Mechanics Research Communications 35(2008)576-582 Contents lists available at Science Direct MECHANICS Mechanics research communications ELSEVIER journalhomepagewww.elsevier.com/locate/mechrescom Energy-based and local approaches to the strength analysis of ceramic laminates with thermal residual stresses through the finite element method P VenaE Bertarelli D Gastaldi, R Contro Dipartimento di ingegneria Strutturale, Laboratory of Biological Structure Mechanics, Politecnico di Milano, Piazza Leonardo da vinci, 32, 20133 Milano, Italy ARTICLE IN FO A BSTRACT Article hist This paper deals with the strength analysis of ceramic laminates subjected to residual d 18 February 2008 stress fields. In particular, alumina/mullite/zirconia ceramic materials have been taken into Available online 12 April 2008 consideration. To this purpose, an energy-based approach and a micromechanical local pproach have been used within the framework of the finite element method. The results obtained through the numerical analyses are consistent with the ex ding a correct estimation of the limit strength lower bound on the external applied loads, below which no crack propagation occurs, can be identified. The local approach has led to a strength distribution that deviates from the typical Weibull Linear elastic fracture mechanics distribution: this is owed to the residual stress field. Indeed, a stress-dependent Weibull Finite element method modulus has been found. e 2008 Elsevier Ltd. All rights reserved. 1 Introduction Ceramic materials show many interesting properties like high thermal shock resistance, chemical inertness and excellent tribological behavior. These properties confer passive biocompatibility, high wear resistance and low friction coefficient that make the based materials suitable for critical applications in many field energy conversion, precision mechanics and engines, cutting tools and biomechanical devices. The main limitation to the use of ceramic materials in structural applications is owed to their low reliability, i.e. their high catter in failure strength and low fracture toughness. This behavior is directly related to the presence of flaws originated during the production processes and the service. The standard deviation of measured strength is often too large to allow a safe desig Many efforts have been made with the purpose to increase the reliability of ceramic materials: a promising approach is the design and manufacturing of laminates with pre-determined residual stress fields. Layers of ceramic composites with different composition are stacked, obtaining a graded ceramic, in order to develop a specific residual stress profile as a result of the sintering process. The thermal stresses are due to the mismatch of the mean coefficient of thermal expansion between the layers and between the different grains in the same layer (Green et al, 1999: Sergo et al, 1997). The particular case of alumina/zirconia laminates is discussed in Vena(2005)from the modeling standpoint. It is well known that a compressive residual stress at the surface of the laminates has beneficial effects on strength. Sim la ly, a compressive layer placed at a certain depth can significantly increase the toughness of the material (vena et al Corresponding author Tel: 23994236;fax:+390223994220 dress: vena(stru polimi it(P. 13/s-see front matter e 2008 Elsevier Ltd. All rights reserved doi: 10.1016/j. mechrescom. 2008.04.003
Energy-based and local approaches to the strength analysis of ceramic laminates with thermal residual stresses through the finite element method P. Vena *, E. Bertarelli, D. Gastaldi, R. Contro Dipartimento di Ingegneria Strutturale, Laboratory of Biological Structure Mechanics, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy article info Article history: Received 18 February 2008 Received in revised form 2 April 2008 Available online 12 April 2008 Keywords: Ceramic laminates Weibull statistic Linear elastic fracture mechanics Finite element method abstract This paper deals with the strength analysis of ceramic laminates subjected to residual stress fields. In particular, alumina/mullite/zirconia ceramic materials have been taken into consideration. To this purpose, an energy-based approach and a micromechanical local approach have been used within the framework of the finite element method. The results obtained through the numerical analyses are consistent with the experimental ones, providing a correct estimation of the limit strength; furthermore, a lower bound on the external applied loads, below which no crack propagation occurs, can be identified. The local approach has led to a strength distribution that deviates from the typical Weibull distribution; this is owed to the residual stress field. Indeed, a stress-dependent Weibull modulus has been found. 2008 Elsevier Ltd. All rights reserved. 