Computational Materials Science 46(2009)614-620 Contents lists available at Science Direct Computational Materials Science ELSEVIER journalhomepagewww.elsevier.com/locate/commatsci Estimation of apparent fracture toughness of ceramic laminates Lubos nahlik a, b, Lucie sestakova a, b Pavel hutar a 6 Institute of Physics of Materials, Academy of Sciences of the Czech Republic. Zizkova 22, 616 62 Brno, Czech Republic Institute of Solid Mechanics, Mechatronics and Biomechanics, Bmo University of Technology, Technicka 2, 616 69 Bmo, Czech Republic ARTICLE INFO ABSTRACT Article history: The paper presented deals with the fracture behaviour of ceramic laminates. The residual stresses in indi Received 14 November 2008 vidual layers of Al2O3/5vol%t-ZrO2(ATZ)and Al2O3/30vol %m-ZrO2(AMZ)are determined. Assumptions form 5 March 2009 ccepted 4 April 2009 oncerning linear elastic fracture mechanics and small scale yielding are considered In this Available online 7 May 2009 procedure based on a generalization of Sihs strain energy density factor to the case of a crae the interfaces between two dissimilar materials is used for determination of effective values of factor on material interfaces. An important increase of fracture toughness at the amzatz inter predicted in comparison to the fracture toughness of individual material components. Predicted re compared with data available in the literature and mutual g ure suggested can be used for estimation of resistance to crack propagation through multilayered , 05 Mh es and its design. The procedure can contribute to enhancing the reliability and safety of struc amics or, more generally, of layered composites with strong interfaces e 2009 Elsevier B.V. All rights reserved. Bi-material 1 Introduction during sintering between adjacent layers, resulting from ence in Youngs moduli, thermal expansion coefficients. In recent decades, remarkable advances have been achieved in reactions and /or phase transformations, generates residu improving the mechanical behaviour of ceramic materials. Re- throughout the material. These residual stresses can be cently, new strategies fundamentally different from the conven- in order to improve mechanical properties. Laminar ceramics de- tional approach (high homogeneity ceramics with very small signed with compressive stresses in the bulk may present a thresh- flaws) have emerged, aiming to achieve"flaw tolerant"materials old strength below which catastrophic failure does not occur [2]. by designing special microstructures that improve the toughne Deriving from the above-mentioned perspectives, zirconia-con- (or apparent toughness)of ceramics. taining laminar ceramics have been employed to develop conn et One of the most attractive proposals for this latter approach sive stresses in the internal layers by means of tetragonal consists of layered architectures that combine materials with dif- monoclinic phase transformations that take place in the cooling ferent properties. As a result, laminates with mechanical behaviour down during the sintering process [2]. Under certain conditions, rior to that of the individual constituents can be fabricated. In these compressive stresses may act as a barrier to crack propagation. this respect, different fracture mechanics approaches have been at- In other cases, crack bifurcation and or deflection phenomena result tempted for the design of ceramic layered co ] in an increase of material fracture toughness and energy absorption In particular, ceramic composites with a layered structure such capability [3]. The procedure for preparation of these composites as alumina-zirconia and mullite-alumina, among others, have and typical crack behaviour can be found elsewhere [1, 4-9. been reported as exhibiting an increased apparent fracture tough For the estimation of apparent fracture toughness the influence ness and energy absorption as well as non-catastrophic failure of the interface between two dissimilar materials (layers )on the ehaviour. For strongly bonded multilayers the elastic mismatch critical values for crack propagation through the interface have to be determined. because the role of interface is fundamental. The existence of two regions with different material properties and ef the cne ch reou bid esk a22. 61 66 2 bsos. zech repub c reemy o the presence of an interface strongly influence the distribution of 532290351;fax:+420541218657 stress in composite bodies To estimate the influence of interfaces on the fracture behaviour of laminated structures the interaction 0927-0256/s- see front matter o 2009 Elsevier B V. All rights reserved. do:101016/ commatsci20090400
Estimation of apparent fracture toughness of ceramic laminates Luboš Náhlík a,b,*, Lucie Šestáková a,b, Pavel Hutarˇ a a Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zˇizˇkova 22, 616 62 Brno, Czech Republic b Institute of Solid Mechanics, Mechatronics and Biomechanics, Brno University of Technology, Technická 2, 616 69 Brno, Czech Republic article info Article history: Received 14 November 2008 Received in revised form 5 March 2009 Accepted 4 April 2009 Available online 7 May 2009 PACS: 02.40.Xx 02.70.Dh 46.50.+a 81.05.Mh Keywords: Strain energy density factor Apparent fracture toughness Ceramic composite Bi-material abstract The paper presented deals with the fracture behaviour of ceramic laminates. The residual stresses in individual layers of Al2O3/5vol.%t-ZrO2 (ATZ) and Al2O3/30vol.%m-ZrO2 (AMZ) are determined. Assumptions concerning linear elastic fracture mechanics and small scale yielding are considered. In this frame the procedure based on a generalization of Sih’s strain energy density factor to the case of a crack touching the interfaces between two dissimilar materials is used for determination of effective values of the stress intensity factor on material interfaces. An important increase of fracture toughness at the AMZ/ATZ interface was predicted in comparison to the fracture toughness of individual material components. Predicted values were compared with data available in the literature and mutual good agreement was found. The procedure suggested can be used for estimation of resistance to crack propagation through multilayered structures and its design. The procedure can contribute to enhancing the reliability and safety of structural ceramics or, more generally, of layered composites with strong interfaces. 