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Design and production of ceramic laminates with high mechanical reliability
Composites: Part B 37 (2006) 481–489 www.elsevier.com/locate/compositesb
Design and production of ceramic laminates with high mechanical reliability Vincenzo M. Sglavo *, Massimo Bertoldi 1 Dipartimento di Ingegneria dei Materiali e Tecnologie Industriali, Universita` di Trento, Via Mesiano, 77, 38050 Trento, Italy Available online 20 March 2006
Abstract A procedure for designing innovative ceramic laminates characterized by high mechanical reliability is proposed in this work. A fracture mechanics approach has been considered to define the stacking sequence, thickness and composition of the different laminae on the basis of the requested strength and of the defect size distribution. Once the different laminae are stacked together a residual stress profile is generated upon cooling after sintering because of the differential thermal expansion coefficient. Such residual stress profile is conceived in order to allow stable growth of surface defects upon bending and guarantee limited strength scatter. As an example, the proposed approach is used to design and produce ceramic laminates in the alumina–zirconia and alumina–mullite system. Mechanical performances of the produced materials are discussed in terms of the generated residual stress profile and compared to parent monolithic ceramics. q 2006 Elsevier Ltd. All rights reserved. Keywords: A. Ceramic-matrix composites; B. Residual/internal stress; B. Strength; B. Fracture toughness; Ceramic laminates
1. Introduction Ceramics are commonly considered as brittle materials. In spite of this, their very attractive physical and chemical properties make such materials suitable for different applications. The limited fracture toughness associated to the presence of flaws generated either upon processing and in service is responsible for their limited mechanical reliability. The resulting strength scatter is usually too large to allow safe design, unless statistical approaches identifying acceptable minimum failure risk are used. In addition, fracture usually occurs in a catastrophic manner in absence of any warning of the incipient rupture [1]. Many efforts have been made in the past to increase the mechanical reliability of glasses and ceramics. Higher fracture toughness have been attained through the exploitation of the reinforcing action of grain anisotropy or second phases or the promotion of crack shielding effects associated to phasetransformation or micro-cracking [1]. In such cases, a precise microstructure control is always required and this is achievable only with a careful control of starting material and process conditions. As an alternative, fracture behavior of ceramics has
* Corresponding author Tel.: C39 461 882468; fax: C39 461 881945. E-mail address: [email protected] (V.M. Sglavo). 1 Now at Eurocoating Spa, Via Al Dos de la Roda 60, 38057 Pergine Valsugana, Italy.
1359-8368/$ - see front matter q 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2006.02.001
been improved by introducing low-energy paths for growing crack in laminated structures [2–6] or by introducing compressive residual stresses [7,8]. Laminates presenting threshold strength have been also successfully produced by alternating thin compressive layers and thicker tensile layers [9]. Unfortunately, the most important limitations of such laminates is that they can be used only with specific orientations with respect to the applied load and, for example, they are not easily suitable to produce real components such as plates, shells or tubes as usually required in typical applications. The idea that surface stresses can hinder the growth of surface cracks has been extensively exploited in the past especially on glasses [10–11]. Sglavo and Green have recently proposed that the creation of a residual stress profile in glass with a maximum compression at a certain depth from the surface can arrest surface cracks and result in higher failure stress and limited strength variability [12–14]. One can point out that surface flaws represent the most typical defects in ceramic and glasses: in fact, once the processing procedures are optimized to reduce or remove heterogeneities that can generate volume defects [15,16], surface flaws are normally created during surface finishing or upon service. In addition, only surface defects become critical when a body is subjected to bending and not to tension, as it is usually the case in ceramic components. Residual stresses in ceramic materials can arise either from differences in the thermal expansion coefficient of the constituting grains or phases, uneven sintering rates or martensitic phase transformations. As described below, if the
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development of the residual stresses in ceramic multilayer is opportunely controlled, materials characterized by high fracture resistance and limited strength scatter can be designed and produced. By varying the nature, thickness and stacking order of the laminae, the residual stress profile developed after sintering can be tailored to promote the growth of surface cracks in a stable manner before final failure. In this way, strength predictable and variable as needed can be obtained by changing the multilayer ‘architecture’. Such approach is presented in the present work and reduced to practice for the production of alumina/zirconia and alumina/mullite composite laminates.
the effect of an external load, flaw with generic size (c1), enclosed in the interval cA ; cB and subjected to KextZT(c1), will propagate instantaneously up to a length within the interval [cA, cB] and then grow stably up to cB for higher Kext values. The arguments proposed so far are absolutely general, regardless the reasons for the non-constant fracture toughness. The presence of residual stresses inside the material can be responsible for a T-curve like that shown in Fig. 1. If the simple model represented in Fig. 2 is considered, which corresponds to a surface crack in a ceramic laminate, residual stresses, sres, are correlated to the stress intensity factor [17,18] ðc
Kres Z sres ðxÞh 2. Theory
(2)
0
The aim of the present work is to set up a design procedure useful to produce ceramic components with high mechanical reliability, i.e. characterized by limited strength scatter and, possibly, high fracture resistance. In order to reach such target, the idea is advanced that stable growth of defects could occur before final failure. In this way, regardless the initial flaw size, an invariant final strength can be attained. For example, stable crack propagation is possible when fracture toughness, T, is a growing function of crack length, c, steeper than the applied stress intensity factor, Kext, which is generally defined as KextZjsc0.5, where j is the shape factor and s the applied stress. Analytically, stable growth occurs when the following condition is satisfied [1,17]: Kext Z TðcÞ
x c ; dx c w
dKext dTðcÞ % dc dc
(1)
where h is a weight function and w is the finite width of the body. In the present analysis discontinuous stepwise stress profile is considered according to the laminated structure subjected to bending loads of relevance here (Fig. 2). Perfect adhesion between different laminae is also hypothesized. In addition, it is assumed that each layer is characterized by a constant fracture toughness value, KCi . Under the influence of the external load (Kext), crack propagation occurs when the sum (KresCKext) equals the fracture toughness, KCi , of the material at the crack tip. If the residual stresses are supposedly considered as a material property, the ‘apparent’ fracture toughness can be defined as: T i Z KCi KKres
(3)
It has been demonstrated elsewhere that the stability range, if any, is finite [12]. This is shown schematically in Fig. 1, where the interval [cA, cB] represents the range, where cracks can grow in a stable fashion under the effect of an external load. As a direct consequence all the defects included in such interval propagate to the same maximum value before final failure upon loading, thus leading to a unique strength value. To be more precise, if kinetic effects are limited or neglected, the stable crack growth interval can be extended down to cA ; in fact, under
It is clear from Eqs. (2) and (3) that for compressive residual stresses (negative) there is a beneficial effect on T. In addition, given a proper residual stress profile, it should be possible to obtain T being a steep growing function of c. Moreover, as surface flaws have been considered, T-curve is unique for any defect and, therefore, it can be considered as fixed with respect to the surface of the body. Consequently, crack length (c) and depth from the surface (x) can be regarded as identical quantities in the subsequent analysis. In order to understand, the effect of residual stress intensity and location on the apparent fracture toughness, it is useful to
Fig. 1. T-curve that allows the stable growth phenomenon in the interval (cA, cB). Straight lines are used to evaluate the stable growth interval and final strength, sF.
Fig. 2. Crack model considered in the present work.
