Subcritical crack growth parameters in glass as a function of environmental conditions

Christopher Brokmann; Stefan Kolling; Jens Schneider

doi:10.1007/s40940-020-00134-6

Subcritical crack growth parameters in glass as a function of environmental conditions

Brokmann, Christopher; Kolling, Stefan; Schneider, Jens 2020-09-27 00:00:00 Glass Struct. Eng. (2021) 6:89–101 https://doi.org/10.1007/s40940-020-00134-6 RESEARCH PAPER Subcritical crack growth parameters in glass as a function of environmental conditions Christopher Brokmann · Stefan Kolling · Jens Schneider Received: 6 February 2020 / Accepted: 21 August 2020 / Published online: 27 September 2020 © The Author(s) 2020 Abstract In the present work, subcritical crack growth failure stress of unindented specimens with a constant in soda–lime silicate glass is investigated under differ- initial crack length was simulated. ent environmental conditions. Crack growth parame- ters as a function of temperature and humidity were Keywords Glass · Subcritical crack growth · determined by dynamic fatigue tests, which has been Environmental inﬂuence · Failure strength · Vickers veriﬁed by using the in-situ method of ﬁlming crack indentation growth during experiments. The specimens were pre- damaged for constant initial crack lengths in all spec- 1 Introduction imens using the Vickers indentation test. The deter- mined parameters were compared with those from lit- The strength of glasses has been studied and discussed erature in order to discuss existing deviations of sub- in a number of publications and books, e.g. Haldimann critical crack growth parameters in literature. These et al. (2008), Wachtman et al. (2009), Quinn (2007) deviations may be caused by environmental conditions and Meyland et al. (2019). Test methods for strength and different chemical compositions of the glass. Arrest are deﬁned in national and international standards. The lines were used to determine the ratio of crack width strength behavior of glass as a function of tempera- to crack depth in Vickers indented specimens. For the ture and humidity is often considered only as a coarse initial crack depth, images of fracture surfaces were approximation. For example, the European standard for taken using an scanning electron microscope. Further- determining the strength of glass by means of a coaxial more, the inﬂuence of humidity and temperature on the ring-on-ring test speciﬁes a relative humidity of 40– 70% during experiments c.f. EN DIN 1288 (2000). Thereby it could already be shown that the environ- C. Brokmann ( )· S. Kolling mental conditions have an enormous inﬂuence on the Institute of Mechanics and Materials, Technische Hochschule Mittelhessen, Wiesenstrasse 14, 35390 failure strength of glass, (Wiederhorn and Bolz 1970; Giessen, Germany Ronchetti et al. 2013). e-mail: christopher.brokmann@me.thm.de The failure strength of glasses and other almost S. Kolling ideally brittle materials is dominated by micro ﬂaws. e-mail: stefan.kolling@me.thm.de These micro ﬂaws are distributed randomly over the J. Schneider entire surface of the glass and grow sub-critically when Institute for Structural Mechanics and Design, Technical an external stress is applied. Depending on the growth University of Darmstadt, Franziska-Braun-Straße 3, 64287 time and growth rate, sub-critical crack growth has a Darmstadt, Germany e-mail: schneider@ismd.tu-darmstadt.de signiﬁcant inﬂuence on the failure stress. Due to their 123 90 C. Brokmann et al. different depth, these ﬂaws cause a statistical distri- over all uniformly pre-damaged specimens. A disad- bution of the failure stresses which has not yet been vantage is that artiﬁcially created cracks may behave adequately investigated, (Ballarini et al. 2016). differently from natural cracks. This problem occurs If the crack growth velocity is represented as a func- particularly due to a plastic zone in the area of the tion of the stress intensity K , it passes through four intentation, which inﬂuences the determination of the growth regions (Wiederhorn 1967). The crack growth growth parameters by induced residual stresses (Fuller takes place above region 0, in which no crack growth is et al. 1983). The crack growth parameters which can be assumed, (Kocer and Collins 2001). The crack growth found in the literature are subjected to some scattering is mainly driven by the humidity at the crack tip in com- which may be attributed to environmental conditions. bination with an external mechanical load. An overview Currently, the inﬂuence of environmental conditions of the phenomenon of sub-critical crack growth in on the strength distribution of glass has not been fully glass is given in Ciccotti (2009). There, all processes investigated, although the inﬂuence on the strength is observed to date for sub-critical crack growth are sum- proven (Wiederhorn 1967). While a higher humidity, marized and examined critically. Possible inﬂuencing and thus more water molecules at the crack tip, leads to factors are also discussed. a higher growth rate, it could be shown that the strength The different regions of crack growth in established of glass increases in hot water (Wiederhorn et al. failure models are often approached by a linear approx- 2013b). It could be shown that some coatings on the imation (Haldimann et al. 2008; Overend and Zammit glass surface prevent the transport of water molecules 2012; Alter et al. 2017; Kinsella and Persson 2018). to the crack tip and thus increasing the failure strength This is particularly useful, since the crack grows only (Mariggiò et al. 2019). The processes at the crack for a comparably short time in the last two growth tip and the results of observation at the crack tip by regions and spends most of its growth in the ﬁrst region, means of modern methods are summarized in Wieder- the linear region. Within this approximation, the crack horn et al. (2013a). However, the behavior of the crack velocity v can be expressed as a function of the stress growth parameters n and v from the linear approxima- intensity factor K by tion with respect to the environmental conditions for engineering applications has not yet been investigated. In this article, samples are pre-damaged using Vick- v(K ) = v = AK (1) I 0 Ic ers indentation testing and then relieved of all resid- Evans and Johnson (1975) and Maugis (1985), where ual stresses by heating and controlled cooling. Vick- K is the critical stress intensity and the parameters ers indentation has become an established method Ic n and v describe the sub-critical crack growth and for artiﬁcially induced cracks. Caution is nevertheless −n the parameter A can be expressed as A = v K .In required, as the lateral cracks created do not always Ic Haldimann (2006), Hilcken (2015) and Schula (2015) form in the geometry and depth under almost iden- a detailed overview and discussion of existing values tical conditions, (Mikowski et al. 2006; Kiefer et al. for crack growth parameters can be found. The present 2020). During indentation, the glass undergoes plas- article will use the sub-critical crack growth formula- tic deformations, which can be divided into densiﬁca- tion with the parameters n and v from the ﬁrst formu- tion and shearﬂow, while both have different effects lation in Eq. (1). There are also empirical approaches on the formation of lateral cracks (Kato et al. 2010). which directly consider the environmental conditions By using Vickers diamonds with different opening (Rodrigues et al. 2017), but which usually have too angles, (Gross 2012), and the numerical simulation of many parameters for the application which are difﬁ- the indentation process, Jebahi et al. (2013) it could cult to determine. For the determination of the crack be shown that the glass mainly undergoes a densiﬁ- growth parameters according to Eq. (1) several meth- cation underneath the indentation zone. Furthermore, ods are available in literature (Lawn et al. 1981; Fuller the relation between Vickers hardness or critical stress et al. 1983; Dwivedi and Green 1995; Wachtman et al. intensity and crack resistance is not clearly recogniz- 2009), where artiﬁcially indented cracks are used in able (Kato et al. 2010). most cases. This has the advantage that the initial crack In the present paper, crack growth parameters are depth and the location of the crack are known. Another determined from Eq. (1) at different temperatures and advantage is that there is a constant initial crack depth humidities using dynamic fatigue tests at different 123 Subcritical crack growth parameters in glass 91 -3 0 10 10 parameter n can be directly determined by the slope 0 I II III 100% in a double-logarithmic representation of the failure -2 -4 stresses via the stress rates. The condition is that the initial crack lengths of all samples are identical. If there -4 are no residual stresses in the test specimen, the crack -5 growth parameter v can be determined by -6 0n -6 2σ c 10 in in -8 v = , (3) λ (n + 1)(n − 2) -7 -10 0n see Fuller et al. (1983), with σ as the inert strength in and c as the indentation crack size including growth in -8 -12 10 10 0.4 0.5 0.6 0.7 0.8 0.2 0.4 0.6 0.8 during aging. The inert strength is given when there is K [MPa m] K [MPa m] I I no sub-critical crack growth before failure. It should be noted that Eq. (3) is only valid if the critical crack Fig. 1 Sub-critical crack velocity in soda–lime–silica glass length c is greater than the initial crack length c .The f i (SLS) according to Schula (2015)onbasis of Wiederhorn (1967) necessary condition is that for various humidites (left) and the linear approximation of sub- critical crack growth for all regions (right) (n−2)/2 (c /c ) 0.01 (4) i f stress rates at an universal testing machine within a has to be fulﬁlled (Wachtman et al. 2009). This is par- climate chamber. These parameters are validated by ticularly important for experiments with low ambient the method of “in-situ” observation of crack growth humidity. In very dry conditions, the initial crack is not according to Dwivedi and Green (1995). The relation subjected to signiﬁcant growth. The result is, that this between crack growth parameters and environmental condition can not be fulﬁlled for certain environments. conditions is shown experimentally. Finally, the inﬂu- A detailed derivation of the solution to determine sub- ence of humidity on failure strength at known crack critical crack growth parameters has already been made depths is calculated numerically. in Lawn et al. (1981), Fuller et al. (1983) and Wachtman et al. (2009). 