Ivan A. Parinov, Eugene V. Rozhkov and Cyril E. Vassil’chenko

Abstract ─ Based on the example of the gradient sintered YBa₂Cu₃O_7-x compositions, a computer microstructural model for crack bridging processes is presented and their effects on the fracture resistance are calculated . Three kinds of restraining bridges are considered , namely: elastic connections, frictional ligaments, and tractions due to the thermal expansion anisotropy. Influence of the initial press-powder porosity on the some parameters of crack bridging is evaluated.

I. INTRODUCTION

Design of superconductive ceramics (SC) behaviour and optimization of their properties are impossible without investigation and estimation of factors defining the fracture resistance. This phenomenon and correspondent alterations of the SC electric and magnetic properties are caused by the microstructural peculiarities and material possibilities to resist applied, internal, physical, and mechanical loadings. As it has been shown in [1], [2], the crack bridging is the main mechanism of the toughening and fracture resistance increase for ceramics of this type. It renders the greatest effect to compared with microcracking and twinning near crack and its branching.

The aim of this paper consists in the following study of the crack bridging in the SC and accompanied trans- and intergranular fractures by the computer simulation methods. These are based on the joint consideration of the material production and fracture processes [1], [3], [4]. Specific attention is devoted to the gradient sintered YBa₂Cu₃O_7-x compositions [5].

 II. MODEL REPRESENTATIONS

Characteristic destructive mechanical phenomena in the YBCO ceramics are connected with existence of the thermal expansion anisotropy (TEA) along the direction of c₀ due to the crystal lattice arameters a₀ and b₀ are closely allied, while differing sharply from c₀ [6].

  These superconductive ceramics possess the toughness growth with crack size, c (so-called effect of T ─ curve) as the other related materials [1]. The process of the formation and subsequent rupture of ligamenting bridges of material, acting as restraining elements behind the crack tip, is the origin of the T ─ curve. The special role in the quantitative interpretation of the T ─ curve is performed by the additions at the intergranular boundaries (here, the material of the intercrystalline layers comprises a BaCuO₂ ─ CuO system with added impurities [5]). This phase has direct and indirect effects on the ceramic strength properties, namely (i) it embrittles the intergranular boundaries and decreases their toughness, and (ii) it lowers the grain boundary mobility during the sintering process and decreases the cooling microcracking of the SC, respectively. The effects of the grain growth inhibition by additions on the SC fracture have been studied [1]. Further, it should be noted that an existence of the experimental observations indicating the crack growth stabilization in the related ceramic [7]: (i) the considerable increase in supported applied load after the elastic limit is exceeded, followed by a load drop to a non-zero stress, (ii) the erratic crack advance is caused by local inhomogeneities in the fracture resistance , (iii) the discontinuous crack traces are regions of unruptured or frictionally locked material which are the restraining ligaments and which centre on large grains, and (iv) the fracture of the grains-bridges is transgranular at the primary intergranular failure. These phenomena confirm the existence of the crack bridging.

Then, we use results of the article [7] and identify three regions of a penny-like crack behaviour with radial size c, namely (i) for small cracks (c < d, d is the interligament spacing) the crack encounters no restraint, and T = T₀, where T₀ = (2g ₀E) is the SC intrinsic toughness, 2g ₀=2g _S-g _b [8], g _S is the surface energy of the bulk solid and g _b is the grain boundary energy, E is Young’s modulus, (ii) for intermediate cracks a zone of restraints is active in the region d Ј c Ј c*, and T > T₀, and (iii) for large cracks (c > c*) a rupture of the restraints occurs at sites distant from crack tip leaving a zone of width c* ─ d expanding outward with the crack, and T = T_Ґ , where T_Ґ is the steady ─ state maximum of toughness. The magnitude of c* is correspondent to T_Ґ and it has form

 c* = b{1 + [4(d/b)² + 1]^1/2}/2,

 where b= (Eu*/YT₀)², 2u* is the ultimate ligament extension, Y = 1.24 is a crack geometry parameter. Next, we consider the constitutive equations ( i.e., stress ─ extension dependencies), for three typical connections

 s(u) = -s*(u/u*),

 for elastic ligaments, where s* is the value of the peak restraining stress exerted by the ligament, and

s(u) = -s*(1 - u/u*)^m,

for restraints caused by the compression of the grains ─ bridges due to the TEA (m = 1), and for frictional connections ( m = 2). Then, the correspondent microstructural contributions are T_m = 0 ( at c < d ), and T_m = T _Ґ - T₀ ( at c > c^*), for all ligaments. The contributions in the intermediate region ( at d Ј c Ј c^*) have forms

  T_m = (T _Ґ - T₀ )[c^*(c² - d²)/c(c^*2 - d²)]^1/2,

for elastic connections, and

T_m = (T _Ґ-T₀ ){1-{1 - [c^*(c²-d²)/c(c^*2 - d²)]^1/2}^m+1},

for compressively stressed ligaments ( m=1) and for frictional ones ( m = 2).

