Strength of Materials, Vol. 36, No. 3, 2004 FRACTURE RESISTANCE OF RESIDUALLY-STRESSED CERAMIC LAMINATED STRUCTURES G. A Gogotsi, N.I. lugovoL, UDC 539.4 and V. N. Slyunvaev We have studied the effect of residual stresses on fracture resistance and crack arrest behavior of asymmetric ceramic laminated Si, N, Si N-TiN structures. Using the compliance method, we assessed the technique of R-curve construction for laminar composites. For laminar structures with layers varying by their elastic characteristics we developed an analytical method for calculation of fracture resistance-crack length"dependence. The method applicability is verified by calculation of stress intensity factors for laminar specimens with an edge crack. The calculated results are compared to the experimental data Keywords: layer structure, fracture toughness, modeling, crack arrest, residual stresses Introduction. Multilayered ceramic-matrix composites(MCMC) have a wide variety of applications in modern technology. Layers comprising ceramic materials are extensively used in engineering structural components with the objective to improve the mechanical, thermal, chemical and tribological performance. Recent research and developments in the area of MCMC seek to utilize the materials in such diverse applications as surface coatings, thermal barrier protection for turboengines, valves in reciprocating engines for automobiles and cutting tools. Despite many attractive properties such as high hardness and high temperature stability, MCMC have the major disadvantage of lacking reliability and sensitivity to surface contact damage. The last factor can lead to strength reduction and even to catastrophic failure A number of strategies have been developed in recent years to design tough and strong MCMC [1]. These include designing weak interfaces for crack deflection [2], using residual compression in surface layers [3], designing crack bifurcation effect in compressive layers [4], and controlling the frontal shape of the transformation zones in zirconia ceramics [5]. These mechanisms should provide an arrest of crack in layered structure, improving consequently its reliability. The reliability of the MCMC can be improved also by controlling the size of flaws introduced into the material during processing. This may be achieved by dispersion of a slurry of the designated power and by its passing through a filter. As a result only heterogeneities with sizes smaller than a critical size can pass through, depending on the filter fineness. Drawback of this procedure is its expensiveness. Moreover, such material is still subject to damage during machining with the reliability degraded accordingly In multilayered materials with strong interfaces, the differences in the coefficients of thermal expan (CTE's)between dissimilar materials or phase transformation in layers inevitably generate thermal residual stresses during subsequent cooling [6]. The essential feature of residual stress distribution in a layered structure is its occurence on a macroscopic scale. The relative thickness of different layers determines the relative magnitudes of compressive and tensile stresses, while the magnitude of the strain mismatch between the layers governs the absolute alues of the residual stresses. Control of the thermal stresses and the accompanying changes in structure are important to ensure the structural integrity of the layered components Pisarenko Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine Translated from Problemy Prochnosti, No. 2, pp. 95-1l1, May -June, 2004. Original article submitted June 12, 2003 0039-23 16/04/3603-0291C 2004 Plenum Publishing Corporation
Strength of Materials, Vol. 36, No. 3, 2004 FRACTURE RESISTANCE OF RESIDUALLY-STRESSED CERAMIC LAMINATED STRUCTURES G. A. Gogotsi, N. I. Lugovoi, UDC 539.4 and V. N. Slyunyaev We have studied the effect of residual stresses on fracture resistance and crack arrest behavior of asymmetric ceramic laminated Si3N4/Si3N4–TiN structures. Using the compliance method, we assessed the technique of R-curve construction for laminar composites. For laminar structures with layers varying by their elastic characteristics we developed an analytical method for calculation of “fracture resistance – crack length” dependence. The method applicability is verified by calculation of stress intensity factors for laminar specimens with an edge crack. The calculated results are compared to the experimental data. Keywords: layer structure, fracture toughness, modeling, crack arrest, residual stresses. Introduction. Multilayered ceramic-matrix composites (MCMC) have a wide variety of applications in modern technology. Layers comprising ceramic materials are extensively used in engineering structural components with the objective to improve the mechanical, thermal, chemical and tribological performance. Recent research and developments in the area of MCMC seek to utilize the materials in such diverse applications as surface coatings, thermal barrier protection for turboengines, valves in reciprocating engines for automobiles and cutting tools. Despite many attractive properties such as high hardness and high temperature stability, MCMC have the major disadvantage of lacking reliability and sensitivity to surface contact damage. The last factor can lead to strength reduction and even to catastrophic failure. A number of strategies have been developed in recent years to design tough and strong MCMC [1]. These include designing weak interfaces for crack deflection [2], using residual compression in surface layers [3], designing crack bifurcation effect in compressive layers [4], and controlling the frontal shape of the transformation zones in zirconia ceramics [5]. These mechanisms should provide an arrest of crack in layered structure, improving consequently its reliability. The reliability of the MCMC can be improved also by controlling the size of flaws introduced into the material during processing. This may be achieved by dispersion of a slurry of the designated power and by its passing through a filter. As a result only heterogeneities with sizes smaller than a critical size can pass through, depending on the filter fineness. Drawback of this procedure is its expensiveness. Moreover, such material is still subject to damage during machining with the reliability degraded accordingly. In multilayered materials with strong interfaces, the differences in the coefficients of thermal expansion (CTE’s) between dissimilar materials or phase transformation in layers inevitably generate thermal residual stresses during subsequent cooling [6]. The essential feature of residual stress distribution in a layered structure is its occurence on a macroscopic scale. The relative thickness of different layers determines the relative magnitudes of compressive and tensile stresses, while the magnitude of the strain mismatch between the layers governs the absolute values of the residual stresses. Control of the thermal stresses and the accompanying changes in structure are important to ensure the structural integrity of the layered components. 0039–2316/04/3603–0291 © 2004 Plenum Publishing Corporation 291 Pisarenko Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 2, pp. 95 – 111, May – June, 2004. Original article submitted June 12, 2003