1. Introduction Ceramic materials show many interesting properties like high thermal shock resistance, chemical inertness and excellent tribological behavior. These properties confer passive biocompatibility, high wear resistance and low friction coefficient that make the ceramic-based materials suitable for critical applications in many fields among which energy conversion, precision mechanics and engines, cutting tools and biomechanical devices. The main limitation to the use of ceramic materials in structural applications is owed to their low reliability, i.e. their high scatter in failure strength and low fracture toughness. This behavior is directly related to the presence of flaws originated during the production processes and the service. The standard deviation of measured strength is often too large to allow for a safe design. Many efforts have been made with the purpose to increase the reliability of ceramic materials: a promising approach is the design and manufacturing of laminates with pre-determined residual stress fields. Layers of ceramic composites with different composition are stacked, obtaining a graded ceramic, in order to develop a specific residual stress profile as a result of the sintering process. The thermal stresses are due to the mismatch of the mean coefficient of thermal expansion between the layers and between the different grains in the same layer (Green et al., 1999; Sergo et al., 1997). The particular case of alumina/zirconia laminates is discussed in Vena (2005) from the modeling standpoint. It is well known that a compressive residual stress at the surface of the laminates has beneficial effects on strength. Similarly, a compressive layer placed at a certain depth can significantly increase the toughness of the material (Vena et al., 0093-6413/$ - see front matter 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2008.04.003 * Corresponding author. Tel.: +39 02 2399 4236; fax: +39 02 2399 4220. E-mail address: vena@stru.polimi.it (P. Vena). Mechanics Research Communications 35 (2008) 576–582 Contents lists available at ScienceDirect Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
2005). Indeed, the compressive stress required to achieve high strength is often very high and localized and delamination between layers(i.e. edge cracking) may occur(Moon et al 2002). Recently, Sglavo et al. (2001) and Sglavo and Bertoldi(2006) have proposed the design and manufacturing of alumina mullite/zirconia multilayer laminates with an engineered residual stress profile with a maximum compression at a certain depth from surface, obtaining a low-scattered strength and a higher reliability in comparison to monolithic ceramics. In the present work both energy-based and local approaches are applied to evaluate the behavior of the aforementioned ceramic laminate, applying the finite element method. First of all, the residual stress field and the effective toughness are put on the role of the residual stress field in the Weibull type statistical andlysis, developed. Particular emphasis will be evaluated. Subsequently a statistical analysis based on the Weibull approach is 2. Finite element model to determine effective toughness and strength of ceramic laminates The residual stress field and the effective toughness of the laminate have been calculated for bars having nominal length of 60 mm and a rectangular cross section(7.5 mm height and 1.45 mm thickness). a nine layer symmetric composite has been considered in this paper. Composite materials with alumina, mullite and zirconia constituents were considered. Thickness and material composi- tions of each layer are reported in Fig. la. Following notation reported in Sglavo and Bertoldi(2006) each layer is identified as AZw/ly or AMwly in which A, Z and M stand for alumina, zirconia and mullite, respectively; w is the volumetric content of zirconia or mullite and y is the layer thickness in um. The elastic constitutive parameters, i.e. the Young modulus and the poisson ratio, as well as the coefficient of thermal expansion of the composites at different volumetric compositions used in Sglavo et al. (2001)have been assumed(see Table 1). The Young modulus and poisson ratio were estimated through the voigt and reuss bounding values. The thermal expan sion coefficient and fracture toughness for AM and az composites were measured on monolithic samples( bertoldi et al 2003) The residual stress field was obtained by solving the finite element equations in which a thermal expansion, proportional to a temperature variation from stress-free temperature to standard conditions AT=-(1200-25)C, was applied. The coef- ficients of thermal expansion of the materials are reported in Table 1 a plane stress model was used to simulate the cooling process after sintering and the standard four-point bending tests: therefore, the finite element mesh of half length of the material sample has been built accounting for symmetry conditions. Second order plane strain displacement based finite elements were used The smallest element size, in the area close to the crack tip along the crack propagation path, is 2.5 um. The size of the largest element is consistent with the smallest layer thickness, so that in the far region, at least one element in the layer thickness is used The finite element analyses are carried out by setting a fictitious thermal expansion coefficient for each layer eft as in which Azo is the reference value, i.e. the coefficient of thermal expansion of pure alumina(azo), with the purpose to avoid unrealistic out-of-plane residual stress values. The residual stresses far from the crack tip are characterized by a homogenous field in each layer As shown in Fig. 1b, the external Az30 layer is subject to a slight tensile stress, whereas the azo (second layer). AM40 and azo ( third layer) are sub- jected to compressive stress field. The middle layer is subjected to tensile residual stress. 400 600
