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

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218 4 Carbon Problem

particles, defects, etc.) [634]. Thus, the relative strength of energy barriers for

dislocation nucleation, produced by the crack tip or/and the other disloca-

tion sources, states whether a crack tip maintains an atomistic sharpness or

emits dislocation loops. Obviously, the carbon segregation in HTSC causes

a complex eﬀect on the dislocation behavior. Then, the occurrence of the

dislocation-screened crack tips is possible in the presence of carbon, depend-

ing on how carbon aﬀects the generation of dislocation at the crack tips and

the other dislocation sources. Besides the crack tip conditions, the strength

of carbon binding at crack surface and intergranular boundaries also controls

the carbon embrittlement of the grain boundaries. It is known that the value

of (H

)

is much higher than that of (H

)

, and they are found by the condi-

tions at interfaces such as solute or carbon coverage, structure and roughness.

Moreover, a high degree of lattice incoherence (e.g., at the interphase bound-

ary [634]) can also state a higher susceptibility to carbon embrittlement of

HTSC. Obviously, there are also other factors controlling the strength of car-

bon binding at interfaces. So, the values of carbon binding need an accurate

experimental foundation for the considered HTSC systems.

The detailed analysis of numerical results for steady-state crack outruns

the book. Note, only the parameters corresponding to the obtained ones for

slow and fast cracks are found to be dependent on the crack velocity and

the carbon diﬀusivities at grain boundaries and crack surfaces. In particular,

the numerical results, obtained for steady-state crack, indicate that the crack

growth rate is higher for smaller grain size, that is as the grain size decreases

the susceptibility to CIIC increases. In total, the above solutions and numeri-

cal results can be used in the ﬁnite element formulations and other numerical

codes by which the stress–strain state distributions, kinetics and parameters

of intergranular defects during HTSC bulk manufacture can be predicted.

General Aspects of HTSC Modeling

The problem of fabrication of the oxide superconductors with high structure-

sensitive properties suggests as one from primary tasks a design and creation of

property monitoring of the HTSC ceramics and composites.

This includes (i)

observation and modeling of microstructure transformations, causing forma-

tion (during processing) and change (during loading) of the material fracture

resistance that renders deﬁning inﬂuence on superconducting properties; (ii)

estimation of change of the microstructure, strength and conducting prop-

erties under diﬀerent loading; and (iii) property prognosis of ﬁnal product,

depending on the sample composition, parameters and features of process-

ing technique. An approximate scheme of numerical monitoring is presented

in Fig. 5.1 and includes speciﬁc stages of the superconductor preparation

and loading, accompanied by microstructure transformations; the modeling

stages of strength, superconducting and other structure-sensitive properties;

and proper processes, determining the ﬁnal sample parameters. Note that

Fig. 5.1d–f shows modeling in square lattice. However, corresponding realiza-

tion of the computational scheme in other lattices (i.e., triangular, hexagonal,

overlapping elements, etc.) does not present principal diﬃculties. Table 5.1

shows possibilities of the monitoring and spectrum of superconductor proper-

ties, which may be deﬁned at the monitoring realization.

As it has been pointed in the previous chapters, the formation of mi-

crostructure defects and weak links during material compaction and sintering

renders a signiﬁcant eﬀect on structure-sensitive properties of HTSC. In this

chapter, two important problems are considered, which relate to the technique

optimization of HTSC systems, namely the yield criteria are stated, which can

describe both displacement into press-powder bulk and its consolidation dur-

ing compaction [805, 806], and the void formation and transformation due

to diﬀusion processes during sintering are studied [801, 805]. The proposed

yield criteria are based on the associated and non-associated ﬂow rules. The

For non-cubic Al

ceramic, a numerical scheme of the monitoring has been

realized in [803].