2 Basics on subcritical crack growth 3 Experimental part The subcritical crack growth undergoes four regions, which can be expressed by a linear approximation The experimental part is divided into dynamic fatigue according to Eq. (1). The original measurements of all tests at different temperatures and humidities to deter- four areas shows a clear humidity dependent behavior mine the crack growth parameters n and v . To validate (Wiederhorn 1967), shown in Fig. 1. the obtained values, crack growth was observed “in- The dependence of sub-critical crack growth on tem- situ” under constant load at two different air humidities perature has already been shown by Wiederhorn and using a light microscope to observe the growth during Bolz (1970). For the often used linear approximation four-point bending tests. this has not been investigated yet. The crack growth parameter n can be determined by dynamic fatigue tests 3.1 Specimen preparation with constant stress rate σ ˙ and identical initial crack lengths with the failure stress σ by the relationship The tested glass is soda lime silicate ﬂoat glass. The chemical composition was determined by inductively n+1 σ σ˙ f1 1 = λ , (2) coupled plasma optical emission spectrometry. The σ σ˙ f2 2 results are shown in Table 1. Glass plates with the see Maugis (1985), where 1/(n + 1) give the slope and dimension of 1480 × 1000 × 1.8 mm were cut into cir- λ the intercept in a double logarithmic failure stress vs cular samples with a diameter of 80 mm and rectangular stress rate plot. The advantage is, that the crack growth samplesof220 × 34 × 1.8 mm. Water enviroment 1% 30% 0.017% 0.2% 10% v = da/dt [m/s] 92 C. Brokmann et al. Table 1 Chemical composition of the considered SLS ﬂoat glass –SiO Na O CaO MgO Al O KOFe O SO TiO 2 2 2 3 2 2 3 3 2 wt% 70.02 14.04 9.49 3.66 1.34 0.58 0.535 0.266 0.021 The obtained specimens were pre-damaged using Vickers indentation test with a indentation force of 9.8 N and a holding time of 3 s to obtain nearly identical crack systems in all specimens. Special care was taken to ensure that all samples were indented with the same force and holding time, as the indentation in glass is rate-dependent (Limbach et al. 2014). All indentations were examined for symmetry of the cracks. Samples that did not develop four cracks perpendicular to each other were rejected. A total of approximately 470 sam- ples were prepared for dynamic fatigue tests, of which 390 could be tested. After the indentation, the samples were stored for at Fig. 2 Microscope (A) with four-point bending setup (C)for direct observation of crack growth. The specimen (B) and one of least 24 h in room climate before they were heated ◦ the humidity and temperature sensors (D) can also be seen. to 520 C in a tempering furnace. Cooling down to room temperature was performed with a maximum of −1 residual stresses caused by the indentation. Several 2Kmin to remove the residual stresses of both the glass and the densiﬁcation zone. An explanation of tests are performed at a temperature of T = 25 C and a relative humidity of H = 40 and 50%. A validation heating inﬂuence on glass material properties can be found in Aronen and Karvinen (2018). The absence of of all temperature and humidity combinations is unfor- tunately not possible due to the setup for the “in-situ” residual stresses was veriﬁed by a scattered light polar- iscope. Although the inﬂuence of the cooling rate on observation device. For the four-point bending test, the density, hardness and Young’s modulus is known from distance between the supporting ﬁns is 155 mm and the Ito and Taniguchi (2004), it was classiﬁed as negligible distance between the load ﬁns is 74 mm. The test setup in this work since the dependency on the environmen- is shown in Fig. 2. In order to assign a geometry factor and stress inten- tal conditions of glass with identical properties will be investigated. The inﬂuence of residual stresses gener- sity to each measured crack length, so-called “arrest lines” are generated. These are created by stopping and ated by Vickers indentation is known. Indented spec- imens obtained 47.6 MPa in Anunmana et al. (2009), restarting the crack growth, (Fréchette 1990). These can be used to determine the ratio of the crack depth while specimens tempered after indentation reached a failure stress of 64.7 MPa. to the observed crack length. This is important because the observation of sub-critical crack growth by the “in- 3.2 In-situ crack growth observation situ” device can only measure the crack width and not the crack depth. The crack width to crack depth ratio The method of “in-situ” observation of sub-critical can also be used to check if the condition according to crack growth is used to validate the results obtained Eq. (4) for each dynamic fatigue test is fulﬁlled. from dynamic fatigue tests. The direct observation of cracks and their growth at constant load is already shown in Dwivedi and Green (1995), together with a 3.3 Dynamic fatigue experiments comparison of the “in-situ” crack growth parameters to those determined at dynamic fatigue experiments. In order to determine the crack growth parameters as In contrast, the specimens in this publication are a function of the environmental conditions, dynamic heat-treated after indentation in order to remove any fatigue tests are performed at several constant stress 123 Subcritical crack growth parameters in glass 93 rates at constant temperature and humidity. Tests are front shape of an arrested, or momentarily-hesitated performed at 15, 25 and 35 C. The humidity is con- crack. Resumed crack propagation occurs under a more stantly regulated for each temperature at 30, 40, 50, or less altered stress conﬁguration (Quinn 2007)onthe 60 and 70% relative humidity. For each tempera- basis of Fréchette (1990). ture with associated humidity, the stress rates σ ˙ = The ratio of the crack depth to the crack length is −1 0.6, 2, 6 and 20 MPas are performed with six tests shown in Fig. 4. The width to depth ratio is ﬁtted by per stress rate. The consideration of even lower stress an 2nd order polynomial. Also the ratio determined rates was rejected due to the large amount of experi- by Dwivedi and Green (1995) is shown in comparison. ments. For the dynamic fatigue tests coaxial ring-on- The variation of both measurements could be explained ring tests are performed with an inner ring radius of due to the fact that the samples in this publication were 6 mm and a support ring radius of 15 mm. A ﬁnite ele- ﬁrst damaged and then heat-treated. This could lead to ment simulation was carried out to validate if a pure a different growth ratio due to the absence of residual biaxial plane stress ﬁeld is present with these coaxial stresses in the indentation area. Furthermore, the envi- ring-on-ring dimensions. ronmental conditions during the tests for determining 0n In order to determine the inert strength σ and asso- the geometry correction factory in Dwivedi and Green in ciated initial crack length c from Eq. (3), 20 speci- (1995) is not given. A different humidity at the crack in mens are sealed with silicone oil to prevent sub-critical tip could lead to different ratios of crack depth to crack crack growth by water at the crack tip during the tests. width. This should be topic of future investigations. The specimens are then tested in coaxial ring-on-ring The geometry correction factor Y can be determined tests with a stress rate for the initial strength σ ˙ of by the solution of Newman and Raju (1981) and the in −1 450 MPas . ratio of crack width to crack depth by All experiments were carried out in a climate 3 2 chamber to ensure constant environmental conditions. Y = 1.418a − 1.826a + 1.016a + 0.7123, (5) Humidity and temperature were measure within a dis- tance of 10 mm to the glass tests surface in order to with a in millimeter. Using this correlation, the stress minimize some inﬂuence of air circulation within the intensity K can be calculated for each determined climate chamber. crack width under constant applied stress and micro- scopically ﬁlmed crack growth. To determine the crack velocity as the ratio of the grown crack between two 4 Results and discussion images, the arithmetic mean of the stress intensity from both measured crack lengths is taken. First, the ratio of crack width to crack depth is shown, which was determined via the generated arrest lines. This ratio is used to determine the crack growth param- eters of the in-situ tests, to validate the parameters of the corresponding dynamic fatigue tests. The crack growth parameters of the dynamic fatigue tests at different temperatures and humidities are then compared for their relationship to each other. Finally, the inﬂuence of humidity and temperature on the failure stress at varying initial crack depths is simulated. 