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Evaluation of crack initiation angle under mixed mode loading at diverse strain rates
Theoretical and Applied Fracture Mechanics 42 (2004) 53–61 www.elsevier.com/locate/tafmec
Evaluation of crack initiation angle under mixed mode loading at diverse strain rates L.H. Herna´ndez-Go´mez *, I. Sauceda-Meza, G. Urriolagoitia-Caldero´n, A.S. Balankin, O. Susarrey Instituto Polite´cnico Nacional, Seccio´n de Estudios de Posgrado e Investigacio´n de la ESIME, Edificio 5, 3er. Piso de la Unidad Profesional Adolfo Lo´pez, Mateos, Col. Lindavista. 07738 Me´xico, DF, Mexico
Abstract Crack initiation angle, under mixed mode loading at several strain rates, is analysed using an experimental–numerical approach. The physical phenomenon for the problem at hand is influenced by the local and global conditions. One of such factors is the strain rate at the crack tip. For this purpose, PMMA plates with centred angled cracks under mixed mode loading were tested. The strain rate at the neighbourhood of the crack tip before crack propagation was evaluated. Considering that this material is strain rate sensitive, the numerical models were calibrated with the modulus of elasticity measured in tension tests at the observed strain rates. Numerical evaluations were performed with the finite element method in conjunction with the volume energy density criterion. An improvement in the evaluation of the crack propagation angle was observed. In order to complete the analysis, the crack initiation angle was also evaluated with the strain energy density factor S, considering the mechanical properties of PMMA, as evaluated at the observed strain rates, and the stress intensity factors k1 and k2. Results are in agreement with those observed experimentally. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Crack propagation stability is an engineering problem that has attracted attention for many years. Much work has been done since the pioneer*
Corresponding author. E-mail address: [email protected] (L.H. Herna´ndezGo´mez).
ing work in [1]. It considered the crack initiation direction for an angled crack which was predicted with the maximum circumferential stress criterion. This work inspired the investigation of several additional factors that are involved in the fracture process. Crack extension stability was studied using the eigenfunction series expansion [2,3]. This approach was also considered in [4] to study the influence of transversely applied stress by
0167-8442/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2004.06.008
54
L.H. Herna´ndez-Go´mez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53–61
including the second term of the series obtained in [2,3]. This was extended for the case of tension and bending [5,6]. Regarding crack path extension, the work in [7] investigated effects caused by the applied load direction, the curvature crack radius and the complete stress and/or energy field. Later, crack growth instability was evaluated numerically [8,9] and experimentally [10]. All this work is based on the single edge notch specimen under biaxial loading. Recently, crack initiation and propagation in beams with edge crack under mixed loading was analysed [11]. This was done by using an energetic approach based on the S-theory [12]. The aforementioned works have shown that there are many factors involved. Some of them are local while others are global. These different factors have to be separated in order to establish their individual contribution to the fracture process. This complicates further when mixed mode loading conditions are developed. In all these cases, quasi-static loading conditions were assumed. However, when the time of load application varies, strain rate conditions are developed at the crack tip. In materials, which are rate sensitive, mechanical properties change with the loading rate. In other words, fracture energy decreases with increasing strain rate. This has been observed by others [13,14]. As a consequence, the evaluation of the crack initiation angle must take into account these factors. Under these considerations, the question being: how will the crack initiation angle be affected under mixed mode loading? This situation is analysed in this work following an experimental–numerical approach. The results are then compared with those obtained by means of the strain energy density factor S [15].
2. Experimental analysis The experimental work has been divided in two parts. The first part evaluates the PMMA modulus of elasticity at different strain rates and the second part evaluates the crack initiation angle. The objective is to first evaluate the mechanical properties under tensile load at the observed strain rates at the crack tip when crack initiation takes place.
Table 1 PMMA modulus of elasticity at different strain rates Strain rate (s1)
Modulus of elasticity (GPa)
Ref.
