Parinov I.A. Microstructure and Properties of High-Temperature Superconductors

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8.2 Twinning Processes in Ferroelastics and Ferroelectrics 371

sintered under thermal gradient at diﬀerent initial press-powder porosittes

[798]. The FC structure is described by lattice with square cell size, δ.Af-

ter ceramic fabrication (i.e., sintering, shrinkage, cooling and poling process)

a microstructure contains the grains, voids, microcracks at the intergranu-

lar boundaries and domain structure. We estimate a microcrcaking by pol-

ing due to domain re-orientations which diﬀer from the 180

◦

ones (otherwise

the residual stresses do not occur). In this case, the spontaneous strains, ε,

for diﬀerent ferrohardness degrees have been deﬁned by experiments [276] as

=14× 10

−3

,ε

=11× 10

−3

and ε

III

=8× 10

−3

(where indexes I, II

and III are concerned with compositions, possessing by ferrohardness, mean

ferrohardness and ferrosoftness, respectively). Then, the critical length of the

cracked boundary, l

, and the facet length subject to cracking, l,foreach

composition are found by (8.4) and (8.5).

After localization of the microcracks at the intergranular boundaries, we

model stress sources at the polydomain grain boundaries by the continuously

distributed eﬀective dislocations and obtain an equilibrium domain width

as [840]

d =



D(1 − ν)σ

4.2GS

(2 − ν)



1/2

, (8.45)

where D is the mean grain size independent of the ferrohardness degree, σ

is the energy of 90

◦

domain boundary, G is the shear modulus, S

is the

spontaneous shear strain and ν is Poisson’s ratio. Note that the critical grain

size, D

∗

, below of which there is no domain twinning, may be found under

condition of D = d (a structure of mono-domain grains) as

∗

(1 − ν)σ

4.2GS

(2 − ν)

. (8.46)

Further, it is known that the FC poling causes a concentration fall of

domain boundaries in the poled sample to compare with the non-poled one.

The structure transition of the twinning type under stress near crack tip

can provoke a reverse process of the domain boundaries density rise, but a

twinning process zone, forming around propagating crack, will drag the crack.

Obviously, the process zone size depends on the composition ferrohardness and

the ceramic piezoelectric properties. Moreover, this size is also deﬁned by the

crack propagation direction. It should be noted that the maximum inﬂuence is

rendered on the crack, advancing along residual poling direction. In contrast,

along normal one this structure change is restrained by the stress state near

crack tip, that is, the crack propagation is realized without an additional

energy loss, and the twinning process zone around the crack does not form.

Around initiated macrocracks favorably oriented to poling direction, there

are twinning process zones with thickness h. Then, a toughening due to the

twinning may be estimated in the crack shielding terms. The shielding, cor-

responding to the crack growth nucleation is represented by the solution for

stationary crack, but the crack nucleation toughness, K

, has form [192]:

=(E

/H)

1/2

, (8.47)

372 8 Computer Simulation of HTSC Microstructure and Toughening

where K

=(2γ

)

1/2

is the critical SIF, required for the crack growth and

reﬂecting a fracture resistance of twin planes, γ

is the fracture energy, H is

Vickers hardness, E

is Young’s modulus for cracked ceramic, calculated by

(8.16), where β

is the microcracking density, formed by poling. Neglecting

the closed porosity eﬀects, the steady-state crack toughening due to twinning

is estimated as [192]





− 1









−

)



1/2

, (8.48)

where K

is the steady-state crack toughness, α

=3/8π, σ

is the yield

stress or the threshold stress, deﬁning the twinning nucleation near crack tip.

The process zone width, h, for diﬀerent ferrohardness degrees is found,

using a calculation of the parameter, 0 <ξ<1, reﬂecting a stability degree

of piezoelectric composition to de-poling (i.e., the ferrohardness degree) [572].

These results are based on the energy balance of the modeled process and on

application of the ﬁnite element method to calculate the zone of feasible mi-

crostructure transformations at the crack tip. For PZT ceramic compositions

I, II and III, respectively, we obtain the values [572]: 4h/D =0.2(forξ =0.3),

0.5 (for ξ =0.5) and 1.2 (for ξ =0.7).

