Prediction of edge and tunnelling crack formation in layered ceramics using a stress-energy fracture criterion

Journal of the European Ceramic Society 43 (2023) 2928–2934 Available online 13 December 2022 0955-2219/© 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Prediction of edge and tunnelling crack formation in layered ceramics using a stress-energy fracture criterion Roman Papˇ sík a , Oldˇ rich ˇ Seveˇ cek b , Anna-Katharina Hofer a , Irina Kraleva a , Josef Kreith a , Raúl Bermejo a , * a Montanuniversit¨ at Leoben, Department of Materials Science, Franz Josef Straße 18, 8700 Leoben, Austria b Brno University of Technology, Faculty of Mechanical Engineering, Institute of Solid Mechanics, Mechatronics and Biomechanics, Technick´ a 2896/2, 616 69 Brno, Czech Republic ARTICLE INFO Keywords: Layered ceramics Coupled criterion Finite fracture mechanics Residual stresses Edge cracks Tunnelling cracks ABSTRACT A coupled stress-energy criterion is utilized to predict initiation of both edge and tunnelling cracks in layered ceramics containing thermal residual stresses. Edge (surface) cracks may originate in layers having high compressive in-plane stresses while tunnelling (internal) cracks may form in layers with high tensile in-plane stresses. This work investigates the influence of both the residual stresses magnitude and layer thickness on the formation of surface cracks and provides a design map defining safe regions where no cracks will be present in the sintered multilayer architecture upon reaching the room temperature. Necessary stress and energy inputs to evaluate the coupled criterion are calculated using the finite element method. Simulation results are validated with experimental observations on sample architectures fabricated with layers of various thicknesses and inplane thermal residual stresses. The good agreement demonstrates the potential of the stress-energy coupled criterion for designing crack-free multi-layered ceramic architectures. 1. Introduction Ceramic materials are used in structural applications requiring high hardness, temperature stability, resistance to oxidation, corrosion or wear. Despite the advantageous properties of ceramics, their use is usually limited in applications where high safety and reliability are required. The main problem is the inherent brittleness due to the low fracture toughness and the significant strength variance caused by the presence of flaws inside the microstructure. Flaws are mostly induced upon processing (pores, inclusions, etc.) or during final machining (notches or scratches). Such flaws may then, under critical conditions, be responsible for the formation and propagation of cracks, causing total component failure. One approach to protecting components against catastrophic failure is embed "protective" layers to arrest the propagation of surface cracks [1]. This might be done, for example, by induction of compressive residual stresses in the critical location/layer, decreasing the stress intensity factor at the crack tip. Thermal residual stresses are induced during the cooling down process from sintering temperature and are primarily caused by a thermal strain mismatch between individual materials. This differential strain may be associated with differences in thermal expansion (CTE), phase transformations or chemical reactions occurring in particular layers. Tailoring the location of in-plane compressive residual stresses in the layered architecture may increase the damage tolerance of the ceramic component [2]. Concurrently, tensile residual stresses are simultaneously induced in the composite and have an opposite effect, i.e., they might promote crack propagation. Therefore, the right balance between compressive and tensile stresses and other geometrical and material parameters must always be found. Two typical crack types associated with residual stresses can be observed in ceramic laminates, i.e. “edge cracks” (Fig. 1a) and “tunnelling cracks” (Fig. 1b) [2]. Edge cracks are associated with out-of-plane tensile stresses at the surface of the compressive layer. In this case, the compressive in-plane residual stress in the bulk vanishes at the surface, and the tensile out-of-plane stress component appears at this location instead. ˇ Seveˇ cek et al. [3] found that the magnitude of tensile out-of-plane stresses and the magnitude of the in-plane compressive stresses are of similar order of magnitude. On the one hand, if the out-of-plane tensile stress component activates a flaw, an edge crack usually develops along the whole perimeter of the layer and extends into * Corresponding author. E-mail address: [email protected] (R. Bermejo). Contents lists available at ScienceDirect Journal of the European Ceramic Society journal homepage: www.elsevier.com/locate/jeurceramsoc https://doi.org/10.1016/j.jeurceramsoc.2022.12.022 Received 1 July 2022; Received in revised form 9 December 2022; Accepted 12 December 2022 Journal of the European Ceramic Society 43 (2023) 2928–2934 2929 the depth perpendicularly to the specimen/component surface. On the other hand, tunnelling