International Journal of Fracture Mechamcs, Vol. 7, No. 2, June 1971 Wolters-Noordhoff Pubhshmg Gromngen Printed in the Netherlands

157

Crack Propagation Under Constant Deformation and Thermal Stress Fracture

D. P. H. H A S S E L M A N

Semor Research Assocu~e, Materials Research Center, Allied Chemical Corp, Morristown, N. J. 07960 (U.S A.)

(Received February 1970; in revised form May 1970)

A B S T R A C T An analogy is drawn for crack propagation m brittle solids under constant deformation (strata) and under thermal shock. Following the original approach by Berry it is shown theoretically that under constant deformation an initially short crack, unstable at a critical strain, propagates to a new length which requires a finite increase m strain before it will become unstable agaln~ Similar behavior is expected under conditions of thermal shock where an initially short crack unstable at a critical temperature difference, propagates to a new length such that a finite increase in temperature difference is required before the crack will again continue to propagate. Relative changes in strength as a result of this type of crack behavior are predicted and verified by experimental results for two industrial polycrystalline aluminum oxides subjected to thermal shock by means of a water quench.

Crack propagation under constant deformation or strain (fixed-grips) is governed only by changes in the elastic energy within the object undergoing fracture. Similar considerations should apply to thermal stress fracture in the absence of surface tractions. Crack propagation for these two conditions should show qualitatively similar behavior.

Berry [1] derived equations of motion for the propagation of Griffith [2, 3] cracks in plates under conditions of constant deformation. It was shown that upon propagation of an initially short crack, the final crack length was subcritical, requiring a finite increase in applied strain before the crack again would continue to propagate.

This phenomenon can be described as follows : After Berry [1], a plate of area A and unit thickness, with Young's modulus E and a central crack of length 2C with a stress (a) applied perpendicular to the length of the crack, exhibits a stress-strain relation:

a = AE8 (A + 2xC z)-I (1)

where ~ is the resulting strain. The total energy (W) in the system is the sum of the elastic energy and the surface energy of

the crack. At a strain a, W equals*:

W = A z E~Z/z(a + 2~C 2) + 47C (2)

where 7 is the surface energy per unit area. After Gtiffith [2, 3] the cracks are unstable whenever:

d W ~ C <= 0 (3)

which is satisfied at a critical strain:

ec > (27/~zE) ~ (1 + 2~zC2/A)(C)- ~. (4)

Eq. 4 is shown in Fig. 1 for two values of A. Crack instability occurs between two values of crack length. At short crack length ec is independent of A. The minima in the curve occur for a value of crack length Cm = (A/6z0~. For a crack with initial crack length Co < C,, requiting a critical strain, ec, upon propagation the elastic energy release rate exceeds the surface energy, the difference being converted into kinetic energy. As shown by Berry [1] this kinetic energy is a maximum when the crack reaches the critical length given by the larger value of the two

* For simplicity, transverse stratus are taken equal to zero.

Int. Journ~ of Fraaure Mech-, 7 (1971) 157 161

158 D. P. H. Hasselman

I I / CRITICAL STRAIN FOR /

IOC CRACK INSTABILITY / IOO CRACK LENGTH RESULTING FROM / ' UNSTABLE CRACK WITH C~,,~C m , /~- 2"r/-/io

/ . . . . . . . . . . . . . . . . . . . . . . . . . . / /

. . . . . . . . . . . . . . . . . . . . . . . . . 21 I / . " ,... ~ c ~ , . / I 1 t " - i o

. . . . . -r--'t-,,,.. /_---r i / .-

J,L ,].L io-2 c;~ c~ io-J c,, % I ct c~ Io

CRACK LENGTH (C) Figure 1. Critical strain required for instabihty of central crack in flat plate under conditaons of constant deformation (A, C, E, 7, e in consistent systems of units).

roots ofEq. 4 for ec. The crack will continue to propagate when all the kinetic energy and further released strain energy is transformed into surface energy. This condition is met at a final crack length (C;) when:

A 2 Ee~ [ {2 (A + 2zcC~) }-I _ {2 (A + 2rcC~) } - 1] = 4y (C¢- Co). (5)

Equations (4) and (5) give:

C~ = A/@zC o - Co/2 (6)

C s is shown in Fig. 1 by the dotted curves. For C o < C,~, C I ~ A/4z~Co, i.e., C I is inversely proportional to C o .

