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

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8.1 YBCO Ceramic Sintering and Fracture 351

Fig. 8.8. (a) Fracture toughness of ceramic vs grain size [294] and (b)thesimulation

results obtained for PbTiO

ceramic

can be estimated, using Cherepanov–Rice’s J-integral [884]. We obtain in the

case of stationary macrocrack [294]

∞

=(1− χ

)



1 −

)

(16/45)(1 − ν

)(10 − 3ν)/(2 − ν)

∞

1 − [(16/45)(1 − ν

)(10 − 3ν)/(2 − ν)] χ



1/2

, (8.26)

where K

∞

are the ceramic fracture toughness at existence and absence

of microcracks, respectively; χ

,χ

∞

are the fraction of cracked boundaries

in the process zone and in the sintered ceramic. The ﬁrst multiplier in (8.26)

deﬁnes a change of local fracture toughness, K

,inthecracktip.In

calculations for χ

and χ

∞

, the following formulas are used:

; χ

∞

, (8.27)

where N

∞

is the microcrack number after spontaneous cracking and the

boundary number in the sample area, S, respectively; L is the length of the

macrocrack path.

352 8 Computer Simulation of HTSC Microstructure and Toughening

The macrocrack shielding caused by microcracking in process zone is esti-

mated, using a model of crack growth under monotonous increasing loading.

Assuming that microcracks are the array of isotropic oriented disk-like cracks,

it may be obtained [608] that

∞



(1 − ν



1/2

. (8.28)

In this case, Young’s modulus, E

, and Poisson’s ratio, ν

, for cracked

material may be estimated using a model based on averaging of strains on mi-

crovolumes, containing misoriented microcracks [892] from (8.16) and (8.17).

The coalescence eﬀect leading to decreasing of the crack growth resistance

is estimated, using a coalescence model of half-inﬁnite crack (having tip in the

point x = 0) with collinear microcrack (having tips in the points: x = a and

x = b, a < b) that is shown in Fig. 8.9. The collinear microcrack is equivalent to

an array of microcracks, distributed in the layer with thickness of h = N

−1/3

which is measured along normal direction to the fracture plane (N is the mean

number of microcracks per volumetric unit). In this case, a/b =1−A,where

A = N

2/3

c

 is the fraction of microcracking square in the fracture plane,

and c

 is the mean cracking square, projected on the fracture plane. Then,

for the condition of quasi-static crack growth, we obtain [896]:

=[aE

/(bE)]

1/2

C, (8.29)

where K

is the SIF, deﬁning the crack coalescence; K

is the intrinsic fracture

toughness of ceramic without microcracks; C = K(k)/E(k); K(k),E(k)are

the elliptic integral of 1 and 2 type, respectively; k =(1− a/b)

1/2

Numerical results obtained for the ceramics without voids, possessing the

perovskite structure (which HTSC also have): BaTiO

(E = 120 GPa,ν =

0.25) and PbTiO

(E =80GPa,ν =0.3) depending on the inhibition parame-

ter of abnormal grain growth (see Table 8.4) lead to the following conclusions.

The increasing of mean grain radius,

R, occurs with the decreasing of the in-

hibition parameter that enhances the fraction of spontaneous-cracked facets

and also (that is not obvious) the diminution of microcracking fraction in the

process zone. The values of h

are shown in Table 8.4 for initial stage

of the macrocrack growth when the process zone size is stated by structure

parameters of sintered ceramic. Taking into account the critical grain size

for Al

ceramic (D

= 400 μm [572]), which has strength properties re-

lated with the BaTiO

ferroelectric ceramic, we obtain the process zone size,

Fig. 8.9. Scheme of interaction of half-inﬁnite crack with collinear microcrack

8.1 YBCO Ceramic Sintering and Fracture 353

Table 8.4. Strength properties of BaTiO

and PbTiO

ferroelectric ceramics [823]

Ceramic I

R/R

∞

BaTiO

1.0 1.00 0.121 0.106 1.53 0.894 0.878 0.910

0.8 1.10 0.161 0.062 1.57 0.938 0.835 0.944

0.6 1.18 0.183 0.059 1.65 0.941 0.810 0.946

0.4 1.33 0.187 0.052 1.70 0.948 0.806 0.952

PbTiO

1.0 1.00 0.121 0. 099 1.56 0.901 0.880 0.912

0.8 1.10 0.161 0.056 1.59 0.944 0.844 0.947

0.6 1.18 0.183 0.056 1.67 0.944 0.821 0.947

0.4 1.33 0.187 0.049 1.73 0.951 0.817 0.951

= 306–340 μm that coincides with analogous test value (h

≈ 300 μm),

ﬁxed for aluminum-oxide ceramic [108].

