Revisiting the strength of micron‐scale ceramic platelets

Individual platelets in nacre‐like ceramics are able to locally withstand stress levels many times larger than the strength of larger specimens made of the same materials. This size effect, usually reported as being relative to the size of pre‐existing defects, is described by considering both stress and energy failure conditions. We show that there is a characteristic length, defined by the material's Young's modulus, fracture energy, and strength, above which failure is governed by a stress criterion and below which energy considerations are dominant. Applying the coupled criterion to three‐point bending of a single alumina platelet allows the identification of the platelet strength and fracture energy. The proposed approach also allows capturing the decrease in failure stress due to a surface defect accounting for its shape and size.


Introduction
The interest towards nacre and nacre-like ceramics has been increasingly growing over the past years [1,2,3]. These bioinspired materials, generally manufactured from brittle constituents, combine high stiffness, strength and fracture toughness [2]. They consist of a dense packing of ordered and oriented micron-scale platelets [2,4,5,6,7,8]. The orientation of the platelets along a specific direction can be achieved during the material manufacturing process by ice templating [2] or by adding ferro-magnetic particles that are sensitive to magnetic fields [5,9]. Other manufacturing approaches can also be employed, such as ceramic layer gluing [10], or 3D printing [11]. The platelets are sintered and a secondary phase acts like an interface that can consist of polymer [4,5,8], metal, ceramic [7], or also a glassy phase [2]. Bouville et al. [2] showed that varying the constituents of the secondary phase could lead to either brittle or non-brittle macroscopic mechanical behavior. The apparent non-brittle macroscopic mechanical behavior is mainly due to fracture mechanisms related to the material microstructure. Rather than propagating directly through the platelets, a crack is more likely to propagate in the secondary phase at the interface between the platelets, which causes crack deflection and branching [2,11,12]. Energy dissipation occurs through these mechanisms since larger crack surfaces are created than in the case of a straight crack that would penetrate through the platelets. An additional energy dissipation may also occur by means of local friction, thus resulting in a non-brittle macroscopic behavior.
The mechanical properties of the platelet and of the interface play a major role in the mechanical response of the material. Controlling the material microstructure is of primary importance in order to obtain mechanical properties that allow an optimal energy dissipation at the macroscopic scale, which could be achieved for instance by varying the platelet size or geometry. To this end, modeling interface and platelet failure may help to obtain guidelines to optimize the manufacturing process [13]. In particular, numerical simulation can be used to set up inverse approaches for the determination of interface strength and fracture toughness [14,15]. Such approaches require the knowledge of the platelet mechanical behavior and properties used in fracture models such as, for instance, cohesive zone models [16,17] phase field approach to fracture [18,19,20] or approaches based on Finite Fracture Mechanics (FFM) [21,22].
The experimental determination of the platelet mechanical properties requires to (i) be able to perform micromechanical tests and (ii) set up post-processing approaches of these tests that provide the properties required as input in the models. Focused ion beam (FIB) allows manufacturing micron-scale specimens which can be mechanically tested thanks to micro-mechanical devices such as nano-indenters [23]. For instance, Pejchal et al. [24] performed flexural tests on microscopic alumina particle C-shaped and cantilever beam specimens. Based on the fracture location observed experimentally, they determined the maximum stress at this location from the finite element (FE) analyses, which was thus considered as the material strength. Maximum stress values as large as 10 GPa were obtained, decreasing to 2 GPa in presence of defects. Similar tests were performed by Mueller et al. [28] in order to determine the strength of silicon particles in an aluminum alloy. Maximum stress levels between 7 and 15 GPa were obtained without apparent defects, and between 2 to 6 GPa in presence of defects. Following the same reasoning, Feilden et al. [3] determined that a single alumina platelet could locally sustain 5 GPa maximum stress under three-point bending.
In the three approaches mentioned above, the strength is defined as the maximum stress that the material can locally sustain before failure. It is obtained based on analytical or FE models that does not account for the possible flaws. In presence of a sharp flaw (crack, V-notch), the stress theoretically tends towards infinity, or at least reaches large values in the vicinity of the flaw. It is thus clear that this given definition of the strength does not allow characterizing a property that is specific to the material, since it depends on features such as, for instance, the specimen geometry or the presence of defects. Size effects in quasibrittle materials have been extensively investigated over the past decades to try to make a link between the strength of large and small structures [25,26,27]. . The objective of this work is to provide insight regarding the high values of failure stress measured for single platelet by employing a framework in which the strength is considered as a characteristic of the material and does not depend on the geometry of the studied specimen. Section 2 is dedicated to the description of a failure prediction approach that combines stress and energy conditions. In Section 3, this approach is applied to obtain the failure stress of a single alumina platelet containing no flaws under tensile loading. Failure of a single platelet under three-point bending is modeled by FE in Section 4 in order to identify the platelet strength and toughness by confrontation to experimental measurements [3]. The influence of a flaw on the failure stress is also studied.