From the curves shown in Fig. 1, it is clear that a crack of final length, C s, which results from a crack of length Co < C,, unstable at a strain ec, is subcritical and requires an increase in the strain to a value ef before it will continue to propagate. Also indicated in Fig. 1 is the crack propagation behavior for two short cracks of dissimilar initial length~ It may be noted that for:

C• < C~ < C,, gives e'c > s;', C} > C} and e} > s j . (7)

Under conditions of thermal shock qualitatively similar behavior is expected. A brittle material subjected to thermal shock involving a temperature difference AT, will be subjected to a strain (s), which can be related [4, 5] directly to ~AT, where e is the coefficient of thermal expansion. Analogous to Eq. 7 it is expected that for:

C~< C~< C,, yields AT'~>AT'j, C'.r > C j and A T e > A T / . (8)

Experimentally because of the dependence of strength of a brittle solid on crack length, changes in crack length due to thermal stress fracture can be monitored by a tensile strength test. In terms of the results of the above discussion, strength is expected to be constant over a range of temperature differences 0 < AT < ATe, will exhibit a discontinuous change at AT = ATe, will again be constant for ATe< AT< AT I and will gradually decrease for AT > A T I. Because of the dependence of strength on crack length and the relationship between C o and C I (Eq. 6) it is also expected that for:

C~< C~ yields S'c >S'c' but S}< S~ (9)

where S¢ and S I are the tensile strength before and after fracture initiation, respectively. In order to investigate if crack propagation behavior under thermal stress and constant

deformation are qualitatively similar and to specifically verify the conclusions of Eqs. 8 and 9, thermal shock tests were carried out. The general procedure of Glenny and Royston [6] and

tint. JourrL of Fracture Mech., 7 (1971) 157-161

Crack propagation and thermal stress fracture of, brittle solids 159

' 0

X

T

(,.9 Z UJ n.'*

CO

6O

5O

4C

30

20

0 0

. . . . S;

r . . . . . . S c

ar,~ ~T; "",t . . I ', [ , I

200 400 600 800

QUENCHING TEMPERATURE DIFFERENCE (DEG, C)

Figure 2. Strength of circular rods of Coors AD-94 alumina as a function of quenching temperature difference.

~0

X

n

t9 Z uJ r r h - cO

40

30

20

I0

0 0

i I I 1

..... S~ ..... J

,J I

I

, II 200

' ii

...... S c - _ _

I

I -

l l , I , I 400 6 ~ 800

Q U E N C H I N G TEMPERATURE DIFFERENCE (DEG,C) Figure 3. Strength of circular rods of Wesgo AL-300 alumina as a functaon of quenching temperature dlf[erence.

TABLE 1 Tensile strength of a quenched aluminum oxide rod as a functton of quenching temperature difference.

Quenching temperature difference (~ C)

Strength psi x 10 -3

Coors AD-94 Wesgo AL-300

0 50.3± 4.6~* (10) ~ 356+13.2% (10) 100 52.2-- 7.9~ (10) 35.7-t- 7.7%(8) 200 48.4± 7.8~ (5) 36.64- 9.8~(10) 250 51.8+ 7.3% (5) 17.2+20.8%(10) 300 49.3+ 7.0% (2) 17.5+ 7.7~ (10) 300 10.5+ 11.3 ~ (8) 400 10.6± 7.2~ (5) 17.9± 5.7,% (10) 500 10.0+ 7.6~ (5) 16.0±12.3~(10) 600 10.0+12.1~ (4) 13.7±15.2~ (10) 700 8.3+12.6~ (5) 13.5+12.2~ (8) 800 4.6+15.3~ (5) 12.9±10.3~ (9)

* Coeffioent of variation. ** Number of specimens.