2005). Indeed, the compressive stress required to achieve high strength is often very high and localized and delamination between layers (i.e. edge cracking) may occur (Moon et al., 2002). Recently, Sglavo et al. (2001) and Sglavo and Bertoldi (2006) have proposed the design and manufacturing of alumina/ mullite/zirconia multilayer laminates with an engineered residual stress profile with a maximum compression at a certain depth from surface, obtaining a low-scattered strength and a higher reliability in comparison to monolithic ceramics. In the present work both energy-based and local approaches are applied to evaluate the behavior of the aforementioned ceramic laminate, applying the finite element method. First of all, the residual stress field and the effective toughness are evaluated. Subsequently a statistical analysis based on the Weibull approach is developed. Particular emphasis will be put on the role of the residual stress field in the Weibull type statistical analysis. 2. Finite element model to determine effective toughness and strength of ceramic laminates The residual stress field and the effective toughness of the laminate have been calculated for bars having nominal length of 60 mm and a rectangular cross section (7.5 mm height and 1.45 mm thickness). A nine layer symmetric composite has been considered in this paper. Composite materials with alumina, mullite and zirconia constituents were considered. Thickness and material compositions of each layer are reported in Fig. 1a. Following notation reported in Sglavo and Bertoldi (2006) each layer is identified as AZw/y or AMw/y in which A, Z and M stand for alumina, zirconia and mullite, respectively; w is the volumetric content of zirconia or mullite and y is the layer thickness in lm. The elastic constitutive parameters, i.e. the Young modulus and the Poisson ratio, as well as the coefficient of thermal expansion of the composites at different volumetric compositions used in Sglavo et al. (2001) have been assumed (see Table 1). The Young modulus and Poisson ratio were estimated through the Voigt and Reuss bounding values. The thermal expansion coefficient and fracture toughness for AM and AZ composites were measured on monolithic samples (Bertoldi et al., 2003). The residual stress field was obtained by solving the finite element equations in which a thermal expansion, proportional to a temperature variation from stress-free temperature to standard conditions DT ¼ ð1200 25Þ C, was applied. The coef- ficients of thermal expansion of the materials are reported in Table 1. A plane stress model was used to simulate the cooling process after sintering and the standard four-point bending tests; therefore, the finite element mesh of half length of the material sample has been built accounting for symmetry conditions. Second order plane strain displacement based finite elements were used. The smallest element size, in the area close to the crack tip along the crack propagation path, is 2.5 lm. The size of the largest element is consistent with the smallest layer thickness, so that in the far region, at least one element in the layer thickness is used. The finite element analyses are carried out by setting a fictitious thermal expansion coefficient for each layer an eff as an eff ¼ an aAZ0 ð1Þ in which aAZ0 is the reference value, i.e. the coefficient of thermal expansion of pure alumina (AZ0), with the purpose to avoid unrealistic out-of-plane residual stress values. The residual stresses far from the crack tip are characterized by a homogenous field in each layer. As shown in Fig. 1b, the external AZ30 layer is subject to a slight tensile stress, whereas the AZ0 (second layer), AM40 and AZ0 (third layer) are subjected to compressive stress field. The middle layer is subjected to tensile residual stress. Fig. 1. Architecture of the AMZ laminate (not to scale): symmetry of the laminate structure and symmetry used in FEA are indicated (a); residual stress profile (b). P. Vena et al. / Mechanics Research Communications 35 (2008) 576–582 577
P. Vena et al/Mechanics Research Communications 35(2008)576-582 Table 1 Kc(MPa mIP) 7.75×10- 010 9492 222 8.37×10 AM40 688×106 0.2450 868×106 ment analyses at increasing crack length from a minimum of o um up to a maximum length of 250um ries of finite ele- The effective toughness of the symmetric laminate subjected to residual stress was obtained through a A two-step procedure is used to determine the effective toughness(Vena et al, 2005): ( i)a series of analysis with increas- ing crack length in which external loads simulating the four-point bending test (spans S1= 20 mm and sz=40 mm accord- ng to the experimental procedure reported in Sglavo and bertoldi, 2006)with no residual stress is simulated and ii)a series of analysis with increasing crack length in which thermal expansion without external load is simulated. The crack propagation along the symmetry plane was simulated by progressively releasing the nodal constraints. Each crack propagation step represented an increase of crack length Aa=5 um; refined analyses with Aa=2.5 um from 150 um to 220 um crack lengths were also performed. In the first series of analysis a unit force is used and a stress intensity factor due to external force is found by calculat the energy release rate g(a)(load controlled test): an aU G(a)= in which n is the total potential energy, U is the elastic energy of the system and a the current crack length. Assuming a mode I crack propagation, the stress intensity factor k for a unit force is K"(a)=VG(a-n2 The above expounded finite element procedure has been validated through a quantitative comparison with results obtained through contour integrals presented on a different ceramic laminate in Chen et al. (2007)and Bertarelli(2007) In the second series of analysis, the stress intensity factor in the laminate subjected to residual stress Kres(a)is calculated through the same procedure based on nodal constraint release as expounded above Due to linearity of the constitutive laws and to the linearity of the strain -displacements relationships, the principle of superposition of the solution from the thermal loading and that from the force loading can be used to determine the total stress intensity factory applied to the laminate subjected to the residual stress field and to the four-point bending external loads k(a K(a=Kres(a)+K(a)P in which P is the force applied in the four-point bending loading condition( Fig. 2). 0.=693MPa g =342MPa 2A230AZ0:AM40Az0A240 Crack length2 [um 21 Fig. 2. Effective fracture toughness of the AMZ laminate Dashed lines represent the applied stress intensity factor to determine laminate strength(slope 693 MPa)and lower threshold(slope 342 MPa)
