Ceramics International xxx (xxxx) xxx–xxx Contents lists available at ScienceDirect Ceramics International journal homepage: www.elsevier.com/locate/ceramint Numerical simulation and experimental study on crack self-healing in BK7 glass ⁎ Chu Wanga, Hongxiang Wanga, , Lu Shena, Jing Houa,b, Qiao Xub, Jian Wangb, Xianhua Chenb a b School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China A R T I C L E I N F O A B S T R A C T Keywords: Glass Self-healing Surface crack Numerical simulation Prediction model Crack self-healing is the result of a combination of multiple eﬀects and involves many factors, such as mechanics, thermodynamics, physics, and chemistry. In this paper, a ﬁnite element simulation model and a modiﬁed cracklength-prediction model for crack self-healing in BK7 glass were proposed and veriﬁed experimentally. The simulation results showed a stress concentration at the tip of the crack at the initial stage of the crack self-healing process. The crack length decreased gradually, and the stress concentration area moved to the surface. The stress concentration area almost disappeared when the crack was healed completely. When the relative humidity was 64% and the compression was 5 MPa, under variable-temperature heating, the required time for a 19.8 µm precrack complete healing was 7.5 h. As the temperature increased, the required time for complete healing decreased, the contact state between the crack boundaries was improved, and thus the cracks were connected and healed. 1. Introduction During the grinding, polishing, and ﬁnishing of brittle optical elements, defects such as breakages, scratches, cracks, dislocations, residual stresses, and impurities are generated in the material surface or subsurface [1]. These defects are precursors of laser damage to optical elements, and initial laser damage points are often formed in their vicinity [2]. Under high-power laser irradiation, the initial laser damage points of the optical elements are tens of microns, and the size of the damage increases exponentially with laser ﬂuence [3]. To improve the laser-induced damage threshold (LIDT) of optical elements, pretreatment is attempted before use. The main pretreatment methods include chemical etching [4], ultraviolet and carbon dioxide laser irradiation [5,6], ion-beam polishing [7], MRF polishing [8], and high-temperature annealing [9]. Although these pretreatment methods have their own advantages, they exhibit some limitations. It is necessary to ﬁnd a new pretreatment method to improve the LIDT of optical elements. When brittle material anneals under appropriate conditions (temperature, humidity, pressure, and residual stress), material in the complex physical and chemical reactions will produce phenomena, such as softening, viscous ﬂow, blunt crack tips, and crack self-healing, and these can mitigate crack [10–13]. Theoretically, this method improves the LIDT of optical elements used in high laser ﬂuence. However, crack self-healing is not only crack closure, disappearance, or the ⁎ recovery of material strength. The temperature, pressure, humidity and heat-treatment time need to be controlled accurately during the process. If the treatment transforms the material from the glass to the molten state, the surface roughness and shape error of the optical elements will be inﬂuenced, which results in the destruction of optical elements. The temperature, pressure, and humidity are important external factors that aﬀect the solid-phase reaction. Many scholars have conducted research on the inﬂuence of these factors on crack self-healing. Zhu et al. investigated the crack healing behavior of (MoNb)Si2 materials with high-temperature oxidation in air. The material bending strength recovered signiﬁcantly after heat treatment and the crackhealed samples exhibited higher bending strengths than the original level after treating in 1200 °C [14]. Doquet et al. found partial crack healing after annealing in air at 425 °C when a 20-MPa compressive stress was applied, and annealing at 450 °C under 20 MPa pressure led to a complete disappearance of the crack. The results also show that a high compressive stress can shorten the crack and promote healing [15]. Nam et al. investigated the crack healing behavior of SiC ceramics. The heat-treatment temperature has a signiﬁcant inﬂuence on the healing