The numerical results for diﬀerent grain sizes of PbTiO

ferroelectric ce-

ramic (see Fig. 8.8b) [796] support qualitative trends of characteristic be-

havior of the fracture toughness depending on microcracking, pointed in

Fig. 8.8a [264]. The spontaneous cracking during cooling is the factor de-

termining the fracture toughness change, K

∞

, for grain sizes which

are greater than the critical size. Numerical results for ferroelectric ceram-

ics show that in account of the frontal zone of microcracking, the value of

the fracture toughness is equal to 0.80–0.88, which is near to 0.9, obtained

for Al

[264]. The local toughening, K

, to the greatest degree de-

ﬁnes the fracture process at minimum spontaneous cracking. Finally, the

trends in the macrocrack shielding by the process zone coincide with the

known results [608]. The decreasing of the stress intensity at the macrocrack

tip due to the microcracking increasing enhances the macrocrack shielding.

As the numerical results show, the analogous trends are proper to HTSC

ceramics.

8.1.6 Crack Branching

The YBa

7−x

ceramic samples heated to the temperature 700

◦

C–930

◦

have demonstrated strong crack branching at intergranular boundaries [905].

The crack branching due to microcracks in process zone, surrounding the

crack tip, enhances the fracture energy. Non-regularity of crack front, causing

an increasing of ceramic toughness, often occurs in the case of very high rate

of the crack tip. A crack branching in Al

ceramic has been observed due

to TEA of the grains [1149] that is also proper to superconducting ceram-

ics. The growing crack can absorb microcracks and, on the other hand, the

microcracking causes crack branching.

354 8 Computer Simulation of HTSC Microstructure and Toughening

In the modeling of macrocrack propagating into superconducting ceramic

structure along intergranular boundaries, the models based on the graph the-

ory can be used, which have been realized in investigation of PZT ferroelectric

ceramics [513, 515, 518]. As has been noted above, a process zone of microc-

racking forms in crack vicinity due to TEA of grains, creating the macrocrack

shielding (the process zone size, 2h

, is found from (8.25)). The simulation

procedure of microcracks into the process zone repeats the algorithm pre-

sented in the previous section. Assuming an existence of a branching crack

zone with width of 2p in the crack tip (see Fig. 8.10b), apparent fracture

energy can be found as [303]



1 −

πβ

2(3 − πβ

)



−1



[2 cos(ϕ/2)]

{log 2/ log[2 cos(ϕ/2)]−1}

−

pβ

(1 − β

π/2)

1/2



, (8.30)

where γ

,γ

are the fracture energy of grain boundary with and without the

macrocrack branching; ϕ is the branching angle. Note that (8.30) includes a

theoretically possible limit case of the microcracking absence (β

= 0)taking

Crack

(a)

(b)

Fig. 8.10. (a) Model of crack branching and (b) macrocrack tip, showing the zones

of crack branching (2p) and microcracking near crack tip (2h

)

8.1 YBCO Ceramic Sintering and Fracture 355

place, in practice, at creation of ﬁne-grain structure due to corresponding

selection of the ceramic processing technique:

=[2cos(ϕ/2)]

{log 2/ log[2 cos(ϕ/2)]−1}

. (8.31)

Moreover, the fracture toughness change may be estimated in the case of

crack branching as

=(γ

/γ

)

1/2

, (8.32)

where K

is the critical SIF, caused by the crack branching; K

is the frac-

ture toughness of the ceramic without crack branching. Note that β

is the

summary density of microcracks formed during cooling and in the process

zone, 2h

. Additionally, in general, the zones of the crack branching and mi-

crocracking near the crack tip do not coincide with each other (h

= p).

Obviously, the actual toughening eﬀects demand fulﬁllment of the condition

>p.

8.1.7 Crack Bridging

Other toughening mechanism for ceramics with TEA of grains connects

with enhancement of fracture resistance to growing crack due to the for-

mation and fracture of bridges behind the crack front (crack bridging) (see

Fig. 8.11). The TEA creates residual compressive stresses, σ

, keeping the

grains that restraint the crack opening in the sites of their disposition.

The toughness change as a function of crack length, c (T-curve) due to

this toughening mechanism may be estimated on the basis of the following

assumptions [62]:

(1) The intergranular fracture is only considered.

