length of each grain so that 2 0 ls V = =const for any elongation.
According to Eq. (19), the grain length l assumes the function of the bar length L in terms of strain so that the strain rate ( ) t ε is equal to According to Eq. (20), length l assumes as well the function of time for a given strain rate because The volume strain rate is zero so that for the grain thickness we get Here t is the thickness of the intergranular layer, and a ρ and b ρ is the mass density of the alloying and base metals.
.or example, in Pb62% Sn eutectic alloy where tin is the base metal, and lead is the alloying ingredient, we get t s = 0.1.

THE THEORY O. SUPERPLASTICITY: ELONGATION TO .AILURE
The large neck-free elongation to failure is the main property describing the superplastic behavior of some materials at low stresses and high temperatures. In general terms, this phenomenon can be understood using the kinetic theory of fracture, see Chapter 2 in [14]. According to this theory, the time to failure of a stretched bar is equal to Here σ is the tensile stress in the bar, . t is the time to failure, .
U is Arrhennius chemical activation energy of the given material substance equal to about 10 2 to 10 3 K J per gram-atom or gram-mole for condensed matter, which characterizes fracturing and is close to the binding energy of the substance, T is the absolute temperature in Kelvins, A v is the bar activation volume per gram-mole or gram-atom, in which the active failure occurs, 0 τ is the characteristic time of an elementary thermal fluctuation, namely, the propagation time of phonons on one interatomic spacing, which is about 10 13 to 10 12 s, and R is the universal gas constant equal to 8.31 J per gram-mole and per Kelvin.
In Eq. (24), Boltzmanns constant is often used instead of the universal gas constant so that in such a case .
U and A v are taken per one atom or molecule of the substance. As a reminder, the universal gas constant equals Boltzmanns constant times the Avogadro number.
The extensive experimental investigation of the long-term strength of bars made of various common materials under stationary tensile loads demonstrated the applicability of Eq. (24) to a wide range of loading times from some microseconds to several months and absolute temperatures from close to zero to the melting point [1417]. Despite of some obvious shortcomings of this equation it provides a good common-sense estimate of time to failure in an exceptionally broad range of time and temperature. However, this theory was done before the era of superplasticity so that its extrapolation for the superplastic state constitutes a hypothesis that should be independently verified.
The strain rate of the steady superplastic flow or creep of a bar is governed by the following empirical constitutive equation [1417] c 0 0 exp .
Here ε is the strain rate, σ is the tensile stress, n is the dimensionless creep index which for most metals usually varies in the range of 2 to 10 and in the superplastic state it can be much greater, c U is the activation energy per gram-atom characterizing the creep or superplastic flow, and 0 r and 0 σ are some fitting constants of the dimension of strain rate and stress, respectively.
Suppose the strain rate and stress do not vary in time during test which is the case in most common test procedures. In this case, from Eq. (21) it follows that Here . l is the grain length at failure which is directly proportional to the specimen length at failure . . L According to Eq. .
The activation energy characterizes the energy barrier which is necessary to overcome in order that a chemical or physical transformation would take place. This barrier is greater for fracturing than for creep or superplastic flow so that the value of ∆U is always positive. .or the superplastic flow, quantity ∆U plays the role of the activation energy of failure/fracturing. Equation (28) provides the specimen length at failure in terms of tensile stress. Let us write down this function as follows When x grows from zero to infinity, y increases from zero to the maximum and then decreases tending to zero at infinity. We accept as a definition that a material is in the superplastic state, if *  if the following inequality is met 1 .
y >> The greater is the value of * , y the more superplastic is the material. When . 0 , L eL = then * 1. y y = = According to Eqs. (29) and (30), the maximum superplastic elongation * y in terms of temperature is given by the following function .unction * * * ( ) y y T = in Eq. (32) has the minimum value at * 1 T n = and is infinite at zero and at infinity. The superplastic state can take place only if * 1 T n > so that, from here and from Eq. (33), it follows that the necessary condition of superplasticity is According to Eq. (25), the tensile stress can be expressed in terms of the strain rate .rom Eqs. (30) and (38), it follows that the maximum elongation is achieved when the strain rate is equal to This strain rate is called the optimum strain rate. The comparison of this theory with test data as well as some consequences and ramifications, are given in the next sections.

CHARACTERIZATION O. SUPERPLASTIC MATERIALS: THE A-NUMBER
To characterize the capability of different materials in yielding the superplastic flow, let us introduce the following dimensionless number (the A-number): . 0 max ln .
According to Eq. (36), the A-number is a function of temperature that can be written as follows: Evidently, the greater the A-number, the more superplastic is the material at a given temperature. .rom Eq. (41), it follows that the A-number increases, i.e. the material becomes more superplastic, when temperature grows.

