A quantitative study of flaw/strength response in ultra-high temperature ceramics based on femtosecond laser method

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Highlights

  • Femtosecond laser method was used to fabricate micro-sized flaws with various sizes and orientations.

  • Different fracture modes were analyzed depending on the flaw sizes and orientations.

  • A new prediction model of strength in ceramics was established, which agreed well with the experimental results.

Abstract

In this paper, a systematic study on the effect of flaws on flexural strength of ultra-high temperature ceramics (UHTCs) was carried out by using three typical materials (i.e., ZrB2, ZrB2-SiC and ZrB2-SiC-G ceramics) containing micro-sized flaws with controllable sizes, shapes as well as orientations. Depending on the flaw sizes and orientations, different fracture modes were analyzed by SEM observation and theoretical calculation. Based on Emmerich’s hole/strength model and the maximum strain-energy release-rate mixed-mode fracture criterion, a universal flaw/strength model was developed to describe the flexural strength of flawed UHTCs after taking both material properties and flaw’s geometries into account, and compared well with the experimental data in both mode I and mixed-mode conditions. Prediction results of the proposed model have further verified that substantially consistent with the flexural strength reports of UHTCs in the literature as well as the theoretical strengths calculated from the elastic modulus, which means that the proposed model has a great potential in guiding the design of high-strength ceramics in full-scale (from atomic-scale to macro-scale).

Introduction

Ceramics are well known to have broad application prospects in aerospace and mechanical manufacturing fields owing to their excellent comprehensive performance [1], [2]. However, it is widely believed that fracture of ceramics at stresses governed by the flaws is far below their theoretical strength [3]. Generally, ceramics are never free of flaws [4], and the flaw sizes (range from 1 [5] to 100 μm [6] in some fine-grained ceramics), shapes (varies from spherical-like shape to elongated shape [7]), and orientations (angles range from 0 to 90° [8]) are always different, resulting in a significant fluctuation of fracture strength during tests. In order to promote the reliable application of ceramics, it is very important to understand the flaw/strength relationship of ceramics comprehensively and accurately, which has become a hot topic since 1913 [9].

Previous studies mainly including two aspects: modeling and experiment. Modeling approaches can be generally divided into mode I and mixed-mode loading conditions. For mode I loading condition, only a few exact solutions for three-dimensional cracked bodies were successfully developed and applied, and the most typical problem of a flat elliptical crack in an infinite solid subjected to uniform tension was first derived by Irwin et al. [10], [11]. For finite bodies, investigators have used approximate analytical methods (alternating method by Smith and Alavi [12], Smith and Sorensen [13] and Kobayashi et al. [14], as well as finite-element method by Kathiresan [15] and Newman and Raju [16], [17]) to obtain stress-intensity factors for surface cracks under tension or bending loads, which in turn can be used for the strength prediction, but these results were only presented in the form of curves or tables. In this wake, for ease of computation, Newman and Raju [18] presented an empirical stress-intensity factor equation for surface cracks in materials with Poisson’s ratio v of 0.3. More recently, various v (0.07–0.4) were considered in the data evaluations and a new stress-intensity factor model was made by Strobl et al [19]. Nevertheless, the application range of the optimized model was very limited (0.4 ≤ a/c ≤ 1.2, 0.01 ≤ a/t ≤ 0.5 and 0.1 ≤ c/b ≤ 0.5, where t is sample thickness, 2b is sample width, 2c is crack width, a is crack depth, respectively), and the calculation process was extremely complicated. Furthermore, all above-mentioned studies were inappropriate in flaw case because these approaches did not depend explicitly on any radius of curvature. The quantification relationship between strengths and different mode I flaws (including flaw length, depth and sharpness, etc.) remains absent in the literature.

It is noteworthy that in practical engineering applications, the stress state around the flaw tip is in most cases of the mixed-mode type. For now, significant progresses of mixed-mode fracture studies have been made, and two of the most classical mixed-mode fracture criteria are: the “maximum normal stress” criterion [20] and the “maximum strain-energy release-rate” criterion [21]. In the former criterion, catastrophic fracture occurred when the tensile stress reached a critical value of the stress-intensity factor. The latter one was based on the assumption that the crack will start to grow when the maximum energy release rate reached a critical value [22], [23]. Both of the theories have been widely utilized in brittle fracture studies, however, their accuracies were still directly relate to the calculation results of the mode I flaw/strength model, which still remains absent in the literature as mentioned in the preceding paragraph.

