materials Article
Comparative Study on Crack Initiation and Propagation of Glass under Thermal Loading Yu Wang, Qingsong Wang *, Haodong Chen, Jinhua Sun and Linghui He State Key Laboratory of Fire Science, University of Science and Technology of China, Hefei 230026, China;
[email protected] (Y.W.);
[email protected] (H.C.);
[email protected] (J.S.);
[email protected] (L.H.) * Correspondence:
[email protected]; Tel.: +86-551-6360-6455; Fax: +86-551-6360-1669 Academic Editor: Mark Whittaker Received: 6 July 2016; Accepted: 14 September 2016; Published: 22 September 2016
Abstract: This paper explores the fracture process based on finite element simulation. Both probabilistic and deterministic methods are employed to model crack initiation, and several commonly used criteria are utilized to predict crack growth. It is concluded that the criteria of maximum tensile stress, maximum normal stress, and maximum Mises stress, as well as the Coulomb-Mohr criterion are able to predict the initiation of the first crack. The mixed-mode criteria based on the stress intensity factor (SIF), energy release rate, and the maximum principal stress, as well as the SIF-based maximum circumferential stress criterion are suitable to predict the crack propagation. Keywords: glass; finite element method; thermal stress; crack initiation; crack growth
1. Introduction Windows are among the weakest parts in a building and have drawn great attention when considering the case of fire. Upon heating, a window glass may break and release the accumulated thermal stress. This opens a gate for oxygen flow from the outside into the room, and thus accelerates the fire development. The breakage of glass exposed to fire includes two stages. First, cracks initiate and propagate in a glass that is under heating. Then they coalesce to form “islands”, followed by the falling out of the fragments. Since this process may exert considerable influence on the development of a fire, understanding this mechanism is of important significance not only for fundamental research but also for practical applications. The edges of a window glass are covered by a frame. When exposed to a fire, the temperatures in the central portion and edge region are different. It is believed that the glass cracks when this temperature difference reaches a critical value. However, the data of the critical temperature difference proposed by different authors exhibit considerable discrepancy. For example, the theoretical predictions given by Keski-Rahkonen [1] and Pagni and Joshi [2–4] are 80 ◦ C and 58 ◦ C, respectively, while the experimental measurements reported by Skelly et al. [5] and Shields et al. [6] vary from 90 to 150 ◦ C, respectively. Klassen et al. [7] examined the effect of sample size on the temperature difference for crack initiation, and found that the critical values for small (305 × 305 mm2 ) and medium (609 × 1219 mm2 ) samples are less than 200 ◦ C and 300 ◦ C, respectively. The non-uniform temperature distribution in a glass exposed to fire may be influenced by various factors, such as the thermal and mechanical properties of the glass, the decay length of flame radiation, and the heating methods [3–5]. In particular, the presence of surface flaws can significantly reduce the practical strength of a glass product [8]. These factors obviously endow some probabilistic characteristics to crack initiation in glass [9]. For this reason, Joshi and Pagni [4] proposed a three-parameter cumulative Weibull function that can be used to model the fracture of ordinary window glass, and Hietaniemi [9] developed a probabilistic approach to evaluate the Materials 2016, 9, 794; doi:10.3390/ma9100794
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glass, and Hietaniemi [9] developed a probabilistic approach to evaluate the breakage condition for a window pane heated by a fire. Nonetheless, research in this direction is still under way, and the current capability predicting propagation in glass can in and should be largely breakage conditionoffor a windowcrack paneinitiation heated byand a fire. Nonetheless, research this direction is still enhanced. thecurrent motivation of theof present study. Based on a finite simulation, this paper under way,This andisthe capability predicting crack initiation andelement propagation in glass can and exploresbethe fracture processThis of is a the window glass of under the thermal the should largely enhanced. motivation the present study. loading Based onofa fire. finiteBoth element probabilisticthis and deterministic are adopted, and the main objective is toloading examine the simulation, paper explores themethods fracture process of a window glass under the thermal of fire. validities of some commonly used criteria of crack for window glasses Both the probabilistic and deterministic methods areinitiation adopted,and and propagation the main objective is to examine under fire conditions. the validities of some commonly used criteria of crack initiation and propagation for window glasses under fire conditions. 2. Thermal Stress Distribution 2. Thermal Stress Distribution In the case of a fire, window glass is normally subjected to a non-uniform heating, due to the In the of a fire, window glass is normally subjected to of a non-uniform heating, due to the fact that thecase temperature of the upper portion is higher than that the lower portion. To investigate fact that the temperature of the upper portion is higher than that of the lower portion. To investigate the glazing behavior, a simple assumption of a three layer rising temperature was adopted. It is the the glazing behavior, a simplethis assumption of aisthree layer rising adopted. is authors’ belief that although assumption not the same as temperature a real fire, itwas is suitable forItthe the authors’ belief that although this assumption is not the same as a real fire, it is suitable for the investigation in the present work, as the objective is to investigate different criteria and determine investigation the present as the objective is to and investigate differentAcriteria determine somea some suitablein methods towork, predict crack initiation propagation. squareand glass pane with suitable methods to ×predict crack propagation. A As square glass with a dimension of dimension of 0.003 0.3 × 0.3 m3initiation was usedand in the simulation. shown in pane Figure 1, the edge of the 3 was used in the simulation. As shown in Figure 1, the edge of the glass pane is 0.003 × 0.3 × 0.3 m glass pane is covered by a frame with a width of 0.02 m. The central portion of the pane is divided covered a frame a width of 0.02 The central portion of the pane is divided vertically into verticallybyinto three with regions in which them. heating rates are different. The sample was heated at 300 K three in which the heating rates are upper, different. The sample was heated 300 K forK,60380 s, and the for 60regions s, and the final temperatures in the middle, and lower regionsatwere 400 K, and final temperatures in the middle, and lower regions K, 380frame, K, andso 345the K, respectively. 345 K, respectively. At upper, the edge, the glass is shaded bywere the 400 window glass rising At the edge, the glass small is shaded thefire. window frame,the so glass the glass rising temperature is quite small temperature is quite in aby real Therefore, edge temperature is assumed to be in a real fire. Therefore, the glass edge temperature is assumed to be invariant. Using an in house invariant. Using an in house software, the thermal stress in the glass was computed via 45 × 45 × 2 software, theelements thermal stress in the glass was computed 45 × 45 × 2 studies hexahedron elements [10], hexahedron [10], with material constants citedvia from previous [11–14]. It is noted with material constants cited from previous studies [11–14]. It isbut noted some glass parameters that some glass parameters are temperature-dependent [15], in that the present simulation the are temperature-dependent [15], but in the present simulation the temperature range is very small temperature range is very small and when the crack occurs, the maximum temperature in glazing is and when occurs, maximum range, temperature in glazing is around 380 K. Inissuch small around 380the K. crack In such a smallthe temperature the variance of material parameters very alimited temperature range, the variance material parameters very limited [16] and isdistributions thus ignored of in the [16] and is thus ignored in the of simulation. Figure 2a,bis reports the calculated simulation. 2a,bMises reports the calculated distributions of thethat temperature and von Mises stress, temperatureFigure and von stress, respectively. It can be seen the von Mises stress attains its respectively. It can thatpart the of von Mises stress attains itsother maximum in the central partthe of maximum value in be theseen central the upper edge. All the stress value components, including the upper stress edge. values All the σother including the principal , σ2 , and σ3 principal 1, σ2, stress and σcomponents, 3 are obtainable. The results are usedstress as a values startingσ1point in the are obtainable. The results areinitiation. used as a starting point in the subsequent analysis of crack initiation. subsequent analysis of crack
Figure 1. statement of the glass under thermal loading. The glass are also reported Figure 1. Problem Problem statement of the glass under thermal loading. Theproperties glass properties are also ([10–13]). reported ([10–13]).
