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Extended MTSN criterion for fracture analysis of soda lime glass
Accepted Manuscript Extended MTSN criterion for fracture analysis of soda lime glass M.M. Mirsayar, V.A. Joneidi, R.V.V. Petrescu, F.I.T. Petrescu, F. Berto PII: DOI: Reference:
S0013-7944(17)30226-6 http://dx.doi.org/10.1016/j.engfracmech.2017.04.018 EFM 5492
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
23 February 2017 10 April 2017 13 April 2017
Please cite this article as: Mirsayar, M.M., Joneidi, V.A., Petrescu, R.V.V., Petrescu, F.I.T., Berto, F., Extended MTSN criterion for fracture analysis of soda lime glass, Engineering Fracture Mechanics (2017), doi: http:// dx.doi.org/10.1016/j.engfracmech.2017.04.018
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Extended MTSN criterion for fracture analysis of soda lime glass M.M. Mirsayar1,*, VA Joneidi2, R.V.V. Petrescu3, F.I.T. Petrescu3, F. Berto4 1
Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843-3136, USA
School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran 3
IFToMM-ARoTMM, Bucharest Polytechnic University, 313 Splaiul Independentei, Bucharest, (CE) Romania 4
Department of Engineering Design and Materials, NTNU, Trondheim, Norway
* Corresponding author: M. M. Mirsayar, E-mail address: [email protected], Tel.: +1 (979) 777-6096
Abstract This paper investigates brittle fracture in soda lime glass subjected to mixed mode I/II loading using different fracture criteria. Different sets of mixed mode I/II fracture test data from literature, conducted by cracked Brazilian disk specimen, are utilized to study brittle fracture in soda lime glass. The fracture initiation conditions in soda lime glass is examined by different traditional fracture criteria including Strain Energy Density, Maximum Tangential Stress, and Maximum Tangential Strain criteria. It is shown that the traditional criteria, which only consider singular stress (strain) terms, are not able to properly predict the fracture test data. The test data are then predicted by an extended version of the maximum tangential strain (EMTSN) criterion which takes into account the effect of first nonsingular strain term as well as the singular strain terms. It is found that the mixed mode fracture toughness of the soda lime glass as well as the crack initiation direction can be predicted successfully by the EMTSN criterion. Keywords: Soda lime glass; strain-based fracture criteria; mixed mode fracture; EMTSN
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Nomenclature a Semi-crack length for centrally cracked specimens. B Biaxiality ratio. CBD Cracked Brazilian disk. E Young’s modulus. EMTSN Extended Maximum tangential strain criterion. F Applied load. fij,k (k≡1, 2, 3) Known strain field functions. gij,k (k≡1, 2, 3) Known stress field functions. G Shear modulus. KI, KII Mode I and II stress intensity factors. KIC Mode I fracture toughness. KIf, KIIf Mode I and II stress intensity factors corresponding to the fracture load. Keff Effective stress intensity factor. * K IC Generalized fracture toughness. LEFM Linear elastic fracture mechanics. MTS Traditional maximum tangential stress criterion. MTSN Traditional maximum tangential strain criterion. Polar coordinate components. r, θ rc Critical distance from crack tip. R, t Radius and thickness of the cracked Brazilian disk specimen. SED Strain energy density criterion. T Coefficient of the first nonsingular strain term. YI(β),YII(β) Mode I and II geometry factors. α Normalized critical distance from crack tip. β Crack inclination angle for cracked Brazilian disk specimen. εΤ Ultimate tensile strain. ε ij Crack tip strain field. θ0 Direction of the crack propagation. Crack tip stress field. σij σΤ Ultimate tensile stress. ν Poisson’s ratio. Φ Strain energy-density factor. Φc Critical strain energy-density factor.
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1. Introduction One of the most important challenges in designing of engineering structures is to predict the conditions in which cracks start to initiate and propagate. In an in-service structure, cracks may be created and developed around the vulnerable area as a result of excessive external loads. In these components, sharp corners and pre-existing cracks are subjected to in-plane (mode I/II) 2
and out of plane (mode III) loading conditions which may finally lead to the structural failure. Although extensive efforts have been carried out so far to investigate brittle fracture behavior in engineering materials, the brittle fracture phenomenon is still considered as a challenging research topic. Linear elastic fracture mechanics (LEFM) has successfully been applied for mixed mode I/II/III fracture analysis of brittle materials by many researchers. Generally, most researchers categorize brittle fracture criteria into energy-based, stress-based and strain-based criteria. G criterion [1] is an example of the energy-based criteria which assumes that the crack initiates to grow in the direction of maximum energy release rate. The maximum tangential stress (MTS) criterion [2], is among the most well-known stress-based fracture criteria which says that the crack propagates along a path where the circumferential stress is maximized. The stress and energy-based criteria are having some advantages and disadvantages. In stress based criteria, the governing equations are simpler, and the effect of each stress term on predicting the fracture behavior is clearly stated. On the other hand, the assumptions in energy-based criteria are more realistic, because they directly relate fracture mechanism to the dissipative fracture energy needed for the crack growth [3]. An example of strain based criterion is MTSN criterion, introduced by Chang [4], which states that a crack begins to grow when the maximum tangential strain at the crack tip attains its maximum value. According to the studies performed in the past, for some materials, the strain-based criteria are able to provide better prediction of fracture behavior than stress-based and energy-based criteria [5-8]. However, according to a direct comparison conducted by Sajjadi et al. [9], for some materials and fracture test specimens, there is a discrepancy between the fracture test data and the predicted values obtained by all traditional fracture criteria (e.g. R criterion [10], Det criterion [11], MTS criterion [2], G criterion [1], S
3