2009 Elsevier B.V. All rights reserved. 1. Introduction In recent decades, remarkable advances have been achieved in improving the mechanical behaviour of ceramic materials. Recently, new strategies fundamentally different from the conventional approach (high homogeneity ceramics with very small flaws) have emerged, aiming to achieve ‘‘flaw tolerant” materials by designing special microstructures that improve the toughness (or apparent toughness) of ceramics. One of the most attractive proposals for this latter approach consists of layered architectures that combine materials with different properties. As a result, laminates with mechanical behaviour superior to that of the individual constituents can be fabricated. In this respect, different fracture mechanics approaches have been attempted for the design of ceramic layered composites [1]. In particular, ceramic composites with a layered structure such as alumina–zirconia and mullite-alumina, among others, have been reported as exhibiting an increased apparent fracture toughness and energy absorption as well as non-catastrophic failure behaviour. For strongly bonded multilayers the elastic mismatch during sintering between adjacent layers, resulting from the difference in Young’s moduli, thermal expansion coefficients, chemical reactions and/or phase transformations, generates residual stresses throughout the material. These residual stresses can be controlled in order to improve mechanical properties. Laminar ceramics designed with compressive stresses in the bulk may present a threshold strength below which catastrophic failure does not occur [2]. Deriving from the above-mentioned perspectives, zirconia-containing laminar ceramics have been employed to develop compressive stresses in the internal layers by means of tetragonal to monoclinic phase transformations that take place in the cooling down during the sintering process [2]. Under certain conditions, these compressive stresses may act as a barrier to crack propagation. In other cases, crack bifurcation and/or deflection phenomena result in an increase of material fracture toughness and energy absorption capability [3]. The procedure for preparation of these composites and typical crack behaviour can be found elsewhere [1,4–9]. For the estimation of apparent fracture toughness the influence of the interface between two dissimilar materials (layers) on the critical values for crack propagation through the interface have to be determined, because the role of interface is fundamental. The existence of two regions with different material properties and the presence of an interface strongly influence the distribution of stress in composite bodies. To estimate the influence of interfaces on the fracture behaviour of laminated structures the interaction 0927-0256/$ - see front matter 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2009.04.005 * Corresponding author. Address: Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zˇizˇkova 22, 616 62 Brno, Czech Republic. Tel.: +420 532 290 351; fax: +420 541 218 657. E-mail address: nahlik@ipm.cz (L. Náhlík). Computational Materials Science 46 (2009) 614–620 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Nahlik et al/Computational Materials Science 46(2009)614-620 615 between cracks and interfaces has to be studied and the critical val- 2. 2. Stress distribution near the crack tip ues for crack propagation through interfaces estimated An analytical expression of the stress distribution around a The stress distribution around a crack tip (in the case of a crack crack terminating at a bi-material interface can be found e.g. in perpendicular to the interface, see Fig. 1)can be expressed in the [10-15]. Attempts at a generalization of fracture criteria of classi al linear elastic fracture mechanics. which would be suitable for H1 Gi rPi(p, a B) (1) face between two materials represents such kind of concentrator ). have also previously been published, e.g. [16, 17- where HI is a generalized stress intensity factor, r radial distance The aim of this paper is to estimate values of the apparent frac- from the crack tip and a, B composite parameters defined at a later ture toughness of a ceramic laminate at interfaces between two stage in the text For given materials and given loadin g condition yers on the basis of the strain energy density concept. he stress distribution around the crack tip is given by the value ues are strongly influenced by residual stresses developed during of the generalized stress intensity factor H. The value of Hy is pro- the sintering process in the layers and by different elastic proper- portional to the applied load and has to be estimated numerically ties of individual layers. a new tentative stability criterion on the For homogeneous materials Hi-Ki is the stress intensity factor. asis of generalization of the strain energy density factor for the and p=1/2 of a crack touching the interface between two materials is for- The exponent of the stress singularity p=1-1 is given by the mulated and used for an estimation of the apparent fracture tough- solution of the equation following from 10: 2(-4x2+4xB)+2x2-2xB+2x-B+1 2. Methodology +(-2+2B-2x+2)cos(i)=0 where composite parameters a, B are defined as: The paper presented deals with an estimation of the apparent fracture toughness of a ceramic laminate or, more generally, lami- E(1+v2)-(1+v1) for plane stress condition nates made from materials with strong interfaces. Layers of the materials mentioned often contain residual stresses developed luring preparation of the composite(e.g. during sintering process, 义= etc. ). As a typical example, the AlzO3-Zroz laminate can be men- plane strain condition. tioned. For an estimation of fracture toughness the following steps E,. E2 are Young moduli and v Poissons ratios of material 1, material 2(see Fig. 2). The solution i of Eg. (2)for different values of a, B is Determination of residual stresses shown in the Fig. 2. Determination of the stress field around the crack tip, especially g(a)is a known function of the material properties [10 in the case of a crack touching the interface between two mate- gR()=2-Cos T rials, by combination of analytical (stress singularity exponent x+2-(1+2x-4x2)cos+(1+2)Cos2m d stress components) and numerical solution (value of the stress intensity factor). 