0.00001 0.0001 0.008 0.065 0.290 20 35
2.6 2.9 2.9 3.4 3.9 3.8 4.3
[16] [16] This work This work This work [17] [17]
This was done with an 8502 Instron machine with several loading rates. Appropriate tension specimens were tested. Table 1 summarises this data. For the comparison of this data, it has to be borne in mind, atleast, two points, namely (1) the compliance of the testing machine and (2) the batch variation characteristics of the raw material. These factors were minimized as much as possible by making direct readings of the specimen deformation on the gage length of the specimen with a strain gage. Besides, all the specimens were produced from the same sheet. For the analysis of crack propagation direction, PMMA plates with central angled cracks were loaded under tension, along its major axis of symmetry. The crack angle with respect to the horizontal plane was set to be 0°, 30°, 45° and 60°. Fig. 1 shows the dimensions of the specimens. In all these cases, the strain field was evaluated with a strain gage located at the crack tip neighbourhood. Specimens were tested with a 8502 Instron machine with the following loading rates: 300 and 3000 mm/min and the strain field variation was recorded with a SYSTEM-6000-6100 of Vishay of Measurement Group, Inc. Reproducibility of the results was checked, by testing 10 specimens with the same geometrical conditions. It is important to keep in mind that, it is difficult to establish the strain rate at the very crack tip. So, an average evaluation was obtained with a 3 mm strain gage located as close as possible to the very crack tip. The case of 0° is well known and the results were used to validate the experimental and numerical results. A typical strain rate history is shown in Fig. 2. This corresponds to a specimen tested at 3000 mm/min with an horizontal crack. As it can
L.H. Herna´ndez-Go´mez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53–61
55
Fig. 1. Dimension of the cracked specimens and strain gage location. In all cases the thickness is 6.35 mm. (a) 0°, (b) 30°, (c) 45° and (d) 60°.
be seen, the strain rate varies as the loading process takes place. Crack initiation is well defined with a curve peak.
3. Numerical analysis The numerical work has been divided in three parts: (1) Model development, (2) Model calibration and (3) Evaluation of crack initiation direction. More specifically, FRANC code [18] was used for this purpose. A finite element model with 4300 nodes was generated. This mesh has rectangular elements of eight nodes and eight quarter point elements were used at the crack tip. The whole plate was simulated because there no sym-
metry prevails when angled cracks are introduced. Crack initiation angle was evaluated with the energy density criterion. In this case, KI and KII are required for these calculations. In order to validate FRANC for this purpose, these parameters were evaluated under quasi-static loading with the same geometry of the cracked specimens. They were loaded quasi-statically in the same range. The numerical results were compared with those obtained with photoelasticity, using polycarbonate specimens. Convergence between the experimental and numerical results was obtained. This suggests that FRANC is adequate for such calculations. In a second step, the model calibration was done. Its purpose was to obtain the numerical reproduction of the strain rate recorded by the
56
L.H. Herna´ndez-Go´mez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53–61
Fig. 2. Strain rate at the crack tip neighbourhood of a specimen with an horizontal crack, and tested with a loading rate of 3000 mm/min.
strain gages bonded near the crack tip. All the calculations were performed with the same loading rate that was applied by the testing machine. Besides, in order to make a comparison, another finite element analysis was done with the typical modulus of elasticity under quasi-static loading (2.9 GPa) commonly reported in the open literature, as in [16]. In Fig. 3, it is shown the typical
divergence between these results. This situation was improved when the value of the modulus of elasticity at the observed strain rate, during the loading process, was introduced in the calculations. As shown in Fig. 4, both results are in agreement. In the third step, the fracture analysis of the centred angled crack plate, loaded under tension,
Fig. 3. Divergence between the numerical evaluation of the strain rate under quasi-static conditions and the experimental results.
L.H. Herna´ndez-Go´mez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53–61
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Fig. 4. Converge between numerical and experimental evaluation of the strain rate at the crack tip neighbourhood, (a) 300 mm/min, (b) 3000 mm/min.
was done. KI and KII were evaluated. In order to make a comparison, this evaluation was made with the mechanical properties at the observed strain rates and under quasi-static loading. In the last case, a modulus of elasticity, equals to 2.9 GPa was also taken into account. Crack initiation angle was evaluated in accordance with the energy criterion with the following equation: 2K I K II h ¼ arc tan 2 ð1Þ K I þ K 2II The results are summarised in Table 2.
