Crack channelling and spalling in a plate due to thermal shock loading

Transcription

1 Journal of the Mechanics and Physics of Solids 48 (2000) 867±897 Crack channelling and salling in a late due to thermal shock loading L.G. Zhao, T.J. Lu*, N.A. Fleck Deartment of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK Received 12 May 1999; received in revised form 24 August 1999 Abstract The roagation of a re-existing edge crack across a nite late subjected to cold shock has been studied. The late, initially at uniform temerature, is exosed to a cold shock on one surface whilst three di erent tyes of heat transfer boundary condition are searately considered for the oosing face: cold shock, thermal insulation and xed temerature. For all three boundary conditions, the late exeriences tensile stress near the cold-shocked surface and comressive stressing near the mid-lane. Consequently, a Mode I edge crack extending into the comressive region may grow in one of three di erent modes: continued extension in lane strain, channelling and salling. The thermal shock conditions governing each failure mode are quanti ed, with a focus on crack channelling and salling. The dislocation method is emloyed to calculate the energy release rates for lane strain cracking and steady-state channelling. For steady-state salling, the energy release rate is obtained by an energy analysis of elastic beams far ahead and far behind the crack ti. Analytical solutions are also obtained in the short crack limit in which the roblem is reduced to an edge crack extending in a half sace; and the arameter range over which the short crack solution is valid for a nite late is determined. Failure mas for the various cracking atterns are constructed in terms of the critical temerature jum and Biot number, and merit indices are identi ed for materials selection against failure by thermal shock. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Crack branching; A. Fracture; B. Thermal stress; B. Crack mechanics * Corresonding author. Tel.: ; fax: address: tjl21@eng.cam.ac.uk (T.J. Lu) /00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S (99)

2 868 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± Introduction In general, two alternative failure criteria can be used to determine the maximum jum in surface temerature that a material can sustain without cracking (i.e. its thermal shock resistance), one based on strength and the other based on fracture toughness (Hasselman, 1969; Nied, 1983; Rizk and Radwan, 1993; Jin and Mai, 1995; Lu and Fleck, 1998). The stress-based fracture arameter s f /Ea is ertinent to the case of severe thermal shock of a solid containing small re-existing cracks, and large values of the Biot number Bi0hH/k. Here, s f, E, a and k are the tensile strength, elastic modulus, coe cient of thermal exansion and thermal conductivity of the material, H is a reresentative length scale (e.g. the samle thickness) and h is the surface heat transfer coe cient. The toughnessbased fracture arameter for severe shock (Bi>>1) of a solid containing large reexisting cracks is K IC /Ea, where K IC is the fracture toughness. Lu and Fleck (1998) exlored both strength-controlled failure and toughness-controlled failure for the full range of Biot number Bi. At low Biot number, the resistance of a material to thermal shock is sensitive to its thermal conductivity, and the aroriate merit index for ranking material erformance is ks f /Ea for strengthcontrolled failure and kk IC /Ea for toughness-controlled failure. The transition aw size a T beyond which a toughness-based aroach takes over from a strength-based aroach is given by a T 1 (1/ )(K IC /s f ) 2, as discussed by Lu and Fleck (1998). Most revious studies on thermal shock failure are restricted to lane strain cracking. For a nite late under cold shock, the usual roblem under consideration is the roagation of a Mode I edge crack towards the centre of the late. Two other failure modes, crack channelling and salling, may nevertheless comete with lane strain cracking, as sketched in Fig. 1 for the case of a cold shock. The channelling and salling of Mode I cracks under remote tensile loading have been analysed in detail by Hutchinson and Suo (1992). As the threedimensional channelling crack extends, a steady state is achieved so that the crack Fig. 1. Three di erent failure modes in cold shock: (a) lane strain cracking; (b) channelling; (c) salling.

3 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± front shae remains constant. The steady-state energy release rate, G t, averaged over the channelling crack front can be calculated directly from lane strain elasticity solutions. Based on this idea, Hutchinson and Suo (1992) and Beuth (1992) investigated the channelling of cracks through a thin lm bonded to a substrate; in similar manner Ho and Suo (1993) and Chan et al. (1993) studied the tunnelling of cracks through constrained layers. Hitherto, crack channelling and salling under transient thermal stressing have not been addressed Scoe of the resent study In the current study, the roagation of a lane strain edge crack and the channelling and salling of the crack within a nite late are analysed in detail for the case of a cold shock. To cover a wide range of cold shock scenarios, three di erent heat transfer conditions are considered for the late, held initially at uniform temerature T i. In Case I, both the to and bottom surfaces of the late are subjected to a cold shock with identical heat transfer coe cients. In Case II, the late is subjected to a cold shock on the to surface, with erfect thermal insulation on the bottom face. In Case III, a cold shock is alied to the to surface of the late while the bottom face is held at the initial temerature. For all three thermal boundary conditions the temerature and stress histories in the uncracked late are obtained over the full range of Biot numbers. The surface subjected to a cold shock always exeriences the highest tensile stress, and hence is the most likely site for the initiation of an edge crack. As discussed above, the crack may grow in one of three di erent modes: it may roagate in lane strain along the thickness direction of the late (Fig. 1(a)), it may channel in the lane of the late (Fig. 1(b)), or it may sall arallel to the surface of the late (Fig. 1(c)). The energy release rate associated with a lane strain edge crack of arbitrary length is calculated as a function of time and Biot number, using the dislocation method. This result is subsequently used to comute the energy release rate associated with steady-state channelling. For steady-state salling the energy release rate is calculated from the elastic energy stored in beams far ahead and far behind the crack ti, with the corresonding mode mix determined from the known solutions for a slit beam (Hutchinson and Suo, 1992). An analytical solution is also obtained in the short crack limit by considering the roagation of an edge crack in a half-sace. The range of validity of this solution is determined by direct comarison with the numerical results for an edge crack in a nite late. Failure mas are constructed to dislay the thermal shock conditions governing lane strain cracking, crack channelling and salling. Finally, merit indices for each mode of failure are obtained in order to aid the selection of engineering materials for thermal shock resistance.