The effective toughness of the symmetric laminate subjected to residual stress was obtained through a series of finite element analyses at increasing crack length from a minimum of 0 lm up to a maximum length of 250 lm. A two-step procedure is used to determine the effective toughness (Vena et al., 2005): (i) a series of analysis with increasing crack length in which external loads simulating the four-point bending test (spans S1 ¼ 20 mm and S2 ¼ 40 mm according to the experimental procedure reported in Sglavo and Bertoldi, 2006) with no residual stress is simulated and (ii) a series of analysis with increasing crack length in which thermal expansion without external load is simulated. The crack propagation along the symmetry plane was simulated by progressively releasing the nodal constraints. Each crack propagation step represented an increase of crack length Da ¼ 5 lm; refined analyses with Da ¼ 2:5 lm from 150 lm to 220 lm crack lengths were also performed. In the first series of analysis a unit force is used and a stress intensity factor due to external force is found by calculating the energy release rate GðaÞ (load controlled test): GðaÞ¼ oP oa ¼ oU oa ð2Þ in which P is the total potential energy, U is the elastic energy of the system and a the current crack length. Assuming a mode I crack propagation, the stress intensity factor Kr for a unit force is Kr ðaÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GðaÞ E 1 m2 r ð3Þ The above expounded finite element procedure has been validated through a quantitative comparison with results obtained through contour integrals presented on a different ceramic laminate in Chen et al. (2007) and Bertarelli (2007). In the second series of analysis, the stress intensity factor in the laminate subjected to residual stress KresðaÞ is calculated through the same procedure based on nodal constraint release as expounded above. Due to linearity of the constitutive laws and to the linearity of the strain–displacements relationships, the principle of superposition of the solution from the thermal loading and that from the force loading can be used to determine the total stress intensity factory applied to the laminate subjected to the residual stress field and to the four-point bending external loads KðaÞ: KðaÞ ¼ KresðaÞ þ Kr ðaÞP ð4Þ in which P is the force applied in the four-point bending loading condition (Fig. 2). Table 1 Material properties Material E (GPa) a (C1 ) KC (MPa m1/2) m AM0/AZ0 394 7:75 106 3.6 0.2300 AZ100 204 – – 0.2900 AM100 229 – – 0.2700 AZ30 322.5 8:37 106 3.9 0.2465 AM40 317 6:88 106 2.4 0.2450 AZ40 302.5 8:68 106 4.5 0.2525 Fig. 2. Effective fracture toughness of the AMZ laminate. Dashed lines represent the applied stress intensity factor to determine laminate strength (slope 693 MPa) and lower threshold (slope 342 MPa). 578 P. Vena et al. / Mechanics Research Communications 35 (2008) 576–582
P. Vena et al /Mechanics Research Communications 35(2008) 576-582 he crack in the layer n propagates when the condition is met, in which Kc is the intrinsic fracture toughness of the composite material in the layer n. Combining the relationships (4)and(5), taking the equality sign, the effective toughness Keff(a) is obtained Keff(a)=ke-Kres(a) (6) nd the relation(5)can be written as lich Per is the force at whi As shown in Fig. 2, the effective stress intensity factor exhibited an increasing trend in the layers characterized by a com- ressive residual stress After having reached a maximum va hen the crack tip is located beyond the interface between he azo and az40 layers the curve decreases as a conseque the tensile residual stress. The critical crack length found through this approach is approximately 169 um which is the result found through the analytical procedures(see Sglavo and Bertoldi, 2006) Through a direct comparison between the effective toughness and the stress intensity factor due to external loads (k(a)=K P). it is possible to determine the stability condition for crack growth In particular, for small magnitude of external loads P, the k(a) curve may intercept the Keff(a) curve for two values of a, ay a1 and a2. This indicates that any crack shorter than al does not propagate and cracks with an a a propagate until they reach the length Through this procedure, a critical value of the external load Pcr for which a2 @cr can be identified. The value of an for Pcr will denote the almax which represents the maximum length of a non-propagating crack. The critical value of external load for the laminate under study is Pcr=47.3 N, which corresponds to a tensile bending strength of 693 MPa. The average strength obtained through the four-point experimental tests is 692+ 25 MPa( Sglavo and bertoldi, 2006). For external loads P> Pcr, all cracks longer than a, will propagate unstably. Moreover, a lower threshold for the external oad Pmin may be identified: the minimum value of P for which al For Pacr)which have no practical interest. The theoretical applied stress at Pmin is 342 