process, and the optimum temperature for SiC healing is 1100 °C [16]. Xu et al. found that the eﬀect of increasing the pressure of heating at low temperature was not obvious, and an unbalanced pressure can increase the time for complete healing [17]. Corresponding author. E-mail address: whx@hit.edu.cn (H. Wang). http://dx.doi.org/10.1016/j.ceramint.2017.10.121 Received 8 September 2017; Received in revised form 18 October 2017; Accepted 18 October 2017 0272-8842/ © 2017 Elsevier Ltd and Techna Group S.r.l. All rights reserved. Please cite this article as: Wang, C., Ceramics International (2017), http://dx.doi.org/10.1016/j.ceramint.2017.10.121 Ceramics International xxx (xxxx) xxx–xxx C. Wang et al. Fig. 1. Sample preparation process. response, and to enable the self-healing method to be used in optical engineering, we need a ﬁnite element method model of the healing process that considers the inﬂuence of multiple physical ﬁelds and covers the entire framework. BK7 glass material was used as the object of the research. Based on the creep characteristics of BK7 glass at a high temperature and the constitutive model of diﬀerent creep stages, a ﬁnite element simulation model for crack self-healing was proposed and established. The inﬂuence of temperature and time on the healing process was analyzed. Finally, the feasibility and the correctness of the model were veriﬁed experimentally. These investigations indicate that reaction and diﬀusion can be enhanced by an increase in temperature, and a high temperature with hot pressing can improve the contact state between crack boundaries and shorten the cracks signiﬁcantly. Transmission between the same materials can cause crack healing and a restoration of material strength [14–17]. The glass viscosity is dependent on the temperature and water content. When a small amount of water is introduced into the melt glass, the material viscosity around the crack can be reduced by water dispersion and glass hydrolysis. Viscous creep ﬂow causes crack healing and the recovery of material strength, and it blunts cracks, which mitigates the stress concentration at the crack tip, which improves the LIDT. However, cracks heal only when the temperature approaches the softening temperature (Ts), and the softening temperature is only tens of degrees lower than the glass transition temperature (Tg), so the temperature control must be very precise during the heat treatment [14–17]. The ﬁnite element method is used to simulate crack self-healing, and the simulation results are intuitive with a low cost. Therefore, many scholars have tried to improve simulations of crack self-healing. Nielsen et al. developed and validated a three-dimensional model to simulate glass tempering, assembled from models of temperature-dependent viscoelasticity and structural relaxation. A prediction of transient and steady-state stresses in complex three-dimensional glass geometries was achieved, and convergence analysis and experimental veriﬁcation were carried out [18]. Barth et al. used the ﬁnite element code Abaqus to simulate cooling of a bulk borosilicate glass. The inﬂuence of thermal gradient on the stress concentration and solidiﬁcation process was analyzed. The dissipation of thermal strain during the transition from liquid to solid reduced the cracking phenomenon during cooling [19]. Nielsen, Barth et al. provided new ideas and reference data for the crack healing of brittle materials. Xu et al. used Abaqus to simulate SCN-1 glass crack closure and healing. The inﬂuence of stress, temperature, and crack morphology on the healing behavior of the glass were investigated and discussed. With an increase in temperature, the increased creep deformation and the grain diﬀusion promoted crack closure and healing [17]. Doquet et al. used the ﬁnite element method to simulate the eﬀect of diﬀerent constraints on the crack healing of inert borosilicate glass. The crack boundary was shortened with an increase in pressure at 400 °C annealing, and eventually the boundary was connected and healed. The tensile strength of the bridging layers was estimated to be 27–39 MPa after vacuum annealing at 400 °C [15]. These results indicate that crack self-healing involves many factors, such as mechanics, thermodynamics, physics, and chemistry, and is the result of multiple eﬀects. Although diﬀerent models have