criterion [12], T criterion [13], M criterion [14], and modified MTS criterion [15]), particularly under pure mode II conditions. Minimizing such discrepancies and achieving more accurate predictions of the fracture initiation conditions have been actuating forces on many researchers to explore for improved fracture criteria. Based on LEFM, the stress and strain fields around the sharp corners (e.g. cracks and notches) can be expressed in terms of a series of infinite terms called Williams series expansion [16]. All traditional fracture criteria (e.g. MTS, MTSN, and G) only take into account the effect of singular terms of the Williams series expansion which are corresponding to stress intensity factors (KI, KII). On the other hand, the significant role of the first nonsingular stress term of the Williams series expansion, called T-stress, on the mixed mode brittle fracture behavior has been pointed out by many investigators including Mirsayar and his colleagues [17-25] and Shahani and Tabatabaei [26]. However, the effect of first nonsingular term of the Williams series expansion on the mixed mode brittle fracture behavior, in a strain-based framework was unrevealed, until Mirsayar [3]. Recently, the traditional MTSN criterion presented by Chang [4] was extended by Mirsayar [3] by taking into account the effect of the first nonsingular strain term, called T-strain, as well as the singular strain terms. The ability of the extended MTSN (EMTSN) criterion has recently been proven by providing accurate prediction of the mixed mode fracture test data obtained for polymethylmethacrylate (PMMA) [3], Polycrystalline graphite [27], and cement mortar [28]. Soda-lime glass (or soda-lime-silica glass) is the most prevalent type of glass, which is used in different industrial applications ranging from windowpanes to glass containers. Because of its brittleness, soda lime glass is exposed to failure and catastrophic fracture particularly in the presence of pre-existing flaws. Crack growth in soda lime glass under various loading conditions 4
has been investigated by many investigators so far. The effect of the indentation load on the fracture resistance of soda lime glass has extensively been explored by the researchers in the past [29 – 33]. Among them, Gong et al. [31] investigated fracture toughness of soda-lime glass using Vickers indentation tests. Abrams et al. [32] studied the fracture strength and multiple cracking behavior of soda-lime silicate glass, and explored the response of this glass type to the contact damage using indented specimens. Crack initiation conditions in soda lime glass under combined loading has also been examined by the researchers [34 – 44]. Using sandwiched beam specimens in water environment, the mixed mode fracture toughness of soda-lime-silica glass have been investigated by Sglavo et al. [34]. Yoda et al. [35] made use of compact tension shear specimens of soda lime glass to study the fracture and subcritical crack growth under mixed mode I/II loading. Li and Sakai [36] studied the mixed mode fracture behavior of soda lime glass using a four point bending technique. However, they reported a considerable discrepancy between their experimental results and theoretical predictions in mode II conditions. Baraki et al. [44] investigated dynamic crack path selection in soda lime glass under external thermal loading using a novel test configuration and measured the mode I fracture toughness of the soda lime glass. They have found that the crack path follows the law of local symmetry (KII = 0), proposed by Gol'Dstein and Salganik [45], for both the quasi-static and dynamic crack propagation. The current paper deals with the evaluation of the mixed mode I/II fracture toughness and the crack initiation angles for the test data on soda lime glass available in the literature. Different mixed mode fracture test data obtained by cracked Brazilian disk specimen are evaluated. Different traditional fracture criteria and an extended version of the MTSN criterion are employed to predict mixed mode I/II crack propagation conditions in soda lime glass. It is shown 5
that the EMTSN criterion which takes into account the effect of T-strain provides better estimates of the test data than all traditional fracture criteria.
2. Fracture criteria 2.1. Crack tip field equations According to Williams [16], the linear elastic stress and strain field equation around the crack tip can be expressed as: = =
√2
, +
√2
√2
, , +
, + . , + √ , / , …
√2
, , + . , , + √ , / , …
(1)
(2)
where , ! ≡ , are polar coordinates with origin at the crack tip, and E is the Young modulus. The first two terms in Eq. (1-2) are singular and depend on the modes I and II stress intensity factors (KI and KII). The parameter T, is the coefficient of the first nonsingular stress (T-stress) or strain (T-strain) terms. The functions ,# and ,# are known functions of and Poisson’s ratio ( ) and are related together by Hooke’s law [3]. The crack tip filed equations can be extended in terms of infinite nonsingular higher order terms. However, the effects of higher order terms are negligible as one approaches the crack tip. 2.2. Traditional fracture criteria In the following, the Strain Energy Density (SED), the Maximum Tangential Stress (MTS), and the Maximum Tangential Strain (MTSN) criteria are briefly described. 6
2.2.1. SED criterion The SED criterion, proposed by Sih [12], is an energy-based fracture criterion which states that crack start to grow in the direction of the minimum value of the strain energy-density factor (Φ), as: %Φ = 0 '( = ) %
% Φ > 0 '( − < ) < %
(3)
(4)
This criterion assumes that crack initiation happens when Φ = Φ- at = ) where Φ- is the critical strain energy-density factor and ) is the direction of the crack initiation. The SED criterion can be expressed in the following form by taking into account only singular terms of the crack tip field equation given in Eq. (1-2): Φ=
1 11 + 2345 − 2346 + 24 782234 − 5 − 19 160
(5)
+ 15 + 11 − 234 + 1 + 2343234 − 16
where, 0 = /821 + ;9 is the shear modulus, and 5 equals to 3 − 4;, and 3 − ;/1 + ; for plane strain and plane stress, respectively. 2.2.2. MTS criterion As a stress-based criterion, the MTS criterion, proposed by Erdogan and Sih [2], assumes that crack initiation occurs from the crack tip in the direction of the maximum tangential stress component (== ), when tangential stress reaches the ultimate tensile strength (> ) of the material. Considering only singular stress terms, the MTS criterion can be formulated as: 7
?
?
A
@A ,1 + AA ,2 B = 0 → )
AD
,1 ) +
AA
AD
,2 ) = 1
(6) (7)
where - is the mode I fracture toughness of the material. 2.2.3. MTSN criterion Originally proposed by Chang [4], the maximum tangential strain criterion postulates that crack initiates from crack tip in direction where the tangential component of the strain field attains its maximum value ( > = > / ). The traditional MTSN criterion only takes into account the singular strain terms and is presented as: ?
?