4.1 Crack shape evolution in subcritical crack growth In order to draw conclusions about the crack depth from the crack width during in-situ observation, so-called “arrest lines” were generated. An example of the gen- Fig. 3 Development of the crack shape in SLS glass caused by erated arrest-lines is shown in Fig. 3. Crack arrest is Vickers indentation test. Double crack width c and crack depth a are highlighted a sharp line on the fracture surface deﬁning the crack 123 94 C. Brokmann et al. measured values Dwivedi & Green, 1995 0 500 1,000 1,500 c[µm] Fig. 5 Scanning electron microscope image of a Vickers inden- Fig. 4 Ratio of crack depth to crack length. In comparison, data −1 tation side after testing with σ ˙ = 450 MPas and silicone from Dwivedi and Green (1995). Solid and dotted lines represent sealing of the crack tip 2nd order polynomial ﬁt with 95% conﬁdential intervals in Cook and Pharr (1990), that below 10 N indentation The geometry correction factor as a function of load, no radial cracks are observable. These ﬁndings the crack depth according to Eq. (5) can also be are in accordance with Fig. 5. used to check if the condition in Eq. (4)isfulﬁlled for each experiment. With known geometry correc- 4.2 In-situ crack growth observation tion factor, failure stress σ and the critical stress intensity K the critical crack depth a can be cal- Ic f The measured sub-critical crack growth velocities culated. The inert strength of the indented specimen and the calculated associated stress intensities for was determined as the arithmetic mean of 20 samples T = 25 C and 40 and 50% relative humidity using 0n of σ = 80.62 MPa with a standard deviation of in in-situ tests are presented in Fig. 6. The crack growth s = 2.24 MPa due to coaxial ring-on-ring tests with a parameters were determined for H = 40% with −1 stress rate of 450 MPas . For the initial crack depth, n = 14.92[13.33; 16.5] and v = 7.07 mm/s [4.569; the fracture surfaces c of the inert strength specimens in 2 9.571] with R = 0.95 using Eq. (1). The param- were examined using a scanning electron microscope. eters for H = 50% were determined analogously to One fracture surface is shown in Fig. 5. The initial n = 14.61[13.43; 15.79] and v = 7.83 mm/s [5.587; crack depth could be determined from these images 9.892] with R = 0.96. The values in brackets belong to a = 54.86 μm. Analogously, the inert crack in to the 95% conﬁdence interval. depth can be calculated with the critical stress inten- sity K = 0.75 MPa m and the geometry correction Ic 10 factor Y = 0.72 to a = 53.14 μm. Since the geome- in try factor for the calculated crack depth is derived from −1 the extrapolation of Eq. (5), the measured initial crack depth is used for further calculation. −2 Previous studies have shown that the Vickers inden- tation creates a half-penny shape crack (Cook and Pharr T = 25 C,H = 50 % 1990; Lawn 1993). However, this was done at an inden- −3 95 % −CI,H = 50 % tation force of 90N and no information was given ◦ T = 25 C,H = 40 % on the existing residual stresses before indentation in 95 % −CI,H = 40 % −4 Lawn (1993). Sglavo and Green (1995) showed, that 0.44 0.48 0.52 0.56 0.6 0.64 at a indentation force of 9.8 N with no residual stress √ K [MPa m] before indentation, no half-penny shape crack system is observable. It is assumed that the median crack and the Fig. 6 Subcritical crack growth velocity v as a function of the lateral cracks connect to a half-penny shape crack sys- stress intensity K determined by in-situ observation. Solid lines represent regressions according to Eq. (1) tem at higher indentation loads. It could also be shown a[µm] v [mm/s] Subcritical crack growth parameters in glass 95 growth parameter v and the relative humidity can be 4.4 expressed by 4.2 ◦ 2 v (25 C) = 0.002236H − 0.1359H + 7.103 (8) ◦ 2 v (35 C) = 0.004236H − 0.2703H + 10.1(9) −10 1 2 3 −1 ln(σ) ln[MPas ] with a coefﬁcient of determination of R = 0.97 and 2 ◦ R = 0.99. For T = 15 C a ﬁt was omitted due Fig. 7 Logarithmic plot of the measured failure stresses versus to only three existing values. With a general compar- the stress rate at 25 C and 50% relative humidity. Linear ﬁt with 2 ◦ R = 0.98 ison of the measured values from 15 to 35 C a com- parable run of the curve with increasing humidity can be observed between temperature and humidity. The polynomials are intended to show the general behav- 4.3 Dynamic fatigue experiments ior of the crack growth parameters as a function of the environmental conditions. The crack growth parameters as a function of temper- The crack growth parameter n for T = 25 Cat ature and humidity were determined from the dynamic H = 40 and 50% from the in-situ observation agree fatigue experiments using Eqs. (2) and (3). An example with the values from the dynamic fatigue tests. The for the failure stress σ vs stress rate σ ˙ curves can be values for v from the in-situ device are smaller then seen in Fig. 7 for T = 25 C and H = 50%. All the values from dynamic fatigue tests. In Dwivedi and determined crack growth parameters are shown in Fig. Green (1995), the parameter v was also slightly lower 8 and Table 2. then the values from dynamic fatigue tests. In general, For T = 15 C and a relative humidity of the measured crack growth parameters from Dwivedi H = 40 and 50% no crack growth parameters could and Green (1995) are lower than the values determined be determined. The reason for this is that condition in in this article. Eq. (4) was not fulﬁlled often enough for some stress Comparing the values of the present work with the rates. This was especially the case for a stress rate of literature values summarized in the appendix Table A1, −1 −1 σ ˙ = 20 MPas and σ ˙ = 6MPas . Accordingly, it can be seen that the values determined here for the there was no sufﬁcient sub-critical crack growth for the parameter n are in good agreement with those from the applied linear approximation. literature. The values shown in Fig. 8 indicate, as expected, The crack growth parameter v is often higher increasing crack growth with increasing temperature than values commonly found in the literature, yet in and humidity, since with increasing temperature also a realistic range. The values for v in Blank (1993) a higher reactivity of the water molecules at the crack ranges between 4.51 mm/s for summer and 8.22 mm/s tip is present and thus an accelerated reaction with the for winter conditions. The subcritical crack growth Si–O–Si molecule chains of the glass can take place. In parameters found in the literature range up to 14.3 order to make a general statement about the inﬂuence mm/s at 45%rH in Sglavo and Green (1995). In of the environmental condition on the crack growth Dwivedi and Green (1995), the comparison of soda– parameters, the parameters were ﬁtted as a function of lime silicate and sodium aluminosilicate glasses shows, humidity. The crack growth parameter n as a function that the crack growth parameter v changes from a of the relative humidity H could be ﬁtted by a 2nd order maximum of 2.6 mm/s for soda–lime silicate glass polynomial to (SiO = 72.3%wt) to 21.8 mm/s for sodium alumi- nosilicate glass (SiO = 62.3%wt). This fact supports ◦ 2 n(25 C) =−0.001564H + 0.06987H + 14.78 (6) the assumption that the chemical composition plays an ◦ 2 n(35 C) =−0.001622H + 0.05754H + 14.19 (7) important role in subcritical crack growth. In comparison to Dwivedi and Green (1995) it can be with a coefﬁcient of determination of R = 0.96 seen that the chemical composition of the glass shown and R = 0.97. The relationship between the crack in Table 1 is different. There are also differences in the ln(σ ) ln[MPa] f 96 C. Brokmann et al. ◦ ◦ T=25 C T=25 C ◦ ◦ 20 insitu at T = 25 C in-situ at T = 25 C 30 40 50 60 70 30 40 50 60 70 ◦ ◦ T=35 C T=35 C 30 40 50 60 70 30 40 50 60 70 relative Humidity [%] relative Humidity [%] Fig. 8 Measured values for subcritical crack growth parameters n (left) and v (right) at different temperatures as a function of humidity. Solid and dotted lines are 2nd order polynomial ﬁts and 95% conﬁdential intervals Table 2 Results for crack growth parameters of dynamic fatigue and in-situ tests in dependence of the environmental conditions ◦ 2 T( C) H (%rH) R n 95%-CI-n v [mm/s] 95%-CI-v (mm/s) 0 0 15 50 0.91 21.361 [18.916; 24.497] 4.86 [3.66; 6.26] 15 60 0.93 20.645 [18.168; 23.851] 5.34 [3.96; 6.98] 15 70 0.94 17.372 [15.477; 19.760] 7.83 [5.98; 9.98] 25 30 0.97 15.431 [14.758; 16.167] 9.54 [8.65; 10.49] 25 40 0.96 15.100 [13.702; 16.794] 10.22 [8.17; 12.56] 25 50 0.98 14.751 [13.966; 15.620] 10.47 [9.28; 11.76] 25 60 0.93 12.961 [11.327; 15.090] 13.95 [10.09; 18.63] 25 70 0.97 12.263 [11.361; 13.306] 15.99 [13.42; 18.85] 35 30 0.95 14.356 [12.812; 16.289] 11.18 [8.56; 14.24] 35 40 0.97 14.013 [12.992; 15.197] 11.18 [9.41; 13.14] 35 50 0.97 13.263 [12.221; 14.482] 13.40 [11.11; 15.97] 35 60 0.98 11.347 [10.506; 12.323] 17.87 [14.96; 21.13] 35 70 0.99 10.453 [9.845; 11.136] 22.30 [19.52; 25.49] Values from in-situ experiments 25 40 0.95 14.92 [13.33; 16.5] 7.07 [4.57; 9.57] 25 50 0.96 14.61 [13.43; 15.79] 7.74 [5.59; 9.89] inert strength of the samples compared to Dwivedi and Since soda–lime silicate glass or glass generally Green (1995) of 15 MPa. This is probably due to the fact exists in various chemical compositions, the chemi- that the residual stress ﬁeld was eliminated by temper- cal composition should always be shown when deter- ing after the Vickers indentation test (Anunmana et al. mining crack growth parameters. The exact inﬂuence 2009). of the chemical components on the subcritical crack n [-] n [-] v [mm/s] v [mm/s] 0 0 