MECHANICS MATERIALS ELSEVIER mechanics of Materials 34(2002)755-772 www.elsevier.com/locate/mechmat Threshold strength predictions for laminar ceramics with cracks that grow straight Kais hbaieb robert m. mcmeeki Department of Materials, College of Engineering ity of California, Santa Barbara, CA 93106, USA Department of Mechanical and Environmental Engineering Received 9 January 2002: received in revised form 27 June 2002 Abstract Finite element analysis is carried out to predict the threshold strength of a laminar ceramic loaded parallel to the layers. These materials are composed of alternate layers of two different ceramics in which residual stress is generated. Strength limiting cracks are trapped by the compressive layers and require a minimum(threshold) applied stress to cause them to fail the laminated ceramic. Predictions of the resulting threshold strength are obtained by finite element analysis of cracks that are assumed to grow straight. The calculations are utilized to study the influence of the elastic modulus mismatch between the alternate tensile and compressive layers. Good agreement is established between nu- merical simulations and theoretical results for materials involving layers with the same elastic properties. Results are obtained for a variety of combinations of different ceramics and suggest that a threshold strength as high as three times the effective residual stress in the compressive layer is achievable in practical situations. The resulting threshold strengths in practical systems could be higher than I GP e 2002 Elsevier Science Ltd. All rights reserved 1. Introduction some components are quite weak and therefore unreliable. The reason for the statistical distribu- Although ceramics have many attractive prop- tion of strength is the existence rties such as hardness and high temperature sta- cracks and crack-like flaws unintentionally intro- bility, they have the major disadvantage of lacking duced during processing or post-processing(such bility. The strength of ceramics obeys a sta- as surface machining)(Green, 1998; Lange, 1989) tistical description(e. g, Weibull) involving a wide Unlike ductile materials such as metals, ceramIcs distribution of values(Green, 1998)meaning that materials lack significant plastic deformation and hence exhibit low resistance to crack propagation Thus, the strength of brittle ceramics correlates Corresponding author. Tel: +1-805-893-4583: fax: +1-805- directly with the presence of flaws and decreases 893-8651 with increasing size of the flaw rmcm(@engineering. ucsb. edu (R. M The reliability of the ceramic could be improved by controlling the size of flaws introduced into the 0167-6636/02/S. see front matter c 2002 Elsevier Science Ltd. All rights reserved. PI:S0167-6636(02)00179-5
Threshold strength predictions for laminar ceramics with cracks that grow straight Kais Hbaieb a , Robert M. McMeeking b,* a Department of Materials, College of Engineering, University of California, Santa Barbara, CA 93106, USA b Department of Mechanical and Environmental Engineering, College of Engineering, University of California, Santa Barbara, CA 93106-5070, USA Received 9 January2002; received in revised form 27 June 2002 Abstract Finite element analysis is carried out to predict the threshold strength of a laminar ceramic loaded parallel to the layers. These materials are composed of alternate layers of two different ceramics in which residual stress is generated. Strength limiting cracks are trapped bythe compressive layers and require a minimum (threshold) applied stress to cause them to fail the laminated ceramic. Predictions of the resulting threshold strength are obtained byfinite element analysis of cracks that are assumed to grow straight. The calculations are utilized to study the influence of the elastic modulus mismatch between the alternate tensile and compressive layers. Good agreement is established between numerical simulations and theoretical results for materials involving layers with the same elastic properties. Results are obtained for a varietyof combinations of different ceramics and suggest that a threshold strength as high as three times the effective residual stress in the compressive layer is achievable in practical situations. The resulting threshold strengths in practical systems could be higher than 1 GPa. 2002 Elsevier Science Ltd. All rights reserved. 1. Introduction Although ceramics have manyattractive properties such as hardness and high temperature stability, they have the major disadvantage of lacking reliability. The strength of ceramics obeys a statistical description (e.g., Weibull) involving a wide distribution of values (Green, 1998) meaning that some components are quite weak and therefore unreliable. The reason for the statistical distribution of strength is the existence of a varietyof cracks and crack-like flaws unintentionallyintroduced during processing or post-processing (such as surface machining) (Green, 1998; Lange, 1989). Unlike ductile materials such as metals, ceramics materials lack significant plastic deformation and hence exhibit low resistance to crack propagation. Thus, the strength of brittle ceramics correlates directlywith the presence of flaws and decreases with increasing size of the flaw. The reliabilityof the ceramic could be improved bycontrolling the size of flaws introduced into the Mechanics of Materials 34 (2002) 755–772 www.elsevier.com/locate/mechmat * Corresponding author. Tel.: +1-805-893-4583; fax: +1-805- 893-8651. E-mail address: rmcm@engineering.ucsb.edu (R.M. McMeeking). 0167-6636/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 6 3 6 ( 0 2 ) 0 0 1 7 9 - 5