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Figure 2. 2. Distributions stress in in thethe glass before crack initiation. (a) Figure Distributionsofoftemperature temperatureand andvon vonMises Mises stress glass before crack initiation. 7” in legend). Temperature distribution in glazing; (b) Von Mises Stress. (“3.8E+07” represents “3.8 × 10 (a) Temperature distribution in glazing; (b) Von Mises Stress. (“3.8E+07” represents “3.8 × 107 ” in legend).
3. Modeling of Crack Initiation
3. Modeling of Crack As the glass crack Initiation initiation shows some stochastic behaviour, a probabilistic criterion is used to predict the glass crack initiation. For comparison, another deterministic criterion is employed in the As the glass crack initiation shows some stochastic behaviour, a probabilistic criterion is used to simulation to predict when and where a crack initiates in the glass pane under the thermal conditions. predict the glass crack initiation. For comparison, another deterministic criterion is employed in the simulation to predict when and where a crack initiates in the glass pane under the thermal conditions. 3.1. Probabilistic Criterion 3.1. Probabilistic Criterion Based on their experiments of the four-point bending test for 59 glass plates, Joshi and Pagni [4] described the distribution of theofbreaking stress by a three-parameter cumulative Weibull function Based on their experiments the four-point bending test for 59 glass plates, Joshi and Pagni [4] as follows: described the distribution of the breaking stress by a three-parameter cumulative Weibull function as follows:
(0 0 F (σb ) = m F ( b ) 1 − exp[−( b σbσ−0σuu )m ]
1 exp[(
b σ u σb < u σb ≥ σu
) ] b u
(1) (1)
where σb is the breaking stress, σ0 and m shape parameters in the Weibull distribution are the scaleand 0 function, and σu is often referred to as the zero strength or lowest strength of the specimen. The values where σb is the breaking stress, σ0 and m are the scale and shape parameters in the Weibull of m, σ0 , and σu were obtained by data fitting and the detailed values are listed in Table 1 [4]. distribution function, and σu is often referred to as the zero strength or lowest strength of the Further experiments by Pagni [17] with different glass thicknesses enable one to assess the dependence specimen. The values of m, σ0, and σu were obtained by data fitting and the detailed values are listed of these parameters on the glass thickness L. The equations characterizing this dependence, written in in Table 1 [4]. Further experiments by Pagni [17] with different glass thicknesses enable one to assess a non-dimensional form, are as follows [9,17]: the dependence of these parameters on the glass thickness L. The equations characterizing this dependence, written in a non-dimensional form, are L as follows [9,17]: σu = 0.5016( ) + 34.59 (2) MPa mm u L 0.5016( ) 34.59 (2) L 2 mm L σ0 M Pa = −0.2919( ) + 2.9309( ) + 27.497 (3) MPa mm mm 0 L 2 L 0.2919(L 2 ) 2.9309( L ) 27.497 (3) mMPa = 0.000815( mm ) − 0.000985( mm ) + 1.205 (4) mm mm L 2 L m 0.000815( ) 0.000985( ) 1.205 (4) mmDistribution Function (4). Table 1. Parameters for two- and mm three-parameter Weibull Weibull Function m σ 0 (MPa) σ u (MPa) Table 1. Parameters for two- and three-parameter Weibull Distribution Function (4). Two-parameter 3.20 74.1 – Weibull Function1.21 m σ0 (MPa) σu (MPa)35.8 Three-parameter 33.0
Two-parameter Three-parameter
3.20 1.21
74.1 33.0
– 35.8
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3.1.1. Two-Parameter Weibull Distribution 3.1.1. Two-Parameter Weibull Weibull Distribution Distribution When σu is taken as zero, Equation (1) becomes the two-parameter Weibull distribution. In this When σuu is is taken taken as as zero, zero, Equation (1) becomes the two-parameter Weibull distribution. In this case, the probability distribution of crack initiation in the glass is shown in Figure 3. It can be noted case, the probability distribution of crack initiation in the glass is shown in Figure 3. It can be noted that the probability is higher in the edge region than in the central portion of the pane. The highest that the probability is higher in the edge region than in the central central portion of the the pane. pane. The highest probability occurs in the portion near the central part of the upper edge, where both the temperature probability occurs in the portion near the central part of the upper edge, where both the temperature and tensile stress attain their maximum values. At the two side edges, the probability becomes and tensile their maximum values. At the the probability becomesbecomes smaller. tensilestress stressattain attain their maximum values. At two the side two edges, side edges, the probability smaller. The smallest probability appears at the lower edge. The Weibull method can predict the The smallest appears atappears the lower edge. The edge. Weibull method canmethod predictcan the probability smaller. The probability smallest probability at the lower The Weibull predict the probability distribution of glass crack initiation, and the maximum value is selected here. In our distribution of glass crack the maximum is selected In our here. simulations, probability distribution of initiation, glass crackand initiation, and thevalue maximum valuehere. is selected In our simulations, we postulate that the glass cracks if the probability is larger than 0.5. As to the random we postulate that the glass cracks if the probability is larger than 0.5. As to the random crack initiation simulations, we postulate that the glass cracks if the probability is larger than 0.5. As to the random crack initiation location, this needs another model for its prediction [18], which is not discussed here. location, this needs another its prediction which is not discussed crack initiation location, thismodel needs for another model for[18], its prediction [18], which ishere. not discussed here.
Figure 3. Probability distribution of crack initiation via the two-parameter Weibull distribution. Figure distribution. Figure 3. 3. Probability Probability distribution distribution of of crack crack initiation initiation via via the the two-parameter two-parameter Weibull Weibull distribution.
3.1.2. Three-Parameter Weibull Distribution 3.1.2. Three-Parameter Weibull Distribution Distribution 3.1.2. Three-Parameter Weibull The three-parameter Weibull distribution is more robust and may provide a better The three-parameter three-parameter Weibull distribution is more and amay provide a better The Weibull distribution more simulation robust androbust may provide characterization of characterization of the data [19]. By setting theissame condition, the better probability distribution characterization of the data [19]. By setting the same simulation condition, the probability distribution thethe data [19]. By setting the same condition, probability thethe three-parameter of three-parameter model is simulation obtained and shownthe in Figure 4. It distribution can be seenof that contour plot of the three-parameter model is shown in Figure It can beplot seen that the to contour plot model is obtained inobtained Figure model. 4. and It can seen that the4.contour similar that of the is similar to that ofand theshown two-parameter Abeslightly different feature is is that the highest value is similar to that of the two-parameter model. A slightly different feature is that the highest value two-parameter model.inA the slightly different feature is thatedge. the highest value of now in the now appears exactly central part of the upper The values theappears critical exactly temperature now appears exactly in the centralvalues part of the upper edge. The values of the for critical temperature central partfor ofthe thefirst upper edge. difference the first crack are difference crack are The predicted asof94the °Ccritical and 93temperature °C by the twoand three-parameter models, differenceas for94the crack predicted as 94 °C and 93 °C bymodels, the twoand three-parameter models, ◦ Cfirst ◦ Care predicted and 93 by the twoand three-parameter respectively. The results exhibit respectively. The results exhibit good coincidence and are close to the value of 90 °C reported by respectively. The and results exhibit coincidence and are close to the good coincidence are close to good the value of 90 ◦ C reported by Skelly [5].value of 90 °C reported by Skelly [5]. Skelly [5].
Figure distribution. Figure 4. 4. Probability Probability distribution distribution of of crack crack initiation initiation via via the the three-parameter three-parameter Weibull Weibull distribution. Figure 4. Probability distribution of crack initiation via the three-parameter Weibull distribution.