4 870 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± Temerature and stress elds 2.1. The thermal model Consider an in nitely large late of thickness 2H and uniform initial temerature T i, and let Cartesian co-ordinates (x, z ) be embedded at the centre of the late, as deicted in Fig. 2. At time t = 0, the late is suddenly exosed at its to surface (z=h ) to a convective cooling medium of temerature T 1 such z, t k ˆ h T T 1, at z ˆ H where k is the thermal conductivity of the material from which the late is made, h is the coe cient of heat transfer, and T(z, t ) is the temerature distribution of the late. At the bottom surface (z= H ), one of the following three di erent heat transfer boundary conditions is assumed to hold, see Fig. 2. Case I is an identical cold shock to the to surface, z, t k ˆ h T T 1, at z ˆ H Case II is thermal insulation, z, t ˆ 0, at z ˆ H and Case III is a constant temerature equal to the initial temerature, Fig. 2. A crack-free late of thickness 2H subjected to three di erent tyes of cold shock where T i > T 1.

5 T ˆ T i, at z ˆ H 2c The transient temerature distribution in the late is governed by the Fourier law of 2 T z, 2 ˆ 1 k L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 z, t, j z j RH where k is the thermal di usivity of the material. For simlicity, the material is assumed to be isotroic, with thermo-hysical roerties indeendent of temerature; the results obtained in the following sections can be extended to orthotroic materials by using the technique of orthotroic scaling (Hutchinson and Suo, 1992; Lu and Fleck, 1998) Evolution of temerature and stress Eq. (3) and the associated initial and boundary conditions can be solved by the standard searation-of-variables technique (Carslaw and Jaeger, 1959). For Case I, the non-dimensional temerature eld is found to be T z, t T z, t T i T i T 1 ˆ 1 2 X1 nˆ1 sin b n cos b n z=h ex b n sin b n cos b n b 2 kt n where b n are the ositive roots of b tan b ˆ Bi; the corresonding solutions for Cases II and III are given in Aendix A [cf Eqs. (A1) and (A3)]. The evolution of temerature in the uncracked late under cold shock (T i > T 1 ), as redicted searately from Eqs. (4), (A1) and (A3), is lotted in Fig. 3 for selected values of dimensionless time t - 0kt/H 2. Here, for brevity, results are resented only for the case of Bi = 10, which is tyical for the quenching of glass in ice water; results over the full range of Biot number (0 < Bi < 1) are qualitatively similar to those shown in Fig. 3. It is interesting to note from Figs. 3(a) and (b) that Cases II and III give almost the same temerature distribution in the uer half of the late (z - > 0) during the initial stage of cold shock (t - < 0.4). This is because, in Cases II and III, the time for the `thermal front' to traverse the late is t (Lu and Chen, 1999). Once the temerature eld has been determined, it is relatively straightforward to determine the transient thermal stress distribution in the late from considerations of force and moment equilibrium. It is assumed that the late is free to exand with vanishing in-lane force resultants H H s xx z, t dz ˆ 0, H H s yy z, t dz ˆ 0 and vanishing normal stress in the through-thickness direction, s zz =0. In addition equilibrium dictates that the bending moment due to the thermal stresses vanishes, giving H 2 4 5

6 872 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 H H s xx z, t z dz ˆ 0 and H H s yy z, t z dz ˆ 0 Note that the temerature distribution is unsymmetrical about the mid-lane z = 0 in Cases II and III, resulting in a nite late curvature. It follows from Eqs. (5) and (6) that the transient stress s xx (z, t ) induced by the temerature distribution T(z, t ) can be written in the following non-dimensional form (cf Manson, 1966) s z, t s xx z, t Ea T i T 1 ˆ T T i T i T 1 1 2H H H T T i dz 3z T i T 1 2H 3 H H 6 T T i z dz T i T 1 7 where E - =E/(1 n 2 ) and a ˆ a 1 n : Here, E, n and a denote Young's modulus, Poisson's ratio, and the coe cient of thermal exansion, resectively. For Case I, substitution of (4) into (7) leads to Fig. 3. Transient temerature distribution in the late under cold shock with Biot number Bi = 10: (a) Case I; (b) Case II; (c) Case III.

7 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± s z, t ˆ 2 X1 sin b n z cos b b n sin b n cos b n sin b n ex b 2 kt n n H b n H 2 nˆ1 For brevity, the stress elds in the late corresonding to Cases II and III boundary conditions are resented in Aendix A. Recall that in the limiting case of a thermally insulated late (Bi = 0) the late is everywhere stress-free, whereas it su ers the most stringent thermal stressing when Bi 4 1. The normalised stress s z, t ˆs z, t = Ea T i T 1 is lotted in Fig. 4 for the choice of Biot number Bi = 10 and for selected values of dimensionless time t ˆ kt=h 2 : At any given instant, the to surface of the late exeriences the largest tensile stress, and is exected to be the site of crack initiation. For Case I, symmetry dictates that both the to and bottom surfaces exerience the same tensile stress history, imlying that edge cracks may initiate 8 Fig. 4. Transient stress distribution in the late under cold shock with Biot number Bi = 10: (a) Case I; (b) Case II; (c) Case III.