Maximum Superplastic Elongation of Grains at Melting Point
Let us estimate the superplastic elongation of a base grain at its melting temperature when the elongation achieves maximum. We assume that the melting point of the base grain is less than that of its alloying boundary, which is usually the case. At this melting point, the solid alloying boundary of a base grain plays a role of the shell that bears all load like the surface tension in liquids or a tensile tension in membrane shells.
Usually, the metal of the boundary layer is more strong and brittle than the base metal of the grain so that at large elongations the boundary layer is torn by a periodic system of fractures, every one of which encircles the cross section of a long prism of the base metal grain (see below). In the process of flow, these fractures become wide gaps or cavities filled by vapor of the base metal. They cannot bear a load so that, in this limiting case of the melting temperature, it is the real surface tension of the liquid base metal that bears the tensile load.
In this limiting case the following equilibrium equation holds for the square cross-section of the liquid base metal grain Let us compare this theory with test data for two typical superplastic alloys taken from the book by K.A. Padmanabhan and G.J. Davis [ 18 ] and from the paper [19] by M. Kawasaki and T.G. Langdon. It is reasonable to choose the dimensionless constants n, δ, and B from the same test data. We normalize the value of strain rate by equation 0 r = ε at the maximum of function y = y(x) where the maximum elongation is achieved. Since its derivative is equal to .or Pb62% Sn alloy [18], first, using Eq. (47), let us calculate the A-number for tin in the Pb62% Sn alloy at T = 252.5, 473, and 505 K, where m T = 505 K is the melting point of tin, the base metal.
We accept the following typical figures for white tin: According to the experimental data [18], the Pb 62% Sn eutectic alloy specimen experiences the maximum length 76.5 times its original length at strain rate 2.12 × 10 4 s 1 and temperature 473 K. At this point the test results well support the theory. According to the theory, in the limit the specimen of this superplastic alloy can extend 1200 times its original length at melting point so that its diameter becomes 35 times less than the initial value before loading.
Let us compare the results of calculation of superplastic elongation using Eq. (49) with other test data given in Ref. [18] for the Pb62% Sn eutectic alloy at T = 473 K ( Table 1). The following values of constants were used in Eq. (49): n = δ = 16, 2.12 × 10 4 s 1 , B = e 16 ln76.5. In this case the maximum superplastic elongation is achieved at 0 . r ε = As seen, the discrepancy between the theory and the test is about 4% in average so that we can conclude the theory is confirmed by the test results because the discrepancy is within the scatter of experimental results caused by many factors ignored either in the test or in the theory. The elongation to failure is very sensitive to these unaccounted factors, which explains wide scatter of data in tests.
.or Zn22% Al alloy [19], first, let us provide some figures characterizing the effect of the value of creep index on the superplastic flow at the lowest temperature m 0.5 T T = of superplasticity (Table 2). Even at the lowest temperature this effect is substantial.
Let us consider the test results of the extension of the Zn22% Al eutectoid alloy specimens at 473 K for several strain rates [19] (Table 3). In these tests, no necking up to failure was observed. The material was originally treated by equal-channel angular processing (ECAP) to produce grains of the size about 0.8 µm.
After ECAP specimens of 2 × 3 × 4 mm size were made to test.
In order to calculate the corresponding theoretical values of elongation to failure we can use Eqs. (54) to (58) where the following figures are valid for this test: 1 x = 10 4 s 1 , m x = 10 2 s 1 , 1 y = 2.04, m y = 3.15. .or these figures, the root of Eq. (56) with a less than 0.5% error is given by n = 23.
(59) In accordance with the theory, when the strain rate grows, the elongation to failure increases until a certain maximum is achieved at the optimum strain rate which is equal to about 10 2 s 1 and then the elongation to failure decreases.
Using the indicated figures and Eqs. (58) and (59), let us calculate the theoretical values of m y y and compare them with the experimental ones (Table 4).
However, despite a very good compliance of this theory with the test results we should keep in mind that Kawasaki and Langdon concluded that grain boundary sliding is the dominant flow process during superplastic elongation while in the present calculation of the strain rate the boundary sliding is ignored and it is only superplastic elongation that is taken into account. This compliance questions the fundamental conclusion of the original authors about the role of the pure stretching versus the boundary sliding in the mechanism of superplastic deformation. Another drastic difference is in the value of parameter n of strain rate sensitivity, which is found here to be equal n = 23 while Kawasaki and Langdon estimate it to be within 2 to 10 at different intervals of the strain rate of grains. The kinetic theory used here takes into account both stretching and sliding mechanisms leading, in sum, to the enormous values of creep index.  These cardinal differences, however, require further studies of the mechanism of superplastic deformation. The Zn22% Al alloy is of special interest for superplasticity because both its components, zinc and aluminum, have ultrafine grain size and are superplastic at 473 K since this test temperature is greater than half of their melting temperatures, 693 and 933 K.

PERIODIC SYSTEM O. .RACTURES IN THE BONDING LAYER
An alloying metal of a thin layer that bonds two neighboring grains of a base metal usually has a greater melting temperature and is less superplastic than the base metal. As a result, at a sufficiently large extension the bonding layer of each grain breaks with many cross-sectional fractures and turns into a periodic system of streaks of length s 2L separated by the fractures. Bonding streaks bridge the base metal grains. We assume that the streak length is much greater than its thickness t, i.e. s . L t >> Suppose the x-axis is chosen along the extension direction, i.e. along the stretched grain and streak, with the origin being at the center of a streak. The streak is subject to the tensile stress σ(x) inside and the shear stress τ(x) on both interfaces of the streak bordering the base metal. These stresses bear the external load of the bar extension; they are tied by the following equilibrium equation Hence, the streak length is about 7.7 times greater than its thickness which sufficiently well complies with the original assumption. However, we should keep in mind that the value of thickness corresponding to the developed superplastic flow is much less than it was initially.

CONCLUSION
The present paper looks into the fracturing, creep and fatigue of polycrystalline superplastic materials at the nano-and microscales of their structure. It is shown that a rule of thumb can be used for a simple estimate of safe cycling loadings. The ultrafine grain size necessary for superplastic flow is estimated in terms of material and physical constants. The maximum elongation to failure under extension of rods is calculated using the kinetic theory and methods of the nanomechanics of fracturing. The dimensionless A-number is introduced to characterize the capability of materials with ultrafine grains to the superplastic flow. .or two popular superplastic alloys, elongations to failure are determined and compared to some test data, with providing a satisfactory agreement of the theory with the experiment.