Availableonlineatwww.sciencedirect.co ScienceDirect Acta materialia ELSEVIER Acta Materialia 55(2007)409-421 www.actamat-journals.com Prediction of the fracture toughness of a ceramic multilayer composite Modeling and experiments C R Chen a. Pascual b F D. Fischer .o. Kolednik d, *R. Danzer b Materials Center Leoben Forschung GmbH, Franz-Josef-Strasse 13, A-8700 Leoben, austria truktur und Funktionskeramik. Montanumiversitat Leoben. Peter. Tunner. Strasse 5.A-8700 Leoben. austria Institute of Mechanics, Montanumirersitat Leoben, Franz-Josef-Strasse 18, A-8700 d erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Jahnstrasse 12, 4-8700 Leoben, Austria Received 24 March 2006: received in revised form 28 June 2006: accepted 2 July 2006 Available online 7 November 2006 Abstract equires experiments to measure the intrinsic fracture toughness of the phases and to determine the required material data. The numerical modeling includes a conventional finite element stress analysis and the calculation of the crack driving force based on the concept of configurational(material)forces. The procedure yields the fracture toughness of the composite as a function of the crack length. a bend- bar consisting of layers made of alumina and an alumina-zirconia composite is investigated The bar has a crack perpendicular to the interfaces. The spatial variations of both the thermal residual stresses and the elastic modulus induce shielding and anti-shielding effects on the crack, which are quantified. The numerically predicted fracture toughness is compared with the experimental values. o 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved Keywords: Ceramics: Multilayer; Fracture toughness; Crack tip shielding: Thermal residual stresses 1. Introduction oxide fuel cells [8] but also in structural applications such as dental restorations or hip replacements [5] To improve the performance against brittle failure in A prerequisite for the application of multilayer ceramic materials, strategies have been developed in recent is the understanding of their resistance against crack initi- years to design tough and strong ceramics consisting of ation and propagation. Different authors have predicted multilayers. These strategies include the tailoring of weak the toughening effect of the residual stress state by means interfaces for crack deflection or the design of multilayers of the weight function method [1, 3, 9, 10]. They used the with compressive residual stresses in the outer layers [1]. classical weight function concept to calculate the stress The beneficial consequences of compressive stresses at the intensity factor, considering an inhomogeneous distribu rface are well known: increases in strength, fracture tion of the residual stresses in a homogeneous body (mostly oughness and reliability [2, 3]. Additionally, the wear resis- with the elastic modulus of the first layer). According to tance is enhanced [4]. Therefore, such ceramics receive seri- Fett et al. [11, 12] an approximate weight function method ous consideration for structural application [5-7]. Ceramic can also be applied to heterogeneous, graded or laminated multilayers have already found functional applications in materials with a variable Youngs modulus. substrates for low-load-bearing integrated circuits or solid The immanent inhomogeneity of the material, however causes implications which are not taken into account by the weight function method: spatially varying material proper- Tel: +43 114: fax: +43 3842804116. ties induce an additional crack driving force term. The nik(aunileoben ac at(O. Kolednik ) propagation of a crack in a direction orthogonal to the 1359-6454/$30.00 O 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:l0.1016 actant200607.046
Prediction of the fracture toughness of a ceramic multilayer composite – Modeling and experiments C.R. Chen a , J. Pascual b , F.D. Fischer c , O. Kolednik d,*, R. Danzer b a Materials Center Leoben Forschung GmbH, Franz-Josef-Strasse 13, A-8700 Leoben, Austria b Institut fu¨r Struktur- und Funktionskeramik, Montanuniversita¨t Leoben, Peter-Tunner-Strasse 5, A-8700 Leoben, Austria c Institute of Mechanics, Montanuniversita¨t Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria d Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Jahnstrasse 12, A-8700 Leoben, Austria Received 24 March 2006; received in revised form 28 June 2006; accepted 2 July 2006 Available online 7 November 2006 Abstract A procedure to predict the fracture toughness of a ceramic multilayer composite made of different phases is presented. The procedure requires experiments to measure the intrinsic fracture toughness of the phases and to determine the required material data. The numerical modeling includes a conventional finite element stress analysis and the calculation of the crack driving force based on the concept of configurational (material) forces. The procedure yields the fracture toughness of the composite as a function of the crack length. A bending bar consisting of layers made of alumina and an alumina–zirconia composite is investigated. The bar has a crack perpendicular to the interfaces. The spatial variations of both the thermal residual stresses and the elastic modulus induce shielding and anti-shielding effects on the crack, which are quantified. The numerically predicted fracture toughness is compared with the experimental values. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Ceramics; Multilayer; Fracture toughness; Crack tip shielding; Thermal residual stresses 1. Introduction To improve the performance against brittle failure in ceramic materials, strategies have been developed in recent years to design tough and strong ceramics consisting of multilayers. These strategies include the tailoring of weak interfaces for crack deflection or the design of multilayers with compressive residual stresses in the outer layers [1]. The beneficial consequences of compressive stresses at the surface are well known: increases in strength, fracture toughness and reliability [2,3]. Additionally, the wear resistance is enhanced [4]. Therefore, such ceramics receive serious consideration for structural application [5–7]. Ceramic multilayers have already found functional applications in substrates for low-load-bearing integrated circuits or solid oxide fuel cells [8], but also in structural applications such as dental restorations or hip replacements [5]. A prerequisite for the application of multilayer ceramics is the understanding of their resistance against crack initiation and propagation. Different authors have predicted the toughening effect of the residual stress state by means of the weight function method [1,3,9,10]. They used the classical weight function concept to calculate the stress intensity factor, considering an inhomogeneous distribution of the residual stresses in a homogeneous body (mostly with the elastic modulus of the first layer). According to Fett et al. [11,12], an approximate weight function method can also be applied to heterogeneous, graded or laminated materials with a variable Young’s modulus. The immanent inhomogeneity of the material, however, causes implications which are not taken into account by the weight function method: spatially varying material properties induce an additional crack driving force term. The propagation of a crack in a direction orthogonal to the 1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.07.046 * Corresponding author. Tel.: +43 3842 804 114; fax: +43 3842 804 116. E-mail address: kolednik@unileoben.ac.at (O. Kolednik). www.actamat-journals.com Acta Materialia 55 (2007) 409–421
C R Chen et al. Acta Materialia 55(2007)409-421 laminate planes can be promoted (anti-shielding) or diamond paste. The procedure yields a notch with a tip retarded(shielding) by the different elastic properties(elas- radius of about 15 um(measured at the lateral surfaces tic mismatch) of the laminae [1]. The spatially varying before fracture). This should assure a reliable fracture residual stress state can have a similar effect [13]. toughness measurement [15](see below ). Four-point-bend- The goals of this paper are, therefore, to present an ing tests were conducted in a commercial Zwick Z010 alternative procedure -the method of configurational (or machine under room conditions (34% relative humidity material) forces- to predict the fracture toughness of a and a temperature of 20C) ceramic multilayer composite which takes these spatial The fracture toughness Kic was determined from the inhomogeneities into account. These numerical predictions expression [14] will then be compared with experimental fracture tough less data. Furthermore, we will outline a generally applica- K F S,-s, 3 (1) ble concept of how to exploit the method of configurational Bh h 2(1-7) forces which would also be applicable for the design of lay- with ered composite materials with improved fracture resistance 2. Materials and processing Tapes of alumina(A)and an alumina-zirconia compos- Y=1.9887-13267 3.49-0687+1.35y2)(1-y) e(az were produced by tape casting at ISTEC Faenza, Italy. The AZ composite consists of 60 vol. Al_O3 and 40 vol %3Y-TZP, which is ZrO, with 3 mol %Y,O3. The In the above relations, F is the fracture load and S, sheets were alternatively stacked, forming a multilayer that (=20 mm) and S2(=10 mm) are the support span lengths was sintered at 1550C. As result, a seven-layer laminate The initial crack length, a, was measured after the tests, ta- was obtained with the structure A/AZ/A/AZ/A/AZ/A, in ken as an average of three measurements on the fracture which the thickness of the individual A- and AZ-layers surface are 190 and 220 um, respectively. More details about pro- Three specimens of the multilayer composite with vari- cessing can be found elsewhere, especially data about the ous initial crack lengths were tested(see Fig. Ib). In the composition of the slurry necessary to produce the tapes [6]. first specimen the crack tip was located in the middle of A complete characterization of the material layers was the first layer, in the second specimen it was shortly behind performed. The results are presented in Table 1. The elastic and in the third shortly beyond the interface with the sec- onstants, Youngs modulus E and Poisson's ratio v, were ond layer (interface 1). The initial geometries and the measured at room temperature by an impulse excitation results of the experiments are listed in Table 2. As well as chnique. The coefficient of thermal expansion( CTE)a the Kic values, the loads at fracture Ffr are also given. was measured between room temperature and the reference Additionally, approximate JIc values are listed which are temperature 1160C by means of a dilatometer (see calculated from the relation J=(1-22)K/E, using vol- below). The average CtE values are listed in Table 1. Also ume-averaged values of Poissons ratio, v=0. 