been proposed to analyze the crack variation at diﬀerent stages during the healing process, it cannot be expressed by a single mechanism, and the results of a single crack model cannot be generalized to multiple cracks by simple linear superposition. Moreover, local variation at the crack boundary is usually unsynchronized and disordered, the crack healing path is unpredictable, and irregular secondary cracks are also involved [17]. To match the healing behavior of a single crack with the macro 2. Sample preparation BK7 glass surface microstructures that were introduced by grinding were observed through cross-sectional microscopy detection technology. [20] The surface crack morphologies were observed by atomic force microscopy (Nanite B, Nanosurf Ltd. contact mode, lateral resolution of 1.7 nm, axial resolution of 0.34 nm, and scanning area of 110 µm × 110 µm × 22 µm). The concrete steps were as follows: (1) Two sides of the BK7 glass (30 mm × 15 mm × 6 mm) were polished (the CeO2 particle diameter was 1.5 µm, the polishing solution concentration was 8 wt%, the polishing time was 4 h). (2) The two polished surface were bonded by melting paraﬃn, and the two parts were clamped by using a vice to provide a uniform and thin adhesivelayer thickness to prevent breakage during polishing, then the sample was placed at room temperature until the paraﬃn cooled. The principle is shown in Fig. 1. (3) The up-surface was lapped by using SiC of W28, with a lapping time of 1 h, and a lapping-ﬂuid concentration of 8 wt%. (4) The BK7 glass was immersed in hot water to melt the paraﬃn, and was placed in an ultrasonic cleaner to remove impurities. The cleaned components were placed into a mixed solution (1.5% hydroﬂuoric acid and 1.5% NH4F) for 30 s to remove the hydrolysis layer for crack exposure and then cleaned again by ultrasonic cleaner. (5) The crosssectional morphology was studied by atomic force microscopy. The surface-crack morphologies and distribution were obtained. Two obvious cracks on two specimens were selected for further simulation and heating experiments. The length of the ﬁrst crack was 19.8 µm, its inclination angle was 85.1°, and its maximum crack width was 235 nm (Fig. 2(a)). The length of the other crack was 26.5 µm, its inclination angle was 86.8°, and its maximum width was 330 nm (Fig. 2(b)). 3. Heat healing experimental procedures and results A heating experiment was carried out to verify the feasibility of the simulation method under variable-temperature heating conditions, and the morphologies of the BK7 glass surface cracks were obtained by optical microscopy. The cracked sample (length of 19.8 µm) was heated in ﬁve steps under a 5-MPa pressure and at a 64% relative humidity in a vacuum 2 Ceramics International xxx (xxxx) xxx–xxx C. Wang et al. Fig. 2. Section proﬁles of the two cracks. Fig. 3. Procedures in variable-temperature heating experiment. Fig. 4. Morphologies of crack healing under variable-temperature heating. In Figs. 4 and 5, the crack length of two surface cracks (lengths of 19.8 µm and 26.5 µm) under a particular compressive stress (5 MPa) and humidity (64% relative humidity) decreased with an increase in heat-treatment temperature and time, and the crack disappeared completely eventually. This result indicates that water can improve the contact state between the crack boundaries and shorten the crack. The material viscosity decreased with an increase in temperature and water diﬀusion, the viscous ﬂow caused the crack length to decrease gradually, and ﬁnally the cracks were healed. heating furnace. The process and parameters of the heating experiment are shown in Fig. 3. The morphologies and distribution of the surface cracks after each heating were observed by optical microscopy, and the images are shown in Fig. 4. We investigated the heat-healing process under constant temperature. The second crack (length of 26.5 µm) was heated in 5 steps (500 °C, compression of 5 MPa, 64% relative humidity, the heating time of each step was 1 h, and cooling was at room temperature for 10 h). The crack morphologies after each heating step are shown in Fig. 5. 