Int. Journ. of Fracture Mech. 7 (1971) 157-161

160 D. P. H. Hasselman

Davidge and Tappin [7] was followed. Circular aluminum oxide rods of length 1.5 inches were placed in a vertical tube furnace with preadjusted temperature. After holding the rods in the furnace for at least 10 minutes to allow the rods to come to thermal equilibrium, they were dropped into water at room temperature. The temperature difference to which the rods were subjected was varied by adjusting the furnace temperature. Under these conditions the thermal stress distribution is such [4] that fracture is nucleated in the surface in the absence of external body forces. Crack propagation can take place only under the influence of the stress field within the specimen analogous to the conditions of "fixed-grips". Following thermal shock the speci- mens were dried and tensile strength determined in four-point bending using a 1~ inch total span in an Instron tester at a cross-head speed of 0.1 inches per minute.

For these tests two industrial aluminum oxide materials were selected: A, Coors AD-94 alumina with relatively high strength S c ~ 50,000 psi, E = 42 x 106 psi and average grain size

10/xm and B, Wesgo AL-300 alumina with Sc ~ 36,000 psi, E ~ 45 x 106 psi and average grain size ~30 to 40/zm. Rod diameters for the two materials were nearly identical (0.187 and 0.195 inches, resp.), to eliminate specimen size effects. An estimate can be made of the average temperature level at which fracture occurs. For the thermal shock conditions employed, a value of heat transfer coefficient, h ~ 1.8 cal. cm-2 sec-1 deg- 1 C, has been reported*. Temperature and stresses as a function of time were calculated by Glenny and Royston [6] and Schneider [8]. For the AD-94 alumina with thermal conductivity '~-k K ~ 0.03 cal. cm- 1 sec- ~ deg- 1 C, at the instant of fracture, the center of the rod is still at the initial temperature with the surface cooled by approx. 200 deg.C. For the AL-300 alumina the corresponding value is approximately 135 deg.C. Since strength is nearly independent [9] of temperature over the temperature range involved, the degree of cooling at the instant of fracture is approximately the same for all initial temperatures.

Experimental results for strength as a function of quenching temperature difference are listed in Table 1 and shown in Figs. 2 and 3. For each set of data points strength is invariant for 0< AT< AT~ and ATe< AT< ATy with a catastrophic decrease at AT~ and a gradual decrease in AT I in accordance with predicted behavior. Comparing the two sets of data for the AD-94 and AL-300 alumina, AT; >ATe" as expected, In addition, for the originally stronger AD-94 alumina the strength after fracture initiation is less than the corresponding strength of the originally weaker AL-300. AT} for the AL-300 alumina is not closely defined because of the rather small decreases in strength for AT > AT}, but AT)>AT} as expected.

It can be concluded from the present results that crack propagation behavior under conditions of thermal shock is similar to crack propagation under conditions of constant deformation and can be predicted qualitatively by the original approach of Berry.

Actual quantitative predictions are expected to be complex. Although thermal stress crack stability has received some attention in the mechanics literature [11, 12], no theories are available for crack stability and propagation in transient non-uniform thermal stress fields as exist under the experimental conditions used in the present study. Approximate attempts were made by the present writer [13, 14]. Because crack propagation occurs under transient conditions and a non-uniform temperature, detailed information is required for the temperature and time dependence of surface energy as determined in a fracture experiment. It is of interest to point out that for C O < Cm Eq. 6 predicts that Cy is independent of material properties. Under thermal stress, however, C ! is expected to be affected by differences in surface energies over the temperature range of the material through which the crack propagates. For poly- crystalline alumina used in the present study over the temperature range involved, little or no change in surface energy is expected as indicated by the results of Gutshall and Gross [10] and the relative independence of strength on temperature [9]. No data appear to be available for the time dependence of surface energy for alumina, but an effect is exlxxzted to exist as observed for glasses by Roesler [15] and polymers by Bennett, Anderson and Williams [16].