K Hbaieb, RM. McMeeking /Mechanics of Materials 34(2002)755-772 material during processing. This can be achieved if a a slurry of the designated powder is dispersed and then passed through a filter(Lange, 1989).De 44444444444 pending on the fineness of the filter, only hetero- geneities with sizes smaller than a critical value Ot+ Oc can pass through. Thus, threshold strength(and hence a guaranteed reliability) can be determined by the size of the filter, i.e., by defining the larg- Ot+ o est faw that can be present in the material lowever; such a material is expensive and isφ中 the reliability degraded accordingly. Recently, Rao et al. (1999)have shown as an alternative that an Fig. 1. A laminar ceramic, involving a through crack in the intrinsic threshold strength can be attained by tensile layer partially penetrating into the compressive layer, is the introduction of a compressive residual stress loaded parallel to the layers. Linearity allows superposition of in the components of the ceramic. As described two known fracture mechanics solutions to account for the below,experiments conducted on two-dimensional total stress intensity factor layered materials having alternating tensile and compressive segments have shown that threshold strengths as high as 500 MPa can be lower the applied stress needed to cause the ma- achieved terial to fail To support and develop the concept of the in such a ceramic body, two sets of alternate layers threshold strength observed in the experiments, a of different materials with different properties were theory was developed (Rao et al, 1999). It was fused together at high temperature(rao et al. assumed that compressive layers of thickness fr 1999). Upon cooling one set of layers has a tensile having a residual stress or, were sandwiched be- residual stress while the other has a compressive tween the tensile layers of thickness t, having a stress, due to thermal expansion differences. This residual stress a2 as shown in Fig. 1. The biaxial arrangement can also arise when the layers un- residual stresses arising in the layers are given by tallographic phase transformation, or undergo a 1=EE (1+E-1 dergo a differing volume increase due to a crys- differing increase in their molar volumes due to a 2E2 chemical reaction. For the experiments carried out 0,=-0=Gr by rao et al. (1999), the layered material was pre- cracked using different indentors and different where e= AAT=(a2-1)AT, a, is the coefficient loads. These indentations were performed in such of thermal expansion, AT is the temperature rela a way, that the resulting pre-cracks were com- tive to a datum at which the thermal residual pletely contained in a tensile layer and perpendic- stresses are zero, E=E/((l-vi), Ei is Youngs ular to the plane direction of the layers. The modulus and vi is Poisson's ratio with the sub- samples were then subjected to 4-point flexural scripts i=1 nating the relevant lay loading tests, such that the top surface of the When E is used without a subscript, it is implied specimens was subjected to an external tensile load that E=El= e2. Such a comment applies also parallel to the layers and perpendicular to the pre- to v cracks. Independent of the size of the pre-crack it In the theoretical analysis, the crack of length was observed that a threshold strength existed and 2a is assumed to span the entire width of the tensile no failure took place at stresses below this level. layer and to penetrate some distance into the This is in contrast to tests on monolithic ceramics compressive. A tensile load oa parallel to the layers where it is observed that the larger the flaw size the and perpendicular to the crack is applied. The
material during processing. This can be achieved if a slurryof the designated powder is dispersed and then passed through a filter (Lange, 1989). Depending on the fineness of the filter, onlyheterogeneities with sizes smaller than a critical value can pass through. Thus, threshold strength (and hence a guaranteed reliability) can be determined bythe size of the filter, i.e., bydefining the largest flaw that can be present in the material. However, such a material is expensive and is still subject to damage during machining with the reliability degraded accordingly. Recently, Rao et al. (1999) have shown as an alternative that an intrinsic threshold strength can be attained by the introduction of a compressive residual stress in the components of the ceramic. As described below, experiments conducted on two-dimensional layered materials having alternating tensile and compressive segments have shown that threshold strengths as high as 500 MPa can be achieved. To impose a biaxial compressive residual stress in such a ceramic body, two sets of alternate layers of different materials with different properties were fused together at high temperature (Rao et al., 1999). Upon cooling one set of layers has a tensile residual stress while the other has a compressive stress, due to thermal expansion differences. This arrangement can also arise when the layers undergo a differing volume increase due to a crystallographic phase transformation, or undergo a differing increase in their molar volumes due to a chemical reaction. For the experiments carried out byRao et al. (1999), the layered material was precracked using different indentors and different loads. These indentations were performed in such a way, that the resulting pre-cracks were completelycontained in a tensile layer and perpendicular to the plane direction of the layers. The samples were then subjected to 4-point flexural loading tests, such that the top surface of the specimens was subjected to an external tensile load parallel to the layers and perpendicular to the precracks. Independent of the size of the pre-crack it was observed that a threshold strength existed and no failure took place at stresses below this level. This is in contrast to tests on monolithic ceramics where it is observed that the larger the flaw size the lower the applied stress needed to cause the material to fail. To support and develop the concept of the threshold strength observed in the experiments, a theorywas developed (Rao et al., 1999). It was assumed that compressive layers of thickness t1, having a residual stress r1, were sandwiched between the tensile layers of thickness t2 having a residual stress r2 as shown in Fig. 1. The biaxial residual stresses arising in the layers are given by: r1 ¼ eE0 1 1 þ t1E0 1 t2E0 2 1 ¼ rc r2 ¼ r1 t1 t2 ¼ rT ð1Þ where e ¼ DaDT ¼ ða2 a1ÞDT , ai is the coefficient of thermal expansion, DT is the temperature relative to a datum at which the thermal residual stresses are zero, E0 i ¼ Ei=ðð1 miÞÞ, Ei is Young’s modulus and mi is Poisson’s ratio with the subscripts i ¼ 1 or 2 designating the relevant layer. When E is used without a subscript, it is implied that E ¼ E1 ¼ E2. Such a comment applies also to m. In the theoretical analysis, the crack of length 2a is assumed to span the entire width of the tensile layer and to penetrate some distance into the compressive. A tensile load ra parallel to the layers and perpendicular to the crack is applied. The Fig. 1. A laminar ceramic, involving a through crack in the tensile layer partially penetrating into the compressive layer, is loaded parallel to the layers. Linearity allows superposition of two known fracture mechanics solutions to account for the total stress intensityfactor. 756 K. Hbaieb, R.M. McMeeking / Mechanics of Materials 34 (2002) 755–772