a key feature that imparts good mechanical properties in the multilayer systems is the ability to be toughened significantly by placing their surfaces in residual compression and to arrest crack. It was shown in [3] that a residual surface compression of -500 MPa in a surface layer of three-layered alumina-zirconia specimen can increase its fracture toughness by a factor of 7.5(up to 30 MPa. m )for edge-crack lengths of the order of the surface-layer thickness. The toughening derived from macroscopic surface compression was, in fact, a crack shielding phenomenon and the fracture toughness increase was equivalent to crack growth resistance (R) behavior [7]. The R-behavior often connected with bridging mechanism. The mechanism is associated with the closure stress field that acts behind the tip of the advancing crack[8. However, there are some differences related to bridging mechanism(this is typical for non-layered ceramics) and the shielding phenomenon in layered structures. Firstly, while bridging mechanism gives rise to dependence of fracture resistance only on crack length increment, the shielding effect results in that fracture resistance depends on overall crack length [3, 7, 9]. Secondly, as a rule the bridging mechanism promotes fracture resistance increasing with crack advance whereas the shielding effect can induce both improvement and deterioration of fracture resistance depending on crack tip location in tensile or compressive layer. Actually layered specimen fracture resistance measured experimentally is the apparent fracture toughnes This is due to superposition of different effects like residual stress shielding and structure inhomogeneity. In fracture mechanics, one usually includes stresses in the crack driving force; however it is sometimes expedient to consider residual stresses as part of the crack resistance. Thus, a higher resistance to failure for layered structure with residual stress is obtained from a reduction of the crack driving force rather than from an increase in the intrinsic material resistance to crack extension [9] Despite numerous experimental and theoretical studies of fracture resistance of MCMC, systematic research of R-behavior and of crack arrest in layered composites are very scarce. a great number of publications deal with symmetrical layered structure. This is an idealized situation. In fact, laminates are characterized by some dissymmetry of their architecture due to random deviations in fabrication process. Moreover, specific non-symmetrical layered structures are important in some engineering applications. Conventional analytical consideration of shielding effect in laminates also neglects difference of elastic moduli of layers [3, 7]. However, effect of different moduli on fracture resistance of laminates is not so negligible. The influence of elastic moduli variation across a layered specimen on R-curve behavior is investigated in [10]. It was shown that the elastic moduli difference affects residual stress distribution and has consequently a significant influence on the measured R-curve behavior. But neither detailed alysis of conditions of crack arresting nor its stable/non-stable growth has been carried out in [10] The effect of macroscopic residual stresses on fracture resistance and crack arresting in non-symmetric Si3N4-based layered structures fabricated in the form of single-edge V-notch-bend(SEVNB)specimens investigated in this study. One of the work goals is application of the compliance technique to study R-curve effect as applied to layered specimens. A special attention is paid to the development of an analytical method to calculate fracture resistance- crack length dependence in layered structures with different elastic moduli of layers. The validit of the method is checked by calculation of the stress intensity factors for edge-cracked layered specimens and comparing the results with the mechanical test data The Model. Figure I shows a scheme of the two-component multilayer specimen analyzed in this study Parameter ti designates thickness of layer number i. The total thickness of specimen of rectangular cross section is w,its width is b, and the total number of layers is N. Choice of coordinate system is important for further consideration. It is the most appropriate to put the coordinate origin on the tensile surface of bending specimen. The geometry of the multilayered material analyzed here is such that the problem can be reduced to one dimension, and that analytically tractable solutions can be used. Here, the parameters of interest in the study of mechanical behavior depend only on coordinate x It was shown in [3, ll] that the stress intensity factor, KI, due to an arbitrary stress distribution in the prospective crack path, in the absence of the crack o(r), can be obtained as a o(r)dx, 292
A key feature that imparts good mechanical properties in the multilayer systems is the ability to be toughened significantly by placing their surfaces in residual compression and to arrest crack. It was shown in [3] that a residual surface compression of ~ 500 MPa in a surface layer of three-layered alumina-zirconia specimen can increase its fracture toughness by a factor of 7.5 (up to 30 MPa m⋅ 1 2/ ) for edge-crack lengths of the order of the surface-layer thickness. The toughening derived from macroscopic surface compression was, in fact, a crack shielding phenomenon and the fracture toughness increase was equivalent to crack growth resistance (R) behavior [7]. The R-behavior is often connected with bridging mechanism. The mechanism is associated with the closure stress field that acts behind the tip of the advancing crack [8]. However, there are some differences related to bridging mechanism (this is typical for non-layered ceramics) and the shielding phenomenon in layered structures. Firstly, while bridging mechanism gives rise to dependence of fracture resistance only on crack length increment, the shielding effect results in that fracture resistance depends on overall crack length [3, 7, 9]. Secondly, as a rule the bridging mechanism promotes fracture resistance increasing with crack advance whereas the shielding effect can induce both improvement and deterioration of fracture resistance depending on crack tip location in tensile or compressive layer. Actually layered specimen fracture resistance measured experimentally is the apparent fracture toughness. This is due to superposition of different effects like residual stress shielding and structure inhomogeneity. In fracture mechanics, one usually includes stresses in the crack driving force; however it is sometimes expedient to consider residual stresses as part of the crack resistance. Thus, a higher resistance to failure for