* Ref. 7, loc. cir. ,~ As reported by the manufacturer.

Int. Journ. of Fracture Mech., 7 (1971) 157-161

Crack propagat ion and thermal stress f rac ture o f britt le solids 161

Acknowledgement

T h e w r i t e r is i n d e b t e d to h i s a s s o c i a t e s J. J. G i l m a n a n d J. C. M. Li for m a n y he lp fu l d i s c u s s i o n s .

R E F E R E N C E S

[1] J. P. Berry, J. Mech. Phys. Solids, 8 (1960) 207-216. [2] A. A~ Griffith, Phil. Trans. Roy. Soc., A221 (1921) 163. [3] A. A. Gfiffith, Proc. lnt. Cong. Appl. Mech, Delft, 55 (1924). [4] B.A. Boley and J. H. Wiener, Theory of Thermal Stress, John Wiley (1960). [5] H. S Carslaw and J. C. Jaeger, Conduction of Heaz in Soluts, 2nd eeL, Clarendon Press (1959) [6] E. Glenny and M. G. Royston, Trans. Brit. Ceram. Soc., 57 (1958) 645~77. [7] tL W. Davidge and G. Tappin, Trans. Brit. Ceram. Soc, 66 (1967) 405-22. [8] P. J. Schneider, Temperature Response Charts, John Wiley (1963). [9] J tL Hague et al, Refractory Ceramics for Aerospace, The American Ceramic Society, Columbus, Ohio (1964)

[10] P. L. Gutshall and G. E. Gross, En 9. Fracture Mechanics, 1 (1969) 463-71. [11] G. C. Sih, J. Appl. Mech., 29 (1962) 587-89. [12] J. N. Goodaer and A. L. Florence, Proc. Xlth lnt. Congr. of Appl. Mech., (1964) 562~58. [13] D. P. H. Hasselman, J. Am. Ceram. Soc, 46 (1963) 535-40. [14] D. P. H. Hasselman, J. Am. Ceram. Soe., 52 (1969) 288-89. [15] F. C. Roesler, Proe. Phys. Soe, 69 (442B) (1956) 981-92. [-16] S. J. Bennett, G. P. Anderson and M. L. Williams, J. AppL PoL Sc, 13 (1970) 735-43.

RI~SUMI~ On 6tablit une analogae entre la propagation des f'tssures clans les solides fragiles sous d~formation constante, et la propagation sous l'effet de chocs thermiques.

Selon la m6thode propos6e originellement par Berry, on montre par vole th6onque que, sous d6formation constante, une fissure de longueur initiale faibl¢, et qui devient instable/t une certaine d6formation critique, se propage jusqu'~t atteindre une nouvelle longueur requ&ant une aceroissement fmi de la d6formation pour devenir instable ~t nouveau.

Un comportement simdaire peut &re artendu dans le cas de chocs thermiques. Ainsi, une ftssure de faible longueur mitiale, et qui devient instable sous un gradient thermique d~termin6, atteint en se propageant une longueur nouvelle telle qu'il faut un accrotssement fml du gradient de temperature pour que reprenne la propagation.

I1 est possible de pr~dire les modifications relative, de r6slstance qui r~sultent de ce type de comportement des fissures. Ces predictions ont Ot6 v6rifi0,es par voie exp~dmentale on soumettant ~ chocs thermiques par trempe h l'eau deux types industriels d'oxydes d'aluminium polycristallins.

Int. Jourm of Fracture Mech., 7 (1971) 157-161