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

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8.4 Small Cyclic Fatigue of YBCO Ceramics 391

8.4 Small Cyclic Fatigue of YBCO Ceramics

It is known that the fracture toughness of YBCO is signiﬁcantly smaller than

that of typical ceramics, for example, Al

, partially stabilized ZrO

or SiC

[1094]. Therefore, it is very important to consider fracture processes in YBCO,

in particular, under cyclic loading. In this section the small cyclic fatigue

fracture model for YBCO is stated, based on the microstructure dissimilitude

eﬀect [135], using joint considerations of the superconducting ceramic manu-

facture and fracture for the YBCO samples [819]. Special attention is devoted

to the correct deﬁnition of speciﬁc fracture energy, taking into account poros-

ity, microcracking and cooling features.

8.4.1 Model Representations

Remind that typical techniques for melt-processed YBCO ceramics consist of

the precursor powder preparation, formation of the so-called “green” sample,

sintering and cooling of the material. The sample cooling and re-crystallization

initiate a formation of defects and microcracks due to the deformation mis-

matches of the 211 and 123 phases, thermal expansion anisotropy of grains

and tetragonal–orthorhombic phase transition. Subsequent electromagnetic

and thermo-mechanical loading of the ceramic in devices contributes to the

damage development (or their initiation) and to corresponding deterioration

of the conductive properties of YBCO.

By modeling of the processes which accompany the ceramic fabrication

and fracture, the general research scheme, presented in the previous sections,

is realized:

(1) The precursor is represented by 2D lattice with 1000 square cells of the

characteristic size, δ. Each cell is either a grain nucleus of the 123 phase or

a void. Initial disposition of the voids is found by Monte-Carlo procedure,

described in Sect. 5.3.

(2) A modeling of the heat front propagation during sintering is carried out.

It is assumed that a temperature in gradient furnace changes only along

one coordinate with constant rate ν.

(3) By description of the ceramic structure re-crystallization, the MonteCarlo

procedure is also used. Here, we neglect the material shrinkage and grain

growth, assuming that during formation of closed porosity there are no

considerable material densiﬁcation (see Fig. 8.31).

(4) The microcracks nucleate at triple junctions during sample cooling (in the

same conditions for all considered structures) from the melting temper-

ature down to the room temperature. The cooling process consists of a

slow decrease (with rate of 1

◦

C/h) of temperature from 1100

◦

Cdownto

970

◦

C and then quenching, for example, in a liquid nitrogen bath [431].

392 8 Computer Simulation of HTSC Microstructure and Toughening

1mm

(a)

(b)

(c)

1mm

Pores

114 14

80 19

19 19 23

4853

23 23

23 23 23

80 97

66 66 68

9999

111

11111

(d)

Fig. 8.31. Micrographs of polished cross-sections of YBCO samples, prepared at

the heat rates: (a)30

◦

C/h(b)20

◦

C/hand(c)10

◦

C/h [642]; (d) fragment of

model microstructure; porosity is shown by gray color, microcrack is present at

intergranular boundary

8.4.2 Microstructure Dissimilitude Eﬀect

It is known that the small fatigue cracks (i.e., cracks that intersect some

grains) frequently grow much faster than large ones under other equal con-

ditions, in particular due to the higher stress concentration factors and the

decreased closing of the cracks. Microstructure dissimilitude is caused by im-

possibility to average the strength and fracture properties on the many grains

and the existence of the dependence on the crack size. As has been demon-

strated, the microstructure dissimilitude can lead to anomalous rapid growth

of the small cracks, altering: (i) the intrinsic fracture toughness and cyclic

crack growth resistance in the process zone and also (ii) the local crack driv-

ing force (ΔK) [135].