layered structure with residual stress is obtained from a reduction of the crack driving force rather than from an increase in the intrinsic material resistance to crack extension [9]. Despite numerous experimental and theoretical studies of fracture resistance of MCMC, systematic research of R-behavior and of crack arrest in layered composites are very scarce. A great number of publications deal with symmetrical layered structure. This is an idealized situation. In fact, laminates are characterized by some dissymmetry of their architecture due to random deviations in fabrication process. Moreover, specific non-symmetrical layered structures are important in some engineering applications. Conventional analytical consideration of shielding effect in laminates also neglects difference of elastic moduli of layers [3, 7]. However, effect of different moduli on fracture resistance of laminates is not so negligible. The influence of elastic moduli variation across a layered specimen on R-curve behavior is investigated in [10]. It was shown that the elastic moduli difference affects residual stress distribution and has consequently a significant influence on the measured R-curve behavior. But neither detailed analysis of conditions of crack arresting nor its stable/non-stable growth has been carried out in [10]. The effect of macroscopic residual stresses on fracture resistance and crack arresting in non-symmetric Si3N4-based layered structures fabricated in the form of single-edge V-notch-bend (SEVNB) specimens is investigated in this study. One of the work goals is application of the compliance technique to study R-curve effect as applied to layered specimens. A special attention is paid to the development of an analytical method to calculate fracture resistance – crack length dependence in layered structures with different elastic moduli of layers. The validity of the method is checked by calculation of the stress intensity factors for edge-cracked layered specimens and comparing the results with the mechanical test data. The Model. Figure 1 shows a scheme of the two-component multilayer specimen analyzed in this study. Parameter ti designates thickness of layer number i. The total thickness of specimen of rectangular cross section is w, its width is b, and the total number of layers is N. Choice of coordinate system is important for further consideration. It is the most appropriate to put the coordinate origin on the tensile surface of bending specimen. The geometry of the multilayered material analyzed here is such that the problem can be reduced to one dimension, and that analytically tractable solutions can be used. Here, the parameters of interest in the study of mechanical behavior depend only on coordinate x. It was shown in [3, 11] that the stress intensity factor, K1, due to an arbitrary stress distribution in the prospective crack path, in the absence of the crack σ( ) x , can be obtained as K h x a x dx a 1 0 = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∫ , () , α σ (1) 292
TABLE 1 Coefficients Ayu in Eq (2)[11] 2μ=3=4 0.498 1.3187 3.067 50806243447-32.7208181214 63-12641519.763 n+1 2 Fig. I Fig. 1. Scheme of the two-component multilayer specimen Fig. 2. Scheme of analyze d crack location in layered specimen. where x is the distance along the crack length measured from the surface of an edge crack, a is the crack length a=a/w, and w is the specimen thickness(Fig. 2). For edge-cracked specimens, Fett and Munz[ll] have developed the following weight function -a)32+∑ of the coefficients Ayu and the exponents v and H in(2)are listed in Table 1 In the case where deformation is a function of coordinate x only, it follows from the strain compatibility [12] that overall deformation E(r) must be linear for elastic material E(x)=Eo +k (3) Here Eo is the deformation at x=0, and k is the specimen curvature. An equal biaxial stressed state is known to be the most appropriate approximation to describe the stressed state in real layered specimens [13]. This is the case of infinite dimensions along directions y and z, but with finite value of the specimen thickness. In the equal biaxial stressed state, we have: E(x)=Ez =E w, o(r)=02=Ow, where Ez, E w, Oz, and o w are strain and stress components along z-and y-axis respectively. Edge effects(occurrence of three-dimensional stresses near the edges of layered composite over a distance from the edge which is approximately equal to the layer thickness [14)can be neglected due to their high-localized character. Then o(x)=E'(x)e(x)-E(x)]
where x is the distance along the crack length measured from the surface of an edge crack, a is the crack length, α = a w, and w is the specimen thickness (Fig. 2). For edge-cracked specimens, Fett and Munz [11] have developed the following weight function: h x a a x a , A ( ) ( ) α ( ) π α α νµ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − − + 2 1 1 1 1 1 2 1 2 3 2 3 2 − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + ∑ x a ν µ α 1 . (2) The values of the coefficients Aνµ and the exponents ν and µ in (2) are listed in Table 1. In the case where deformation is a function of coordinate x only, it follows from the strain compatibility [12] that overall deformation ε( ) x must be linear for elastic material: ε ε () . x kx = + 0 (3) Here ε 0 is the deformation at x = 0, and k is the specimen curvature. An equal biaxial stressed state is known to be the most appropriate approximation to describe the stressed state in real layered specimens [13]. This is the case of infinite dimensions along directions y and z, but with finite value of the specimen thickness. In the equal biaxial stressed state, we have: ε εε ( ) x = = zz yy , σ σσ () , x = = zz yy where ε zz, ε yy , σ zz, and σ yy are strain and stress components along z- and y-axis respectively. Edge effects (occurrence of three-dimensional stresses near the edges of layered composite over a distance from the edge which is approximately equal to the layer thickness [14]) can be neglected due to their high-localized character. Then σ εε ( ) ( )[ ( ) ~ x Ex x x = ′ − ( )], (4) 293 TABLE 1. Coefficients Aνµ in Eq. (2) [11] ν Aνµ µ = 0 µ = 1 µ = 2 µ = 3 µ = 4 0 0.498 2.4463 0.07 1.3187 − 3.067 1 0.54165 − 5.0806 24.3447 − 32.7208 18.1214 2 − 0.19277 2.55863 − 12.6415 19.763 − 10.986 Fig. 1 Fig. 2 Fig. 1. Scheme of the two-component multilayer specimen. Fig. 2. Scheme of analyzed crack location in layered specimen