1+2x+2x2-2(x+x2)c0si-4x22 consideration of residual stresses and existence of interfaces 2.3. Generalized strain energy density factor concept Calculation of apparent fracture toughness values with respect to the existence of the interface The strain energy density factor concept in fracture mechanics was originally proposed for a crack in a homogeneous material The assumptions of linear elastic fracture mechanics(LEFM) (i.e. for power of singularity p= 1/ 2). On the basis of strain energy and small scale yielding conditions are considered in the sections density Sih [18,19] introduced a strain energy density factor S, following. which is independent of a distance r ahead of the crack tip 2.1. Crack touching the interface For estimation of the apparent fracture toughness of a ceramic laminate the values for crack propagation through the material Material 1 Material 2 interface must be determined. However on interfaces the classical approaches of LEFM cannot be used, due to the different stress sin- E 1,V1 E2,V2 gularity exponent p of a crack touching the interface. In this case, the stress singularity exponent lies in the interval 0<p*1/2<1 The apparent fracture toughness cannot be estimated directly due to different stress singularity exponents and a special ap- proach taking this fact into account should be applied for the esti- crack mation of apparent fracture toughness Kaptc on material interfaces. The knowledge of Kaptc on interfaces allows us to determine the fracture toughness for all crack lengths considered and to estimate the apparent fracture toughness of the laminate, since extreme ( minimal or maximal) values of Kaptc can be found close to or di- rectly on the interfaces. Thus Kapte values for interfaces play a key role in the fracture behaviour of these laminates and its deter- mination is important for the description of fracture behaviou Fig. 1. Crack touching the interface between two dissimilar material
between cracks and interfaces has to be studied and the critical values for crack propagation through interfaces estimated. An analytical expression of the stress distribution around a crack terminating at a bi-material interface can be found e.g. in [10–15]. Attempts at a generalization of fracture criteria of classical linear elastic fracture mechanics, which would be suitable for general singular stress concentrators (a crack touching the interface between two materials represents such kind of concentrator), have also previously been published, e.g. [16,17]. The aim of this paper is to estimate values of the apparent fracture toughness of a ceramic laminate at interfaces between two layers on the basis of the strain energy density concept. These values are strongly influenced by residual stresses developed during the sintering process in the layers and by different elastic properties of individual layers. A new tentative stability criterion on the basis of generalization of the strain energy density factor for the case of a crack touching the interface between two materials is formulated and used for an estimation of the apparent fracture toughness of a ceramic laminate. 2. Methodology The paper presented deals with an estimation of the apparent fracture toughness of a ceramic laminate or, more generally, laminates made from materials with strong interfaces. Layers of the materials mentioned often contain residual stresses developed during preparation of the composite (e.g. during sintering process, etc.). As a typical example, the Al2O3–ZrO2 laminate can be mentioned. For an estimation of fracture toughness the following steps are necessary: – Determination of residual stresses. – Determination of the stress field around the crack tip, especially in the case of a crack touching the interface between two materials, by combination of analytical (stress singularity exponent and stress components) and numerical solution (value of the stress intensity factor). – Calculation of effective values of the stress intensity factor with consideration of residual stresses and existence of interfaces. – Calculation of apparent fracture toughness values with respect to the existence of the interface. The assumptions of linear elastic fracture mechanics (LEFM) and small scale yielding conditions are considered in the sections following. 2.1. Crack touching the interface For estimation of the apparent fracture toughness of a ceramic laminate the values for crack propagation through the material interface must be determined. However, on interfaces the classical approaches of LEFM cannot be used, due to the different stress singularity exponent p of a crack touching the interface. In this case, the stress singularity exponent lies in the interval 0 < p – 1/2 < 1. The apparent fracture toughness cannot be estimated directly due to different stress singularity exponents and a special approach taking this fact into account should be applied for the estimation of apparent fracture toughness Kapt;c on material interfaces. The knowledge of Kapt;c on interfaces allows us to determine the fracture toughness for all crack lengths considered and to estimate the apparent fracture toughness of the laminate, since extreme (minimal or maximal) values of Kapt,c can be found close to or directly on the interfaces. Thus Kapt;c values for interfaces play a key role in the fracture behaviour of these laminates and its determination is important for the description of fracture behaviour. 