8 874 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 Fig. 5. Maximum tensile stress, s max, at the to surface of the stri and the corresonding time elase, t - max, lotted as a function of Biot number Bi. from either surface. It is known from a comarison of the stress intensity factor solutions for a single edge crack, and for two oosing edge cracks of equal length that the initiation of a single edge crack is energetically more favourable than the initiation of two symmetrical edge cracks (Hutchinson and Suo, 1992); hence, the roagation of a single edge crack will be the focus of our study. It is also noted that, in the early stages of a cold shock (t - < 0.4), almost identical stress elds are generated in Cases II and III in the region z - > 0 as a direct consequence of the nearly identical temerature distributions [recall Figs. 3(b) and (c)]. For all three cases considered, the maximum tensile stress s max attained at the to surface of the late together with its time of occurrence t - max are lotted against 1/Bi in Fig. 5. The values for s max and t - max in Cases II and III are nearly identical. The following aroximations for s max and t - max have the correct asymtotic behaviours and are found to describe closely the comuted values shown in Fig. 5, s max 0, t max ˆ 1:5 3:25 Bi 0:5 e 16=Bi 1, t max ˆ 0:42 1 1:18 Bi 9a for Case I and s max 0, t max ˆ 1:5 5:55 Bi 0:5 e 44=Bi 1, t max ˆ 0:21 1 0:88 Bi 9b for Cases II and III. Similar curve- tting formulae have been roosed by Manson (1966). It is clear that the magnitude of s max increases with increasing Biot number Bi; in the limit Bi 4 1, we nd that s max ˆ 1 and t - max=0. Of the

9 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± three cases studied, the constraint imosed by the Case I boundary condition roduces the largest s max Ð although the corresonding time of occurrence t - max is also the longest. 3. Plane strain cracking under thermal shock The results resented above indicate that, under cold shock, cracks are most likely to initiate from the to surface of the late. Once such an edge crack has formed, it may grow in one of three di erent modes as sketched in Fig. 1. We begin the crack analysis by considering the idealised case of an isolated Mode I edge crack of deth a (Fig. 6(a)). The analysis resented here builds uon that given already by Lu and Fleck (1998) for an edge crack under Case I boundary conditions. It is assumed that the late contains a distribution of aws on the scale of the structural dimension, and that the crack commences from the ``worst aw'' having the largest driving force, i.e., the greatest energy release rate. In this section, lane strain cracking is analysed to rovide baseline solutions for channelling (Section 4) and salling (Section 5); results for the short crack limit of an edge crack in a half sace are resented in Section 6. For lane strain cracking, channelling and salling in a nite late, emhasis is laced on nding the energy release rate maximised with resect to crack length and time as a function of the Biot number. As will become evident later, these solutions are no longer valid as the late thickness 2H 4 1 (i.e. Bi 4 1); short crack solutions must then be used. For an edge crack roagating in lane strain (Fig. 1(a)), the energy release rate Fig. 6. (a) An in nite late embedded with a re-existent crack subjected to three di erent tyes of cold shock, and (b) energy release rate G as a function of crack size and elased time for lane strain cracking (Case I with Bi = 10).

10 876 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 G at the crack ti is given by G ˆ K 2 I E 10 where the Mode I stress intensity factor K I can be determined by the dislocation method. Here the Mode I edge crack is modelled as a continuous distribution of edge dislocations. The crack face traction boundary condition is enforced by integrating the traction contribution on the crack face (x = 0) due to the distribution of dislocations, resulting in the singular integral equation, a 2A s a z s ds A s H s, z ds ˆ s z, t, 0 < z < a where s(z, t ) is the thermal stress distribution along the crack line, A(s ) is the density of edge dislocations, and the kernel function H(s, z ) is given by Civelek (1985). Following Erdogan et al. (1973), the dislocation density function A(s ) is 1=2, 1=2 exanded as a series of Jacobi olynomials P x, such that s 1 x A s ˆA x ˆ 1 x X N kˆ1 uon making use of the changes of variables k 1 1=2, 1=2 C k P k 1 x 12 1 x s ˆ a, 1 < x < By selecting N collocation oints on the crack surface to match the boundary condition (11), a system of linear equations is obtained to determine the coe cients C k for k in the range 1 to N. Once the solution has been obtained, the stress intensity factor K I and the oening dislacement d(a') at any oint a' on the crack face are calculated from the exressions and K I ˆ 2 a a X N kˆ1 1=2, 1=2 C k P k d a 0 ˆ A s ds a 0 For the Case I temerature boundary condition with Bi = 10, the normalised energy release rate G G = Ea 2 H T i T 1 2 redicted by the dislocation method is lotted in Fig. 6(b) against dimensionless crack lengths ā 0 a/h, for selected values of dimensionless time t - 0 kt/h 2. As a check, we have solved the same roblem by using the weight function method (Tada et al., 1985; Lu and Fleck, 1998). The redictions are in close agreement with those obtained from the

11 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± dislocation method Ð the latter method is more accurate but is comutationally more intensive. In the remainder of this aer, unless otherwise stated, all calculations have been erformed by the dislocation method. We note that G - attains a eak value for all values of crack length at t ; the global maximum of G - is attained at t and ā For Cases II and III and for Biot numbers over the full range 0 < Bi < 1, the results are qualitatively similar to those shown in Fig. 6(b) for Case I. Consider the case of a re-existing lane strain edge crack of length a in a late with a given Biot number Bi; then G attains its eak value (G ) eak after a time t eak. When (G ) eak equals the toughness G IC, lane strain cracking occurs; otherwise, the edge crack either arrests or advances in other modes such as Fig. 7. Failure ma for lane strain cracking, channelling and salling, with Bi = 10. (a) Case I; (b) Cases II and III (Bi = 10).

