HIGH-TEMPERATURE

DEFORMATION

OF HOT-PRESSED

SILICON NITRIDE IN CREEP

A. A. Postnikov, M. L. Mel'nikova,

UDC 621.002.3:666.3

and V. G. Verevka

Mathematical Model. High-temperature deformation of ceramic materials is characterized by a large number of special features. In contrast to metals, the inelastic aftereffect is accompanied by recovery which greatly exceeds the residual strains. The creep rate of ceramics in' tensile loading is usually higher than in compressive loading. These nonlinearity features are greatly determined by the structure of the ceramic materials [1], whose model can be represented in the form of packing of elastic grains with a viscous glass at the faces. Different mechanisms of grain sliding, formation and subsequent growth of voids, and the presence of microcracks at their joints [21 make it possible to explain the nature of residual strains and differences in the creep rate in tension and compression [3]. In constructing a mathematical model of inelastic behavior of ceramics it was assumed that inelastic strain was completely restored; the nature of restored strain does not depend on creep resistant to deformation; damage cumulation is linear and is governed by Bailey's law; damage forms only in stretched zones. Thus, the total strain can be represented in the form of the sum of elastic, inelastic restored, and strain-resistant creep. Assuming that the nature of deformation in reverse creep and inelastic deformation is identical and associated with grain sliding, the curve of unsteady creep can be described by an equation of MacVetty type [4] in = A (Ka -- 'n),

(1)

where eR is the reversed creep strain; a is stress, K is a parameter of the material. Since creep is a thermally activated process, the variation of inelastic strain, associated with hydrostatic tensile loading, can be ignored. Consequently, Eq. (1) assumes the form ~Rq --= A [Ko exp (--

On/RT)

(3/2)S~] - - emil:

(2)

where eRij is the deviator of reversible creep strain, A and K o are constants of the material, QR is the activation energy of recovery, R is the gas constant, Sij is the stress deviator. Steady creep is also a thermally activated process. Using the hypothesis on the relationship of creep strain with the stress deviator, the equation of kinetics of steady creep can be written as follows ~'1 = (3/2) Bo'exp

(-- Q~/ RT) ~7-' S~i exp

( , m , ~ ),

(3)

where Bo, m, n are parameters of the material, cri is the stress intensity Qv is the activation energy of steady creep. The hypothesis of the relationship of steady creep with the formation and growth of damage (parameter o~) shows that stress a in Eq. (3) is "effective" and is linked with the relation a o by the relation a = ao/1-,0. According to the accepted hypothesis, this assumption is used 0nly for tensile-loaded zones. The curves of steady creep to rupture can be used to construct damage curves described by the Kachanov-Rabotnov type equation: o = D [(rl/(1--co) ] h,

(4)

Tekhnologiya Scientific and Production Association, Obninsk. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, No. 3, pp. 84-88, May-June, 1992. Original article submitted June 21, 1991. 284

0038-5565/92/2803-0284512.50 9

Plenum Publishing Corporation

a

te

/tr f, h

Fig. 1. Deformation curve of HPSN: 1) reversible strain; 2) residual strain.

TABLE 1. Test Conditions of Specimens* T,K

q

P, N 6, MPa

147a [ 681 350

157a 583 300

681 350

t 107a 778 400

681 350

*Specimens were tested by A. I. Zabotko at the Institute of Physics and Technology of Instruments and Experiments of the Academy of Sciences of the Lithuanian Republic.

TABLE 2. Recovery Test Conditions

r. K P0, N P, N Wr~, ~m

'1

147a 140 1300 10

i 1493 [ ,60a 100 1510 140

100 1730 170

100 1510 10

where D and k are parameters of the material, o"1 is the first main stress. The hypothesis on the relationship of steady creep strain with damage formation shows that the activation energy of steady creep is required for rupturing bonds between the grains. Consequently, the relation (3) is transformed to the form

g -- D o e x p ( - - Q ~ / R T ) [o / (I - - o ) ] L

(5)

Integrating Eq. (4) with an allowance made for the boundary conditions t = 0, oa = 0 and t = tR, co = 1, we obtain the duration of the damage process

t = [1-(l

-~)~+'1 [D~'(n + l ) l - '

(6)

and the time to rupture

tze = [D, exp ( - - Q~ / R r ) ok 1~-'~.