220 5 General Aspects of HTSC Modeling

Structure Elements and Causes, Forming Properties of HTSC Ceramics

(10) Aggregates of Grains

(11) Admixtures and Additives

(12) Macrocrack

(13) Fibers and Secondary Phase

Inclusions

(1) Raw Powder

(2) Crystallographic

Properties

(3) Boundaries

(4) Triple Junctions

(5) Domain Structure

(6) Dislocations

(7) Microcracks

(8) Porosity

(9) Grains

(a) (b)

(c)

(d)

(e)

(f)

3 4 4 1 1

6 4 4 7 7

6 6 5 5 7

6 6 5 2 2

34411

3441

64477

6655

66522

Fig. 5.1. General scheme of computational monitoring of HTSC structure-sensitive

properties: (a)initialpowder,(b)sintering,(c) cooling, (d) macrocrack propaga-

tion, (e) presentation of structure fragment in PC, (f) model structure for study of

percolation properties

5 General Aspects of HTSC Modeling 221

Table 5.1. Possibilities of computer monitoring and spectrum of estimated

properties

Investigated processes and eﬀects Estimated and accounted properties and

characteristics

(1) Thermal and mechanical loading (1) Elastic anisotropy

(2) Solidiﬁcation (2) Thermal expansion anisotropy

(3) Shrinkage (3) Deformation and temperature

mismatch of phases and components

(4) Secondary re-crystallization (grain

growth)

(4) Residual (internal) stresses

(5) Phase transitions (5) External physical and mechanical

(6) Superconducting transition (6) Fracture conditions and types

(7) Non-superconducting ﬁbers,

inclusions and additives into

superconducting matrix

(7) Microcracking

(8) Superconducting inclusions into

non-superconducting matrix

(8) Phase transformations

(9) Defect structures, in particular

carbon eﬀect

(9) Twinning and domain re-orientation

(10) Dislocation structures (10) Crack deﬂection and twisting

(11) Percolation (11) Crack branching

(12) Structure eﬀect on magnetic ﬂux

pinning

(12) Crack interaction with structure

heterogeneity (pores, dislocations,

microcracks, inclusions, etc.)

(13) Crack bridging

(14) R-(orT -)curvebehavior

(15) Pushing of grains (or ﬁbers) by crack

surfaces

(16) Crack coalescence

(17) Crack healing

(18) Crack cohesion

(19) Scaling factor

(20) Eﬀective superconducting properties

(21) Percolation properties

(22) Magnetic ﬂux pinning

microstructure transformations of porosity are investigated in the framework

of phenomenological models of the shrinkage, coarsening and coalescence of

intergranular voids, and also their separation from grain boundaries and dis-

placement to interior of grains is possible. Second part of the chapter is de-

voted to modeling of micro- and macrostructure processes during processing

and fracture of HTSC-ceramic, taking into account heating, shrinkage and

cooling of material, grain growth and sample microcracking.

222 5 General Aspects of HTSC Modeling

5.1 Yield Criteria and Flow Rules

for HTSC Powders Compaction

Obviously, the optimization of the process of the HTSC powders compaction

plays important role in the preparation of superconductors with optimum

structure-sensitive properties. As it has been demonstrated in the previous

chapters, the numerous techniques can be used to prepare oxide HTSC ma-

terials and propose an application of the complex thermal and mechanical

treatments of powdered precursors (e.g., cold one-axis pressing [19, 851, 1043],

cold isostatic pressing [282, 479, 851, 1157, 1160], hot isostatic pressing [131],

hot forging [1043], etc.). Final aim of the pointed techniques consists of the

preparation of strong coupled and aligned structures of grains in the ﬁnal

product. Nevertheless, a formation of microstructure defects and weak links is

unavoidable due to intrinsic brittleness of oxide superconductors. In this case,

it could be noted that selection of thermal and mechanical loading as rule

is badly caused. Therefore, in order to solve the above problem, understand-

ing of densiﬁcation mechanisms and features of HTSC powder deformation is

required. The critical mechanical behavior during HTSC powder compaction

may be described by using yield criterion and ﬂow rule, taking into account

microstructure properties, material parameters, applied loading, etc.

Below, consider the problem of improved compaction technique for pow-

dered HTSC precursors, coupled with selection of optimum yield criterion

with the associated (or non-associated) ﬂow rule, which can describe both

displacement into bulk and consolidation of the powder during compaction

process. Moreover, a validity of the non-associated ﬂow rules for description

of HTSC powder compaction is discussed, taking into account the dissipation

energy due to particle re-arrangement and friction.