12 878 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 channelling and salling. On recalling the connection G A (DT ) 2 where DT 0 T i T 1 is the temerature jum at the to surface of the late, the fracture criterion (G ) eak =G IC can be used to evaluate the critical surface temerature jum (DT ) eak to cause lane strain cracking as a function of aand Bi. For illustration, Fig. 7(a) lots the redicted D T eak DT eak = G IC = Ea 2 H against the dimensionless crack size ā=a/h for Bi = 10 and Case I boundary condition. We emhasise that (DT ) eak is the maximum temerature jum that can be alied to the late without causing lane strain cracking, over all time, for any xed crack length and Biot number. Similar calculations are reorted below for steady-state channelling and salling, and the corresonding critical temerature jums are included in Fig. 7(a) as dashed lines (channelling) and dotted lines (salling). For Cases II and III, the results are summarised in Fig. 7(b), where again a value Bi = 10 is assumed. Fig. 7 can be used as a failure ma to determine which one of the failure modes is most likely to occur when a material embedded with a re-existent aw is subjected to cold shock. Deending uon the size of the aw, this may haen in one of the six crack atterns indicated in Fig. 7: (A) no cracking; (B) lane strain cracking; (C) channelling; (D) channelling and salling; (E) lane strain cracking and channelling; (F) lane strain cracking, channelling and salling. Note that, in general, lane strain cracking tends to dominate when the aw size a is small, whereas channelling and/or salling is referred when a is large. For a given Biot number Bi, G (t, a ) achieves a global maximum value (G ) max after a time t c and for a articular crack length a c. Non-dimensional values of (G ) max have been obtained as functions of Bi for the three boundary conditions, Cases I±III. It is found that the following simle curve- tting formulas are in good agreement with the numerical values G max G max ˆ 0: :2 2, Case I 16 G 0 Bi G max G max ˆ 0: :9 2, Cases II and III 17 G 0 Bi where G 0 ˆ Ea 2 H T i T 1 2 : We note that the limiting value of the energy release rate for Case I is G max ˆ 0:187 Ea 2 H T i T 1 2 at a/h 1 1/4 and kt/h 2 =0.06. This energy release rate, attained under the most severe thermal shock boundary condition Bi=1, exceeds the largest values obtained for Cases II and III. A comarison with the energy release rates obtained below for channelling and salling also reveals that this value of (G ) max is the highest achieved for the three cracking modes under any boundary condition considered. However, we emhasise that the limit Bi=1, when alied to Eqs. (16) and (17), should be interreted as h 4 1 at nite H and not H 4 1 at nite h, i.e., Eqs. (16) and (17) do not aly in the latter case. The limiting case of an edge crack extending in a semi-in nite solid (i.e., H 4 1) will be discussed later in Section 6.

13 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± On recalling that G G = Ea 2 T i T 1 2, Eqs. (16) and (17) can be used together with the fracture criterion, G max ˆ G IC K 2 IC = E to determine the maximum surface temerature jum that can be alied to the late without incurring lane strain cracking, (DT ) max, as a function of the Biot number Bi. The resulting dimensionless maximum temerature jum D T max DT max a EH = G IC and the corresonding values of t c kt c =H 2 and a c a c =H at which (DT - ) max occurs are lotted in Fig. 8 as a function of Bi for Cases I±III. Curve- tting formulae for these results are Ea H D T max DT max ˆ 2:31 5:05 18a K IC Bi Fig. 8. (a) Maximum surface temerature jum (thermal shock resistance) as a function of a Biot number for all three cases. Critical crack size and time elase for (b) Case I and (c) Cases II and III.

14 880 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 t c kt c H 2 ˆ 0:06 0:43 1 1:5 Bi 18b a c a c H 10:3 for Case I, and Ea H D T max DT max ˆ 3:18 9:2 K IC Bi t c kt c 0:3 ˆ 0:02 H 2 1 Bi 18c 19a 19b a c a c H 10:22 19c for Cases II and II. We note in assing that Eqs. (18a) and (b) are slightly more accurate than the revious curve tting results of Lu and Fleck (1998). 4. Channelling under thermal shock Consider the roagation of a lane strain edge crack in a late under cold shock. For all three boundary conditions considered (Fig. 6(a)) we redict that the lane strain crack roagates unstably at short crack lengths, whereas for long cracks the energy release rate decreases with increasing a. This decreasing driving force is due to the resence of comressive stress in the central ortion of the late. In addition to advance in the through-thickness direction, a crack can advance in other modes, such as by channelling or salling. The channelling of an edge crack is a comlicated, three-dimensional rocess di cult to analyse. However, for a su ciently long channelling crack, a steady state is reached whereby the crack channels at xed deth, with a constant ti shae and constant energy release rate. The steady-state energy release rate, G t, averaged over the deth of the channelling crack front can be directly calculated from the lane strain elasticity solutions resented in the revious section. If it is assumed that the toughness of the solid G IC is indeendent of the mode mix around the erihery of the curved crack front then a fracture criterion for channelling can be written as G t =G IC. Considerations of energy balance lead to two alternative but equivalent formulas for G t (Hutchinson and Suo, 1992; Beuth, 1992; Ho and Suo, 1993; Chan et al., 1993), as G t ˆ 1 a G a 0 da 0 20 a and 0

15 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± G t ˆ 1 2a H a H s a, z d a, z dz 21 Eq. (20) requires the solution of G for a lane strain crack of arbitrary length a' (0 < a' < a ). An evaluation of (21) makes use of the thermal stress distribution s(a, z ) on the rosective crack lane rior to cracking and the associated oening ro le d(a, z ) of the crack. Both Eqs. (20) and (21) can be used to calculate G t. For Eq. (20) the weight function method should be used to determine G (a'), whereas for Eq. (21) the dislocation method is recommended to determine the crack oening ro le d(a, z ). The results for G t reorted below are calculated from Eq. (21). For illustration, consider Case I boundary conditions with Bi = 10. The normalised energy release rate for steady-state channelling G - =G t /G 0 is lotted in Fig. 9 as a function of normalised crack length ā=a/h and normalised time t - =kt/h 2, where G 0 Ea 2 H T i T 1 2 : The trends exhibited by G t for steady-state channelling are similar to those of G for lane strain cracking, with the eak value of G t at any crack deth occurring at t [comare Figs. 6(b) and 9]. The result for G t as a function of ā at xed t - are qualitatively similar for all three tyes of boundary condition, and so lots are omitted for the boundary conditions of Cases II and III. Consider a channelling crack in steady state, with a xed deth a and a given Biot number Bi. Then G t attains its eak value (G t ) eak after a time t eak. The associated surface temerature jum (DT t ) eak is determined by equating (G t ) eak with the toughnessg IC. The non-dimensional temerature for channelling, D T t eak DT t eak = G IC = Ea 2 H, is lotted in Fig. 7(a) against the dimensionless crack size ā=a/h, for Bi = 10 and the Case I boundary condition. The results for Cases II and III, with Bi = 10, are summarised in Fig. 7(b). We note from Figs. 7(a) and (b) that the critical temerature jum for lane strain Fig. 9. Energy release rate G t as a function of crack size and time elase for steady-state channelling (Case I with Bi = 10).