Stress and energy failure conditions
Following the approach proposed by Leguillon [21], we consider that the nucleation of a crack requires the simultaneous fulfillment of two conditions. The first condition describes the need for a sufficiently large local stress state to trigger crack initiation. It states that the stress normal to the crack path σ nn must overcome the material strength σ c over the entire 3 crack path before its nucleation. The second condition describes the need for a sufficiently large stored potential energy prior to crack initiation. It is derived from an energy balance (1) G inc is the incremental energy release rate, which is usually an increasing function of the crack surface. Therefore, the second condition in Eqn.
(1) provides a lower bound for the admissible crack surfaces at initiation. On the other side, the stress is generally a decreasing function of the distance to a stress concentrator in its vicinity, therefore the first condition in Eqn. (1) defines an upper bound for the admissible initiation crack surfaces. For a too small imposed loading, the stress criterion can be fulfilled for crack surfaces for which the incremental energy release rates are smaller than G c . Moreover, the energy criterion can be fulfilled for crack surfaces for which the stress is smaller than σ c . Therefore, the admissible crack initiation ranges provided by both criteria do not match and no crack surfaces satisfy both criteria simultaneously, as illustrated in Fig. 1a). Increasing the loading allows (i) increasing the crack surface upper bound provided by the stress criterion and (ii) decreasing the crack surface lower bound provided by the energy criterion, both bounds matching for a sufficiently large imposed loading which corresponds to the initiation loading (Fig. 1b)).
The initiation loading is thus obtained as the minimum loading for which both criteria are simultaneously fulfilled for a given crack surface.

Application of the coupled criterion
The application of the coupled criterion (CC) for failure prediction requires several features. The stress state must be calculated for the configuration prior to crack initiation.
This can be performed either based on analytical solutions [29,30,31] or employing Finite Element (FE) calculations (cf. Section 4). Thus, the stress condition can be computed, which allows determining the minimum loading required to fulfill the stress condition P s (S) 4 as a function of the crack surface. The energy condition can be obtained by calculating the potential energy variation as a function of the crack surface, which allows determining the minimum loading required to fulfill this condition P e (S). The loading required to initiate a crack of a given surface S is determined as P (S) = max(P s (S), P e (S)). The initiation surface S * minimizes the loading required to fulfill both conditions: The initiation loading P * can finally be determined as P * = P (S * ) = max(P s (S * ), P e (S * )).
The input parameters required to apply the CC are the material constitutive behavior (Young's modulus and Poisson's ration for a linear elastic isotropic behavior) and its fracture energy G c and strength σ c . Therefore, it raises the question of a strength definition that is specific to the material, and does not depends on features such as the specimen geometry or the presence of external defects.