K Hbaieb, R M. MeMeeking Mechanics of Materials 34(2002)755-772 intensity factor is calculated by superposing the estimate of the threshold strength given in stress fields applied to the same crack as (3). They also found that in some cases, unstable shown in Fig. 1. The first stress field is a tensile growth commences before the crack tip reaches the stress of magnitude(oa-Oc)applied to the whole interface with the tensile layer. In other consider specimen, while the second stress field is a tensile ations, McMeeking and Hbaieb(1999)provided stress of magnitude (oc + or)applied only across results that optimize the threshold strength for an the tensile layer. The total stress intensity factor elastically homogenous system. They found that for the crack in the tensile layer extending into the the threshold strength is maximized by selection of compressive layer is determined by summing the a good combination of materials to give the stress intensity factors for each of the stress field highest values for Kc, E(where E=EI=E, due mentioned above. In Rao et al. (1999), the stress to homogeneity)and e. Further maximization of intensity factors were approximated by results for the threshold strength is possible by choosing the an elastically homogenous system. This result is thicknesses of the tensile and compressive layers given by McMeeking and Hbaieb(1999)demonstrated that optimization is achieved by choosing the layers as K=(a-0)√a thin as possible. However, they assumed that there is a technological limit to how thin the layers can +(+ar)√d|=sin (2) be and therefore considered the case where one oth of de at the In the rest of this paper, this equation will be re- technological limit of thinness. They found that ferred to as the theoretical model result for K the for high toughness materials, the optimal thresh threshold strength, the stress needed for the crack old strength is associated with layers of equal to extend unstably, was assumed in Rao et al. (1999)to occur when it had penetrated through the ness). In contrast, for low toughness materials the compressive layer, i. e, when 2a= t2 +21 and optimal threshold strength occurs when the tensile K=Kc. The threshold strength, therefore, was layers in the system are thicker than the com- given by pressive layers by a ratio lying in the range 1-2.8 Furthermore, McMeeking and Hbaieb (1999) provided estimates of the optimal threshold 示(+2 strength, which they found to be at least -0.3 E'E and to be significantly higher than this for high toughness materials This paper is an extension of the work done by McMeeking and Hbaieb (1999)to develop the theoretical basis for the experimental observation Similarly, we refer later on to this equation as the made by rao et al. (1999). However, the effort theoretical model result for the threshold strength. described here is based on finite element modeling McMeeking and Hbaieb(1999) have shown that in rather than analytical calculations. Consequently an elastically homogenous system, the value of the the heterogeneous case where the tensile layer has threshold strength for a flaw initially in the tensile a different elastic modulus from the compres- layer is lower than its value when the flaw is ini- sive layer can be analyzed properly. In the tially in the compressive layer. As a consequence, a work presented here, the cracks are assumed to crack in the tensile layer is considered to account grow straight. Predictions are given for the for the true threshold strengt threshold strength as dependent on the compres Furthermore, McMeeking and Hbaieb(1999) sive layer toughness, the ratio of the elastic mod- ve the conditions that must be met for stable ulus of the tensile layer to the elastic modulus of growth to persist until the crack tip penetrates the compressive layer and the thicknesses of the through the compressive layer, thereby validating
stress intensityfactor is calculated bysuperposing two stress fields applied to the same crack as shown in Fig. 1. The first stress field is a tensile stress of magnitude ðra rcÞ applied to the whole specimen, while the second stress field is a tensile stress of magnitude ðrc þ rTÞ applied onlyacross the tensile layer. The total stress intensity factor for the crack in the tensile layer extending into the compressive layer is determined by summing the stress intensityfactors for each of the stress fields mentioned above. In Rao et al. (1999), the stress intensityfactors were approximated byresults for an elasticallyhomogenous system. This result is given by: K ¼ ðra rcÞ ffiffiffiffiffi pa p þ ðrc þ rTÞ ffiffiffiffiffi pa p 2 p sin1 t2 2a ð2Þ In the rest of this paper, this equation will be referred to as the theoretical model result for K. The threshold strength, the stress needed for the crack to extend unstably, was assumed in Rao et al. (1999) to occur when it had penetrated through the compressive layer, i.e., when 2a ¼ t2 þ 2t1 and K ¼ Kc. The threshold strength, therefore, was given by: rth ¼ Kc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p t2 2 1 þ 2t1 t2 r þ rc 1 " 1 þ t1 t2 2 p sin1 1 1 þ 2t1 t2 !# ð3Þ Similarly, we refer later on to this equation as the theoretical model result for the threshold strength. McMeeking and Hbaieb (1999) have shown that in an elasticallyhomogenous system, the value of the threshold strength for a flaw initiallyin the tensile layer is lower than its value when the flaw is initiallyin the compressive layer. As a consequence, a crack in the tensile layer is considered to account for the true threshold strength. Furthermore, McMeeking and Hbaieb (1999) gave the conditions that must be met for stable growth to persist until the crack tip penetrates through the compressive layer, thereby validating the estimate of the threshold strength given in Eq. (3). Theyalso found that in some cases, unstable growth commences before the crack tip reaches the interface with the tensile layer. In other considerations, McMeeking and Hbaieb (1999) provided results that optimize the threshold strength for an elasticallyhomogenous system. Theyfound that the threshold strength is maximized byselection of a good combination of materials to give the highest values for Kc, E0 (where E0 ¼ E0 1 ¼ E0 2 due to homogeneity) and e. Further maximization of the threshold strength is possible bychoosing the thicknesses of the tensile and compressive layers. McMeeking and Hbaieb (1999) demonstrated that optimization is achieved bychoosing the layers as thin as possible. However, theyassumed that there is a technological limit to how thin the layers can be and therefore considered the case where one or other or both of the sets of layers are made at the technological limit of thinness. Theyfound that for high toughness materials, the optimal