Consider a small crack which violates microstructure similitude and uses

the modiﬁed Barenblatt–Dugdale’s (BD) crack tip model that is based on the

linear elastic fracture mechanics. The classical BD-crack with process zones

(presenting extended cohesive parts at the crack tips) may be replaced by the

superposition of two loading conﬁgurations (see Fig. 8.32), reﬂecting a local

8.4 Small Cyclic Fatigue of YBCO Ceramics 393

∞

Fig. 8.32. Barenblatt–Dugdale’s crack, presenting small crack, which disturbs mi-

crostructure similitude. One may be studied in the framework of linear elastic frac-

ture mechanics, introducing additional stresses at the crack surfaces (σ

= σ

−σ

)

that are caused by microstructure dissimilitude

cleavage and general fracture. Then, the fracture toughness increment of large

cracks (ΔK

) can be divided into two terms for cyclic loading:

ΔK

=ΔK

+ΔK

, (8.94)

where ΔK

is the fracture toughness increment of small cracks, and ΔK

represents the local increment of the fracture toughness, induced as a result

of microstructure dissimilitude and grain misorientation. Then, ΔK

for an

elastic crack, subjected to a partial tensile load increment on the crack surfaces

(Δσ

), is given as [1033]

ΔK

√

Δσ

√

πb , (8.95)

where Δσ

is taken to be 2σ

,sothat

Δσ

=2σ

(1 − σ

/σ

) . (8.96)

In (8.95) and (8.96), b is the length of the region over which the crack tip

cleavage occurs (the process zone size), σ

is the macroscopic fracture strength

of the specimen, σ

is the local cleavage strength in a grain, containing the

crack tip. Substituting (8.95) and (8.96) in (8.94), we obtain

ΔK

=ΔK

[1 +

√

(σ

/Δσ

∞

)



b/a(1 − σ

/σ

)] , (8.97)

where a is the half-length of the crack without process zone, and Δσ

∞

is the

increment of the applied remote stress. Similarly, it can be estimated that

= K

[1 +

√

∞



b/a(1 − σ

/σ

)] . (8.98)

394 8 Computer Simulation of HTSC Microstructure and Toughening

Assuming that for small and large cracks there are relationships:



(1)

1 − ν



1/2

and



(0)

1 − ν



1/2

respectively, we obtain from (8.97) and (8.98)

ΔK

=ΔK

[1 + 2(S − 1)σ

∞

/Δσ

∞

] , (8.99)

where

S =



(0)

(1 − ν

)

(1)

(1 − ν

)



1/2

. (8.100)

Eﬀective elastic modules E

and ν

for solid with pores and microcracks

can be estimated, for example, using the dependencies of strain characteristics

for elastic bodies on the concentrations and sizes of spherical pores and disc-

like cracks in the self-consistent diﬀerential method as [239]

= E

exp(−α

); ν

= ν

exp(−α

) ; (8.101)

= E

(1 − η

)

; ν

=0.2+(ν

− 0.2)(1 − η

)

, (8.102)

where η

=(a

)

,η

=1− ρ/ρ

are the damage parameters, determined

through the average size of the disc-like cracks (a

) and the average distance

between them (r

); ρ and ρ

are the material and crystalline cell density,

respectively; α

,β

(i =1, 2) are the given constants; E

,ν

are the elastic

modules of the material without defects; E

,ν

are the elastic modules of

sample with voids.

8.4.3 Fracture Energy and Microstructure Features

Obviously, the correct deﬁnition of fracture energy, taking into account a

microstructure of superconductor, has a signiﬁcance for understanding of the

dynamical crack problem, generally, and the cyclic fractures, particularly. We

present the energetic balance condition in the usual form:

J =2γ, (8.103)

where

J =





(U + T )n

− σ

∂u

∂x



dS ;2U = σ

;2T =˙u

˙u

(8.104)