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analyze some special cases. First of all, if the reference model (Fig. 2) is thought to correspond to an edge crack in a semiinfinite body, Eq. (2) can be simplified as Y Kres Z ðpcÞ0:5
ðc 0
sres ðxÞ
2c c2 Kx2
0:5 dx
(4)
where Yz1.1215. One could point out that such simplification is not rigorous as Y maintains a slight dependence on x/c [18]. Nevertheless, this allows to perform the calculations in closed form without loosing of generality. One very simple situation corresponds to the step profile shown in Fig. 3(a) defined as: ( sres Z 0 0! x! x1 (5) sres ZKsR x1 ! x!CN In this case, T can be analytically calculated as: 8 0! x! x1 T Z KC > > > > < 0 10:5 0 0 11 c p x > > > T Z KC C 2Y @ A sR @ Karcsin@ 0 AA x1 ! x!CN > : p 2 c (6)
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As shown in Fig. 3(b) a stability range exists between x1 and the tangent point between Kext and T. One can observe that for increasing x1 the strength decreases and the stability interval width increases. Since, both high strength and large stable growth interval are desirable, an intermediate value of x1 has to be considered in the perspective laminate design. On the other side, an increase of sR is useful to increase both the stable growth range and the maximum stress. In addition, if KC increases, the maximum stress is higher but the stability range decreases though one must considered that KC is a parameter that depends on the material selection and it is not usually modified in the design procedure. A more realistic residual stress shape is the square-wave profile (Fig. 4) defined as: 8 0! x! x1 > < sres Z 0 sres ZKsR x1 ! x! x2 (7) > : sres Z 0 x2 ! x!CN In this case, the T-curve can be calculated both analytically and by using the principle of superposition [1,17]. The squarewave profile can be considered in fact as the sum of two simple step profiles with stresses of identical amplitude but opposite sign placed at different depths (x1 and x2). The apparent fracture toughness becomes:
8 T Z KC 0! x! x1 > > > > > > 0 10:5 2 0 13 > > > > > c p x < x1 ! x! x2 T Z KC K2Y @ A sR 4 Karcsin@ 1 A5 p 2 c > > > 0 10:5 2 0 1 0 13 > > > > c x x > > T Z KC K2Y @ A sR 4arcsin@ 2 A Karcsin@ 1 A5 x2 ! x!CN > > : p c c
(8)
Fig. 3. Step residual stress profile (a) and corresponding apparent fracture toughness (b). The effects of intensity (left) and location (right) of the residual stress are shown.
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Fig. 4. Square-wave stress profile and relative apparent fracture toughness.
This special case is useful to discuss an important point. Depending on the width (x2Kx1) of the compressive layer, the tangent point can fall beyond the position x2. In this case, the stable growth range is automatically defined by the interval [x1, x2] and the maximum stress is lower than the tangent stress. Strength and instability point become, therefore, mutually independent within a certain degree. A single compressive layer of proper thickness placed at a certain depth from the surface is, therefore, suitable to generate a stable growth range for surface defects. Unfortunately, this simple solution is not actually practicable because the forces equilibrium in the component is not satisfied. In addition, in order to achieve elevated strength, the required compressive stress is usually very high and localized intense interlaminar shear stresses can be generated; these can be then responsible for delamination between layers. Edge cracking can also arise at the interface between highly compressed laminae. This phenomenon was analyzed in a previous work [19] to occur when the layer thickness is larger than a critical value, tc Z KC2 =½0:34ð1C nÞs2c , sc being the compressive stress and n the Poisson’s ratio. Layer thickness and compressive stress are,
therefore, mutually dependent also for this reason and it is not possible to design the desired mechanical behavior by using a square-wave stress (single layer) profile, only. Fortunately, almost all these problems can be overcome by considering a multilayered structure. Before moving towards a more complex profile, it is useful to analyze another simple case. Consider two stress profiles obtained by the combination of simple square-wave profiles of different (double, for simplicity) amplitude and identical extension (Fig. 5). This situation corresponds to laminates with two layers of different composition and identical thickness. The actual order of the two layers is the only difference between the two examined profiles. It is clear from Fig. 5 that the order of the compressive layers is important either for the final strength and the stability interval. Such consideration is general and the final conclusion can be drawn that the compression intensity in successive layers must be continuously growing to obtain a properly designed T-curve. At this point, the principle of superposition can be used to calculate the T-curve for a general multi-step profile. The proposed approach can be extended in fact to n layers provided that n step profiles with amplitude Dsj (Fig. 2), equal to the stress increase of layer j with respect to the previous one, are considered. A general equation, which defines the apparent fracture toughness for layer i in the interval [xiK1, xi] (Fig. 2), can be obtained T
Z KCi K
i X jZ1
2Y
c 0:5 p
Dsres;j
x i jK1 Karcsin 2 c
hp
(9)
xiK1 ! x! xi where i indicates the layer rank and xj is the starting depth of layer j. Eq. (9)represents a short notation of n different equations, the sum being calculated for different number of terms for each i. This represents a mathematical translation of the ‘memory’ effect of stress history that deeper layers maintain with respect to the layer previously encountered by the propagating crack. Regardless the layer order, since 2n parameters (xi, Dsi) are now available and two conditions have to be satisfied (forces equilibrium and equivalence between the sum of single layer
Fig. 5. Residual stress and corresponding T-curve for two simple square-wave profiles placed in different order.
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thickness and the total laminate thickness), 2nK2 are the remaining degrees of freedom suitable to define the desired T-curve. It is important to point out that in the calculations carried out to obtain Eq. (9) the approximation is made that the elastic modulus of the different layers is constant. It has been demonstrated elsewhere that the approximation in T estimate does not exceeds 10% if the Young’s modulus variation is less than 33% [20,21].
3. Laminates design Eq. (9) suggests some considerations about the conditions that a proper stress profiles should possess to promote the stable growth of surface cracks. The stable propagation of surface defects is possible only when the T-curve is a monotonic increasing function of c and this requires a continuous increase of the compressive stresses from the surface towards internal layers. A stress-free or slightly tensile stressed layer is also preferred on the surface, since this allows to move the lower boundary of the stable growth interval towards the surface. It is important to point out that, according to Eq. (9), the effect of the surface layer is transferred to all internal laminae. The surface tensile layer has in fact a reducing effect on the T-curve for any crack length and for this reason its depth and intensity must be limited, the maximum stress being, otherwise, too low. In addition, by using multi-step profiles it is possible to reduce the thickness of the most stressed layer with the introduction of intermediate layers before and beyond it. The risk of edge cracking and delamination phenomena are reduced accordingly. The residual stress profile that develops within a ceramic laminate is related either to the composition/microstructure and thickness of the laminae and to their stacking order. According to the theory of composite plies [22], in order to maintain flatness during in-plane loading, as in the case of biaxial residual stresses developed upon processing, laminate structure has to satisfy some symmetry conditions. If each layer is isotropic, like in ceramic laminae with fine and randomly oriented crystalline microstructure, and the stacking order is
symmetrical, the laminate remains flat upon sintering and, being orthotropic, its response to loading is similar to that of a homogeneous plate [22]. Regardless the physical source of residual stresses, their presence in co-sintered multilayer is related to constraining effect. Under the condition of perfectly adherent layers, every lamina must deform similarly and at the same rate of the others. The difference between free deformation or free deformation rate of the single lamina with respect to the average value of the whole laminate accounts for the creation of residual stresses. Such stresses can be either viscous or elastic in nature and can be relaxed or maintained within the material depending on temperature, cooling rate and material properties. With the exception of the edges, if thickness is much smaller than the other dimensions, each layer can be considered to be in a biaxial stress state. At this point the fundamental task to properly design a symmetric multilayer is the estimate of the biaxial residual stresses. In the common case of stresses developed from differences in thermal expansion coefficients only, the following conditions (related to forces equilibrium, compatibility and constitutive model) must be satisfied Xn s t Z 0 3i Z ei C ai DT Z 3 si Z Ei ei (10) iZ1 res;i i ai being the thermal expansion coefficient, Ei Z Ei =ð1Kni Þ (niZPoisson’s ratio, EiZYoung modulus), ei the elastic strain, 3i the deformation. The system defined by Eq. (10) represent a set of 3nC1 equations and 3nC1 unknowns (si, 3i, ei,3). The solution of such linear system allows to calculate the residual stress in the generic layer i (among n layers) as sres;i Z Ei ðaKa i ÞDT
(11)
where DTZTSFKTRT (TSFZstress free temperature, TRTZ room temperature) and a is the average thermal expansion coefficient of the whole laminate defined as Xn .Xn a Z E ta E t (12) 1 i i i 1 i i ti being the layer thickness. In this specific case, the residual stresses are, therefore, generated upon cooling after sintering. It has been shown in previous works that TSF represents the temperature below which the material can be considered to behave as a perfectly elastic body and visco–elastic relaxation phenomena do not occur [23]. It must be pointed out that the reported analysis, corresponding to the development of stresses from differences in thermal expansion coefficients only, can be easily generalized when other differential strain developers are active like those associated to martensitic phase transformations. In this case, the compatibility equation in Eq. (10) becomes 3i Z ei C ai DT C 3T Z 3
Fig. 6. Architecture of the AZ-1 and AM-1 laminates. Layers thickness and composition are reported (dimensions are not in scale).