25, and presented are the total thickness of the four A-layers (A, Youngs modulus, E=375 GPa nd the three Az-layers tAz in the multilayer are given The intrinsic fracture toughness Ko of the A and aZ which are needed in Appendix I material was determined by testing multilayered homoge- neous specimens consisting of only A-an 3. Fracture mechanics experiments respectively. Indentation toughness values following [16] were determined, and these are presented in Table 1. In The fracture toughness of the composite was measured addition, VAMAS-ESIS experiments were also performed on single edge V-notched beams, following the VAMAs- For the A-layers the VAMAS-ESIS procedure gave exactly ESIS procedure [14]. Bar-shaped specimens with length the same value as the indentation procedure: K0=3.8+0.3 L=28 mm, width h= 1.42 mm and thickness B 2.5 mm MPa vm. For the AZ-layers, however, the VAMAS-ESIS were cut from the original plates using a diamond saw(see values show a big scatter and higher mean value: KAZ Fig. la). The notches were machined in a home-made auto- 5.4+ 1.0 MPa vm compared with K0=4.3+0.1 matic device which uses a razor blade sprinkled with MPa vm from the indentation tests. There are two possi- Table l propertIes f(mm) E(GPa) x(10-6K-) Ko(MPa vm) AlO3(A 392±5 0.24±0.04 8.64±0.03 3.8±0.2 35±2 AlO -ZrO(AZ 0.66 305±4 0.26±0.03 9.24±0.02 4.3±0.1 7士3
laminate planes can be promoted (anti-shielding) or retarded (shielding) by the different elastic properties (elastic mismatch) of the laminae [1]. The spatially varying residual stress state can have a similar effect [13]. The goals of this paper are, therefore, to present an alternative procedure – the method of configurational (or material) forces – to predict the fracture toughness of a ceramic multilayer composite which takes these spatial inhomogeneities into account. These numerical predictions will then be compared with experimental fracture toughness data. Furthermore, we will outline a generally applicable concept of how to exploit the method of configurational forces which would also be applicable for the design of layered composite materials with improved fracture resistance. 2. Materials and processing Tapes of alumina (A) and an alumina–zirconia composite (AZ) were produced by tape casting at ISTEC Faenza, Italy. The AZ composite consists of 60 vol.% Al2O3 and 40 vol.% 3Y-TZP, which is ZrO2 with 3 mol.% Y2O3. The sheets were alternatively stacked, forming a multilayer that was sintered at 1550 C. As result, a seven-layer laminate was obtained with the structure A/AZ/A/AZ/A/AZ/A, in which the thickness of the individual A- and AZ-layers are 190 and 220 lm, respectively. More details about processing can be found elsewhere, especially data about the composition of the slurry necessary to produce the tapes [6]. A complete characterization of the material layers was performed. The results are presented in Table 1. The elastic constants, Young’s modulus E and Poisson’s ratio m, were measured at room temperature by an impulse excitation technique. The coefficient of thermal expansion (CTE) a was measured between room temperature and the reference temperature 1160 C by means of a dilatometer (see below). The average CTE values are listed in Table 1. Also presented are the total thickness of the four A-layers tA, and the three AZ-layers tAZ in the multilayer are given which are needed in Appendix 1. 3. Fracture mechanics experiments The fracture toughness of the composite was measured on single edge V-notched beams, following the VAMAS– ESIS procedure [14]. Bar-shaped specimens with length L = 28 mm, width h = 1.42 mm and thickness B 2.5 mm were cut from the original plates using a diamond saw (see Fig. 1a). The notches were machined in a home-made automatic device which uses a razor blade sprinkled with diamond paste. The procedure yields a notch with a tip radius of about 15 lm (measured at the lateral surfaces before fracture). This should assure a reliable fracture toughness measurement [15] (see below). Four-point-bending tests were conducted in a commercial Zwick Z010 machine under room conditions (34% relative humidity and a temperature of 20 C). The fracture toughness KIC was determined from the expression [14] KIC ¼ F B ffiffiffi h p S1 S2 h 3 ffiffi c p 2ð1 cÞ 1:5 Y ð1Þ with c ¼ a h and Y ¼ 1:9887 1:326c ð3:49 0:68c þ 1:35c2Þcð1 cÞ ð1 þ cÞ 2 ð2Þ In the above relations, F is the fracture load and S1 (=20 mm) and S2 (=10 mm) are the support span lengths. The initial crack length, a, was measured after the tests, taken as an average of three measurements on the fracture surface. Three specimens of the multilayer composite with various initial crack lengths were tested (see Fig. 1b). In the first specimen the crack tip was located in the middle of the first layer, in the second specimen it was shortly behind and in the third shortly beyond the interface with the second layer (interface 1). The initial geometries and the results of the experiments are listed in Table 2. As well as the KIC values, the loads at fracture Ffr are also given. Additionally, approximate JIC values are listed which are calculated from the relation J ¼ ð1 m2ÞK2 =E, using volume-averaged values of Poisson’s ratio, m ¼ 0:25, and Young’s modulus, E ¼ 375 GPa. The intrinsic fracture toughness K0 of the A and AZ material was determined by testing multilayered homogeneous specimens consisting of only A- and AZ-layers, respectively. Indentation toughness values following [16] were determined, and these are presented in Table 1. In addition, VAMAS–ESIS experiments were also performed. For the A-layers the VAMAS–ESIS procedure gave exactly the same value as the indentation procedure: KA 0 ¼ 3:8 0:3 MPa pm. For the AZ-layers, however, the VAMAS–ESIS values show a big scatter and higher mean value: KAZ 0 ¼ 5:4 1:0 MPa pm compared with KAZ 0 ¼ 4:3 0:1 MPa pm from the indentation tests. There are two possiTable 1 Material properties Material t (mm) E (GPa) m (–) a (106 K1 ) K0 (MPa pm) J0 (J/m2 ) Al2O3 (A) 0.76 392 ± 5 0.24 ± 0.04 8.64 ± 0.03 3.8 ± 0.2 35 ± 2 Al2O3–ZrO2 (AZ) 0.66 305 ± 4 0.26 ± 0.03 9.24 ± 0.02 4.3 ± 0.1 57 ± 3 410 C.R. Chen et al. / Acta Materialia 55 (2007) 409–421
C R Chen et al. Acta Materialia 55(2007)409-421 919922 h=1.42 mm aX Cross-section F/2 symmetry plane Longitudinal section z=0 Fig. 1. Geometry of the four-point-bending test arrangement of the laminate specimen consisting of layers of A- and AZ-material Results of the fracture toughness tests on the multilayer composite Tip location Fr(N IC(M 117 n AZ 8.4 ble explanations for this difference. The first is that some lie in planes parallel to the y-z plane. The arrangement cutting-induced phase transformation occurs when produc- and thickness of the laminae can be taken from the long ing the notch. In the machine used, the force used while tudinal section at z=0(Fig. lb). In this longitudinal sec cutting the notch cannot be controlled. The second possible tion, an integration path I is marked surrounding a reason for the discrepancy is that the notch radius of about rectangle Q2far with the area hLy. The length Ly will be var 5 um is certainly small enough for the alumina, with its ied to obtain different paths T. The six interfaces, num- grain size of approximately 5 um, but it is quite large for bered from 1 to 6, intersect the integration path I at the AZ-microstructure, which has a grain size of approxi- y=0 and y= Ly. The normal unit vectors to the interfaces mately 0.7 um. This could cause an overestimation of the as well as to the integration path I are designated n. The fracture toughness. Therefore, to be on the safe side, we crack with variable length a is located in the plane y=0; decided to use the lower value of the indentation measure- the crack front is assumed to be parallel to the line x=0 ment in the paper. Fig. Ic presents a cross-section at y=0. The average Ko values and the corresponding Jo values are also listed in Table 1. It is seen that the fracture tough- 4.2. The residual stress state ness values of the composite are significantly larger than the corresponding intrinsic values of the A and Az-materials The specimens are fabricated at high temperatures. Due It should be noted here that the Jo values characterize the to the difference in the Cte, a cooling from the sintering fracture initiation toughness of the materials; in the ceramic temperature to the room temperature 20C introduces a ommunity the term"fracture energy is often used residual stress state(see e.g. [I]. At high temperatures, relaxation processes prevent the development of residual 4. Numerical modeling stresses. The main reason is that the az-material exhibits extensive plasticity between 1200C and the sintering tem- 4. 1. Description of the model perature [17]. Therefore, an upper reference temperature is the effective temperature The global setting is depicted in Fig. la. The laminate is calculated. This reference temperature was determined in beam is supported at a distance S, of 20 mm and loaded [18]as Tref=1160C, leading to an effective temperature by a pair of loads at a distance S2 of 10 mm. The laminae difference AT=-l140C