16 882 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 crack growth is less than that for channelling, for small values of crack length [ā < 0.5 in Fig. 7(a), and ā < 0.3 in Fig. 7(b)], whereas channelling is exected to occur in reference to lane strain crack growth at greater crack deths. The global maximum (G t ) max for any given Biot number is obtained by searching for the maximum value of G t over all values of crack length and time. As in the case of lane strain cracking, simle curve- t formulae are obtained for the global maximum energy release rate for steady-state channelling, (G t ) max,with G t max G t max ˆ 0: :2 2, Case I 22a G 0 Bi G t max G t max ˆ 0:08 1 2:9 2, Case II and III 22b G 0 Bi These results can be used together with the fracture criterion (G t ) max =G IC to estimate the maximum temerature jum (DT t ) max that can be alied without channelling. [Again, note that Eq. (22) cannot be alied in the limiting case of H 4 1 at nite h.] The redicted (DT - t) max, together with the corresonding crack deth a c ˆ a c =H and elased time t c ˆ kt c =H 2, are shown in Fig. 8 as dashed lines and are adequately described by the following curve- ts: Ea H D T t max DT t max ˆ 2:54 5:4 23a K IC Bi t c kt c H 2 ˆ 0:06 0:43 1 1:5 Bi 23b a c a c H 10:54 for Case I and Ea H D T t max DT t max ˆ 3:53 10:0 K IC Bi 23c 24a t c kt c 0:3 ˆ 0:02 H 2 1 Bi 24b a c a c H 10:41 24c for Cases II and III. Di erentiation of (20) with resect to a gives

17 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± dg t da ˆ 1 a G G t 25 and so G equals G t at the local maximum dg t /da = 0, with t held xed. Consequently, DT t for channelling equals DT for lane strain cracking at this local maximum. The global maximum for channelling, G t =(G t ) max occurs at a crack length ā c and a time t - c, and the DT for lane strain cracking equals (DT t ) max at this time. But, for this value of crack length ā c, there exists a (DT ) eak which occurs at some other time and is less than the value DT. Consequently, the curve of (DT ) eak versus ā does not cross the (DT t ) eak versus ā trajectory at the oint where (DT t ) eak attains the turning value (DT t ) max. 5. Salling under thermal shock Consider again a Mode I edge crack extending normal to the to surface of the late (Fig. 6(a)). Initially, G increases with a and crack growth is dynamically unstable (Fig. 6(b)). However, when the ti of the crack enters a zone of comressive stress in the central ortion of the late, the elastic energy available for continued crack extension decreases Ð crack extension is now dynamically stable (Fig. 6(b)). The crack may then choose to channel as discussed in Section 4, or to de ect into a salling ath as shown in Fig. 1(c) and analysed below E ect of T-stress on directional stability of crack extension The ath along which a Mode I crack may choose to grow is sensitive to the T- stress. For a Mode I crack extending in an isotroic material, the well-known Williams' asymtotic exansion of in-lane stress comonents near the crack ti is KI s ab ˆ ~s ab y T d 1a d 1b O r 1=2, a, b ˆ1, r where (r, y ) are olar co-ordinates centred at the ti of the crack, d ab is the Kronecker delta, and T is a non-singular stress acting arallel to the crack lane, namely, the T-stress. For anisotroic materials, Eq. (26) can still be used to de ne the stress intensity factor K I but the dimensionless functions ~s ab now deend on material roerties as well as y (Sih et al., 1965). Some insight into the value of T is obtained by recalling that, for a semi-in nite crack in a full sace, the T-stress equals the negative of the normal traction acting on the cracking lane at the location of the crack ti (Lu, 1996). Once the T-stress is known, the directional stability of crack extension is determined: stable if T is negative and unstable if T is ositive (Cotterell and Rice, 1980). Physically, this may be interreted as follows: when straight-ahead advance of a Mode I crack is erturbed at the crack ti due to micro-heterogeneity such as ores and inclusions, a ositive T-stress will render the crack to veer away from the straight ath while a negative T-stress will kee the crack in its original straight trajectory. Overall force equilibrium in a

18 884 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 thermally shocked late dictates that a tensile thermal stress eld is always accomanied with a comressive thermal stress eld; hence, a Mode I crack advancing normal to this stress eld has the tendency to kink away from the straight ath once its ti enters the comressive zone Initiation and steady-state growth of salling cracks Referring to Fig. 10(a), assume that a crack of length a, initially advancing in Mode I in a direction erendicular to the surface of the late, branches into two identical cracks arallel to the surface. The doubly de ected crack ath is chosen not only for its simlicity, but also because the associated energy release rate and mode mix are, to an excellent aroximation, identical to those associated with a singly de ected ath (He and Hutchinson, 1994). (The incorrect results for the doubly de ected crack roblem contained in the original aer of He and Hutchinson, 1989, are corrected in He et al., 1994.) Although this symmetry is idealised, it is exected to cature the essence of all de ection behaviours. The analysis below will be further restricted to the initiation of kinking (c/a < 0.1) and steady-state salling (c/a > 1), with the transient stage ignored. The solutions obtained for steady-state salling will aly so long as the de ected crack length c is larger than a (Hutchinson and Lu, 1995; Lu, 1996) Initiation of kinking (c<<a) Consider a utative kink of length c at right-angles to a arent Mode I crack of length a (>>c ). The Mode I and II stress intensity factors at the ti of the kink (K 1, K 2 ) are related to the stress intensity factors at the ti of the arent crack (K I, K II ) by (Hutchinson and Suo, 1992; He et al., 1994; Lu, 1996) K 1 ˆ 0:398K I 1:845T c Fig. 10. (a) Model for steady-state salling, and (b) equivalent edge force and moment.