threshold strength is associated with layers of equal thickness (but at the technological limit of thinness). In contrast, for low toughness materials the optimal threshold strength occurs when the tensile layers in the system are thicker than the compressive layers by a ratio lying in the range 1–2.8. Furthermore, McMeeking and Hbaieb (1999) provided estimates of the optimal threshold strength, which theyfound to be at least 0:3 E0 e and to be significantlyhigher than this for high toughness materials. This paper is an extension of the work done by McMeeking and Hbaieb (1999) to develop the theoretical basis for the experimental observation made byRao et al. (1999). However, the effort described here is based on finite element modeling rather than analytical calculations. Consequently the heterogeneous case where the tensile layer has a different elastic modulus from the compressive layer can be analyzed properly. In the work presented here, the cracks are assumed to grow straight. Predictions are given for the threshold strength as dependent on the compressive layer toughness, the ratio of the elastic modulus of the tensile layer to the elastic modulus of the compressive layer and the thicknesses of the layers. K. Hbaieb, R.M. McMeeking / Mechanics of Materials 34 (2002) 755–772 757
K Hbaieb, R M. McMeeking/ Mechanics of Materials 34(2002)755-772 2. Overview external rightmost lateral and top surfaces. The crack surface, which lies along the bottom surface To model a crack in the laminar material of the model extending from the lower left corner omposed of alternate tensile layers fused together is free of traction. An external tensile load was with compressive layers, a finite element analysis is applied on the top surface. The path independent J carried out. It is assumed that the crack has al integral(Rice, 1968)is calculated for several ready tunnelled down the tensile layer so that the cracks with differing lengths and hence the rela- configuration analyzed is a through crack as de- tionship of the stress intensity factor vs. the crack picted in Fig. 1. It is assumed that the crack grows length is investigated. When layers of different aight, without deviating or bifurcating. For elastic moduli are considered, the mesh near the mplicity only a quarter of the specimen is mod- interface between the layers is refined to make sure eled which accounts for the whole body by way of that the J integral is path independent and accu symmetry. Several tensile and compressive layers rate. Only crack tips displaced from the layer in- are present in the finite element model, as shown in terface are investigated to avoid non-square root Fig 2, to represent adequately the specimen used singular crack tip fields arising for crack tips ex- in the experiments. The number of compressive actly at the material interface( Cook and Erdogan layers in the model varies from 4 to 6 depending on 1972). However, since the mesh is very fine, the the geometrical configuration of the layers. Be- accuracy of the stress intensity factor for crack tips ause of the way symmetry is enforced, the number near the interface is good of tensile layers is equal to the number of com- To have results that are generally valid, several pressive layers plus one half. The length of the parameters are varied. The ratio of elastic modulus model is more than three times larger than the in the tensile layer, E2, to the elastic modulus in the width; the latter is in turn much larger than compressive layer, El, was varied from 1/10 to 10 the crack, so that the finite element calculations It was assumed that layer thicknesses would typi effectively simulate an infinite body fracture anal- cally be of the order 10 um-l mm. However, in sis. Displacement boundary conditions are im- view of the results for E1= Ey obtained by posed on the symmetry lines of the model, while, McMeeking and Hbaieb (1999), only thickness referring to Fig. 2, no constraint is applied to the ratios t2/ equal to 1, 3/2, 2 and 5/2 are accounted for. The expectation is that optimal threshold strengths will occur within this thickness ratio Oa range. The strain mismatch 8=0.357% was as- f sumed to be typical of the contrast between the compressive and tensile layers, so that the effective residual stress-cE would typically be of the order of 30 MPa-3 GPa. Finally, with the assumption that the toughness of the ceramic materials com- posing the layers would fall in the range 1-10 MPavm, the threshold strength dependence on toughness, thickness ratio, elastic modulus ratio and residual strain is investigated 3. Model description Fig. 2. Finite element mesh of the plane-strain model. The The computer simulation is carried out using different shades of gray indicate the ceramic layers. The crack the finite element code ABAQUS (1998)to per- es along the bottom surface extending from the lower left hand form linear elastic calculations. Isotropy is as- corner. The mesh is refined in the region around the crack tip. sumed for all materials so that the only mechanical
2. Overview To model a crack in the laminar material composed of alternate tensile layers fused together with compressive layers, a finite element analysis is carried out. It is assumed that the crack has alreadytunnelled down the tensile layer so that the configuration analyzed is a through crack as depicted in Fig. 1. It is assumed that the crack grows straight, without deviating or bifurcating. For simplicityonlya quarter of the specimen is modeled which accounts for the whole bodybywayof symmetry. Several tensile and compressive layers are present in the finite element model, as shown in Fig. 2, to represent adequatelythe specimen used in the experiments. The number of compressive layers in the model varies from 4 to 6 depending on the geometrical configuration of the layers. Because of the waysymmetryis enforced, the number of tensile layers is equal to the number of compressive layers plus one half. The length of the model is more than three times larger than the width; the latter is in turn much larger than the crack, so that the finite element calculations effectivelysimulate an infinite bodyfracture analysis. Displacement boundary conditions are imposed on the symmetry lines of the model, while, referring to Fig. 2, no constraint is applied to the external rightmost lateral and top surfaces. The crack surface, which lies along the bottom surface of the model extending from the lower left corner, is free of traction. An external tensile load was applied on the top surface. The path independent J integral (Rice, 1968) is calculated for several cracks with differing lengths