3 Ceramics International xxx (xxxx) xxx–xxx C. Wang et al. Fig. 5. Morphologies of crack healing under constant-temperature heating. Fig. 6. Variations of creep strain of glass material at diﬀerent times. Table 1 BK7 glass parameters. Parameter Value Elastic modulus E (MPa) Poisson's ratio μ Thermal expansion coeﬃcient α (/K) A B k n Qc (kJ/mol) γc 82000 0.206 8.3×10−6 0.00132 0.00529 0.0169 0.609 610.7 49.2 Fig. 7. Finite element model with a V-type edge crack. stable creep stage in which the strain rate remains constant, and the third is the accelerating creep stage in which the strain rate increases with time [21]. The creep equation of BK7 glass is expressed by the creep strain rate [22]: 4. Numerical simulation of crack self-healing 4.1. Constitutive model ε ̇ ∝ f (T )⋅f (σ )⋅f (S ) Brittle glass material that is heated at a certain stress and high temperature will produce creep, and the creep process is divided into three stages, as shown in Fig. 6. The ﬁrst stage is the deceleration creep stage in which the strain rate decreases with time, the second is the (1) where ε ,̇ T, σ , and S are the creep strain rate, temperature, compression stress, and structure factor, respectively, which characterize the eﬀects of density, porosity, and other internal structures on the creep strain 4 Ceramics International xxx (xxxx) xxx–xxx C. Wang et al. Fig. 8. Stress distributions after each heating step in variable-temperature simulation. compression stress (MPa), n is the average stress index, Qc is the average apparent activation energy (kJ/mol), R is the gas constant, T is the absolute temperature (K), and A exp(−Qc / RT ) describes the BK7 glass viscosity dependence on temperature. The creep of BK7 glass at high temperature is determined by temperature and compression, and the internal material structure changes rate. The creep strain rate in the stable creep stage can be described as [23–25]: εċ st = Aσ n exp(−Qc / RT ) where εċ st (2) is the creep strain rate, A is the material constant, σ is the 5 Ceramics International xxx (xxxx) xxx–xxx C. Wang et al. Fig. 9. Stress distributions in constant-temperature heating simulation. signiﬁcantly when it is near the transition temperature (Tg). This change inﬂuences the creep strain rate of the stable creep stage. Therefore, we need to correct Eq. (2) to a modiﬁed equation [26]: where (T / Tg ) γc describes the eﬀect of temperature near Tg on the creep strain rate and γc is the index of structure in the stable creep stage. The constitutive equation of BK7 glass in the stable creep stage is obtained by integrating time: εċ st = Aσ n exp(−Qc / RT )(T / Tg ) γc εcst (σ , T , t ) = Btσ n exp(−Qc / RT )(T / Tg ) γc (3) 6 (4) Ceramics International xxx (xxxx) xxx–xxx C. Wang et al. obtained previously (length of 19.8 µm, inclination angle of 85.1°, and maximum width of 235 nm). The simulation pressure was set to 5 MPa; the heating process was divided in 5 steps; the heating temperatures were 400 °C, 425 °C, 450 °C, 475 °C, and 500 °C; and the time for each heating step was 1.5 h. The stress distributions after each heating step in the simulation are shown in Fig. 8. A simulation under constant temperature was also carried out. The model dimensions were set as the second crack we obtained previously (length of 26.5 µm, inclination angle of 86.8°, and maximum width of 330 nm). The same model was heated at the same temperature but for a diﬀerent heating time. The heat-healing simulation parameters were as follows: the pressure was 5 MPa; the heating temperature was 500 °C; and the heating times were 1 h, 2 h, 3 h, 4 h, and 5 h, respectively. The stress distributions in the simulation for diﬀerent heating times are shown in Fig. 9. Table 2 Calculation results for slope k. Experiment number P (MPa) T (°C) PT P2 T2 k 1 2 3 4 5 6 7 8 9 10.5 10.5 4.5 4.5 12.5 2.5 7.5 7.5 7.5 485 415 485 415 450 450 500 400 450 5092.5 4357.5 2182.5 1867.5 5625 1125 3750 3000 3375 110.25 110.25 20.25 20.25 156.25 6.25 56.25 56.25 56.25 235225 172225 235225 172225 202500 202500 250000 160000 202500 4.2 3.5 3.5 2.9 4.4 1.9 4.7 2.9 3.8 where B is the material constant. In the deceleration creep stage, the creep strain rate decreases with time, and ﬁnally tends to stability. The process is short, and the constitutive equation is expressed as: εcin (σ , T , t ) = A (1 − exp(−kt )) σ n exp(−Qc /RT )(T /Tg ) γc 5. Prediction model of crack-healing length (5) Greil proposed a model of the crack length with time at a given temperature and pressure [27]. When a pressure is applied to the specimen under constant-temperature