where E'(x)=E(x)/1-v(x) In Eqs. (4),(5), E(r) and v(x)are the elastic modulus and Poisson ratio distributions along x-axis, respectively. Value of E(x) is the strain non-associated with stress. It is associated with thermal expansion or/and with a volume change due to a crystallographic phase transformation. The static balance conditions [12] in the chosen coordinate system result in a system of linear equations with Inknown values Eo and k Fa+bo(x, Eo, k )dk=0, ,k) here Fa is the applied axial force and Ma is the applied bending moment. Solution of the system is [13] I2Jo-Fa/ b)+I01-Malb Eo 1012 11Jo-Fa/b)-lo(1-Ma/b) k whe L=rE'(x)dx (=0, 1, 2), (8) 0,1) Note that the superposition principle is valid for this problem. It permits to express the stress variation along rack path in a specimen as 0(x)=0a(x)+G(x), (10) where ga(x) is the bending stress in the prospective crack path in the absence of any residual stresses, and o(r)is the macroscopic residual stress distribution In [3], the bending stress oa(x) was expressed as follows 0a(x)=0m where om is the applied stress on tensile surface of bending specimen. It is well known that 15Ps 6M Here P is the critical load(applied bending load corresponding to the specimen failure)and s is the support span lowever, the differences in the elastic moduli of the layers were not taken into account in [ 3]. Difference in elastic
where E x Ex x ′( ) ( ) [ ( )]. = −1 ν (5) In Eqs. (4), (5), E x( ) and ν( ) x are the elastic modulus and Poisson ratio distributions along x-axis, respectively. Value of ~ε ( ) x is the strain non-associated with stress. It is associated with thermal expansion or/and with a volume change due to a crystallographic phase transformation. The static balance conditions [12] in the chosen coordinate system result in a system of linear equations with unknown values ε 0 and k: F b x k dk M bx x k dx a w a w + = + = ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ∫ ∫ σ ε σ ε (, , ) , (, , ) , 0 0 0 0 0 0 (6) where Fa is the applied axial force and Ma is the applied bending moment. Solution of the system is [13]: ε 0 2 0 11 1 2 0 2 = − +− − I J Fb IJ M b I II a a ( )( ) , (7a) k IJ Fb I J M b I II a a = −−− − 10 01 1 2 0 2 ( )( ) , (7b) where I x E x dx j j j w = ′ = ∫ ( ) ( , , ), 012 0 (8) J x x E x dx j j j w = ′ = ∫ 0 0 1 ~ε ( ) ( ) ( , ). (9) Note that the superposition principle is valid for this problem. It permits to express the stress variation along the crack path in a specimen as σσ σ ( ) ( ) ( ), xxx = + a r (10) where σa ( ) x is the bending stress in the prospective crack path in the absence of any residual stresses, and σr ( ) x is the macroscopic residual stress distribution. In [3], the bending stress σa ( ) x was expressed as follows: σ σ a m x x w () , = − ⎛ ⎝ ⎜ ⎞ ⎠ 1 ⎟ 2 (11) where σ m is the applied stress on tensile surface of bending specimen. It is well known that σ m Ps a bw M bw = = 1 5 6 2 2 . . (12) Here P is the critical load (applied bending load corresponding to the specimen failure) and s is the support span. However, the differences in the elastic moduli of the layers were not taken into account in [3]. Difference in elastic 294
residual stress -neutral axis G1(x) pplied stre applled stress Ior homogeneous specimen with elastic modulus e,ef surface Fig. 3. Schematic of residual and applied stress distribution in layered specimen moduli of the layers result in specific distribution of the applied stress along x-direction. Elastic material demonstrates continuous linear distribution of the applied strain under bending. This promotes piecewise-linear distribution of the applied stress, shown schematically in Fig. 3. To derive the applied stress distribution under bending we can use expressions (3),(4),(7a), (7b), and(12), taking into account that in this case Fa =0. If only the applied stress considered, we can take E(r)=0. Then it follows that the applied stress acting in the layer with number i is E [Lor-lu, ≤x≤ 6 Here xi is the coordinate of upper boundary of ith layer, E=E/(-vi), and E and vi are the elastic modulus and Poisson ratio of layer number i, respectively. Values of IL can be obtained from expression(8)accounting for re(Fig. 1) Residual stress distribution can be found from Eqs. (3),(4),(7a), and(7b) taking into account that Fa =0, E 0,(x)= [nJn1-112Jo+(uJLo-loJn)x],x-1≤x≤x 111-110/12 (15) where Jy can be obtained from the expressions(9)accounting for layered structure ∑Exr1-x) Here Ei is the strain of ith layer non-associated with stress. The thermal expansion or/and a volume change due to a crystallographic phase transformation can be the source of this strain. However, the case of phase transformation is out of the scope of this paper. In case of thermal expansion E,=B, (T)dT, where, (T)is the thermal expansion
moduli of the layers result in specific distribution of the applied stress along x-direction. Elastic material demonstrates continuous linear distribution of the applied strain under bending. This promotes piecewise-linear distribution of the applied stress, shown schematically in Fig. 3. To derive the applied stress distribution under bending we can use expressions (3), (4), (7a), (7b), and (12), taking into account that in this case Fa = 0. If only the applied stress is considered, we can take ~ε ( ) x = 0. Then it follows that the applied stress acting in the layer with number i is: σ σ a i L L L mL L i i x E w I II ( ) IxI x xx ( ) = [ ], . ′ − − ≤≤ − 2 1 2 0 2 01 1 6 (13) Here xi is the coordinate of upper boundary of ith layer, E E ii i ′ = − ( ), 1 ν and Ei and νi are the elastic modulus and Poisson ratio of layer number i, respectively. Values of I Lj can be obtained from expression (8) accounting for layered structure (Fig. 1): I j Ex x j Lj i i j i j i N = + ′ − = + − + = ∑ 1 1 012 1 1 1 1 ( ) ( , , ). (14) Residual stress distribution can be found from Eqs. (3), (4), (7a), and (7b) taking into account that Fa = 0, Ma = 0 (Fig. 3): σr i L L L LL L L LL L L x E I II () [ ( ) = IJ I J IJ I J ′ − −+ − 1 2 0 2 11 2 0 10 01 x], x xx i i −1 ≤ ≤ , (15) where JLj can be obtained from the expressions (9) accounting for layered structure: J j Ex x j Lj i i i j i j i N = + ′ − = + − + = ∑ 1 1 0 1 1 1 1 1 ~ε ( ) ( , ). (16) Here ~εi is the strain of ith layer non-associated with stress. The thermal expansion or/and a volume change due to a crystallographic phase transformation can be the source of this strain. However, the case of phase transformation is out of the scope of this paper. In case of thermal expansion ~ε β () , i i T T T dT j = ∫ 0 where βi ( ) T is the thermal expansion 295 Fig. 3. Schematic of residual and applied stress distribution in layered specimen
coefficient of ith layer at temperature T, and To and Ti are actual and "joining"temperature, respectively. During cooling of the sample the deformation difference, due to the different thermal expansion coefficients, is accommodated by creep as long as the temperature is high enough. Below a certain temperature, called the "joining temperature, the different components become bonded together and internal stresses appear. The "joining temperature is usually the value that is known only approximately. In practice, T is generally accepted to lie somewhat below the sintering temperature. If Bi(T)is a linear function, Ei sPi >AT, where AT=T-To 阝(70)+B1(T) is the average value of the thermal expansion coefficient in the temperature range from To Using the condition of crack growth(K1=Klc, where K,l is the fracture toughness of material) and(1), (10), we obtain K Ic (17) Using Eq.(13)the first integral in(17) can be expressed for a layered material as o(a,a,(r)dx=-Onwy2 6(21-102) where n is the number of layers broken by the crack (or notch) completely(Fig. 2). Using Eq(15), the second integral in(17) for a layered material takes the form Kr=h-, a o, (r)dx E#1J42,a|ua/4-120+(Jo-10x (19) Here, K is the stress intensity due to the residual stresses The following formula is given for the stress intensity of an edge crack in the specimen under bending as being accurate to +0. 2% in the range a=0 to 1 [15] f0(), where fo(a) is a nondimensional stress intensity factor given by the following expression [15] 1.5a2[1.99-0(1-0)(215-3930+270 fo(0)= +2x)(1-a)32 (21) Taking into account Eqs. (12),(21), expression(20) can be transformed to the form KI=Y(a)o a12 (22)