and hence the relationship of the stress intensityfactor vs. the crack length is investigated. When layers of different elastic moduli are considered, the mesh near the interface between the layers is refined to make sure that the J integral is path independent and accurate. Onlycrack tips displaced from the layer interface are investigated to avoid non-square root singular crack tip fields arising for crack tips exactlyat the material interface (Cook and Erdogan, 1972). However, since the mesh is veryfine, the accuracyof the stress intensityfactor for crack tips near the interface is good. To have results that are generallyvalid, several parameters are varied. The ratio of elastic modulus in the tensile layer, E2, to the elastic modulus in the compressive layer, E1, was varied from 1/10 to 10. It was assumed that layer thicknesses would typicallybe of the order 10 lm–1 mm. However, in view of the results for E1 ¼ E2 obtained by McMeeking and Hbaieb (1999), onlythickness ratios t2=t1 equal to 1, 3/2, 2 and 5/2 are accounted for. The expectation is that optimal threshold strengths will occur within this thickness ratio range. The strain mismatch e ¼ 0:357% was assumed to be typical of the contrast between the compressive and tensile layers, so that the effective residual stress eE0 1 would typically be of the order of 30 MPa–3 GPa. Finally, with the assumption that the toughness of the ceramic materials composing the layers would fall in the range 1–10 MPa ffiffiffiffi mp , the threshold strength dependence on toughness, thickness ratio, elastic modulus ratio and residual strain is investigated. 3. Model description The computer simulation is carried out using the finite element code ABAQUS (1998) to perform linear elastic calculations. Isotropyis assumed for all materials so that the onlymechanical Fig. 2. Finite element mesh of the plane-strain model. The different shades of grayindicate the ceramic layers. The crack lies along the bottom surface extending from the lower left hand corner. The mesh is refined in the region around the crack tip. 758 K. Hbaieb, R.M. McMeeking / Mechanics of Materials 34 (2002) 755–772
K Hbaieb, RM. McMeeking Mechanics of Materials 34(2002)755-772 are the elastic modulus and top surface. Linearity of the solutions allows ons ratio. All layers are given the value of scaling and superposition of the residual stress 0.32 for Poisson's ratio loading and the applied loading in arbitrary ways a different coefficient of thermal expansion is The model used is two dimensional and plane given to the alternating layers. To the tensile layers strain. Eight noded plane strain quadrilateral ele- a value of 9x 10-K- is assigned, whereas the ments are used in the mesh such as the one shown compressive layers are given the value of 6.025x in Fig. 2. The elements near the crack tip are small, 10-K. These values were chosen to represent as shown in Fig. 3, especially in the compressive the materials alumina and alumina/mullite mixture layer containing the tip. Such an arrangement at 70% mullite fraction, respectively. However, we permits the accurate calculations of J by the do- stress that our numerical results are not confined main integral method (Moran and Shih, 1987)as to these materials. The same thermal expansion well as accurate solution of the near tip stresses coefficient values are used throughout the analysis For calculating the residual stresses, a temperature 1200 oC lower than the stress-free state is used and 4. simulation results ABAQUS employed to compute the thermal stresses in the layers. As a consequence, the layers For crack lengths with a tip in the material with with thickness t, have a biaxial residual compres- modulus Ei, the stress intensity factor is calculated sive stress of magnitude a. and the layers of from the value of J through K=VJE/(1-v2 thickness t2 have a biaxial tensile stress oT, relieved and then plotted vs. the crack length. The calcu only by the presence of the crack, as depicted in lations are first conducted for layers having the Fig. 1. As a result, a value of s equal to 0. 357% is same elastic properties. This is intended to verify generated. The externally applied stress in the the finite element results in comparison with the material is introduced by an applied tensile load exact theoretical model for an infinite body on the top surface of the model. It is simulated by(McMeeking and Hbaieb, 1999). The finite ele- a traction of 400 MPa applied uniformly on the ment results for the elastically homogeneous case along with the theoretical model results are plotted in Fig. 4 for a thickness ratio t2/t 1. To obtain accurate trends for results having crack tips near 0 the interface between the compressive and tensile layers, many calculations were carried out for tips located in the vicinity of the interface. Rest shown separately in Fig. 4 for the stress intensity factor Residual due to the residual stress caused by thermal expansion mismatch and the stress inten sity factor Applied due to the externally applied load. Specific results are indicated by the symbols and lines are drawn through them for illustration. The good agreement between the finite element results and the theoretical model results shown in Fig. 4 implies reliability of the finite element model for the solution of the crack problems and the calculation of J Fig. 3. Detail of the finite element mesh showing compres Results for the stress intensity factor for cracks ding) and tensile layers (dark shading). The in a heterogeneous material with E2/E1=1.7 are impressive layer on the left contains the crack tip and the crack surface lies on the bottom surface of the mesh extending plotted in Fig. 5. In this case, crack tips very om the bottom left corner. The elements in this layer are sma all close to the layer interface are avoided because r higher accuracy of the non-square root singularity (Cook and