2.2. Stress distribution near the crack tip The stress distribution around a crack tip (in the case of a crack perpendicular to the interface, see Fig. 1) can be expressed in the general form rij ¼ H1 ffiffiffiffiffiffi 2p p rpfijðp; a; bÞ; ð1Þ where H1 is a generalized stress intensity factor, r radial distance from the crack tip and a,b composite parameters defined at a later stage in the text. For given materials and given loading conditions the stress distribution around the crack tip is given by the value of the generalized stress intensity factor H1. The value of H1 is proportional to the applied load and has to be estimated numerically. For homogeneous materials H1 = KI is the stress intensity factor, and p = 1/2. The exponent of the stress singularity p =1– k is given by the solution of the equation following from the boundary conditions [10]: k2 ð4a2 þ 4abÞ þ 2a2 2ab þ 2a b þ 1 þ ð2a2 þ 2ab 2a þ 2bÞ cosðkpÞ ¼ 0; ð2Þ where composite parameters a,b are defined as: a ¼ E1 E2 ð1 þ m2Þð1 þ m1Þ 4 ; b ¼ E1 E2 for plane stress condition; a ¼ E1 E2 1þm1 1þm2 1 4ð1 m1Þ ; b ¼ E1 E2 1 m2 1 1 m2 2 : for plane strain condition: E1, E2 are Young moduli and m Poissons ratios of material 1, material 2 (see Fig. 2). The solution k of Eq. (2) for different values of a,b is shown in the Fig. 2. gR(k) is a known function of the material properties [10]: gRðkÞ ¼ k cos kp b½a þ 2k ð1 þ 2a 4ak2 Þ cos kp þ ð1 þ aÞ cos 2kp 1 þ 2a þ 2a2 2ða þ a2Þ cos kp 4a2k2 : 2.3. Generalized strain energy density factor concept The strain energy density factor concept in fracture mechanics was originally proposed for a crack in a homogeneous material (i.e. for power of singularity p = 1/2). On the basis of strain energy density Sih [18,19] introduced a strain energy density factor S, which is independent of a distance r ahead of the crack tip: Fig. 1. Crack touching the interface between two dissimilar materials. L. Náhlík et al. / Computational Materials Science 46 (2009) 614–620 615
16 ience 46(2 入=0.2 16 入=0.3 入=0.7 Fig. 2. Curves of constant eigen value i for different values of composite parameters z, S=auki +2a12Kk+a (3) where H1, H2 [MPa. m] are generalized stress intensity factors and A,. Ar and az are known functions. the distance r ahead of the where K Ku are stress intensity factors corresponding to modes I, I crack tip corresponds to the failure mechanism. of loading and an, a1 and azz are known functions. Note that S can If the crack is perpendicular to a bi-material interface and mode be related to K when only mode i of loading is considered I load prevails, the value of generalized strain energy density factor 2 can be estimated from relation(only one stress singularity expo- (1+v)(1-2v) ane strain condition and nent p exists in this case (r0)=r1-2A1H1 (6 where (1-v) 2E for plane stress co Al 1-=P.(41-2y)+(8A)-p2 2IE for plane strain condition and The relation (3)can be analogically written for a general singu lar stress concentrator Generalization of strain energy density fac tor S for V-notches, i.e. for singular(sharp)stress concentrators in works [16, 17] and used in the case of bi-material notch e.g. in An-(1+2)(1-p2. (4(1-v+(8R()-p) with stress singularity exponents different from 1, was suggested 1+V2 [20]. The generalization of S for a crack touching the interface be- for plane stress condition ween two materials was suggested in work [21. The generalized strain energy density factor is defined the On the basis of the idea of equality S(K)=2(Hi, it is possible to find relat (r0)=r1-A1H+r1=p-p2A12H1H2+r1-2A2H2 ween H, and effective value of a stress in (5) factor on an interface Kler on a material interface 30 mm crack path 60 mm Fig 3. Considered AlzO3-Zro ceramics laminate
S ¼ a11K2 I þ 2a12KIKII þ a22K2 II; ð3Þ where KI, KII are stress intensity factors corresponding to modes I, II of loading and a11, a12 and a22 are known functions. Note that S can be related to KI when only mode I of loading is considered: S ¼ ð1 þ mÞð1 2mÞK2 I 2pE for plane strain condition and ð4aÞ S ¼ ð1 mÞK2 I 2pE for plane stress condition: ð4bÞ The relation (3) can be analogically written for a general singular stress concentrator. Generalization of strain energy density factor S for V-notches, i.e. for singular (sharp) stress concentrators with stress singularity exponents different from ½, was suggested in works [16,17] and used in the case of bi-material notch e.g. in [20]. The generalization of S for a crack touching the interface between two materials was suggested in work [21]. The generalized strain energy density factor is defined there as Rðr; hÞ ¼ r12p1A11H2 1 þ r1p1p2 2A12H1H2 þ r12p2A22H2 2; ð5Þ where H1, H2 [MPa.mp ] are generalized stress intensity factors and A11, A12 and A22 are known functions. The distance r ahead of the crack tip corresponds to the failure mechanism. If the crack is perpendicular to a bi-material interface and mode I load prevails, the value of generalized strain energy density factor R can be estimated from relation (only one stress singularity exponent p exists in this case) Rðr; hÞ ¼ r12pA11H2 1; ð6Þ where A11 ¼ ð1 þ m2Þð1 pÞ 2 2pE 4ð1 2m2ÞþðgRðkÞ pÞ 2 for plane strain condition and ð7aÞ A11 ¼ ð1 þ m2Þð1 pÞ 2 2pE 4ð1 m2Þ 1 þ m2 þ ðgRðkÞ pÞ 2 for plane stress condition: On the basis of the idea of equality S(KI) = R (H1), it is possible to find relation between H1 and effective value of a stress intensity factor on an interface KI;eff on a material interface: Fig. 2. Curves of constant eigen value k for different values of composite parameters a,b. Fig. 3. Considered Al2O3–ZrO2 ceramics laminate. 616 L. Náhlík et al. / Computational Materials Science 46 (2009) 614–620