19 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± K 2 ˆ 0:323K I 0:212T c 27 where the T-stress, T, is calculated in Aendix B. The corresonding energy release rate G s is given by G s ˆ 1 E K 2 1 K Note that the mode mix at the ti of the de ected crack is fairly large, with c 0 tan 1 (K 2 /K 1 )=39.18, for the case of a arent crack in ure Mode I. Thus, the kink has a tendency to deviate in ath back towards the original direction of growth of the Mode I arent crack. However, if microstructural inhomogeneities are resent to kee the kink advancing arallel to the free surface of the late, a steady-state is achieved for the salling crack with a new xed value of crack ti energy release rate and mode mix Steady-state salling (c>>a) When c>>a, the de ected crack advances in steady-state as shown in Fig. 10(b); the crack is driven by a constant bending moment M and a constant in-lane axial force P er unit deth, related to the thermal stress eld by P P 1 EaH T i T 1 ˆ s z=h, t d z=h 1 a=h M M 1 EaH 2 T i T 1 ˆ z=h a=2h 1 s z=h, t d z=h 1 a=h 29 Now, the steady-state energy release rate G s due to (P, M ) and the associated stress intensity factors (K 1, K 2 ) can be obtained by an energy analysis of the elastic beams far ahead and far behind the crack ti, couled with arguments based on linearity and dimensionality (Hutchinson and Suo, 1992). The desired results for the steady-state values are ( ) G s ˆ Gs ˆ 1 P 2 G 0 2 A a=h M 2 I a=h 3 2 P M AI a=h 2 sin g 30a and K 1 Ea H Ti T 1 ˆ P M cos o q sin o g 2A a=h 2I a=h 3 K 2 Ea H Ti T 1 ˆ P M sin o q cos o g 30b 2A a=h 2I a=h 3

20 886 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 where 1 A ˆ 1 4Z 6Z2 3Z 3, 1 I ˆ 12 1 Z3, sin g ˆ 6Z 2 1 Z AI Z ˆ a=h, o 52:18 38Z 31 2 a=h The hase angle c of mode mix at the ti of the de ected crack is de ned by c ˆ tan 1 K 2 =K 1 32 For Case I temerature boundary condition with Bi = 10, the normalised energy release rate G - s and the associated hase angle c for steady-state salling are lotted in Figs. 11(a) and (b) as a function of the normalised salling deth ā, at selected values of normalised time t -. For all values of crack deth ā, G - s attains a maximum value at a time t (Fig. 11(a)). The global maximum in G - s is attained at t and at ā=1. A straightforward hysical exlanation is that at t ˆ kt=h 2 ˆ 0:1, the two thermal fronts commencing searately from the to and bottom surfaces meet at the mid-oint (Lu and Chen, 1999). The hase angle c deends strongly on both ā and t -, as shown in Fig. 11(b), but a few general remarks can be made. The mid-lane ā=1 is always a Mode I ath, such that c=0. For t - less than about 0.1, there are three ossible Mode I aths: the mid-lane and two aths symmetrically disosed to the mid-lane. For t - greater than about 0.1, a single Mode I ath exists along the mid-lane. Recall that a Mode I ath is likely to roagate in a stable con guration, without deviation in ath, is negative, as discussed by Fleck et al. (1991) and as summarised as follows. Consider a crack located arallel to the ath of c=0 but laced a small distance dā below it and assume is negative; then, for such a erturbed ath, c is negative and the crack will tend to kink back towards the Mode I ath. Examination of Fig. 11(b) shows that for t - r 0.1 the mid-lane is a con gurationally stable Mode I ath, whereas for t - < 0.1 the midlane is con gurationally unstable but the adjacent two Mode I aths are stable. Results for G - s and c for Cases II and III are resented in Figs. 11(c) and (d). Again, a value Bi = 10 is assumed and the deendence of G - s and c uon the nondimensional crack deth ā is lotted for selected values of scale time, t -. At any given time, there exists a single Mode I ath, its location shifting gradually with time from near the to surface towards the bottom surface of the late; the Mode I aths are stable in the sense <0. For a xed Biot number, the maximum temerature jum to withstand steadystate salling, (DT s ) eak, for any deth ā of a salling crack is calculated by equating the maximum value of G s over all time with the toughness G IC, for that value of ā. The resulting values of D T s eak DT s eak = G IC = Ea 2 H are included in Figs. 7(a) and (b) for the case Bi = 10. Let (G s ) max denote the global maximum energy release rate for steady-state salling at any xed value of Bi. Simle curve-

21 t formulas for (G s ) max are obtained as G s max L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± G s max ˆ 0: :1 2, Case I 33a G 0 Bi G s max G s max ˆ 0: :4 2, Cases II and III 33b G 0 Bi Note that (G s ) max corresonds to a Mode I ath in Case I (Figs. 11(a) and (b)) and a nearly Mode I ath in Cases II and III (Figs. 11(c) and (d)). From the fracture criterion (G s ) max =G IC, the redicted maximum temerature jum (DT s ) max without salling is shown in Fig. 8 as dotted lines together with a c ˆ a c =H and t c ˆ kt=h 2 : The related curve- tting formulas are Ea H D T s max DT s max ˆ 3:60 7:5 34a K IC Bi Fig. 11. (a) Energy release rate, (b) hase angle as functions of salling deth and time for Case I (Bi = 10), (c) energy release rate, and (d) hase angle as functions of salling deth and time for Cases II and III (Bi = 10).