ble explanations for this difference. The first is that some cutting-induced phase transformation occurs when producing the notch. In the machine used, the force used while cutting the notch cannot be controlled. The second possible reason for the discrepancy is that the notch radius of about 15 lm is certainly small enough for the alumina, with its grain size of approximately 5 lm, but it is quite large for the AZ-microstructure, which has a grain size of approximately 0.7 lm. This could cause an overestimation of the fracture toughness. Therefore, to be on the safe side, we decided to use the lower value of the indentation measurement in the paper. The average K0 values and the corresponding J0 values are also listed in Table 1. It is seen that the fracture toughness values of the composite are significantly larger than the corresponding intrinsic values of the A- and AZ-materials. It should be noted here that the J0 values characterize the fracture initiation toughness of the materials; in the ceramic community the term ‘‘fracture energy’’ is often used. 4. Numerical modeling 4.1. Description of the model The global setting is depicted in Fig. 1a. The laminate beam is supported at a distance S1 of 20 mm and loaded by a pair of loads at a distance S2 of 10 mm. The laminae lie in planes parallel to the y z plane. The arrangement and thickness of the laminae can be taken from the longitudinal section at z =0 (Fig. 1b). In this longitudinal section, an integration path C is marked surrounding a rectangle Xfar with the area hLy. The length Ly will be varied to obtain different paths C. The six interfaces, numbered from 1 to 6, intersect the integration path C at y = 0 and y = Ly. The normal unit vectors to the interfaces as well as to the integration path C are designated n. The crack with variable length a is located in the plane y = 0; the crack front is assumed to be parallel to the line x = 0. Fig. 1c presents a cross-section at y = 0. 4.2. The residual stress state The specimens are fabricated at high temperatures. Due to the difference in the CTE, a cooling from the sintering temperature to the room temperature 20 C introduces a residual stress state (see e.g. [1]). At high temperatures, relaxation processes prevent the development of residual stresses. The main reason is that the AZ-material exhibits extensive plasticity between 1200 C and the sintering temperature [17]. Therefore, an upper reference temperature is introduced from which the effective temperature difference is calculated. This reference temperature was determined in [18] as Tref = 1160 C, leading to an effective temperature difference DT = 1140 C. Fig. 1. Geometry of the four-point-bending test arrangement of the laminate specimen consisting of layers of A- and AZ-material. Table 2 Results of the fracture toughness tests on the multilayer composite Specimen B (mm) h (mm) a (mm) Tip location Ffr (N) KIC (MPa pm) J appr IC ðJ=m2 Þ 1 2.72 1.42 0.10 In A 117 6.1 98 2 2.64 1.42 0.21 In AZ 110 8.4 188 3 2.64 1.42 0.18 In A 103 7.3 142 C.R. Chen et al. / Acta Materialia 55 (2007) 409–421 411
C R Chen et al. Acta Materialia 55(2007)409-421 The calculation procedure to find the residual stress and aaz= aAZ -A=0.6x 10-K- as the substitute state in laminate specimens has been heavily investigated CTE in the AZ-material. The second 2D computation used (see e.g. [19, 20]. We use a three-dimensional (3D) model the corresponding plane stress model of the specimen, consisting of a slice of length 8 in the y- direction and covering the area0≤x≤M2,0≤z≤B/2.4.3. Beam bending Symmetry conditions are applied along 2=0 and x= h/ 2, and the y-displacements are fixed on the bottom of the As outlined above, four-point-bend tests are performed layer. Only in the end regions far from the crack plane will on rather slender beams. The classical beam-bending the the residual stress state depend on the y-coordinate. Since ory could be used to evaluate the stress state the specimen is very long in y-direction, the residual stres- uncracked composite beam; for details, see e.g ses can be assumed to be independent of the y-coordinate. Since we need to determine the stress state in the specimen Therefore, an unknown but spatially constant displace- with a crack of length a, finite element calculations are per ment u] is assumed along the upper boundary y=8. The formed. The beam is replaced by a two-dimensional plane finite element program system ABAQUS(htp!∥ strain model covering the area0≤x≤ h and y≥0.Note www.hks.com)isengagedforthecomputationusingthattheplanestrainmodelcanbetreatedasaplanestress eight-node 3D solid elements model by replacing Youngs modulus E by E= E/(1-v) It is well known that the behavior of a crack is deter- Only x-displacements ux are allowed at y=0. The speci mined not only by the stresses perpendicular to the crack men is fixed at the point P, in the x-direction; the load F plane y=0. The in-plane and out-of-plane constraints also is applied at point P2(see Fig. 1). The mesh consists of play a role, i.e. the stresses in x-and z-directions. The mate- eight-node plane strain elements rial can freely move in x-direction, so that no correspond To model a realistic stress state of the fracture mechan- g residual stresses will appear In the z-direction residual ics specimens, the following procedure is applied.First, the stresses o, Ar will appear which infiuence the strain energy uncracked and unloaded specimen is subjected to a thermal loading by a temperature difference AT=-1140C to cal The materials of the A- and AZ-layers are modeled as culate the thermal residual stresses. Then in the unloaded linear-elastic. The corresponding material properties are specimen a crack of length a is introduced by a node release taken from Table 1. The longitudinal residual stresses o, Ar technique. Subsequently, the specimen is loaded by pre in the symmetry plane z=0 and in the side-surface plane scribing the load Fat the load application point. The final z= B/2 are plotted between 0<x<h/2 in Fig. 2. Fig. 3 stress and strain distribution within the specimen is used shows the variation of o, Ar from the midsection to the for the evaluation of the crack driving force, which is side-surface for sections in the middle of the various described beloy laminae In addition to the 3D computation, two simple 2D com- 4. 4. Calculation of the crack driving force putations were also performed. The first one used a plane strain model covering the area0≤x≤h,0≤y≤ s with The study of stress intensity factors, as well as of their assumes no displacement in the 2-direction, u,=0. To mechanics research in composite materials (see e. g Simha, Kolednik et al. [13, 26-28]. References to the open formulations can be taken from these extensive papers Specifically, Section 3 of [28] provides the corresponding equations which are reshaped below in the specific form a composites with constant material properties within each lamina 。-atz=1.25of3 D model The concept of configurational forces considers a mate- rial inhomogeneity as an additional defect in the material (besides the crack) which induces an additional contribu- tion to the crack driving force. This contribution has been plane stress b called the material inhomogeneity term Cinh. The thermo- dynamic force at the crack tip, denominated as the local near-tip crack driving force Jtip, is the sum of the nominally x[ mm] applied far-field crack driving force Far and the material inh [28]: Fig. 2. Thermal residual stresses o,. AT along the x-direction at 2=0 and 25 mm for a plane stress, plane strain and three-dimensional (3D)model Jtip=Far Ci
The calculation procedure to find the residual stress state in laminate specimens has been heavily investigated (see e.g. [19,20]). We use a three-dimensional (3D) model of the specimen, consisting of a slice of length d in the ydirection and covering the area 0 6 x 6 h/2, 0 6 z 6 B/2. Symmetry conditions are applied along z = 0 and x = h/ 2, and the y-displacements are fixed on the bottom of the layer. Only in the end regions far from the crack plane will the residual stress state depend on the y-coordinate. Since the specimen is very long in y-direction, the residual stresses can be assumed to be independent of the y-coordinate. Therefore, an unknown but spatially constant displacement uy is assumed along the upper boundary y = d. The finite element program system ABAQUS (http:// www.hks.com) is engaged for the computation, using eight-node 3D solid elements. It is well known that the behavior of a crack is determined not only by the stresses perpendicular to the crack plane y = 0. The in-plane and out-of-plane constraints also play a role, i.e. the stresses in x- and z-directions. The material can freely move in x-direction, so that no corresponding residual stresses will appear. In the z-direction residual stresses rz,DT will appear which influence the strain energy density (see Appendix 1). The materials of the A- and AZ-layers are modeled as linear-elastic. The corresponding material properties are taken from Table 1. The longitudinal residual stresses ry,DT in the symmetry plane z = 0 and in the side-surface plane z = B/2 are plotted between 0 6 x 6 h/2 in Fig. 2. Fig. 3 shows the variation of ry,DT from the midsection to the side-surface for sections in the middle of the various laminae. In addition to the 3D computation, two simple 2D computations were also performed. The first one used a plane strain model covering the area 0 6 x 6 h, 0 6 y 6 d with unit thickness in the z-direction. The plane strain model assumes no displacement in the z-direction, uz ” 0. To avoid any stresses due to the global shrinkage of the specimen, we set a A ¼ 0 as the substitute CTE in the A-material and a AZ ¼ aAZ aA ¼ 0:6 106 K1 as the substitute CTE in the AZ-material. The second 2D computation used the corresponding plane stress model. 