485
(13)
where 3T represents the strain associated to phase transformation [8,24]. Eq. (9) represents the fundamental tool for the design of a ceramic laminate with pre-defined mechanical properties. Different ceramic layers can be stacked together in order to develop after sintering a specific residual stress profile that can
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be evaluated by Eq. (11) if elastic constants, thermal expansion coefficient and thickness of each layer are known. Since, the stress level in Eq. (11) does not depend on stacking order, the sequence of laminae can be still changed provided the symmetry condition is maintained to tailor the T-curve and promote stable growth of defects. Once the stress profile is defined, the apparent fracture toughness can be estimated by Eq. (9) and strength and fracture behavior are directly defined. By changing the stacking order and composition of the layers, i.e. the laminate architecture, it is therefore possible to produce a material with unique and predefined failure stress. As examples, ceramic laminates composed of layers belonging to the alumina/zirconia and alumina/mullite systems have been designed. The thermal expansion coefficient required for the development of the residual stress profile was tailored by changing the composition of the single laminae. The architecture of the engineered laminates is reported in Fig. 6. In the used notation, ‘AZw/y’ or ‘AMw/y’, A stays for alumina, Z for zirconia, M for mullite, w corresponds to the volume percent content of zirconia or mullite and y to the layer thickness in microns. The composition and thickness of the layers and their stacking order were selected to produce ceramic laminates with a ‘constant’ strength, sF, equal to z500 and z400 MPa in the AZ and AM system, respectively. The apparent fracture toughness curve and corresponding residual stress profile were also tailored in order to promote the stable growth of surface defects as deep as z150 and z180 mm in the AZ and AM system, respectively. On the basis of the aforementioned analysis, once the Young modulus, Poisson’s ratio, thermal expansion coefficient and fracture toughness for each layer are determined, the residual stress distribution and the corresponding apparent fracture toughness curve for each laminate can be estimated. In this study room temperature equal to 25 8C and stress-free temperature equal to 1200 8C were established as indicated in previous works [23–25]. The properties of the materials required for the calculation are summarized in Table 1 and in Fig. 7. Young modulus and Poisson’s ratio values for AM and AZ composites shown in Table 1 correspond to Voigt-Reuss bounds [17]; according to previous results [26], Young modulus and Poisson’s
Fig. 7. Thermal expansion coefficient for AZ and AM composites.
ratio equal to 229 GPa and 0.27, respectively, were considered for pure mullite. The elastic modulus for pure alumina and zirconia was measured on monolithic samples as reported elsewhere [27]. The difference between the elastic modulus bounds is lower than 7 and 10%, respectively for mullite and zirconia content below 40%. For the Poisson’s ratios the difference is lower than 1.5%. Therefore, the average of the values reported in Table 1 has been used for the evaluation of Eqs. (11) and (12), thus accepting an error equal to z5% at the highest for E*. The thermal expansion coefficient and fracture toughness for AM and AZ composites were measured on monolithic samples as reported in a previous work [27]. The residual stress profile and the KC;i curve for the AZ-1 and AM-1 engineered laminates are shown in Figs. 8 and 9,
Table 1 Materials properties used to estimate the stress distribution and the apparent fracture toughness Material
E (GPa)
KC (MPam0.5)
n
AM0/AZ0 AM10 AM20 AM30 AM40 AZ10 AZ20 AZ30 AZ40 AZ100
394 (14) 378O368 361O344 345O324 328O306 375O360 356O332 337O308 318O287 204 (8)
3.6 (0.2) 3.3 (0.2) 3.1 (0.3) 2.6 (0.2) 2.4 (0.2) 3.5 (0.3) 3.6 (0.2) 3.9 (0.3) 4.5 (0.3) –
0.23a 0.234O0.233 0.238O0.237 0.242O0.241 0.246O0.244 0.236O0.235 0.242O0.240 0.248O0.245 0.254O0.251 0.29a
Numbers between parentheses correspond to the standard deviation. Elastic modulus values correspond to calculated Voigt-Reuss bounds for AM10-AM40 and AZ10-AZ40 composites. a Ref. [26].
Fig. 8. Residual stress profile of the AZ-1 (a) and AM-1 (b) engineered laminates.
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respectively. The applied stress intensity factor corresponding to the predefined maximum stress (strength) and the cracks depth interval are also shown in Fig. 9. Since, KC;i was calculated step by step Eq. (9), the corresponding diagram is discontinuous at the boundary between layers, this reflecting the discontinuities in the sres diagram (Fig. 8). One can easily suppose that the real apparent fracture toughness trend is continuous and that the discontinuities in Fig. 9 correspond to mathematical artifacts only. 4. Production and properties of the designed laminates Ceramic laminates corresponding to the materials designed in the previous section were produced and characterized. As previously pointed out, the thermal expansion coefficient as required for the development of the residual stress profile was tailored by considering composites in the alumina/mullite and alumina/zirconia systems for the production of the single laminae. Alpha–alumina (ALCOA, A-16SG, D50Z0.4 mm) was considered as the fundamental starting material. High purity mullite (KCM Corp., KM101, D50Z0.77 mm) and yttria (3 mol%) stabilised zirconia (TOSOH, TZ-3YS, D50Z 0.4 mm) powders were chosen as second phases to vary the thermal expansion coefficient with respect to pure alumina. Green laminae were produced by tape casting water-based slurries. Suspensions were prepared by using NH4-PMA (Darvan Cw, R.T. Vanderbilt Inc.) as dispersant and acrylic
Fig. 9. Apparent fracture toughness for AZ-1 (a) and AM-1 (b) engineered laminates.