cracks appear primarily in layers containing in-plane tensile residual stresses. Tensile bi-axial in-plane residual stress acting inside the tensile layer may activate a flaw, which usually results in a crack formation and further propagation until it appears at the free surface perpendicularly to the interfaces with other layers. Edge and tunnelling cracks in brittle layers are particular cracks observed and studied for almost three decades [4,5]. These cracks are not limited to multi-material ceramic components but can also be observed for instance in glass fibre epoxy laminates [6]. An external mechanical load might also induce this type of cracks without the contribution of residual stresses. Empirical observations [5] show that under the same magnitude of internal residual stresses edge cracks only initiate in layers thicker than a certain value (critical thickness). There are two common approaches to explain this: (i) the flaw statistics approach based on the Weibull theory suggests that, due to smaller effective volume, the apparent strength increases; (ii) the energy approach states that a crack in the thin layer cannot propagate since the steady-state energy release rate is too low. These explanations are based on the theory of linear elastic fracture mechanics (LEFM), which always assumes the presence of a crack. Such crack shall propagate as long as Griffith’s energy criterion is fulfilled, i.e. the energy release rate G overcomes the fracture energy G c (given by the fracture toughness of the corresponding material). However, when the initial crack does not exist, LEFM cannot predict its onset. Instead, the Finite Fracture Mechanics (FFM) approach which employs a coupled stress-energy criterion (CC) needs be implemented. FFM considers the formation of a crack as a fracture event [7] and does not regard the history of its formation. The moment of crack initiation is determined by the “coupled criterion”, which requires that both a stress and energy criteria must be fulfilled simultaneously for a crack to nucleate [8]. The size, shape or location of a critical flaw (a crack) is usually not known prior to fracture; hence, it needs to be assumed. It was demonstrated by Leguillon et al. [9] and ˇ Seveˇ cek et al. [3,10] for edge cracks and by Garcia et al. [11,12] for tunnelling cracks that the coupled criterion governs crack initiation and can predict the size effect. However, in both cases, systematic validation of the used models was lacking. In this work, finite element models for predicting edge crack and tunnelling crack formation in layered ceramics was developed, based on a coupled stress-energy criterion. Two materials with different thermoelastic constants (i.e. Young’s modulus and coefficient of thermal expansion (CTE)), are used to induce residual stresses in dissimilar adjacent layers. The edge crack model exploits the axial symmetry and the edge crack initiation is assumed to always occur in only one layer. The model for prediction of the tunnelling crack onset was created as a 3D model, but due to assumptions about the initial crack shape, a 3-fold symmetry could be exploited. In both cases, a parametric analysis considering various thicknesses of cracked layers and different magnitude of residual stresses in the layers was performed to determine the critical conditions necessary for the onset of both types of cracks. To verify the predictions of the employed models, ceramic laminated samples with distinct materials and various thicknesses of particular layers were manufactured and analysed for the presence of edge and tunnelling cracks. 2. Experimental 2.1. Fabrication of samples Samples were manufactured by the tape casting technology. Fig. 2 depicts two symmetric architectures of specimens manufactured for investigation of edge cracks (Fig. 2a) or tunnelling cracks (Fig. 2b). Using the tape casting technique, dried tapes of square shape and 40 ×40 mm size were stacked and warm-pressed by 20 MPa at 75 ◦C for 15 min, followed by the iso-static lamination at 20 MPa/75 ◦C for 30 min, and by binder burn-out at 600 ◦C for 2 h. Afterwards, the stacked plates were cold-isostatic pressed with 100 MPa for 15 min and sintered at 1550 ◦C (heating rate 10 ◦C/min) for 2 h to achieve high relative density. Three different materials were used for manufacturing the samples: (i) pure alumina (A0), (ii) alumina with 50 vol% of (stabilized) tetragonal zirconia (A50TZ) and (iii) alumina with 15 vol% of (non-stabilized) monoclinic zirconia (A15MZ). The material properties of used ceramics are summarized in Table 1. The choice of A15MZ ceramic was based on previous experimental observations of samples containing layers with either 20 vol% or 10 vol% of monoclinic zirconia (A20MZ and A10MZ, respectively), where cracks were observed in all A0 layers (combined with A20MZ), and no cracks were found in any A0 layer (combined with A10MZ). To investigate edge cracks, the A0 layers were embedded between the ZTA50 layers (Fig. 2a) to induce tensile out-of-plane residual stresses at the free surface of the A0 layers (alumina layers). The