@A ,1 , + AA ,2 , B = 0 → )
==, , 0 + ==, , 0 = -
(8) (9)
2.3. EMTSN criterion The Extended MTSN (EMTSN) criterion is recently proposed by Mirsayar [3] and includes the first nonsingular strain term (T-strain) as well as the singular strain terms. According to EMTSN criterion [4], a crack starts to grow at a critical distance, rc, from the crack tip in direction, θ0, when the elastic tangential strain reaches the ultimate tensile strain of the material. FG E HH F=
I
JKJL ,=K=M
F = 0, E
NG
HH
F=N
== O , ) = > = > /
I
JKJL ,=K=M
<0
(10) (11)
The critical distance rc, equals to the size of the fracture process zone and is considered as a material property [46]. Physically, the critical distance rc, represents the size of an area around 8
the crack tip which is discontinues and contains micro-sized cracks. The theory of critical distance postulates that crack starts to initiate at the boundary of this area instead of crack tip. Assuming that the critical distance is independent from mode mixity, it can be evaluated as a function of mode I fracture toughness and the ultimate tensile stress as [47]:
O =
1 - 2 2
(12)
By substituting Eq. (2) into Eq. (10) and Eq. (11), the EMTSN criterion is given as,
% 1 S S S E == P , ) + ==, , ) B + . ==, , ) = 0 =0⇒ @ ==, % JKJL ,=K=M R2 O
(13)
⇒ )
> = > =
1
R2 O
@ ==, , ) + ==, , ) B + . ==, , )
F S , ) = E ==, , I where, ==, F=
=K=M
(14)
. The EMTSN criterion can be rewritten in the following
form in terms of normalized parameters as, S S S , ) = 0 ⇒ ) ==, , ) + , ) + VW ==, TUU TUU ==,
==, , ) + ==, , ) = > R2 O − VWTUU ==, , )
(15) (16)
where TUU = R + , W = R2 O /' , and a is the crack length for edge cracked specimens and semi-crack length for centrally cracked specimens. The parameter B is called biaxiality ratio and is defined as [47]:
V=
√ ' TUU
(17)
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According to Mirsayar [3], the fracture loci for the EMTSN criterion can be presented in ∗ ∗ U /− U /diagram by the following formulation:
U ∗ = U ∗ = -
1 − 1 + VW
==, , ) 81 + D
− YZ1 + 9
\ 81 − 1 + VW9 ==, , ) 81 + D − YZ1 + 9 [
(18)
(19)
where parameters C and H are defined as: D=
==, , ) ==, , )
Y = −VW
==, , ) ==, , )
(20)
(21)
∗ The parameter , called generalized fracture toughness, is defined in Eq. (22) and takes into
consideration the effect of specimen geometry as well as the Poisson’s ratio (see Mirsayar [3] for ∗ ∗ more details about U /− U /diagram). ∗ =
> R2 O 1 − 1 + VW
(22)
3. Cracked Brazilian disk (CBD) specimen The CBD specimen has been employed by Shetty and co-workers [42, 43], and Awaji and Kato [38], separately, to obtain mixed mode fracture toughness of soda lime glass. As shown in Fig. 1, the CBD specimen is made of a centrally cracked disk of radius R subjected to a
10
compressive load F along its diameter. The specimen has a thickness t, and the total length of the central crack equals 2a.
Fig. 1. Scheme of CBD specimen subjected to a diametrally compressive load.
As shown in Fig. 1, one can conduct different mixed mode I/II fracture tests using CBD specimen by altering the loading angle β. For CBD specimens, the stress intensity factors (KI, KII) can be rewritten in normalized forms as:
= ] ^
_ ' Z `(
= ] ^
_ ' Z `(
(23)
(24)
where ] ^ and ] ^ are the normalized forms of KI, KII, and T, respectively. For a fixed √'/` ratio, these geometry factors are only functions of inclination angle β. These parameters can be obtained using finite element simulation of the specimen at different values of β. For the CBD specimens employed by Shetty and co-workers [42, 43], and those of used by Awaji and Kato 11
[38], the normalized crack length (a/R) was 0.25 (R=25mm), and 0.4 (R=20mm), respectively. Plane stress condition is assumed since the specimen thickness was relatively smaller than other dimensions of the CBD specimens in all three sets. A typical finite element mesh used for simulation of the CBD specimens and an expanded view of the crack tip elements are shown in Fig. 2. Due to high stress/strain gradient resulting from the singular stress/strain field near the notch tip, a very fine mesh was used in the region close to the crack tip. The auxiliary asymptotic displacement fields used with J-integral to determine the stress intensity factors as well as the T parameter at different geometry and loading configuration.
Fig. 2. Typical finite element mesh used for modeling the CBD specimens (β=15°)
Using finite element simulation, ] ^ and ] ^ for these geometries can be found for different inclination angle β, as illustrated in Fig. 3. Also shown in Fig. 3 is the variation of the
12
normalized T parameter, V^ =
> √ab cdee
, versus β. The pure mode II happens for a/R = 0.25 and
a/R = 0.4 at β = 28ο and 25ο, respectively.
a)
b)
Fig.3. Normalized crack tip parameters for CBD specimen obtained by finite element simulation for: a) a/R = 0.25, b) a/R = 0.4
By substituting] ^, ] ^ and V^ given in Fig. 3 into the Eq. (17, 23, 24), one can obtain crack initiation angle and onset of fracture using different fracture criteria. 4. Analytical predictions The predictions performed by different fracture criteria are compared and discussed in this section for the reported fracture test data for soda lime glass conducted by the CBD specimens. Fig. 4 shows predictions presented by different traditional fracture criteria (SED, MTS, and MTSN) for the crack initiation direction in soda lime glass tested by Shetty et al. [42]. It is needed to be pointed out that among all three sets of the fracture tests ([38, 42, and 43]), the crack initiation direction is reported only by Shetty et al. [42]. It is seen that all traditional
13
criteria fail to properly predict crack initiation angle in soda lime glass under mixed mode loading and they all overestimate the test data. Among these traditional criteria, the MTSN criterion presents the best prediction at pure mode II (β = 28o). However, like other criteria, the MTSN criterion deviates from the experimental data in mixed mode conditions.
Fig.4. Direction of the crack initiation obtained by Shetty et al. [42], as well as the prediction curves provided by traditional fracture criteria (SED, MTS, and MTSN)
The theoretical prediction curves for SED, MTS, and MTSN criteria are plotted in Fig. 5 for mixed mode fracture toughness of the soda lime glass. Results are plotted as normalized
c
effective fracture toughness (TUU /- ) versus mode mixity parameter (fT = a tanj c k ). The kk
parameter fT represents contribution of each crack tip displacement mode and equals to unity at pure mode I and zero at pure mode II. As depicted in Fig. 3, the crack inclination angles (β) corresponding to the pure mode II conditions are 25o, and 28o for the CBD specimens tested by Awaji and Kato [38], and Shetty and co-workers [42, 43], respectively. Therefore, the mixed mode fracture test data and the theoretical curves are plotted separately in order to clearly show 14
the corresponding crack inclination angle. It is obvious from Fig. 5 that none of the traditional criteria is able to reasonably estimate mixed mode fracture toughness of the soda lime glass. Furthermore, the discrepancy between the mixed mode fracture test data and the theoretical predictions provided by traditional criteria increases as one approaches mode II conditions. a)
b)
Fig.5. Theoretical predictions provided by different traditional criteria for mixed mode fracture test data of soda lime glass tested by a) Awaji and Kato [38], b) Shetty and co-workers [42, 43]
In the current analysis, the ultimate tensile stress of 20.6 MPa, the Young’s modulus of 70GPa, and the Poisson’s ratio of ν = 0.22 are assumed as for soda lime glass [38, 42, 48, 49]. The critical distance of rc = 0.2mm is also selected which is in agreement with the theoretical investigations performed by stress-based criteria [50]. The predictions obtained by EMTSN criterion is plotted in Fig. 6 for the crack initiation angles in soda lime glass tested by Shetty et al. [42]. It can be observed that adding the first nonsingular strain term (T-strain) significantly improves estimates presented by MTSN criterion. Except for
15
pure mode II, the predictions results obtained by EMTSN criterion are remarkably better that traditional MTSN criterion under mixed mode conditions. The mixed mode fracture toughness of soda lime glass tested by Shetty and co-workers [42, 43], and Awaji and Kato [38] are evaluated in Fig. 7 by MTSN and EMTSN criteria. As shown in Fig. 7, the EMTSN presents very accurate evaluation of the current state for the mixed mode fracture toughness of soda lime glass. According to the Fig. 7, the traditional MTSN criterion always underestimates the mixed mode fracture toughness of the soda lime glass, conducted by CBD specimens.