4. Analysis with the strain energy density concept Another way to evaluate the crack initiation angle is with the fracture theory based on the field strength of the local strain energy density proposed in [15]. In this case, the energy release rate is not required and mixed mode crack extension problems may be treated. The fundamental parameter of this theory, the strain energy density factor S, is direction sensitive and it is evaluated in two dimensions with the following relation:
L.H. Herna´ndez-Go´mez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53–61
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Table 2 Crack initiation angle evaluated with the energy criterion under quasi-static and dynamic loading Crack angle (°)
Load speed (mm/min)
KI quasi-static (MPa m1/2)
KI observed strain rate (MPa m1/2)
KII quasi-static (MPa m1/2)
KII observed strain rate (MPa m1/2)
Crack initiation angle quasi-static (°)
Crack initiation angle at observed strain rate (°)
0 0 30 30 45 45 60 60
300 3000 300 3000 300 3000 300 3000
1.811 1.811 1.209 1.209 0.8429 0.8429 0.2079 0.2079
1.699 1.634 1.240 1.190 0.842 0.840 0.2093 0.2069
0.005411 0.005411 0.4140 0.4140 0.8202 0.8202 1.2935 1.2935
0.005081 0.005363 0.3909 0.3929 0.8437 0.8444 1.3900 1.4912
0.342 0.342 31.50 31.50 44.98 44.99 62.39 62.39
0.342 0.376 29.83 30.76 44.99 45.00 61.40 60.22
S ¼ a11 k 21 þ 2a12 k 1 k 2 þ a22 k 22
ð2Þ
where k1 and k2 are the stress intensity factors under loading mode I and II, respectively. These parameters are related to the energy release rate, in generalized plane stress, there results GI ¼ ðpk 21 Þ=E and GII ¼ ðpk 22 Þ=E. The coefficients aij (i, j = 1, 2) are given by 1 ½ð3 4m cos hÞð1 þ cos hÞ 16l 1 2 sin h½cos h ð1 2mÞ a12 ¼ 16l 1 ½4ð1 mÞð1 cos hÞ a22 ¼ 16l þ ð1 þ cos hÞð3 cos h 1Þ
a11 ¼
ð3Þ Fig. 5. Parameters involved in the calculation of the crack initiation angle.
where h is the polar angle, which varies around the crack tip, m is the PoissonÕs ratio and l is the shear modulus of elasticity. Fig. 5 shows the parameters involved in the crack initiation angle evaluation at the crack tip neighbourhood. Once S is established, crack initiation will take place in a radial direction h, from the crack tip, along which the strain energy density is minimum. Hence the crack initiation angle determined from
ulus is strain rate dependent, therefore it was calculated from the modulus of elasticity observed during the experimental tests. The results obtained are shown in Table 3 and they are compared with those obtained with the maximum circumferential stress criterion. In all the cases reported in Table 3, the load speed was 3000 mm/min. These are the situations, which are more sensitive to the strain rate.
oS ¼0 oh
5. Discussion of the results
ð4Þ
In this work, the crack inclination angle is taken into account in the calculations by means of the values of the SIF k1 and k2, because their values are function of the orientation of the crack plane. These parameters were calculated numerically with the finite element method. Besides, the shear mod-
The tension tests have shown that PMMA becomes more rigid as the strain rate increases. This is reflected with the increment of the modulus of elasticity as the strain rate growths. Comparing KI and KII evaluations at the observed strain rates
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Table 3 Crack initiation angle evaluated with the strain energy density factor S and the maximum circumferential stress criterion Crack inclination angle
0° 30° 45° 60°
Crack initiation angle This work
Strain energy density concept
Maximum circumferential stress
0.376° 30.76° 45° 60.22°
0° 31.2° 46.5° 62.7°
0° 26.346° 45.207° 62.618°
with those made under quasi-static conditions (Table 2) a divergence of results is found. In fact, the biggest difference of KI values is for angles lesser than 45°. This is the range in which KI is a dominant factor in the fracture process. On the other hand, the upper divergence of KII lies on the range of angles greater than 45°. This is the case in which this parameter plays an important role in crack direction. In the case of the specimen with an horizontal crack, KII should be zero. Nonetheless, the numer-
ical results show that this value is small. This can be explained by the fact that the whole plate was modelled and some nodes may not lay on the crack plane, especially with those that are used in the fracture parameters calculation. Regarding the crack initiation angle, it is nearly 0°. This was expected. In spite of these differences, it can be considered that the numerical evaluations are in line with reality. With respect to the other cases, the specimens have been made in such a way that the crack runs along the horizontal plane, because
Fig. 6. Fracture surface of a 60° cracked specimen tested at 3 mm/min.