properties needed are the elastic modulus and Poisson’s ratio. All layers are given the value of 0.32 for Poisson’s ratio. A different coefficient of thermal expansion is given to the alternating layers. To the tensile layers a value of 9 106 K1 is assigned, whereas the compressive layers are given the value of 6:025 106 K1. These values were chosen to represent the materials alumina and alumina/mullite mixture at 70% mullite fraction, respectively. However, we stress that our numerical results are not confined to these materials. The same thermal expansion coefficient values are used throughout the analysis. For calculating the residual stresses, a temperature 1200 C lower than the stress-free state is used and ABAQUS employed to compute the thermal stresses in the layers. As a consequence, the layers with thickness t1 have a biaxial residual compressive stress of magnitude rc and the layers of thickness t2 have a biaxial tensile stress rT, relieved onlybythe presence of the crack, as depicted in Fig. 1. As a result, a value of e equal to 0.357% is generated. The externallyapplied stress in the material is introduced byan applied tensile load on the top surface of the model. It is simulated by a traction of 400 MPa applied uniformlyon the top surface. Linearityof the solutions allows scaling and superposition of the residual stress loading and the applied loading in arbitraryways. The model used is two dimensional and plane strain. Eight noded plane strain quadrilateral elements are used in the mesh such as the one shown in Fig. 2. The elements near the crack tip are small, as shown in Fig. 3, especiallyin the compressive layer containing the tip. Such an arrangement permits the accurate calculations of J bythe domain integral method (Moran and Shih, 1987) as well as accurate solution of the near tip stresses. 4. Simulation results For crack lengths with a tip in the material with modulus Ei, the stress intensityfactor is calculated from the value of J through K ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi JEi=ð1 m2Þ p and then plotted vs. the crack length. The calculations are first conducted for layers having the same elastic properties. This is intended to verify the finite element results in comparison with the exact theoretical model for an infinite body (McMeeking and Hbaieb, 1999). The finite element results for the elasticallyhomogeneous case along with the theoretical model results are plotted in Fig. 4 for a thickness ratio t2=t1 ¼ 1. To obtain accurate trends for results having crack tips near the interface between the compressive and tensile layers, many calculations were carried out for tips located in the vicinityof the interface. Results are shown separatelyin Fig. 4 for the stress intensity factor Kresidual due to the residual stress caused by thermal expansion mismatch and the stress intensityfactor Kapplied due to the externallyapplied load. Specific results are indicated bythe symbols and lines are drawn through them for illustration. The good agreement between the finite element results and the theoretical model results shown in Fig. 4 implies reliabilityof the finite element model for the solution of the crack problems and the calculation of J. Results for the stress intensityfactor for cracks in a heterogeneous material with E2=E1 ¼ 1:7 are plotted in Fig. 5. In this case, crack tips very close to the layer interface are avoided because of the non-square root singularity(Cook and Fig. 3. Detail of the finite element mesh showing compressive layers (light shading) and tensile layers (dark shading). The compressive layer on the left contains the crack tip and the crack surface lies on the bottom surface of the mesh extending from the bottom left corner. The elements in this layer are small for higher accuracy. K. Hbaieb, R.M. McMeeking / Mechanics of Materials 34 (2002) 755–772 759
K Hbaieb, R M. McMeeking/ Mechanics of Materials 34(2002)755-772 → compressive layer 2→ tensile layer G r2 △aMTE (simulation) Aa△TE Crack Length 2a/t Fig. 4. Comparison of simulation results with theoretical model results for a homogeneous material. Both tensile and compressive layers have same thickness. Erdogan, 1972)arising when the crack tip is ex- starting with a length 2a just greater than t2 so that actly at the interface and because of path de- the crack tip is just inside a compressive layer. The pendence of J when the domain for its calculation total crack tip stress intensity factor, K, is the sum encompasses the neighboring layers. It is worth of the applied load stress intensity factor Kapplied noting here that the trend of our results in the and the residual stress intensity factor Residual close vicinity of the interface is consistent with However, for the crack to grow, the total stress the results of Cook and Erdogan(1972). A com- intensity factor, K must be equal to the fracture parison between the finite element results and the toughness, Ke, of the compressive layer. This al- theoretical model results is also given. In the lows us to calculate the applied stress intensity theoretical model results, it is assumed that both factor needed to sustain crack growth as tensile and compressive layers have the same Applied=Ke-Kresidual elastic properties, E2 and v2. It can 5 that the theoretical model is somewhat in error Defining a parameter S=S(a/t2, t2/t1, E2/E1as makes it clear that the finite element results are s、K的,m叫 d stress Gcrackgrowth nec when the ceramic layers are heterogeneous. This needed for a proper treatment of the threshold strength problem we can The results of the finite element simulation are essary to sustain crack growth by combining Eqs used as follows to obtain the threshold strength. (4)and(5)to give Values of△a△TE, tI and t2 are chosen so that Kresidual for each crack length is fixed. The crack i K=K Ocrackgrow then considered to grow through the material S√2
Erdogan, 1972) arising when the crack tip is exactlyat the interface and because of path dependence of J when the domain for its calculation encompasses the neighboring layers. It is worth noting here that the trend of our results in the close vicinityof the interface is consistent with the results of Cook and Erdogan (1972). A comparison between the finite element results and the theoretical model results is also given. In the theoretical model results, it is assumed that both tensile and compressive layers have the same elastic properties, E2 and m2. It can be seen in Fig. 5 that the theoretical model is somewhat in error when the ceramic layers are heterogeneous. This makes it clear that the finite element results are needed for a proper treatment of the threshold strength problem. The results of the finite element simulation are used as follows to obtain the threshold strength. Values of DaDTE0 1, t1 and t2 are chosen so that Kresidual for each crack length is fixed. The crack is then considered to grow through the material starting with a length 2a just greater than t2 so that the crack tip is just inside a compressive layer. The total crack tip stress intensityfactor, K, is the sum of the applied load stress intensityfactor Kapplied and the residual stress intensityfactor Kresidual. However, for the crack to grow, the total stress intensityfactor, K must be equal to the fracture toughness, Kc, of the compressive layer. This allows us to calculate the applied stress intensity factor needed to sustain crack growth as: Kapplied ¼ Kc Kresidual ð4Þ Defining a parameter S ¼ Sða=t2; t2=t1; E2=E1Þ as S ¼ Kapplied rapplied ffiffiffiffiffiffi p 2 t2 p ð5Þ we can compute the applied stress rcrackgrowth necessaryto sustain crack growth bycombining Eqs. (4) and (5) to give rcrackgrowth ¼ Kc Kresidual S ffiffiffiffiffiffi p 2 t2 p ð6Þ Fig. 4. Comparison of simulation results with theoretical model results for a homogeneous material. Both tensile and compressive layers have same thickness. 760 K. Hbaieb, R.M. McMeeking / Mechanics of Materials 34 (2002) 755–772