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3.2. Deterministic Criterion 3.2. Deterministic Criterion Another way to predict the crack initiation is to use a deterministic criterion. To perform this, Another way to predict the crack initiation is to use a deterministic criterion. To perform this, we employ the following five strength theories: we employ the following five strength theories: 3.2.1. Maximum Normal Stress Criterion 3.2.1. Maximum Normal Stress Criterion The maximum normal stress criterion, also known as Coulomb’s criterion, states that a crack The maximum normal stress criterion, also known as Coulomb’s criterion, states that a crack occurs when the maximum normal stress reaches the ultimate strength of the material [20]. The ultimate occurs when the maximum normal stress reaches the ultimate strength of the material [20]. The tensile strength is smaller than the compressive strength in most cases. We assume the same ultimate ultimate tensile strength is smaller than the compressive strength in most cases. We assume the same strength Sut in tension and compression, which means that the crack forms in the multi-axial state ultimate strength Sut in tension and compression, which means that the crack forms in the multi-axial of stress, when the maximum principal normal stress exceeds the ultimate tensile or compressive state of stress, when the maximum principal normal stress exceeds the ultimate tensile or strength. This is the safer way to predict the crack initiation, and can be written as: compressive strength. This is the safer way to predict the crack initiation, and can be written as: max(σ11 ,, σ max( 22, σ , 3 )3 ≥ ) Sut S ut
(5) (5)
ut is the ultimate tensile stress of the glass pane. Here Sut ut was where Sut was taken taken as as 40 MPa for the float glass without withoutconsideration considerationofof flaw distribution. The distribution of is 3 1)/S , ut ) / S ut in flaw distribution. The distribution of max(σmax( is 3plotted 1 , σ2 , σ 2 , Figure results given by the probabilistic criterion, the maximum is seen at the plotted5.inSimilar Figure to 5. the Similar to the results given by the probabilistic criterion, the value maximum value is center of the upper edge. At the center of the glass pane, there are some grades which are not visible in seen at the center of the upper edge. At the center of the glass pane, there are some grades which are Figures 3 and 4. These3differences aredifferences caused by are the caused calculation methods. not visible in Figures and 4. These by the calculation methods.
Figure 5. Distribution of of before crack initiation. max( ) / utS utbefore Figure 5. Distribution max(σ11,,σ22,,σ3 )3/S crack initiation.
3.2.2. Coulomb-Mohr Criterion The Coulomb-Mohr criterion asserts that fracture occurs when the principal stresses satisfy the following condition [21]: σ1 σ − 3 ≥1 (6) Sut1 Suc3 1 (6) S ut compressive S uc with Sut and Suc being the ultimate tensile and strengths, respectively. Note that both σ3 and Suc are always negative. As can be noted in Figure 6, the maximum value of σ1 /Sut − σ3 /Suc with Sut and Suc being the ultimate tensile and compressive strengths, respectively. Note that both σ3 is located at the center of the upper edge, meaning that the cracking of the glass initiates from and Suc are always negative. As can be noted in Figure 6, the maximum value of 1 / S ut 3 / S u c is that area. This case is consistent with the foregoing predictions. Considerable differences between located at the center of the upper edge, meaning that the cracking of the glass initiates from that area. Figures 5 and 6 appear in the exposed part of the pane, but have no significant effect on crack initiation This case is consistent with the foregoing predictions. Considerable differences between Figures 5 because the stress there is much smaller than that in the edge region. and 6 appear in the exposed part of the pane, but have no significant effect on crack initiation because the stress there is much smaller than that in the edge region.
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Figure 6. Distribution of 1 / S ut 3 / S uc before crack initiation. Figure before crack initiation. 11/S / Sutut − σ33 /S / Suc Figure 6. 6. Distribution Distributionof of σ uc before crack initiation.
3.2.3. Maximum Principal Stress Criterion 3.2.3. Maximum Criterion 3.2.3. Maximum Principal Principal Stress Stress Criterion The maximum principal stress criterion can be expressed by: The maximum maximum principal principal stress stress criterion criterion can can be be expressed expressed by: by: The
σ111 1 ≥ 11 SS Sutut
(7) (7) (7)
ut
This means that the crack occurs when the largest principal stress σ σ1 exceeds uniaxial tensile exceeds the This means that the crack occurs when the largest principal stress σ11 exceeds the uniaxial tensile strength. Although this criterion allows for aa quick and simple comparison with experimental data, Although this criterion allows for quick and simple comparison with experimental data, strength. Although this criterion allows for a quick and simple comparison with experimental data, it is rarely suitable for design purposes [22]. The distribution of σ1 /S is depicted in Figure 7. is depicted in Figure 7. As in S u t is depicted in Figure 7. As it is rarely suitable for design purposes [22]. The distribution of 11 // ut As S ut Figures 5 and 6, the maximum value is also located at the of the edge.edge. At the time, in Figures 5 and 6, the maximum value is also located at top the center top center ofglass the glass Atsame the same in Figures 5 and 6, the maximum value is also located at the top center of the glass edge. At the same it is interesting to see that the distribution in the exposed region of the glass resembles a combination time, it is interesting to see that the distribution in the exposed region of the glass resembles a time, it is interesting to see that the distribution in the exposed region of the glass resembles a of those shown in Figures 5 and 6. Since5the involved principle stress σ1 is always this combination of those shown in Figures andonly 6. Since the only involved principle stresspositive, σ1 is always combination of those shown in Figures 5 and 6. Since the only involved principle stress σ1 is always criterion can be used only if the glass is under expansion. positive, this criterion can be used only if the glass is under expansion. positive, this criterion can be used only if the glass is under expansion.
Figure 7. 7. Distribution ofofσ1 /S beforecrack crackinitiation. initiation. Figure Distribution Figure 7. Distribution of 11 // SSutuutt before before crack initiation.
3.2.4. Maximum Criterion 3.2.4. Maximum Mises Mises Stress Stress 3.2.4. Maximum Mises Stress Criterion Criterion Also known as or criterion Also as the the shear-energy shear-energy theory theory or the the maximum maximum distortion distortion energy energy theory, theory, this this Also known known as the shear-energy theory or the maximum distortion energy theory, this criterion criterion states that aamaterial starts to crack at a at location wherewhere the von Mises stress attains the yieldthe strength states that material starts to crack a location the von Mises stress attains yield states that a material starts to crack at a location where the von Mises stress attains the yield strength [23]: strength [23]: [23]:
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vm 0.577S ut
1
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(8)
vm
1 σvm Sof / 0 .5 7 7 S appears at the top center,(8)in As shown in Figure 8, the maximum0value .577 ut (8) ut≥ 1v m 0.577Sut accordance with the results given by the other criteria mentioned above. However, clear gradation the top center, in As shownininFigure Figure 8, the maximum value v m / ut 0 .5appears 7 7 S u t appears Asseen shown 8, the maximum of σvmof/0.577S at the topatcenter, in accordance can be in the three regions where thevalue temperatures are different due to unequal heating rates. In accordance withgiven the the occurs other criteria mentioned However, clear gradation with the regions, results by thegiven otherby criteria mentioned However, clear gradation can be for seen all three the results lowest Mises stress at the above. central part,above. implying that it is difficult a can beto seen in the three regions where the temperatures aredue different due toheating unequalrates. heating rates. In in the three regions where the temperatures are different to unequal In all three crack initiate there. all three the regions, theMises lowest Mises stressatoccurs at thepart, central part, implying it is difficult forto a regions, lowest stress occurs the central implying that it is that difficult for a crack crack tothere. initiate there. initiate
Figure 8. Distribution of vm / 0.577S ut before crack initiation. Distribution beforecrack crackinitiation. initiation. 0.577Sutut before Figure Distributionofofσvm vm //0.577S 3.2.5. Maximum ShearFigure Stress8.8. Criterion
maximum shear stress criterion is also known as the Tresca yield criterion, and it predicts 3.2.5.The Maximum Shear Stress Criterion failure of a material when the absolute maximum of the shear stress reaches the strength of simple The[23], maximum shear stress criterion criterion is also also known known as as the Tresca yield criterion, and it predicts tension i.e., failure of a material when the absolute maximum of the shear stress reaches the strength of simple max( xy , xz , yz ) tension [23], i.e., 1 (9) max(σ0xy.5, σ xz , σyz ) S ut ≥1 (9) max(0.5S xy , xz , yz ) ut 1 (9) 0.5SutSututisisshown max(( xzxz,,σyzyz))/0.5S / 0.5 The shownFigure Figure9.9. The The pattern pattern is is dramatically dramatically The distribution distribution of of max σxyxy,,σ different from those obtained from any other criteria, and the largest value appears near the top-left The distribution of max( xy , xz , yz ) / 0.5Sut is shown Figure 9. The pattern is dramatically corner. However, However, this corner. this does does not mean that cracks will initiate from that position as well, because under different from those obtained any are other criteria, and thethe largest nearthis thecriterion top-left Therefore, the thermal loading the normalfrom stresses much larger than shearvalue stress.appears Therefore, corner. However, doescrack not mean that cracks will initiate from that position as well,stresses. because under used to this predict initiation as it the from cannot be it excludes excludes the contribution contribution from normal normal stresses. the thermal loading the normal stresses are much larger than the shear stress. Therefore, this criterion cannot be used to predict crack initiation as it excludes the contribution from normal stresses.