MPa. 3. Statistical analysis of laminates with residual stress As opposed to the energy approach, a local approach based on the weibull theory is presented here. In particular, the stress fields obtained through the finite element analyses were post-processed with the purpose to determine the Weibull stress defined (P) (omax))"dv (omax))"sin addr 8 (omax)being the positive part of the maximum principal stress, m the Weibull modulus, vo a reference volume, v the volume of the process zone and t the thickness of the sample (beremin, 1983: Esposito et al. 2007). The Weibull modulus is, incidentally, a stress exponent that describes the relation between the weibull stress(directly related to the failure probability) and the relevant applied stress. The reference volume Vo should be chosen consistently with the material microstructure; nevertheless, from a numerical point of view, it can be considered as a scale factor and it can be hosen arbitrarily (potentially the unit volume) but kept constant for all the analyses (munz and fett, 1999; Lei et al, 1998). In this application, the process zone is assumed to be circular with radius Imax and Vo=tL L being a characteristic length. a polar coordinate system was used to compute the volume integral(see Fig 3). Due to the symmetry of the model, the inte- gral over half process zone was calculated roces F/2 Fig 3. Graphical representation of the process zone: increasing radii are shown(not to scale)
The crack in the layer n propagates when the condition KðaÞ P Kn c ð5Þ is met, in which Kn c is the intrinsic fracture toughness of the composite material in the layer n. Combining the relationships (4) and (5), taking the equality sign, the effective toughness KeffðaÞ is obtained: KeffðaÞ ¼ Kn c KresðaÞ ð6Þ and the relation (5) can be written as Kr ðaÞPcr P Kn c KresðaÞ ð7Þ in which Pcr is the force at which crack propagates. As shown in Fig. 2, the effective stress intensity factor exhibited an increasing trend in the layers characterized by a compressive residual stress. After having reached a maximum value when the crack tip is located beyond the interface between the AZ0 and AZ40 layers the curve decreases as a consequence of the tensile residual stress. The critical crack length found through this approach is approximately 169 lm which is the same result found through the analytical procedures (see Sglavo and Bertoldi, 2006). Through a direct comparison between the effective toughness and the stress intensity factor due to external loads ðKextðaÞ ¼ Kr PÞ, it is possible to determine the stability condition for crack growth. In particular, for small magnitude of external loads P, the KextðaÞ curve may intercept the KeffðaÞ curve for two values of a, say a1 and a2. This indicates that any crack shorter than a1 does not propagate and cracks with a1 Pcr, all cracks longer than a1 will propagate unstably. Moreover, a lower threshold for the external load Pmin may be identified: the minimum value of P for which a1 ¼ a2. For P acrÞ which have no practical interest. The theoretical applied stress at Pmin is 342 MPa. 3. Statistical analysis of laminates with residual stress As opposed to the energy approach, a local approach based on the Weibull theory is presented here. In particular, the stress fields obtained through the finite element analyses were post-processed with the purpose to determine the Weibull stress defined as rWðPÞ ¼ 1 V0 Z V ð Þ hrmaxi mdV 1=m ¼ 2t V0 Z rmax 0 Z p 0 ð Þ hrmaxi m sin hdhdr 1=m ð8Þ hrmaxi being the positive part of the maximum principal stress, m the Weibull modulus, V0 a reference volume, V the volume of the process zone and t the thickness of the sample (Beremin, 1983; Esposito et al., 2007). The Weibull modulus is, incidentally, a stress exponent that describes the relation between the Weibull stress (directly related to the failure probability) and the relevant applied stress. The reference volume V0 should be chosen consistently with the material microstructure; nevertheless, from a numerical point of view, it can be considered as a scale factor and it can be chosen arbitrarily (potentially the unit volume) but kept constant for all the analyses (Munz and Fett, 1999; Lei et al., 1998). In this application, the process zone is assumed to be circular with radius rmax and V0 ¼ tL2 , L being a characteristic length. A polar coordinate system was used to compute the volume integral (see Fig. 3). Due to the symmetry of the model, the integral over half process zone was calculated. Fig. 3. Graphical representation of the process zone: increasing radii are shown (not to scale). P. Vena et al. / Mechanics Research Communications 35 (2008) 576–582 579
580 P. Vena et al/Mechanics Research Communications 35(2008)576-582 The linearity of the governing equations allows one to calculate the stress field due to residual stresses and applied exter- nal forces by making use of the superposition principle in which ols and og are the Cauchy components of the stress field due to the temperature change and to a unit external force (for a four-point bending configuration), respectively For the sake of computational accuracy, a round notch is considered at the tip of the crack, with notch radius of 4 um. this notch size is assumed to be consistent with the microstructures of the materials. As discussed in Lei et al. (1998). the size of the finite elements should be based on the requirement of statistical independence thus requiring that the smallest element be of the order of a few grain size. However, mathematically, this seems to be an unnecessary requirement as in the case of a non-singular stress field