5.1.1 HTSC Compaction and Yield Criterion

Eﬀective application of ﬁnite element method and other numerical methods

to modeling of microstructure transformations is needed in detail information

about material properties, critical mechanical characteristics and applied load-

ing and densiﬁcation mechanisms. In particular, density distribution, damage

initiation and growth depend on the stress–strain state during HTSC powder

compaction. In this case, the densiﬁcation mechanism consists of two pro-

cesses, namely displacement into powdered volume and plastic deformation of

particles. Because the aggregate includes multiple particles, one may be con-

sidered as uniform continuum. Then, it is necessary to state a critical condition

or a yielding criterion in order to describe the powder strain and densiﬁca-

tion. Successive selection of the yielding criterion and ﬂow rule are capable of

helping in creation of optimum HTSC composition and microstructure. The

deformation, caused by sliding of grains, leads to increase in volume during

compaction; at the same time it leads to decrease in volume during consoli-

dation of grains. At the so-called critical state (transition point), there is no

5.1 Yield Criteria and Flow Rules for HTSC Powders Compaction 223

tendency for volume change. Diﬀerent yield criteria for free powder ﬁlling,

porous and granular specimens have been discussed (see, e.g., [461]). Theories

of yielding with an increase in density have been developed in [894]. On the

other hand, theories of plasticity with a decrease or no change in density have

been discussed in [1008] but stated numerous issues. However, very often, the

yield criteria do not take into account the density eﬀect on the sample strain.

Moreover, due to volume changes during sample compaction, an eﬀect of hy-

drostatic pressure should be included into yield criterion. At the same time,

the limiting envelopes stated by the Mohr–Coulomb’s criterion and similar

criteria suggest an inﬁnitely large shear stress to cause slip at compressive

stresses. This does not apply to the powder compaction. Therefore, the yield

criterion for powder aggregate should be in the form of closed curve (e.g.,

an ellipse) with asymmetric conditions for compressive and tensile stresses,

because the powders are not able to sustain signiﬁcant tensile stresses. Con-

sidering the isotropic case of HTSC powder compaction, a 3D yield criterion

has the form [140]:

f = α(I

+ s)

+ J

= βY

, (5.1)

where α, β and s are functions of relative density, I

= σ

+ σ

is the ﬁrst

invariant of the stress tensor, J

=(1/6)[(σ

−σ

)

+(σ

−σ

)

+(σ

−σ

)

]

is the second invariant of the deviatoric stress tensor (σ

are the principle

stresses) and Y is the yield stress of the solid material.

Using the yield function, f , as the plastic potential, the associated ﬂow

rule is obtained by imposing the normality condition between the plastic strain

rate and the yield surface:

˙e

= λ

∂f(σ

)

∂σ

= λ[2α(σ

+ s)δ

+ s

] , (5.2)

where δ

is Kronecker’s delta function, σ

and s

are the hydrostatic and

deviatoric stress, respectively. A dot above a symbol implies the material time

derivative, and upper superscript p signs plastic component.

Considering the principal strain rates, ˙e

, and determining the volumetric

strain rate, ˙e

=˙e

+˙e

, we obtain from (5.1) and (5.2) the positive

constant λ, deﬁning the strain amount at a given point, as

λ =



(˙e

)

/(18α)+



1/2



2βY ) , (5.3)

where the deviatoric deformation rate has the form:

=˙e

− (1/3) ˙e

Then, (5.1) can be presented by using normal stress, σ, and shear stress, τ:

(σ + s/3)





(1 + 12α)/9α

√

βY





√

βY



=1. (5.4)

The material constants α, β and s can be determined by using shear tests

[1086]. For this, the yield locus (Fig. 5.2) is described in the σ − τ plane by

224 5 General Aspects of HTSC Modeling

–S/3

Y√β

Normal Stress (–σ)

Shear Stress (

)

Fig. 5.2. The proposed yield locus for a shear test

an ellipse represented by (5.4). Due to the assumption of an isotropic defor-

mation in a shear test, the transition point almost coincides with the apex of

the minor axis of the ellipse. Then, the inclined angle, ψ, of the critical line

against the abscissa, σ,andtheratio,R, of the major axis to the minor axis

of the ellipse can be considered as material constants, and by using (5.4), they

have the forms:

tgψ =



βY/(s/3) ; (5.5)

R =



(1 + 12α)/9α. (5.6)

Finally, we select the relation between the density of a powder compact and

pressure, needed to achieve that density, as remaining third equation [402]:

KP =ln[(1− ρ

)/(1 − ρ)] , (5.7)

where P is the applied pressure, ρ

and ρ are the average densities of the loose

powder and the compressed powder, respectively, and K is the test constant.