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emulsions (B-1235, DURAMAXw) as binder. A lower-Tg acrylic emulsion (B-1000, DURAMAXw) was also added in a 1:2 by weight ratio with respect to the binder content as plasticizer to increase green flexibility and to reduce cracks occurrence in the dried tape. The slurries were obtained using a two-stage process as detailed elsewhere [27,28]. All suspensions were produced with a powder content of 39 vol% and a final organic content around 21 vol%. Just before casting, slurries were filtered at 60 mm to ensure the elimination of any bubble or cluster of flocculated polymer. Tape casting was carried out using a double doctor-blade assembly (DDB-1-6, 6 in. wide, Richard E. Mistler Inc., USA) at a speed of 1 m/min for a length of about 1000 mm. A composite three-layer film (PET12/Al7/LDPE60, BP Europack, Italy) was used as substrate in order to make the removal of the dried green tape easier. For this reason the polyethylene hydrophobic side of the film was placed side-up. The substrate was placed on a rigid float glass plate in order to ensure a flat surface. The relative humidity of the over-standing environment was controlled and set to about 80% during casting and successive drying to avoid fast evaporation of the solvent and possible cracking of green tapes due to shrinkage stresses. Suspensions casting was carried out using two different blade heights, 250 and 100 mm, respectively. Drops of a 10 wt% wetting agent water solution (NH4-lauryl sulphate, code 09887, FLUKA CHEMIE AG, CH) were added to the slurries to help the casting tape spreading on the substrate when needed, particularly in the case of thinner tapes. Green tapes of nominal dimension 60!45 mm were punched using a hand-cutter, stacked together and thermocompressed at 70 8C and 30 MPa for 15 min applied by a universal mechanical testing machine (MTS Systems, mod. 810, USA). Two 100 mm thick PET layers were placed between the laminate and the die to make the removal easier. For mechanical characterisation bars of nominal dimensions 60! 6!1–2 mm were cut after the thermo-compression and then re-laminated [29] before final thermal treatment to avoid any delamination promoted by localised shear stresses developed upon cutting. Samples were finally sintered at 1600 8C for 2 h. After sintering the edges were slightly chamfered to remove macroscopic defects and geometrical irregularities. No further polishing and finishing operations were performed on the sample surfaces or edges to avoid any artificial reduction of flaws severity. Alumina (AM0) and AZ40 monolithic samples (thickness z1.5 mm) were produced in the same way for the measurement of thermal expansion coefficient, a, elastic modulus, E, and fracture toughness, KC. The thermal expansion coefficient was measured in the range 30–1000 8C by using a silica dilatometer and a heating rate of 2 8C/min. Elastic modulus was measured by 4-points bending tests (spans equal to 40 and 20 mm) by using a calibrated extensometer. Fracture toughness was measured by the conventional indentation fracture method [30]. The structure of the AZ-1 and AM-1 laminates is shown as an example in Fig. 10. The perfect adhesion between layers is evident as well as the absence of any edge cracks.
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The mechanical properties of the engineered AM and AZ laminates are compared to those of monolithic alumina and AZ40 composite in Table 2. The comparison is thought to be effective, since the surface layer of the engineered laminated is made by pure alumina and AZ40 composite in the AM-1 and AZ-1 laminate, respectively, and is subjected to quite small tensile stresses. The average bending strength measured on the AZ-1 and AZ40 samples is equal to 594G59 and 741G86 MPa, respectively. One can easily observe that the strength values measured on the AZ-1 laminate are in good agreement with the design value. Conversely, the strength scatter is not substantially reduced. This can be explained by considering the initial size of the flaws on the surface of the engineered AZ-1 laminate as it can be estimated from the strength on monolithic AZ40 samples ranging from 586 to 892 MPa. Semi-circular and through-thickness flaws can be considered as opposite situations. In the former case sizes ranging from 16 to 37 mm are calculated [17]; if through-thickness flaws are considered, their length ranges from 6 to 15 mm. The comparison between calculated sizes and T-curve shown in Fig. 9(a) points out that most of the surface defects in AZ-1 laminate are shorter than the lower boundary of the stability interval. Consequently, most of the flaws (i.e. the smallest ones) on the AZ-1 surface do not undergo to the desired stable growth before final failure but propagate unstably producing a scatter similar to homogeneous laminate. It is anyway interesting to observe that in this case the designed stress (z500 MPa) almost exactly corresponds to the minimum strength value. For the AM-1 laminate either there is an agreement between the design and measured failure stress and the strength scatter
Table 2 Fracture stress of the engineered laminates and monolithic samples sF (MPa) Monolithic AM0 Monolithic AZ40 AZ-1 laminate AM-1 laminate
418G45 741G86 594G59 457G32
is reduced. In this case, most of the initial cracks (ranging from 36 to 68 mm and from 14 to 28 mm if semi-circular or throughthickness cracks are considered, respectively) fall within the stability interval (Fig. 9(b)). Also in this case the minimum measured strength (405 MPa) is substantially identical to the design failure stress. One final comment regards the reliability of the laminates designed and produced in this work. If the mechanical reliability is intended as the possibility of defining a fracture strength for a given material associated to a relatively high safety factor, both engineered laminates considered here can be defined as reliable. In fact, since the measured failure stress is always larger than the design strength, it is definitely safe to use such value as the trusted mechanical resistance of the material. In addition, for the AM-1 laminate the condition is also fulfilled that the material fails when subjected to applied stresses equal to the design strength G7% with a probability larger than 60%.
5. Conclusions Innovative ceramic laminates with high mechanical reliability can be designed on the basis of a fracture mechanics approach. The residual stress that develops after sintering in ceramic multilayer due to differential thermal expansion is used to generate an apparent fracture toughness curve that increases with crack length and allows the stable growth of surface flaws. A certain number of requirements correlated to the laminate architecture have to be considered for the proper laminate design. The thickness and composition of different layers, that are though to be perfectly adherent, are responsible for the development of the residual stress; the stacking sequence determines the final stress profile. In addition, in order to avoid the initiation of delamination or edge cracks, the stress within successive layers must be continuously growing or decreasing. The examples reported in the present work, corresponding to laminates in the alumina–zirconia and alumina–mullite systems, show the suitability of the proposed design approach to obtain mechanically reliable ceramic characterized by a ‘constant’ strength or, at least, by a minimum mechanical resistance. References
Fig. 10. SEM micrograph showing the surface structure of the (a) AZ-1 and (b) AM-1 laminates.
[1] Lawn BR. Fracture of brittle solids. Cambridge: Cambridge University Press; 1993.
V.M. Sglavo, M. Bertoldi / Composites: Part B 37 (2006) 481–489 [2] Davis JB, Kristoffersson A, Carlstrom E, Clegg WJ. Fabrication and crack deflection in ceramic laminates with porous interlayer. J Am Ceram Soc 2000;83(10):2369–74. [3] Clegg WJ, Kendall K, Alford M. A simple way to make tough ceramics. Nature 1990;347:455–7. [4] Kriven WM, Kuo D. High-strength, flaw-tolerant, oxide ceramic composite. US Patient No. 5,948,516; 1999. [5] Mistler RE. Strengthening alumina substrates by incorporating grain growth inhibitor in surface and promoter in interior. US Patient No. 3,652,378; 1972. [6] Harmer MP, Chan NM, Miller GA. Unique opportunities for microstructural engineering with duplex and laminar ceramic composites. J Am Ceram Soc 1992;75(7):1715–28. [7] Russo CJ, Harmer P, Chan HM, Miller GA. Design of laminated ceramic composite for improved strength and toughness. J Am Ceram Soc 1992; 75(12):3396–400. [8] Lakshminarayanan R, Shetty DK, Cutler RA. Toughening of layered ceramic composites with residual surface compression. J Am Ceram Soc 1996;79(1):79–87. [9] Rao MP, Sa´nchez-Herencia AJ, Beltz GE, McMeeking RM, Lange FF. Laminar ceramics that exhibit a threshold strength. Science 1999;286: 102–5. [10] Donald IW. Methods for improving the mechanical properties of oxide glasses. J Mater Sci 1989;24:4177–208. [11] Bartolomew RF, Garfinkel HM. In: Uhlmann DR, Kreidl NJ, editors. Glass science and technology, vol. 5. New York: Academic Press; 1980. [12] Sglavo VM, Larentis L, Green DJ. Flaw-insensitive ion-exchanged glass: I, theoretical aspects. J Am Ceram Soc 2001;84(8):1827–31. [13] Sglavo VM, Green DJ. Flaw-insensitive ion-exchanged glass: II, production and mechanical performance. J Am Ceram Soc 2001;84(8): 1832–8. [14] Green DJ, Tandon R, Sglavo VM. Crack arrest and multiple cracking in glass using designed residual stress profiles. Science 1999;283:1295–7. [15] Lange FF. Powder processing science and technology for increased reliability. J Am Ceram Soc 1989;72(1):3–15. [16] Yu BC, Lange FF. Colloidal isopressing: a new shape forming method. Adv Mater 2001;13(4):276–80.