Fig.6. Direction of the crack initiation obtained by Shetty et al. [42], as well as the prediction curves provided by MTSN and EMTSN criteria
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Fig.7. Theoretical predictions provided by MTSN and EMTSN criteria for mixed mode fracture test data of soda lime glass tested by a) Awaji and Kato [38], b) Shetty and co-workers [42, 43]
Based on the results shown in Figs. 4 – 7, it can be realized that the first non-singular strain term significantly influences on the predictions obtained from strain-based criteria and must be considered in the fracture prediction model. The role of first nonsingular term of the Williams series expansion in the brittle fracture of the engineering materials has widely been investigated in stress-based and energy-based frameworks. However, adding the first nonsingular term to the fracture criterion does not always guarantee for more accurate predictions. As an example, the generalized MTS criterion [47], which revises the traditional MTS criterion by adding the T-stress term, fails to provide accurate predictions near mode II conditions in many cases, and requires adding higher order terms (i.e. second or third nonsingular terms) to achieve an acceptable accuracy [3]. Malikova [51] has recently indicated that at least three or four higher order terms of the Williams series expansion should be considered to present acceptable accuracy by MTS criterion. It is also needed to be pointed out that the role of 17
Poisson’s ratio in the mixed mode fracture behavior is not taken into account in many fracture criteria, including generalized MTS criterion. The effect of Poisson’s ratio on the onset of fracture is taken into account in some energy-based criteria (e.g. SED [12] criterion). However, the energy-based criteria are relatively more complex than stress-based and strain-based criteria in terms of mathematical formulation. Therefore, it can be considered as an advantage for the EMTSN criterion to be able to involve effects of Poisson’s ratio in a relatively simple formulation.
5. Conclusion Mixed mode fracture behavior of soda lime glass was investigated in this paper using different fracture criteria: SED, MTS, MTSN, and EMTSN. Traditional fracture criteria (SED, MTS, and MTSN), which only take into account the effect of singular terms of the strain (stress) field equations, are not able to reasonably predict crack initiation angle and onset of the fracture in soda lime glass. Theoretical predictions for different fracture test data obtained by CBD specimens demonstrate the significant role of T-strain in the mixed mode fracture toughness as well as the crack initiation angle in soda lime glass. Depending on the specimen geometry, the traditional MTSN criterion may underestimate or overestimate the fracture test data. The discrepancies between the MTSN predictions and the test data is resolved when employing EMTSN criterion which takes into account the T-strain as well as the singular strain terms. Since the value of the T-strain depends on the specimen geometry (based on Fig. 3), it is highly recommended to consider effect of T-term when applying any fracture criterion.
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Acknowledgements The research presented in this paper was supported by Zachry Department of Civil Engineering at Texas A&M University. Any opinions, findings, conclusions, and recommendations expressed in this paper are those of the authors alone and do not necessarily reflect the views of the sponsoring agency.
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[28] M.M. Mirsayar, P. Park, Mixed mode brittle fracture analysis of high strength cement mortar using strain-based criteria, Theor Appl Fract Mech 86 (2016): 233-238. [29] T. Rouxel , J.C. Sanglebúuf, The brittle to ductile transition in a soda lime silica glass, Journal of Non-Crystalline Solids. 271 (2000) 224-235 [30] G.A. Gogotsi, S.P. Mudrik, Glasses: New approach to fracture behavior analysis, Journal of Non-Crystalline Solids. 356 (2010) 1021–1026 [31] J. Gong, Y. Chen, C. Li, Statistical analysis of fracture toughness of soda-lime glass determined by indentation, Journal of Non-Crystalline Solids. 279 (2001) 219–223. [32] M.B. Abrams, D.J. Green, S. Jill Glass, Fracture behavior of engineered stress profile soda lime silicate glass, Journal of Non-Crystalline Solids. 321, (2003) 10–19. [33] S. Deriano, A. Jarry, T. Rouxel, J.C. Sangleboeuf, S. Hampshire, The indentation fracture toughness (KC) and its parameters: the case of silica-rich glasses, Journal of Non-Crystalline Solids. 344 (2004) 44–50. [34] V.M. Sglavo, T. Micheletti, S. Di Cuonzo, Influence of crack geometry and loading mode on fracture and fatigue of soda-lime–silica glass, Physics and Chemistry of Glasses – European Journal of Glass Science and Technology. Part B 44 (2003) 337–344. [35] M. Yoda, Y. Yoshikawa, K. Ono, M. Ide, Fracture and subcritical crack growth in soda-lime glass under combined modes I and II loading, Journal of the American Ceramic society. 78 (1995) 161–165. [36] M. Li, M. Sakai, Mixed-mode fracture of ceramics in asymmetric four-point bending: effect of crack-face grain interlocking/bridging, Journal of the American Ceramic Society. 79 (1996) 2718–2726. [37] S.W. Freiman, A.C. Gonzalez, J.J. Mecholsky, Mixed-mode fracture in soda-lime glass, Journal of the American Ceramic society. 62 (1979) 206–208. [38] H. Awaji, T. Kato, Criterion for combined mode I-II brittle fracture. Materials Transactions, JIM. 40(1999) 972-979. [39] H. Awaji, T. Kato, Griffith criterion for mode II fracture of ceramics. In: 11 th International Conference on Experimental Mechanics. (1998) 1199-1204. [40] H. Kordisch, J. Riedmuller, E. Sommer, The strain energy density criterion – investigations for its applicability. Absorbed specific energy and/or strain energy density criterion. (1982) 3343. [41] V.V. Panasyuk, L.T. Berezhnitskiy, S.Y. Kovchik, Propagation of an arbitrary oriented rectilinear crack during extension of a plate. Pirkladnaya Mekhanika 1(1965) 48–55. [42] D.K. Shetty, A.R. Rosenfield, W.H. Duckworth, Mixed-mode fracture in biaxial stress state: application of the diametral compression (Brazilian disk) test, Eng Fract Mech. 26 (1987) 825– 840. 21