4.3. Beam bending As outlined above, four-point-bend tests are performed on rather slender beams. The classical beam-bending theory could be used to evaluate the stress state in the uncracked composite beam; for details, see e.g. [21,22]. Since we need to determine the stress state in the specimen with a crack of length a, finite element calculations are performed. The beam is replaced by a two-dimensional plane strain model covering the area 0 6 x 6 h and y P 0. Note that the plane strain model can be treated as a plane stress model by replacing Young’s modulus E by E* = E/(1 m 2 ). Only x-displacements ux are allowed at y = 0. The specimen is fixed at the point P1 in the x-direction; the load F is applied at point P2 (see Fig. 1). The mesh consists of eight-node plane strain elements. To model a realistic stress state of the fracture mechanics specimens, the following procedure is applied. First, the uncracked and unloaded specimen is subjected to a thermal loading by a temperature difference DT = 1140 C to calculate the thermal residual stresses. Then in the unloaded specimen a crack of length a is introduced by a node release technique. Subsequently, the specimen is loaded by prescribing the load F at the load application point. The final stress and strain distribution within the specimen is used for the evaluation of the crack driving force, which is described below. 4.4. Calculation of the crack driving force The study of stress intensity factors, as well as of their relevance for crack growth, has been a topic of fracture mechanics research in composite materials (see e.g. [1,23–25]). In the current investigation the concept of con- figurational forces is used. Here we refer to the works by Simha, Kolednik et al. [13,26–28]. References to the open literature with respect to this concept and other related formulations can be taken from these extensive papers. Specifically, Section 3 of [28] provides the corresponding equations which are reshaped below in the specific form for composites with constant material properties within each lamina. The concept of configurational forces considers a material inhomogeneity as an additional defect in the material (besides the crack) which induces an additional contribution to the crack driving force. This contribution has been called the material inhomogeneity term Cinh. The thermodynamic force at the crack tip, denominated as the local, near-tip crack driving force Jtip, is the sum of the nominally applied far-field crack driving force Jfar and the material inhomogeneity term Cinh [28]: Jtip ¼ Jfar þ Cinh: ð3Þ Fig. 2. Thermal residual stresses ry,DT along the x-direction at z = 0 and 1.25 mm for a plane stress, plane strain and three-dimensional (3D) model. 412 C.R. Chen et al. / Acta Materialia 55 (2007) 409–421
C R Chen et al. Acta Materialia 55(2007)409-421 plane strain ri plane stress plane strain z[mm Fig 3. Thermal residual stresses , AT along the z-direction for various values of y for a plane stress, plane strain and three-dimensional (3D) model. Jfar is the classical J-integral of fracture mechanics. For a done by specifying the set of nodes on the interface as crack growing in the x-direction, Jfar is virtual crack tip nodes. Even a contour directly adjacent to the interface yields very accurate results. For the eval- (4) uation of the J-integral, the virtual crack growth direc tion must be specified; this is the(,0)-direction, as for The components t;(t;=fr,fy)of the traction vector t along he evaluation of Je the contour I follow from the stress tensor g with the rel-. Generally, both Far and Cinh depend on the crack length evant components ox, Oy, txy as I=g n. Note that the a. They also depend on Ly, but produce Jtip values stress components are the sum of their contributions due according to Eq .(3)which are independent of L,; for to bending and the residual stress state, e.g. 0,=0x, b+ details see Appendix 2 At. The components u;(u;=ux, ly) of the displacement vector u are differentiated with respect to the crack growth After the finite element stress analysis, the material i direction, i. e. the x-direction. The quantity o is the specific mogeneity term Cinh is calculated from Eq. (5)by a elastic strain energy and nx is the x-component of the unit processing procedure. The integration along the interface normal vector n to the integration path I is performed using the trapezoid formula. For this, the The material inhomogeneity term can be evaluated by node values of the stress and strain components and the strain energy density are taken, which are extrapolated val ues from the Gauss integration points. The far-field J-inte- Cnh=>Cinh i, Cinh =-2/(l,-(g)le)dy (5) gral Jar is calculated using the virtual crack extension method of ABAQUS. Then the near-tip crack driving force The jump [b] and the average (b)of a quantity b at an Jtip is calculated from Eq (3). The numerical results will be interface are defined presented in the following section [b]=b-b,(b)=(b+b) It should be noted that Sun and Wu[29]have calculated the effective crack driving force by replacing the region far where br and b denote the limiting values of the quantity b by subregions, each including only one layer, and applying on the right and left side of the interface, respectively. The the J-integral procedure for each individual layer. The index i refers to the individual interface; the integer I de- strength of the configurational forces concept lies in its gen notes the total number of interfaces in the specimen, in eral applicability. The material inhomogeneity can be our case I =6 either a sharp interface with a discrete jump of the material The following comments may be useful properties or a region where the material properties change continuously. The Cinh-evaluation procedure can be The multiplier 2 in Eqs. (4)and(5)points to the fact that applied to any arbitrary spatial distribution of these mate- only the upper half of a symmetric configuration with rial inhomogeneities in both elastic and elastic-plastic respect to I is considered. materials. In general, the evaluation of Cinh can be per The material inhomogeneity terms Cinh. i can be also formed very accurately. This enables us to evaluate Jtip as found via the J-integral calculation routine provided the sum of Far and Cinh more accurately than it would be by the finite element code by evaluating the J-integral possible from the calculation of Jtip using the conventional around the ith interface Jint. i[26]. In ABAQUS, this is J-evaluation procedures, especially for cases when the
Jfar is the classical J-integral of fracture mechanics. For a crack growing in the x-direction, Jfar is Jfar ¼ 2 Z C /nx ti oui ox ds: ð4Þ The components ti (ti = tx,ty) of the traction vector t along the contour C follow from the stress tensor r with the relevant components rx, ry, sxy as t = r Æ n. Note that the stress components are the sum of their contributions due to bending and the residual stress state, e.g. ry = ry,b + ry,DT. The components ui (ui = ux,uy) of the displacement vector u are differentiated with respect to the crack growth direction, i.e. the x-direction. The quantity / is the specific elastic strain energy and nx is the x-component of the unit normal vector n to the integration path C. The material inhomogeneity term can be evaluated by [28] Cinh ¼ XI i¼1 Cinh;i; Cinh;i ¼ 2 Z Ly 0 ð½½/i hrii½½eiÞdy ð5Þ The jump [[b]] and the average Æbæ of a quantity b at an interface are defined as ½½b ¼ br bl; hbi¼ðbl þ brÞ=2; ð6Þ where br and bl denote the limiting values of the quantity b on the right and left side of the interface, respectively. The index i refers to the individual interface; the integer I denotes the total number of interfaces in the specimen, in our case I = 6. The following comments may be useful: The multiplier 2 in Eqs. (4) and (5) points to the fact that only the upper half of a symmetric configuration with respect to C is considered. The material inhomogeneity terms Cinh,i can be also found via the J-integral calculation routine provided by the finite element code by evaluating the J-integral around the ith interface Jint,i [26]. In ABAQUS, this is done by specifying the set of nodes on the interface as virtual crack tip nodes. Even a contour directly adjacent to the interface yields very accurate results. For the evaluation of the J-integral, the virtual crack growth direction must be specified; this is the (1, 0)-direction, as for the evaluation of Jfar. Generally, both Jfar and Cinh depend on the crack length a. They also depend on Ly, but produce Jtip values according to Eq. (3) which are independent of Ly; for details see Appendix 2. After the finite element stress analysis, the material inhomogeneity term Cinh is calculated from Eq. (5) by a postprocessing procedure. The integration along the interface is performed using the trapezoid formula. For this, the node values of the stress and strain components and the strain energy density are taken, which are extrapolated values from the Gauss integration points. The far-field J-integral Jfar is calculated using the virtual crack extension method of ABAQUS. Then the near-tip crack driving force Jtip is calculated from Eq. (3). The numerical results will be presented in the following section. It should be noted that Sun and Wu [29] have calculated the effective crack driving force by replacing the region Xfar by subregions, each including only one layer, and applying the J-integral procedure for each individual layer. The strength of the configurational forces concept lies in its general applicability. The material inhomogeneity can be either a sharp interface with a discrete jump of the material properties or a region where the material properties change continuously. The Cinh-evaluation procedure can be applied to any arbitrary spatial distribution of these material inhomogeneities in both elastic and elastic-plastic materials. In general, the evaluation of Cinh can be performed very accurately. This enables us to evaluate Jtip as the sum of Jfar and Cinh more accurately than it would be possible from the calculation of Jtip using the conventional J-evaluation procedures, especially for cases when the Fig. 3. Thermal residual stresses ry,DT along the z-direction for various values of y for a plane stress, plane strain and three-dimensional (3D) model. C.R. Chen et al. / Acta Materialia 55 (2007) 409–421 413