Figure 9.9. Distribution ofofmmax beforecrack crack initiation. ax((σxyxy,,σxz / 0.5 Sutut before Figure Distribution σyzyz))/0.5S initiation. xz ,, Figure 9. Distribution of m ax( xy , xz , yz ) / 0.5 S ut before crack initiation.
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All the methods mentioned above were applied to examine the time to first crack, and the results All theinmethods above demonstrated were applied toabove, examine time to first during crack, and results are listed Table 2.mentioned For the reasons no the crack appears thethe period of are listed in Table 2. For the reasons demonstrated above, no crack appears during the period of simulation when the maximum shear stress criterion is used. This exception aside, one can see that simulation when the maximum stress criterion is used. exception one canmethods see that the deterministic methods No. 3shear to 6 give shorter times to firstThis crack than theaside, probabilistic the methods No. 3 to 6 give shorter times to first crackand thanMaximum the probabilistic methods No.deterministic 1 to 2. Particularly, the Maximum normal stress, Coulomb-Mohr, principle criteria No. 1 to 2. Particularly, the Maximum normal stress, Coulomb-Mohr, and Maximum principle criteria lead to an identical result of 23.5 s. This is the shortest time to first crack and can be used as a criterion lead to aninitiation. identical result of 23.5 s. This is the shortest first crack andresults can be in used as a criterion for crack In comparison, the maximum von time Misestostress criterion a slightly longer for crack initiation. In comparison, the maximum von Mises stress criterion results in a slightly longer time of 25 s. time of 25 s. Table 2. Time to first crack predicted by various criteria. Table 2. Time to first crack predicted by various criteria. σvm, MPa σ1, MPa No. Criteria ΔT, °C σy, MPa No. Criteria σ 1 , MPa 1 Two-parameter Weibull ∆T, ◦ C 94 σ y , MPa 67.78 σ vm , MPa 67.23 67.79 Three-parameter Weibull 94 93 60.50 12 Two-parameter Weibull 67.78 60.50 67.2359.80 67.79 23 Three-parameter Weibullstress 93 59.8039.48 60.50 Maximum normal 76.8 60.50 40.03 40.03 34 Maximum normal stress 76.8 76.8 40.03 40.03 39.4839.48 40.03 Coulomb-Mohr 40.03 4 Coulomb-Mohr 76.8 40.03 39.48 40.03 5 Maximum principle stress 76.8 40.03 39.48 40.03 5 Maximum principle stress 76.8 40.03 39.48 40.03 Maximum von Mises 40.73 66 Maximum von Mises stress stress 78.0 78.0 40.72 40.72 40.1440.14 40.73 Maximum no crack – – 77 Maximum shearshear stressstress no crack – – – –
Time, s Time, 44.0s
43.0 44.0 43.0 23.5 23.5 23.5 23.5 23.5 23.5 25.0 25.0 ––
4. Modeling 4. Modelingof ofCrack CrackPropagation Propagation Three linearly linearly independent cracking modes are categorized as Mode I, II, and III, as shown shown in in Figure 10. 10. Mode I is an opening mode where the crack surfaces surfaces move move directly directly apart apart and and itit is is the the most Figure common load type type encountered encountered in in engineering engineering design. design. Mode II is a sliding mode where the crack crack surfaces slide slide over overone oneanother anotherinina adirection directionperpendicular perpendicular leading edge of the crack. Mode surfaces toto thethe leading edge of the crack. Mode III IIIaistearing a tearing mode where crack surfaces moverelative relativetotoone oneanother anotherand andparallel parallelto tothe the leading leading is mode where thethe crack surfaces move edge of of the the crack. crack. Different subscripts are used to designate the stress intensity intensity factor (K) and and the the energy release releaserate rate(G) (G)for forthe thethree threedifferent differentmodes. modes. The and G for mode I are designated KI and energy The KK and G for mode I are designated KI and GI GI when applied to the crack opening mode. mode II stress intensity factor and energy release when applied to the crack opening mode. TheThe mode II stress intensity factor and energy release rate, rate, KIIGand GIIapplied , are applied the sliding crack sliding and the III mode IIIintensity stress intensity factor and K to theto crack mode,mode, and the mode stress factor and energy II and II , are energy rate, release III, are applied to the tearing release KIIIrate, andKGIIIIIIand , areGapplied to the tearing mode. mode.
Figure 10. 10. Crack Crack Mode Mode I, I, Mode Mode II, II, and and Mode Mode III. III. Figure
Numerical simulation of crack propagation has been a challenging problem for many years, and Numerical simulation of crack propagation has been a challenging problem for many years, and discrete crack propagation modeling approaches using remeshing techniques are usually adopted. discrete crack propagation modeling approaches using remeshing techniques are usually adopted. The moving tip remesh method is utilized here to refine the tip field of the crack. The following six The moving tip remesh method is utilized here to refine the tip field of the crack. The following six criteria are chosen to determine whether a crack grows or is arrested. criteria are chosen to determine whether a crack grows or is arrested. 4.1. Mixed-Mode Mixed-Mode Criterion Criterion Based Based on on SIFs SIFs 4.1. The SIFs-based SIFs-based mixed-mode mixed-mode criterion criterion is is commonly commonly used used to to predict predict crack crack growth. growth. It It assumes assumes that that The cracks start to grow once the stress intensity factors K I , K II , and K III satisfy the following condition cracks start to grow once the stress intensity factors KI , KII , and KIII satisfy the following condition [24]: [24]: γ K α K β K ( I ) + ( II ) + ( III ) = 1 (10) KIC KIIC KIIIC
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KI K II K III ) 1 K( IK ) K( IIK ) K( III ( )IC ( )IIC ( K IIIC ) 1 K IC K IIC K IIIC
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where KIC, KIIC, and KIIIC denote the values of fracture toughness of the I, II and III modes, respectively, IC,β, Kand IIC, and KIIICempirical denote the values ofChoosing fracture toughness respectively, where Kα, and γ are constants. and ignoring the term of the tearing of 2of,the where K theI,I,II IIand andIII IIImodes, modes, respectively, IC , KIIC , and KIIIC denote the values of fracture toughness andmode α, and are constants. Choosing and ignoring ignoring the the term term of of the the tearing tearing β = 22,, and as γ suggested by Wu [25], yields and α, β, β, γ, and γ are empirical empirical constants. Choosing α= γ, as suggested by Wu [25], yields mode γ,
K K ( 2I ) 2 K( 2II ) 2 1 K (( KIIK))IC2+(( KIIIIK))IIC2=11 KIC KKIIC K IC IIC
(11) (11) (11)
Based on this criterion, the crack propagation path is predicted and shown in Figure 11. The Based on the crack is predicted and and shown in Figure 11. center The on this thiscriterion, criterion, the crack propagation path is propagates predicted shown intoFigure 11.crack The crack starts from the center of the propagation upper edge, path and then downwards the of the starts from the center of the of upper edge, and then downwards center oflarger the upper crack starts from the upper edge, and then propagates downwards to the center ofthan the KII I is initially upper region inthe thecenter pane. During this process, aspropagates illustrated in Figure 12,toKthe region in pane. this process, as direction. illustrated in Figurein12, is 12, initially larger than initially larger than KIIIII upper inthus theDuring pane. During process, as illustrated Figure KI is and region KIIIthe and dominates thethis crack Thereafter, KIIKIbecomes larger than KIK, IIsoand theKcrack andchanges thus dominates the crack direction. Thereafter, KII becomes larger than KI , so theKcrack changes KIII and thus dominates the cracktodirection. larger than I, so the crack the propagation direction the right Thereafter, of the glass.KII becomes the propagation directiondirection to the right of the glass. changes the propagation to the right of the glass.