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[17] Green DJ. An introduction to the mechanical properties of ceramics. Cambridge: Cambridge University Press; 1998. [18] Munz D, Fett T. Ceramics: mechanical properties, failure behaviour, materials selection, vol. 36. Berlin-Heidelberg (D): Springer Series in Material Science; 1999. [19] Ho S, Hillman C, Lange FF, Suo Z. Surface cracking in layers under biaxial, residual compressive stress. J Am Ceram Soc 1995;78(9):2353–9. [20] Moon RJ, Hoffman M, Hilden J, Bowman KJ, Trumble KP, Ro¨del J. Rcurve behaviour in alumina/zirconia composites with repeating graded layers. Eng Fract Mech 2002;69:1647–65. [21] Chung TJ, Neubrand A, Ro¨del J, Fett T. Fracture toughness and R-curve behaviour of Al2O3/Al FGMs. Ceram Trans 2001;114:789–96. [22] Halpin JC. Primer on composite materials analysis. 2nd ed. Lancaster, PA: Technomic Publishing Company Inc.; 1992. [23] Rao MP, Ro¨del J, Lange FF. Residual stress induced R-curves in laminar ceramics that exhibit a threshold strength. J Am Ceram Soc 2001;84(11): 2722–4. [24] Pontin MG, Rao MP, Sa´nchez-Herencia AJ, Lange FF. Laminar ceramics utilizing the zirconia tetragonal-to-monoclinic phase transformation to obtain a threshold strength. J Am Ceram Soc 2002;85(12):3041–8. [25] Sergo V, Wang X, Becher PF, Clarke DR. Residual stresses in alumina/ceria stabilized zirconia composites. J Am Ceram Soc 1995; 78(8):2213–4. [26] Ledbetter H, Kim S, Crudele DB, Kriven W. Elastic properties of mullite. J Am Ceram Soc 1998;81(4):1025–8. [27] Sglavo VM, Paternoster M, Bertoldi M. Tailored residual stresses in high reliability alumina-mullite ceramic laminates. J Am Ceram Soc; 2005; 88[10]:2826–32. [28] Bertoldi M, Paternoster M, Sglavo VM. Production of multilayer ceramic laminates with improved mechanical properties. Adv Ceram Matrix Compos IX Ceram Trans 2003;153:89–102. [29] Plucknett KP, Caceres CH, Hughes CH, Wilkinson DS. Processing of tape-cast laminates prepared from fine alumina/zirconia powders. J Am Ceram Soc 1994;77(8):2145–53. [30] Anstis GR, Chantikul P, Lawn BR, Marshall DB. A critical evaluation of indentation techniques for measuring fracture toughness: I. Direct crack measurement. J Am Ceram Soc 1981;64:533–9.
Design and production of ceramic laminates with high mechanical reliability Vincenzo M. Sglavo *, Massimo Bertoldi 1 Dipartimento di Ingegneria dei Materiali e Tecnologie Industriali, Universita` di Trento, Via Mesiano, 77, 38050 Trento, Italy Available online 20 March 2006
Abstract A procedure for designing innovative ceramic laminates characterized by high mechanical reliability is proposed in this work. A fracture mechanics approach has been considered to define the stacking sequence, thickness and composition of the different laminae on the basis of the requested strength and of the defect size distribution. Once the different laminae are stacked together a residual stress profile is generated upon cooling after sintering because of the differential thermal expansion coefficient. Such residual stress profile is conceived in order to allow stable growth of surface defects upon bending and guarantee limited strength scatter. As an example, the proposed approach is used to design and produce ceramic laminates in the alumina–zirconia and alumina–mullite system. Mechanical performances of the produced materials are discussed in terms of the generated residual stress profile and compared to parent monolithic ceramics. q 2006 Elsevier Ltd. All rights reserved. Keywords: A. Ceramic-matrix composites; B. Residual/internal stress; B. Strength; B. Fracture toughness; Ceramic laminates
1. Introduction Ceramics are commonly considered as brittle materials. In spite of this, their very attractive physical and chemical properties make such materials suitable for different applications. The limited fracture toughness associated to the presence of flaws generated either upon processing and in service is responsible for their limited mechanical reliability. The resulting strength scatter is usually too large to allow safe design, unless statistical approaches identifying acceptable minimum failure risk are used. In addition, fracture usually occurs in a catastrophic manner in absence of any warning of the incipient rupture [1]. Many efforts have been made in the past to increase the mechanical reliability of glasses and ceramics. Higher fracture toughness have been attained through the exploitation of the reinforcing action of grain anisotropy or second phases or the promotion of crack shielding effects associated to phasetransformation or micro-cracking [1]. In such cases, a precise microstructure control is always required and this is achievable only with a careful control of starting material and process conditions. As an alternative, fracture behavior of ceramics has
* Corresponding author Tel.: C39 461 882468; fax: C39 461 881945. E-mail address: [email protected] (V.M. Sglavo). 1 Now at Eurocoating Spa, Via Al Dos de la Roda 60, 38057 Pergine Valsugana, Italy.
1359-8368/$ - see front matter q 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2006.02.001
been improved by introducing low-energy paths for growing crack in laminated structures [2–6] or by introducing compressive residual stresses [7,8]. Laminates presenting threshold strength have been also successfully produced by alternating thin compressive layers and thicker tensile layers [9]. Unfortunately, the most important limitations of such laminates is that they can be used only with specific orientations with respect to the applied load and, for example, they are not easily suitable to produce real components such as plates, shells or tubes as usually required in typical applications. The idea that surface stresses can hinder the growth of surface cracks has been extensively exploited in the past especially on glasses [10–11]. Sglavo and Green have recently proposed that the creation of a residual stress profile in glass with a maximum compression at a certain depth from the surface can arrest surface cracks and result in higher failure stress and limited strength variability [12–14]. One can point out that surface flaws represent the most typical defects in ceramic and glasses: in fact, once the processing procedures are optimized to reduce or remove heterogeneities that can generate volume defects [15,16], surface flaws are normally created during surface finishing or upon service. In addition, only surface defects become critical when a body is subjected to bending and not to tension, as it is usually the case in ceramic components. Residual stresses in ceramic materials can arise either from differences in the thermal expansion coefficient of the constituting grains or phases, uneven sintering rates or martensitic phase transformations. As described below, if the
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development of the residual stresses in ceramic multilayer is opportunely controlled, materials characterized by high fracture resistance and limited strength scatter can be designed and produced. By varying the nature, thickness and stacking order of the laminae, the residual stress profile developed after sintering can be tailored to promote the growth of surface cracks in a stable manner before final failure. In this way, strength predictable and variable as needed can be obtained by changing the multilayer ‘architecture’. Such approach is presented in the present work and reduced to practice for the production of alumina/zirconia and alumina/mullite composite laminates.