Subcritical crack growth parameters in glass 97 Fig. 9 Flowchart of the Input Parameters: algorithm to calculate ˙ ,dt,K ,K ,a , ,Y, th Ic in 0 failure stresses out of initial m = 1 ﬂaws undergoing sub-critical crack growth next cycle = + ˙ dt m m−1 m=m+1 K = Y a Im m m K < K K ≥ K I th I Ic check if K < K or K ≥ K I th I Ic K ≤ K < K th I Ic da = v(K ;v ,n)dt m 0 Im a = a + da m m−1 m fail m growth has not been investigated according to the cur- steps, care must be taken that K is not signiﬁcantly Ic rent knowledge of the authors. exceeded. The threshold limit below which no sub- critical crack growth occurs was assumed to be K = th 4.4 Simulation of environmental inﬂuence on failure 0.25 MPa m, the critical stress intensity to K = Ic 0.75 MPa m. The geometry correction factor Y, to In order to demonstrate the inﬂuence of humidity on the present state of knowledge of the authors unknown the failure strength of glass, the growth of cracks of for natural ﬂaws, was set to the constant value Y = 1 initial depths a = 1, 10 and 100 μm were simulated for the input at the ﬁrst time step in cycle m = 0. numerically using MatLab. This was done for a tem- A biaxial plane stress ﬁeld during a coaxial ring-on- perature of T = 25 C and relative humidity of 30, 50 ring test was assumed, so that the crack orientation can and 70%, respectively. A ﬂowchart of the algorithm is be neglected. The specimen radius was set to 40 mm, the radius of the support ring to 15 mm and of the load shown in Fig. 9. As initial values, the stress rate σ ˙,the time step dt, the crack growth threshold K , the critical ring to 6 mm. The thickness of the glass is set to 1.8 mm, th stress intensity K , an initial crack depth a , the initial the Poisson’s ratio was assumed to be 0.23. Ic in stress σ and the geometry correction factor Y for the During the simulation, the applied stress is ﬁrst stress intensity formulation are set. updated at the current time step in the current cycle −1 We consider a constant stress rate of σ ˙ = 2MPas m. Then the stress intensity K is calculated to check and a constant time step of dt = 0.01 ms. This whether crack growth is present or the critical stress small time step was chosen because with higher time intensity has already been reached. Finally, the crack growth velocity according to Eq. (1) is calculated. With Table 3 Inﬂuence of humidity on numerical failure stresses at the crack velocity and the time step, the crack growth in different initial crack depths at T = 25 C the current time step da is added to the existing crack Flaw size 1 μm10 μm 100 μm depth a for the updated crack depth a . m−1 m H = 30% 184.82 MPa 72.06 MPa 28.11 MPa Table 3 shows the resulting numerical failure stresses H = 50% 177.53 MPa 69.79 MPa 27.47 MPa as a function of the initial crack depths and environ- H = 70% 151.7 MPa 60.12 MPa 24.62 MPa mental conditions. It can be seen that the failure stress at a humidity difference of H = Δ40% differs up Maximum deviation 17.92% 16.57% 12.42% to 17.92%. This shows that the permitted difference in 123 98 C. Brokmann et al. environmental conditions in e.g. the European standard composition of the glass. The crack growth param- are to high and when comparing strength distributions, eters from Dwivedi and Green (1995) were com- the environmental conditions during the experiments pared with those determined here, who also exam- must always be included. ined soda–lime silicate glass. The chemical com- position of both publication differs slightly. 4. Arrest-lines were used to determine the ratio of 5 Conclusions crack width to crack depth in Vickers indented glass specimen. This differs slightly from the val- 1. It could be shown that the temperature and humidity ues available in the literature. This may be due to dependence of the subcritical crack growth parame- the tempering after indentation of the specimens ters n and v of the linear approximation by Maugis 0 within the scope of this publication. Scanning elec- (1985) exists as expected. At a humidity above 50% tron microscope images of the Vickers indentation the crack growth rate for all three investigated tem- fracture surface were also shown, to validate crack peratures increased more than in the area before depth and crack width in the initial stadium after 50%. The parameter v determined here is gener- 0 Vickers indentation and tempering. ally higher than the values available in the literature. Acknowledgements The presented work is based on results of A possible explanation could be that often Vickers the research project named 18295N “Stochastisches Bruchver- indented specimens are used with residual stresses halten von Glas” which has been funded by the AiF within the due to indentation. This is contradicted by the fact programme for sponsorship by Industrial Joint Research (IGF) that Dwivedi and Green (1995) has taken this into of the German Federal Ministry of Economic Affairs and Energy based on an enactment of the German Parliament. The research account and also obtained lower values for v .The project was carried out in co-operation with Forschungsvereini- parameter n is in the range of the literature values. gung Automobiltechnik e.V.—FAT. It has been shown that the crack growth parameters The authors would also like to thank P. Paulus from Pilkington as a function of humidity can be represented by a for providing the glass samples. 2nd order polynomial. Funding Open Access funding provided by Projekt DEAL. 2. By simulating double ring bending tests with the determined crack growth parameters it could be Compliance with ethical standards shown that at e.g. 25 C the failure stress at con- Conﬂict of interest The authors declare that they have no con- stant initial crack length varies by up to 18% in ﬂict of interest. the range of 30–70% relative humidity for initial cracks of 1, 10 and 100 μm. This effect increases Open Access This article is licensed under a Creative Com- mons Attribution 4.0 International License, which permits use, with variations in temperature. A possible future sharing, adaptation, distribution and reproduction in any medium study could investigate a shift for the strength dis- or format, as long as you give appropriate credit to the original tributions of glass as a function of environmental author(s) and the source, provide a link to the Creative Com- conditions, as it already exists for size effects. mons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s 3. When comparing the determined sub-critical crack Creative Commons licence, unless indicated otherwise in a credit growth parameters with literature values, it is line to the material. If material is not included in the article’s Cre- noticeable that some scatters of the literature values ative Commons licence and your intended use is not permitted by can be traced back to scatter of ambient conditions. statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view Nevertheless, unexplainable variations in literature a copy of this licence, visit http://creativecommons.org/licenses/ values of the sub-critical crack growth parameters by/4.0/. remain. These may be possible due to the chemical 123 Subcritical crack growth parameters in glass 99 Appendix Table A1 Literature review of subcritical crack growth parameters for SLS ﬂoat glass Environment Test method n v [mm/s] References 22.7 C, 50%rH Dynamic fatigue 14.22 2.2 Hilcken (2015) Water Mod. double beam cantilever* 15.44 2.92 Gehrke et al. (1987) 50%rH Mod. double beam cantilever* 16.66 0.83 Gehrke et al. (1987) Values from Dwivedi and Green (1995) Soda–lime silicate glass 27 C, 65%rH In-situ, Vickers indented 19.7–21.2 0.2–0.4 27 C, 65%rH Dynamic fatigue 21.8 2.6 27 C, 65%rH Dynamic fatigue, Vickers indented 21.1 2.4 Sodium aluminosilicate glass: 27 C, 65%rH In-situ, Vickers intended 25.6–26.0 11.6–21.8 27 C, 65%rH Dynamic fatigue 25.9 2.3 27 C, 65%rH Dynamic fatigue, Vickers indented 22.1 6.1 Extract of the summary from Haldimann (2006) Water In-situ 16.0 50.1 Kerkhof et al. (1981) Air, 50%rH In-situ 18.1 2.47 Kerkhof et al. (1981) Laboratory, summer Derived from Kerkhof et al. (1981) 16.0 4.51 Blank (1993) Laboratory, winter, 2 C Derived from Kerkhof et al. (1981) 16.0 8.22 Blank (1993) Water Values from 9 laboratories and 2000 specimens 17.7 10.7 Ritter et al. (1985) Water Dynamic fatigue 26 ±73.7 × 10 Sglavo and Bertoldi (2006) Water Dynamic fatigue 18 ±119 ± 4 Sglavo et al. (1997) Water Dynamic fatigue, intended 20.1 ± 0.7 28.8 ± 6.4 Sglavo and Green (1999) Water Dynamic fatigue, annealed 19.9 ± 0.7 6.4 ± 1.4 Sglavo and Green (1999) Values from Wiederhorn (1967), converted by Schula (2015): 25 C, Water Double-cantilever cleavage 17.4 3.8 25 C, 100%rH Double-cantilever cleavage 20.8 3.6 25 C, 30%rH Double-cantilever cleavage 22.6 1.7 25 C, 10%rH Double-cantilever cleavage 21.4 0.6 25 C, 0.017%rH Double-cantilever cleavage 27.2 0.09 Vacuum Double-cantilever cleavage 93.3 0.13 Extract of the summary from Schula (2015) Water Unkown 13.0 1.1 Gehrke and Ullner (1988) 50%rH Unkown 14.3 0.16 Gehrke and Ullner (1988) Water Unkown 18.4 17.1 Ullner and Höhne (1993) 50%rH Unkown 19.7 2.8 Ullner and Höhne (1993) 25 C, 45%rH Dynamic fatigue, Vickers intended 18.8 14.3 Sglavo and Green (1995) *Fitted from displayed data 123 100 C. Brokmann et al. 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Publisher: Springer Journals
Copyright: Copyright © The Author(s) 2020
ISSN: 2363-5142
eISSN: 2363-5150
DOI: 10.1007/s40940-020-00134-6
Publisher site: See Article on Publisher Site

Abstract