Figure 11. Crack path predicted by the mixed-mode criterion based on SIFs. Figure criterion based based on on SIFs. SIFs. Figure 11. 11. Crack Crack path path predicted predicted by by the the mixed-mode mixed-mode criterion 7
7x10
KI
7
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SIFs (Pam ) 0.5 SIFs (Pam )
7 7x10 6x10
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KII
KII KIII
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Time step (s) 10 15 20 Time step (s)
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25 25
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Figure 12. 12. Variation of SIFs associated with thethe mixed-mode criterion based on SIFs. Figure Variation of SIFs associated with mixed-mode criterion based on SIFs. Figure 12. Variation of SIFs associated with the mixed-mode criterion based on SIFs.
Mixed-Mode Criterion Based Energy Release Rates 4.2.4.2. Mixed-Mode Criterion Based on on Energy Release Rates 4.2. Mixed-Mode Criterion Based on Energy Release Rates In this case, cracks assumed to grow if the energy release rates II, and III satisfy In this case, cracks areare assumed to grow if the energy release rates GI , GIII,,Gand GIIIGsatisfy [24][24] thethe In thiscondition: case, cracks are assumed to grow if the energy release rates GI, GII, and GIII satisfy [24] the following condition: following γ following condition: G α G β G ( I ) + ( II ) + ( III ) = 1 (12) GIC GIIC GIIIC
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GI GII GIII G G G (( G I )) (( G II )) (( G III )) 11 IIC IIIC GICIC GIIC GIIIC
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(12) (12)
where GIIC and G GIIIC arethe thecritical criticalenergy energyrelease releaserates ratesin inthe theI,I, IIII and and III III modes, modes, respectively, respectively, IIC, ,and IIICare IC,, G where G GIC IC, GIIC, and GIIIC are the critical energy release rates in the I, II and III modes, respectively, where G and α, β, and γ are constants. By setting G = G = G = G and α = β = γ = 1, the above condition IIIc= Gc cand 1 , the above condition and α, α, β, β, and and γγ are are constants. constants. By By setting setting G GIcIcIc == G GIIc IIc = GIIIc and IIc = GIIIc = Gc and 1 , the above condition is reduced to the is reduced reduced to to the the fracture fracture criterion criterion based based on on the the total total energy energyrelease releaserate rate[26]: [26]: is fracture criterion based on the total energy release rate [26]: G G G=G GII+G GIIII+GGIIIIII=GGcc G I GII GIII Gc
(13) (13) (13)
This criterion is different from the SIFs-based mixed-mode criterion in the power value of α, α, β, β, This criterion is different from the SIFs-based mixed-mode criterion in the power value of α, β, and γ. The K is ignored in the SIFs-based mixed-mode criterion and G is considered in the energy III and γ. γ. The K KIIIIII III is ignored in the SIFs-based GIII III is considered and is ignored in the SIFs-based mixed-mode criterion and G is considered in the energy release rates based mixed-mode mixed-mode criterion. criterion.The Thesimulated simulatedcrack crackpath pathusing using this criterion is plotted this criterion is plotted plotted in release rates based mixed-mode criterion. The simulated crack path using this criterion is in in Figure 13. In comparison with the case of the SIF criterion, a difference is that the crack changes Figure 13. 13. In In comparison comparison with with the the case case of of the the SIF SIF criterion, criterion, aa difference difference is is that that the the crack crack changes changes its its Figure its direction once again when it reaches the right edge of the pane and then propagates upwards. direction once again when it reaches the right edge of the pane and then propagates upwards. For direction once again when it reaches the right edge of the pane and then propagates upwards. For For elucidation, the variations theand SIFsenergy and energy rates the during theprocess whole are process are elucidation, the variations variations of the theof SIFs releaserelease rates during during whole depicted elucidation, the of SIFs and energy release rates the whole process are depicted depicted in Figures 14 and 15. It is seen that the growth of the crack is first dominated by K or G in Figures Figures 14 14 and and 15. 15. It It is is seen seen that that the the growth growth of of the the crack crack is is first first dominated dominated by by K KIIII or or G GIIII,, and and II then II , in then and GIsituation , as in the situation shown Figure 12.in However, in the lastthe stage when the by K KIthen I or Gby asKin in the shown in Figure Figure 12.inHowever, However, the last last stage stage when crack reaches I or by or GII,, as the situation shown in 12. in the when the crack reaches crack reaches the right edge of the glass, the values of K and K come very close to each other, and the right right edge edge of of the the glass, glass, the the values values of of K KIIII and and K KII come come to each each other, other, and and neither neither can can II very close I the very close to neither canthe dominate the crack direction. A possible result in this circumstance iscrack the vertical crack dominate crack direction. A possible result in this circumstance is the vertical propagation, dominate the crack direction. A possible result in this circumstance is the vertical crack propagation, propagation, because the horizontal path isby constrained by the protecting frame. because the the horizontal horizontal path is constrained constrained the protecting protecting frame. because path is by the frame.
Figure 13. 13. Crack path predicted by the the mixed-mode mixed-mode criterion criterion based based on on energy energy release release rates. Figure Crack path path predicted predicted by by Figure 13. Crack the mixed-mode criterion based on energy release rates. rates. 7
7x107 7x10
K KII K K IIII K K IIIIII
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0.5 SIFs SIFs(Pa (Pam m ) )
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Time step step (s) (s) Time Figure 14. Variation of SIFs associated with the mixed-mode criterion based on energy release rates. Figure release rates. rates. Figure 14. 14. Variation Variation of of SIFs SIFs associated associated with with the the mixed-mode mixed-mode criterion criterion based based on on energy energy release
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Energy release rate (Nm ) -1 Energy release rate (Nm )
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GI GII GIII
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Figure 15. Variation of energy release rates associated with the mixed-mode criterion based on energy Figure Variation 15. Variation of energy release associated mixed-mode criterion based on energy Figure of energy release ratesrates associated withwith the the mixed-mode criterion based on energy release 15. rates. release rates. release rates.