8.4 Small Cyclic Fatigue of YBCO Ceramics 395

where σ

,ε

are the components of the stress and strain tensors,

vector of the displacement and external normal to surface of S, surrounding

a crack tip, respectively; γ is Griﬃth’s parameter. A dot above a symbol

implies the material time derivative. There is summation on the repeated

indexes from 1 to 3 in (8.104). In the static case, the condition (8.103) reduces

to the algebraic relation between physical parameters. The energetic balance

condition in the form (8.103) is also applied to the dynamical case. However,

the assumption of the J-integral as critical parameter in the criterion of crack

advancing [93] reﬂects only a static nature of the condition (8.103). Therefore,

in the dynamical problem of the rectilinear ﬂaw with size, 2L(t), it is necessary

to take into account that the J-integral depends on the time t and is caused by

the selection of the L(t). The J-integral is equal to constant value of 2γ only

when a law of the ﬂaw growth for the L(t) satisﬁes (8.103) in any time. After

calculation of the J-integral, this condition reduces to non-linear diﬀerential

equation, which deﬁnes the crack growth during time. Thus, a statement of the

dynamical crack problem must include the condition (8.103), which identiﬁes

the ﬂaw growth with crack advancing during fracture of solid. Then, speciﬁc

fracture energy, γ

, to being a limit ratio of the work on separation of atom

couples at the fracture surface F to its square at decreasing of this surface to

zero, demonstrates a local character. The value of γ

should be signiﬁcantly

distinct from the Griﬃth’s parameter γ, which is a result of the γ

averaging

on the fracture surface. Therefore, the physical proper statement demands to

replace γ through γ

in condition (8.103). The energetic balance condition as

dynamical condition for brittle fracture can be represented as [511]

J =

∞

π(1 − ν

)sin

L(t)





L(t)



sin



L(t)



cos



=2γ

, (8.105)

where E and ν are the elastic modules; σ

∞

is the remote tension; Θ

the angle between the load and crack direction at the time t; c

is the rate

of transversal waves; F

and F

are the monotonically decreasing functions,

determined in the non-stationary problem. It may be shown [511] that the

crack growth is irregularly accelerated and gives the relationship:

L(t)=L(0) +

−

L(0)



2L(0)



+ O





2L(0)





, (8.106)

where the acceleration a

is found as

π(1 − ν)

8ργ

∞

sin

(ν)sin

+ α

(ν)cos

; L(0) =

2Eγ

∞

π(1 − ν

)

(8.107)

396 8 Computer Simulation of HTSC Microstructure and Toughening

The parameters α

(ν)andα

(ν) are also found, solving a non-stationary

problem. Note that (8.105) deﬁnes all features of sub-critical crack behavior,

in particular, the crack tilt, twist and branching.

The value of γ

is calculated in the case of fatigue fracture, using both

material microstructure and cyclic loading. Estimate this parameter taking

into account the structural elements of YBCO (i.e., voids and microcracks),

which are small to compare with macroscopic scale of stress–strain state. Be-

cause the microcracks and voids, as a rule, are localized at the grain boundaries

which are regions of a lesser dense atomic packing compared with grain phase,

the curve of interatomic interactions can be approximated by the sinusoid [24]:

σ(ε)=



sin 2πε ε ∈ [0; 0.5]

0 ε ∈ [0.5; ∞]

, (8.108)

where ε = r/r

− 1,r

is the mean equilibrium distance between atoms,

= E

/2π is the theoretical strength of atomic link. Then, we obtain for

transgranular fracture:

(0)

∞



σ[ε(r)]dr =



sin 2π(r/r

− 1)dr =

4π

. (8.109)

In order to deﬁne the mean interatomic spacing at intergranular boundary,

, we take into account a proportional dependence between adjacent atoms

and the absolute temperature T . Then, r/r

∗

−1=αT ,whereα is the thermal

expansion factor, r

∗

is the equilibrium spacing between the atoms at the

T = 0 K. Further, in order to deﬁne the temperature ﬁeld in the ceramic

sample during cooling from the sintering temperature (T

)downtotheroom

temperature (T

), with no destroying of generality, we study the next 1D

initial boundary problem:

∂T

∂t

= b

∂

∂x

ρc

; (8.110)