choice of layer thicknesses was based on preliminary analytical calculations of stresses. For investigation of tunnelling cracks, the A0 layer was embedded between A15MZ layers (Fig. 2b) to induce tensile in-plane residual stress in the A0 alumina layers. Young’s modulus E of individual layers was determined on bulk specimens from loading curves of displacement controlled 3-point bending tests. Universal testing machine (Messphysik, Microstrain, Fürstenfeld, Austria) with a 100 N load cell and a fixture with 30 mm outer span was used. Experiment followed the EN 843–2 standard [13] with the crosshead speed 0.5 mm/min. Ambient conditions were 24 ◦C temperature and 42% humidity. A 1 N preload and maximum load of 35 N were selected for alternate loading/unloading of three samples per material. Common values of Poisson’s ratios ν were assumed. Secant coefficients of thermal expansion α (CTE) were determined from dilatation curves of monolithic prismatic bars. Specimens with Fig. 1. Examples of (a) the circumferential edge crack; (b) the tunnelling crack [2]. R. Papˇ sík et al. Journal of the European Ceramic Society 43 (2023) 2928–2934 2930 standardized length of 25 mm were heated from 30 ◦C up to 900 ◦C using a dilatometer (Netzsch 402E, 95100 Selb, Germany). The change of length was registered during heating with 1 h long holding segments at every 100 ◦C with 5 ◦C/min heating rate in between. The CTE was calculated between the room temperature 25 ◦C and an estimated stressfree temperature 1470 ◦C [14] from an extrapolation of length at holding segments. During the cooling from sintering, typically around 1100 ◦C, zirconia in A15MZ material undergoes a phase transformation [15], from the tetragonal to the monoclinic phase, with an associated volume increase. This phase change cannot be observed and taken into account when dilatation is measured on already sintered specimen in a temperature range below 900 ◦C. The fracture toughness K Ic was determined using the single edge Vnotched beam method (SEVNB) according to the ISO 23146 standard [16]. A universal testing machine (Zwick 010, Zwick/Roell Ulm, Germany) with a 200 N load cell was used at ambient conditions of 24 ◦C temperature and 32% humidity. Specimens were tested using a 4-point bending fixture with a 40 mm outer span and a 20 mm inner span with applied displacement rate of 0.5 mm/min. Surfaces of specimens were sputtered with a gold using the Agrar Sputter Coater and investigated using a SEM (JEOL JCM-6000Plus, Neoscope, JEOL Ltd., Tokyo, Japan). Edge and tunnelling crack were sought in all layers. The strength of A0 was measured by 4-point-bending method using a universal testing machine Zwick Z010 (Zwick/Roell, Ulm, Germany). For statistical significance, 26 samples were used. Ambient conditions were 23 ◦C temperature and 37% relative humidity. A cross beam speed of 1.5 mm/min and a pre-load force of 10 N was chosen. The strength of A50TZ and A15MZ was not measured, since it is not needed for the calculations; cracking is expected only in the A0 layers. 3. Theory and calculations In this section, the computational model for an edge crack and a tunnelling crack is described. The finite fracture mechanics is briefly introduced as a complementary approach to the linear elastic fracture mechanics and the methodology for prediction of crack initiation by the coupled stress-energy criterion (CC) is summarized. 3.1. Residual stresses in laminates Edge and tunnelling cracks in ceramic laminates originate solely due to a presence of residual stresses caused by a mismatch in thermal strains between layers of different materials. No external mechanical load is required for the formation of these types of cracks. The magnitude of tensile and compressive residual stresses in the corresponding layers can be controlled by changing the volume ratio of materials [17]. The volume ratio  Vi of the i-th material in a composite of N materials is defined as:  Vi=Vi ∑N i=1Vi (1) where Vi is the volume of the i-th material. Figs. 3a and 3c illustrate the manufactured specimens in crosssections exploiting the 3-fold symmetry. The residual stresses are depicted along paths A and B in the innermost layer from the free surface to the bulk. Normal stresses σ along paths A and B are plotted in Figs. 3b and 3c, respectively. Path lengths a are normalised by corresponding layer thickness tinner A0 and stresses are normalised by the corresponding inplane (x-y) residual stresses σ in. In the case of path A, the out-of-plane stress σ zz can be as high as the in-plane stress σ xx or σ yy and thus promotes crack initiation at the surface. In the case of path B, the in-plane stress σ yy is homogeneous and constant in the bulk but decreases towards the surface. Thus, we infer that the most favourable location for initiation of tunnelling cracks is not at the surface, but below the surface. 