4.3. SIF-Based Maximum Circumferential Stress Criterion SIF-Based Maximum Circumferential Stress Criterion 4.3.4.3. SIF-Based Maximum Circumferential Stress Criterion This criterion states that a crack grows in the direction of the maximum circumferential stress This criterion states that a crack grows in the direction of the maximum circumferential stress This criterion states that [24,27] which is determined by: a crack grows in the direction of the maximum circumferential [24,27] which is determined by:by: stress [24,27] which is determined
3 K e cos 0 (K0 Ieff cos 2 02 30 K3II sin 0 ) K IC θ θ 02 K cos0 ( Kcos sinθ K II sin 0 ) K IC Ieff 2cos Ke =e cos 2 (K −22K 0 ) = KIC I I2 2 Ieff
(14) (14) (14)
KKIeff =KKII + BB||KKIIIIII| | Ieff KIeff KI B | KIII |
(15) (15) (15)
2
2
2
where KeKise is the effective stress intensity factor, KIC K isICthe fracture toughness, and where BBisisan anempirical empiricalfactor, factor, the effective stress intensity factor, is the fracture toughness, where B is of ancrack empirical factor, Ke is the stress intensity factor, KIC is the fracture θand the growth. According to effective Equations (14) and (15), simulated crack path istoughness, shown 0 isangle the angle of crack growth. According to Equations (14) andthe (15), the simulated crack path is 0 is θ and θ016. is the angle of crack growth. According to Equations (14) and (15), theofsimulated crack path is in Figure Again, a significant difference can only be found at the later stage crack propagation: shown in Figure 16. Again, a significant difference can only be found at the later stage of crack shown Figure once 16. Again, significant difference can only atThis thephenomenon later stage ofcan crack the crack isin arrested runs ainto the it edge region by be thefound frame. propagation: the crack is itarrested once runs into protected the edge region protected by the frame. This propagation: the crack is arrested once it runs into the edge region protected by the frame. be understoodcan with the aid of Figure 17 in the 17 variations the SIFs are given. Although KIIThis phenomenon be understood with the aidwhich of Figure in whichofthe variations of the SIFs are given. phenomenon can be understood with the aid of Figure 17 in which the variations of the SIFs are given. is larger than KIlarger at thatthan stage, not stage, sufficient drive furthertogrowth of the crack arriving the Although KII is KI itatisthat it is to not sufficient drive further growth of the at crack Although KII is larger than KI at that stage, it is not sufficient to drive further growth of the crack edge region. arriving at the edge region. arriving at the edge region.
Figure 16. path predicted by SIF-based maximum circumferential circumferential stress stress criterion. criterion. Figure 16. Crack Crack pathpath predicted by the the SIF-based maximum Figure 16. Crack predicted by the SIF-based maximum circumferential stress criterion.
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0.50.5 SIFs (Pa SIFs (Pa m )m ) ) SIFs (Pa m
7x1077 6x10 7x10 7 6x1077 5x10 6x10 7 5x1077 4x10 5x10 7 4x1077 3x10 4x10 7 3x1077 2x10 3x10 7 2x1077 1x10 2x10 7 1x1007 1x10 7 -1x100 0 7
-1x107 -1x10
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KII K III KIII
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10 15 20 Time step (s) 10 15 20
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Time step (s)
Figure17. 17.Variation VariationofofSIFs SIFsassociated associated with with the maximum criterion. Time (s) circumferential Figure thestep maximum circumferentialstress stress criterion. Figure 17. Variation of SIFs associated with the maximum circumferential stress criterion. Figure 17.on Variation of SIFs associated with the maximum circumferential stress criterion. 4.4. Criterion Based Maximum Principal Stress
4.4. Criterion Based on Maximum Principal Stress 4.4. Criterion Based on Maximum Principal Stress This criterion states that a Principal crack grows 4.4. Criterion Based on Maximum Stressin the direction where the maximum principal stress This criterion states that a crack grows in the direction where the maximum principal stress exceeds tensile strength of the material. Theinimplementation relies on detailed stress distribution Thisthe criterion states that a crack grows the direction where thea maximum principal stress criterion states that a crack grows the direction where principal stress exceedsThis the tensile strength of the material. Theinimplementation reliesthe on amaximum detailed stress distribution and thus requires very fine meshes in the vicinity of the crack tip in the finite element simulation. exceeds the tensile strength of the material. The implementation relies on a detailed stress distribution exceeds the tensile strength of the material. The implementation relies on a detailed stress distribution and thus18 requires very fine meshes inof the of the crack tip in the finite element simulation. Figure shows the propagation path thevicinity crack. The result is almost that predicted via and thus requires very fine meshes in the vicinity of the crack tip in the same finite as element simulation. and thus requires very fine meshes inofthe vicinity of the crack tip in the finite element simulation. Figure 18 shows the propagation path the crack. The result is almost the same as that predicted the mixed-mode criterion based on theofenergy release rate. The corresponding variation of the SIF, Figure 18 shows the propagation path the crack. The result is almost the same as that predicted viavia Figure 18 showscriterion the propagation path ofenergy the crack. The result is almost the same as variation that predicted viaSIF, theplotted mixed-mode based on the release rate. The corresponding of the in Figurecriterion 19, is almost same as that given in The Figure 14 as well. variation Thereforeofit the canSIF, be the mixed-mode based the on the energy release rate. corresponding the mixed-mode criterion based on the energy release rate. The corresponding variation of the SIF, plotted in Figure 19, is almost the same as that given in path Figure 14 as well. it can beitconcluded concluded both result in same almost same crack propagation. plotted in that Figure 19,criteria is almost the asthe that given inofFigure 14 asTherefore well. Therefore can be plotted in Figure 19, in is almost almost the the same path as that crack givenpropagation. in Figure 14 as well. Therefore it can be that both criteria concluded that result both criteria result insame almost the of same path of crack propagation. concluded that both criteria result in almost the same path of crack propagation.
Figure 18. Crack path predicted by the criterion based on maximum principal stress. Figure 18.Crack Crack pathpredicted predicted by by the the criterion criterion based stress. Figure 18. basedon onmaximum maximumprincipal principal stress. 7 Figure 18. Crackpath path 7x10predicted by the criterion based on maximum principal stress.
SIFs (Pa SIFs (Pa m )m) SIFs (Pa m
0.50.5 0.5
)
7
7x1077 6x10 7x10 7 6x1077 5x10 6x10 7 5x1077 4x10 5x10 7 4x1077 3x10 4x10 7 3x1077 2x10 3x10 7 2x1077 1x10 2x10 7 1x1007 1x10 0 0
KI K KIII K K KIIIIII K KIIIII KIII
0
5
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20
25
30
0 0
5 5
10 15 20 Time step (s) 10 15 20
15
25 25
30 30
Time step (s)
Figure 19. Variation of SIFs associated with Time the criterion based on maximum principal stress. step (s) Figure 19. Variation of SIFs associated with the criterion based on maximum principal stress. Figure 19. Variation of SIFs associated with the criterion based on maximum principal stress. Figure 19. Variation of SIFs associated with the criterion based on maximum principal stress.
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4.5. Criterion Based on Crack Tip Opening Angle (CTOA) 4.5. Criterion Based on Crack Tip Opening Angle (CTOA) 4.5. Criterion Based on Crack Tip Opening Angle (CTOA) According to this criterion, thethecrack if the theCTOA CTOAexceeds exceeds a critical value According to this criterion, crackisisextended extended if a critical value [28]. [28]. The According to this criterion, theon crack is extended if the CTOA exceeds a critical value [28]. The The evaluation of the CTOA is based the angles of sub-cracks and in the simulation of uniform evaluation of the CTOA is based on the angles of sub-cracks and in the simulation of uniform crack crack evaluation of the CTOA is based on the angles of sub-cracks andof inall thethe simulation ofangles. uniform growth, and the actual 3D crack considered as the average Bycrack setting growth, and the actual 3D angle crack is angle is considered as the average of sub-crack all the sub-crack angles. By growth, and the actual 3D crack angle is considered as the average of all the sub-crack angles. By 20. the critical CTOA value as 0.00001 for the glass [29], the crack path is obtained and shown in Figure setting the critical CTOA value as 0.00001 for the glass [29], the crack path is obtained and shown in the critical CTOA valuefrom as 0.00001 for the glass [29], the crack path is downwards obtained andtoshown in It issetting visible the starts center of the runs the center Figurethat 20. Itthe is crack visiblestarts that the crack from theupper centeredge, of thethen upper edge, then runs downwards Figure 20. It is visible that the crack starts from the center of the upper edge, then runs downwards of thetoupper regionofwith temperature. Aftertemperature. some frequent butsome short-lived transitions in the the center the higher upper region with higher After frequent but short-lived to the center of the upper region with higher temperature. After some frequent but short-lived direction, the crack goes back before reaching the before boundary between the upper and middle regions, transitions in the direction, the crack goes back reaching the boundary between the upper and transitions in the direction, the crack goes back before reaching the boundary between the upper and middle regions, finally an island the variations upper region. Theassociated variations SIFs of theare associated SIFs finally forming an island informing the upper region.inThe of the provided in middle regions, finally forming an island in the upper region. The variations of the associated SIFs are Figure It can to bedecrease seen thatwith KI tends to decrease crack growth, and KII Figure 21.provided It can beinseen that21. K tends the crack growth,with and the KII exhibits considerable are provided in Figure 21. ItI can be seen that KI tends to decrease with the crack growth, and KII exhibitswhile considerable fluctuation, KIII remains nearlystage constant. the initial K I is ,larger fluctuation, KIII remains nearly while constant. In the initial KI is In larger than Kstage and K II KI is larger III thus exhibits considerable fluctuation, while KIII remains nearly constant. In the initial stage than K II and K III , thus dominating the crack growth. After about 13 time steps K I becomes smaller, and dominating the After time steps KII and KIII start I becomes than KII and KIIIcrack , thusgrowth. dominating theabout crack 13 growth. After K about 13 timesmaller, steps KIand becomes smaller, and to K II and KIII start to dominate the growth alternatively. This explains why the crack changes its dominate explains why theThis crackexplains changeswhy its direction and KII and the KIII growth start to alternatively. dominate the This growth alternatively. the crackfrequently changes its direction frequently and leaves behind a island. closed trajectory to form an island. leaves behind a closed trajectory to form an direction frequently and leaves behind a closed trajectory to form an island.