K Hbaieb, R M. MeMeeking Mechanics of Materials 34(2002)755-772 E/E1=1.7,t2/t Compressive Layer l→ compressive layer △aATE"1Jm △aTE12 Fig. 5. Simulation results for the case where the elastic modulus in the tensile layer E2 is 1.7 times higher than the elastic modulus in the compressive layer El. The theoretical model results for homogeneous material is also plotted for comparison. Clearly S is given as a function of a t2 for the case all the way to the interface with the tensile layer. In of t2/+1=l, E2/E1=1.7 by the plot of Kapplied/ this case, unstable crack growth will set in while (applied vit/2)for the finite element simulation the tip is still in the compressive layer. However, in results given in Fig. 5. The stress crackgrowth(nor general malized by△a△TE1), needed to sustain crack growth is plotted against crack length in Fig. 6 for the case t2/1=1,E2/E1=1.7andK/(△△TE1 maxS t VIt/2)=0.123 with Kc assumed to be the same in both the tensile and compressive layers. It can where max[ indicates the maximum value of the be seen that this stress rises with crack length until term inside the brackets for crack lengths lying in the crack tip reaches the interface with the tensile the range t2< 2a< t2+ 2t1 yer, indicating stable crack growth to that extent As noted above, in some cases stable growth If the toughness of the tensile layer is equal to or occurs at least until the crack tip is almost at the less than Kc, the crack will begin to propagate interface between the compressive layer and the unstably under load control once it has reached adjacent tensile layer (i.e, as in Fig. 6). Since we the interface. We take this situation to denote the do not have a suitable model for what happens strength of the layered ceramic and define the when a crack tip grows through the interface stress level at that stage to be oth, the threshold( Cook and Erdogan, 1972)we prefer to define the trength of the material, thereby ignoring the threshold strength in this case as the stress neces- possibility of a tougher tensile layer leading to an sary to drive the crack to a position just short of ength. In this interface. The actual threshold strength reaches a maximum before the crack tip penetrates be higher than this due to at least two possibilities
Clearly S is given as a function of a=t2 for the case of t2=t1 ¼ 1, E2=E1 ¼ 1:7 bythe plot of Kapplied= ðrapplied ffiffiffiffiffiffiffiffiffiffiffi pt2=2 p Þ for the finite element simulation results given in Fig. 5. The stress rcrackgrowth (normalized by DaDTE0 1), needed to sustain crack growth is plotted against crack length in Fig. 6 for the case t2=t1 ¼ 1, E2=E1 ¼ 1:7 and Kc=ðDaDTE0 ffiffiffiffiffiffiffiffiffiffiffi 1 pt1=2 p Þ ¼ 0:123 with Kc assumed to be the same in both the tensile and compressive layers. It can be seen that this stress rises with crack length until the crack tip reaches the interface with the tensile layer, indicating stable crack growth to that extent. If the toughness of the tensile layer is equal to or less than Kc, the crack will begin to propagate unstablyunder load control once it has reached the interface. We take this situation to denote the strength of the layered ceramic and define the stress level at that stage to be rth, the threshold strength of the material, therebyignoring the possibilityof a tougher tensile layer leading to an even higher strength. In some cases, rcrackgrowth reaches a maximum before the crack tip penetrates all the wayto the interface with the tensile layer. In this case, unstable crack growth will set in while the tip is still in the compressive layer. However, in general, rth ¼ max Kc Kresidual S ffiffiffiffiffiffi p 2 t2 p " # ð7Þ where max½ indicates the maximum value of the term inside the brackets for crack lengths lying in the range t2 6 2a 6 t2 þ 2t1. As noted above, in some cases stable growth occurs at least until the crack tip is almost at the interface between the compressive layer and the adjacent tensile layer (i.e., as in Fig. 6). Since we do not have a suitable model for what happens when a crack tip grows through the interface (Cook and Erdogan, 1972) we prefer to define the threshold strength in this case as the stress necessaryto drive the crack to a position just short of this interface. The actual threshold strength may be higher than this due to at least two possibilities. Fig. 5. Simulation results for the case where the elastic modulus in the tensile layer E2 is 1.7 times higher than the elastic modulus in the compressive layer E1. The theoretical model results for homogeneous material is also plotted for comparison. K. Hbaieb, R.M. McMeeking / Mechanics of Materials 34 (2002) 755–772 761
K Hbaieb, R M. McMeeking/ Mechanics of Materials 34(2002)755-772 Compressive Layer =0.123 1→ compressive layer 2- tensile layer Crack Length 2a/t, Fig. 6. Plot of the stress needed to sustain crack growth vs. crack length for the case of t2/n=l, E2/Et= 1.7 and Kc/(AzATEVit/2)=0.123. The stress is increased monotonically until the crack tip is almost at the interface between the com- One possibility is that it takes a higher stress to residual stress -EE=-AzATE and the fracture drive the crack tip through the interface. The other toughness is normalized by(-EE1)VIt/2.Note possibility is that the crack tip goes through the that the symbols in the figures do not correspond interface easily but then the tensile layer has a to finite element calculations; rather they are ob- higher fracture toughness than the compressive tained from Eq (7)as a result of inserting values layer, requiring a higher stress to sustain crack for K ranging from I to 10 MPavm after values growth Since we cannot readily quantify the stress for S have been established. However, six finite to drive the crack through the interface, we take element calculations are performed for each value the condition just prior to the crack tip reaching of E2/El for different crack lengths before Eq.(7) the interface to be the point of crack growth in- is employed. Fig. 7 shows the case where the stability. Consequently, Eq.