3.2. Computational model for edge cracking The edge cracking was simulated on a circular five-layer disc (Fig. 4). Since the edge crack is formed all along the whole circumference of the disc, this task can be solved as an axisymmetric problem. Although the manufactured specimen was a plate, similar conditions for crack formation at the edges exist on the side of the disc, with less demanding computational time. The inner layer and two outer layers have characteristics of material A0 (see Table 1), which is a material with smaller CTE than that of the other material of the disc, resulting in compressive in-plane residual stresses in A0 layers. Remaining layers were made of A50TZ material, where tensile residual in-plane stresses were induced. At the free surface of A0 layers the in-plane compressive residual stresses transform into the out-of-plane tensile stresses and vanish, which might be (upon certain conditions) responsible for the formation of circumferential edge cracks [3]. The geometry of the disc is depicted in Fig. 4a. The disc radius was fixed at R =20 mm, the inner layer thickness was variable in the range from 10 µm to 500 µm, the A50TZ layers were at least 400 µm thick and they were adjusted together with outer most layers to reach the desired level of residual stresses. The crack depth a was simulated from 0 µm up to the depth where tensile out-of-plane stresses vanished. The meridian cross-section of the disc was discretized by quadratic 2D elements (PLANE183) having the axisymmetric option activated. The mesh was coarse in general (element edge length was set approximately to 5 µm), but a refined mesh of approximately 50 nm (Fig. 4b) was used in the crack vicinity. The crack was incrementally opened by disconnecting coincident nodes on opposite faces along the expected crack path. The mesh did not change during the crack growth to avoid Fig. 2. Architectures of specimens manufactured for analyses of (a) edge crack formation and; (b) tunnelling crack formation. Table 1 Material properties of ceramics used for manufacturing of samples. Material property Material A0 A50TZ A15MZ E [GPa] 398 ±3 292 ±3 354 ±4 ν [–] 0.23 0.23 0.23 α [K −1 ] 8.3⋅10 −6 9.6⋅10 −6 7.5⋅10 −6 σ 0 [MPa] ~350 n/a n/a K Ic [MPa⋅m 1/2 ] 3.40 ±0.4 4.8 ±0.3 4.3 ±0.1 G c [J⋅m −2 ] 27.9 ±6.5 74.7 ±9.4 48.3 ±1.3 R. Papˇ sík et al. Journal of the European Ceramic Society 43 (2023) 2928–2934 2931 spurious variations of the energy. On the symmetry axis displacements in the radial direction were fixed (u r (r=0) =0) and additionally in one node of the symmetry axis (at z=0) a zero displacement in the axial direction was prescribed to avoid rigid motion upon the simulation. Correctness of the axisymmetric simplification and corresponding values of the calculated energy release rates were verified by a comparison with a full 3D disc model, where a perfect match was reached. 3.3. Computational model for tunnelling cracking Initiation of a tunnelling crack was simulated on a symmetric plate composed of three layers. The outer layers were made of material with smaller CTE (A15MZ), to induce compressive stresses and the inner layer with larger CTE (A0) resulting in a formation of tensile residual stresses (Table 1). The thickness of outer layers was kept at 500 µm and the thickness of the inner layer varied between 20 µm and 300 µm. Length and width were adjusted to avoid influence of edges on the layer centre. By exploiting the threefold symmetry, only 1/8 of the actual geometry was sufficient for the modelling of the problem. It was assumed that the crack has a circular or elliptic shape (the so called “penny-shape crack”), located in the middle of the inner layer with faces oriented in parallel with the x-y cross-section (Fig. 5a). The two axes describing the crack size are a x and a z . The location of the crack was in the region of homogeneous tensile stress far from the free surface, where the tensile stress slightly decreased (but not vanished). The geometry was discretized by 3D hexahedral quadratic elements (SOLID186). The vicinity of the crack front was swept by elements with shifted nodes closer to the crack front to better capture the singularity (Fig. 5b). Identical mesh topology for models with and without crack was kept again to avoid spurious energy variations. Besides the temperature change only displacement boundary conditions on the symmetry planes were prescribed. Namely, displacements on all nodes on each symmetry plane were fixed in the normal direction (except those which are inside the elliptical crack in y-z plane to enable opening of the crack). Fig. 3. (a, c) Schematics of the multilayer designs. (b, d) Residual stress profiles along paths A and B in manufactured specimens (b,d) of size L×W×H. Stresses σ are normalised by the in-plane residual stress σ in and the path lengths a are normalised by the associated layer thickness tinner A0. Fig. 4. (a) Cross-section of the specimen with radius R in the plane of crack with length a; (b) detail of the finite element mesh with a detail of the tensile (red) out-ofplane stress σ zz at the free surface of inner layer. R. Papˇ sík et al. Journal of the European Ceramic Society 43 (2023) 2928–2934 2932 3.4. Finite fracture mechanics and coupled criterion To find conditions upon which the crack will initiate, the