L Nahlik et al/Computational Materials Science 46(2009)614-620 3.1. Numerical determination of residual stresses -p)4(1-2)+(g)-p)7h The type of laminate studied is prepared by sintering and The critical distance rc ahead of crack tip in Eq. 8)can be esti- mainly due to different coefficients of the thermal expansion of mated from relation(see e.g.[17, 22, 23) ceramics: its layers contain rather high compressive and tensile residual stresses which significantly influence the fracture behav (9) lour of the laminate Residual stresses that develop during the sin- tering process were determined by numerical calculations in commercial code ANSYS 11.0 as 2D calculation with approximation ar is the ultimate strength of material where the crack will of plane strain conditions. The sintering temperature of 1250C propagate. The influence of critical distance rc on values of Kleff is was considered as a residual stress free temperature. The compos- eak and is sufficient to determine only the order of magnitude ite specimen was then subjected to cooling to room temperature of the critical distance. The value of the generalized stress intensity (20%C). The resulting residual stresses in individual layers of the factor Hy must be determined by a numerical method (e.g. by finite composite are shown in Fig.4 The tensile residual stresses of value 110 MPa in the ATz layers and compressive stresses of value -715 MPa in the AMZ layers 3. Numerical example were numerically determined. These results are in good agreement with previously mentioned experimental work [ 2, 24 For further considerations concerning the estimation of appar- ent fracture toughness the Al2O3-ZrOz ceramic laminate can be 3. 2. Apparent fracture toughness used as an example. The material characteristics and geometry of the composite body considered were taken(by courtesy Bermejo) The presence of a crack in the laminate was assumed in subse- from Refs. [2, 24 to provide a compar th the published data. quent investigations. The edge crack growing from the surface The composite studied was composed of nine layers of Al203/ through the first four layers of laminate( to the middle of the spec 5voL%t-ZrO2(alumina with tetragonal zirconia, noted as ATz) imen)was considered, see Fig. 1. This kind of crack propagation and Al2O3/30vol96m-ZrO2 (alumina with monoclinic zirconia, was assumed on the basis of symmetry of geometry and loading noted as AMZ), see Fig. 3. The particle size of individual material conditions of the model and on the basis of experimental observa- components was about 0.3 um [2]. The thickness of ATZ layers tions performed on a very similar ceramics laminate published in was considered as tATz=0.52 mm and thickness of AMz tAMz- the literature see Fig5 0.1 mm. All material properties used for numerical simulations The values of stress intensity factor Kires were determined for are summarized in table 1 different crack lengths. This procedure is usually termed K-calibra- on. The specimen was loaded by thermal residual stresses only The finite element model contained more then 100.000 8-node iso- parametric elements. Around the crack tip the mesh was very fine Material properties of Al203-zroz laminate [2.24]- in order to describe singular stresses near the crack tip. The density of the mesh employed was sufficient for most crack lengths and ATZ AMz further refinements of the mesh had no influence on the results Youngs modulus E Details of the numerical model and opening stress pattern are oisson's rato v 1.22 shown in Fig. 6. Results derived from numerical calculations are Coefficient of thermal expansion r 9.82 Fracture toughness Kc MPa√m 8.02 shown in Fig. 7. Layer thickness 0.1 Positive values of Kires were calculated for tensile loaded layers of ATZ. In atz cracks propagate easily due to the existence of ter 西0 -100 400 distance in thickness direction(mm) Fig. 4. Values of residual stresses through the layers of composit
KI;eff ¼ 1 2m2 ð1 pÞ 2 ð4ð1 2m2ÞþðgRðkÞ pÞ 2 Þ !1 2 r 1 2p c H1: ð8Þ The critical distance rc ahead of crack tip in Eq. (8) can be estimated from relation (see e.g. [17,22,23]) rc ¼ K2 Ic 2pr2 f ; ð9Þ rf is the ultimate strength of material where the crack will propagate. The influence of critical distance rc on values of KI,eff is weak and is sufficient to determine only the order of magnitude of the critical distance. The value of the generalized stress intensity factor H1 must be determined by a numerical method (e.g. by finite element method). 3. Numerical example For further considerations concerning the estimation of apparent fracture toughness the Al2O3–ZrO2 ceramic laminate can be used as an example. The material characteristics and geometry of the composite body considered were taken (by courtesy Bermejo) from Refs. [2,24] to provide a comparison with the published data. The composite studied was composed of nine layers of Al2O3/ 5vol.%t-ZrO2 (alumina with tetragonal zirconia, noted as ATZ) and Al2O3/30vol.%m-ZrO2 (alumina with monoclinic zirconia, noted as AMZ), see Fig. 3. The particle size of individual material components was about 0.3 lm [2]. The thickness of ATZ layers was considered as tATZ = 0.52 mm and thickness of AMZ tAMZ = 0.1 mm. All material properties used for numerical simulations are summarized in Table 1. 3.1. Numerical determination of residual stresses The type of laminate studied is prepared by sintering and mainly due to different coefficients of the thermal expansion of ceramics; its layers contain rather high compressive and tensile residual stresses which significantly influence the fracture behaviour of the laminate. Residual stresses that develop during the sintering process were determined by numerical calculations in commercial code ANSYS 11.0 as 2D calculation with approximation of plane strain conditions. The sintering temperature of 1250 C was considered as a residual stress free temperature. The composite specimen was then subjected to cooling to room temperature (20 C). The resulting residual stresses in individual layers of the composite are shown in Fig. 4. The tensile residual stresses of value 110 MPa in the ATZ layers and compressive stresses of value 715 MPa in the AMZ layers were numerically determined. These results are in good agreement with previously mentioned experimental work [2,24]. 