the effect of an external load, flaw with generic size (c1), enclosed in the interval cA ; cB and subjected to KextZT(c1), will propagate instantaneously up to a length within the interval [cA, cB] and then grow stably up to cB for higher Kext values. The arguments proposed so far are absolutely general, regardless the reasons for the non-constant fracture toughness. The presence of residual stresses inside the material can be responsible for a T-curve like that shown in Fig. 1. If the simple model represented in Fig. 2 is considered, which corresponds to a surface crack in a ceramic laminate, residual stresses, sres, are correlated to the stress intensity factor [17,18] ðc
Kres Z sres ðxÞh 2. Theory
(2)
0
The aim of the present work is to set up a design procedure useful to produce ceramic components with high mechanical reliability, i.e. characterized by limited strength scatter and, possibly, high fracture resistance. In order to reach such target, the idea is advanced that stable growth of defects could occur before final failure. In this way, regardless the initial flaw size, an invariant final strength can be attained. For example, stable crack propagation is possible when fracture toughness, T, is a growing function of crack length, c, steeper than the applied stress intensity factor, Kext, which is generally defined as KextZjsc0.5, where j is the shape factor and s the applied stress. Analytically, stable growth occurs when the following condition is satisfied [1,17]: Kext Z TðcÞ
x c ; dx c w
dKext dTðcÞ % dc dc
(1)
where h is a weight function and w is the finite width of the body. In the present analysis discontinuous stepwise stress profile is considered according to the laminated structure subjected to bending loads of relevance here (Fig. 2). Perfect adhesion between different laminae is also hypothesized. In addition, it is assumed that each layer is characterized by a constant fracture toughness value, KCi . Under the influence of the external load (Kext), crack propagation occurs when the sum (KresCKext) equals the fracture toughness, KCi , of the material at the crack tip. If the residual stresses are supposedly considered as a material property, the ‘apparent’ fracture toughness can be defined as: T i Z KCi KKres
(3)
It has been demonstrated elsewhere that the stability range, if any, is finite [12]. This is shown schematically in Fig. 1, where the interval [cA, cB] represents the range, where cracks can grow in a stable fashion under the effect of an external load. As a direct consequence all the defects included in such interval propagate to the same maximum value before final failure upon loading, thus leading to a unique strength value. To be more precise, if kinetic effects are limited or neglected, the stable crack growth interval can be extended down to cA ; in fact, under
It is clear from Eqs. (2) and (3) that for compressive residual stresses (negative) there is a beneficial effect on T. In addition, given a proper residual stress profile, it should be possible to obtain T being a steep growing function of c. Moreover, as surface flaws have been considered, T-curve is unique for any defect and, therefore, it can be considered as fixed with respect to the surface of the body. Consequently, crack length (c) and depth from the surface (x) can be regarded as identical quantities in the subsequent analysis. In order to understand, the effect of residual stress intensity and location on the apparent fracture toughness, it is useful to
Fig. 1. T-curve that allows the stable growth phenomenon in the interval (cA, cB). Straight lines are used to evaluate the stable growth interval and final strength, sF.
Fig. 2. Crack model considered in the present work.
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analyze some special cases. First of all, if the reference model (Fig. 2) is thought to correspond to an edge crack in a semiinfinite body, Eq. (2) can be simplified as Y Kres Z ðpcÞ0:5
ðc 0
sres ðxÞ
2c c2 Kx2
0:5 dx
(4)
where Yz1.1215. One could point out that such simplification is not rigorous as Y maintains a slight dependence on x/c [18]. Nevertheless, this allows to perform the calculations in closed form without loosing of generality. One very simple situation corresponds to the step profile shown in Fig. 3(a) defined as: ( sres Z 0 0! x! x1 (5) sres ZKsR x1 ! x!CN In this case, T can be analytically calculated as: 8 0! x! x1 T Z KC > > > > < 0 10:5 0 0 11 c p x > > > T Z KC C 2Y @ A sR @ Karcsin@ 0 AA x1 ! x!CN > : p 2 c (6)
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As shown in Fig. 3(b) a stability range exists between x1 and the tangent point between Kext and T. One can observe that for increasing x1 the strength decreases and the stability interval width increases. Since, both high strength and large stable growth interval are desirable, an intermediate value of x1 has to be considered in the perspective laminate design. On the other side, an increase of sR is useful to increase both the stable growth range and the maximum stress. In addition, if KC increases, the maximum stress is higher but the stability range decreases though one must considered that KC is a parameter that depends on the material selection and it is not usually modified in the design procedure. A more realistic residual stress shape is the square-wave profile (Fig. 4) defined as: 8 0! x! x1 > < sres Z 0 sres ZKsR x1 ! x! x2 (7) > : sres Z 0 x2 ! x!CN In this case, the T-curve can be calculated both analytically and by using the principle of superposition [1,17]. The squarewave profile can be considered in fact as the sum of two simple step profiles with stresses of identical amplitude but opposite sign placed at different depths (x1 and x2). The apparent fracture toughness becomes:
8 T Z KC 0! x! x1 > > > > > > 0 10:5 2 0 13 > > > > > c p x < x1 ! x! x2 T Z KC K2Y @ A sR 4 Karcsin@ 1 A5 p 2 c > > > 0 10:5 2 0 1 0 13 > > > > c x x > > T Z KC K2Y @ A sR 4arcsin@ 2 A Karcsin@ 1 A5 x2 ! x!CN > > : p c c
(8)
Fig. 3. Step residual stress profile (a) and corresponding apparent fracture toughness (b). The effects of intensity (left) and location (right) of the residual stress are shown.
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Fig. 4. Square-wave stress profile and relative apparent fracture toughness.
This special case is useful to discuss an important point. Depending on the width (x2Kx1) of the compressive layer, the tangent point can fall beyond the position x2. In this case, the stable growth range is automatically defined by the interval [x1, x2] and the maximum stress is lower than the tangent stress. Strength and instability point become, therefore, mutually independent within a certain degree. A single compressive layer of proper thickness placed at a certain depth from the surface is, therefore, suitable to generate a stable growth range for surface defects. Unfortunately, this simple solution is not actually practicable because the forces equilibrium in the component is not satisfied. In addition, in order to achieve elevated strength, the required compressive stress is usually very high and localized intense interlaminar shear stresses can be generated; these can be then responsible for delamination between layers. Edge cracking can also arise at the interface between highly compressed laminae. This phenomenon was analyzed in a previous work [19] to occur when the layer thickness is larger than a critical value, tc Z KC2 =½0:34ð1C nÞs2c , sc being the compressive stress and n the Poisson’s ratio. Layer thickness and compressive stress are,
therefore, mutually dependent also for this reason and it is not possible to design the desired mechanical behavior by using a square-wave stress (single layer) profile, only. Fortunately, almost all these problems can be overcome by considering a multilayered structure. Before moving towards a more complex profile, it is useful to analyze another simple case. Consider two stress profiles obtained by the combination of simple square-wave profiles of different (double, for simplicity) amplitude and identical extension (Fig. 5). This situation corresponds to laminates with two layers of different composition and identical thickness. The actual order of the two layers is the only difference between the two examined profiles. It is clear from Fig. 5 that the order of the compressive layers is important either for the final strength and the stability interval. Such consideration is general and the final conclusion can be drawn that the compression intensity in successive layers must be continuously growing to obtain a properly designed T-curve. At this point, the principle of superposition can be used to calculate the T-curve for a general multi-step profile. The proposed approach can be extended in fact to n layers provided that n step profiles with amplitude Dsj (Fig. 2), equal to the stress increase of layer j with respect to the previous one, are considered. A general equation, which defines the apparent fracture toughness for layer i in the interval [xiK1, xi] (Fig. 2), can be obtained T
Z KCi K
i X jZ1
2Y
c 0:5 p
Dsres;j
x i jK1 Karcsin 2 c
hp
(9)
xiK1 ! x! xi where i indicates the layer rank and xj is the starting depth of layer j. Eq. (9)represents a short notation of n different equations, the sum being calculated for different number of terms for each i. This represents a mathematical translation of the ‘memory’ effect of stress history that deeper layers maintain with respect to the layer previously encountered by the propagating crack. Regardless the layer order, since 2n parameters (xi, Dsi) are now available and two conditions have to be satisfied (forces equilibrium and equivalence between the sum of single layer
Fig. 5. Residual stress and corresponding T-curve for two simple square-wave profiles placed in different order.
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thickness and the total laminate thickness), 2nK2 are the remaining degrees of freedom suitable to define the desired T-curve. It is important to point out that in the calculations carried out to obtain Eq. (9) the approximation is made that the elastic modulus of the different layers is constant. It has been demonstrated elsewhere that the approximation in T estimate does not exceeds 10% if the Young’s modulus variation is less than 33% [20,21].