∂T

∂x

= ±h(T − T

) , for x = ±l ; (8.111)

T = T

, for t =0, (8.112)

where c is the speciﬁc heat, k is the thermal conductivity factor, h is the

thermal expansion factor, 2l is the sample length. The solution of this problem

is determined as

T (x, t)=T

∞

m=1

exp(−λ

t)cos λ

x, (8.113)

8.4 Small Cyclic Fatigue of YBCO Ceramics 397

where A

=2sinλ

l/(λ

l +sin 2λ

l) and eigenvalues, λ

,havethe

following approximate values: λ

= mπ(1/l − h/π

). Retaining only the ﬁrst

term of the quickly convergent series (8.113), we have ﬁnally:

(1)

/γ

(0)

= E

=exp(−α

)(1 − η

)

[1 + α(T

− T

)

× 2sinλ

l/(λ

l +sin 2λ

l)] . (8.114)

When λ

l is small, (8.114) reduces to

(1)

/γ

(0)

=exp(−α

)(1 − η

)

[1 + 2α(T

− T

)/3] . (8.115)

8.4.4 Some Numerical Results

The computer simulation results have been obtained for known parameters

of YBCO [239, 431]: T

= 970

◦

C,T

=20

◦

C,α =2× 10

−5

−1

,α

=1.65,

=1.47,β

=1.96,β

=0.23,ν

=0.22,E

= 64 GPa. Additionally, the val-

ues of η

and η

are stated by the microstructure modeling, using the formulae:

= Na

and η

= C

,whereN is the crack number per volume unit, C

the closed porosity. The numerical results are shown in Fig. 8.33 for diﬀerent

values of the initial porosity of C

and the heat rate of the sample, ν [819].

The increased rate at the same initial porosity gives a more ﬁne-grain struc-

ture, smaller microcracking of grain boundaries and greater value of γ

(1)

/γ

(0)

Generally, the decreasing heat rate leads to ﬁner sizes of voids and to their

homogeneity in sintered samples that improve the YBCO conducting proper-

ties (see Fig. 8.31). However, excessive rate decreasing may increase the liquid

phase loss. This would be undesirable to a highest degree from stoichiometry

0.65

0.60

0.55

0.50

0.45

0.40

0.35

010 20 30 40

(1)

/γ

(0)

ΔK

/ΔK

3.00

2.75

2.50

2.25

2.00

1.75

1.50

, %

Fig. 8.33. Relative values of speciﬁc fracture energies, γ

(1)

/γ

(0)

, and fracture tough-

ness increments, ΔK

/ΔK

,vsceramicporosity,C

,at2σ

∞

=Δσ

∞

.Fortwoheat

rates, ν =10

◦

C(1) and ν =20

◦

C(2)

398 8 Computer Simulation of HTSC Microstructure and Toughening

consideration [642]. The increasing initial porosity (at the same heat rate)

deﬁnes the ﬁner-grain structure with corresponding decreasing of a sponta-

neous microcracking of intergranular boundaries during cooling. At the same

time, this leads to the smaller values of γ

(1)

/γ

(0)

and proper increasing of

ΔK

/ΔK

. Obviously, the closure of the small cracks under unloading com-

pared with the large crack case is insigniﬁcant. Then, the maximum toughen-

ing, based on the steady-state crack conditions, can be achieved after a ﬁnite

crack extension [1012]. This can lead to an overestimation of the toughening

in consideration of the cyclic fatigue and, therefore, it should be calculated

directly, taking into account the microstructure dissimilitude eﬀects.