In experimental aspect, progress in this field is hampered by a lack of flaw’s fabrication methods owing to the technical limitations. At the present stage, the most commonly used methods are indentation [24], [25], [26] and firing green compact containing organic particles of known size [27], [28], [29]. Compared to the latter one, indentation method appeared more convenient to study the relationship between the characteristics of strength and flaw. Previous studies on the strength characteristics using indentation flaws in various typical ceramics (such as Al2O3 [24], Si3N4 [24], soda-lime glass [25] and SiC [26]) indicated that the strengths of these ceramics remained stable first, then decreased gradually as the indentation load increased. However, indentation method has some obvious drawbacks: unclear effect of the residual stress and the subcritical cracks as well as uncontrollable crack size [30]. Thus, in order to fill the experimental gaps for the effect of micron-sized flaws on fracture strength, it is essential to provide a more accurate and controllable flaw’s introduction method. Due to the great superiority in high-precision processing, femtosecond laser technique attracted our interests and has been successfully used for the fabrication of micro-sized flaws in our previous studies [31].

In this paper, we described a systematic study of the effect of flaws on flexural strength using three types of typical UHTCs (i.e., porous ZrB2, fully dense ZrB2-SiC, and reinforced ZrB2-SiC-G ceramics) containing flaws with controllable sizes, shapes as well as orientations, fabricated by a femtosecond laser notching approach. A universal flaw/strength model was developed and compared with the quantitative experimental results. Taking flaw control as the starting point, the possibility of expanding the proposed model to high-strength ceramic design was also discussed.

Section snippets

Materials and test samples

Commercial ZrB2 (mean particle size is 2 μm, purity > 99.5%, Northwest Institute for Non-ferrous Metal Research, China), SiC (mean particle size is 0.5 μm, purity > 99%, Aladdin Industrial Corporation, China) and graphite flake (mean diameter and thickness is 15 μm and 2 μm, respectively, purity > 99.0%, Qingdao Tiansheng Graphite Co.Ltd., Qingdao, China) were used, and the preparation procedure of ZrB2 (Z), ZrB2-20 vol% SiC (ZS) and ZrB2-20 vol% SiC-15 vol% Graphite (ZSG) ceramics were

Irwin model for mode I flaw

The most typical problem of a flat elliptical crack in an infinite solid subjected to uniform tension, was first derived by Irwin [10] using an exact stress analysis by Green and Sneddon [11].KIc=πKK2=1.2σ2aΦ2-0.212(σ/σs)2where KIc is the fracture toughness, K is the stress-intensity factor, the term σS represents the ultimate strength and Φ is the elliptic integral:Φ=0π/2[sin2ϕ+(ac)2cos2ϕ]1/2dϕwhere ϕ is crack angle. The experimental results of Srawley [34] were encouraging with regard to the

A new model derived from the Emmerich model

By considering and connecting microstructure and atomicity, and using an experimentally proved maximum stress criterion for fracture, Emmerich obtained an expression for the strength of the brittle materials, where an effective local cohesive stress (σeff) was innovatively defined [38]. By using the criterion that fracture begins when the local force of cohesion between the critical unit cells attained its elastic maximum value [39], [40], Emmerich assumed that the rupture will start at a

Results

In mode I conditions, as shown in Fig. 6, crack paths can be divided into two parts: (i) linear propagation and (ii) deflection and/or branching at a critical crack length, 2cb, and 2cb increased gradually with increasing flaw length (2c) and flaw depth (a). This interesting phenomenon is consistent with the reports of Field et al. [45], [46], [47], [48], which have shown that the behavior may be caused by the dynamic instability during high speed cracking. They claimed that when a running

Discussion

As the proposed model predicts the flexural strength of Z, ZS and ZSG ceramics containing various flaws fairly well, it could be used to predict flexural strength for any desired microstructure. Generally, the most typical flaw in ceramics can be assumed as spherical or hemi-spherical, where c = a = ρ [55]. As shown in Fig. 11, 2c = 2a is defined as the flaw size, and the black solid lines, which indicate the relationship between flexural strength and hemi-spherical surface flaws for Z and ZS

Conclusions

In this paper, the effect of flaws on fracture behavior and flexural strength of typical UHTCs (i.e., porous Z, fully dense ZS, and reinforced ZSG ceramics) was systematically studied. The following are the main conclusions:

  • Micron-scale flaws with controllable sizes (length of 5–500 μm, depth of 1–51 μm), shapes (tip radius of 0.5–6 μm) and orientations (angle of 0–90°) in UHTCs can be successfully fabricated by femtosecond laser method.

  • In mode I condition, catastrophic fracture continued along

CRediT authorship contribution statement

Anzhe Wang: Conceptualization, Writing - original draft, Project administration, Funding acquisition. Xinyuan Zhao: Data curation, Writing - original draft. Mingxu Huang: Investigation, Writing - original draft. Zhen Zhang: Resources, Writing - review & editing. Lishuai Xie: Funding acquisition, Writing - review & editing.

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.

Acknowledgement

Financial support was provided by the Surface Projects of Natural Science Research in Jiangsu Province (No. 19KJB430021), the Special Talent Introduction for “Double Innovation Plan” in 2019, Natural Science Fund project in Jiangsu Province (No. BK20191020) and the University Research Foundation of Nanjing Institute of Technology (No. YKJ201806).

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