conditions, crack healing is accelerated. After a given time, the crack length can be predicted by the following formula: where k is the material constant and t is the time. The constitutive equation of the total creep process of BK7 glass at high temperature can be obtained by combining Eqs. (4) and (5) as: εc (σ , T , t ) = σ n exp(−Qc /RT )(T /Tg ) γc⋅[A (1 − exp(−kt )) + Bt ] (6) cheal = cdam − Fσ mt (7) where cheal is the crack length after heating, cdam is the crack length before heating, t is the heating time, F and m are constants that describe the crack healing of the material at a certain temperature and stress, F varies from 0 to 1, m varied from 0 to 4, σ is the stress, and Fσm can describe the healing rate under certain conditions. However, F and the m are not the same at diﬀerent temperatures, and the formula does not give direct correspondence between these two parameters and the temperature. Therefore, we cannot use the formula proposed by Greil to calculate the crack length. Through our ﬁnite element simulation of the crack-healing process, combined with the experimental design and regression analysis, the slope of the curve of crack length change with time at a certain temperature and pressure can be obtained. The experimental design and calculation results are shown in Table 2, and the slope of the curve can be given by: 4.2. Material parameters and boundary conditions BK7 glass is a typical hard and brittle material with parameters as shown in Table 1. To investigate the inﬂuence of compression and temperature on the BK7 glass healing process, it is necessary to exert pressure on the surface of the element (shown in Fig. 5) and to apply a temperature ﬁeld to the model. Because the sample was small, and the heat-transfer eﬀect was very rapid, the entire model was set at the heating temperature. All degrees of freedom of the right surface in the model were constrained to prevent movement and rotation during the simulation. When the two sides of the crack make contact, the crack is deﬁned as healed. The variation in crack length with time and the complete healing time can be obtained by real-time monitoring of the crack. k = 44.95576 + 2.027267P − 0.24593T − 0.00348PT − 0.01426P 2 4.3. Numerical simulation + 0.000327T 2 It is necessary to establish the crack geometric model according to the actual size for the simulation. A two-dimensional planar model (0.5 mm × 0.5 mm) was established by using the ﬁnite element code Abaqus. A simulation of heating at variable temperature was carried out. The model dimensions (Fig. 7) were set according to the ﬁrst crack that we (8) where k is the slope of the curve, P is the pressure (MPa), and T is the temperature (°C). An improved crack-length prediction model during healing can be obtained from: cheal = cdam − kt Fig. 10. Variations in crack length obtained experimentally and by prediction (a) under variable-temperature heating and (b) under constant-temperature heating. 7 (9) Ceramics International xxx (xxxx) xxx–xxx C. Wang et al. Fig. 11. Relationships of crack length with time (a) for diﬀerent pre-crack lengths, pressures of 5 MPa, and a heating temperature of 500 °C; (b) for diﬀerent temperatures, a pressure of 5 MPa, and a crack length of 20 µm; (c) for diﬀerent pressures, a heating temperature of 500 °C and a pre-crack length of 20 µm. brittle materials. As the temperature increases, the viscosity decreases rapidly and viscous ﬂow is accelerated, therefore, the time for complete healing is reduced. However, when the temperature exceeds the glasstransition temperature (556 °C), the internal structure of the glass material will change from the glass to the molten state, which will aﬀect the optical element surface accuracy and roughness directly. Fig. 11(c) shows that the time was reduced signiﬁcantly when the pressure exceeded 5 MPa. This occurs because the applied pressure can counteract the residual tensile stress that is caused by grinding and/or polishing and reduce the crack width simultaneously, which promotes viscous ﬂow. When the pressure is too large or exceeds the material strength limit, the glass will break with more cracks, and produce permanent plastic deformation. Therefore, the pressure that is applied should be limited, so that crack healing can be more eﬃcient and prevent additional damage and plastic