the computed Weibull stress value should be independent of the finite element mesh used. The Weibull stress ow(di) calculated for the critical crack length acr =169 um, and m=35(referring to Sglavo and Ber- toldi, 2006 )is reported in Fig. 4a. The values for different sizes of the process zone(radius from 5 um to 100 um)are pre- nted. In this figure ai is the nominal applied stress which is linearly related to the It can be observed that for ai greater than 330 MPa a linear relationship between the Weibull stress and the external the oretical stress is found. This linear relationship is independent of the process zone size. Whereas, for of lower than 330 MPa, the Weibull stress is not linearly related to the external loads and an appreciable dependence on the process zone size found. For the largest process zone size (rmax=100 um), a bilinear ow(oi)relationship is found. This response is owed to the residual stress field; indeed, for low magnitude external loads(Gi 330 MPa), the tensile esidual stress acting in the direction perpendicular to the interface planes into the wake of the crack gives the predominant contribution into the integral 8), see Fig 4b. The high value of m makes this effect particularly remarkable When the external load increases, the most relevant contribution to the integral 8)is the combination of residual stress and the stress due to the external forces acting along the direction perpendicular to the crack propagation ahead of the crack tip, see Fig 4c. This type of stress is the one that leads to the failure mode experimentally observed (ie crack propagate along the direction perpendicular to the interfaces). For di >330 MPa the relationship between the Weibull stress and the applied theoretical stress is linear and can be writ ten in the following form in which a is dependent on the parameter m used in the integral()and do is dependent on the residual stress field. For the AMZ laminate studied in this paper one has a= 20.3 and oo=283 MPa. The probability of failure for a given set of Weibull parameters m and awo, is expressed as( beremin, 1983: Esposito et a 2007) Taking logarithms twice the function f[In(oi)] can be defined mdh|(以)=m[(=) (12) which represents the Weibull plot used to determine the Weibull parameters for a series of experimental data. Note that, in case of no residual stress field, the Weibull plot( 12)is a linear function with slope m. a4000 3000 AZ40 1000 AM40 AM40 100200300400500 1 Fig 4. Weibull stress versus nominal stress at different radii (acr= 169 um. m=35)(a). Sketch of the dominant stress components for low (b)and high(c)
The linearity of the governing equations allows one to calculate the stress field due to residual stresses and applied external forces by making use of the superposition principle: rijmaxðPÞ ¼ rres ij þ Prr ij max ð9Þ in which rres ij and rr ij are the Cauchy components of the stress field due to the temperature change and to a unit external force (for a four-point bending configuration), respectively. For the sake of computational accuracy, a round notch is considered at the tip of the crack, with notch radius of 4 lm. This notch size is assumed to be consistent with the microstructures of the materials. As discussed in Lei et al. (1998), the size of the finite elements should be based on the requirement of statistical independence thus requiring that the smallest element be of the order of a few grain size. However, mathematically, this seems to be an unnecessary requirement as in the case of a non-singular stress field, the computed Weibull stress value should be independent of the finite element mesh used. The Weibull stress rWðrT 1Þ calculated for the critical crack length acr ¼ 169 lm, and m ¼ 35 (referring to Sglavo and Bertoldi, 2006) is reported in Fig. 4a. The values for different sizes of the process zone (radius from 5 lm to 100 lm) are presented. In this figure, rT 1 is the nominal applied stress which is linearly related to the applied force. It can be observed that for rT 1 greater than 330 MPa a linear relationship between the Weibull stress and the external theoretical stress is found. This linear relationship is independent of the process zone size. Whereas, for rT 1 lower than 330 MPa, the Weibull stress is not linearly related to the external loads and an appreciable dependence on the process zone size is found. For the largest process zone size ðrmax ¼ 100 lmÞ, a bilinear rWðrT 1Þ relationship is found. This response is owed to the residual stress field; indeed, for low magnitude external loads ðrT 1 330 MPa the relationship between the Weibull stress and the applied theoretical stress is linear and can be written in the following form: rW ¼ a rT 1 r0 ð10Þ in which a is dependent on the parameter m used in the integral (8) and r0 is dependent on the residual stress field. For the AMZ laminate studied in this paper one has a ¼ 20:3 and r0 ¼ 283 MPa. The probability of failure, for a given set of Weibull parameters m and rW0, is expressed as (Beremin, 1983; Esposito et al., 2007): F ¼ 1 exp rW rW0 m ð11Þ Taking logarithms twice the function f½lnðrT 1Þ can be defined: f lnðrT 1Þ ¼ ln ln 1 1 F ¼ m ln a rW0 eln rT 1 r0 ð12Þ which represents the Weibull plot used to determine the Weibull parameters for a series of experimental data. Note that, in case of no residual stress field, the Weibull plot (12) is a linear function with slope m. Fig. 4. Weibull stress versus nominal stress at different radii ðacr ¼ 169 lm; m ¼ 35Þ (a). Sketch of the dominant stress components for low (b) and high (c) external loads. 580 P. Vena et al. / Mechanics Research Communications 35 (2008) 576–582