[43] D.K. Shetty, A.R. Rosenfield. Slow crack growth in glass in combined mode I and mode II loading. Scripta metallurgica et materialia 25 (1991) 997-1002. [44] O. Barkai, T. Menouillard, J.H. Song, T. Belytschko, D. Sherman, Crack initiation and path selection in brittle specimens: A novel experimental method and computations, Engineering Fracture Mechanics, 89 (2012) 65-74. [45] R.V. Gol’Dstein, R.L. Salganik, Brittle fracture of solids with arbitrary cracks, International journal of Fracture, 10(1974) 507–523. [46] D. Taylor. The theory of critical distances. Eng. Fract. Mech. 75(7) 2008, 1696–1705. [47] Smith DJ, Ayatollahi MR, Pavier MJ. The role of T-stress in brittle fracture for linear elastic materials under mixed mode loading. Fatigue Fract Eng Mater Struct 2001; 24(2):137–50. [48] Gong J, Miao H, Peng Z. On the contact area for nanoindentation tests with Berkovich indenter: case study on soda-lime glass. Materials letters. 2004; 58(7):1349-53. [49] Oliver WC, Pharr GM. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. Journal of materials research. 1992; 7(06):1564-83. [50] Ayatollahi MR, Aliha MR. Mixed mode fracture in soda lime glass analyzed by using the generalized MTS criterion. International Journal of Solids and Structures. 2009; 46(2):311-21. [51] L. Malikova, Multi-parameter fracture criteria for the estimation of crack propagation direction applied to a mixed-mode geometry, Engineering Fracture Mechanics, 143(2015) 32– 46.
22
Highlights •
Mixed mode fracture of soda lime glass is evaluated by different criteria.
•
Traditional criteria cannot provide proper predictions of the fracture test data.
•
EMTSN criterion presents more accurate predictions than traditional criteria.
23
S0013-7944(17)30226-6 http://dx.doi.org/10.1016/j.engfracmech.2017.04.018 EFM 5492
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
23 February 2017 10 April 2017 13 April 2017
Please cite this article as: Mirsayar, M.M., Joneidi, V.A., Petrescu, R.V.V., Petrescu, F.I.T., Berto, F., Extended MTSN criterion for fracture analysis of soda lime glass, Engineering Fracture Mechanics (2017), doi: http:// dx.doi.org/10.1016/j.engfracmech.2017.04.018
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Extended MTSN criterion for fracture analysis of soda lime glass M.M. Mirsayar1,*, VA Joneidi2, R.V.V. Petrescu3, F.I.T. Petrescu3, F. Berto4 1
Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843-3136, USA
School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran 3
IFToMM-ARoTMM, Bucharest Polytechnic University, 313 Splaiul Independentei, Bucharest, (CE) Romania 4
Department of Engineering Design and Materials, NTNU, Trondheim, Norway
* Corresponding author: M. M. Mirsayar, E-mail address: [email protected], Tel.: +1 (979) 777-6096
Abstract This paper investigates brittle fracture in soda lime glass subjected to mixed mode I/II loading using different fracture criteria. Different sets of mixed mode I/II fracture test data from literature, conducted by cracked Brazilian disk specimen, are utilized to study brittle fracture in soda lime glass. The fracture initiation conditions in soda lime glass is examined by different traditional fracture criteria including Strain Energy Density, Maximum Tangential Stress, and Maximum Tangential Strain criteria. It is shown that the traditional criteria, which only consider singular stress (strain) terms, are not able to properly predict the fracture test data. The test data are then predicted by an extended version of the maximum tangential strain (EMTSN) criterion which takes into account the effect of first nonsingular strain term as well as the singular strain terms. It is found that the mixed mode fracture toughness of the soda lime glass as well as the crack initiation direction can be predicted successfully by the EMTSN criterion. Keywords: Soda lime glass; strain-based fracture criteria; mixed mode fracture; EMTSN
1
Nomenclature a Semi-crack length for centrally cracked specimens. B Biaxiality ratio. CBD Cracked Brazilian disk. E Young’s modulus. EMTSN Extended Maximum tangential strain criterion. F Applied load. fij,k (k≡1, 2, 3) Known strain field functions. gij,k (k≡1, 2, 3) Known stress field functions. G Shear modulus. KI, KII Mode I and II stress intensity factors. KIC Mode I fracture toughness. KIf, KIIf Mode I and II stress intensity factors corresponding to the fracture load. Keff Effective stress intensity factor. * K IC Generalized fracture toughness. LEFM Linear elastic fracture mechanics. MTS Traditional maximum tangential stress criterion. MTSN Traditional maximum tangential strain criterion. Polar coordinate components. r, θ rc Critical distance from crack tip. R, t Radius and thickness of the cracked Brazilian disk specimen. SED Strain energy density criterion. T Coefficient of the first nonsingular strain term. YI(β),YII(β) Mode I and II geometry factors. α Normalized critical distance from crack tip. β Crack inclination angle for cracked Brazilian disk specimen. εΤ Ultimate tensile strain. ε ij Crack tip strain field. θ0 Direction of the crack propagation. Crack tip stress field. σij σΤ Ultimate tensile stress. ν Poisson’s ratio. Φ Strain energy-density factor. Φc Critical strain energy-density factor.
1
1. Introduction One of the most important challenges in designing of engineering structures is to predict the conditions in which cracks start to initiate and propagate. In an in-service structure, cracks may be created and developed around the vulnerable area as a result of excessive external loads. In these components, sharp corners and pre-existing cracks are subjected to in-plane (mode I/II) 2
and out of plane (mode III) loading conditions which may finally lead to the structural failure. Although extensive efforts have been carried out so far to investigate brittle fracture behavior in engineering materials, the brittle fracture phenomenon is still considered as a challenging research topic. Linear elastic fracture mechanics (LEFM) has successfully been applied for mixed mode I/II/III fracture analysis of brittle materials by many researchers. Generally, most researchers categorize brittle fracture criteria into energy-based, stress-based and strain-based criteria. G criterion [1] is an example of the energy-based criteria which assumes that the crack initiates to grow in the direction of maximum energy release rate. The maximum tangential stress (MTS) criterion [2], is among the most well-known stress-based fracture criteria which says that the crack propagates along a path where the circumferential stress is maximized. The stress and energy-based criteria are having some advantages and disadvantages. In stress based criteria, the governing equations are simpler, and the effect of each stress term on predicting the fracture behavior is clearly stated. On the other hand, the assumptions in energy-based criteria are more realistic, because they directly relate fracture mechanism to the dissipative fracture energy needed for the crack growth [3]. An example of strain based criterion is MTSN criterion, introduced by Chang [4], which states that a crack begins to grow when the maximum tangential strain at the crack tip attains its maximum value. According to the studies performed in the past, for some materials, the strain-based criteria are able to provide better prediction of fracture behavior than stress-based and energy-based criteria [5-8]. However, according to a direct comparison conducted by Sajjadi et al. [9], for some materials and fracture test specimens, there is a discrepancy between the fracture test data and the predicted values obtained by all traditional fracture criteria (e.g. R criterion [10], Det criterion [11], MTS criterion [2], G criterion [1], S
3
criterion [12], T criterion [13], M criterion [14], and modified MTS criterion [15]), particularly under pure mode II conditions. Minimizing such discrepancies and achieving more accurate predictions of the fracture initiation conditions have been actuating forces on many researchers to explore for improved fracture criteria. Based on LEFM, the stress and strain fields around the sharp corners (e.g. cracks and notches) can be expressed in terms of a series of infinite terms called Williams series expansion [16]. All traditional fracture criteria (e.g. MTS, MTSN, and G) only take into account the effect of singular terms of the Williams series expansion which are corresponding to stress intensity factors (KI, KII). On the other hand, the significant role of the first nonsingular stress term of the Williams series expansion, called T-stress, on the mixed mode brittle fracture behavior has been pointed out by many investigators including Mirsayar and his colleagues [17-25] and Shahani and Tabatabaei [26]. However, the effect of first nonsingular term of the Williams series expansion on the mixed mode brittle fracture behavior, in a strain-based framework was unrevealed, until Mirsayar [3]. Recently, the traditional MTSN criterion presented by Chang [4] was extended by Mirsayar [3] by taking into account the effect of the first nonsingular strain term, called T-strain, as well as the singular strain terms. The ability of the extended MTSN (EMTSN) criterion has recently been proven by providing accurate prediction of the mixed mode fracture test data obtained for polymethylmethacrylate (PMMA) [3], Polycrystalline graphite [27], and cement mortar [28]. Soda-lime glass (or soda-lime-silica glass) is the most prevalent type of glass, which is used in different industrial applications ranging from windowpanes to glass containers. Because of its brittleness, soda lime glass is exposed to failure and catastrophic fracture particularly in the presence of pre-existing flaws. Crack growth in soda lime glass under various loading conditions 4
has been investigated by many investigators so far. The effect of the indentation load on the fracture resistance of soda lime glass has extensively been explored by the researchers in the past [29 – 33]. Among them, Gong et al. [31] investigated fracture toughness of soda-lime glass using Vickers indentation tests. Abrams et al. [32] studied the fracture strength and multiple cracking behavior of soda-lime silicate glass, and explored the response of this glass type to the contact damage using indented specimens. Crack initiation conditions in soda lime glass under combined loading has also been examined by the researchers [34 – 44]. Using sandwiched beam specimens in water environment, the mixed mode fracture toughness of soda-lime-silica glass have been investigated by Sglavo et al. [34]. Yoda et al. [35] made use of compact tension shear specimens of soda lime glass to study the fracture and subcritical crack growth under mixed mode I/II loading. Li and Sakai [36] studied the mixed mode fracture behavior of soda lime glass using a four point bending technique. However, they reported a considerable discrepancy between their experimental results and theoretical predictions in mode II conditions. Baraki et al. [44] investigated dynamic crack path selection in soda lime glass under external thermal loading using a novel test configuration and measured the mode I fracture toughness of the soda lime glass. They have found that the crack path follows the law of local symmetry (KII = 0), proposed by Gol'Dstein and Salganik [45], for both the quasi-static and dynamic crack propagation. The current paper deals with the evaluation of the mixed mode I/II fracture toughness and the crack initiation angles for the test data on soda lime glass available in the literature. Different mixed mode fracture test data obtained by cracked Brazilian disk specimen are evaluated. Different traditional fracture criteria and an extended version of the MTSN criterion are employed to predict mixed mode I/II crack propagation conditions in soda lime glass. It is shown 5
that the EMTSN criterion which takes into account the effect of T-strain provides better estimates of the test data than all traditional fracture criteria.
2. Fracture criteria 2.1. Crack tip field equations According to Williams [16], the linear elastic stress and strain field equation around the crack tip can be expressed as: = =
√2
, +
√2
√2
, , +
, + . , + √ , / , …
√2
, , + . , , + √ , / , …
(1)
(2)
where , ! ≡ , are polar coordinates with origin at the crack tip, and E is the Young modulus. The first two terms in Eq. (1-2) are singular and depend on the modes I and II stress intensity factors (KI and KII). The parameter T, is the coefficient of the first nonsingular stress (T-stress) or strain (T-strain) terms. The functions ,# and ,# are known functions of and Poisson’s ratio ( ) and are related together by Hooke’s law [3]. The crack tip filed equations can be extended in terms of infinite nonsingular higher order terms. However, the effects of higher order terms are negligible as one approaches the crack tip. 2.2. Traditional fracture criteria In the following, the Strain Energy Density (SED), the Maximum Tangential Stress (MTS), and the Maximum Tangential Strain (MTSN) criteria are briefly described. 6
2.2.1. SED criterion The SED criterion, proposed by Sih [12], is an energy-based fracture criterion which states that crack start to grow in the direction of the minimum value of the strain energy-density factor (Φ), as: %Φ = 0 '( = ) %
% Φ > 0 '( − < ) < %
(3)
(4)
This criterion assumes that crack initiation happens when Φ = Φ- at = ) where Φ- is the critical strain energy-density factor and ) is the direction of the crack initiation. The SED criterion can be expressed in the following form by taking into account only singular terms of the crack tip field equation given in Eq. (1-2): Φ=
1 11 + 2345 − 2346 + 24 782234 − 5 − 19 160
(5)
+ 15 + 11 − 234 + 1 + 2343234 − 16
where, 0 = /821 + ;9 is the shear modulus, and 5 equals to 3 − 4;, and 3 − ;/1 + ; for plane strain and plane stress, respectively. 2.2.2. MTS criterion As a stress-based criterion, the MTS criterion, proposed by Erdogan and Sih [2], assumes that crack initiation occurs from the crack tip in the direction of the maximum tangential stress component (== ), when tangential stress reaches the ultimate tensile strength (> ) of the material. Considering only singular stress terms, the MTS criterion can be formulated as: 7
?
?
A
@A ,1 + AA ,2 B = 0 → )
AD
,1 ) +
AA
AD
,2 ) = 1
(6) (7)
where - is the mode I fracture toughness of the material. 2.2.3. MTSN criterion Originally proposed by Chang [4], the maximum tangential strain criterion postulates that crack initiates from crack tip in direction where the tangential component of the strain field attains its maximum value ( > = > / ). The traditional MTSN criterion only takes into account the singular strain terms and is presented as: ?
?
@A ,1 , + AA ,2 , B = 0 → )
==, , 0 + ==, , 0 = -
(8) (9)
2.3. EMTSN criterion The Extended MTSN (EMTSN) criterion is recently proposed by Mirsayar [3] and includes the first nonsingular strain term (T-strain) as well as the singular strain terms. According to EMTSN criterion [4], a crack starts to grow at a critical distance, rc, from the crack tip in direction, θ0, when the elastic tangential strain reaches the ultimate tensile strain of the material. FG E HH F=
I
JKJL ,=K=M
F = 0, E
NG
HH
F=N
== O , ) = > = > /
I
JKJL ,=K=M
<0
(10) (11)
The critical distance rc, equals to the size of the fracture process zone and is considered as a material property [46]. Physically, the critical distance rc, represents the size of an area around 8
the crack tip which is discontinues and contains micro-sized cracks. The theory of critical distance postulates that crack starts to initiate at the boundary of this area instead of crack tip. Assuming that the critical distance is independent from mode mixity, it can be evaluated as a function of mode I fracture toughness and the ultimate tensile stress as [47]:
O =
1 - 2 2
(12)
By substituting Eq. (2) into Eq. (10) and Eq. (11), the EMTSN criterion is given as,
% 1 S S S E == P , ) + ==, , ) B + . ==, , ) = 0 =0⇒ @ ==, % JKJL ,=K=M R2 O
(13)
⇒ )
> = > =
1
R2 O
@ ==, , ) + ==, , ) B + . ==, , )
F S , ) = E ==, , I where, ==, F=
=K=M
(14)
. The EMTSN criterion can be rewritten in the following
form in terms of normalized parameters as, S S S , ) = 0 ⇒ ) ==, , ) + , ) + VW ==, TUU TUU ==,
==, , ) + ==, , ) = > R2 O − VWTUU ==, , )
(15) (16)
where TUU = R + , W = R2 O /' , and a is the crack length for edge cracked specimens and semi-crack length for centrally cracked specimens. The parameter B is called biaxiality ratio and is defined as [47]:
V=
√ ' TUU
(17)
9
According to Mirsayar [3], the fracture loci for the EMTSN criterion can be presented in ∗ ∗ U /− U /diagram by the following formulation:
U ∗ = U ∗ = -
1 − 1 + VW
==, , ) 81 + D
− YZ1 + 9
\ 81 − 1 + VW9 ==, , ) 81 + D − YZ1 + 9 [
(18)
(19)
where parameters C and H are defined as: D=
==, , ) ==, , )
Y = −VW
==, , ) ==, , )
(20)
(21)
∗ The parameter , called generalized fracture toughness, is defined in Eq. (22) and takes into
consideration the effect of specimen geometry as well as the Poisson’s ratio (see Mirsayar [3] for ∗ ∗ more details about U /− U /diagram). ∗ =
> R2 O 1 − 1 + VW
(22)
3. Cracked Brazilian disk (CBD) specimen The CBD specimen has been employed by Shetty and co-workers [42, 43], and Awaji and Kato [38], separately, to obtain mixed mode fracture toughness of soda lime glass. As shown in Fig. 1, the CBD specimen is made of a centrally cracked disk of radius R subjected to a
10
compressive load F along its diameter. The specimen has a thickness t, and the total length of the central crack equals 2a.
Fig. 1. Scheme of CBD specimen subjected to a diametrally compressive load.
As shown in Fig. 1, one can conduct different mixed mode I/II fracture tests using CBD specimen by altering the loading angle β. For CBD specimens, the stress intensity factors (KI, KII) can be rewritten in normalized forms as:
= ] ^
_ ' Z `(
= ] ^
_ ' Z `(
(23)
(24)
where ] ^ and ] ^ are the normalized forms of KI, KII, and T, respectively. For a fixed √'/` ratio, these geometry factors are only functions of inclination angle β. These parameters can be obtained using finite element simulation of the specimen at different values of β. For the CBD specimens employed by Shetty and co-workers [42, 43], and those of used by Awaji and Kato 11
[38], the normalized crack length (a/R) was 0.25 (R=25mm), and 0.4 (R=20mm), respectively. Plane stress condition is assumed since the specimen thickness was relatively smaller than other dimensions of the CBD specimens in all three sets. A typical finite element mesh used for simulation of the CBD specimens and an expanded view of the crack tip elements are shown in Fig. 2. Due to high stress/strain gradient resulting from the singular stress/strain field near the notch tip, a very fine mesh was used in the region close to the crack tip. The auxiliary asymptotic displacement fields used with J-integral to determine the stress intensity factors as well as the T parameter at different geometry and loading configuration.