C R Chen et al. Acta Materialia 55(2007)409-421 crack tip comes close to an interface. All these points show Homogeneous materal with and without residual stress the advantages of the configurational force concept over the conventional J-integral approach 5. Results and discussion In the following, the results of the numerical analyses for our multilayer composite are presented. Corresponding of the results are shown in Appendix In Appendix ion strain energy density and the thermal residual stresses are ●a=0.18mm derived for the composite under plane strain conditions. a=0. In Appendix 2 useful analytical estimates of the crack driv ing force are given 51. Thermal residual stresses crack driving force materia A )t wis nd In Figs. 2 and 3 the results of the 3D and 2D computa tions are compared. As already experienced in a study by Shan et al. [30] nearly 80% of the cross-section measured specific crack lengths, a=0.18 mm (crack tip 0.01 mm in the z-direction shows residual stresses o,Ar according hind the first interface) and a=0.20 mm(crack tip to the plane strain model. In the side-surface region, say 0.01 mm beyond of the first interface). For comparison, within 1.00. An analytical esti- pare Table 1). The fracture load can be estimated as the mate of the strain energy density and the thermal residual intersection point of these horizontal lines with the Jtip ver- stresses for the actual configuration under plane strain con- sus F-curve. Note that the material is rather brittle; it thus ditions is given in Appendix 1. For the data at hand the exhibits only a small process zone in front of the crack tip evaluation yields GAar=-144 MPa and oMAr=166 MPa. where the microscopic processes of micro-crack formation and growth take place, which lead to brittle fracture 5.2. Elastically homogeneous material with inhomogeneity in Therefore, it can be assumed that the fracture resistance CTE of the composite is primarily determined by the material in which the process zone is located. Due to the residual There are two sources of shielding and anti-shielding stre esses specific fracture load of the specimen with effects in our multilayer composite: the spatially varying a=0.20 mm increases from Fr 2.5N/mm to residual stresses and the different elastic moduli of the lam- Ffr A 20 N/mm. For the specimen with a=0.18 mm the inae. To separate the two effects, we will first present the shift of the fracture load is even larger, from results for an elastically homogeneous material with spa- Ffr R 10 N/mm to Ffr R 22 N/mm. tially varying thermal residual stresses. It is assumed that The influence of the crack length a on the crack driving the whole specimen has the elastic properties of material force at a constant loading is shown in Fig 5a and b Plot A. The CtE shows a spatial variation with values of a ted are the effective crack driving force Jtip and the term as defined in Section 4.2. To get in the elastically homoge- Far -Jar(0). Far denotes the far-field J-integral for the leous composite exactly the same residual stresses as composite with crack length a, and the expression Faro) ppear in the elastically inhomogeneous composite, describes the far-field J-integral for the composite with zero Mar =-144 MPa and d,ar=166 MPa, the effective tem- crack length(see Appendix A2. 2). Note that for a loaded perature difference was set to△T=-1007.3°C component which contains residual stresses Faro) is non- In Fig 4 the crack driving force Jtip is plotted against the zero and depends on the length of the integration path specific load F=F/(2B) Presented are the curves for two L,, see Appendix A2. 2 and Eq(A2.3). For a component
crack tip comes close to an interface. All these points show the advantages of the configurational force concept over the conventional J-integral approach. 5. Results and discussion In the following, the results of the numerical analyses for our multilayer composite are presented. Corresponding analytical evaluations which are helpful for the discussion of the results are shown in Appendix. In Appendix 1 the strain energy density and the thermal residual stresses are derived for the composite under plane strain conditions. In Appendix 2 useful analytical estimates of the crack driving force are given. 5.1. Thermal residual stresses In Figs. 2 and 3 the results of the 3D and 2D computations are compared. As already experienced in a study by Shan et al. [30], nearly 80% of the cross-section measured in the z-direction shows residual stresses ry,DT according to the plane strain model. In the side-surface region, say within 1.0 6 z 6 1.25 mm, the residual stresses can be roughly approximated by the plane stress distribution. Directly at the side-surface the absolute values of ry,DT can, however, become significantly smaller than the plane stress values. It should be mentioned that at the positions where the interfaces 1–6 impinge the surface, weak stress singularities for ry,DT occur which can be ignored. Since 80% of the specimen is controlled by plane strain conditions, we keep this simple eigenstress distribution for ry,DT as the relevant one for our further fracture mechanics studies. The thermal residual stresses in the A- and AZ-layers are denoted rA y;DT 0. An analytical estimate of the strain energy density and the thermal residual stresses for the actual configuration under plane strain conditions is given in Appendix 1. For the data at hand the evaluation yields rA y;DT ¼ 144 MPa and rAZ y;DT ¼ 166 MPa. 5.2. Elastically homogeneous material with inhomogeneity in CTE There are two sources of shielding and anti-shielding effects in our multilayer composite: the spatially varying residual stresses and the different elastic moduli of the laminae. To separate the two effects, we will first present the results for an elastically homogeneous material with spatially varying thermal residual stresses. It is assumed that the whole specimen has the elastic properties of material A. The CTE shows a spatial variation with values of a* as defined in Section 4.2. To get in the elastically homogeneous composite exactly the same residual stresses as appear in the elastically inhomogeneous composite, rA y;DT ¼ 144 MPa and rAZ y;DT ¼ 166 MPa, the effective temperature difference was set to DT = 1007.3 C. In Fig. 4 the crack driving force Jtip is plotted against the specific load F b ¼ F =ð2BÞ. Presented are the curves for two specific crack lengths, a = 0.18 mm (crack tip 0.01 mm behind the first interface) and a = 0.20 mm (crack tip 0.01 mm beyond of the first interface). For comparison, the curves of the completely homogeneous specimen without residual stresses are also given, and these exhibit the common quadratic dependency on the load. The residual stresses shift the origin of the curves. Since the residual stress state in the first layer is a compressive one and the bending stresses ry,b are tensile, the crack will open at a load F0 when the bending stresses balance the residual stresses ry;bmax ¼ rA y;DT ; compare Eq. (A2.6). From this equation, the quadratic dependency of Jtip on (F F0) can be also deduced. In Fig. 4 the intrinsic fracture toughness values of the A- and AZ-material are indicated (compare Table 1). The fracture load can be estimated as the intersection point of these horizontal lines with the Jtip versus F b-curve. Note that the material is rather brittle; it thus exhibits only a small process zone in front of the crack tip where the microscopic processes of micro-crack formation and growth take place, which lead to brittle fracture. Therefore, it can be assumed that the fracture resistance of the composite is primarily determined by the material in which the process zone is located. Due to the residual stresses, the specific fracture load of the specimen with a = 0.20 mm increases from F bfr ¼ 12:5 N=mm to F bfr 20 N=mm. For the specimen with a = 0.18 mm the shift of the fracture load is even larger, from F bfr 10 N=mm to F bfr 22 N=mm. The influence of the crack length a on the crack driving force at a constant loading is shown in Fig. 5a and b. Plotted are the effective crack driving force Jtip and the term Jfar Jfar(0). Jfar denotes the far-field J-integral for the composite with crack length a, and the expression Jfar(0) describes the far-field J-integral for the composite with zero crack length (see Appendix A2.2). Note that for a loaded component which contains residual stresses Jfar(0) is nonzero and depends on the length of the integration path Ly; see Appendix A2.2 and Eq. (A2.3). For a component 0 2 4 6 8 10 12 14 16 18 20 22 24 0 5 10 15 20 25 30 35 40 45 50 55 60 J0 (A) J0 (AZ) with residual stress a = 0.18 mm a = 0.20 mm Homogeneous material with and without residual stress no residual stress a = 0.18 mm a = 0.20 mm Jtip [J/m2 ] F/2B [N/mm] Fig. 4. Effective crack driving force Jtip as a function of the specific loading F/2B for the homogeneous specimen (material A) with and without residual stresses. 414 C.R. Chen et al. / Acta Materialia 55 (2007) 409–421
C R Chen et al. Acta Materialia 55(2007)409-421 Homogeneous material with and without residual stress fold effect of the residual stresses the first effect is that 6=20 Nmm compared with the fully homogeneous material, the term Jfar faro) is generally reduced. This is due to the com- pressive residual stresses in the outer layer which restrain the opening of the crack. The second effect arises from with residual stress the shielding(Fig. 5a)or anti-shielding(Fig. 5b) of the crack tip due to the inhomogeneity of the residual stress distribution in the specimen. This makes the effective crack driving force Jtip differ from the term Far -Far(0). Eq. (3) can be extended to the relation J tip-Ufar-Jfar())=Cinh-Cinh(0) where Cinh(0) denotes the material inhomogeneity term for a component with zero crack length. Note that the terms amy o-15020025 Far(0)and Cinh(0) are used in the figures only to make the Far and Cinh values path-independent, and that the relation Far(0)=-Cinh(O) holds, see Appendix A2.2 and Homogeneous materi and without residual stress Eq(A2.5). The material inhomogeneity term Cinh reaches F2B= 10 N/mm a local extremum if the crack just penetrates an interface The material inhomogeneity term Cinh is negative and J and J crack tip shielding occurs for a crack located at interface l; a positive Cinh and strong anti-shielding occurs for a crack located at interfac 5.3. Multilayer composite with inhomogeneity in elastic modulus and CTE In this section the numerical results of the actual multi layer composite are presented and compared with the experimental results. Besides the spatially var stresses due to the spatial CTE-variation, also the different 025 elastic moduli of the a-and Az-laminae influence the frac- ture behavior. Fig. 6 shows the Jtip vs. F curves for speci- Fig. 5. n and the path-independent far-field J-integral term J -Jao) mens with a=0. 18 mm and a=0.20 mm. The curves of a function of the crack length a for the homogeneous specimen with and the elastically inhomogeneous specimen without residual without residual stresses. (a) For F/B=20N/mm;(b) for F/2B= stresses are also given. The comparison with Fig 4 delivers the following findings: The inhomogeneity of the elastic modulus does not influence the origin of the curves however, it generally increases the slopes of the curves so with a crack, Jfar(a) depends also on Ly, but the term Jfar -Far(O) is path-independent; see Eq(A2.6) For comparison, the Jtip and Far VS a curves of the com pletely homogeneous specimen without residual stresses are Iso given. The two curves coincide and Far(0)=0. Fig 5a lows the curves for a crack close to interface 1 and F=20 N/mm; Fig 5b shows the curves for a crack close to interface 2 and F=10 N/mm. (The loads were chosen so that j in has a realistic size. not far from the size of the intrinsic fracture toughness values. )As deduced in Eq (A2.7), the effective crack driving force tip shows an approximately linear dependence on the crack length a The small deviation from non-linearity appears since the 亠a= parameter K(see Appendix A2.3, Eqs.(A2.7) and(A2. 12) is slightly dependent on the crack length a. Note that 1618202224 Jtip Far t Cinh does not depend on the length of the inte- gration path Ly, and that, for a component with zero crack Fig. 6. Effective crack driving force Juip as a function of F/2B for the length, JtiplO)=0; see Eq(A2.5). The curves reveal a two- elastically inhomogeneous composite with and without residual stresses
with a crack, Jfar(a) depends also on Ly, but the term Jfar Jfar(0) is path-independent; see Eq. (A2.6). For comparison, the Jtip and Jfar vs. a curves of the completely homogeneous specimen without residual stresses are also given. The two curves coincide and Jfar(0) = 0. Fig. 5a shows the curves for a crack close to interface 1 and F b ¼ 20 N=mm; Fig. 5b shows the curves for a crack close to interface 2 and F b ¼ 10 N=mm. (The loads were chosen so that Jtip has a realistic size, not far from the size of the intrinsic fracture toughness values.) As deduced in Eq. (A2.7), the effective crack driving force Jtip shows an approximately linear dependence on the crack length a. The small deviation from non-linearity appears since the parameter j (see Appendix A2.3, Eqs. (A2.7) and (A2.12) is slightly dependent on the crack length a. Note that Jtip = Jfar + Cinh does not depend on the length of the integration path Ly, and that, for a component with zero crack length, Jtip(0) = 0; see Eq. (A2.5). The curves reveal a twofold effect of the residual stresses: the first effect is that, compared with the fully homogeneous material, the term Jfar Jfar(0) is generally reduced. This is due to the compressive residual stresses in the outer layer which restrain the opening of the crack. The second effect arises from the shielding (Fig. 5a) or anti-shielding (Fig. 5b) of the crack tip due to the inhomogeneity of the residual stress distribution in the specimen. This makes the effective crack driving force Jtip differ from the term Jfar Jfar(0). Eq. (3) can be extended to the relation Jtip ð Þ¼ Jfar Jfarð0Þ Cinh Cinhð0Þ; ð7Þ where Cinh(0) denotes the material inhomogeneity term for a component with zero crack length. Note that the terms Jfar(0) and Cinh(0) are used in the figures only to make the Jfar and Cinh values path-independent, and that the relation Jfar(0) = Cinh(0) holds, see Appendix A2.2 and Eq. (A2.5). The material inhomogeneity term Cinh reaches a local extremum if the crack just penetrates an interface. The material inhomogeneity term Cinh is negative and crack tip shielding occurs for a crack located at interface 1; a positive Cinh and strong anti-shielding occurs for a crack located at interface 2. 5.3. Multilayer composite with inhomogeneity in elastic modulus and CTE In this section, the numerical results of the actual multilayer composite are presented and compared with the experimental results. Besides the spatially varying residual stresses due to the spatial CTE-variation, also the different elastic moduli of the A- and AZ-laminae influence the fracture behavior. Fig. 6 shows the Jtip vs. F b curves for specimens with a = 0.18 mm and a = 0.20 mm. The curves of the elastically inhomogeneous specimen without residual stresses are also given. The comparison with Fig. 4 delivers the following findings: The inhomogeneity of the elastic modulus does not influence the origin of the curves; however, it generally increases the slopes of the curves so 0.00 0.05 0.10 0.15 0.20 0.25 0 20 40 60 80 100 120 140 160 180 interface 1 no residual stress Jtip and Jfar Homogeneous material with and without residual stress F/2B = 20 N/mm with residual stress Jtip Jfar - Jfar(0) Jtip and Jfar - Jfar(0) [J/m2 ] a [mm] 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0 20 40 60 80 100 120 140 160 180 interface 2 interface 3 no residual stress Jtip and Jfar Homogeneous material with and without residual stress F/2B = 10 N/mm with residual stress Jtip Jfar - Jfar(0) Jtip and Jfar - Jfar(0) [J/m2 ] a [mm] Fig. 5. Jtip and the path-independent far-field J-integral term Jfar Jfar(0) as a function of the crack length a for the homogeneous specimen with and without residual stresses. (a) For F/2B = 20 N/mm; (b) for F/2B = 10 N/mm. 0 2 4 6 8 10 12 14 16 18 20 22 24 0 5 10 15 20 25 30 35 40 45 50 55 60 with residual stress a = 0.18 mm a = 0.20 mm J0 (A) J0 (AZ) Inhomogeneous material with and without residual stress no residual stress a = 0.18 mm a = 0.20 mm Jtip [J/m2 ] F/2B [N/mm] Fig. 6. Effective crack driving force Jtip as a function of F/2B for the elastically inhomogeneous composite with and without residual stresses. C.R. Chen et al. / Acta Materialia 55 (2007) 409–421 415
C R Chen et al. Acta Materialia 55(2007)409-421 that all the fracture loads are decreased by approximately values become singular: Cinh oo and Jtip -oo. Near 10%. interface 2, the shielding effect of the compliant-stiff transi- The dependency of the effective crack driving force Jtip tion induces a negative material inhomogeneity term. For and the term Far -Far(0) on the crack length a at a con- the crack ending directly at the interface, we get stant load is presented in Fig. 7a and b. Fig. 7a shows Cinh --Far and Jtip -0[26,31,32]. For a discussion see the curves for a crack near to interface I and also [33]. F=20 N/mm; Fig. 7b shows the curves for a crack near When comparing the curves of the real composite with to interface 2 and F=10 N/mm. The corresponding E- and CTE- inhomogeneity to those with only the E- curves of the elastically inhomogeneous specimen without inhomogenity and, therefore, without residual stresses, it residual stresses are also plotted. These Jfar vs. a curves can be seen that the thermal residual stresses provoke a are continuous curves which are only slightly bent. The general decrease of the apparent crack driving force comparison with Fig. 5a and b shows that these curves The inhomogeneity of the elastic modulus and the Cte ie above the corresponding curves of the completely inhomogeneity have opposite effects on the material inho- homogeneous specimen. The inhomogeneity of the elastic mogeneity term, but obviously the thermal residual stres modulus induces a material inhomogeneity term which is ses have a much stronger influence on the shielding/anti- positive near interface 1, since the stiff-compliant transition shielding behavior than the modulus inhomogeneity. A (EA>E1%) induces an anti-shielding effect and Jtip >Jfar comparison of Fig. 5 with Fig. 7 shows that, compared [26, 28]. For the crack ending directly at the interface, the with the elastically homogeneous specimen, the Jtip vs. a mogeneous material with and without residual stress F/2B=20 N/mm residual stress with residual stress …d(0) 20 a [mm] Inhomogeneous material with and without residual stress F2B= 10 Nmm no residual stress 8 with residual stress 0.3 35 Fig. 7. Jtip and Far -(0)as a function of the crack length a for the elastically inhomogeneous composite with and without residual stresses. (a)For F/2B=20 N/mm;(b) for F/2B=10N/mm
that all the fracture loads are decreased by approximately 10%. The dependency of the effective crack driving force Jtip and the term Jfar Jfar(0) on the crack length a at a constant load is presented in Fig. 7a and b. Fig. 7a shows the curves for a crack near to interface 1 and F b ¼ 20 N=mm; Fig. 7b shows the curves for a crack near to interface 2 and F b ¼ 10 N=mm. The corresponding curves of the elastically inhomogeneous specimen without residual stresses are also plotted. These Jfar vs. a curves are continuous curves which are only slightly bent. The comparison with Fig. 5a and b shows that these curves lie above the corresponding curves of the completely homogeneous specimen. The inhomogeneity of the elastic modulus induces a material inhomogeneity term which is positive near interface 1, since the stiff-compliant transition (EA > EAZ) induces an anti-shielding effect and Jtip > Jfar [26,28]. For the crack ending directly at the interface, the values become singular: Cinh ! 1 and Jtip ! 1. Near interface 2, the shielding effect of the compliant-stiff transition induces a negative material inhomogeneity term. For the crack ending directly at the interface, we get Cinh ! Jfar and Jtip ! 0 [26,31,32]. For a discussion see also [33]. When comparing the curves of the real composite with E- and CTE-inhomogeneity to those with only the Einhomogenity and, therefore, without residual stresses, it can be seen that the thermal residual stresses provoke a general decrease of the apparent crack driving force. The inhomogeneity of the elastic modulus and the CTE inhomogeneity have opposite effects on the material inhomogeneity term, but obviously the thermal residual stresses have a much stronger influence on the shielding/antishielding behavior than the modulus inhomogeneity. A comparison of Fig. 5 with Fig. 7 shows that, compared with the elastically homogeneous specimen, the Jtip vs. a 0.00 0.05 0.10 0.15 0.20 0.25 0 20 40 60 80 100 120 140 160 180 200 220 240 interface 1 no residual stress Jtip Jfar Inhomogeneous material with and without residual stress F/2B = 20 N/mm with residual stress Jtip Jfar - Jfar(0) Jtip and Jfar - Jfar(0) [J/m2 ] a [mm] 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0 20 40 60 80 100 120 140 160 180 interface 2 interface 3 no residual stress Jtip Jfar Inhomogeneous material with and without residual stress F/2B = 10 N/mm with residual stress Jtip Jfar - Jfar(0) Jtip and Jfar - Jfar(0) [J/m2 ] a [mm] Fig. 7. Jtip and Jfar Jfar(0) as a function of the crack length a for the elastically inhomogeneous composite with and without residual stresses. (a) For F/2B = 20 N/mm; (b) for F/2B = 10 N/mm. 416 C.R. Chen et al. / Acta Materialia 55 (2007) 409–421