Glass Struct. Eng. (2021) 6:89–101 https://doi.org/10.1007/s40940-020-00134-6 RESEARCH PAPER Subcritical crack growth parameters in glass as a function of environmental conditions Christopher Brokmann · Stefan Kolling · Jens Schneider Received: 6 February 2020 / Accepted: 21 August 2020 / Published online: 27 September 2020 © The Author(s) 2020 Abstract In the present work, subcritical crack growth failure stress of unindented specimens with a constant in soda–lime silicate glass is investigated under differ- initial crack length was simulated. ent environmental conditions. Crack growth parame- ters as a function of temperature and humidity were Keywords Glass · Subcritical crack growth · determined by dynamic fatigue tests, which has been Environmental inﬂuence · Failure strength · Vickers veriﬁed by using the in-situ method of ﬁlming crack indentation growth during experiments. The specimens were pre- damaged for constant initial crack lengths in all spec- 1 Introduction imens using the Vickers indentation test. The deter- mined parameters were compared with those from lit- The strength of glasses has been studied and discussed erature in order to discuss existing deviations of sub- in a number of publications and books, e.g. Haldimann critical crack growth parameters in literature. These et al. (2008), Wachtman et al. (2009), Quinn (2007) deviations may be caused by environmental conditions and Meyland et al. (2019). Test methods for strength and different chemical compositions of the glass. Arrest are deﬁned in national and international standards. The lines were used to determine the ratio of crack width strength behavior of glass as a function of tempera- to crack depth in Vickers indented specimens. For the ture and humidity is often considered only as a coarse initial crack depth, images of fracture surfaces were approximation. For example, the European standard for taken using an scanning electron microscope. Further- determining the strength of glass by means of a coaxial more, the inﬂuence of humidity and temperature on the ring-on-ring test speciﬁes a relative humidity of 40– 70% during experiments c.f. EN DIN 1288 (2000). Thereby it could already be shown that the environ- C. Brokmann ( )· S. Kolling mental conditions have an enormous inﬂuence on the Institute of Mechanics and Materials, Technische Hochschule Mittelhessen, Wiesenstrasse 14, 35390 failure strength of glass, (Wiederhorn and Bolz 1970; Giessen, Germany Ronchetti et al. 2013). e-mail: christopher.brokmann@me.thm.de The failure strength of glasses and other almost S. Kolling ideally brittle materials is dominated by micro ﬂaws. e-mail: stefan.kolling@me.thm.de These micro ﬂaws are distributed randomly over the J. Schneider entire surface of the glass and grow sub-critically when Institute for Structural Mechanics and Design, Technical an external stress is applied. Depending on the growth University of Darmstadt, Franziska-Braun-Straße 3, 64287 time and growth rate, sub-critical crack growth has a Darmstadt, Germany e-mail: schneider@ismd.tu-darmstadt.de signiﬁcant inﬂuence on the failure stress. Due to their 123 90 C. Brokmann et al. different depth, these ﬂaws cause a statistical distri- over all uniformly pre-damaged specimens. A disad- bution of the failure stresses which has not yet been vantage is that artiﬁcially created cracks may behave adequately investigated, (Ballarini et al. 2016). differently from natural cracks. This problem occurs If the crack growth velocity is represented as a func- particularly due to a plastic zone in the area of the tion of the stress intensity K , it passes through four intentation, which inﬂuences the determination of the growth regions (Wiederhorn 1967). The crack growth growth parameters by induced residual stresses (Fuller takes place above region 0, in which no crack growth is et al. 1983). The crack growth parameters which can be assumed, (Kocer and Collins 2001). The crack growth found in the literature are subjected to some scattering is mainly driven by the humidity at the crack tip in com- which may be attributed to environmental conditions. bination with an external mechanical load. An overview Currently, the inﬂuence of environmental conditions of the phenomenon of sub-critical crack growth in on the strength distribution of glass has not been fully glass is given in Ciccotti (2009). There, all processes investigated, although the inﬂuence on the strength is observed to date for sub-critical crack growth are sum- proven (Wiederhorn 1967). While a higher humidity, marized and examined critically. Possible inﬂuencing and thus more water molecules at the crack tip, leads to factors are also discussed. a higher growth rate, it could be shown that the strength The different regions of crack growth in established of glass increases in hot water (Wiederhorn et al. failure models are often approached by a linear approx- 2013b). It could be shown that some coatings on the imation (Haldimann et al. 2008; Overend and Zammit glass surface prevent the transport of water molecules 2012; Alter et al. 2017; Kinsella and Persson 2018). to the crack tip and thus increasing the failure strength This is particularly useful, since the crack grows only (Mariggiò et al. 2019). The processes at the crack for a comparably short time in the last two growth tip and the results of observation at the crack tip by regions and spends most of its growth in the ﬁrst region, means of modern methods are summarized in Wieder- the linear region. Within this approximation, the crack horn et al. (2013a). However, the behavior of the crack velocity v can be expressed as a function of the stress growth parameters n and v from the linear approxima- intensity factor K by tion with respect to the environmental conditions for engineering applications has not yet been investigated. In this article, samples are pre-damaged using Vick- v(K ) = v = AK (1) I 0 Ic ers indentation testing and then relieved of all resid- Evans and Johnson (1975) and Maugis (1985), where ual stresses by heating and controlled cooling. Vick- K is the critical stress intensity and the parameters ers indentation has become an established method Ic n and v describe the sub-critical crack growth and for artiﬁcially induced cracks. Caution is nevertheless −n the parameter A can be expressed as A = v K .In required, as the lateral cracks created do not always Ic Haldimann (2006), Hilcken (2015) and Schula (2015) form in the geometry and depth under almost iden- a detailed overview and discussion of existing values tical conditions, (Mikowski et al. 2006; Kiefer et al. for crack growth parameters can be found. The present 2020). During indentation, the glass undergoes plas- article will use the sub-critical crack growth formula- tic deformations, which can be divided into densiﬁca- tion with the parameters n and v from the ﬁrst formu- tion and shearﬂow, while both have different effects lation in Eq. (1). There are also empirical approaches on the formation of lateral cracks (Kato et al. 2010). which directly consider the environmental conditions By using Vickers diamonds with different opening (Rodrigues et al. 2017), but which usually have too angles, (Gross 2012), and the numerical simulation of many parameters for the application which are difﬁ- the indentation process, Jebahi et al. (2013) it could cult to determine. For the determination of the crack be shown that the glass mainly undergoes a densiﬁ- growth parameters according to Eq. (1) several meth- cation underneath the indentation zone. Furthermore, ods are available in literature (Lawn et al. 1981; Fuller the relation between Vickers hardness or critical stress et al. 1983; Dwivedi and Green 1995; Wachtman et al. intensity and crack resistance is not clearly recogniz- 2009), where artiﬁcially indented cracks are used in able (Kato et al. 2010). most cases. This has the advantage that the initial crack In the present paper, crack growth parameters are depth and the location of the crack are known. Another determined from Eq. (1) at different temperatures and advantage is that there is a constant initial crack depth humidities using dynamic fatigue tests at different 123 Subcritical crack growth parameters in glass 91 -3 0 10 10 parameter n can be directly determined by the slope 0 I II III 100% in a double-logarithmic representation of the failure -2 -4 stresses via the stress rates. The condition is that the initial crack lengths of all samples are identical. If there -4 are no residual stresses in the test specimen, the crack -5 growth parameter v can be determined by -6 0n -6 2σ c 10 in in -8 v = , (3) λ (n + 1)(n − 2) -7 -10 0n see Fuller et al. (1983), with σ as the inert strength in and c as the indentation crack size including growth in -8 -12 10 10 0.4 0.5 0.6 0.7 0.8 0.2 0.4 0.6 0.8 during aging. The inert strength is given when there is K [MPa m] K [MPa m] I I no sub-critical crack growth before failure. It should be noted that Eq. (3) is only valid if the critical crack Fig. 1 Sub-critical crack velocity in soda–lime–silica glass length c is greater than the initial crack length c .The f i (SLS) according to Schula (2015)onbasis of Wiederhorn (1967) necessary condition is that for various humidites (left) and the linear approximation of sub- critical crack growth for all regions (right) (n−2)/2 (c /c ) 0.01 (4) i f stress rates at an universal testing machine within a has to be fulﬁlled (Wachtman et al. 2009). This is par- climate chamber. These parameters are validated by ticularly important for experiments with low ambient the method of “in-situ” observation of crack growth humidity. In very dry conditions, the initial crack is not according to Dwivedi and Green (1995). The relation subjected to signiﬁcant growth. The result is, that this between crack growth parameters and environmental condition can not be fulﬁlled for certain environments. conditions is shown experimentally. Finally, the inﬂu- A detailed derivation of the solution to determine sub- ence of humidity on failure strength at known crack critical crack growth parameters has already been made depths is calculated numerically. in Lawn et al. (1981), Fuller et al. (1983) and Wachtman et al. (2009). 