P. Vena et al /Mechanics Research Communications 35(2008) 576-582 b10 n 0 10 0 100 283MP 30m=10 -200 5.6 6.4 n(o1) n(o1) Fig. 5. Weibull plots for m= 35 at different values of Go (a): Weibull plots for o= 283 MPa at different values of m(b 525.66646.8 6.4 6.5 6.6 Fig. 6. Numerical Weibull and experimental(squares) results for pure alumina and the AMz laminate(m=20)(a); enlargement of the plot showing numerical Weibull (solid line) and experimental results(circles) for the AMZ laminate(b) In the case of laminates with oo #0 a non-linear plot is obtained and it is singular for In(d)=In(oo)(i. e. it exhibits a ve ical asymptote). In Fig 5a, Weibull plots for different values of the parameter do are reported. The Weibull plots, as obtained by using(8)and(12). for three different values of the parameter m, are reported in Fig 5b. The plots for all values of the parameter m exhibit a bend at oi=330 MPa, which is the stress level beyond which the rela tionship (10) does not hold anymore. This behavior is consistent with the results outlined in the recent work dealing with tatistical numerical simulations for materials with residual stress fields(danzer et al, 2007 The slope(m1)of the Weibull plot calculated for f=0(i.e. a failure probability of approximately 63. 2%), which is a mea- sure of the reliability of the material strength, can be easily related to the Weibull parameter m in(11)and( 12)according to the following relationship For ao=0(i.e no residual stress), mi= m is obtained. This implies that experimental results exhibiting slope mi=35 in the Weibull plot of AMZ laminates can be fit by using, according to(13), m= 20. 1(Fig. 6). In this paper, a numerical method suitable to study and design high-reliability AMZ ceramic laminates characterized by residual stresses is presented. It involves both energy-based(global) and statistical (local)approaches, with the purpose to determine the fracture strength in a four-point bending stress and to discuss the effect of the thermal residual stress. The energy approach in the finite element model allowed to estimate the effective toughness during crack propagation, ointing out the stable propagation of cracks from the surface of the specimen subjected to the four-point bending test along the direction perpendicular to the interfaces
In the case of laminates with r0 6¼ 0 a non-linear plot is obtained and it is singular for lnðrT 1Þ ¼ lnðr0Þ (i.e. it exhibits a vertical asymptote). In Fig. 5a, Weibull plots for different values of the parameter r0 are reported. The Weibull plots, as obtained by using (8) and (12), for three different values of the parameter m, are reported in Fig. 5b. The plots for all values of the parameter m exhibit a bend at rT 1 ¼ 330 MPa, which is the stress level beyond which the relationship (10) does not hold anymore. This behavior is consistent with the results outlined in the recent work dealing with statistical numerical simulations for materials with residual stress fields (Danzer et al., 2007). The slope ðm1Þ of the Weibull plot calculated for f ¼ 0 (i.e. a failure probability of approximately 63.2%), which is a measure of the reliability of the material strength, can be easily related to the Weibull parameter m in (11) and (12) according to the following relationship: m1 m ¼ 1 þ a r0W r0 ð13Þ For r0 ¼ 0 (i.e. no residual stress), m1 ¼ m is obtained. This implies that experimental results exhibiting slope m1 ¼ 35 in the Weibull plot of AMZ laminates can be fit by using, according to (13), m ¼ 20:1 (Fig. 6). 4. Conclusions In this paper, a numerical method suitable to study and design high-reliability AMZ ceramic laminates characterized by residual stresses is presented. It involves both energy-based (global) and statistical (local) approaches, with the purpose to determine the fracture strength in a four-point bending stress and to discuss the effect of the thermal residual stress. The energy approach in the finite element model allowed to estimate the effective toughness during crack propagation, pointing out the stable propagation of cracks from the surface of the specimen subjected to the four-point bending test along the direction perpendicular to the interfaces. Fig. 5. Weibull plots for m ¼ 35 at different values of r0 (a); Weibull plots for r0 ¼ 283 MPa at different values of m (b). Fig. 6. Numerical Weibull and experimental (squares) results for pure alumina and the AMZ laminate ðm ¼ 20Þ (a); enlargement of the plot showing numerical Weibull (solid line) and experimental results (circles) for the AMZ laminate (b). P. Vena et al. / Mechanics Research Communications 35 (2008) 576–582 581
582 P Vena et al/ mechanics research communications 35(2008)576-582 The calculated laminate strength is consistent with the experimental results taken from the literature. Moreover, the en- rgy analysis evidenced the existence of a threshold lower limit on the external applied load below which the fracture will not occur except for very long cracks, which have no practical interest. This particular aspect cannot be obtained through mechanical laboratory tests unless a large number of specimens are considered. Indeed, the experimental tests presented in Sglavo and Bertoldi (2006)did not exhibit such threshold behavior. The micromechanical analyses have shown that the ceramic AMz laminate, because of the presence of residual stresses, does not exhibit a typical Weibull statistics behavior. The Weibull modulus is stress-dependent and a threshold stress below which Weibull statistic does not hold can be identified The AMZ laminate can be described using a modified Weibull approach, where additional parameters are needed. There- fore, further theoretical developments are required for a refined formulation of the Weibull statistics for materials with esidual stresses Acknowledgement The financial support of the Italian Ministry of University and research is kindly acknowledged. References Beremin, F.M., 1983. A local criterion for cleavage fracture of a nuclear pressure vessel steel Metallurgical Transactions A 14A, 2277-2287 Bertoldi, M. Paternoster, M, Sglavo, V M, 2003. Production of multilayer ceramic laminates with improved mechanical properties. Ceramic Transactions R, Pascual, ], Fischer, FD, Kolednik, O, Danzer, R, 2007. Prediction of the fracture toughness of a ceramic multilayer composite -modelling and ments. Acta Materialia 55. 