Then taking into account the relation

P = s/3+R



βY , (5.8)

the test constants ψ, R and K can be used to deﬁne the material constants

α, β and s:

α =

(9R

− 12)

; s =

K(1 + Rtgψ)

1 − ρ

;

β =



tgψ

K(1 + Rtgψ)

1 − ρ



. (5.9)

A validity of the proposed yield criterion for description of the HTSC

powder compaction can be veriﬁed by using, for example, a 3D compaction

5.1 Yield Criteria and Flow Rules for HTSC Powders Compaction 225

test (Fig. 5.3a), in which three pairs of anvils compress the powder in three

mutually perpendicular directions. However, note that tests of granular ma-

terials (e.g., soils) or cemented materials (e.g., concrete or rock), presented

in Fig. 5.3b and c, have shown that associated ﬂow rule cannot depict test

data satisfactorily [1114]. At the same time, satisfactoriness of associated (or

non-associated) plasticity has not been stated for compacted HTSC powders.

Therefore, it is very important to additionally consider non-associated ﬂow

rules for which the plastic strain rate is not orthogonal to the yield surface.

5.1.2 Non-Associated Plasticity of HTSC Powders

Base hypothesis of normality [224], which forms a basis of associated plas-

ticity and is very successful in the description of metallic compositions, can

be broken for non-metallic granular materials in free ﬁlling, in particular for

HTSC powders. The key circumstance is that the associated plasticity can-

not validly describe shear dilatancy of granular material (i.e., the change in

volume that is associated with shear distortion of an element in the material,

consisting of multiple particles as microelements). As has been shown [1114],

the material, retaining constant volume under plasticity, reacts otherwise on

loading, demonstrating plastic expansion. The diﬀerences are related to both

the “load–deformation” curve and the value of critical loading. In order to

characterize a dilatant material, dilatancy angle Ψ is introduced, presenting

the ratio of plastic volume change over plastic shear strain.

An ideal triaxial test should permit independent control of all three princi-

pal stresses so that general states of stress could be examined. Typical results

in a standard triaxial test of granular material in free ﬁlling are shown in

Fig. 5.4. In elastic region (I), as usual, the strains are reversible at loading.

In hardening regime (II), the strain of granular material becomes more and

more inelastic due to particle sliding. Here, non-linear elasticity predicts con-

tinuing contraction of the specimen under continued loading in compression.

However, such a prediction is disproved by experimental evidence (Fig. 5.5),

which shows a dilatant volume increase at subsequent loading [1114]. This

phenomenon takes place due to frictional sliding along particles. The elastic

strain rate in the hardening regime is almost zero. Moreover, there exists a

linear relation between the volume change and the change of the axial strain

near the end of the hardening regime (II) and in the softening regime (III)

(see Fig. 5.4c). Then, following [1114], the constant dilatancy angle, Ψ, may

be introduced as

sin Ψ =

˙e

−2˙e

+˙e

. (5.10)

Equation (5.10) deﬁnes constant rate of dilatation, and it is valid under

conditions of triaxial compression. For broad row of granular materials, the

dilatancy angle, Ψ , approximately equals 0–20

◦

; at the same time, the inter-

nal friction angle, Φ = 15–45

◦

[1114]. Because the dilatancy angle can be

considerably smaller than internal friction angle, it is necessary to apply for

226 5 General Aspects of HTSC Modeling

(a)

(b)

(c)

Fig. 5.3. Diﬀerent schemes of loading for triaxial compression. Third pair of anvils

compresses the powder in the direction perpendicular to the ﬁgure plane. The num-

bers sign anvils (1) and powder (2). Arrows show applied loads

5.1 Yield Criteria and Flow Rules for HTSC Powders Compaction 227

•

III

•

III

•

⏐σ

−

⏐

⏐σ

−

⏐

(a)

−e

(b)

•

III

•

(c)

−e

–e

Fig. 5.4. Typical triaxial test results for loose granular material

Rate

Fig. 5.5. Sliding between groups of particles, leading to dilatation