coupled stress-energy criterion was employed [18]. First, the stress condition must be fulfilled all along the prospective crack path (stresses normal to the crack plane must be higher than the tensile strength σ c) and subsequently the incremental energy release rate Ginc must be higher than its critical value G c . The incremental energy release rate is generally calculated as the difference between the potential energy Π of the body with and without a crack of size/surface A, respective, divided by this surface as stated in the following relation: Ginc := − Π(A) − Π(0) A(2) In the case of circumferential edge cracks the incremental energy release rate was calculated as follows: Ginc = − Π(a) − Π(0) π R2− π (R−a)2(3) where Π(a)and Π(0)are the potential energies [in J] of the disc with and without a crack, respectively. The area of the crack (given in the denominator of equation (3)) is that of an annulus. For each combination of the volume ratio and the layer thickness, the coupled criterion was evaluated according to a graphical demonstration shown in Fig. 6. As the temperature decreases after the sintering process, residual stress in the out-of-plane direction σ zz arise on the free edges of the laminate. In Fig. 6a, σ zz and Ginc are plotted for two temperature differences. At −700 ◦C, the stress represented by the (green) dashed curve does not overcome the strength, nor does the Ginc represented by (violet) dotted curve overcome GC. The coupled criterion is fulfilled only after significant decrease of temperature. In Fig. 6b, the Ginc are plotted for crack paths normalised by the layer thickness in which the crack is located and propagates. Although the σ zz stresses along the prospective crack path are the same in all 3 layers, only in the thickest (purple) layer the energy condition is favourable for the crack onset since its values exceed G c (or 1 in the normalized graph) for the same crack length as the stress condition exceeds σ c value. Since, for this particular case, it is not important at which temperature a crack initiates, but only if it initiates, we can evaluate the coupled criterion only for the worst-case state reached with the maximum temperature difference ΔTmax = − 1450℃. In the case of tunnelling crack, the incremental energy release rate was calculated as follows: Ginc = − Π(A) − Π(0) π axaz (4) where Π(A)and Π(0)are the energies [in J] of the whole body (multiplied from the eight volume of the 3-fold symmetric model) with and without a crack respectively. The expression in the denominator of equation (4) is the area of an elliptic crack with major and minor axis a x and a z , respective. Fig. 7 illustrates the evaluation of the coupled criterion for two layers with thickness 100 µm (Fig. 7a) and 300 µm (Fig. 7b), respectively. The stress normalised by strength is represented by the green surface and the red and violet surfaces represents normalised Ginc. In Fig. 7a the stress criterion is fulfilled in the thicker layer at a temperature difference that is smaller than which can be achieved by cooling from sintering and at the same time the energy criterion is fulfilled also (both surfaces are above value 1) and thus an elliptic crack may form. In Fig. 7b the strength criterion has been fulfilled in the thinner layer at maximal achievable temperature but despite that, Ginc is unable to reach Gc (violet surface overcoming value 1) even in an extreme case, when the Fig. 5. (a) A cross-section sketch of geometry with a potential elliptic crack. (b) Detail of mesh around the crack vicinity exploiting 3-fold symmetry. Fig. 6. Evaluation of the coupled criterion for edge crack: (a) in layer of constant thickness and constant volume ratio for increasing temperature and (b) in layers of different thicknesses at constant volume ratio and maximal achievable temperature difference. R. Papˇ sík et al. Journal of the European Ceramic Society 43 (2023) 2928–2934 2933 initial elliptic crack would extend across the whole layer thickness and had extreme aspect ratio. Thus, formation of tunnelling cracks is prevented. Necessary inputs into the coupled criterion, such as stresses along the prospective crack path or the incremental energy release rate, as a function of the crack length, were calculated using the finite element simulation in Ansys Mechanical [19]. All essential results were exported for the subsequent post-processing in the mathematical software Matlab [20], where the fulfilment of CC was evaluated. 