coefficient of ith layer at temperature T, and T0 and T j are actual and “joining” temperature, respectively. During cooling of the sample the deformation difference, due to the different thermal expansion coefficients, is accommodated by creep as long as the temperature is high enough. Below a certain temperature, called the “joining” temperature, the different components become bonded together and internal stresses appear. The “joining” temperature is usually the value that is known only approximately. In practice, T j is generally accepted to lie somewhat below the sintering temperature. If βi ( ) T is a linear function, ~ε β i i ≤ >∆T, where ∆TT T = −j 0 , < ≥ + β β β i i ij () () T T 0 2 is the average value of the thermal expansion coefficient in the temperature range from T0 to T j . Using the condition of crack growth (K K 1 1 = c , where K1c is the fracture toughness of material) and (1), (10), we obtain: K h x a x dx h x a x dx c a a a 1 r 0 0 = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∫ ∫ , () , () . ασ ασ (17) Using Eq. (13) the first integral in (17) can be expressed for a layered material as h x a x dx w I II E h x a a a m L L L n , () ( ) α σ , σ α ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = − ′ ∫ + 0 2 1 2 0 2 1 6 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − + ′ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − − [ ] ,[ ] I x I dx E h x a I x I dx LL i LL x x i 01 01 1 α i n i n x a ∫ ∑ ∫ = ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ 1 ⎪ , (18) where n is the number of layers broken by the crack (or notch) completely (Fig. 2). Using Eq. (15), the second integral in (17) for a layered material takes the form: K h x a x dx I II E h x a r a r L L L = n ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = − ′ ⎛ ⎝ ⎜ ∫ + , () , ασ α 0 1 2 0 2 1 1 ⎞ ⎠ ⎟ ⎧ ⎨ ⎪ ⎩ ⎪ −+ − ∫ x a LL L L LL L L n [ ( )] I J I J I J I J x dx 11 2 0 10 01 + ′ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ −+ − − ∑ ∫ = E h x a IJ I J IJ i x x i n LL L L LL i i ,[ ( α 1 1 11 2 0 10 I J x dx L L 0 1 )] . ⎫ ⎬ ⎪ ⎭ ⎪ (19) Here, Kr is the stress intensity due to the residual stresses. The following formula is given for the stress intensity of an edge crack in the specimen under bending as being accurate to ±0.2% in the range α = 0 to 1 [15]: K Ps bw f 1 3 2 = 0 ( ), α (20) where f 0 ( ) α is a nondimensional stress intensity factor given by the following expression [15]: f 0 12 2 1 5 1 99 1 2 15 3 93 2 7 1 2 ( ) . [ . ( )( . . . )] ( )( α α αα α α α = −− − + + 1 3 2 − α) . (21) Taking into account Eqs. (12), (21), expression (20) can be transformed to the form: KY a 1 m 1 2 = () , α σ (22) 296
where 199-a(1-a)(215-3930+270x2) Y() (23) (1+20) /2 It was shown in [3] that Eqs.(20),(22)can be successfully used to determine fracture toughness of ceramic matrix layered materials. However, it should be noted that as applied to inhomogeneous(particularly, layered) materials the equations give the so-called apparent fracture toughness. In bending test, this is the fracture toughness of some effective homogeneous specimen that meets the following conditions: 1)to have the same dimensions as real layered specimen; 2)to have notch depth equal to that of real layered specimen; 3) under the same loading conditions to demonstrate the same critical load as that for real layered specimen. In spite of relativity of this value, it is a useful characteristic allowing contributions of such factors as residual stresses and material inhomogeneity to be accounted for. Thus, experimental value of the apparent fracture toughness of layered specimen can be found using expression Kom =Y(o)om, a/2 (24) It follows from Eqs.(17)(19)and(24)that apparent fracture toughness of layered composite K written as 6(a)2(2-1012)kx(-k,) K a[Lox-lu x where Ki is the fracture toughness of ith layer material. Expression(25)suggests that the higher resistance to fracture is derived from a reduction of the crack driving force rather than from an increase in the intrinsic resistance to crack extension Experimental. The choice of composition for Si3N4 based ceramics laminates is determined by the coefficient of thermal expansion and Young's modulus of the compounds. Three compositions of composite layers were used: 1)Si,N4(MIl, Starck, Germany); 2) Si3N4-20 wt TiN (grade C, Starck, Germany); 3)Si3N--70 wt Tin (grade C, Starck, Germany). Youngs moduli, Poisson ratios, joining temperature T and average values of coefficients of thermal expansion of compositions under study are given in [16]. Mean values of intrinsic fracture toughness of monolith materials are evaluated in the work to be approximately the same for all layer compositions, being 5 MPa. m. Note that the intrinsic fracture toughness corresponds to fracture toughness of layer material Milling of mixtures of certain compositions was done in the ball mill for 5 h. The formation of a thin ceramic layer is of specific importance, as the sizes of residual stress zones(tensile and compressive)are directly connected with the thickness of layers. Green tapes were manufactured with rolling. Rolling permits to control thickness of green layers, to obtain high green density and a rather low amount of solvent and organic additives in comparison with other methods such as a tape casting [17]. However there is a problem to produce thin tapes(<100 um) with a small amount of plasticizer and sufficient strength and elasticity to handle green layers after rolling Crude rubber (4 wt %)was added to the mixture of powders as a plastisizer through a 3% solution in petrol Then the powders were dried up to the 2 wt% residual amount of petrol in the mixture. After sieving powders with 500 um sieve, granulated powders were dried up to the 0.5 wt. residual petrol. A roll mill with 40 mm rolls used for rolling. The velocity of rolling was 1. 5 m/min. Working pressure varied from 10 MPa for the relative density of 64% to 100 MPa for 74% density. The thickness of green tapes was either 0.4-0.5 mm or 0.8-1.0 mm, the width was 60-65 mm. Samples of ceramics were prepared by hot pressing of tapes stacked together. The hot pressi was performed at the temperature 1780-1820%C, with duration of 20 min and under the pressure of 30 MPa Green tapes were stacked together to form desirable layered structure. The graphite dies were used for the hot pressing without protective atmosphere. After hot pressing, the thickness of the Si3N4 layers was 160-960 um, and the thickness of the SiN \ayers with Tin additive varied from 160 to 480Aim In the range of