2 Basics on subcritical crack growth 3 Experimental part The subcritical crack growth undergoes four regions, which can be expressed by a linear approximation The experimental part is divided into dynamic fatigue according to Eq. (1). The original measurements of all tests at different temperatures and humidities to deter- four areas shows a clear humidity dependent behavior mine the crack growth parameters n and v . To validate (Wiederhorn 1967), shown in Fig. 1. the obtained values, crack growth was observed “in- The dependence of sub-critical crack growth on tem- situ” under constant load at two different air humidities perature has already been shown by Wiederhorn and using a light microscope to observe the growth during Bolz (1970). For the often used linear approximation four-point bending tests. this has not been investigated yet. The crack growth parameter n can be determined by dynamic fatigue tests 3.1 Specimen preparation with constant stress rate σ ˙ and identical initial crack lengths with the failure stress σ by the relationship The tested glass is soda lime silicate ﬂoat glass. The chemical composition was determined by inductively n+1 σ σ˙ f1 1 = λ , (2) coupled plasma optical emission spectrometry. The σ σ˙ f2 2 results are shown in Table 1. Glass plates with the see Maugis (1985), where 1/(n + 1) give the slope and dimension of 1480 × 1000 × 1.8 mm were cut into cir- λ the intercept in a double logarithmic failure stress vs cular samples with a diameter of 80 mm and rectangular stress rate plot. The advantage is, that the crack growth samplesof220 × 34 × 1.8 mm. Water enviroment 1% 30% 0.017% 0.2% 10% v = da/dt [m/s] 92 C. Brokmann et al. Table 1 Chemical composition of the considered SLS ﬂoat glass –SiO Na O CaO MgO Al O KOFe O SO TiO 2 2 2 3 2 2 3 3 2 wt% 70.02 14.04 9.49 3.66 1.34 0.58 0.535 0.266 0.021 The obtained specimens were pre-damaged using Vickers indentation test with a indentation force of 9.8 N and a holding time of 3 s to obtain nearly identical crack systems in all specimens. Special care was taken to ensure that all samples were indented with the same force and holding time, as the indentation in glass is rate-dependent (Limbach et al. 2014). All indentations were examined for symmetry of the cracks. Samples that did not develop four cracks perpendicular to each other were rejected. A total of approximately 470 sam- ples were prepared for dynamic fatigue tests, of which 390 could be tested. After the indentation, the samples were stored for at Fig. 2 Microscope (A) with four-point bending setup (C)for direct observation of crack growth. The specimen (B) and one of least 24 h in room climate before they were heated ◦ the humidity and temperature sensors (D) can also be seen. to 520 C in a tempering furnace. Cooling down to room temperature was performed with a maximum of −1 residual stresses caused by the indentation. Several 2Kmin to remove the residual stresses of both the glass and the densiﬁcation zone. An explanation of tests are performed at a temperature of T = 25 C and a relative humidity of H = 40 and 50%. A validation heating inﬂuence on glass material properties can be found in Aronen and Karvinen (2018). The absence of of all temperature and humidity combinations is unfor- tunately not possible due to the setup for the “in-situ” residual stresses was veriﬁed by a scattered light polar- iscope. Although the inﬂuence of the cooling rate on observation device. For the four-point bending test, the density, hardness and Young’s modulus is known from distance between the supporting ﬁns is 155 mm and the Ito and Taniguchi (2004), it was classiﬁed as negligible distance between the load ﬁns is 74 mm. The test setup in this work since the dependency on the environmen- is shown in Fig. 2. In order to assign a geometry factor and stress inten- tal conditions of glass with identical properties will be investigated. The inﬂuence of residual stresses gener- sity to each measured crack length, so-called “arrest lines” are generated. These are created by stopping and ated by Vickers indentation is known. Indented spec- imens obtained 47.6 MPa in Anunmana et al. (2009), restarting the crack growth, (Fréchette 1990). These can be used to determine the ratio of the crack depth while specimens tempered after indentation reached a failure stress of 64.7 MPa. to the observed crack length. This is important because the observation of sub-critical crack growth by the “in- 3.2 In-situ crack growth observation situ” device can only measure the crack width and not the crack depth. The crack width to crack depth ratio The method of “in-situ” observation of sub-critical can also be used to check if the condition according to crack growth is used to validate the results obtained Eq. (4) for each dynamic fatigue test is fulﬁlled. from dynamic fatigue tests. The direct observation of cracks and their growth at constant load is already shown in Dwivedi and Green (1995), together with a 3.3 Dynamic fatigue experiments comparison of the “in-situ” crack growth parameters to those determined at dynamic fatigue experiments. In order to determine the crack growth parameters as In contrast, the specimens in this publication are a function of the environmental conditions, dynamic heat-treated after indentation in order to remove any fatigue tests are performed at several constant stress 123 Subcritical crack growth parameters in glass 93 rates at constant temperature and humidity. Tests are front shape of an arrested, or momentarily-hesitated performed at 15, 25 and 35 C. The humidity is con- crack. Resumed crack propagation occurs under a more stantly regulated for each temperature at 30, 40, 50, or less altered stress conﬁguration (Quinn 2007)onthe 60 and 70% relative humidity. For each tempera- basis of Fréchette (1990). ture with associated humidity, the stress rates σ ˙ = The ratio of the crack depth to the crack length is −1 0.6, 2, 6 and 20 MPas are performed with six tests shown in Fig. 4. The width to depth ratio is ﬁtted by per stress rate. The consideration of even lower stress an 2nd order polynomial. Also the ratio determined rates was rejected due to the large amount of experi- by Dwivedi and Green (1995) is shown in comparison. ments. For the dynamic fatigue tests coaxial ring-on- The variation of both measurements could be explained ring tests are performed with an inner ring radius of due to the fact that the samples in this publication were 6 mm and a support ring radius of 15 mm. A ﬁnite ele- ﬁrst damaged and then heat-treated. This could lead to ment simulation was carried out to validate if a pure a different growth ratio due to the absence of residual biaxial plane stress ﬁeld is present with these coaxial stresses in the indentation area. Furthermore, the envi- ring-on-ring dimensions. ronmental conditions during the tests for determining 0n In order to determine the inert strength σ and asso- the geometry correction factory in Dwivedi and Green in ciated initial crack length c from Eq. (3), 20 speci- (1995) is not given. A different humidity at the crack in mens are sealed with silicone oil to prevent sub-critical tip could lead to different ratios of crack depth to crack crack growth by water at the crack tip during the tests. width. This should be topic of future investigations. The specimens are then tested in coaxial ring-on-ring The geometry correction factor Y can be determined tests with a stress rate for the initial strength σ ˙ of by the solution of Newman and Raju (1981) and the in −1 450 MPas . ratio of crack width to crack depth by All experiments were carried out in a climate 3 2 chamber to ensure constant environmental conditions. Y = 1.418a − 1.826a + 1.016a + 0.7123, (5) Humidity and temperature were measure within a dis- tance of 10 mm to the glass tests surface in order to with a in millimeter. Using this correlation, the stress minimize some inﬂuence of air circulation within the intensity K can be calculated for each determined climate chamber. crack width under constant applied stress and micro- scopically ﬁlmed crack growth. To determine the crack velocity as the ratio of the grown crack between two 4 Results and discussion images, the arithmetic mean of the stress intensity from both measured crack lengths is taken. First, the ratio of crack width to crack depth is shown, which was determined via the generated arrest lines. This ratio is used to determine the crack growth param- eters of the in-situ tests, to validate the parameters of the corresponding dynamic fatigue tests. The crack growth parameters of the dynamic fatigue tests at different temperatures and humidities are then compared for their relationship to each other. Finally, the inﬂuence of humidity and temperature on the failure stress at varying initial crack depths is simulated. 