409-42 ramics- Weibu Fracture Mechanics 74. 2919-29 posito, L Gentile, D, Bonora, N, 2007. Investigation on the Weibull parameters identification for local approach application in the ductile to brittle Gre od Cai, P Z, Messing, G L, 1999. Residual stresses in alumina-zirconia laminates. Journal of the European Ceramic Society 19. 2511-2517 Hilden, J, Bowman, K Trumble, KP, Rodel. ]- 2002. R-curve behavior in alumina-zirconia composites with repeating graded layers. Engineering Fracture Mechanics 69. 1647-1665 ett, T, 1999. Ceramics. Springer, Berlin. ty80.1633-1638 V M, Bertoldi, M., 2006. Design and production of ceramic laminates with high mechanical resistance and reliabilit Sglavo, VM Larentis, L, Green, JL, 2001. Flaw insensitive ion-exchanged glass: 1. Theoretical aspects. Journal of America Ceramic Society 84, 1827-1831 Vena, P, 2005. Thermal residual stresses in graded ceramic composites: a microscopic computational model versus homogenized models. Meccanica 40, Vena, P, Gastaldi, D, Contro, R, 2005. Effects of the thermal residual stress field on the crack propagation in graded alumina-zirconia ceramics Material Science Forum 492-493. 177-182
The calculated laminate strength is consistent with the experimental results taken from the literature. Moreover, the energy analysis evidenced the existence of a threshold lower limit on the external applied load below which the fracture will not occur except for very long cracks, which have no practical interest. This particular aspect cannot be obtained through mechanical laboratory tests unless a large number of specimens are considered. Indeed, the experimental tests presented in Sglavo and Bertoldi (2006) did not exhibit such threshold behavior. The micromechanical analyses have shown that the ceramic AMZ laminate, because of the presence of residual stresses, does not exhibit a typical Weibull statistics behavior. The Weibull modulus is stress-dependent and a threshold stress below which Weibull statistic does not hold can be identified. The AMZ laminate can be described using a modified Weibull approach, where additional parameters are needed. Therefore, further theoretical developments are required for a refined formulation of the Weibull statistics for materials with residual stresses. Acknowledgement The financial support of the Italian Ministry of University and Research is kindly acknowledged. References Beremin, F.M., 1983. A local criterion for cleavage fracture of a nuclear pressure vessel steel. Metallurgical Transactions A 14A, 2277–2287. Bertarelli, E., 2007. Master’s Thesis, Politecnico di Milano (in Italian). Bertoldi, M., Paternoster, M., Sglavo, V.M., 2003. Production of multilayer ceramic laminates with improved mechanical properties. Ceramic Transactions 153, 89–102. Chen, C.R., Pascual, J., Fischer, F.D., Kolednik, O., Danzer, R., 2007. Prediction of the fracture toughness of a ceramic multilayer composite – modelling and experiments. Acta Materialia 55, 409–421. Danzer, R., Suspancic, P., Pascual, J., Lube, T., 2007. Fracture statistics of ceramics – Weibull statistics and deviations from Weibull statistics. Engineering Fracture Mechanics 74, 2919–2932. Esposito, L., Gentile, D., Bonora, N., 2007. Investigation on the Weibull parameters identification for local approach application in the ductile to brittle transition regime. Engineering Fracture Mechanics 74, 549–562. Green, D.J., Cai, P.Z., Messing, G.L., 1999. Residual stresses in alumina–zirconia laminates. Journal of the European Ceramic Society 19, 2511–2517. Lei, Y., O’Dowd, N.P., Busso, E.P., Webster, G.A., 1998. Weibull stress solutions for 2D cracks in elastic and elastic–plastic materials. International Journal of Fracture 89, 245–268. Moon, R.J., Hoffman, M., Hilden, J., Bowman, K.J., Trumble, K.P., Rödel, J., 2002. R-curve behavior in alumina–zirconia composites with repeating graded layers. Engineering Fracture Mechanics 69, 1647–1665. Munz, D., Fett, T., 1999. Ceramics. Springer, Berlin. Sergo, V., Lipkin, M., De Portu, G., Clarke, D.R., 1997. Edge stresses in alumina–zirconia laminates. Journal of American Ceramic Society 80, 1633–1638. Sglavo, V.M., Bertoldi, M., 2006. Design and production of ceramic laminates with high mechanical resistance and reliability. Acta Materialia 54, 4929–4937. Sglavo, V.M., Larentis, L., Green, J.L., 2001. Flaw insensitive ion-exchanged glass: I. Theoretical aspects. Journal of America Ceramic Society 84, 1827–1831. Vena, P., 2005. Thermal residual stresses in graded ceramic composites: a microscopic computational model versus homogenized models. Meccanica 40, 163–179. Vena, P., Gastaldi, D., Contro, R., 2005. Effects of the thermal residual stress field on the crack propagation in graded alumina–zirconia ceramics. Material Science Forum 492–493, 177–182. 582 P. Vena et al. / Mechanics Research Communications 35 (2008) 576–582