where Y ( ) . ( )( . . . ) ( )( ) α . αα α α α α = −− − + + − 1 99 1 2 15 3 93 2 7 12 1 2 3 2 (23) It was shown in [3] that Eqs. (20), (22) can be successfully used to determine fracture toughness of ceramic matrix layered materials. However, it should be noted that as applied to inhomogeneous (particularly, layered) materials the equations give the so-called apparent fracture toughness. In bending test, this is the fracture toughness of some effective homogeneous specimen that meets the following conditions: 1) to have the same dimensions as real layered specimen; 2) to have notch depth equal to that of real layered specimen; 3) under the same loading conditions to demonstrate the same critical load as that for real layered specimen. In spite of relativity of this value, it is a useful characteristic allowing contributions of such factors as residual stresses and material inhomogeneity to be accounted for. Thus, experimental value of the apparent fracture toughness of layered specimen can be found using expression (22): KY a app m = () . α σ 1 2 (24) It follows from Eqs. (17)–(19) and (24) that apparent fracture toughness of layered composite Kapp can be written as K Y a I II K K wE h x a app L L L c i r n = − − ′ ⎛ + 6 1 2 1 2 0 2 1 2 1 ( ) ( )( ) , ( ) α α ⎝ ⎜ ⎞ ⎠ ⎟ − + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − = ∫ [ ] ,[ ] I x I dx h ∑ x a I x I dx LL LL i n x a n 01 01 1 α ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ , (25) where K c i 1 ( ) is the fracture toughness of ith layer material. Expression (25) suggests that the higher resistance to fracture is derived from a reduction of the crack driving force rather than from an increase in the intrinsic resistance to crack extension. Experimental. The choice of composition for Si3N4 based ceramics laminates is determined by the coefficient of thermal expansion and Young’s modulus of the compounds. Three compositions of composite layers were used: 1) Si3N4 (M11, Starck, Germany); 2) Si3N4–20 wt.% TiN (grade C, Starck, Germany); 3) Si3N4–70 wt.% TiN (grade C, Starck, Germany). Young’s moduli, Poisson ratios, joining temperature T j and average values of coefficients of thermal expansion of compositions under study are given in [16]. Mean values of intrinsic fracture toughness of monolith materials are evaluated in the work to be approximately the same for all layer compositions, being 5 MPa m⋅ 1 2/ . Note that the intrinsic fracture toughness corresponds to fracture toughness of layer material. Milling of mixtures of certain compositions was done in the ball mill for 5 h. The formation of a thin ceramic layer is of specific importance, as the sizes of residual stress zones (tensile and compressive) are directly connected with the thickness of layers. Green tapes were manufactured with rolling. Rolling permits to control thickness of green layers, to obtain high green density and a rather low amount of solvent and organic additives in comparison with other methods such as a tape casting [17]. However there is a problem to produce thin tapes (<100 µm) with a small amount of plasticizer and sufficient strength and elasticity to handle green layers after rolling. Crude rubber (4 wt.%) was added to the mixture of powders as a plastisizer through a 3% solution in petrol. Then the powders were dried up to the 2 wt.% residual amount of petrol in the mixture. After sieving powders with a 500 µm sieve, granulated powders were dried up to the 0.5 wt.% residual petrol. A roll mill with 40 mm rolls was used for rolling. The velocity of rolling was 1.5 m/min. Working pressure varied from 10 MPa for the relative tape density of 64% to 100 MPa for 74% density. The thickness of green tapes was either 0.4–0.5 mm or 0.8–1.0 mm, the width was 60–65 mm. Samples of ceramics were prepared by hot pressing of tapes stacked together. The hot pressing was performed at the temperature 1780–1820°C, with duration of 20 min and under the pressure of 30 MPa. Green tapes were stacked together to form desirable layered structure. The graphite dies were used for the hot pressing without protective atmosphere. After hot pressing, the thickness of the Si3N4 layers was in the range of 160–960 µm, and the thickness of the Si3N4 layers with TiN additive varied from 160 to 480 µm. 297
0=168 MP Lo=176 M Fig. 4. Optical photograph of layered specimen with a typical notch tip The specimens for mechanical tests were prepared from hot-pressed plates. SEVNB specimens were used for testing. The test data have confirmed that the SEVNB method can be easier applied in practice and used for the majority of advanced ceramics and ceramic particulate composites [18]. The V-notches with tip radii of an order of 10-15 um were made in the specimens by a two-stage technique. In the first stage, the specimens were notched by a diamond saw. Then sharp tip of notch was obtained using stainless steel blade and diamond abrasive. Optical photograph of a typical notch tip along with average values of calculated residual stresses acting in the layers is presented in Fig. 4. The depth of the notches was about 60-80% of the specimen thickness being 3.37-4.0 mm Specimens of larger thickness were ground down to such sizes A stiff load cell ensuring the rigid loading of specimens under three-point bending with a 16 mm span was used for mechanical tests [18]. This"autonomous"cell is equipped with specific rigid dynamometer providing an ultimate load of 2000 N with a specimen deflection measuring system using a deflectometer suspended on the specimen. The testing machine used is designed only for the displacement of a loading crosshead and control of its speed. To study R-curve effect the compliance technique was used. Notched specimen was placed into the hard load cell. Then loading of the specimen was performed up to crack growth onset followed by unloading. In addition to recording of the load-deflection diagrams, after each unloading of specimen, its polished lateral surface was examine by an optical microscope(x1000) to measure crack length. After measurement of crack length the next loading unloading cycle was made. The operations were repeated up to the total failure of specimen. Apparent fracture toughness was calculated by using expressions(12),(23), and (24) Results and Discussion. Asymmetric structure of layered specimens under study results in linear variation of residual stresses within each layer. The critical issue to analyze fracture behavior of laminates is a choice of the coordinate system. Calculated values of apparent fracture toughness Kapp in layered specimens under study are analyzed depending on crack length parameter a, where a=Y(a)aV2. The crack length parameter a is the most appropriate to demonstrate critical conditions of crack growth. One of advantages of this parameter is that the stre intensity factor of an edge crack for fixed value of the applied stress om is depicted in the coordinate system Kapp -a as a straight line from the coordinate origin. Indeed, it follows from(22) that K1=oma, therefore, the slope of straight line equals to the applied stress om. The conditions for unstable crack growth in the internal stress field are as follows [9]: K,(om, a)=kapp(a), dK,(om, a)/da 2 dK app(a)/da. Using parameter a, these conditions become:oma=Kapp(a),om>dAnn(a ) /da. The last two conditions can be reduced to v(a)a≥dK app(a)/da (26)