The strength to be used in the coupled criterion
The CC requires the material strength as an input to predict failure, which rises the question of defining the ceramic platelet strength, since it is well know that ceramic failure al. [34] and Chantikul et al. [35], Leguillon et al. [32] recently proposed a definition for the strength to be used in the CC in the case of polycrystalline ceramics and studied the influence of a surface defect on the failure stress under tensile loading [33]. By studying the influence of an extrinsic (surface) flaw on the ceramic failure, they showed that (illustration of the results are given in Fig. 2): ❼ For a given grain size, the failure stress is governed by the flaw size for flaws larger than the grain size (Fig. 2a). However, a constant failure stress is obtained for extrinsic flaws smaller than the grain size, which allows defining an "intrinsic strength", i.e a material property that does not depends on the extrinsic flaw size. The intrinsic strength is however governed by the grain size and the instrinsic flaws such as, e.g., porosities (which size is, in first approximation, in the same order of magnitude as the grain size).
❼ For large grains in the absence of extrinsic flaws, the intrinsic strength follows a Hall-Petch-like law, i.e. the instrinsic strength varies as 1 √ Φ G , where Φ G is the grain size (Fig. 2b).
❼ The intrinsic strength does not follow Hall-Petch for small grains (which would lead to high strength values) but reaches a plateau towards a value (called the "intrinsic "intrinsic intrinsic strength" intrinsic strength" [32]), which corresponds to the failure stress for a ceramic containing very small grains and flaws.
The ceramic platelet in nacre-like ceramics is a micron scale object containing relatively small (nanometric) intrinsic flaws. Hence, based on the abovementionned considerations, it appears that the strength value of a ceramic platelet that must be employed for the CC should be at least equal to the "intrinsic intrinsic strength", i.e. the failure stress corresponding to the smallest possible grain size and intrinsic flaws. In the following, the influence of the strength on platelet failure prediction with the CC is studied and an estimate of both the platelet strength and fracture energy are identified by confrontation to experimental results available in the literature.

Failure of a single alumina platelet containing no flaws
Let us consider a single platelet subjected to uniaxial tensile loading (with a stress intensity σ 0 ). For the sake of simplicity, we consider a parallepipedic platelet of length 2L and section S = wt with a linear elastic behavior, cf Fig. 3. We first consider a platelet that does not contain any flaw (Fig. 3a). The stress state in the platelet is homogeneous and equal to the imposed stress σ 0 . Applying the stress conditions described in Section 2, we obtain a first necessary condition for the platelet failure: where σ c is the strength of the platelet constitutive material. Since the stress is homogeneous in the whole platelet, there is no preferential location for a crack to nucleate, such as in presence of a stress raiser (e.g hole) or a singularity (e.g crack, V-notch). Therefore, we assume that the platelet fails by suddenly breaking into two parts. It means that the potential energy after failure W (S) is zero, thus the potential energy difference ∆W can be obtained from the potential energy of the platelet before failure W (0): where E is the platelet Young's modulus in the loading direction. Therefore, the energy condition given in Section 2 reads: which leads to a second condition for the platelet failure: Fig. 4 shows the variation of the imposed stresses required to fulfill both criteria as a function of the platelet length. For a given platelet length, the failure stress σ f is thus the maximum between both values, which corresponds to the minimum loading for which both criteria are simultaneously fulfilled. For long enough platelets, it can be seen that the platelet failure stress is equal to its constitutive material strength because there is more available energy than that necessary to fulfill the energy criterion. Therefore, the platelet breaks into two parts and the energy not consumed for the platelet failure is dissipated, for instance as kinetic energy The strength is first chosen equal to the "intrinsic intrinsic strength" described in previous Section for illustrative purpose, an inverse identification approach for its determination is then proposed in Section 4. The corresponding material characteristic length is L mat = 14 ➭m. Therefore, the failure stress of platelets larger than 14 ➭m would be equal to the strength σ c . Since we consider platelets smaller than this characteristic length, failure is governed by the energy condition and thus depends on the platelet Young's modulus and  can be noted that this value is almost twice the material strength. This is not only due to the fact that the smaller the platelet, the lower the number and the size of flaws within it, which is a classical explanation to the outstanding platelet (apparent) strength measured experimentally. Indeed, we show here that the sole consideration of the platelet size allows explaining the increase in the failure stress, considering the same material properties for the platelet independently of its size. It can be noted that the platelet can locally withstand stress levels larger than its material constitutive strength. Indeed, in this case, failure cannot occur for stress levels corresponding to the characteristic strength because not enough energy is stored in the material.

Finite element model of platelet three-point bending
This section is dedicated to the platelet failure prediction under three-point bending using the criterion presented in Section 2. is refined in the vicinity of the specimen center, near the crack path corresponding to the location where the stress variation is computed in order to determine the stress criterion.