Figure 20. Crack path predicted by the CTOA-based crack growth criterion. Figure20. 20.Crack Crackpath pathpredicted predicted by by the the CTOA-based Figure CTOA-basedcrack crackgrowth growthcriterion. criterion. 6
KI
0.5
SIFs (Pam ) 0.5 SIFs (Pam )
6x10 6 6x10 6 5x10 6 5x10 6 4x10 6 4x10 6 3x10 6 3x10 6 2x10 6 2x10 6 1x10 6 1x10 0 0 6 -1x10 6 -1x10 6 -2x10 6 -2x10 0
KII KIII
0
2
2
4
4
6
6
8
8
10
KI KII KIII
10 12 14 16 18 12 14 16 18
Time step (s) Time step (s)
Figure 21. Variation of SIFs associated with the CTOA-based crack growth criterion. Figure the CTOA-based CTOA-basedcrack crackgrowth growthcriterion. criterion. Figure21. 21.Variation VariationofofSIFs SIFsassociated associated with with the
4.6. Criterion Based on Crack Tip Opening Displacement (CTOD) Criterion BasedononCrack CrackTip TipOpening OpeningDisplacement Displacement (CTOD) 4.6.4.6. Criterion Based This criterion postulates the growth of a crack when the current CTOD reaches a critical value, This criterionpostulates postulatesthe thegrowth growth of of aa crack crack when the CTOD a critical value, This thecurrent current CTODreaches reaches a critical value, alongcriterion the direction that causes either the opening when or the shearing component of the CTOD at the new along the direction thatcauses causeseither eitherthe theopening opening or or the shearing component ofofthe CTOD at at thethe new along the direction that the shearing component the CTOD new crack tip to be the absolute maximum. In the simulation of uniform crack growth, the actual 3D crack crack theabsolute absolutemaximum. maximum.In In the the simulation simulation of uniform crack the actual 3D3D crack crack tiptip to to bebe the crackgrowth, growth, the actual crack displacement is considered as the average of those forof alluniform the sub-crack angles [30]. L.C.S. Nunes and displacement is considered as the average of those for all the sub-crack angles [30]. L.C.S. Nunes and displacement is considered as the average of those for all the sub-crack angles [30]. L.C.S. Nunes and J.M.L. Reis [31] experimentally studied the CTOD and crack extension of glass using the digital image J.M.L. Reis [31] experimentally studied the CTOD and crack extension of glass using the digital image
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correlation method. It was found that the glass fiber starts to extend when the CTOD is larger than J.M.L. Reis [31] experimentally studied CTOD and starts crack extension using digital correlation method. Iton was found that glass CTOD fiber when the CTOD is largerimage than 0.00001 m. Based this work, thethe critical valueto isextend taken of asglass 0.00001 m the to predict the glass correlation method. It was found that the glass fiber starts to extend when the CTOD is larger than 0.00001 m. Based work, the critical value is taken 0.00001 m to predict variations the glass in extension here.on Thethis computed crack path CTOD is shown in Figure 22, as with the corresponding 0.00001 m.here. Based on this work, the crack critical CTODin is 22, taken asof 0.00001 m to predict the glass extension Thein computed path isinitiates shown Figure with the corresponding variations in the SIFs given Figure 23.crack The atvalue the top center the glass, and propagates down extension here. The computed crack path is shown in Figure 22, with the corresponding variations the towards SIFs given in Figure 23. The crack initiates at the top center of the glass, and propagates down the center of the upper region. After KII becomes larger than KI, the crack turns to the right in the in of Figure 23. The crack initiates theless toplarger center of the and propagates down towards center the upper region. After KII at becomes than KIglass, , m. the From crack turns to theon, right andSIFs isthegiven then arrested when the CTOD becomes than 0.00001 that time large towards the center of the upper region. After K becomes larger than K , the crack turns to the right and andfluctuations is then arrested when the CTOD becomes less than 0.00001 m. From that time on, large II I of the SIFs occur and the magnitude of KI is always smaller than those of KII and KIII. Thus is then arrested when CTOD becomes less than 0.00001 m. smaller From that time on, large fluctuations fluctuations of the occur and the magnitude of I is formation always than those of K II and KIIIthe . Thus the direction ofSIFs thethe crack is changed, leading to Kthe of an island at the center of upper of SIFs When occur and theismagnitude of KI0.00001 istoalways smaller than those of K0.0001 KIII . the Thus the thethe direction of the crack changed, the m, formation an island at the center ofthe upper II and region. the CTOD is larger leading than such as of 0.00002 m and m, simulation direction ofshow thethe crack isthe changed, leading to grows the formation island atsame the center ofm, the upper region. region. When CTOD is larger than 0.00001 m, such of as an 0.00002 m and 0.0001 the simulation results that crack no longer anymore under the thermal loading condition. When the CTOD is larger than 0.00001 m, such as 0.00002 m and 0.0001 m, the simulation results show by results show that the crack no longer grows anymore under the same thermal loading condition. When the CTOD is smaller than 0.00001 m, such as 0.000005 m, the crack also grows. However, that thethe crack no longer grows anymore under theas same thermal loading condition. When thepredict CTOD When CTOD iswith smaller 0.00001 m, Nune’s such 0.000005 the crack also grows. However, by the comparing this the than other cases and work [31],m,0.000005 m is not suitable to is smaller than 0.00001 m, such as 0.000005 m, the crack also grows. However, by comparing this with comparing this with the other cases and Nune’s work [31], 0.000005 m is not suitable to predict the crack growth. the other cases and Nune’s work [31], 0.000005 m is not suitable to predict the crack growth. crack growth.
Figure 22. Crack path predicted by the CTOD-based crack growth criterion. Figure by the the CTOD-based CTOD-based crack crack growth growth criterion. criterion. Figure 22. 22. Crack Crack path path predicted predicted by KI
7
3x10 7 7 3x10 2x10 7 2x10 7 1x10 7 1x10 0
KI
0.5
SIFs (Pam ) 0.5 SIFs (Pam )
KII KII KIII KIII
0 7 -1x10 7 7 -1x10 -2x10 7 -2x10 7 -3x10 7 7 -3x10 -4x10 7 -4x10 7 -5x10 7 -5x10
0 0
20 20
40 40
60 60
80 80
Time step (s) Time step (s)
100 100
Figure Variation of SIFs associated the CTOD-based crack growth criterion. Figure 23. 23. Variation of SIFs associated withwith the CTOD-based crack growth criterion. Figure 23. Variation of SIFs associated with the CTOD-based crack growth criterion.