C R Chen et al. Acta Materialia 55(2007)409-421 curves of the real composite are generally shifted to F= Ffr can be taken as the apparent fracture toughness higher Jtip values. In addition, the shapes of the curves Jfr of the specimen which would be measured in the differ when the crack approaches the interfaces: near experiment. For an elastically inhomogeneous material interface I, Jtip is enhanced; near interface 2, Jtip is without residual stresses, the far-field J-integral of the specimen with zero crack length Far(0)=0 and, therefore, As discussed in Section 5.2, it can be assumed that the Far is path-independent if the path I crosses all the inter- specimen fractures when Jtip reaches the value of the intrin- faces and if Ly is not too small. In the case of a residual sic fracture toughness Jo of the material where the crack tip stress discontinuity, the size of Far(0) depends on the load is located. With this condition, the fracture loads Frr of F and therefore Jfar becomes strongly path-dependent pecimens with arbitrary crack lengths can be evaluated. This is the reason why the Jfar value which correspond Fr= Fr/(2B)on the crack length a. For comparison, apparent fracture toughness of the specimen. Instead the corresponding curves of the elastically inhomogeneous we use an"apparent far-field J-integral"Jfr which is specimen without residual stresses are also given. Note that determined as follows: first the fracture load Ffr is deter in a homogeneous specimen Ffr should be proportional to mined so that for the real composite Jtip equals Jo. This a-1/. It is seen that the compressive residual stresses in Fr value is then applied to the model without the thermal layer I greatly increase Fir. This is so not only when the residual stresses and, subsequently, the path-independent crack tip is located in layer l, but also for some crack value of Far is evaluated. This Far value represents the length after crossing interface l For the crack tip in layer apparent fracture toughness Jfr of the specimen with the 2, the tensile residual stresses in the layer make the slope of residual stress state the Ffr vS a curve distinctly larger than that of the model In Fig. 9 the apparent Jfr values are plotted as a function without the residual stresses. When the crack tip has passed of the crack length a. For comparison, the intrinsic fracture the middle of layer 2, the fracture load falls below the Frr toughness values Jo of the A- and AZ-material and the Jfr value of the model without the residual stresses. After the values of the elastically inhomogeneous specimen without crack tip has penetrated interface 2, the compressive resid- residual stresses are also included. The latter demonstrate ual stresses in layer 3 quickly enhance Ffr over that of the the effect of the inhomogeneity of the elastic modulus on model without the residual stresses. According to this the apparent fracture toughness. This effect is rather small urve, even a stable crack extension should be possible compared with the effect of the thermal residual stresses for an initial crack with its tip close to interface 2. For however, for a crack approaching interface 2, it still leads all other initial crack lengths, unstable crack growth will to an increase of the fracture toughness of about 35% occur and the specimen will fail catastrophically when the The benefit of the compressive thermal residual stresses crack starts to grow in layer I is largest for a crack approaching interface 1 The experimentally measured fracture loads of the spec- for the crack tip in the layer 1, Jfr increases almost line imens 1, 2, and 3 are also indicated in Fig. 8. The experi- with increasing a, reaching a final value more than four mental data match the numerically predicted values that of the intrinsic fracture toughness. After having pene- s For a specimen without residual stresses, the far-field trated interface 1, the apparent fracture toughness J-integral of the specimen evaluated at the fracture load decreases sharply, reaching a minimum value for the crack homogeneous material with and without residual stress material with and without residual stress 4-with residual stress J, with residual stress 去 000.050.100.150200250.300.350.40045050055060 000050.100.150200250.300.35040045050055060 Fig. 8. Numerically predicted and experimentally measured specific Fig. 9. Numerically predicted fracture toughness Jfr, computed for the fracture loads F/2B which make Juip equal to the intrinsic fracture composite with and without thermal residual stresses, and experimentally toughness Jo of the individual layers. measured JIc values
curves of the real composite are generally shifted to higher Jtip values. In addition, the shapes of the curves differ when the crack approaches the interfaces: near interface 1, Jtip is enhanced; near interface 2, Jtip is reduced. As discussed in Section 5.2, it can be assumed that the specimen fractures when Jtip reaches the value of the intrinsic fracture toughness J0 of the material where the crack tip is located. With this condition, the fracture loads Ffr of specimens with arbitrary crack lengths can be evaluated. Fig. 8 shows the dependency of the specific fracture load F bfr ¼ F fr=ð2BÞ on the crack length a. For comparison, the corresponding curves of the elastically inhomogeneous specimen without residual stresses are also given. Note that in a homogeneous specimen Ffr should be proportional to a1/2. It is seen that the compressive residual stresses in layer 1 greatly increase Ffr. This is so not only when the crack tip is located in layer 1, but also for some crack length after crossing interface 1. For the crack tip in layer 2, the tensile residual stresses in the layer make the slope of the Ffr vs. a curve distinctly larger than that of the model without the residual stresses. When the crack tip has passed the middle of layer 2, the fracture load falls below the Ffr value of the model without the residual stresses. After the crack tip has penetrated interface 2, the compressive residual stresses in layer 3 quickly enhance Ffr over that of the model without the residual stresses. According to this curve, even a stable crack extension should be possible for an initial crack with its tip close to interface 2. For all other initial crack lengths, unstable crack growth will occur and the specimen will fail catastrophically when the crack starts to grow. The experimentally measured fracture loads of the specimens 1, 2, and 3 are also indicated in Fig. 8. The experimental data match the numerically predicted values. For a specimen without residual stresses, the far-field J-integral of the specimen evaluated at the fracture load F = Ffr can be taken as the apparent fracture toughness Jfr of the specimen which would be measured in the experiment. For an elastically inhomogeneous material without residual stresses, the far-field J-integral of the specimen with zero crack length Jfar(0) = 0 and, therefore, Jfar is path-independent if the path C crosses all the interfaces and if Ly is not too small. In the case of a residual stress discontinuity, the size of Jfar(0) depends on the load F and therefore Jfar becomes strongly path-dependent. This is the reason why the Jfar value which corresponds to the critical force Ffr cannot be directly taken as the apparent fracture toughness of the specimen. Instead, we use an ‘‘apparent far-field J-integral’’ Jfr which is determined as follows: first the fracture load Ffr is determined so that for the real composite Jtip equals J0. This Ffr value is then applied to the model without the thermal residual stresses and, subsequently, the path-independent value of Jfar is evaluated. This Jfar value represents the apparent fracture toughness Jfr of the specimen with the residual stress state. In Fig. 9 the apparent Jfr values are plotted as a function of the crack length a. For comparison, the intrinsic fracture toughness values J0 of the A- and AZ-material and the Jfr values of the elastically inhomogeneous specimen without residual stresses are also included. The latter demonstrate the effect of the inhomogeneity of the elastic modulus on the apparent fracture toughness. This effect is rather small compared with the effect of the thermal residual stresses; however, for a crack approaching interface 2, it still leads to an increase of the fracture toughness of about 35%. The benefit of the compressive thermal residual stresses in layer 1 is largest for a crack approaching interface 1: for the crack tip in the layer 1, Jfr increases almost linearly with increasing a, reaching a final value more than fourfold that of the intrinsic fracture toughness. After having penetrated interface 1, the apparent fracture toughness decreases sharply, reaching a minimum value for the crack Fig. 8. Numerically predicted and experimentally measured specific fracture loads Ffr/2B which make Jtip equal to the intrinsic fracture toughness J0 of the individual layers. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0 20 40 60 80 100 120 140 160 180 200 220 Experimental JIC-data Inhomogeneous material with and without residual stress interface 1 interface 2 interface 3 Jfr with residual stress Jfr without residual stress J0 J-integral [J/m2 ] a [mm] Fig. 9. Numerically predicted fracture toughness Jfr, computed for the composite with and without thermal residual stresses, and experimentally measured JIC values. C.R. Chen et al. / Acta Materialia 55 (2007) 409–421 417