4.1 Crack shape evolution in subcritical crack growth In order to draw conclusions about the crack depth from the crack width during in-situ observation, so-called “arrest lines” were generated. An example of the gen- Fig. 3 Development of the crack shape in SLS glass caused by erated arrest-lines is shown in Fig. 3. Crack arrest is Vickers indentation test. Double crack width c and crack depth a are highlighted a sharp line on the fracture surface deﬁning the crack 123 94 C. Brokmann et al. measured values Dwivedi & Green, 1995 0 500 1,000 1,500 c[µm] Fig. 5 Scanning electron microscope image of a Vickers inden- Fig. 4 Ratio of crack depth to crack length. In comparison, data −1 tation side after testing with σ ˙ = 450 MPas and silicone from Dwivedi and Green (1995). Solid and dotted lines represent sealing of the crack tip 2nd order polynomial ﬁt with 95% conﬁdential intervals in Cook and Pharr (1990), that below 10 N indentation The geometry correction factor as a function of load, no radial cracks are observable. These ﬁndings the crack depth according to Eq. (5) can also be are in accordance with Fig. 5. used to check if the condition in Eq. (4)isfulﬁlled for each experiment. With known geometry correc- 4.2 In-situ crack growth observation tion factor, failure stress σ and the critical stress intensity K the critical crack depth a can be cal- Ic f The measured sub-critical crack growth velocities culated. The inert strength of the indented specimen and the calculated associated stress intensities for was determined as the arithmetic mean of 20 samples T = 25 C and 40 and 50% relative humidity using 0n of σ = 80.62 MPa with a standard deviation of in in-situ tests are presented in Fig. 6. The crack growth s = 2.24 MPa due to coaxial ring-on-ring tests with a parameters were determined for H = 40% with −1 stress rate of 450 MPas . For the initial crack depth, n = 14.92[13.33; 16.5] and v = 7.07 mm/s [4.569; the fracture surfaces c of the inert strength specimens in 2 9.571] with R = 0.95 using Eq. (1). The param- were examined using a scanning electron microscope. eters for H = 50% were determined analogously to One fracture surface is shown in Fig. 5. The initial n = 14.61[13.43; 15.79] and v = 7.83 mm/s [5.587; crack depth could be determined from these images 9.892] with R = 0.96. The values in brackets belong to a = 54.86 μm. Analogously, the inert crack in to the 95% conﬁdence interval. depth can be calculated with the critical stress inten- sity K = 0.75 MPa m and the geometry correction Ic 10 factor Y = 0.72 to a = 53.14 μm. Since the geome- in try factor for the calculated crack depth is derived from −1 the extrapolation of Eq. (5), the measured initial crack depth is used for further calculation. −2 Previous studies have shown that the Vickers inden- tation creates a half-penny shape crack (Cook and Pharr T = 25 C,H = 50 % 1990; Lawn 1993). However, this was done at an inden- −3 95 % −CI,H = 50 % tation force of 90N and no information was given ◦ T = 25 C,H = 40 % on the existing residual stresses before indentation in 95 % −CI,H = 40 % −4 Lawn (1993). Sglavo and Green (1995) showed, that 0.44 0.48 0.52 0.56 0.6 0.64 at a indentation force of 9.8 N with no residual stress √ K [MPa m] before indentation, no half-penny shape crack system is observable. It is assumed that the median crack and the Fig. 6 Subcritical crack growth velocity v as a function of the lateral cracks connect to a half-penny shape crack sys- stress intensity K determined by in-situ observation. Solid lines represent regressions according to Eq. (1) tem at higher indentation loads. It could also be shown a[µm] v [mm/s] Subcritical crack growth parameters in glass 95 growth parameter v and the relative humidity can be 4.4 expressed by 4.2 ◦ 2 v (25 C) = 0.002236H − 0.1359H + 7.103 (8) ◦ 2 v (35 C) = 0.004236H − 0.2703H + 10.1(9) −10 1 2 3 −1 ln(σ) ln[MPas ] with a coefﬁcient of determination of R = 0.97 and 2 ◦ R = 0.99. For T = 15 C a ﬁt was omitted due Fig. 7 Logarithmic plot of the measured failure stresses versus to only three existing values. With a general compar- the stress rate at 25 C and 50% relative humidity. Linear ﬁt with 2 ◦ R = 0.98 ison of the measured values from 15 to 35 C a com- parable run of the curve with increasing humidity can be observed between temperature and humidity. The polynomials are intended to show the general behav- 4.3 Dynamic fatigue experiments ior of the crack growth parameters as a function of the environmental conditions. The crack growth parameters as a function of temper- The crack growth parameter n for T = 25 Cat ature and humidity were determined from the dynamic H = 40 and 50% from the in-situ observation agree fatigue experiments using Eqs. (2) and (3). An example with the values from the dynamic fatigue tests. The for the failure stress σ vs stress rate σ ˙ curves can be values for v from the in-situ device are smaller then seen in Fig. 7 for T = 25 C and H = 50%. All the values from dynamic fatigue tests. In Dwivedi and determined crack growth parameters are shown in Fig. Green (1995), the parameter v was also slightly lower 8 and Table 2. then the values from dynamic fatigue tests. In general, For T = 15 C and a relative humidity of the measured crack growth parameters from Dwivedi H = 40 and 50% no crack growth parameters could and Green (1995) are lower than the values determined be determined. The reason for this is that condition in in this article. Eq. (4) was not fulﬁlled often enough for some stress Comparing the values of the present work with the rates. This was especially the case for a stress rate of literature values summarized in the appendix Table A1, −1 −1 σ ˙ = 20 MPas and σ ˙ = 6MPas . Accordingly, it can be seen that the values determined here for the there was no sufﬁcient sub-critical crack growth for the parameter n are in good agreement with those from the applied linear approximation. literature. The values shown in Fig. 8 indicate, as expected, The crack growth parameter v is often higher increasing crack growth with increasing temperature than values commonly found in the literature, yet in and humidity, since with increasing temperature also a realistic range. The values for v in Blank (1993) a higher reactivity of the water molecules at the crack ranges between 4.51 mm/s for summer and 8.22 mm/s tip is present and thus an accelerated reaction with the for winter conditions. The subcritical crack growth Si–O–Si molecule chains of the glass can take place. In parameters found in the literature range up to 14.3 order to make a general statement about the inﬂuence mm/s at 45%rH in Sglavo and Green (1995). In of the environmental condition on the crack growth Dwivedi and Green (1995), the comparison of soda– parameters, the parameters were ﬁtted as a function of lime silicate and sodium aluminosilicate glasses shows, humidity. The crack growth parameter n as a function that the crack growth parameter v changes from a of the relative humidity H could be ﬁtted by a 2nd order maximum of 2.6 mm/s for soda–lime silicate glass polynomial to (SiO = 72.3%wt) to 21.8 mm/s for sodium alumi- nosilicate glass (SiO = 62.3%wt). This fact supports ◦ 2 n(25 C) =−0.001564H + 0.06987H + 14.78 (6) the assumption that the chemical composition plays an ◦ 2 n(35 C) =−0.001622H + 0.05754H + 14.19 (7) important role in subcritical crack growth. In comparison to Dwivedi and Green (1995) it can be with a coefﬁcient of determination of R = 0.96 seen that the chemical composition of the glass shown and R = 0.97. The relationship between the crack in Table 1 is different. There are also differences in the ln(σ ) ln[MPa] f 96 C. Brokmann et al. ◦ ◦ T=25 C T=25 C ◦ ◦ 20 insitu at T = 25 C in-situ at T = 25 C 30 40 50 60 70 30 40 50 60 70 ◦ ◦ T=35 C T=35 C 30 40 50 60 70 30 40 50 60 70 relative Humidity [%] relative Humidity [%] Fig. 8 Measured values for subcritical crack growth parameters n (left) and v (right) at different temperatures as a function of humidity. Solid and dotted lines are 2nd order polynomial ﬁts and 95% conﬁdential intervals Table 2 Results for crack growth parameters of dynamic fatigue and in-situ tests in dependence of the environmental conditions ◦ 2 T( C) H (%rH) R n 95%-CI-n v [mm/s] 95%-CI-v (mm/s) 0 0 15 50 0.91 21.361 [18.916; 24.497] 4.86 [3.66; 6.26] 15 60 0.93 20.645 [18.168; 23.851] 5.34 [3.96; 6.98] 15 70 0.94 17.372 [15.477; 19.760] 7.83 [5.98; 9.98] 25 30 0.97 15.431 [14.758; 16.167] 9.54 [8.65; 10.49] 25 40 0.96 15.100 [13.702; 16.794] 10.22 [8.17; 12.56] 25 50 0.98 14.751 [13.966; 15.620] 10.47 [9.28; 11.76] 25 60 0.93 12.961 [11.327; 15.090] 13.95 [10.09; 18.63] 25 70 0.97 12.263 [11.361; 13.306] 15.99 [13.42; 18.85] 35 30 0.95 14.356 [12.812; 16.289] 11.18 [8.56; 14.24] 35 40 0.97 14.013 [12.992; 15.197] 11.18 [9.41; 13.14] 35 50 0.97 13.263 [12.221; 14.482] 13.40 [11.11; 15.97] 35 60 0.98 11.347 [10.506; 12.323] 17.87 [14.96; 21.13] 35 70 0.99 10.453 [9.845; 11.136] 22.30 [19.52; 25.49] Values from in-situ experiments 25 40 0.95 14.92 [13.33; 16.5] 7.07 [4.57; 9.57] 25 50 0.96 14.61 [13.43; 15.79] 7.74 [5.59; 9.89] inert strength of the samples compared to Dwivedi and Since soda–lime silicate glass or glass generally Green (1995) of 15 MPa. This is probably due to the fact exists in various chemical compositions, the chemi- that the residual stress ﬁeld was eliminated by temper- cal composition should always be shown when deter- ing after the Vickers indentation test (Anunmana et al. mining crack growth parameters. The exact inﬂuence 2009). of the chemical components on the subcritical crack n [-] n [-] v [mm/s] v [mm/s] 0 0 Subcritical crack growth parameters in glass 97 Fig. 9 Flowchart of the Input Parameters: algorithm to calculate ˙ ,dt,K ,K ,a , ,Y, th Ic in 0 failure stresses out of initial m = 1 ﬂaws undergoing sub-critical crack growth next cycle = + ˙ dt m m−1 m=m+1 K = Y a Im m m K < K K ≥ K I th I Ic check if K < K or K ≥ K I th I Ic K ≤ K < K th I Ic da = v(K ;v ,n)dt m 0 Im a = a + da m m−1 m fail m growth has not been investigated according to the cur- steps, care must be taken that K is