4.7. Discussion 4.7. Discussion 4.7. Discussion Several criteria for crack initiation and propagation are examined in the simulation. It should be Several criteria for crack initiation and propagation are examined in the simulation. It should Several crack and propagation examined in very the simulation. It should be noted thatcriteria in a firefor the heatinitiation transfer between glass andare smoke or fire is complicated, and includes be noted that in a fire the heat transfer between glass and smoke or fire is very complicated, and noted that in a fire the heat transfer between glass and smoke or fire is very complicated, and includes radiation and convention, and the interaction of the fire and glass may markedly change the enclosure includes radiation and convention, and the interaction of the fire and glass may markedly change radiation and convention, and theas interaction of the fire and characteristics glass may markedly change the enclosure fire dynamic. These factors well as the inherent in brittle materials, such as the enclosure fire dynamic. These factors as well as the inherent characteristics in brittle materials, firemicrocracks dynamic. These factors asthe well as initiation the inherent in the brittle materials, such as [32], may cause crack to be characteristics very random. On other hand, from previous microcracks [32], may cause the crack initiation to be very random. On the other hand, from previous experiments it was found that despite the tests that were conducted under nearly identical conditions, experiments it was found that despite the tests that were conducted under nearly identical conditions,
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such as 2016, microcracks [32], may cause the crack initiation to be very random. On the other hand,15from Materials 9, 794 of 17 previous experiments it was found that despite the tests that were conducted under nearly identical conditions, the critical breakage stressesrandomly distributed [4]. Thus,exists uncertainty in crack the critical breakage stresses distributed [4].randomly Thus, uncertainty in crackexists initiation and initiation and whenthe considering the fire and glass it properties, it is predict glass when considering fire scenario andscenario glass properties, is difficult to difficult predict to glass breakage breakage very accurately. crack path, according to the numerical results occurrenceoccurrence very accurately. RegardingRegarding the crack the path, according to the numerical results there are there are essentially kindspath. of crack path.is The first ispath the that crackstarts pathfrom that the starts fromupper the central essentially two kindstwo of crack The first the crack central edge upper edge and thenafter terminates turning nearly a right-angle the right of the upper and then terminates turningafter nearly a right-angle at the rightatedge of theedge upper region, as region, as predicted by using criteria basedenergy on SIFs, energy release and maximum predicted by using the criteriathe based on SIFs, release rates, andrates, maximum principalprincipal stresses, stresses, as the wellSIF-based as the SIF-based maximum circumferential stress criterion. The is other the crack as well as maximum circumferential stress criterion. The other the iscrack path path initiating from the upper center of the pane edge and stopping in the central part of the upper initiating from the upper center of the pane edge and stopping in the central part of the upper region region with the formation of an as island, as predicted by the CTOAand CTOD-based crack criteria. growth with the formation of an island, predicted by the CTOAand CTOD-based crack growth criteria. If the conditions in the experiments are controlled very strictly to generate the crack If the conditions in the experiments are controlled very strictly to generate the crack path,path, the the phenomenon in crack propagation, such as oscillatory behavior, may be deterministic rather than phenomenon in crack propagation, such as oscillatory behavior, may be deterministic rather than a astatistical statisticalprocess process[33,34], [33,34],while whilealso alsodue duetotothe the microcrack microcrack and and thermal thermal gradient, gradient, the crack path presents probabilistic characteristics in a fire. For comparison, a typical experimental result of the crack path in a heated glass pane that was observed in our previous previous study [35] [35] is is shown shown in in Figure Figure 24. 24. The glass critical strain decreases with increasing microcrack size, so the edge of the glass has been polished to avoid the strength experimental condition condition is similar similar to that used in the strength reduction. reduction. The experimental simulation, and and for formore moreinformation informationplease pleaserefer refertoto [35]. The crack in Figure 24 initiates from the [35]. The crack in Figure 24 initiates from the top top center ofglass, the glass, and propagates the upper to the right left. Relatively, the center of the and propagates along along the upper edge toedge the right and left.and Relatively, the situation situation seems more consistent with theoffirst kind of in crack path in the Itsimulation. is therefore seems more consistent with the first kind crack path the simulation. is thereforeItbelieved that believed that the criteria based on SIFs, energy rates, andprincipal maximum principal as well the criteria based on SIFs, energy release rates,release and maximum stresses, as stresses, well as the SIFas the SIF-based circumferential stress criterion, more for suitable use in predicting crack based maximummaximum circumferential stress criterion, are moreare suitable use infor predicting crack growth growth in glazing to rather a fire, rather thanCTOAthe CTOAand CTOD-based criteria. Unfortunately, in glazing exposedexposed to a fire, than the and CTOD-based criteria. Unfortunately, the the program used here currently only model one crackpropagation, propagation,and andfurther furthereffort effort is is needed needed to program used here cancan currently only model one crack develop a code which can simulate the complicated process of crack initiation and growth in a real fire scenario.
(a)
(b)
Figure 24. Figure 24. The The (a) (a) edge edge condition condition and and (b) (b) aa typical typical crack crack pattern pattern in in aa glass glass pane pane under under thermal thermal loading. loading.
5. Conclusions 5. Conclusions Due to the stochastic nature, prediction of fracture and fall-out of window glasses exposed to Due to the stochastic nature, prediction of fracture and fall-out of window glasses exposed to fire fire is still an unsolved problem. In an effort to solve this, crack initiation and propagation in a model is still an unsolved problem. In an effort to solve this, crack initiation and propagation in a model glass pane under a fire condition has been investigated using the finite element method. Both the probabilistic and deterministic methods were adopted to examine crack initiation, while several commonly used criteria were utilized to predict crack propagation. Through detailed data analysis and comparison with the experimental observations, the following conclusions can be made.
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glass pane under a fire condition has been investigated using the finite element method. Both the probabilistic and deterministic methods were adopted to examine crack initiation, while several commonly used criteria were utilized to predict crack propagation. Through detailed data analysis and comparison with the experimental observations, the following conclusions can be made. The two- and three-parameter Weibull distributions give almost the same location and temperature difference for crack initiation. However, the predicted critical temperature difference is much larger than that determined via the criteria of maximum tensile stress, maximum normal stress, and maximum Mises stress, as well as the Coulomb-Mohr criterion. These last four criteria can be used to model the initiation of the crack in the glass pane. In contrast, the maximum shear stress criterion predicts no crack initiation and thus should be abandoned. The mixed-mode criteria based on SIF, energy release rate, and the maximum principal stress, as well as the SIF-based maximum circumferential stress criterion, can provide relatively consistent crack propagation paths with the experiment. At the same time, the predictions via the CTOA- and CTOD-based crack growth criteria show considerable differences. Consequently, the first four criteria are more suitable for modeling the crack propagation. The actual case of glass crack in fire is very complex, and includes bifurcations and multi-cracks. The fracture theory and technology to simulate such a complicated phenomenon are still under development. At present, only one crack is simulated, and the simulation of multi-cracks will have to be performed in the future. Once the multi-crack can be simulated, it will be easier to compare the simulated results with experimental results. Acknowledgments: This work was supported by the National Natural Science Foundation of China (Grant No. 51578524, 51120165001), the National Basic Research Program of China (973 Program, Grant No. 2012CB719703), and the Youth Innovation Promotion Association CAS (Grant No. 2013286). Author Contributions: Qingsong Wang and Yu Wang conceived and designed the numerical simulation; Yu Wang and Haodong Chen performed the experiments; Qingsong Wang, Jinhua Sun and Linghui He contributed analysis tools and provided guidance of this work; Qingsong Wang and Yu Wang wrote and revised the paper. Conflicts of Interest: The authors declare no conflict of interest.
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