(7)

The relations (4)-(7) were obtained for static loading. In alternating loading we can use the following relation: t l*

to = j t-~ 1 dr. 0

(8)

Using Eq. (7), we have 285

i 0

~.

~

8

0

,'0

fZ

f~E.fO~'

Fig. 2. Dependence of strain rate on creep strain: 1) a = 350 MPa, T = 1473 K; 2) a = 300 MPa; T = 1573 K; 3) = 350 MPa, T = 1573 K; 4) ~ = 350 MPa; T = 1673 K.

./]l

/o

-

/X

I

~00 9O 8O

7O

i T 8 ; ; 6 8 , , f l l r . l O ~'/i( _ Fig. 3

#00 $50 6, MPa Fig. 4

Fig. 3. Dependence of the "reduced rate" of unsteady creep on temperature at a stress of ~r = 350 MPa. Fig. 4. Dependence of "temperature-compensated rate" of unsteady creep on stress at temperature T = 1573 K. t

to := .f Do exp (-- Q~/RT)(k + 1) <~dt. 0

(9)

The possibility of using these equations for describing high-temperature deformation and fracture of ceramics has been confirmed by investigations of long-term strength and creep of hot-pressed silicon nitride (HPSN) at 1473-1673 K in air. The experiments also confirm the hypothesis on the possibilities of restoring creep strain in the unsteady stage (Fig. 1). Experiments. To confirm the model of high-temperature deformation and determine the parameters of high-temperature creep in different stages of deformation of HPSN, tests were carried out on 7 x 5 • 70 mm specimens in three-point bending (the distance between the support frames was 60 mm) in air at constant load P and temperature T (the test conditions are given in Table 1). The results show (Fig. 1) that the deformation curves consist of two sections: I) unsteady conditions with a rapid reduction of creep rate, and 1I) quasistationary conditions with a slow reduction of the creep rate. As reported previously in [5], the only reliable method of detecting the quasistationary, conditions is to construct the curve of the dependence of the strain rate on the strain (Fig. 2). The deformation curve in section I was described by an equation of the MacVetty type (2), whereas Eq. (3) was used to describe section 11. The unsteady creep parameters, included in Eq. (2), were determined at time t = 10 h, because the experimental results show that the duration of the first stage is around 10-15 h. The constructed graphs (Figs. 2 and 3) wereused to 286

-e,,Y -g2S -8

-6 8

8.e5

Fig. 5

" ga r ?/r.to*.!/K

Fig. 6

Fig. 5. Dependence of residual deflection on load. Fig. 6. Dependence of residual deflection on temperature.

0

18

(,_7 ,Y2

.

48

.

.

.

g~, ~, h

Fig. 7. Programmed deformation of a HPSN specimen: 1) P = 1300 N, Po = 149 N, T = 1473 K, nitrogen; 2) P = 1510 N, Po = 100 N, T = 1473 K, nitrogen; 3) P = 1730 N, Po = 100 N, T = 1493 K, nitrogen; 4) P = 1510 N, Po = 100 N, t = 1693 K, vacuum.

determine the values of the parameters of "reciprocal time" A and activation energy QR: A = 0.19-0.23 h -1, QR = 270 N/mole. To determine the parameter Ko, it is necessary to introduce the "temperature-compensated rate" of unsteady creep Z = exp (Qa/R T) (s

ca)/A = Koa.

(10)

Using the straight line which expresses the dependence of the "temperature-compensated rate" of unsteady creep (Fig. 4), we determine parameter Ko, which is numerically equal to the tangent of the angle of inclination of the straight line: Ko = 0.35.104 M P a - l . h - 1 Investigations were not conducted at a constant stress, but the authors of [4-9] recommend use of the following dependence in the quasisteady stage ~ = a o exp (--me), where % is the "initial" stress, m is a positive constant of the material. The parameters of quasisteady creep were determined at t = 100 h. To compute the parameter n, we use the following expression for the maximum deflection in creep: w~"=--

) n +2

'