4. Results and discussion 4.1. Edge cracks Results of the performed parametric study are shown in Fig. 8a. It shows a chart of two regions separated by a black solid curve defining the critical combination of layer thickness and level of residual compressive stresses leading to initiation of edge cracking. Position and shape of this curve depend primarily on the values of material fracture mechanics characteristics – namely of the tensile strength and fracture toughness – see also [3]. In the upper-left region, the coupled criterion is fulfilled and conditions for the edge crack formation are thus favourable. In the lower-right region, neither stress or energy criterion are fulfilled, hence, edge cracks should not initiate. The architecture in Fig. 8b was chosen such that layer thicknesses corresponded to full symbols (coloured markers) in Fig. 8a. One can see that edge cracks indeed initiate in thick layers (represented by a red triangle) – as the coupled criterion is fulfilled and no edge cracks were observed in the thin layer (represented by the green circle) even though the compressive stress magnitudes in both layers were the same. In the layer with an intermediate thickness (depicted by a yellow square) no cracks were observed, since the tensile strength of the layer was not reached. This crack extended along the whole free surface and into a certain depth which is of the order of the layer thickness, as experimentally evidenced by subsequently polishing the specimens from the side. 4.2. Tunnelling cracks The result of performed simulations analysing conditions for the tunnelling crack formation is depicted in Fig. 9a. The graph area is divided by a vertical line corresponding to 63rd quantile of the measured strength (63% probability of failure). In-plane stresses left from this line are so high that it is almost guaranteed that the stress criterion be fulfilled. Right from these lines, the stresses are so low that it is very unlikely that the stress criterion be satisfied. The chart is further split by a thick black curve, above which the energy criterion is fulfilled. A cross-section showing the free surface of the manufactured specimen is shown in Fig. 9b. The tunnelling crack is clearly visible in the thickest layer (depicted by a red circle), while the other (depicted by yellow squares and blue triangles) remain without cracks. Surface layers (depicted by green stars) were also fractured, however prediction of these surface cracks was not modelled in this work. By comparing Figs. 9a and 9b a good agreement between predictions and empirical observations was found, which demonstrates that the coupled criterion is a powerful and applicable tool for designing crackFig. 7. Evaluation of the coupled criterion for the tunnelling crack (a) in a 300 µm thick layer at lower than maximal achievable temperature; (b) in a 100 µm thick layer at maximal achievable temperature. Fig. 8. (a) Regions of fulfilment/non-fulfilment of CC for edge crack, (b) SEM images of the specimen with evidence of edge cracking. Symbols of different shape and colours represent different layer thicknesses. R. Papˇ sík et al. Journal of the European Ceramic Society 43 (2023) 2928–2934 2934 free ceramic components. We caution the reader that the model for tunnelling cracks developed here does not study crack formation in surface layers, where stress decreases significantly; this will be addressed in the future work. 5. Conclusion This work demonstrates the ability of the coupled stress-energy criterion to predict the initiation of edge or tunnelling cracks in a bimaterial layered ceramic architecture. The size effect (thickness) in individual layers is governed by the fulfilment of the coupled stress-energy criterion, not only by the stress or the energy criterion alone. For both edge and tunnelling cracks, there exists a region where crack may not initiate as a consequence of the energy criterion not being fulfilled even if the stress reached the strength that corresponds to 99% probability of failure (99th quantile of the Weibull strength distribution). An advantage of the coupled criterion is that it only requires the fracture toughness, the tensile strength and the elastic material properties of layers, where these cracks are investigated. Results herein can be used as a guide for designing components having no processing cracks induced upon the cooling down process from the sintering temperature. Experimental observations showed a good agreement with the presented numerical models and confirm the ability of the coupled stress energy criterion in predicting crack formation in layered ceramics designed with residual stresses. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements Funding for this research was provided by the European Research Council (ERC) excellent science grant “CERATEXT” through the Horizon 2020 program under contract 817615. References [1] R. Bermejo, Z. Chlup, L. ˇ Sest´ akov´ a, O. ˇ Seveˇ cek a R. Danzer, Strategies to optimize the strength and fracture resistance of ceramic laminates,“ v Mechanical Properties and Performance of Engineering Ceramics and Composites VII, 2012. [2] R. Bermejo, Toward seashells under stress, J. Eur. Ceram. Soc. vol. 37 (2017) 3823–3839. [3] O. ˇ Seveˇ cek, M. 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Mech. vol. 86 (1) (2016) 375–401, 1. [19] Ansys Mechanical, Release 2020 R2. [20] MATLAB. The Mathworks, Inc, Natick, Massachusetts, 2021. Fig. 9. (a) Chart illustrating regions of fulfilment of CC for tunnelling cracks. (b) SEM images of the specimen with evidence of tunnelling cracks (circled). Symbols of different shapes and colours represent different layer thicknesses. R. Papˇ sík et al.