22 888 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 t c kt c H 2 ˆ 0:08 0:4 1 1:35 Bi 34b a c a c H ˆ 1:0 for Case I and Ea H D T s max DT s max ˆ 6:08 14:8 K IC Bi t c kt c 0:35 ˆ 0:04 H 2 1 Bi a c a c H ˆ 0:44 0:35 1 0:25 Bi 34c 35a 35b 35c for Cases II and III. It is clear from Fig. 8(a) that the maximum temerature that can be alied without steady state salling exceeds that for lane strain cracking and channelling, for all three cases of thermal boundary condition, and for all values of Biot number. Finally, we note in assing that Eqs. (33)±(35) do not aly in the case of H 4 1 at nite h. 6. Solutions in the short crack limit At su ciently short times after thermal shock loading of a late, the temerature and stress distributions decay raidly a small distance from the surface of the late, and the late behaves as a semi-in nite solid. In this section, we study the resonse of a lane strain crack, a channelling crack and a salling crack within a semi-in nite solid when its surface is subjected to a thermal shock. These asymtotic solutions are good aroximations to the nite late case rovided the crack length a is small comared to the secimen thickness 2H, and rovided the time t is su ciently short. The range of validity of these short crack solutions will be quanti ed for various values of h. The limiting case h 4 1 for lane strain cracking of an in nite solid has already been solved by Hutchinson and Suo (1992); no analytical solutions aear to exist for crack channelling and salling within an in nite solid Plane strain cracking In the limit a/h 4 0, the lane strain cracking roblem is reduced to that of an edge crack of length a in a semi-in nite solid subjected to a cold shock on the surface [see Fig. 12(a), where z = 0 is now laced on the surface of the solid]. Then, the boundary condition on the surface becomes

23 k ˆ h T T 1 at z ˆ 0 36 and the resulting temerature distribution in the block is given by (Carslaw and Jaeger, 1959) kt a, Bi ( ex Bi z ) 2 a Bi kt erfc a z T a, L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± T T i T i T 1 ˆ 1 erf z a a 2 kt z! 37 a a 2 Bi kt kt a where erf x ˆ 2= x 0 ex z 2 dz is the error function, erfc(x )=1 erf x, and Bi =ha/k is a new Biot number based on the crack length a (comare with Bi=hH/k for the nite late). The thermal stress eld due to the temerature transients is (Hutchinson and Suo, 1992) z s a, kt a, Bi s xx z, t z Ea T i T 1 ˆ T a, kt a, Bi 38 and the normalised stress intensity factor at the ti of the lane strain edge crack of length a is (Tada et al., 1985) kt ^K I a, K I Bi Ea T i T 1 a ˆ T z q f1:3 0:3 z=a 5=4 gd 0 1 z=a 2 a Fig. 12. (a) An edge crack in a half sace under cold shock, and (b) steady-state salling in a half sace.

24 890 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 Consequently, the normalised energy release rate for lane strain cracking is kt ^G a, Bi G Ea 2 T i T 1 2 a ˆ ^K I 2 40 It is instructive Ð and also ractically relevant Ð to determine the ranges over which the solution (37)±(40) is valid for an edge crack of the same length extending in a late of nite thickness 2H. For the Case I cold shock, the normalised stress intensity factor for an edge crack in a late of thickness 2H is lotted in Fig. 13(a) as a function of normalised time kt/a 2,atBi =ha/k =1. Results are lotted for selected ratios of crack length a to late semi-thickness H, including the limiting case of an edge crack in a half-sace a/h = 0. The solutions for an edge crack with thermal boundary conditions given by Case II and III, and for channelling cracks with all three tyes of thermal boundary condition, show similar trends to those shown in Fig. 13(a). For any given a/h ratio and Biot number Bi, the stress intensity factor at the ti of an edge crack in a half-sace is nearly identical to that at the ti of an edge crack in a nite late, rovided the normalised time kt/a 2 is smaller than a critical value kt /a 2. In general, the value of this critical time increases with decreasing a/h and with decreasing Bi. For de niteness, let kt /a 2 denote the time interval over which the asymtotic solution for an edge crack of length a in a half-sace is within 10% of the solution for an edge crack, of length a, in a late of given thickness 2H, and subjected to the same cold shock. The interval kt /a 2 is lotted in Fig. 13(b) as a function of a/h Fig. 13. (a) Stress intensity factor as a function of dimensionless time kt/a 2 for selected value of a/h, subjected to Case I cold shock with Bi =1, and (b) range of validity for the short crack solutions.

25 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± and Bi for all three cases of thermal boundary condition. Observe that kt /a 2 deends strongly on a/h, and only weakly on Bi Steady-state channelling For steady-state channelling of an edge crack in a half-sace the normalised energy release rate is related to that for lane strain cracking by kt ^G t a, G t Bi Ea 2 T i T 1 2 a ˆ 1 a ^G a 0 da 0 41 a 0 Fig. 14(a) lots GÃ and GÃ t as functions of dimensionless time kt/a 2 and Biot number Bi =ha/k. Note that both GÃ and GÃ t reach their global maximum values ^G 1 ˆ 2 ^G 1 t ˆ 3:96 in the limit kt/a 2 4 1; this corresonds to a state of uniform thermal stress s xx ˆ 1 over the full deth of the crack, with ^K 1 I ˆ 1:1215: These limiting values are indeendent of the surface heat transfer coe cient h (or Biot number Bi ). On alying the toughness-based fracture criterion G =G IC and G t =G IC, the maximum surface temerature jum which the half-sace can sustain is DT max ˆ 0:5 G IC = Ea 2 a for lane strain cracking, and DT max ˆ 0:71 G IC = Ea 2 a for channelling. The half-sace solution give in Fig. 14(a) is valid for a nite late of su cient thickness 2H in relation to the crack length a. Since the channelling solution is derived directly from the lane strain cracking solution via (41), the minimum Fig. 14. (a) Energy release rate as a function of dimensionless time kt/a 2 for selected values of Biot number Bi for lane strain cracking G and channelling G t, and (b) energy release rate along the Mode I salling ath and the corresonding salling deth as functions of Bi.