deformation. Under a given temperature and pressure condition, the crack length at any time can be calculated directly by using Eq. (9). For the cracks that we selected previously (length of 19.8 µm and 26.5 µm), variations in crack length under variable and constant temperature can be obtained (Fig. 10). By using Eq. (9), we can obtain the required time for complete healing of cracks of diﬀerent lengths, and the results are shown in Fig. 11(a). Similarly, variations in crack length with time under a 5MPa pressure and various heating temperatures can be obtained (Fig. 11(b)). The variations in crack length with time are shown in Fig. 11(c) at 500 °C and 2.5 MPa, 5 MPa, 7.5 MPa, 10 MPa, and 12.5 MPa. 6. Results and discussion In Figs. 8 and 9, an obvious stress concentration exists at the tip of the crack. The crack was ﬁlled by material viscous ﬂow, and the crack length and width were reduced gradually. At the same time, the stress concentration area moved gradually to the left surface of the model. When the crack is healed completely, the stress concentration area almost disappears. Fig. 10 shows that the experimental results and prediction model are in good agreement, which veriﬁes the correctness of the proposed prediction model. From Fig. 11(a), the healing rate of cracks of diﬀerent lengths was the same under certain pressure and temperature conditions, and the length was linear with time. When the pressure was constant, a higher temperature yielded a higher healing rate and a lower time for the same crack length to heal completely, which agrees with the thermodynamic theory above. Fig. 11(b) shows that the required time for complete healing decreased with temperature. This occurs because the crackhealing reaction needs to overcome the activation barrier by reaching a certain temperature, and the temperature inﬂuences the viscosity of 7. Conclusions A ﬁnite element simulation of BK7 glass crack self-healing was proposed and the process was simulated numerically. The eﬀect of temperature on crack healing was analyzed, and the correctness and feasibility of the simulation model were veriﬁed experimentally. The simulation results show an obvious stress concentration at the tip of the crack at the initial stages of crack healing. With crack healing, the length and width of the crack were reduced gradually, and the stress concentration area moved gradually to the left surface of the model. Finally, the crack was healed completely, and the stress concentration area almost disappeared. An improved crack-length prediction model was proposed and veriﬁed experimentally. The inﬂuences of crack length, temperature, and pressure on the crack healing rate were analyzed. Under diﬀerent pressure and temperature conditions, the healing rate was the same for a diﬀerent crack length, and the healing length was linear with time. 8 Ceramics International xxx (xxxx) xxx–xxx C. Wang et al. Laboratory (LLNL), Livermore, CA, 2006. [9] Y. Jiang, X. Xiang, C. Liu, H. Wang, W. Liao, H. Lv, X. Yuan, R. Qiu, Y. Yang, W. Zheng, Eﬀect of residual stress on laser-induced damage characterization of mitigated damage sites in fused silica, J. Non-Cryst. Solids 410 (2015) 88–95. [10] K. Ando, M.-C. Chu, K. Tsuji, T. Hirasawa, Y. Kobayashi, S. 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The healing rate increases with temperature and pressure. When the pressure exceeded 5 MPa, the required time for complete crack healing was reduced signiﬁcantly. The activation energy barrier of the healing process is high, which requires a certain temperature to activate and overcome. This occurs because the brittle-material viscosity exhibits a viscosity-temperature characteristic; as the temperature increases, the viscosity decreases and the viscous ﬂow is accelerated, so the required time is reduced. The applied pressure is able to counteract the residual tensile stress that is caused by grinding and polishing. As the pressure increases, the crack boundary is shortened, and thus, the required time is also reduced. Acknowledgements This work was supported by the Science Challenge Project of China (Grant No.JCKY2016212A506-0503) and National Natural Science Foundation of China (Grant No.51475106). References [1] T. Suratwala, R. Steele, M. Feit, L. Wong, P. Miller, J. Menapace, P. 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