(7)is taken to always thicknesses of the layers are the same and E2/Er give the threshold strength with the caveat that the ranges from l to 10. The threshold strength in- condition is imposed for crack lengths up to just creases with increasing toughness and also in- lightly less than t2+ 2t1. Therefore, the fracture creases with increasing elastic modulus ratio oughness used in our calculations of the threshold E2/E1. Fig 8 also shows the results for the case Furthermore, the threshold strengths for cracks the elastic modulus ratio E2/E, is now rangin trength is always that of the compressive layer. where the layer thicknesses are identical; however, that grow straight cannot be lower than the values from 1/10 to 1/2. Similarly to Fig. 7, the threshold we have predicted strength increases with increasing toughness The resulting threshold strengths oth are plotted However, the increase of the threshold strength in Figs. 7-14 against the fracture toughness Ke of with elastic modulus ratio E2/En is only present for the compressive layer for a variety of layer thick- E2/E bigger than 1/3. For smaller E2/El, this ness ratios and elastic modulus ratios. The trend is only valid for smaller toughnesses. For threshold strength is normalized by the effective higher toughnesses and E2/E1 smaller than 1/3, the
One possibilityis that it takes a higher stress to drive the crack tip through the interface. The other possibilityis that the crack tip goes through the interface easilybut then the tensile layer has a higher fracture toughness than the compressive layer, requiring a higher stress to sustain crack growth. Since we cannot readilyquantifythe stress to drive the crack through the interface, we take the condition just prior to the crack tip reaching the interface to be the point of crack growth instability. Consequently, Eq. (7) is taken to always give the threshold strength with the caveat that the condition is imposed for crack lengths up to just slightlyless than t2 þ 2t1. Therefore, the fracture toughness used in our calculations of the threshold strength is always that of the compressive layer. Furthermore, the threshold strengths for cracks that grow straight cannot be lower than the values we have predicted. The resulting threshold strengths rth are plotted in Figs. 7–14 against the fracture toughness Kc of the compressive layer for a variety of layer thickness ratios and elastic modulus ratios. The threshold strength is normalized bythe effective residual stress eE0 1 ¼ DaDTE0 1 and the fracture toughness is normalized by ðeE0 1Þ ffiffiffiffiffiffiffiffiffiffiffi pt1=2 p . Note that the symbols in the figures do not correspond to finite element calculations; rather theyare obtained from Eq. (7) as a result of inserting values for K ranging from 1 to 10 MPa ffiffiffiffi mp after values for S have been established. However, six finite element calculations are performed for each value of E2=E1 for different crack lengths before Eq. (7) is employed. Fig. 7 shows the case where the thicknesses of the layers are the same and E2=E1 ranges from 1 to 10. The threshold strength increases with increasing toughness and also increases with increasing elastic modulus ratio E2=E1. Fig. 8 also shows the results for the case where the layer thicknesses are identical; however, the elastic modulus ratio E2=E1 is now ranging from 1/10 to 1/2. Similarlyto Fig. 7, the threshold strength increases with increasing toughness. However, the increase of the threshold strength with elastic modulus ratio E2=E1 is onlypresent for E2=E1 bigger than 1/3. For smaller E2=E1, this trend is onlyvalid for smaller toughnesses. For higher toughnesses and E2=E1 smaller than 1/3, the Fig. 6. Plot of the stress needed to sustain crack growth vs. crack length for the case of t2=t1 ¼ 1, E2=E1 ¼ 1:7 and Kc=ðDaDTE0 1 ffiffiffiffiffiffiffiffiffiffiffi pt1=2 p Þ ¼ 0:123. The stress is increased monotonicallyuntil the crack tip is almost at the interface between the compressive layer and the tensile layer. 762 K. Hbaieb, R.M. McMeeking / Mechanics of Materials 34 (2002) 755–772
K Hbaieb, R M. MeMeeking Mechanics of Materials 34(2002)755-772 r:/t=1 l→ compressive layer → tensile layer E:/E1=7 E:/E1=4 K Fig. 7. Threshold strength vs compressive layer toughness for the ratio E2/En of the elastic modulus of the tensile layer to the elastic modulus of the compressive layer in the range 1-10. The thicknesses of the tensile and compressive layers are identical. 2→ tensile laver E:/E1=14 Fig. 8. Threshold strength vs. compressive layer toughness for the ratio E2/En of the elastic modulus of the tensile layer to the elastic modulus of the compressive layer in the range 1/10-1/2. The thicknesses of the tensile and compressive layers are identical
Fig. 7. Threshold strength vs. compressive layer toughness for the ratio E2=E1 of the elastic modulus of the tensile layer to the elastic modulus of the compressive layer in the range 1–10. The thicknesses of the tensile and compressive layers are identical. Fig. 8. Threshold strength vs. compressive layer toughness for the ratio E2=E1 of the elastic modulus of the tensile layer to the elastic modulus of the compressive layer in the range 1/10–1/2. The thicknesses of the tensile and compressive layers are identical. K. Hbaieb, R.M. McMeeking / Mechanics of Materials 34 (2002) 755–772 763
K Hbaieb, R M. McMeeking/ Mechanics of Materials 34(2002)755-772 1→ compressive layer 2→ tensile layer K (EE )xt/2 Fig. 9. Threshold strength vs. compressive layer toughness for the ratio E2/En of the elastic modulus of the tensile layer to the elast modulus of the compressive layer in the range 1-10. The thickness of the tensile layer is 1.5 times the thickness of the compressive layer E:/E1=1/5 K Fig. 10. Threshold strength vs compressive layer toughness for the ratio E2/Ei of the elastic modulus of the tensile layer to the elastic modulus of the compressive layer in the range 1/10-1/2. The thickness of the tensile layer is 1.5 times the thickness of the compressive
Fig. 9. Threshold strength vs. compressive layer toughness for the ratio E2=E1 of the elastic modulus of the tensile layer to the elastic modulus of the compressive layer in the range 1–10. The thickness of the tensile layer is 1.5 times the thickness of the compressive layer. Fig. 10. Threshold strength vs. compressive layer toughness for the ratio E2=E1 of the elastic modulus of the tensile layer to the elastic modulus of the compressive layer in the range 1/10–1/2. The thickness of the tensile layer is 1.5 times the thickness of the compressive layer. 764 K. Hbaieb, R.M. McMeeking / Mechanics of Materials 34 (2002) 755–772