3. Laminates design Eq. (9) suggests some considerations about the conditions that a proper stress profiles should possess to promote the stable growth of surface cracks. The stable propagation of surface defects is possible only when the T-curve is a monotonic increasing function of c and this requires a continuous increase of the compressive stresses from the surface towards internal layers. A stress-free or slightly tensile stressed layer is also preferred on the surface, since this allows to move the lower boundary of the stable growth interval towards the surface. It is important to point out that, according to Eq. (9), the effect of the surface layer is transferred to all internal laminae. The surface tensile layer has in fact a reducing effect on the T-curve for any crack length and for this reason its depth and intensity must be limited, the maximum stress being, otherwise, too low. In addition, by using multi-step profiles it is possible to reduce the thickness of the most stressed layer with the introduction of intermediate layers before and beyond it. The risk of edge cracking and delamination phenomena are reduced accordingly. The residual stress profile that develops within a ceramic laminate is related either to the composition/microstructure and thickness of the laminae and to their stacking order. According to the theory of composite plies [22], in order to maintain flatness during in-plane loading, as in the case of biaxial residual stresses developed upon processing, laminate structure has to satisfy some symmetry conditions. If each layer is isotropic, like in ceramic laminae with fine and randomly oriented crystalline microstructure, and the stacking order is
symmetrical, the laminate remains flat upon sintering and, being orthotropic, its response to loading is similar to that of a homogeneous plate [22]. Regardless the physical source of residual stresses, their presence in co-sintered multilayer is related to constraining effect. Under the condition of perfectly adherent layers, every lamina must deform similarly and at the same rate of the others. The difference between free deformation or free deformation rate of the single lamina with respect to the average value of the whole laminate accounts for the creation of residual stresses. Such stresses can be either viscous or elastic in nature and can be relaxed or maintained within the material depending on temperature, cooling rate and material properties. With the exception of the edges, if thickness is much smaller than the other dimensions, each layer can be considered to be in a biaxial stress state. At this point the fundamental task to properly design a symmetric multilayer is the estimate of the biaxial residual stresses. In the common case of stresses developed from differences in thermal expansion coefficients only, the following conditions (related to forces equilibrium, compatibility and constitutive model) must be satisfied Xn s t Z 0 3i Z ei C ai DT Z 3 si Z Ei ei (10) iZ1 res;i i ai being the thermal expansion coefficient, Ei Z Ei =ð1Kni Þ (niZPoisson’s ratio, EiZYoung modulus), ei the elastic strain, 3i the deformation. The system defined by Eq. (10) represent a set of 3nC1 equations and 3nC1 unknowns (si, 3i, ei,3). The solution of such linear system allows to calculate the residual stress in the generic layer i (among n layers) as sres;i Z Ei ðaKa i ÞDT
(11)
where DTZTSFKTRT (TSFZstress free temperature, TRTZ room temperature) and a is the average thermal expansion coefficient of the whole laminate defined as Xn .Xn a Z E ta E t (12) 1 i i i 1 i i ti being the layer thickness. In this specific case, the residual stresses are, therefore, generated upon cooling after sintering. It has been shown in previous works that TSF represents the temperature below which the material can be considered to behave as a perfectly elastic body and visco–elastic relaxation phenomena do not occur [23]. It must be pointed out that the reported analysis, corresponding to the development of stresses from differences in thermal expansion coefficients only, can be easily generalized when other differential strain developers are active like those associated to martensitic phase transformations. In this case, the compatibility equation in Eq. (10) becomes 3i Z ei C ai DT C 3T Z 3
Fig. 6. Architecture of the AZ-1 and AM-1 laminates. Layers thickness and composition are reported (dimensions are not in scale).
485
(13)
where 3T represents the strain associated to phase transformation [8,24]. Eq. (9) represents the fundamental tool for the design of a ceramic laminate with pre-defined mechanical properties. Different ceramic layers can be stacked together in order to develop after sintering a specific residual stress profile that can
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be evaluated by Eq. (11) if elastic constants, thermal expansion coefficient and thickness of each layer are known. Since, the stress level in Eq. (11) does not depend on stacking order, the sequence of laminae can be still changed provided the symmetry condition is maintained to tailor the T-curve and promote stable growth of defects. Once the stress profile is defined, the apparent fracture toughness can be estimated by Eq. (9) and strength and fracture behavior are directly defined. By changing the stacking order and composition of the layers, i.e. the laminate architecture, it is therefore possible to produce a material with unique and predefined failure stress. As examples, ceramic laminates composed of layers belonging to the alumina/zirconia and alumina/mullite systems have been designed. The thermal expansion coefficient required for the development of the residual stress profile was tailored by changing the composition of the single laminae. The architecture of the engineered laminates is reported in Fig. 6. In the used notation, ‘AZw/y’ or ‘AMw/y’, A stays for alumina, Z for zirconia, M for mullite, w corresponds to the volume percent content of zirconia or mullite and y to the layer thickness in microns. The composition and thickness of the layers and their stacking order were selected to produce ceramic laminates with a ‘constant’ strength, sF, equal to z500 and z400 MPa in the AZ and AM system, respectively. The apparent fracture toughness curve and corresponding residual stress profile were also tailored in order to promote the stable growth of surface defects as deep as z150 and z180 mm in the AZ and AM system, respectively. On the basis of the aforementioned analysis, once the Young modulus, Poisson’s ratio, thermal expansion coefficient and fracture toughness for each layer are determined, the residual stress distribution and the corresponding apparent fracture toughness curve for each laminate can be estimated. In this study room temperature equal to 25 8C and stress-free temperature equal to 1200 8C were established as indicated in previous works [23–25]. The properties of the materials required for the calculation are summarized in Table 1 and in Fig. 7. Young modulus and Poisson’s ratio values for AM and AZ composites shown in Table 1 correspond to Voigt-Reuss bounds [17]; according to previous results [26], Young modulus and Poisson’s
Fig. 7. Thermal expansion coefficient for AZ and AM composites.
ratio equal to 229 GPa and 0.27, respectively, were considered for pure mullite. The elastic modulus for pure alumina and zirconia was measured on monolithic samples as reported elsewhere [27]. The difference between the elastic modulus bounds is lower than 7 and 10%, respectively for mullite and zirconia content below 40%. For the Poisson’s ratios the difference is lower than 1.5%. Therefore, the average of the values reported in Table 1 has been used for the evaluation of Eqs. (11) and (12), thus accepting an error equal to z5% at the highest for E*. The thermal expansion coefficient and fracture toughness for AM and AZ composites were measured on monolithic samples as reported in a previous work [27]. The residual stress profile and the KC;i curve for the AZ-1 and AM-1 engineered laminates are shown in Figs. 8 and 9,
Table 1 Materials properties used to estimate the stress distribution and the apparent fracture toughness Material
E (GPa)
KC (MPam0.5)
n
AM0/AZ0 AM10 AM20 AM30 AM40 AZ10 AZ20 AZ30 AZ40 AZ100
394 (14) 378O368 361O344 345O324 328O306 375O360 356O332 337O308 318O287 204 (8)
3.6 (0.2) 3.3 (0.2) 3.1 (0.3) 2.6 (0.2) 2.4 (0.2) 3.5 (0.3) 3.6 (0.2) 3.9 (0.3) 4.5 (0.3) –
0.23a 0.234O0.233 0.238O0.237 0.242O0.241 0.246O0.244 0.236O0.235 0.242O0.240 0.248O0.245 0.254O0.251 0.29a
Numbers between parentheses correspond to the standard deviation. Elastic modulus values correspond to calculated Voigt-Reuss bounds for AM10-AM40 and AZ10-AZ40 composites. a Ref. [26].
Fig. 8. Residual stress profile of the AZ-1 (a) and AM-1 (b) engineered laminates.