(2) The grain structure is unimodal and uniform with equal probability of

distribution of the compressive and tensile internal stresses (compressed

grains form bridges, and tensile grains deﬁne a matrix).

(3) The solution is constructed in approximation of a “weak shielding”, that

is, the eﬀect of compressive stresses on crack is taken into account in the

SIF balance, but is neglected in the relationship for displacements of crack

surfaces at its opening.

(4) The conception of a “geometric similarity” is used, that is, YBCO ceramic

structure changes only in scale without signiﬁcant alteration of the grain

geometry, but parameters, characterizing the geometrical features of the

microstructure, are invariant.

Then, T -curve can be constructed as a function of growing crack length,

c, as [62]

356 8 Computer Simulation of HTSC Microstructure and Toughening

Crack

Grain

(a)

(b)

De-

bonding Pushing

2u∗ 2u

(c)

(d )

Crack

Fig. 8.11. Formation and fracture of bridges behind growing crack front: (a)

schematic structure fragment; (b) two interaction stages of advancing crack with

grain-bridge (de-bonding with matrix and pushing of grain by the crack surfaces);

(d) penny-shaped crack into region of the crack bridging (sight with side and upper)

8.1 YBCO Ceramic Sintering and Fracture 357

T (c)=T

− ψσ

1/2

c ≤ d

T (c)=T

− ψσ

1/2



1 −

3σ



∗



1/4



1 −



3/4



d ≤ c ≤ c

T (c)=T

− ψσ

1/2



1 −



1 −



1/2





1/2



1 −



1/2



1 −

3σ



∗



1/4



1 −



1/4



+ ψp

1/2





1 −



1/2

−





1/2



1 −



1/2





1 −





∗



1/2



1 −



1/2



∗



1/2



1 −



1/2



≤ c ≤ c

∗

T (c)=T

+0.5ψp

1/2

∗

c ≥ c

∗

(8.33)

where p

=2(∈

μσ

)

1/2

are the shear stresses, forming at the grain

de-bonding; c

is the crack length, corresponding to a cross of the regions

of de-bonding and pushing from matrix of the grains, restraining the crack;

ψ =1.24 is the geometrical parameter, corresponding to a penny-shaped

surface crack; ∈

is the deformation at the fracture of the links; μ is the

factor of the sliding friction. The value of c

is calculated from the equation:

− c

∗









+ σ

)



1/2

− 1



− d

=0. (8.34)

There are two parameters of the T -curve that are need for maximization

to attain necessary tolerance of the ceramic to crack growth. These are T

∞

and c

∗

/d,whereT

=(2γ

)

1/2

is the intrinsic toughness of ceramic, but a

toughness at the steady-state crack growth (T

∞

) and the crack length in which

the far most links from the crack tip begin fracture (c

∗

) are calculated as

∞

= T

+0.5ψp

1/2

∗

; (8.35)

∗



∈

2ψT



, (8.36)

where d is the mean spacing between bridging grains, selected as equal to

characteristic grain size of the ceramic structure; p

=2∈

μσ

are the

stresses caused by the sliding friction; γ

= γ

− γ

(1 − f

)/2 is the speciﬁc

surface energy; γ

is the surface energy in transgranular fracture; γ

is the

358 8 Computer Simulation of HTSC Microstructure and Toughening

fracture energy of grain boundary; f

is the summary fraction of the cracked

facets during the ceramic cooling and in the process zone.

Source of the T -curve is a successive formation and fracture of bridges in

material that act as inhibiting links behind the crack tip. Secondary phases

at grain boundaries (in our case, the BaCuO

–CuO system with admixture

inclusions [61]) play a special role in quantitative interpretation of the T -curve.

On one hand, they lead to brittlement of grain boundaries and to decreasing

their toughness, but on the other hand they inhibit grain growth, leading to

the increasing of ceramic strength. In addition, for related ceramics, there are

phenomena which point to a possible stabilization of the crack growth [162]:

(i) signiﬁcant increase of bore applied load after the elastic limit is exceeded,

followed by a load drop to non-zero stress (see Fig. 8.12b); (ii) the erratic

crack advance is caused by local inhomogeneities in the fracture resistance

(see Fig. 8.13); (iii) the discontinuous crack traces are regions of unruptured

or frictionally locked material which are the restraining ligaments and which

center on large grains (see Fig. 8.14); and (iv) the fracture of the grains-bridges

occurs in transgranular mechanism at the primary intergranular failure. These

phenomena witness the existence of the processes of the bridges formation

behind the fracture front that states an eﬀective toughening mechanism. Due

to this, we use results of [162] below and identify three regions of behavior of

the penny-shaped crack with the radial coordinate c (see Fig. 8.11), namely

(i) the crack motion does not restraint, and T = T

for short cracks (c<d),

20 ms

Time

Applied Load

(a)

(b)

20 ms

Time

Applied Load

Fig. 8.12. Applied loading as function of time: (a) typical response for brittle

material, showing spontaneous fall of the loading to zero after exceeding of the

elastic threshold (E), and (b) response for material at stable crack motion, showing

signiﬁcant increasing bore loading out of elastic limits with following fall of the

applied stress down to some non-zero level (R) [162]

8.1 YBCO Ceramic Sintering and Fracture 359

•

Fracture

Applied Stress

Crack Length

Fig. 8.13. Crack growth vs applied stress (continuous line) for Al

sample,

demonstrating erratic crack advance, caused by local heterogeneities in fracture

resistance. Dashed curve corresponds to material with uniform fracture resistance

(this line is reduced to the same fracture stress) [162]

(a)

(b)

Crack

Fig. 8.14. (a) Ruptures of crack path, localized at big grains in Al

ceramic,

which are regions of unfractured material, forming links that limit the crack growth;

S the crack surfaces and BB the continuous crack into bulk, depicted by dashed

curve; the stresses, which close the crack and are caused by bridges at the crack

surfaces, are shown by arrows;(b) grains connected to one another by friction forces;

the stresses, closing the crack, and are caused by the crack surface friction are also

depicted by arrows [162]

360 8 Computer Simulation of HTSC Microstructure and Toughening

where d is the average distance between adjacent grains-bridges; (ii) the zone

of restraints is active in the region d ≤ c ≤ c

∗

,andT>T

for intermediate

cracks; (iii) the rupture of the restraints occurs at sites distant from the crack

tip, leaving a zone of width c

∗

− d for long cracks (c>c

∗

). The latter zone

expands outward together with the crack. In this case, the toughness reaches

a maximum value for steady-state crack growth. This toughness corresponds

to the crack length, c

∗

, which is calculated as [800]

∗

⎧

⎨

⎩



1+4







1/2

⎫

⎬

⎭

, (8.37)

where β =



∗

ψT



;2u

∗

is the opening of the crack with length c

∗

Then, we consider the constitutive equations (the stress-extension depen-

dencies) for inhibiting ligaments of three types [821], namely (i) σ(u)= − σ

∗

(u/u

∗

) for elastic ligaments, where σ

∗

is the value of the peak restraining stress

exerted by the ligament; (ii) σ(u)=−σ

∗

(1 − u/u

∗

)

for restraints caused by

the compressive grains-bridges due to TEA (m = 1) and frictional ligaments

(m = 2). Then, the corresponding microstructure contributions to toughness

values are estimated as T

=0atc<d,andT

= T

∞

− T

at c>c

∗

(for

all three types of the ligaments). In the intermediate zone (d ≤ c ≤ c

∗

), the

corresponding contributions are estimated for elastic ligaments as

=(T

∞

− T

)



∗

− d

)

c(c

∗

− d

)



1/2

, (8.38)

for ligaments due to TEA (m = 1) and frictional ligaments (m =2)as

=(T

∞

− T

)

1 −



1 −



∗

− d

)

c(c ∗

−d

)



1/2



m+1

. (8.39)

Following [163], we use superposition method in the framework of the force

approach and put a local internal stress, caused by microstructure eﬀects of

fracture resistance on uniform driving force of crack. Then, total SIF is K =

+ K

,whereK

= ψσ

1/2

is the applied SIF from indentation; σ

is the

homogeneous external stress. The local SIF is deﬁned as K

= χP/c

3/2

,where

χ =0.004(E

/H)

1/2

is the constant, depending on the contact geometry and

elastic–plastic properties; H is the material hardness; P is the contact loading.

We assume a power dependence of toughness on the crack length [163], T =

(c/d)

,where0<τ <0.5 (the toughening is absent for τ =0,andthe

catastrophic fracture does not reach for τ =0.5). Then, from equilibrium

condition K = T and the transition condition from stable crack growth to

unstable one, dK/dc =dT/dc, we ﬁnd the critical crack length, c

cat

,andthe