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

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8.1 YBCO Ceramic Sintering and Fracture 361

applied loading, σ

cat

, at which the fracture becomes unstable or catastrophic

[163, 657]:

cat



4χP d

(1 − 2τ)



2/(2τ+3)

; (8.40)

cat



(2τ +3)

4ψd



(2τ −1)/2

cat

. (8.41)

Then, the inert strength, σ

(P ), may be estimated in the kinetic eﬀects

absence, considering a condition of unlimited fracture of the ceramic under

loading of P due to indentation by using Vickers pyramid [62]. Then, the

applied tensile stress, σ

(c), as function of the crack length, c,iscalculated,

using the toughness magnitude, T ,as

(c)=

ψc

1/2



T (c) −

χP

3/2



. (8.42)

Generally (for non-cubic, ferroelectric and superconducting ceramics), the

function of σ

(see Fig. 8.15b) [803, 811]. The ﬁrst maximum (for c<d) is associated with

the residual contact ﬁeld that dominates. The second maximum (for c>d)is

associated with the bridging that dominates. The appropriate barrier heights

depend on the external load P . When crack overcomes the ﬁrst barrier (for

c<d), the crack becomes unstable. However, the crack grows spontaneously

up to catastrophic failure if the second barrier (for c>d) turns out below the

ﬁrst barrier. Otherwise, the macroscopic fracture is possible only if the load to

be suﬃcient that the crack overcomes the second barrier. Hence, the ceramic

strength, σ

, is found by the greater maximum. Note also that the transition

point to the horizontal section of the curve T (c) in Fig. 8.15a corresponds to

the crack length, c = c

∗

[814].

Crack Length, c Crack Length, c

Toughness, T

c = d

Applied Stress, σ

(a) (b)

Fig. 8.15. Qualitative trends in the toughness (a) and in the applied loading (b)

as functions of crack length. Solid and dashed curves demonstrate the dependences

before and after grain growth, respectively

362 8 Computer Simulation of HTSC Microstructure and Toughening

8.1.8 Some Numerical Results

In order to obtain numerical results, the known parameters of YBa

7−x

and some data for related ceramics have been used. A necessary number of

the computer realization has been established on the basis of stereological ap-

proach. The corresponding procedure for deﬁnition of unbiased estimation of

any stereological characteristic [145] has been presented in Sect. 5.5. We con-

sider the mean grain radius as this stereological characteristic in the computer

simulations.

Young’s modulus is deﬁned as E =3K(1−2ν), using the volume modulus,

K = γc/(βV ), where γ =1.35 is Gr¨uneisen’s constant; c/V =2.72 MPa/K

is the characteristic volume heat; β =5.86 × 10

−5

−1

is the mean volume

thermal expansion factor; ν =0.21; K

=1MPam

1/2

[335]. Selecting δ = l

we estimate an eﬀect of the initial sample porosity before sintering, C

,and

the inhibition parameter of grain growth, I

, on diﬀerent strength charac-

teristics. The obtained numerical results are shown in Figs. 8.16–8.18 and in

Table 8.5 [814].

The calculations show a growth of the mean grain size with the decreasing

of C

; at the same time, the insigniﬁcant alteration of closed porosity is ob-

served in the considered cases. This causes a formation of longer microcracks

at smaller values of C

. However, the eﬀect of grain growth stagnation and

existence of a smaller number of triple points (nuclei of microcracks) state

the insigniﬁcant change of the microcracking density β

(1)

(after cooling) with

attainment of its minimum value for C

= 30%.

0.13

0.11

0.09

0.07

0.05

2.4

2.3

2.2

2.1

2.0

05040302010

p(%)

/σ

√l

/σ

√l

/σ

√l

/σ

√l

/σ

√l

; K

/σ

√l

Fig. 8.16. SIF vs initial porosity of precursor sample, C

. The deviation of results

corresponding to interval change of magnitude of the abnormal grain-growth inhi-

bition parameter (I

=0.4 − 1.0) is shown

8.1 YBCO Ceramic Sintering and Fracture 363

260

240

220

200

180

1300

1200

1100

1000

900

(MPa)

(μm)

05040302010

C p(%)

Fig. 8.17. Fracture stress, σ

, and critical grain size, D

, causing the further crack

growth vs initial porosity of precursor sample, C

=0.4 −1.0)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

2.5

2.0

1.5

1.0

0.5

0.0

p/a

= 1/16

1/8

1/4

1/2

ϕ/2 = 10°

Fig. 8.18. Fracture toughness, K

, vs microcracking density, β

,fordiﬀerent

sizes of process zone for crack branching (p/a

)atI

=2.0

364 8 Computer Simulation of HTSC Microstructure and Toughening

Table 8.5. Fracture toughness change at crack branching (I

=2.0)

Properties C

=0%C

= 10% C

= 20% C

= 30% C

= 40% C

= 50%

(1)

0.17 0.17 0.13 0.12 0.15 0.17

,p/a

=1/8 1.04 1.04 1.03 1.03 1.03 1.04

,p/a

=1/3 0.99 0.99 1.00 1.00 1.00 0.99

,p/a

=1/2 0.96 0.96 0.97 0.97 0.97 0.96

(2)

0.28 0.31 0.32 0.33 0.39 0.44

,p/a

=1/8 1.08 1.09 1.10 1.10 1.14 1.19

,p/a

=1/3 0.98 0.98 0.98 0.97 0.96 0.92

,p/a

=1/2 0.90 0.88 0.87 0.86 0.77 0.62

(1)

is the microcracking density after cooling; β

(2)

is the summary density of

microcracks, formed during cooling and in the process zone near macrocrack.

Joint eﬀects of microcracking and grain size on strength properties are

found by the ratio, S/R. This parameter owing to above results prevails com-

pared to dependence on the closed porosity of the sample. This explains, at

ﬁrst view, the paradoxical result of the fracture stress, σ

, increasing with the

initial porosity, C

(see Fig. 8.16). The last result follows from the concept

that the fracture stress increases with decreasing of the grain size. In addi-

tion, the same trend retains in the dependence of σ

on S/R (at ﬁxed value of

the grain size) [583]. The obtained values of the fracture stress are near the

test data observed by the electronic microscopy method for YBa

7−x

ceramic (σ

= 300 MPa) [567]. The SIF dependencies (K

)andthe

critical grain size for further crack growth (D

) as function of C

,presented

in Figs. 8.16 and 8.17, reach extremes in intermediate points of the interval

values of C

. This coincides with the observed trends for non-cubic [583] and

ferroelectric [849] ceramics. It should be noted that the SIF, corresponding

to the tangential component of thermal stresses (K

) is smaller than the SIF

caused by the radial component (K

). Due to K

cannot be independently

used to deﬁne a critical condition of cracking. Similarly, the spontaneous

microcracking during cooling to be signiﬁcant phenomenon of the I mode of

fracture in the ceramics with non-cubic symmetry, which demonstrate brittle

fracture [609].

The numerical results (Fig. 8.18 and Table 8.5) show that fracture tough-

ness increases together with the crack branching angle and at interaction of

macrocrack with microcracks in the short process zone of the crack branching.

However, in the case of the process zone increasing, the dominating inﬂuence

of the crack branching is replaced by the microcracking eﬀect, decreasing the

fracture toughness. The transition from toughening to the crack ampliﬁcation

is deﬁned by the process zone size, p/a

≈ 1/3 (for the branching angle,

ϕ/2=10

◦

). The weak dependence of K

on C

at ﬁxed value of p/a

is caused by the small microcracking density, β

=0.12–0.17 (after the sam-

ple cooling). An increase of β

=0.28–0.44 (on account of damage into the

8.1 YBCO Ceramic Sintering and Fracture 365

microcracking process zone) leads to formation of the fracture toughness de-

pendence on the initial sample porosity. The analysis of the dependence of

on β

(see Fig. 8.18) shows that an eﬀect of the crack ampliﬁcation

< 1) takes place for all sizes of the crack branching zone and ex-

ists in the case of the microcrack coalescence, at β

≈ 0.5. The last value is

usually selected as the critical one for initiation of the crack coalescence in

ceramics [294].

In order to estimate the eﬀect of the crack bridging, we use the next data

[62, 162, 796]: ∈

=0.1; μ =1.8; γ

≈ 2γ

=6J/m

; σ

= 100 MPa; H =

19.1 GPa. We take into account in the present case only microcracks formed

during cooling. The microcracking due to macrocrack advancing changes

quantitative results, but qualitative ones, presented in Fig. 8.15, are re-

tained. In modeling of abnormal grain growth, the characteristic values of

=2.20 and 2.84 (the case of no porous ceramic) have been esti-

mated preliminarily for two values of the grain growth inhibition parameter,

=2.0and1.2(D

is the grain size before beginning of the grain growth).

As is followed from Fig. 8.15a, the eﬀect of the grain size on the change of

T -curve is not simple. This is explained by an additional microcracking ef-

fect of intergranular boundaries. In the ﬁeld c<d, an enhancement of the

value d will lead to corresponding diminishing of left part of the T (c)curve

and at some value of parameter d,theT (c) curve crosses the c-axis before

fulﬁllment of the condition: c = d. Then, primarily existing microcracks and

residual stresses, formed during the ceramic processing, will lead to unstable

macrocrack advancing without dependence on applied load. In this case, the

spontaneous microcracking prevails over the toughening eﬀects, caused by the

crack bridging, and states the ceramic strength. The critical value of the pa-

rameter d is estimated from the condition: T =0,atc = d. Hence, for the

sizes of grains-bridges

d ≥ (T

/ψσ

)

, (8.43)

a positive eﬀect of crack bridging is exceeded by the eﬀects of “deleterious”

microcracking.

Other condition for deﬁnition of the critical value of the parameter d can

be stated from (8.4). Assuming that the sizes of grains and facets per grain are

subjected to normal distribution, we obtain [176]: D

≈ 2l

. Then, the mean

separation between bridging grains, d, coinciding with the characteristic grain

size can be estimated as

d<2β

(1 + ν)/(Eε)]

. (8.44)

The selection of the condition for d as well as for ﬁxing of the grains-

bridges into ceramic microstructure should be carried out on the basis of

experimental data. Then note that the de-bonding region (d ≤ c ≤ c

)is

very short (c

/d ≤ 1.01) and is not observed practically in Fig. 8.15. Thus,

pushing of grain by the surfaces of the growing crack mainly contributes to the

increasing of fracture resistance on account of existence of the residual thermal

366 8 Computer Simulation of HTSC Microstructure and Toughening

stresses. This conclusion is supported by theoretical analysis and numerical

results of [62, 1113].

The parameters characterizing a capability of the YBCO ceramics to frac-

ture resistance in the considered cases (D

=2.20 and 2.84) are, respec-

tively, equal to T

∞

=2.01 and 2.17 (from (8.35)); c

∗

/d =15.6 and 17.1

(from (8.36)). They deﬁne a signiﬁcant eﬀectiveness of present process as the

toughening mechanism of superconducting ceramics. Generally, these parame-

ters increase with grain size growth (see Fig. 8.15) for the same initial porosity,

, that corresponds to experimental observations [1113].

Using the above computational approach, the microstructure parameters

required to estimate trans- and intergranular fracture can be calculated [800,

821]. The numerical data, characterizing the YBCO microstructure at diﬀer-

ent initial porosity of C

, are given in Table 8.6. The computational procedure

includes the next stages. First, we deﬁne the grain-boundary intercept length,

λ, and ratio of the largest grain size in the material to the average one, η.

Then, the grains-bridges are selected at the lattice directly (in every case

there are 10% of grains from those, that are similar to the experimental data

for related compositions [1113]). These are the greatest grains and we select

the value of d so that it is equal to the average size of the grains-bridges.

Next, the transgranular fracture of restraints and failure of the intergranular

boundaries between bridges are simulated (see Fig. 8.19b), using the graph

theory for deﬁnition of the shortest crack route at the lattice of the intergran-

ular boundaries [72]. After that, the proportion of transgranular failure, A

is found. The sections of the transgranular failure are determined using the

characteristic size of the bridge, but the sections of the intergranular fracture

are deﬁned using the direct crack path along grain boundaries. Further, we

choose the parameter u

∗

for the above ligaments, namely: u

∗

/λ ∼ 0.01 for

elastic and frictional restraints [162] and u

∗

/λ ∼ η

for compressive trac-

tion [1029], where the strain for YBCO due to the TEA is ε

=3.1 × 10

−3

(see Sect. 8.1.2). Finally, we determine the values of c

∗

and T

/(T

∞

−T

)from

(8.37)–(8.39).

The computations demonstrate the value λ ≈ 1.83δ without dependence

on the initial porosity of C

. Then, the magnitudes of u

∗

/λ,presentedin

Table 8.6 for the case of compressive ligaments, coincide well with the cases of

Table 8.6. Some model parameters for YBCO ceramic

Properties C

=0% C

= 20% C

= 40% C

= 60%

η 1.498 1.636 1.758 1.842

0.312 0.293 0.274 0.442

d/δ 3.423 3.270 3.028 2.646

∗

/λ 0.0070 0.0083 0.0096 0.0105

∗

/d 10.56 11.05 11.92 13.62

5.32 7.66 10.98 15.00

8.1 YBCO Ceramic Sintering and Fracture 367

Fig. 8.19. (a) Representation of YBCO ceramic structure fragment in PC (voids

are denoted by 1 and depicted by gray color), and (b) macrocrack propagation (gray

line) into model structure between two grains-bridges depicted by black color

the elastic and frictional restraints. The ratio of the largest grain to the aver-

age one (η) rises with initial porosity, showing a growth of the heterogeneity

of grain sizes. This is due to the outstripping decrease of the average grain

size compared with the highest one. Further, there is growth of the grain size

and grain-bridge with the initial porosity decrease. Moreover, the numerical

data give dependence, d = kλ(1 <k<2), as being similar to alumina ce-

ramics [162]. The intergranular fracture is decreased with the initial porosity

growth, because high sinuosity of a crack trajectory corresponds to the coarse-

grained structures. However, the fraction of the transgranular fracture (A

)

does not change monotonously due to the relative equalizing of the transgran-

ular failures in the various cases of C

and owing to a rise of the structural

parameter (η) with the initial porosity. Thus, the most ﬁne-grained structure

with C

= 60%, but possessing the greatest value of η, has the increased

toughening eﬀects, which correlate with increased area fractions of transgran-

ular failure [162]. On the other hand, this result supports a known thesis about

necessary to characterize a polycrystalline structure by a distribution of grain

sizes, but no their maximum (or minimum) value.

The parameters which characterize the ability of superconducting ceramics

to resist failure are ratios of T

/(T

∞

−T

)andc

∗

/d.Thenumericalresultsfor

the latter (obtained from (8.37)) are shown in Table 8.6 where the upper line

corresponds to the elastic and frictional ligaments, but lower line conforms to

the compressive traction due to TEA. The observed growth of c

∗

/d together

with the initial porosity for all restraints is caused by appropriate rise of the

368 8 Computer Simulation of HTSC Microstructure and Toughening

1.0

0.5

0.0

(μm)

/(T

∞

– T

)

200 600 1000 1400 1800

Fig. 8.20. Microstructure contributions of T

/(T

∞

−T

) to the HTSC ceramic from

elastic (1), frictional (2) and compressive (3) ligaments vs crack length, c

structure heterogeneity parameter, η, and coincides with experimental data

for diﬀerent ceramics [62, 162, 1113]. In Fig. 8.20, there are microstructure

contributions of T

/(T

∞

−T

) due to the crack bridging mechanism as function

of the crack length c for the above types of ligaments, and C

=0%.Obviously,

the main contribution to the toughening is found by the compressive traction

due to TEA, and the least one corresponds to the elastic links. These trends

are supported by the experiments [62, 1113] as well.

The critical parameters c

cat

,andσ

cat

(see (8.40) and (8.41)) are shown

in Fig. 8.21 depending on the load P and on diﬀerent toughening powers τ

for characteristic grain size, D

/δ =2.20. The results show an increasing of

the critical crack length, c

cat

, and a decreasing of the critical applied stress,

P (N)

200

150

100

cat

(μm)

cat

(MPa)

–1

Fig. 8.21. Critical parameters σ

cat

and c

cat

vs contact load P for two toughening

powers: τ =0.25(1) and τ =0.10(2)

8.2 Twinning Processes in Ferroelastics and Ferroelectrics 369

cat

, with growth of the contact load, P , without dependence on features of

abnormal grain growth [814]. However, the changes of these parameters from

the toughening power have diﬀerent characters.

Thus, the inhibition parameter of grain growth (I

) and the mean spac-

ing between bridging grains (d) are parameters which should be taken into

account in the calculation and optimization of microstructure and strength

parameters of HTSC ceramics. The ﬁrst parameter is stated by the history

of the ceramic processing and by its composition (i.e., thermal treatment,

admixture additions in press-powders). The second one is the key parame-

ter of the bridging mechanism, which states a transition from the toughening

eﬀects to the ampliﬁcation crack. At the same time, the account of these pa-

rameters does not reject but, on the contrary, proposes to use in calculations

other characteristics, in particular, the toughness of intergranular boundaries

), residual stresses (σ

) and coeﬃcient of the sliding friction at pushing

of grains by surfaces of growing crack (μ). Hence, a design of YBCO ce-

ramic microstructure, which is optimum from the view of material strength,

is connected with introducing of the grains-bridges on the path of the possible

macrocrack propagation at corresponding suppression of “deleterious” micro-

cracking in this zone. This assumes a requirement to form superconducting

grains with maximum permissible sizes, which do not exceed the critical value

of the spontaneous cracking, and with distribution, demonstrating a maxi-

mum possible parameter of the structure heterogeneity, η. The “deleterious”

microcracking is formed owing to the grain growth that can be regulated by

admixture phases, pulled out during sintering at intergranular boundaries. As

a result, it is possible to predict properties of the sintered ceramics even at the

earlier stages of the ceramic fabrication in dependence on, for example, the

sintering technique, thermal treatment and parameters of secondary phases.

8.2 Twinning Processes in Ferroelastics

and Ferroelectrics

It is well known that YBCO ceramics possess ferroelastic properties owing to

domain (platelet) structure. Therefore, a suﬃciently great attention has been

spared to description of microstructure and residual stresses of YBCO poly-

crystals in the framework of the numerical method for description of twinning

processes [27, 270, 840] and also to computer simulation of twins [1123].

Single crystallites in polycrystalline ferroelastics (YBCO) and ferroelectrics

(BaTiO

, PZT, etc.) at structure phase transition are subjected to twinning,

which is a result of the elastic energy minimization. The twin formation leads

to signiﬁcant decreasing of quasi-uniform stresses in grain volume at simul-

taneous formation of considerable heterogeneous stresses localized near inter-

granular and interphase boundaries [27, 261, 840]. A twinning as a sequence

of phase transformation, causing a small structure change of unit cell, can be

described as mechanical twinning [27, 517] or as martensitic transformation

370 8 Computer Simulation of HTSC Microstructure and Toughening

[192, 262]. The transition accompanied by diﬀusion, in some cases, also leads

to the twinning.

The mechanical twinning in ferroelastics can be compared

with formation of ferroelectric domains. In the latter case, an energy decrease

for electric ﬁeld is accompanied by the energy diminution for domain walls

that leads to minimization of total energy of the system. In the ferroelastics,

this summary energy, including elastic energy and energy of domain walls,

also becomes minimum. The domain walls displacement can be accompanied

by change of the crystallite shape and serves the sources of residual stresses.

From the view of inﬂuence on fracture toughness, a change of domain struc-

ture in vicinity of crack under mechanical loading presents a special interest.

This process aﬀects the crack growth. A relationship of domain structures (see

Fig. 8.22) and twinning features [27, 840] prompts a possibility of compara-

tive analysis of the ferroelectrics and ferroelastics behavior. An inﬂuence of

the twinning processes on fracture toughness of these samples and appropriate

eﬀects on fracture resistance are estimated below.

8.2.1 Domain Structure and Fracture of Ferroelectric Ceramic

It is well known that the fracture toughness anisotropy of ferroelectric ce-

ramic (FC) is the sequence of structure changes during poling. There are

three main causes of formation of the fracture toughness anisotropy [849]: (i)

the toughness anisotropy of single crystallites (due to the crack interaction

with the boundaries of twins), the greater part of which is oriented along the

poling direction; (ii) the anisotropy of internal (residual) stresses into micro-

volume of piezoelement, leading to the crack advancing to higher values of

in poling plane and to lower values in perpendicular plane; (iii) the plas-

tic deformation of twinning near crack tip. The investigations of (i) and (ii)

cases have been carried out using computer simulations in [517, 813]. Here,

consider the twinning eﬀects on the strength properties, taking into account

the ceramic ferrohardness and initial porosity on example of PZT ceramics,

(a) (b)

Fig. 8.22. (a) Simple platelet twinning in YBCO and (b) strip structure of twins

in BaTiO

In particular, the tetragonal–orthorhombic phase transition in YBCO supercon-

ductor demands oxygen diﬀusion into crystalline lattice [277].