III. DISCUSSION AND CONCLUSIONS

Computer simulation results are obtained by using the SC microstructural representations at the two-dimensional lattice consisting of the 1000 square cells with characteristic size, d [1]. It should be noted, that while the microstructural processes in the present work are studied by using the square lattice, however we wait that the models for other kinds of cells give close results. It has been shown [9] by the simulations of the secondary recrystallization that at the high temperatures the kinetics of the grain growth and microstructural morphologies on the square lattice are identical to the corresponding data found on the triangular lattice. Similar results for simulations on the honeycomb lattice have been obtained, too. These suggest the presence of universal (or lattice independent) kinetics at elevated temperatures. Therefore, our modelling on the square lattice is justified. Then, we simulated the variants with doubled number of the lattice cells. The numerical results confirm the insignificant effects of the microstructural alterations.

The present approach allows to compute the microstructural parameters required for evaluations of the trans─ and intergranular fractures and bridging characteristics. The numerical data vs. initial presspowder porosity of C_p⁰ are given in Table 1. First, we define the grain ─ boundary intercept length, l, and ratio of the largest grain size in the material to the average one, h. Then, the grains ─ bridges are selected at the lattice, directly (in every case there are 10% of grains from all ones that to be similar to the experimental data for related compositions [10]). 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.

TABLE 1

Some Simulated Parameters 

Next, the transcrystallite fracture of restraints and failure of the intergranular boundaries between bridges are simulated (see Fig. 1). The separate sections of transgranular failure are found by the corresponding grains ─ bridges sizes. We use graph theory for definition of the shortest crack route at the lattice of the intergranular boundaries [11]. A necessary number of the computer modelling iterations is established on the basis of the stereological method [1], [12]. After that, the proportion of transgranular failure, A_{T ,}

Further, we choose the parameter of u* for above ligaments, namely u*/l » 0.01, for elastic and frictional restraints [7] and u*/l » h²e_T for compressive tractions [13], where the strains due to the TEA is e_T = DaDT » 3.1ґ10^-3 [1], for SC [ Da = (a_max ─ a_min)/2 is the half difference between thermal expansion coefficients, DT is the temperature difference covered at cooling ]. Finally, we determine the values of c* and T_m/(T_Ґ ─ T₀). The initial parameters for computer simulations are found by the experiments [1].

The computations demonstrate the value l/d »1.83 without dependence on the initial porosity of C_p⁰. Then, the magnitudes of u*/l are represented in the Table 1 for the case of compressive ligaments and they coincide well with the cases of the elastic and frictional restraints. The ratio of the largest grain to the average one (h) 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 to compare with greatest one. Further, there is growth of the grain size and grain-bridge one with the initial porosity decrease. Moreover, the numerical data give dependence d = kl (1< k < 2) to being similar to alumina ceramics [7]. Next, the intergranular fracture is decreased with the initial porosity growth due to more sinuosity of a crack trajectory correspondent to the coarse ─ grained structures. But the proportion of the transgranular fracture, A_T, has not monotonous character due to the relative equalizing of the transgranular failures in the various cases of C_p⁰ and growth of the structural parameter of n with the initial porosity. So, the most fine ─ grained structure with C_p⁰= 60% and possessing the greatest value of h has the increased toughening effects which correlate with increased area fractions of transgranular failure [7]. The parameters which characterize the ability of superconductive ceramics to resist failure are ratios of T_Ґ/T₀ and c*/d. The numerical results for the latter are shown in Table 1 where upper line corresponds to the elastic and frictional ligaments but lower line conforms to the compressive tractions. The observed growth of c*/d with initial porosity for all restraints is caused by appropriate rise of the structural heterogeneity parameter of h and this coincides with experimental data [7], [8], [10]. In Fig. 2 there are microstructural contributions of T_m/(T_Ґ ─ T₀) due to crack bridging by the considered ligaments vs. crack size of c for case C_p⁰= 0%. It is obvious, that the main contribution to the toughening is due to the compressive tractions, and the least one corresponds to the elastic ligaments. These trends are supported by the experiments [8], [10], as well.

Finally, note, that the secondary recrystallization and following cooling during SC production cause the grain growth, intergranular boundaries microcracking and significant alterations of the ceramic strength and fracture toughness. The studies of the bridging effects [1] show an absence of the simple dependencies of the toughness T vs. grain size. These results are explained by additive microcracks influences localized at the intergranular boundaries. In the region of c < d the effect of increasing d will be to extend corresponding portion of the T(c) curve downward due to the increased microcracking by cooling. If we scale up d such that T(c) intersects the c ─ axis before the condition c = d is satisfied, then the pre ─ existing flaws would become amenable to unstable crack extension without any external load applied. In this case, the spontaneous microcracking will prevail over the effects of crack bridging and defines the ceramic strength. In the intermediate region (d Ј c Ј c*) there is a transition to the outstripping rise of the T ─ curve for the greater grain sizes and at the long scale crack these grains ensure the increased data for toughness. Similar fracture behaviour has been obtained in the related ceramics [7].

Thus, for creation of the SC with optimum toughness behaviour, it is necessary to obtain the microstructures with grain distributions making sure the bridging effects and having the maximum parameter of the heterogeneity, h. However, the grain sizes are not allowed to exceed the critical ones in order to the microcracking during cooling could be suppressed.

 Fig. 1. Fragment of modelled SC microstructure with propagated crack (circulars) between two grains ─ bridges (black), the voids are shaded.

 Fig.2. Microstructural contributions of T_m/(T_Ґ ─ T₀) to the SC toughening from elastic (1), frictional (2) and compressive (3) ligaments vs. crack size, c.

REFERENCES