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L.H. Herna´ndez-Go´mez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53–61
Fig. 7. Fracture surface of a 60° cracked specimen tested at 3000 mm/min.
it is perpendicular to the applied tension load. This situation is observed when all the calculations are made with the parameters obtained with the calibrated model at the observed strain rate. When quasi-static conditions are applied it can be seen there is some divergence. The crack surfaces were microscopically analysed. Typical results are shown in Figs. 6 and 7, in which the real image is magnified 35 times. In both cases, the initial crack front is on the top of each picture. This is the limit between the mark left by tool that completed the crack front, and the crack surface. In the last area, different sort of marks are depicted. In the case of the specimen tested at low strain rate (Fig. 6), the crack surface close to the crack front has many ribs. On the other hand, the crack loaded with the biggest strain rate (Fig. 7), has a surface which is ‘‘clean’’. This reflects that the latter has used less energy. As the crack propagates, there is a bright region and at a far distance from the crack front the surface becomes rough. Nonetheless, in the case where a
lower strain rate was applied (Fig. 6), there are deeper marks. In other words, as the strain rate increases, the crack surface is smoother. It is important to mention that this evaluation reflects the fact that less fracture energy is required as the strain rate increases. To complete the discussion, the results with the highest strain rate were compared with those obtained with the strain energy density factor S and the maximum circumferential stress. As it can be seen, the results of this work and those obtained with the S factor are in agreement. Besides, the case of the horizontal crack (0°), the crack initiation angle is 0°, as it was expected. In accordance with the authorÕs opinion, two points have to be kept in mind when this approach is followed. Namely, (1) the material mechanical properties at the observed strain rates are easily introduced in the calculations by means of the coefficients a11, a12 and a22 and (2) the SIFÕs have to be evaluated as accurate as possible. These should lead to an exact evaluation of the crack angle initiation under
L.H. Herna´ndez-Go´mez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53–61
mixed loading mode, as the material is strain rate dependent. Finally, the work in [15] has shown that the predictions based on the maximum stress criterion do deviate from those obtained by the strain energy density criterion. The results reported in Table 3 are in agreement with this statement.
6. Conclusions The accurate evaluation of the stress intensity factors of a crack under mixed mode loading is relevant in order to establish the crack initiation angle. However, when the strain rate increases at the crack tip, a quasi-static evaluation may no be valid. Under this scheme, the crack direction angle of propagation calculated varies in relation with the one observed in quasi-static conditions. This is confirmed with the values of fracture parameters, such as KI and KII. The calculations are improved when the changes of mechanical properties with strain rate are taken into account. This situation is also valid when the strain energy density factor S is applied. The crack initiation angle was calculated along the radial direction on which the strain energy density is a minimum. The obtained results are in agreement with those observed experimentally. Also, this confirms the fact that the stationary value of Smin can be used as an intrinsic material parameter, and from this a mixed mode fracture criterion can be stated. Besides, the expected divergence with maximum circumferential stress was also observed.
Acknowledgments The grant 34950-U awarded by Consejo Nacional de Ciencia y Tecnologı´a and the support given to this project by COFAA and CGEPI of Instituto Polite´cnico Nacional are grateful acknowledged. Also, the authors thank Mr. Ca´ndido Zamora for the final numerical calculations.