C R Chen et al. Acta Materialia 55(2007)409-421 tip at interface 2 which is far below the intrinsic toughness. Appendix 1. Analytical estimate of the thermal residual When the crack tip is in layer 3, Jfr increases again with the stresses increase of a In Fig. 9 the experimental values of the fracture tough Since the layers in our composite lie parallel to the y-z less are also plotted. These values were evaluated from the plane, very low stresses in the x-direction o can be measured fracture loads of the specimens, using the above- expected therefore, we can assume ox=0 in the following mentioned procedure to determine Far. These values fit context. In the case of plane strain, the conditions &==0 well to the numerically predicted values and ox=0 yield the stresses in the z-direction G2=Vy-Ex△T 6. Conclusion where a denotes the substitute value of the CtE(see Sec o In this paper, a procedure has been presented to predict tion 4.2). The elastic strain components ef and ef follow as Gy+wa△T =-x△T experiments to measure the intrinsic fracture toughness of the single phases and to determine the material data The elastic strain energy density can then be calculated (Youngs modulus, Poissons ratio, CTE)which are neces- sary for the stress and strain analysis and the calculation of =5(arey +o26g +E(x△T)2)(A13 the residual stress state. The numerical modeling includes a conventional finite element stress analysis and the calcula- Eq.(Al. 3) is valid for any material point under thermal ion of the crack driving force based on the concept of con- and/or mechanical loading if plane strain conditions pre- figurational forces. The procedure yields the fracture vail and the stresses in the x-direction are zero. The strain toughness of the composite as a function of the crack energy density o consists of two parts, one depending on length. The spatial variation of both the elastic modulus the stress in the y-direction y and the other due to the ther and the thermal residual stresses influence the fracture mal expansion or contraction. Note that the stresses in the y-direction o, depend on both the external loading and the In the investigated bending bar, consisting of layers thermal expansion/contraction made of alumina and an alumina-zirconia composite If no external loading applies and only the eigenstrain the variations of the residual stresses produce much lar- aAT is active, apart from the end regions, the strain state ger shielding and anti-shielding effects than the variation Ey is independent of the position vector, say ey. AT=c with c of the elastic modulus. In particular, the compressive being a constant. The stress o, Ar follows as esidual stress state in the surface layer is of high rele vance for inhibiting the stable crack growth of surface yAT= Ec Er'AT (A1.4) cracks, since the fracture toughness of the composite is several times higher than the intrinsic fracture toughness. The constant c is found by utilizing the global equilibrium The numerical predictions fit well to the experimentally condition, Jo y Rdx=0. This integral can easily be evalu- measured fracture toughness data. The described proce- ated, since E, aAT, v are constant in the individual layers dure can be used in future to improve the design of mul- (1+v)(EAAA + EAZAZlAZ tilayer composites with arbitrary shape and loading △T (A15) conditions (EALA EAzIaz) The parameters (A and (Az denote the total thickness of all Acknowledgements A-and AZ-layers, respectively. Poissons ratio v is hereby considered to be the same for both materials which is a The work was partly supported by the European Com- good approximation for most structural ceramics. In our munity,s Human Potential Programme under contract case v= VA=VAZ A 0.25 HPRN-CT-2002-2003. J. Pascual acknowledges the finan The evaluation yields for the data of our laminate com- cial support provided through the European Commu- posite v AT=-144 and +166 MPa for the A-and az- nity's Human Potential Programme under contract material, respectively, which is identical to the numerical HPRN-CT-2002-00203.R. Danzer and J. Pascual express results of Fig. 2 for plane strain conditions their thanks to francis chalvet and goffredo de portu from ISTEC Faenza for providing the specimens. C Appendix 2. Analytical estimate of the crack driving force Chen, F.D. Fischer, and O. Kolednik gratefully acknowl- edge the financial support by the Osterreichische A2. 1. Expressions for Jfar and Cinh Forschungsforderungsgesellschaft mbh. the Province of Styria, the Steirische Wirtschaftsforderungsgesellschaft To evaluate the crack driving force according to Eq. (3) mbH and the Municipality of Leoben under the frame both an estimate of the far-field J-integral Jfar and the of the Austrian Kplus Programme material inhomogeneity term Cinh are needed First we con-
tip at interface 2 which is far below the intrinsic toughness. When the crack tip is in layer 3, Jfr increases again with the increase of a. In Fig. 9 the experimental values of the fracture toughness are also plotted. These values were evaluated from the measured fracture loads of the specimens, using the abovementioned procedure to determine Jfar. These values fit well to the numerically predicted values. 6. Conclusions In this paper, a procedure has been presented to predict and optimize the fracture toughness of a multilayer ceramic composite made of different phases. The procedure requires experiments to measure the intrinsic fracture toughness of the single phases and to determine the material data (Young’s modulus, Poisson’s ratio, CTE) which are necessary for the stress and strain analysis and the calculation of the residual stress state. The numerical modeling includes a conventional finite element stress analysis and the calculation of the crack driving force based on the concept of con- figurational forces. The procedure yields the fracture toughness of the composite as a function of the crack length. The spatial variation of both the elastic modulus and the thermal residual stresses influence the fracture behavior. In the investigated bending bar, consisting of layers made of alumina and an alumina–zirconia composite, the variations of the residual stresses produce much larger shielding and anti-shielding effects than the variation of the elastic modulus. In particular, the compressive residual stress state in the surface layer is of high relevance for inhibiting the stable crack growth of surface cracks, since the fracture toughness of the composite is several times higher than the intrinsic fracture toughness. The numerical predictions fit well to the experimentally measured fracture toughness data. The described procedure can be used in future to improve the design of multilayer composites with arbitrary shape and loading conditions. Acknowledgements The work was partly supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-2003. J. Pascual acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2002-00203. R. Danzer and J. Pascual express their thanks to Francis Chalvet and Goffredo de Portu from ISTEC Faenza for providing the specimens. C.R. Chen, F.D. Fischer, and O. Kolednik gratefully acknowledge the financial support by the O¨ sterreichische Forschungsfo¨rderungsgesellschaft mbH, the Province of Styria, the Steirische Wirtschaftsfo¨rderungsgesellschaft mbH and the Municipality of Leoben under the frame of the Austrian Kplus Programme. Appendix 1. Analytical estimate of the thermal residual stresses Since the layers in our composite lie parallel to the y z plane, very low stresses in the x-direction rx can be expected; therefore, we can assume rx ” 0 in the following context. In the case of plane strain, the conditions ez ” 0 and rx ” 0 yield the stresses in the z-direction: rz ¼ mry Ea DT ðA1:1Þ where a* denotes the substitute value of the CTE (see Section 4.2). The elastic strain components eel y and eel z follow as e el y ¼ ð1 m2Þ E ry þ ma DT ; e el z ¼ a DT ðA1:2Þ The elastic strain energy density / can then be calculated as / ¼ 1 2 ðry e el y þ rze el z Þ ¼ 1 2 1 m2 E r2 y þ Eða DT Þ 2 ðA1:3Þ Eq. (A1.3) is valid for any material point under thermal and/or mechanical loading if plane strain conditions prevail and the stresses in the x-direction are zero. The strain energy density / consists of two parts, one depending on the stress in the y-direction ry and the other due to the thermal expansion or contraction. Note that the stresses in the y-direction ry depend on both the external loading and the thermal expansion/contraction. If no external loading applies and only the eigenstrain a*DT is active, apart from the end regions, the strain state ey is independent of the position vector, say ey,DT = c with c being a constant. The stress ry,DT follows as ry;DT ¼ Ec 1 m2 EaDT 1 m ðA1:4Þ The constant c is found by utilizing the global equilibrium condition, R h 0 ry;DT dx ¼ 0. This integral can easily be evaluated, since E, a*DT, m are constant in the individual layers: c ¼ ð1 þ mÞðEAa AtA þ EAZa AZtAZÞ ðEAtA þ EAZtAZÞ DT ðA1:5Þ The parameters tA and tAZ denote the total thickness of all A- and AZ-layers, respectively. Poisson’s ratio m is hereby considered to be the same for both materials, which is a good approximation for most structural ceramics. In our case m = mA = mAZ 0.25. The evaluation yields for the data of our laminate composite ry,DT = 144 and +166 MPa for the A- and AZmaterial, respectively, which is identical to the numerical results of Fig. 2 for plane strain conditions. Appendix 2. Analytical estimate of the crack driving force A2.1. Expressions for Jfar and Cinh To evaluate the crack driving force according to Eq. (3), both an estimate of the far-field J-integral Jfar and the material inhomogeneity term Cinh are needed. First we con- 418 C.R. Chen et al. / Acta Materialia 55 (2007) 409–421