3.2. Apparent fracture toughness The presence of a crack in the laminate was assumed in subsequent investigations. The edge crack growing from the surface through the first four layers of laminate (to the middle of the specimen) was considered, see Fig. 1. This kind of crack propagation was assumed on the basis of symmetry of geometry and loading conditions of the model and on the basis of experimental observations performed on a very similar ceramics laminate published in the literature, see Fig. 5. The values of stress intensity factor KIres were determined for different crack lengths. This procedure is usually termed K-calibration. The specimen was loaded by thermal residual stresses only. The finite element model contained more then 100.000 8-node isoparametric elements. Around the crack tip the mesh was very fine in order to describe singular stresses near the crack tip. The density of the mesh employed was sufficient for most crack lengths and further refinements of the mesh had no influence on the results. Details of the numerical model and opening stress pattern are shown in Fig. 6. Results derived from numerical calculations are shown in Fig. 7. Positive values of KIres were calculated for tensile loaded layers of ATZ. In ATZ cracks propagate easily due to the existence of tenTable 1 Material properties of Al2O3–ZrO2 laminate [2,24]. Property Units ATZ AMZ Young’s modulus E GPa 390 280 Poisson’s ratio m – 0.22 0.22 Coefficient of thermal expansion at 106 K1 9.82 8.02 Fracture toughness KIC MPa ffiffiffiffiffi mp 3.2 2.6 Layer thickness t mm 0.52 0.1 Fig. 4. Values of residual stresses through the layers of composite. L. Náhlík et al. / Computational Materials Science 46 (2009) 614–620 617
18 L Nahlik et aL/ Computational Materials Science 46(2009)614-620 b AL, O Zro, ALO, 20m 20 ILn (a) (b) sile residual stresses there. Negative values of Kires(Fig. 7)are phys- AMZ/ATZ interfaces is caused by compressive residual stress and cally impermissible, but the correct interpretation is that the crack the presence of interfaces between two dissimilar materials is closed and crack faces are loaded by pressure keeping the crack closed. For the crack propagation an additional remote tensile 3.3. Apparent fracture toughness of the interface loading is required. The values of apparent fracture toughness can be calculated Finite element calculations were performed to determine corre- from the following expression sponding H, values on interfaces and relations(8)and(9)were Kaptc=Klc-kires used for estimation of Kleff. Constants included in the Eq(8)for (10) individual interfaces are displayed in Table 2. The value of the gen- where Ki is the fracture toughness of each individual layer, see Ta- eralized stress intensity factor H, was determined by direct meth- ble 1. The apparent fracture toughness of the multilayered laminate od, i.e. by extrapolation of tangential stress ahead of the crack tip is shown in Fig. 8 (for ooe=0)to the crack tip. The values of Kinter ace for interfaces Negative values of Kapt. c represent intervals where the loading of are then given by expression layers due to residual tensile stresses exceed the fracture tough ness K of individual layers and crack propagates without addi- tional external loading. The negative theoretical values of Kapte Resulting values of kinterace obtained for interfaces are summa- have no other physical meaning and from a practical point of view rized in Table 3 Here the calculated values kn and results obtained by ana- toughness of the laminate. The increase of Kap. c in the vicinity of lytical solution through weight function analysis introduced in [2]
sile residual stresses there. Negative values of KIres (Fig. 7) are physically impermissible, but the correct interpretation is that the crack is closed and crack faces are loaded by pressure keeping the crack closed. For the crack propagation an additional remote tensile loading is required. The values of apparent fracture toughness can be calculated from the following expression Kapt;c ¼ KIc KIres; ð10Þ where KIc is the fracture toughness of each individual layer, see Table 1. The apparent fracture toughness of the multilayered laminate is shown in Fig. 8. Negative values of Kapt,c represent intervals where the loading of layers due to residual tensile stresses exceed the fracture toughness KIc of individual layers and crack propagates without additional external loading. The negative theoretical values of Kapt,c have no other physical meaning and from a practical point of view Kapt,c = 0. Positive values of Kapt,c represent the apparent fracture toughness of the laminate. The increase of Kapt,c in the vicinity of AMZ/ATZ interfaces is caused by compressive residual stress and the presence of interfaces between two dissimilar materials. 3.3. Apparent fracture toughness of the interface Finite element calculations were performed to determine corresponding H1 values on interfaces and relations (8) and (9) were used for estimation of KI;eff . Constants included in the Eq. (8) for individual interfaces are displayed in Table 2. The value of the generalized stress intensity factor H1 was determined by direct method, i.e. by extrapolation of tangential stress ahead of the crack tip (for rhh ¼ 0) to the crack tip. The values of Kinterface apt;c for interfaces are then given by expression Kinterface apt;c ¼ KIc KI;eff : ð11Þ Resulting values of Kinterface apt;c obtained for interfaces are summarized in Table 3. Here the calculated values Kinterface apt;c and results obtained by analytical solution through weight function analysis introduced in [2] Fig. 5. Microphotographs of indentation cracks in alumina/zirconia layered composite propagated perpendicular to the interface of alumina and zirconia (crack initiated in (a) alumina and in (b) zirconia) – by courtesy of H. Hadraba [5]. Fig. 6. Detail of FE model (a) of layered composite and stress distribution (stress component normal to the crack faces) around the crack tip touching the ATZ/AMZ interface (b). Due to symmetry only one half of the laminate was modeled. 618 L. Náhlík et al. / Computational Materials Science 46 (2009) 614–620