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respectively. The applied stress intensity factor corresponding to the predefined maximum stress (strength) and the cracks depth interval are also shown in Fig. 9. Since, KC;i was calculated step by step Eq. (9), the corresponding diagram is discontinuous at the boundary between layers, this reflecting the discontinuities in the sres diagram (Fig. 8). One can easily suppose that the real apparent fracture toughness trend is continuous and that the discontinuities in Fig. 9 correspond to mathematical artifacts only. 4. Production and properties of the designed laminates Ceramic laminates corresponding to the materials designed in the previous section were produced and characterized. As previously pointed out, the thermal expansion coefficient as required for the development of the residual stress profile was tailored by considering composites in the alumina/mullite and alumina/zirconia systems for the production of the single laminae. Alpha–alumina (ALCOA, A-16SG, D50Z0.4 mm) was considered as the fundamental starting material. High purity mullite (KCM Corp., KM101, D50Z0.77 mm) and yttria (3 mol%) stabilised zirconia (TOSOH, TZ-3YS, D50Z 0.4 mm) powders were chosen as second phases to vary the thermal expansion coefficient with respect to pure alumina. Green laminae were produced by tape casting water-based slurries. Suspensions were prepared by using NH4-PMA (Darvan Cw, R.T. Vanderbilt Inc.) as dispersant and acrylic
Fig. 9. Apparent fracture toughness for AZ-1 (a) and AM-1 (b) engineered laminates.
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emulsions (B-1235, DURAMAXw) as binder. A lower-Tg acrylic emulsion (B-1000, DURAMAXw) was also added in a 1:2 by weight ratio with respect to the binder content as plasticizer to increase green flexibility and to reduce cracks occurrence in the dried tape. The slurries were obtained using a two-stage process as detailed elsewhere [27,28]. All suspensions were produced with a powder content of 39 vol% and a final organic content around 21 vol%. Just before casting, slurries were filtered at 60 mm to ensure the elimination of any bubble or cluster of flocculated polymer. Tape casting was carried out using a double doctor-blade assembly (DDB-1-6, 6 in. wide, Richard E. Mistler Inc., USA) at a speed of 1 m/min for a length of about 1000 mm. A composite three-layer film (PET12/Al7/LDPE60, BP Europack, Italy) was used as substrate in order to make the removal of the dried green tape easier. For this reason the polyethylene hydrophobic side of the film was placed side-up. The substrate was placed on a rigid float glass plate in order to ensure a flat surface. The relative humidity of the over-standing environment was controlled and set to about 80% during casting and successive drying to avoid fast evaporation of the solvent and possible cracking of green tapes due to shrinkage stresses. Suspensions casting was carried out using two different blade heights, 250 and 100 mm, respectively. Drops of a 10 wt% wetting agent water solution (NH4-lauryl sulphate, code 09887, FLUKA CHEMIE AG, CH) were added to the slurries to help the casting tape spreading on the substrate when needed, particularly in the case of thinner tapes. Green tapes of nominal dimension 60!45 mm were punched using a hand-cutter, stacked together and thermocompressed at 70 8C and 30 MPa for 15 min applied by a universal mechanical testing machine (MTS Systems, mod. 810, USA). Two 100 mm thick PET layers were placed between the laminate and the die to make the removal easier. For mechanical characterisation bars of nominal dimensions 60! 6!1–2 mm were cut after the thermo-compression and then re-laminated [29] before final thermal treatment to avoid any delamination promoted by localised shear stresses developed upon cutting. Samples were finally sintered at 1600 8C for 2 h. After sintering the edges were slightly chamfered to remove macroscopic defects and geometrical irregularities. No further polishing and finishing operations were performed on the sample surfaces or edges to avoid any artificial reduction of flaws severity. Alumina (AM0) and AZ40 monolithic samples (thickness z1.5 mm) were produced in the same way for the measurement of thermal expansion coefficient, a, elastic modulus, E, and fracture toughness, KC. The thermal expansion coefficient was measured in the range 30–1000 8C by using a silica dilatometer and a heating rate of 2 8C/min. Elastic modulus was measured by 4-points bending tests (spans equal to 40 and 20 mm) by using a calibrated extensometer. Fracture toughness was measured by the conventional indentation fracture method [30]. The structure of the AZ-1 and AM-1 laminates is shown as an example in Fig. 10. The perfect adhesion between layers is evident as well as the absence of any edge cracks.
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The mechanical properties of the engineered AM and AZ laminates are compared to those of monolithic alumina and AZ40 composite in Table 2. The comparison is thought to be effective, since the surface layer of the engineered laminated is made by pure alumina and AZ40 composite in the AM-1 and AZ-1 laminate, respectively, and is subjected to quite small tensile stresses. The average bending strength measured on the AZ-1 and AZ40 samples is equal to 594G59 and 741G86 MPa, respectively. One can easily observe that the strength values measured on the AZ-1 laminate are in good agreement with the design value. Conversely, the strength scatter is not substantially reduced. This can be explained by considering the initial size of the flaws on the surface of the engineered AZ-1 laminate as it can be estimated from the strength on monolithic AZ40 samples ranging from 586 to 892 MPa. Semi-circular and through-thickness flaws can be considered as opposite situations. In the former case sizes ranging from 16 to 37 mm are calculated [17]; if through-thickness flaws are considered, their length ranges from 6 to 15 mm. The comparison between calculated sizes and T-curve shown in Fig. 9(a) points out that most of the surface defects in AZ-1 laminate are shorter than the lower boundary of the stability interval. Consequently, most of the flaws (i.e. the smallest ones) on the AZ-1 surface do not undergo to the desired stable growth before final failure but propagate unstably producing a scatter similar to homogeneous laminate. It is anyway interesting to observe that in this case the designed stress (z500 MPa) almost exactly corresponds to the minimum strength value. For the AM-1 laminate either there is an agreement between the design and measured failure stress and the strength scatter
Table 2 Fracture stress of the engineered laminates and monolithic samples sF (MPa) Monolithic AM0 Monolithic AZ40 AZ-1 laminate AM-1 laminate
418G45 741G86 594G59 457G32
is reduced. In this case, most of the initial cracks (ranging from 36 to 68 mm and from 14 to 28 mm if semi-circular or throughthickness cracks are considered, respectively) fall within the stability interval (Fig. 9(b)). Also in this case the minimum measured strength (405 MPa) is substantially identical to the design failure stress. One final comment regards the reliability of the laminates designed and produced in this work. If the mechanical reliability is intended as the possibility of defining a fracture strength for a given material associated to a relatively high safety factor, both engineered laminates considered here can be defined as reliable. In fact, since the measured failure stress is always larger than the design strength, it is definitely safe to use such value as the trusted mechanical resistance of the material. In addition, for the AM-1 laminate the condition is also fulfilled that the material fails when subjected to applied stresses equal to the design strength G7% with a probability larger than 60%.
5. Conclusions Innovative ceramic laminates with high mechanical reliability can be designed on the basis of a fracture mechanics approach. The residual stress that develops after sintering in ceramic multilayer due to differential thermal expansion is used to generate an apparent fracture toughness curve that increases with crack length and allows the stable growth of surface flaws. A certain number of requirements correlated to the laminate architecture have to be considered for the proper laminate design. The thickness and composition of different layers, that are though to be perfectly adherent, are responsible for the development of the residual stress; the stacking sequence determines the final stress profile. In addition, in order to avoid the initiation of delamination or edge cracks, the stress within successive layers must be continuously growing or decreasing. The examples reported in the present work, corresponding to laminates in the alumina–zirconia and alumina–mullite systems, show the suitability of the proposed design approach to obtain mechanically reliable ceramic characterized by a ‘constant’ strength or, at least, by a minimum mechanical resistance. References
Fig. 10. SEM micrograph showing the surface structure of the (a) AZ-1 and (b) AM-1 laminates.
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