not signiﬁcantly Ic rent knowledge of the authors. exceeded. The threshold limit below which no sub- critical crack growth occurs was assumed to be K = th 4.4 Simulation of environmental inﬂuence on failure 0.25 MPa m, the critical stress intensity to K = Ic 0.75 MPa m. The geometry correction factor Y, to In order to demonstrate the inﬂuence of humidity on the present state of knowledge of the authors unknown the failure strength of glass, the growth of cracks of for natural ﬂaws, was set to the constant value Y = 1 initial depths a = 1, 10 and 100 μm were simulated for the input at the ﬁrst time step in cycle m = 0. numerically using MatLab. This was done for a tem- A biaxial plane stress ﬁeld during a coaxial ring-on- perature of T = 25 C and relative humidity of 30, 50 ring test was assumed, so that the crack orientation can and 70%, respectively. A ﬂowchart of the algorithm is be neglected. The specimen radius was set to 40 mm, the radius of the support ring to 15 mm and of the load shown in Fig. 9. As initial values, the stress rate σ ˙,the time step dt, the crack growth threshold K , the critical ring to 6 mm. The thickness of the glass is set to 1.8 mm, th stress intensity K , an initial crack depth a , the initial the Poisson’s ratio was assumed to be 0.23. Ic in stress σ and the geometry correction factor Y for the During the simulation, the applied stress is ﬁrst stress intensity formulation are set. updated at the current time step in the current cycle −1 We consider a constant stress rate of σ ˙ = 2MPas m. Then the stress intensity K is calculated to check and a constant time step of dt = 0.01 ms. This whether crack growth is present or the critical stress small time step was chosen because with higher time intensity has already been reached. Finally, the crack growth velocity according to Eq. (1) is calculated. With Table 3 Inﬂuence of humidity on numerical failure stresses at the crack velocity and the time step, the crack growth in different initial crack depths at T = 25 C the current time step da is added to the existing crack Flaw size 1 μm10 μm 100 μm depth a for the updated crack depth a . m−1 m H = 30% 184.82 MPa 72.06 MPa 28.11 MPa Table 3 shows the resulting numerical failure stresses H = 50% 177.53 MPa 69.79 MPa 27.47 MPa as a function of the initial crack depths and environ- H = 70% 151.7 MPa 60.12 MPa 24.62 MPa mental conditions. It can be seen that the failure stress at a humidity difference of H = Δ40% differs up Maximum deviation 17.92% 16.57% 12.42% to 17.92%. This shows that the permitted difference in 123 98 C. Brokmann et al. environmental conditions in e.g. the European standard composition of the glass. The crack growth param- are to high and when comparing strength distributions, eters from Dwivedi and Green (1995) were com- the environmental conditions during the experiments pared with those determined here, who also exam- must always be included. ined soda–lime silicate glass. The chemical com- position of both publication differs slightly. 4. Arrest-lines were used to determine the ratio of 5 Conclusions crack width to crack depth in Vickers indented glass specimen. This differs slightly from the val- 1. It could be shown that the temperature and humidity ues available in the literature. This may be due to dependence of the subcritical crack growth parame- the tempering after indentation of the specimens ters n and v of the linear approximation by Maugis 0 within the scope of this publication. Scanning elec- (1985) exists as expected. At a humidity above 50% tron microscope images of the Vickers indentation the crack growth rate for all three investigated tem- fracture surface were also shown, to validate crack peratures increased more than in the area before depth and crack width in the initial stadium after 50%. The parameter v determined here is gener- 0 Vickers indentation and tempering. ally higher than the values available in the literature. Acknowledgements The presented work is based on results of A possible explanation could be that often Vickers the research project named 18295N “Stochastisches Bruchver- indented specimens are used with residual stresses halten von Glas” which has been funded by the AiF within the due to indentation. This is contradicted by the fact programme for sponsorship by Industrial Joint Research (IGF) that Dwivedi and Green (1995) has taken this into of the German Federal Ministry of Economic Affairs and Energy based on an enactment of the German Parliament. The research account and also obtained lower values for v .The project was carried out in co-operation with Forschungsvereini- parameter n is in the range of the literature values. gung Automobiltechnik e.V.—FAT. It has been shown that the crack growth parameters The authors would also like to thank P. Paulus from Pilkington as a function of humidity can be represented by a for providing the glass samples. 2nd order polynomial. Funding Open Access funding provided by Projekt DEAL. 2. By simulating double ring bending tests with the determined crack growth parameters it could be Compliance with ethical standards shown that at e.g. 25 C the failure stress at con- Conﬂict of interest The authors declare that they have no con- stant initial crack length varies by up to 18% in ﬂict of interest. the range of 30–70% relative humidity for initial cracks of 1, 10 and 100 μm. This effect increases Open Access This article is licensed under a Creative Com- mons Attribution 4.0 International License, which permits use, with variations in temperature. A possible future sharing, adaptation, distribution and reproduction in any medium study could investigate a shift for the strength dis- or format, as long as you give appropriate credit to the original tributions of glass as a function of environmental author(s) and the source, provide a link to the Creative Com- conditions, as it already exists for size effects. mons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s 3. When comparing the determined sub-critical crack Creative Commons licence, unless indicated otherwise in a credit growth parameters with literature values, it is line to the material. If material is not included in the article’s Cre- noticeable that some scatters of the literature values ative Commons licence and your intended use is not permitted by can be traced back to scatter of ambient conditions. statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view Nevertheless, unexplainable variations in literature a copy of this licence, visit http://creativecommons.org/licenses/ values of the sub-critical crack growth parameters by/4.0/. remain. These may be possible due to the chemical 123 Subcritical crack growth parameters in glass 99 Appendix Table A1 Literature review of subcritical crack growth parameters for SLS ﬂoat glass Environment Test method n v [mm/s] References 22.7 C, 50%rH Dynamic fatigue 14.22 2.2 Hilcken (2015) Water Mod. double beam cantilever* 15.44 2.92 Gehrke et al. (1987) 50%rH Mod. double beam cantilever* 16.66 0.83 Gehrke et al. (1987) Values from Dwivedi and Green (1995) Soda–lime silicate glass 27 C, 65%rH In-situ, Vickers indented 19.7–21.2 0.2–0.4 27 C, 65%rH Dynamic fatigue 21.8 2.6 27 C, 65%rH Dynamic fatigue, Vickers indented 21.1 2.4 Sodium aluminosilicate glass: 27 C, 65%rH In-situ, Vickers intended 25.6–26.0 11.6–21.8 27 C, 65%rH Dynamic fatigue 25.9 2.3 27 C, 65%rH Dynamic fatigue, Vickers indented 22.1 6.1 Extract of the summary from Haldimann (2006) Water In-situ 16.0 50.1 Kerkhof et al. (1981) Air, 50%rH In-situ 18.1 2.47 Kerkhof et al. (1981) Laboratory, summer Derived from Kerkhof et al. (1981) 16.0 4.51 Blank (1993) Laboratory, winter, 2 C Derived from Kerkhof et al. (1981) 16.0 8.22 Blank (1993) Water Values from 9 laboratories and 2000 specimens 17.7 10.7 Ritter et al. (1985) Water Dynamic fatigue 26 ±73.7 × 10 Sglavo and Bertoldi (2006) Water Dynamic fatigue 18 ±119 ± 4 Sglavo et al. (1997) Water Dynamic fatigue, intended 20.1 ± 0.7 28.8 ± 6.4 Sglavo and Green (1999) Water Dynamic fatigue, annealed 19.9 ± 0.7 6.4 ± 1.4 Sglavo and Green (1999) Values from Wiederhorn (1967), converted by Schula (2015): 25 C, Water Double-cantilever cleavage 17.4 3.8 25 C, 100%rH Double-cantilever cleavage 20.8 3.6 25 C, 30%rH Double-cantilever cleavage 22.6 1.7 25 C, 10%rH Double-cantilever cleavage 21.4 0.6 25 C, 0.017%rH Double-cantilever cleavage 27.2 0.09 Vacuum Double-cantilever cleavage 93.3 0.13 Extract of the summary from Schula (2015) Water Unkown 13.0 1.1 Gehrke and Ullner (1988) 50%rH Unkown 14.3 0.16 Gehrke and Ullner (1988) Water Unkown 18.4 17.1 Ullner and Höhne (1993) 50%rH Unkown 19.7 2.8 Ullner and Höhne (1993) 25 C, 45%rH Dynamic fatigue, Vickers intended 18.8 14.3 Sglavo and Green (1995) *Fitted from displayed data 123 100 C. Brokmann et al. 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Technial (2013b) Report, Bundesanstalt für Materialforschung und-Prüfung (BAM) in Cooperation with Fraunhofer-Institut für Werk- Publisher’s Note Springer Nature remains neutral with regard stoffmechanik, Berlin (1993) to jurisdictional claims in published maps and institutional afﬁl- Wachtman, J.B., Cannon, W.R., Matthewson, M.J.: Mechanical iations. Properties of Ceramics. Wiley, Hoboken (2009)

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Subcritical crack growth parameters in glass as a function of environmental conditions

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Subcritical crack growth parameters in glass as a function of environmental conditions

Subcritical crack growth parameters in glass as a function of environmental conditions

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