L Nahlik et al/Computational Materials Science 46(2009)614-620 619 ATZ AMZ ATZ ATZ -8 k length a(mm) Fig. 7. Values of stress intensity factor Kres caused by thermal residual stresses calculated along crack path leading through the thickness of the specimen from surface to the middle ATZ ATZ ATZ 8 E crack length a(mm) for nearly the same geometry and conditions are evidently in good meaning(practically Kaptc -0). Extreme values can be found on agreement. These values are 2.5-3 times higher than the fracture material interfaces, which determine the fracture properties of toughness of individual layers and the increase in toughness of the laminate studied. The fracture toughness of the amzjatZ inter d conditions the maximal value of Kaptc of AMZ/ATZ interface ness of indiVidu higher in comparison with the fracture tough 8.6 MPam was found in the literature, see 1. interface can grow through the interface( for Klet >Kant.r c )or can In Fig. 9 the dependence of apparent fracture toughness Kap.c of stay arrested in front of the AMZ/ATZ interface. The AMZ/ATZ inter the Al2O3-ZrO2 laminate on crack length is shown. The dependence face creates an effective barrier to crack propagation. The values of tentioned includes the influence of residual stresses and value ness of ATZ in ATZ layers and therefore the apparent fracture Table 3 oughness in these regions can be considered as having no practical Calculated apparent fracture toughness values on interfaces. Crack length a(mm)Km( MPam) K-"(MPa√而“m÷Ma Table tants of analytical solution used in Eq(10) 0.62(AMZJATZ) Type of interface v(-) r(mm) 1.24(AMZ/ATz) ATZ/ AMZ 0.5360 22 0.54513 15E-05 AMZIATZ 046452 0.22 7.0E-06 Values published in Ref. [2] predicted analytically through weight function analysis for room temperature
for nearly the same geometry and conditions are evidently in good agreement. These values are 2.5–3 times higher than the fracture toughness of individual layers and the increase in toughness of the laminate is substantial. For a similar material configuration and conditions the maximal value of Kapt,c of AMZ/ATZ interface 8.6 MPa ffiffiffiffiffi mp was found in the literature, see [1]. In Fig. 9 the dependence of apparent fracture toughness Kapt,c of the Al2O3–ZrO2 laminate on crack length is shown. The dependence mentioned includes the influence of residual stresses and values Kinterface apt;c for interfaces as well. Values of KIres exceed fracture toughness of ATZ in ATZ layers and therefore the apparent fracture toughness in these regions can be considered as having no practical meaning (practically Kapt,c = 0). Extreme values can be found on material interfaces, which determine the fracture properties of the laminate studied. The fracture toughness of the AMZ/ATZ interface is significantly higher in comparison with the fracture toughness of individual layers. A crack propagating close to the AMZ/ATZ interface can grow through the interface (for KI;eff > Kinterface apt;c ) or can stay arrested in front of the AMZ/ATZ interface. The AMZ/ATZ interface creates an effective barrier to crack propagation. The values of Fig. 7. Values of stress intensity factor KIres caused by thermal residual stresses calculated along crack path leading through the thickness of the specimen from surface to the middle. Fig. 8. Theoretical values of apparent fracture toughness Kapt,c containing residual stresses in the laminate. Table 2 Constants of analytical solution used in Eq. (10). Type of interface p (–) m (–) gR (–) rc (mm) ATZ/AMZ 0.53609 0.22 0.54513 1.5E05 AMZ/ATZ 0.46452 0.22 0.47242 7.0E06 Table 3 Calculated apparent fracture toughness values on interfaces. Crack length a (mm) KI;effðMPa ffiffiffiffiffi mp Þ Kinterface apt;c ðMPa ffiffiffiffiffi mp Þ a Kinterface apt;c ðMPa ffiffiffiffiffi mp Þ 0.52 (ATZ/AMZ interface) 2.5 0.1 – 0.62 (AMZ/ATZ) 4.8 8.0 7.1 1.14 (ATZ/AMZ) 2.2 0.4 – 1.24 (AMZ/ATZ) 5.1 8.3 8.1 a Values published in Ref. [2] predicted analytically through weight function analysis for room temperature. L. Náhlík et al. / Computational Materials Science 46 (2009) 614–620 619
L Nahlik et aL/ Computational Materials Science 46(2009)614-620 ATZ AMZ crack length a (mm Fig 9. Apparent fracture toughness of Al2O3-Zroz laminate(solid line) in dependence on crack length and theoretical values calculated by Fem(doted). apparent fracture toughness in the case studied are quite indepen- Acknowledgement dent of absolute crack length in comparison to cracks in homoge neous materials because the opening of crack faces is strongly This research was supported through the Grant No. GAAV influenced by the presence of compressive residual stresses in 00410803 of the Grant Agency of the Academy of Sciences of the aMz layers, which close the crack approximately in the middle zech Republic. of each ATZ layer Kapt.c values are influenced mainly by the magni- lde of residual stresses and distance of the crack tip from the References interfaces. Only the maximal values of Kapt.=Kantrace icance from the practical point of view. These values determine the [11 herencia. L llanes, C. baudin. Acta materialia transversal resistance to crack propagation of the ceramic [3] w.J. Clegg, K Kendall, N M. Alford, T.w. Button, J D. Birchall, Nature 347(1990) 4. Conclusions [4 H. Hadraba, J Klimes, K Maca, Journal of Materials Science 42(2007)6404- The paper presented focuses on"flaw tolerant" ceramic lami- [5| H. Hadraba, K Maca, J. Cihlaf, Ceramics International 30(2004)853-863 nates. In the framework of linear elastic fracture mechanics resid [6]S Bueno, R Moreno, C Baudin, Journal of European Ceramic Society 25(2005 847-856. ual stresses in layers of the Al2O3-ZrO2 composite were Baudin, Journal of European Ceramic Society 27(2007)1455-1462. umerically determined. So-called K-calibration was performed for a crack growing through the thickness of the laminate speci- 19L. Marsavina. T. sadows men. A new, tentative procedure for estimation of the effective stress intensity factor on material interfaces based on a generalized [10] KY Lin, J.w. Mar, International Journal of Fracture 12 (4)(1976)521-531 strain energy density factor was presented. The concept referred to 12jTS Cook, F Erdogan, International Journal of Engineering Sciences 10(1972) generalizes approaches of classic LEFM in the area of general singu- 677-697 lar stress concentrators. Then the apparent fracture toughness [13] D.B. Bogy. Journal