The specimens for mechanical tests were prepared from hot-pressed plates. SEVNB specimens were used for testing. The test data have confirmed that the SEVNB method can be easier applied in practice and used for the majority of advanced ceramics and ceramic particulate composites [18]. The V-notches with tip radii of an order of 10–15 µm were made in the specimens by a two-stage technique. In the first stage, the specimens were notched by a diamond saw. Then sharp tip of notch was obtained using stainless steel blade and diamond abrasive. Optical photograph of a typical notch tip along with average values of calculated residual stresses acting in the layers is presented in Fig. 4. The depth of the notches was about 60–80% of the specimen thickness being 3.37–4.0 mm. Specimens of larger thickness were ground down to such sizes. A stiff load cell ensuring the rigid loading of specimens under three-point bending with a 16 mm span was used for mechanical tests [18]. This “autonomous” cell is equipped with specific rigid dynamometer providing an ultimate load of 2000 N with a specimen deflection measuring system using a deflectometer suspended on the specimen. The testing machine used is designed only for the displacement of a loading crosshead and control of its speed. To study R-curve effect the compliance technique was used. Notched specimen was placed into the hard load cell. Then loading of the specimen was performed up to crack growth onset followed by unloading. In addition to recording of the load-deflection diagrams, after each unloading of specimen, its polished lateral surface was examined by an optical microscope (×1000) to measure crack length. After measurement of crack length the next loading– unloading cycle was made. The operations were repeated up to the total failure of specimen. Apparent fracture toughness was calculated by using expressions (12), (23), and (24). Results and Discussion. Asymmetric structure of layered specimens under study results in linear variation of residual stresses within each layer. The critical issue to analyze fracture behavior of laminates is a choice of the coordinate system. Calculated values of apparent fracture toughness Kapp in layered specimens under study are analyzed depending on crack length parameter ~a, where ~aY a = α( ) 1 2 . The crack length parameter ~a is the most appropriate to demonstrate critical conditions of crack growth. One of advantages of this parameter is that the stress intensity factor of an edge crack for fixed value of the applied stress σ m is depicted in the coordinate system K a app − ~ as a straight line from the coordinate origin. Indeed, it follows from (22) that K a 1 = σ m ~, therefore, the slope of straight line equals to the applied stress σ m . The conditions for unstable crack growth in the internal stress field are as follows [9]: K aK a 1 m app ( , ) ( ), σ = dK a da dK a da 1 m app ( , ) () σ ≥ . Using parameter ~a, these conditions become: σ m app aK a ~ ( ~ = ), σ m app ≥ dK a da ( ~) ~. The last two conditions can be reduced to: K a a dK a da app app ( ~) ~ ( ~) ~ ≥ . (26) 298 Fig. 4. Optical photograph of layered specimen with a typical notch tip
ity facto at constant ap stress kAppa fracture Kappa=om Fig. 5. Condition of unstable crack growth in the internal stress field It follows from Eq(26) and Fig. 5 [9 that unstable crack growth occurs if the slope of straight line corresponding to the stress intensity factor at constant applied stress is no less than the slope of tangent line to the fracture resistance curve at the same point igure 6 shows dependence of the apparent fracture toughness on crack length parameter a in laminate Si3N4/Si3N4-20 wt TIN, specimen 1(solid curve). The areas corresponding to compressive and tensile layers are shown in gray and white, respectively. The fracture toughness of layer material is shown as horizontal straight line The dependence of apparent fracture toughness on a is non-monotonous. The apparent fracture toughness increases in the compressive layers and decreases in the tensile layers. The peak values of K app correspond to interfaces between layers. The apparent fracture toughness of the layered composite varies from 2 to 10 MPa m" depending on the crack length. The initial notch tip is in tenth layer that is under residual tension. Measured value of the apparent fracture toughness corresponding to the initial notch is 5.57 MPa m"that is in accord with the calculated value Unloading was made after small advance of crack from the initial notch. Crack arrest occurred in the 12th layer of specimen. The length of arrested crack was measured. Then the next loading resulted in the total failure of specimen. Measured value of the apparent fracture toughness corresponding to arrested crack is 7.42 MPa m"that is also in accord with the calculated value Figure 7 shows dependence of the apparent fracture toughness on crack length parameter a in specimen 2 Designations are the same as in Fig. 6. The dependence of the apparent fracture toughness on crack length parameter is non-monotonous as well. The fracture toughness behavior in compressive and tensile layers in specimen 2 is qualitatively similar to that in specimen 1. However, difference of specimen geometry results in some difference of the apparent fracture toughness range. Specifically, the apparent fracture toughness of specimen 2 varies from 3 to 11 MPa- m". The initial notch tip in the specimen is also in tenth layer that is under residual tension. In this case, measured value of the apparent fracture toughness corresponding to the initial notch is 6.39 MPa.m". That is in accord with the calculated value too. After unloading crack was arrested in 12th layer like specimen 1. Specimen 2 with arrested crack demonstrates the apparent fracture toughness value of 6.27 MPa m". This is in good accord with the Additionally to Si3 N//Si3 N4-20 wt TiN layered specimens, the mechanical behavior of Si3N4/Si3N4- 70 wt. TIN laminates was studied by the compliance technique. Figure 8 shows cyclic load -displacement diagram of layered specimen Si3N4/Si3N4-70 wt TiN with crack. An interesting feature of this diagram is a number of hysteresis loops recorded during specimen unloading and its further loading. It can be connected with some energy dissipation during unloading-loading cycle. A similar effect was also observed, e.g., in the studies of R-curves for graphite [19]. It was connected with the amount of energy dissipated by plastic strains. The microscopic analysis of fractured specimens demonstrated that tensile-stressed layers containing 70% Tin display multiple channel cracks 299
It follows from Eq. (26) and Fig. 5 [9] that unstable crack growth occurs if the slope of straight line corresponding to the stress intensity factor at constant applied stress is no less than the slope of tangent line to the fracture resistance curve at the same point. Figure 6 shows dependence of the apparent fracture toughness on crack length parameter ~a in laminate Si3N4/Si3N4–20 wt.% TiN, specimen 1 (solid curve). The areas corresponding to compressive and tensile layers are shown in gray and white, respectively. The fracture toughness of layer material is shown as horizontal straight line. The dependence of apparent fracture toughness on ~a is non-monotonous. The apparent fracture toughness increases in the compressive layers and decreases in the tensile layers. The peak values of Kapp correspond to interfaces between layers. The apparent fracture toughness of the layered composite varies from 2 to 10 MPa m⋅ 1 2/ depending on the crack length. The initial notch tip is in tenth layer that is under residual tension. Measured value of the apparent fracture toughness corresponding to the initial notch is 5.57 MPa m⋅ 1 2/ that is in accord with the calculated value. Unloading was made after small advance of crack from the initial notch. Crack arrest occurred in the 12th layer of specimen. The length of arrested crack was measured. Then the next loading resulted in the total failure of specimen. Measured value of the apparent fracture toughness corresponding to arrested crack is 7.42 MPa m⋅ 1 2/ that is also in accord with the calculated value. Figure 7 shows dependence of the apparent fracture toughness on crack length parameter ~a in specimen 2. Designations are the same as in Fig. 6. The dependence of the apparent fracture toughness on crack length parameter is non-monotonous as well. The fracture toughness behavior in compressive and tensile layers in specimen 2 is qualitatively similar to that in specimen 1. However, difference of specimen geometry results in some difference of the apparent fracture toughness range. Specifically, the apparent fracture toughness of specimen 2 varies from 3 to 11 MPa m⋅ 1 2/ . The initial notch tip in the specimen is also in tenth layer that is under residual tension. In this case, measured value of the apparent fracture toughness corresponding to the initial notch is 6.39 MPa m⋅ 1 2/ . That is in accord with the calculated value too. After unloading crack was arrested in 12th layer like specimen 1. Specimen 2 with arrested crack demonstrates the apparent fracture toughness value of 6.27 MPa m⋅ 1 2/ . This is in good accord with the calculated value. Additionally to Si3N4/Si3N4–20 wt.% TiN layered specimens, the mechanical behavior of Si3N4/Si3N4– 70 wt.% TiN laminates was studied by the compliance technique. Figure 8 shows cyclic load – displacement diagram of layered specimen Si3N4/Si3N4–70 wt.% TiN with crack. An interesting feature of this diagram is a number of hysteresis loops recorded during specimen unloading and its further loading. It can be connected with some energy dissipation during unloading-loading cycle. A similar effect was also observed, e.g., in the studies of R-curves for graphite [19]. It was connected with the amount of energy dissipated by plastic strains. The microscopic analysis of fractured specimens demonstrated that tensile-stressed layers containing 70% TiN display multiple channel cracks 299 Fig. 5. Condition of unstable crack growth in the internal stress field
Kapp, MPa.//2 Kann MPa. m2 4 (, m 0 04 0.2 04 Fis Fig. 7 Fig. 6. Dependence of the apparent fracture toughness on the crack length parameter a in laminate Si3 N4/Si3 N-20 wt. TiN(specimen 1). Areas of compressive layer are shown in grey. Solid curve is the calculated dependence, horizontal line is the fracture toughness of layer material. Dashed line is the tress intensity factor at constant applied stress of crack growth onset. Open circle corresponds to the initial notch, filled circle corresponds to arrested crack. Fig. 7. Dependence of the apparent fracture toughness on the crack length parameter a in laminate Si3N4/Si3N4-20 wt TIN(specimen 2). Designations are the same as in Fig. 6 P N X, Hm Fig 8. Cyclic load -displacement diagram of layered specimen Si3N4/ Si3N-70 wt %TiN with crack. formed during specimen sintering that is probably due to the insufficient strength of these layers. However, propagating crack, fracturing the specimens, did not always pass through channel cracks. It was established specimens of such composition layer cracks did not practically propagate in the direction of loading, and they did not even always start from the tip of a v-notch. As a whole, the fracture pattern of Si3 N4/Si3 N4-70 wt TiN specimens appeared to be very. Therefore, theoretical analysis and calculation of the apparent fracture toughness - crack length dependence of such laminates was not carried out in the present work. To describe crack behavior in layered specimens containing 70% TiN in tensile layer properly, it is necessary to take into account contributions of such factors as crack branching, microcracking, multiple channel cracks formation, crack kinking, etc
formed during specimen sintering that is probably due to the insufficient strength of these layers. However, the propagating crack, fracturing the specimens, did not always pass through channel cracks. It was established that specimens of such composition layer cracks did not practically propagate in the direction of loading, and they did not even always start from the tip of a V-notch. As a whole, the fracture pattern of Si3N4/Si3N4–70 wt.% TiN specimens appeared to be very. Therefore, theoretical analysis and calculation of the apparent fracture toughness – crack length dependence of such laminates was not carried out in the present work. To describe crack behavior in layered specimens containing 70% TiN in tensile layer properly, it is necessary to take into account contributions of such factors as crack branching, microcracking, multiple channel cracks formation, crack kinking, etc. 300 Fig. 6 Fig. 7 Fig. 6. Dependence of the apparent fracture toughness on the crack length parameter ~a in laminate Si3N4/Si3N4–20 wt.% TiN (specimen 1). Areas of compressive layer are shown in grey. Solid curve is the calculated dependence, horizontal line is the fracture toughness of layer material. Dashed line is the stress intensity factor at constant applied stress of crack growth onset. Open circle corresponds to the initial notch, filled circle corresponds to arrested crack. Fig. 7. Dependence of the apparent fracture toughness on the crack length parameter ~a in laminate Si3N4/Si3N4–20 wt.% TiN (specimen 2). Designations are the same as in Fig. 6. Fig. 8. Cyclic load – displacement diagram of layered specimen Si3N4/ Si3N4–70 wt.% TiN with crack