26 892 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 thickness 2H for which the channelling solution in Fig. 14(a) is accurate can be read from Fig. 13(b), for given values of (a, t, Bi ) Steady-state salling Consider the case of a salling crack arallel to the free surface of a half-sace, after a cold shock has been imosed with boundary condition (36). The hysical roblem is illustrated in Fig. 12(b). Far ahead of the crack ti, the stress and temerature distributions are given by (37) and (38). Hutchinson and Suo (1992) have already considered this roblem for the case of ideal heat transfer, h=1, and their ndings are summarised here before extending their results to include the case when h is nite. For the case h=1, a Mode I crack ath exists arallel to the free surface, at any value of time, t. Ast increases, both the energy release rate and the deth of the Mode I salling crack increase. Hutchinson and Suo (1992) argue that the salling deth and the associated time to salling are set by alying the fracture criterion G s =G IC. Now consider the case of steady-state salling, with a nite value of heat transfer h at the surface of the half-sace. The equivalent edge force and moment indicated in Fig. 12(b) are ^P P Eaa T i T 1 ˆ 1 0 z s a, kt z a, Bi d a ^M M Eaa 2 T i T 1 ˆ 1 z s a, kt 1 a, Bi 2 z z d a a 0 42 and the corresonding energy release rate and stress intensity factors are given by ^G s ˆ G s Ea 2 a T i T 1 2 ˆ 1 2 f ^P 2 12 ^M 2 g 43 and K 1 Ea a Ti T 1 ˆ ^P cos o ^M sin o 2 =6 K 2 Ea a Ti T 1 ˆ ^P sin o ^M cos o 44 2 =6 where o= Extensive calculations have been carried out to determine GÃ s and the hase angle c=tan 1 (K 2 /K 1 ) as functions of kt/a 2 and Bi. It is found that both GÃ s and c increase with increasing time t and with increasing heat transfer coe cient h, although the deendence of c on h is weak; in the limit kt/ a 2 4 1, GÃ s and c In Fig. 14(b), the values of GÃ s and a s = kt

27 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867± corresonding to the Mode I salling ath (K 2 =0) are lotted as functions of Bi. Note that a s = kt is relatively insensitive to the Biot number Bi, esecially when Bi < 1. In the limit Bi 4 1, the results reduce to those obtained by Hutchinson and Suo (1992), i.e., G s ˆ 0:036a s Ea 2 DT 2 (or, equivalently, K 1 ˆ 0:190 EaDT a s and a s ˆ 6:82 kt : For illustration, consider the same choice of glass as made by Hutchinson and Suo (1992), with K IC =0.7 MPa m, EaDT ˆ 100 MPa and k= m 2 /s. The redicted salling deth for Bi =1 is a s =1.4 mm, associated with a time elase of t = 0.06 s. In view of Fig. 14(b), this time will increase slightly when the idealised boundary condition Bi =1 is relaced by one having a nite Biot number. The salling crack solution given in Fig. 14(b) for a semi-in nite solid is best understood as follows. For assumed values of crack deth a, heat transfer coe cient h and material toughness G IC, the value of Biot number Bi is calculated, and the corresonding values of a s = kt and G Ã s are read from the gure. On assuming that cracking occurs when G s =G IC, the required temerature jum for salling is deduced directly from the de nition ^G s G s = Ea 2 a T i T 1 2 Š, and the time for salling is deduced from the value of a s = kt : We exect the asymtotic solution for salling of the semi-in nite solid to adequately reresent the solution for a nite late rovided the salling time is less than t, as lotted in Fig. 13(b). 7. Merit indices for thermal shock The results resented above can be analysed to obtain aroriate material indices that govern thermal shock fracture in the full range of Biot numbers. Two failure criteria may be used, one based on strength and the other based on fracture toughness. The stress-based fracture criterion assumes that s max (H, t ) attains the fracture strength of the solid s f, whereas the toughness-based fracture criterion assumes that K max (a c, t c ) attains the fracture toughness of the solid K IC. In the case of ideal heat transfer (Bi=1), the maximum surface temerature jum DT that can be tolerated without causing strength-controlled failure follows directly from (9a) and (b) as DT1 s f 45 Ea which is valid for Cases I to III. Here, the distinctions between E and E -, and between a and a have been droed, as only those roerties of rimary signi cance are considered for selecting materials. In the limit of small Biot number (Bi < 1), DT is again obtained from Eqs. (9a) and (b) as DT ˆ A 1 s f Ea k hh where A for Case I and A for Cases II and III. With relations (45) 46

28 894 L.G. Zhao et al. / J. Mech. Phys. Solids 48 (2000) 867±897 and (46) in hand, the otimal materials to withstand strength-dominated fracture by thermal shock can be selected from a lot with s f /Ea and ks f /Ea as axes. Lu and Fleck (1998) have dislayed such a lot for a wide range of engineering ceramics. A similar aroach can be adoted for materials selection against crack roagation under a cold shock. We assume that the maximum surface temerature jum sustainable by a cracked late of thickness 2H is associated with lane strain cracking; a worst aw is considered, with the crack deth equal to the critical value a c to give the highest energy release rate. Then the temerature jum DT equals (DT ) max, as seci ed in Eqs. (18a). Since (DT ) max is a function of the Biot number Bi, the worst case is taken with Bi 4 1 and DT given by K IC DT ˆ A 2 Ea 47 H Here, A for the Case I thermal boundary condition, and A for Cases II and III. Similarly, in the limit of small Biot number (Bi < 1), the thermal shock fracture resistance becomes K IC DT ˆ A 3 Ea H k hh where A for the Case I boundary condition and A for Cases II and III. The otimal materials to withstand toughness-controlled fracture by thermal shock can be selected from a lot with K IC /Ea and kk IC /Ea as axes. Lu and Fleck (1998) have already dislayed a large number of engineering ceramics on a materials selection ma of this tye. 48 Acknowledgements The authors are grateful for the nancial suort from the ONR Grant J Aendix A. Temerature and stress elds (Cases II and III) For a late of nite thickness subjected to Case II boundary condition, the nondimensional temerature eld is given by T z, t ˆ 1 2 X1 sin b n cosfb n z H =2H g ex b2 n kt b n sin b n cos b n 4 H 2 A1 nˆ1 where b n are the ositive roots of b tan b=2 Bi. The corresonding stress eld is