of Applied Mechanics(1971)911-918 (with influence of residual stresses and presence of material inter- 14N Fenner, International Joumal of Fracture 12 (5)01945205-721 face)was determined. The interface values of the fracture tough- 116i Z Knesl, Acta Technica &SAV 38(1993)221- ness are crucial for estimation of the fracture behaviour of [171 G.C. Sih, J.W. Ho, Theoretical and Applied Fracture Mechanics 16(1991)179- in the AMz/ATz interface was predicted in comparison with the 18 frcture. Nsprdiaofh toe of ioanck propagation in c sn ed.). Mechanics of fracture toughness of individual material components. Predicted [19] G.C. Sih, Mechanics of Fracture L Method and Solutions of crack values were compared with the data available in the literature and mutual good agreement was found 1] L Nahlik, Z Knesl, ) Klusak, Engineering 008)99-114 The procedure suggested can be used for the estimation of resis- [22] G.C. Sih, E T. Moyer Jr. E.E. Gdoutos, Engineer tance to crack propagation through multilayered structures and for 1983)731 their design. The procedure can contribute to enhancement of the [23 A Seweryn, A Lukaszewics, Engineering Fracture Mechanics 69(2002)1487- reliability and safety of structural ceramics or, more generally, of [24]. Bermejo, L Llanes, M. Anglada, P. Supancic. T. Lube, Key Engineering layered composites with strong interfaces. terials290(2005)191-1
apparent fracture toughness in the case studied are quite independent of absolute crack length in comparison to cracks in homogeneous materials because the opening of crack faces is strongly influenced by the presence of compressive residual stresses in the AMZ layers, which close the crack approximately in the middle of each ATZ layer. Kapt,c values are influenced mainly by the magnitude of residual stresses and distance of the crack tip from the interfaces. Only the maximal values of Kapt;c ¼ Kinterface apt;c have significance from the practical point of view. These values determine the transversal resistance to crack propagation of the ceramic laminate. 4. Conclusions The paper presented focuses on ‘‘flaw tolerant” ceramic laminates. In the framework of linear elastic fracture mechanics, residual stresses in layers of the Al2O3–ZrO2 composite were numerically determined. So-called K-calibration was performed for a crack growing through the thickness of the laminate specimen. A new, tentative procedure for estimation of the effective stress intensity factor on material interfaces based on a generalized strain energy density factor was presented. The concept referred to generalizes approaches of classic LEFM in the area of general singular stress concentrators. Then the apparent fracture toughness (with influence of residual stresses and presence of material interface) was determined. The interface values of the fracture toughness are crucial for estimation of the fracture behaviour of ceramic laminates. An important increase in fracture toughness in the AMZ/ATZ interface was predicted in comparison with the fracture toughness of individual material components. Predicted values were compared with the data available in the literature and mutual good agreement was found. The procedure suggested can be used for the estimation of resistance to crack propagation through multilayered structures and for their design. The procedure can contribute to enhancement of the reliability and safety of structural ceramics or, more generally, of layered composites with strong interfaces. Acknowledgement This research was supported through the Grant No. GAAV KJB200410803 of the Grant Agency of the Academy of Sciences of the Czech Republic. References [1] R. Bermejo, A.J. Sánchez-Herencia, L. Llanes, C. Baudín, Acta Materialia 55 (2007) 4891–4901. [2] R. Bermejo, Y. Torres, C. Baudín, A.J. Sánchez-Herencia, J. Pascual, M. Anglada, L. Llanes, Journal of the European Ceramic Society 27 (2007) 1443–1448. [3] W.J. Clegg, K. Kendall, N.M. Alford, T.W. Button, J.D. Birchall, Nature 347 (1990) 455–461. [4] H. Hadraba, J. Klimeš, K. Máca, Journal of Materials Science 42 (2007) 6404– 6411. [5] H. Hadraba, K. Máca, J. Cihlárˇ, Ceramics International 30 (2004) 853–863. [6] S. Bueno, R. Moreno, C. Baudín, Journal of European Ceramic Society 25 (2005) 847–856. [7] S. Bueno, C. Baudín, Journal of European Ceramic Society 27 (2007) 1455–1462. [8] C. Kohnle, O. Mintchev, D. Brunner, S. Schmauder, Computational Materials Science 19 (2000) 261–266. [9] L. Marsavina, T. Sadowski, Computational Materials Science (2008), doi:10.1016/j.commatsci.2008.06.005. [10] K.Y. Lin, J.W. Mar, International Journal of Fracture 12 (4) (1976) 521–531. [11] S.A. Meguid, M. Tan, Z.H. Zhu, International Journal of Fracture 73 (1995) 1–23. [12] T.S. Cook, F. Erdogan, International Journal of Engineering Sciences 10 (1972) 677–697. [13] D.B. Bogy, Journal of Applied Mechanics (1971) 911–918. [14] D.N. Fenner, International Journal of Fracture 12 (5) (1976) 705–721. [15] C.R. Chiang, International Journal of Fracture 47 (1991) 55–58. [16] Z. Knésl, Acta Technica CˇSAV 38 (1993) 221–234. [17] G.C. Sih, J.W. Ho, Theoretical and Applied Fracture Mechanics 16 (1991) 179– 214. [18] G.C. Sih, A special theory of crack propagation, in: G.C. Sih (Ed.), Mechanics of Fracture, Noordhoff International Publishing, Leyden, 1977. [19] G.C. Sih, Mechanics of Fracture I. Methods of Analysis and Solutions of Crack Problems, Noordhoff International Publishing, Leyden, 1973. [20] J. Klusák, Z. Knésl, Computational Materials Science 39 (2007) 214–218. [21] L. Náhlík, Z. Knésl, J. Klusák, Engineering Mechanics 2 (2008) 99–114. [22] G.C. Sih, E.T. Moyer Jr., E.E. Gdoutos, Engineering Fracture Mechanics 18 (3) (1983) 731–733. [23] A. Seweryn, A. Lukaszewics, Engineering Fracture Mechanics 69 (2002) 1487– 1510. [24] R. Bermejo, L. Llanes, M. Anglada, P. Supancic, T. Lube, Key Engineering Materials 290 (2005) 191–198. Fig. 9. Apparent fracture toughness of Al2O3–ZrO2 laminate (solid line) in dependence on crack length and theoretical values calculated by FEM (doted). 620 L. Náhlík et al. / Computational Materials Science 46 (2009) 614–620