(11)

where B = B0 exp (-Qv/RT). 287

From the tangent of the angle of inclination of the dependence In Wvm a x - In P (Fig. 5) we determine the creep factor n = 2.8-3.2. The activation energy of quasisteady creep Qv was determined by means of the tangent of inclination of the straight line of the dependence In Wvmax- 1/T (Fig. 6): Qv = 355 kJ/mole [4, 6, 8-11]. Taking the logarithm of Eq. (11) and using the resultant values of n and Qv we obtain the values of the parameter Bo in testing the specimens in three-point bending: I30 = 0.8964-1.0186 MPa-2"S.h -1. Reversibility of Creep Strain in the Unsteady State. In examining the inelastic behavior of HPSN over a short period of time (no more than 7 h) followed by unloading in nitrogen and in vacuum three-point bend tests were carried out on specimens 7 x 7 x 70 mm in size. The experiment temperature was 1473, 1493, and 1693 K, the distance between the supports 54 mm. Constant load P was established several minutes after applying initial load Po. After 5-7 h the load decreased relatively rapidly to Po (Table 2). The experimental data confirmed the hypothesis according to which the strain in the unsteady stage is restored almost completely (Fig. 7). CONCLUSIONS A model of high-temperature deformation of hot-pressed silicon nltride in creep conditions was formulated. The mathematical model was confirmed by experiment, and the creep parameters of the material were determined. Main results can be described as follows: 1. Creep strain of HPSN consists of unsteady (disappearing after removing the load) and residual creep strains. 2. Unsteady creep and reversible creep strain are of the same nature, associated with grain sliding, and are described by the same law of MacVetty type. 3. The suitability of the law of the MacVetty type is confirmed by the linear form of the section of the ~R--ea dependence (see Fig. 2) and by the linear form of the ln(~s - AeR)-- 1/T dependence (see Fig. 3). 4. The residual creep strain represents "logarithmic creep" (see Fig. 1) and is described by the model of strain hardening. The suitability of the model was confirmed by experiment (see Figs. 5 and 6). 5. The assumption on division of the total strain into two components is confirmed by the creep tests with breaks between loading cycles (see Fig. 7). LITERATURE CITED .

2. 3.

4. 5. 6.

.

8. 9. 10. 11.

288

J. L. Chermant, R. Moussa, and H. Osterstock, "Thermomechanical properties and static analysis of SiC, Si3N4, SiAION, and SiliCOMPS Material," Rev. Int. Hautes Temp. Refract., 18, 5-55 (1981). W. R. Cannon and T. G. Langdon, "Creep of ceramics," J. Mater. Sci., 18, No. 1, 1-50 (1983). A. Venkateswaran, D. P. H. Hasselman, "Creep analysis of bend specimens subjected to tensile cracking," J. Am. Ceram. Soe., 7, 144-155 (1984). F. Garofalo, Laws of Creep and Long-Term Strength of Metals and Alloys [Russian translation], Metallurgiya, Moscow (1968). R. M. Arons, J. K, Tien, and E. M. Lenoe, "Observation of viscoelastic strain recovery in hot-pressed silicon nitride," J. Am. Ceram. Sot., 62, No. 3-4, 216-217 (1979). V. G. Verevka, M. L. Mel'nikova, A. A. Postnikov, and I. I. Tkacheva, "Nonlinear deformation of ~tructural ceramics on the basis of silicon nltride," in: Proceedings of 6th All-Union Conference, Physics of Fracture, Part 1, Institute of Powder Metallurgy, Academy of Sciences of the Ukraine, Kiev (1989), p. 105. G. G. Pisarenko and N. S. Mozharovskii, The Equations and Boundary Problems of the Theory of Plasticity and Creep. A Handbook [in Russian], Naukova Dumka, Kiev (1981). S. Taira and A. Otanl, Theory of High-Temperature Strength of Materials [Russian translation], Metallurgiya, Moscow (1986). Ono, Murakami, and Ueno, "A controlling creep model describing recovery and hardening of the material during changes of the loading direction," Trans. ASME J. Basic Eng., 107, No. 1, 1-7 (1985). R. F. Pabst, "Creep in SiaN4 and SiC materials," in: Creep Behavior of Crystalline Solids, Swansea (1985), pp. 255310. G. P. Poirier, Creep of Crystals. Deformation Mechanisms of Metals, Ceramics, and Minerals at High Temperatures [Russian translation], Mir, Moscow (1988).