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References [1] F. Erdogan, G.C. Sih, On crack extension in plates under plane loading and transverse shear, J. Basic Eng. Trans. ASME 85(D) (1963) 519–527. [2] B. Cotterell, On brittle fracture paths, Int. J. Fract. Mech. 1 (1965) 96–103. [3] B. Cotterell, Notes on the paths and stability of cracks, Int. J. Fract. Technol. 2 (1966) 526–533. [4] J.G. Williams, P.D. Ewing, Fracture under complex stress: The angled crack problem, Int. J. Fract. Mech. 8 (1972) 441–446. [5] I. Finnie, A. Saith, A note on the angled crack problem and directional stability of a crack, Int. J. Fract. 9 (1973) 484–496. [6] P.D. Ewing, J.L. Swedlow, Further results on the angled crack problem, Int. J. Fract. 12 (1976) 85–93. [7] M.E. Kipp, G.C. Sih, The strain energy density failure criterion applied to notched elastic solids, Int. J. Solids Struct. 11 (1975) 153–173. [8] G. Urriolagoitia-Caldero´n, L.H. Herna´ndez-Go´mez, Evaluation of crack propagation stability with the Williams stress function. Part I, Int. J. Comput. Struct. 61 (4) (1996) 775–780. [9] G. Urriolagoitia-Caldero´n, L.H. Herna´ndez-Go´mez, Evaluation of crack propagation stability with the Williams stress function. Part II, Int. J. Comput. Struct. 63 (5) (1997) 1007–1014. [10] G. Urriolagoitia-Caldero´n, L.H. Herna´ndez Go´mez, Experimental analysis of crack propagation stability in single edge notch specimens, Theor. Appl. Fract. Mech. 28 (1997) 57–68. [11] L. Nobile, Mixed mode crack initiation and direction in beams with edge crack, Theor. Appl. Fract. Mech. 33 (2000) 107–116. [12] G.C. Sih, Strain density factor applied to mixed mode crack problems, Int. J. Fract. 10 (1974) 305–321. [13] G.C. Sih, D.Y. Tzou, Dynamic fracture rate of Charpy Vnotch specimen, Theor. Appl. Fract. Mech. 5 (1986) 189– 203. [14] L.H. Herna´ndez-Go´mez, C. Ruiz, Assessment of data for dynamic crack initiation under shock pressure loading: Part I––Experiment, Theor. Appl. Fract. Mech. 19 (1993) 75–83. [15] G.C. Sih, Mechanics of Fracture Initiation and Propagation, Kluwer Academic Publishers, Dordrecht, 1991, pp. 3– 13. [16] I.C.I. The properties of ‘‘Pespex’’ acrylic materials, Plastics Division, Welwyn. [17] L.H. Herna´ndez-Go´mez, C. Ruiz, Assessment of data for dynamic crack initiation under shock pressure loading: Part II––Analysis, Theor. Appl. Fract. Mech. 19 (1993) 85– 94. [18] D. Swenson, J. Mark, FRANC2D/L: A crack propagation simulator for plane layered structures, version 1.3 UserÕs guide, Kansas State University.
Evaluation of crack initiation angle under mixed mode loading at diverse strain rates L.H. Herna´ndez-Go´mez *, I. Sauceda-Meza, G. Urriolagoitia-Caldero´n, A.S. Balankin, O. Susarrey Instituto Polite´cnico Nacional, Seccio´n de Estudios de Posgrado e Investigacio´n de la ESIME, Edificio 5, 3er. Piso de la Unidad Profesional Adolfo Lo´pez, Mateos, Col. Lindavista. 07738 Me´xico, DF, Mexico
Abstract Crack initiation angle, under mixed mode loading at several strain rates, is analysed using an experimental–numerical approach. The physical phenomenon for the problem at hand is influenced by the local and global conditions. One of such factors is the strain rate at the crack tip. For this purpose, PMMA plates with centred angled cracks under mixed mode loading were tested. The strain rate at the neighbourhood of the crack tip before crack propagation was evaluated. