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