On account of a necessary number of computer realizations, deﬁned on the

basis of stereological method [145], in order to obtain numerical results, we use

known data for PZT ceramics and related materials: E =70GPa,ν =0.25

[849], H = 3 GPa [848], G =20GPa,S

=0.01,σ

=3× 10

−3

J/m

[840].

The cell size, δ, is selected to be equal to the critical size, l

, for composition

with mean ferrohardness. Then, selecting γ

= γ

− γ

(1 − f

)/2, where γ

is the surface energy for bulk body (γ

≈ 2γ

=6J/m

[796]) and f

the fraction of facets cracked in the poling process, we calculate the values

of D, β

and f

using computer simulation. Finally, the threshold stress,

= 20 MPa, is used which has been estimated in the experiments on im-

pact loading of BaTiO

ceramic [578]. The numerical results show that the

observed decreasing of the grain size, D, with increasing of the initial poros-

ity, C

, leads to a decreasing of the width of equilibrium domain, d,andthe

process zone size, h. The latter also decreases at rise of the FC ferrohardness.

In the toughening due to twinning, the value of K

, deﬁning start of the

crack growth, changes in the limits: 4.06 ± 0.15, 4.30 ± 0.15 and 4.74 ± 0.08,

for I, II and III compositions, respectively, at the initial porosity C

=0−60%.

Similar dependencies of toughening, K

, on the ferrohardness and the

initial porosity in the case of the steady-state crack, are shown in Fig. 8.23

[798]. The results for the case of mean ferrohardness are close to the ratio of

the fracture toughness in existence of twinning to its absence in γ-TiAl [192].

In this case, the known growth of the twinning process zone and fracture

toughness with increasing of FC ferrosoftness is supported [849]. FC toughen-

ing, caused by the twinning process, is a more eﬀective toughening mechanism

compared to microcracking near crack tip [518], to crack branching and crack

8.2 Twinning Processes in Ferroelastics and Ferroelectrics 373

3.0

2.5

2.0

1.5

1.0

III

05040302010

C p ,%

Fig. 8.23. Fracture toughness (K

) at steady-state crack vs initial porosity

) for diﬀerent ferrohardness of FC

bridging [797], and also for phase transformations due to coexistence of rhom-

bohedral and tetragonal ferroelectric phases in the region of morphotropic

transitions. The obtained maximum toughening in the range of change of the

initial porosity (at C

= 20%) could be explained by mutual superposition

of complex microstructure eﬀects and is qualitatively supported by similar

results for the toughening mechanism of microcracking near crack tip [849].

8.2.2 Fracture Features in Domain Structure of Ferroelectric

In order to consider fracture features in the domain structure of ferroelectric,

we study a possibility of sub-critical crack in PZT monocrystal, growing par-

allel to internal 180

◦

domain or phase boundary in the structure of bi-material

[799, 811]. Three mechanisms of the crack braking at 90

◦

-domain boundaries

are studied in [512, 517], namely (i) due to change of the crack orientation,

(ii) due to the crack interaction with non-coherent boundary and (iii) due to

the crack advancing through the boundary with absorbed impurities.

The crystallite with layer domain structure or co-existing phases will be

present as a composite system, which consists of homogeneous layer with

thickness h and half-inﬁnite substrate (see Fig. 8.24). Let their elastic prop-

erties (E,ν) be the same and the homogeneous tensile stresses, acting in the

layer, σ

= Eε, are found by microstrain ε, caused by the thermal and phase

properties of the FC material. The stress analysis shows that the most possi-

ble crack initiation occurs in substrate and it extends parallel to the internal

interface of the composite. We assume a stable (sub-critical) character of the

crack growth. When the crack is localized in substrate at the depth λh below

the interface, then the layer above the crack is subjected to a bend and some

elastic strain energy retains after crack propagation. In order to estimate this

energy, we take into account that the contributions of the strain energy due

to the compressive force P and bending moment M are additive [1059]. Then,

374 8 Computer Simulation of HTSC Microstructure and Toughening

λh

Layer

Substrate

Interface

Neutral Axis

Crack

(a) (b)

Fig. 8.24. Used composite structure: (a)tensionσ

is caused by microstrain ε in

the homogeneous layer; (b) equivalent system with applied force P and bending

moment M

using the theory of composite beam and based on the approach of [221], we

obtain the asymptotic strain energy release rate, G, for steady-state crack with

length c as

G =

∂

∂c



2EA

2EIh



, (8.49)

where the dimensionless moment of inertia, I, of the beam per unit layer

thickness and the eﬀective cross-section, A,are

I =(λ +1)

/12 ; A = h(1 + λ); λ = z/h . (8.50)

The load P and the bending moment M (per unit thickness) caused by

uniform tensile stress σ

are deﬁned as

P = σ

h ; M = σ

λ/2 . (8.51)

Then, we have from (8.49) to (8.51):

G =

h(4λ

+2λ +1)

2E(λ +1)

. (8.52)

Further, the dimensional analysis gives relations for modes I and II stress

intensity factors, K

and K

, assuming that each of them has own contribu-

tions from the force P and the moment M [221]:

= C

−1/2

f(λ)+C

−3/2

g(λ) ; (8.53)

= C

−1/2

f(λ)+C

−3/2

g(λ) , (8.54)

where C

are unknown constants and f (λ),g(λ) are unknown functions. An

assumption about simplicity of the functions f(λ)andg(λ) is supported by

the computational results of [221]. Comparing (8.49) and (8.52) with the re-

lationship: GE = K

+ K

, connecting the strain energy release rate with the

SIFs, we obtain:

+ C

=1/2I ;(C

+ C

= h/2A. (8.55)

8.2 Twinning Processes in Ferroelastics and Ferroelectrics 375

Assume that the factors C

contribute only to the constant terms of (8.55),

while the functions f(λ)andg(λ) contribute to the variables; then (8.55) are

reduced to

+ C

)=1/2; (C

+ C

)=1/2; g = I

−1/2

; f =(h/A)

1/2

(8.56)

In order to deﬁne ﬁnally the constants of C

, it is necessary to suggest

a strict solubility of the problem for the case of either P =0orM =0.

The validity of the last condition in the present formulation of the problem

and also the construction and computational solution of the corresponding

integral equation for SIF at half-inﬁnite crack propagation near free surface

have been represented in [1059]. The unknown constants have the values: C

0.434 and C

=0.558. Besides, there is an additional condition for unknown

constants: C

+ C

= 0. Then, we obtain from (8.56), C

= C

=0.558;

= −C

= −0.434; and the relationships for K

and K

are obtained from

(8.53) and (8.54), substituting the corresponding parameters, as

1/2

(1 + λ)

1/2



0.434 + 0.966

λ +1



; (8.57)

1/2

(1 + λ)

1/2



0.558 − 0.752

λ +1



. (8.58)

As is followed from test data for diﬀerent bimaterial systems and the load-

ing schemes [221, 1059], the crack path, corresponding to the simple Mode

I stress intensity demonstrates a surprising stability, and fracture resistance

values, obtained under conditions of the Mode I, have a ﬁne reproducibil-

ity to compare with the results for mixed mode. Furthermore, as the condi-

tion of steady-state crack into ferroelectric domain structure, we select the

following one:

=0. (8.59)

Hence, from (8.58), the depth λ

∗

, at which the crack has a steady-state

trajectory, namely: λ

∗

= z

∗

/h =2.876 is found. The existence of the asymp-

totic limit permits to calculate the critical layer thickness, h

∗

,belowwhichthe

complete fracture is inhibited. This value is obtained by equating the asymp-

totic value of K

to the corresponding value of the fracture toughness, K

.In

the case of the crack growth into substrate along the steady-state trajectory,

= 0, (8.58) provides the prediction:

∗

=0.755(1 + λ

∗

)(K

/σ

)

. (8.60)

The above assumption about stable crack growth in FC crystallites de-

mands additional discussions and grounds. First, in [343], it has been shown

that a stress concentration near crack tip in the region of the morphotropic

boundary at existence of the tetragonal (T ) and rhombohedral (R) phases can

initiate phase transitions in the ceramic grains. It has been proved that for

376 8 Computer Simulation of HTSC Microstructure and Toughening

short cracks (c<c

∗

), at any great stresses, an initiation of phase transfor-

mations near crack tip is more advantageously compared to its catastrophic

growth. For long cracks (c>c

∗

), the calculation gives a higher critical load

compared to Griﬃth’s formula. Secondly, in [105], the mechanism of local

phase transition near heterogeneities has been considered, in particular at

crack tip. It is shown that an interaction of order parameter with strains leads

to the fact that the stresses, concentrating at heterogeneities, can transform

the media in other phases [105]. Thus, an occurring of local transformational

plasticity in the vicinity of the crack tip is possible, dragging its growth.

Thirdly, a non-linear decreasing of the mechanical strength has been observed

at a tension with an increasing of a poling ﬁeld in the poling of the FC on the

base of PbTiO

[350]. This non-linear behavior is explained by non-stationary

processes of microcracks nucleation and growth due to their interaction with

re-constructing domain structure and defects. In particular, it is assumed that

the vacancies, released from connections after disappearance of domain walls

during poling, can form “clouds” near tips of the cracks with critical size and

restrain their growth. Finally, the cracks in BaTiO

monocrystals which have

been observed near indenter imprint at the (001) facet and grown along the

cleavage planes of {100}, {110} and {111} can restrain up to total stop at

strong ﬁxed 90

◦

domain boundaries which are similar to the twin boundaries

[512]. Hence, it is seem to be correct a conclusion about principle possibility

of stable, sub-critical crack growth into domain structure of FC.

As has been shown by experiments [588], the strains inside crystallites in

paraelectric phase change from 5 × 10

−5

to 2 × 10

−4

in the morphotropic

transition regions in dependence on the sintering temperature of PZT ceram-

ics. Similar strains in grains of the T -phase are independent of temperature

within test error. They exceed the microstrains in the paraelectric phase, at-

taining the values ∼ (6 ± 3) × 10

−4

. The microstrains in the crystallites of

T -andP -phases coincide in the order of magnitude. Therefore, in computer

simulations, we consider the microstrain range: ε =5× 10

−5

− 9 × 10

−4

,for

deﬁnition of the uniform tensile stress of σ

, leading to the crack advancing.

Select the values of Young’s modulus, E = 63–100 GPa [849], and also

the fracture toughness of PZT monocrystals, K

. In relation to the latter,

note that the poling processes lead to the fracture toughness anisotropy as

of whole ceramic, as of single crystallites. In the FC ceramics, the value of

attains maximum value along the poling direction and minimum value in

the perpendicular direction [813, 849]. However, an anisotropy of the tetrag-

onal ferroelectric monocrystals shows another character, namely: the fracture

toughness in a plane perpendicular to the direction of spontaneous poling ex-

ceeds signiﬁcantly the toughness values in other directions. Fracture in this

plane is not advantageous energetically, because it leads to a greater sur-

face density of charges with diﬀerent signs at the opposite surfaces of the

crack [512]. This fracture character is experimentally supported by prevail-

ing growth of microcracks through grain bulk in the poling direction [350].

These features of anisotropic behavior can be explained: in the polarized

8.2 Twinning Processes in Ferroelastics and Ferroelectrics 377

ceramic, a signiﬁcant part of domains remains oriented in arbitrary direction,

not contributing to the anisotropy of K

. Moreover, for ﬁne-grain ceramics,

an intergranular fracture is proper, which excludes the crack interaction with

domain structure [849]. Due to complex behavior of the fracture toughness,

we use in calculations the experimental values: K

=0.4–1.4MPa m

1/2

[849].

Finally, we deﬁne the layer thickness as h = nh

,whereh

≈ 0.2 μmisthe

equilibrium domain width for grain size, D =10μm, based on the simulation

of the stress sources at the boundaries of polydomain grains, using contin-

uously distributed dislocations [840]; n is the domain number in crystallite

under uniform tension, σ

. Obviously, in order for the coordinate of z

∗

for

steady-state crack path to be in grain, it is necessary to select h

max

=3.4 μm,

then n

max

= 17.

Further, (8.57) and (8.60) give K

=0.585σ

1/2

,andh

∗

=2.926

/σ

)

,whereK

is the SIF for steady-state crack growth at the depth of

∗

h,andh

∗

is the limit layer thickness, deﬁning a restraint fracture. The se-

lection of the parameters, corresponding to the maximum of K

and minimum

of h

∗

,givesε =9×10

−4

; K

=0.4MPa m

1/2

; h =3.4 μm; E = 100 GPa. As a

result, we obtain K

=0.097 MPa m

1/2

∗

=57.8 μm. Obviously, K

<< K

and h

∗

>> D for the considered PZT parameters. Hence, the stable (i.e., sub-

critical) crack growth is impossible parallel to or along the interfaces in the

considered crystallites. The nucleated crack at this boundary will propagate

catastrophically until it meets a ﬁxed 90

◦

-domain boundary. Thus, there may

be stable crack growth in FC crystallites only in the cases of crack braking by

the 90

◦

-domain boundaries [512, 517].

8.2.3 Thermodynamics of Martensitic Transformation in HTSC

Now, investigate the twinning processes in YBCO superconductor [815]. As

has been noted above, due to the grain TEA, the cracking of HTSC ceramics

occurs during cooling that deteriorates mechanical and strength properties. At

the same time, a twinning process zone, forming around an advancing crack,

as may be suggested similar to ferroelectric ceramics, should lead to the crack

shielding and to the fracture toughness increasing. In this case, a relaxation

of internal stresses of second type, nucleated in the HTSC, occurs in two

ways [277]: (i) due to the crack nucleation, which is proper for temperatures,

T ≤ 500

◦

C, and (ii) owing to the martensitic process of re-construction of the

grain domain structure, that is most intensive at T ≥ 500

◦

During oxygen annealing of HTSC, two characteristic sites are observed in

the curve of oxygen parameter (x) depending on time (t). At the initial “fast”

stage of the process, where an accumulation of oxygen by cuprate follows to

diﬀusive regime, the dependence “x–t” is subjected to parabolic law. At the

next “slow” stage, the kinetics of reaction are found by relaxation of elastic

stresses, and oxidation follows to ∂x/∂t ≈ const regime, that is, to linear law

[277, 278]. As has been shown in [277], YBa

7−x

compound possesses

the “shape memory eﬀect” that speaks about martensitic mechanism of the

378 8 Computer Simulation of HTSC Microstructure and Toughening

stress relaxation at the slow stage of the material oxidation. Therefore, it may

be assumed that deletion of the internal stresses of second type initiated near

growing macrocrack can occur, according to the martensitic mechanism in

account of the energetically advantageous re-construction of domain structure.

We shall assume that the stresses, introduced in the twinning process,

to be shear. The normal stresses, deﬁning a boundary microcracking, give a

secondary contribution in the energy of plastic deformation and do not take

into account here.

Estimate the critical number of twins, η

, in spherical grain with radius R,

corresponding to the martensitic transformation. With this aim, consider al-

terations in the thermodynamic potential, which accompany formation of the

twinned martensite. They include the increment of the mechanical potential,

ΔΦ

, consisting of change of the strain energy and interaction energy, the

increment of the surface energy, ΔΦ

, and also the increment of the chemical

potential, ΔF

. The last is independent of geometrical parameters of twins,

and for spherical particle with radius R, the increment of the chemical free

energy per crystallite is found by [863]

ΔF

= −(4/3)πR

ΔF

, (8.61)

where

ΔF

= P

Δe

= P

ΔV, (8.62)

where P

is the stress, leading to the transformation; Δe

is the diﬀerence

of components of the strain tensor; ΔV is the volumetric deformation at the

transformation; and Kronecker’s delta-function is found as



1 i = j

0 i = j

(8.63)

The surface work is deﬁned by changes occurring at the martensite inter-

face and during formation of the twin boundaries. The total increment of the

surface energy, ΔΦ

, for spherical grain is calculated as [261]

ΔΦ

=4πR

ΔΓ

+(2/3)πR

(η − 1)(η +1)Γ

/η , (8.64)

where ΔΓ

is the diﬀerence of the surface energy; Γ

is the energy of twin

boundary; η =2R/d is the number of twins per grain; d is the twin extent

(see Fig. 8.25).

The diﬀerence of the mechanical energy has two terms, namely (i) due

to the twinning, ΔΦ

, and (ii) owing to macroscopic change of the grain

shape, ΔΦ

. The component caused by the twinning is calculated using

Eshelby’s method [259] for study of the tangential stress state, developing in

the martensite plates, consisting of twins. Then, for Poisson’s ratio (ν =0.2),

we have [261]

ΔΦ

1.6πR

Gγ

3η

, (8.65)

8.2 Twinning Processes in Ferroelastics and Ferroelectrics 379

(a)

(b)

(c)

Fig. 8.25. (a) No restraint shape of particle before start of twinning; the direc-

tions of tension and compression along main axes, characterizing transformation are

shown by arrows;(b) the same particle after twinning (γ

is the shear angle, d is

the twin extent); (c) restoration of initial shape, using uniform tangential stresses

(τ = Gγ

/2), shown by arrows

where G is the shear modulus, γ

is the shear deformation at the twinning.

This term is determined by the tangential stresses, an action which is re-

strained by the zone close to the interface “parent phase–twinning product”.

Moreover, due to neglectfulness by the normal stresses, it may be assumed

that (8.65) presents lower boundary for possible values of ΔΦ

Macroscopic component includes the strain energy and the energy of inter-

action with applied stress p

. We are limited by only dilatation component of

the applied stress, p

(neglecting deviator component) as while the fraction

of transformed particle contains a set of planes, formed by shear, there are

altogether alternative planes subjected to shear of opposite sign. Therefore,

appreciable macroscopic shear strains apparently are diﬃcult expected even

in the case of ﬁnite number of twins. Then, ΔΦ

is found as [265]

ΔΦ

=(4/3)πR

ΔV (0.14EΔV − p

) , (8.66)

where E =2G(1 + ν) is Young’s modulus.

380 8 Computer Simulation of HTSC Microstructure and Toughening

Taking into account (8.61), (8.64)–(8.66), we obtain a change of the total

potential due to the transformation as

ΔΦ

(4/3)πR

= −ΔF

6ΔΓ

ηd

(η

− 1)

+0.14EΔV

− p

ΔV +

0.4

Gγ

(8.67)

Thus, the transformation will be deﬁned by four main parameters, namely

(i) the chemical free energy (ΔF

), (ii) the grain radius (R), (iii) the extent

of twins (d) and (iv) the applied hydrostatic component of stress (p

In order to estimate a critical value of η

=2R

/d, corresponding to the

transformation (where R

is the critical grain radius), we equate the change

of the total potential, ΔΦ, to zero. In this condition, there is minimum value

of the critical number domains per grain, η

, at which the transformation will

start. Then, we have from (8.67):

−ΔF

6ΔΓ

(η

− 1)

+0.14EΔV

− p

ΔV +

0.4

Gγ

=0. (8.68)

Next, estimate the twin extent, d, using a modeling of the stress source at

the boundaries of polydomain grains by the continuously distributed disloca-

tions as [840]

d =



12.6GS



1/2

, (8.69)

where S

is the spontaneous shear strain; σ

is the energy of 90

◦

-domain

wall. Note that the square root dependence d ∼ D

1/2

has also been predicted

for partially stabilized ZrO

at D>>d[261]. Moreover, the critical grain

size, D

∗

, lesser than which the domain twinning does not occur, is estimated

from the condition, D = d. Hence

∗

12.6GS

. (8.70)

This value corresponds to the grain, which has no suﬃcient elastic energy

to render signiﬁcant inﬂuence on the agreement of domain wall, possessing

the energy σ

Substituting (8.69) into (8.68) and assuming p

= 0 (i.e., an absence of

applied stresses), we obtain ﬁnally the equation for calculation of the critical

value of η

8.2.4 About Toughening of Superconducting Ceramics

Today, a special value has consideration of stable (sub-critical) crack growth

at study of fracture processes in the framework of microstructure fracture

mechanics. Diﬀering from unstable (catastrophic) fracture, submitting to the

classic Griﬃth–Irwin’s theory, it can take into account internal mechanisms of