FE model without flaws
This mesh refinement is required in order to catch the crack initiation length, which is not known a priori. Nevertheless, it can be shown that this length is a fraction of the material characteristic length L mat = EGc

Numerical prediction of failure stress
The FE model presented in Section 4.1 is now used in order to predict single alumina platelet failure under three-point bending. Experimental measurements taken from [3] are considered herein. The dimensions of ten tested platelets are given in Tab. 1 as well as the measured failure forces. Local stress levels as large as 5 GPa may be reached at the platelet surface, which is consistent with the results obtained in [3]. Of course, since the upper half of the platelet undergoes compression, the stress is null in the platelet center in the thickness direction. Fig. 6 displays the variation of the stress to strength and incremental energy release rate to fracture energy ratios as a function of the crack length corresponding to specimen #10  ❼ For small enough strength values (σ c ≤900 MPa, Fig. 6a-b), it can be seen that the stress state in the platelet overcomes the material strength over almost all the area undergoing tension (i.e. for l < t 2 ≈ 0.4 µm). Therefore, failure is driven by the energy condition since the stress criterion could be fulfilled for smaller imposed loadings, however the energy criterion would not be fulfilled and hence failure would not occur.
Such a configuration has already been encountered in other situations, such as, for instance, in the case of transverse cracking in composite materials [36,37]. In this case, the failure load F c corresponds to the loading required to fulfill the energy criterion, which is larger than that required to fulfill the stress criterion. The failure load thus mainly depends on the fracture energy and almost does not depend on the strength.
In this case, micromechanical tests such as three-point bending of micron-scale platelet allows determining the material fracture energy rather than its strength.
❼ For larger strength values (e.g σ c =2 GPa, Fig. 6c), the failure load predicted by the CC both depends on the material fracture energy and strength. This is consistent with results obtained in Section 3 since larger strength value leads to smaller values of L mat and hence smaller ranges of platelet length for which the failure is only driven by the energy condition.
In both cases, stress levels larger than σ c are locally attained. Nevertheless, it is shown herein that this local stress level does not define the platelet strength. Such stress levels are reached because the imposed loading must be sufficiently high so that there is enough stored elastic energy available to be dissipated for the platelet failure. Therefore, the platelet failure is not only driven by its strength but also by its fracture energy.

Fracture energy and strength identification
The CC allows the prediction of the failure force providing the knowledge of the material strength and fracture energy. If these parameters are not known, it can also be used as a way to determine it by indirect confrontation to experimental measurements. The identification approach consists in determining the parameter couple (G c ,σ c ) that minimizes the residuals R quantifying the difference between predicted and measured failure forces obtained for N specimens: where F n i (G c ,σ c ) is the failure force predicted numerically with the coupled criterion for the specimen #i employing the strength σ c and fracture energy G c , and F e i is the force at failure measured experimentally for the same specimen. The minimization problem that must be solved writes: Solving this minimization problem requires the calculation of the failure force corresponding to each specimens for several couples (G c , σ c ) either to set up a gradient descend approach or to compute the residuals variation for a priori chosen couples of parameters. The failure force calculation is obtained employing the coupled criterion, which requires some FE calculations to compute the incremental energy release rate and the stress variation (cf. Section 4.1). It can be noted that a major advantage of the CC is the possibly to vary (G c ,σ c ) without any extra FE calculations since these parameters are only implemented during the post-processing step. The minimization procedure thus only require post-processing FE calculations, which is computationally very efficient compared to other approaches such as, e.g., cohesive zone modeling [16,17] or phase field [18,19,20], which would require as many FE calculations as the number of chosen (G c , σ c ) couples. Fig. 7a displays the residuals variation as a function of G c and σ c . The black dots indicate the residuals minimum for a fixed strength (corresponding to the fracture energy that minimize the difference between predicted and measured failure stress for this strength), which are also reported in Fig. 7b. It can be observed first that for small enough strength value (below around 2 GPa), the residuals minimum is reached for the same value of G c . As already explained previously (cf., e.g. Fig. 6a-b), it corresponds to a regime for which failure is mainly driven by the fracture energy, therefore similar fracture energy (G c between 20 and 25 J/m 2 ) is obtained regardless the strength. For larger strengths, it appears that failure is driven both by fracture energy and strength, thus the fracture energy corresponding to the residuals minimum obtained for a given strength decreases with increasing strength (Fig 7a). As observed in Fig. 7b It can be seen that the identified strength value is a slightly larger than the alumina "intrinsic intrinsic strength". This is not surprising since the latter corresponds to a configuration with very small grains and flaws, however still possibly including intrinsic microstructural features such as grain boundaries, small porosities that could be privileged locations for crack initiation. The forces at failure predicted numerically employing the coupled criterion F n c for the ten specimens are summarized in Table 1 for (G c =22 J/m 2 ,σ c =1.1 GPa). It can be seen that a good estimate of the failure force is obtained employing the proposed approach. The deviation between numerically predicted and experimentally measured failure force is within 20% for all the considered specimens.