8.5 Residual Thermal Stresses in YBCO/Ag Composite

As has been shown in previous chapters, the silver inclusions are widely used

to improve both superconducting and mechanical properties of HTSC. Con-

sider eﬀects of residual stresses in this section which form around inclusions

in YBCO/Ag composite after cooling from the peritectic temperature down

to room temperature and are caused by the great diﬀerence in thermal ex-

pansion factors of the metal particles and the ceramic matrix. Plastic defor-

mations in ductile silver inclusions should be considered in an estimation of

residual stresses around silver particles. The ﬁnite element method (FEM)

has been applied for evaluating such diﬃcult problem [652]. The using of AN-

SYS general-purpose software helps to estimate eﬀects of the distribution of

the residual stresses on features of macrocrack growth in this composite. For

the matrix and inclusion properties given in Table 8.8 and also for the ther-

mal diﬀerence ΔT between the peritectic temperature of YBCO/Ag (970

◦

and room temperature (20

◦

C), two silver round inclusions with a diameter

D =10μm and the distance between them, l =5μm, are considered on the

isotropic ab-plane of the YBCO matrix (see Fig. 8.34a). Figure 8.34b shows

the distribution of the residual stresses calculated by the FEM and indicates

that expansive (compressive) stresses create after cooling of the composite

due to the TEA of the matrix and inclusions.

Figure 8.35 explains the mechanism of the crack deﬂection between two

inclusions. Tensile and compressive stresses remain around the silver particles

in the r-axis and θ-axis of the polar coordinate system, respectively. First,

the tensile stress along the r-axis assists the crack growth. However, when it

Table 8.8. Material properties used in FEM computations

Material Thermal expansion

factor (×10

−6

−1

)

Young’s modulus

(GPa)

Yield strength

(MPa)

Maximum

elongation

YBCO (a-axis) 13 [524] 182 [338] – –

Ag 21 [698] 80 [377] 50 [377] 0.5 [377]

8.5 Residual Thermal Stresses in YBCO/Ag Composite 399

100 MPa

(b)

(a)

r x

Fig. 8.34. (a) Mesh pattern for FEM calculation around silver particles dispersed

in YBCO matrix and (b) maximum and minimum principal stress distributions

around the silver particles in the 123 matrix [652]

reaches a gap between the silver particles, the angle between the vectors of the

residual stresses and the crack growth is changed. This means that the tensile

(compressive) residual stress promotes (suppresses) the crack growth along

the y-axis (x-axis). Therefore, the direction of the crack growth is altered

by the residual stresses around the silver particles, according to the energy

minimum condition. This also means that the I mode of fracture is changed

to the mixing mode (both I and II modes) by this deﬂection. Such crack

deﬂections ensure the possibility of toughening of the brittle YBCO matrix

and increasing its fracture toughness due to the increasing of the deﬂected

crack path.

400 8 Computer Simulation of HTSC Microstructure and Toughening

Ag Ag

Crack

Direction

Load

Residual

Stresses

Fig. 8.35. Schematic illustration of the relationship between residual stress and

crack propagation [652]

8.6 Toughening of Bi-2223 Bulk, Fabricated

by Hot-Pressing Method

Models for processing and fracture of Bi-2223 ceramic which is obtained by

hot pressing are discussed in this section. Computer simulation is applied to

phenomena occurring during sintering, cooling and following fracture due to

the macrocrack growth. The eﬀects of Ag particles dispersed into the Bi-2223

matrix on some strength properties are studied [577].

8.6.1 Microstructure Formation by Processing

The sintering and cooling models for oxide superconductor have been discussed

in detail in the previous sections. Therefore, we discuss in some detail the

model of abnormal grain growth, caused by the sintering conditions (i.e.,

pressure and temperature [1034]) and by existence of secondary phases (e.g.,

CaCuO

and CuO), deﬁning the Bi-2223 microstructure formation [700].

For diﬀerent ceramics [796, 814], the Wagner–Zlyosov–Hillert’s model [1]

can be used. However, this model does not take into account the eﬀects

of texture on grain growth, which in this case is controlled by the inhi-

bition parameter, depending on volume fraction and size of the secondary

phase particles. At the same time, experiments have shown that the primary

re-crystallization, as a rule, initiates a texture [2]. Therefore, in modeling

of secondary re-crystallization, we consider the dependence of intergranular