Fig. 2. Typical finite element mesh used for modeling the CBD specimens (β=15°)
Using finite element simulation, ] ^ and ] ^ for these geometries can be found for different inclination angle β, as illustrated in Fig. 3. Also shown in Fig. 3 is the variation of the
12
normalized T parameter, V^ =
> √ab cdee
, versus β. The pure mode II happens for a/R = 0.25 and
a/R = 0.4 at β = 28ο and 25ο, respectively.
a)
b)
Fig.3. Normalized crack tip parameters for CBD specimen obtained by finite element simulation for: a) a/R = 0.25, b) a/R = 0.4
By substituting] ^, ] ^ and V^ given in Fig. 3 into the Eq. (17, 23, 24), one can obtain crack initiation angle and onset of fracture using different fracture criteria. 4. Analytical predictions The predictions performed by different fracture criteria are compared and discussed in this section for the reported fracture test data for soda lime glass conducted by the CBD specimens. Fig. 4 shows predictions presented by different traditional fracture criteria (SED, MTS, and MTSN) for the crack initiation direction in soda lime glass tested by Shetty et al. [42]. It is needed to be pointed out that among all three sets of the fracture tests ([38, 42, and 43]), the crack initiation direction is reported only by Shetty et al. [42]. It is seen that all traditional
13
criteria fail to properly predict crack initiation angle in soda lime glass under mixed mode loading and they all overestimate the test data. Among these traditional criteria, the MTSN criterion presents the best prediction at pure mode II (β = 28o). However, like other criteria, the MTSN criterion deviates from the experimental data in mixed mode conditions.
Fig.4. Direction of the crack initiation obtained by Shetty et al. [42], as well as the prediction curves provided by traditional fracture criteria (SED, MTS, and MTSN)
The theoretical prediction curves for SED, MTS, and MTSN criteria are plotted in Fig. 5 for mixed mode fracture toughness of the soda lime glass. Results are plotted as normalized
c
effective fracture toughness (TUU /- ) versus mode mixity parameter (fT = a tanj c k ). The kk
parameter fT represents contribution of each crack tip displacement mode and equals to unity at pure mode I and zero at pure mode II. As depicted in Fig. 3, the crack inclination angles (β) corresponding to the pure mode II conditions are 25o, and 28o for the CBD specimens tested by Awaji and Kato [38], and Shetty and co-workers [42, 43], respectively. Therefore, the mixed mode fracture test data and the theoretical curves are plotted separately in order to clearly show 14
the corresponding crack inclination angle. It is obvious from Fig. 5 that none of the traditional criteria is able to reasonably estimate mixed mode fracture toughness of the soda lime glass. Furthermore, the discrepancy between the mixed mode fracture test data and the theoretical predictions provided by traditional criteria increases as one approaches mode II conditions. a)
b)
Fig.5. Theoretical predictions provided by different traditional criteria for mixed mode fracture test data of soda lime glass tested by a) Awaji and Kato [38], b) Shetty and co-workers [42, 43]
In the current analysis, the ultimate tensile stress of 20.6 MPa, the Young’s modulus of 70GPa, and the Poisson’s ratio of ν = 0.22 are assumed as for soda lime glass [38, 42, 48, 49]. The critical distance of rc = 0.2mm is also selected which is in agreement with the theoretical investigations performed by stress-based criteria [50]. The predictions obtained by EMTSN criterion is plotted in Fig. 6 for the crack initiation angles in soda lime glass tested by Shetty et al. [42]. It can be observed that adding the first nonsingular strain term (T-strain) significantly improves estimates presented by MTSN criterion. Except for
15
pure mode II, the predictions results obtained by EMTSN criterion are remarkably better that traditional MTSN criterion under mixed mode conditions. The mixed mode fracture toughness of soda lime glass tested by Shetty and co-workers [42, 43], and Awaji and Kato [38] are evaluated in Fig. 7 by MTSN and EMTSN criteria. As shown in Fig. 7, the EMTSN presents very accurate evaluation of the current state for the mixed mode fracture toughness of soda lime glass. According to the Fig. 7, the traditional MTSN criterion always underestimates the mixed mode fracture toughness of the soda lime glass, conducted by CBD specimens.
Fig.6. Direction of the crack initiation obtained by Shetty et al. [42], as well as the prediction curves provided by MTSN and EMTSN criteria
16
Fig.7. Theoretical predictions provided by MTSN and EMTSN criteria for mixed mode fracture test data of soda lime glass tested by a) Awaji and Kato [38], b) Shetty and co-workers [42, 43]
Based on the results shown in Figs. 4 – 7, it can be realized that the first non-singular strain term significantly influences on the predictions obtained from strain-based criteria and must be considered in the fracture prediction model. The role of first nonsingular term of the Williams series expansion in the brittle fracture of the engineering materials has widely been investigated in stress-based and energy-based frameworks. However, adding the first nonsingular term to the fracture criterion does not always guarantee for more accurate predictions. As an example, the generalized MTS criterion [47], which revises the traditional MTS criterion by adding the T-stress term, fails to provide accurate predictions near mode II conditions in many cases, and requires adding higher order terms (i.e. second or third nonsingular terms) to achieve an acceptable accuracy [3]. Malikova [51] has recently indicated that at least three or four higher order terms of the Williams series expansion should be considered to present acceptable accuracy by MTS criterion. It is also needed to be pointed out that the role of 17
Poisson’s ratio in the mixed mode fracture behavior is not taken into account in many fracture criteria, including generalized MTS criterion. The effect of Poisson’s ratio on the onset of fracture is taken into account in some energy-based criteria (e.g. SED [12] criterion). However, the energy-based criteria are relatively more complex than stress-based and strain-based criteria in terms of mathematical formulation. Therefore, it can be considered as an advantage for the EMTSN criterion to be able to involve effects of Poisson’s ratio in a relatively simple formulation.
5. Conclusion Mixed mode fracture behavior of soda lime glass was investigated in this paper using different fracture criteria: SED, MTS, MTSN, and EMTSN. Traditional fracture criteria (SED, MTS, and MTSN), which only take into account the effect of singular terms of the strain (stress) field equations, are not able to reasonably predict crack initiation angle and onset of the fracture in soda lime glass. Theoretical predictions for different fracture test data obtained by CBD specimens demonstrate the significant role of T-strain in the mixed mode fracture toughness as well as the crack initiation angle in soda lime glass. Depending on the specimen geometry, the traditional MTSN criterion may underestimate or overestimate the fracture test data. The discrepancies between the MTSN predictions and the test data is resolved when employing EMTSN criterion which takes into account the T-strain as well as the singular strain terms. Since the value of the T-strain depends on the specimen geometry (based on Fig. 3), it is highly recommended to consider effect of T-term when applying any fracture criterion.
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Acknowledgements The research presented in this paper was supported by Zachry Department of Civil Engineering at Texas A&M University. Any opinions, findings, conclusions, and recommendations expressed in this paper are those of the authors alone and do not necessarily reflect the views of the sponsoring agency.
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Highlights •
Mixed mode fracture of soda lime glass is evaluated by different criteria.
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Traditional criteria cannot provide proper predictions of the fracture test data.
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EMTSN criterion presents more accurate predictions than traditional criteria.
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