Possible surface flaws
The common explanation for the large measured platelet failure stress is based on the fact that the smaller the platelet the smaller the flaws, that may be privileged locations for crack initiation. We demonstrated in the previous sections that due to its small dimensions, according to the CC, the platelet failure stress is actually driven by both its strength and fracture energy and reaches large values because of its size that is smaller than the material characteristic length. This section is dedicated to the influence of a surface flaw on platelet failure. We consider three possible kinds of flaws: a crack, a V-notch or a semi-circular flaw.
The same method as described in Section 4.1 can be employed in order to predict failure in presence of a flaw. The stress along the presupposed crack path and the potential energy for several crack lengths are computed, which allows applying the coupled criterion so as to determine the failure load. the flaw shape is highlighted since the failure force is smaller in the case of a crack than for a V-notch or for a semi-circular hole. It is consistent with the conclusions drawn by Martin et al. [33] who highlighted the small influence of flaw shape for flaws smaller than Lmat 2 under tensile loading. We also display on Fig. 8 the force at failure obtained using the classical explanation based on linear elastic fracture mechanics that the smaller the defects, the higher the failure stress (gray dashed line). This failure stress is computed as σ f = K Ic √ πa . Then, the failure force is estimated as the force for which the maximum stress reached locally in the platelet is σ f , following the approach described for instance in [3,24,28]. This approach provides failure forces in the same order of magnitude as those obtained with the CC for large enough flaw size. However, the predicted failure force deviates from that obtained with the CC for small flaws since it tends towards infinity for infinitesimal flaws.
The order of magnitude of the flaw size estimated based on experimental data in alumina platelets is around 30 nm [3]. For the platelets under investigation, it leads to a flaw size to platelet thickness ratio between 0.032 and 0.073, which corresponds to a maximum failure force decrease predicted by the CC smaller than 10% compared to the failure force obtained for a platelet containing no flaw.

Conclusion
The high failure stress of a single alumina platelet results from its dimensions compared to its constitutive material characteristic length. Considering both energy and stress failure conditions allows predicting the platelet failure loading under three-point bending. The proposed approach requires the knowledge of the platelet Young's modulus, fracture energy and strength, these parameters being considered as intrinsic to the constitutive material and independent from the studied structure size or the possible presence of external flaws. The platelet failure is driven either by only the material fracture energy, or by both its strength and fracture energy. In the case of specimens smaller than the material characteristic length, the energy condition requires substantially high loading to be met compared to the loading required to fulfill the stress condition. The platelet can thus undergo local stress levels several times larger than its constitutive material strength.
The coupled criterion allows the inverse identification of a single alumina platelet strength and fracture energy by confrontation of predicted and measured failure stress under threepoint bending. The proposed approach also allows evaluating the influence of a surface defect on the failure force. It is worth pointing out that the proposed approach is not inconsistent with statistical approaches to failure, such as Weibull theory. Both approaches can even be combined so as to predict failure [38].
It finally appears that micromechanical tests, such as three-point bending of a single ceramic platelet, can be employed in order to determine the platelet fracture energy and possibly its strength. The platelet smallness allows it to withstand high local stress levels compared to the strength of their constitutive material because failure requires a sufficient amount of energy to be stored so that it can be released into crack surface creation energy.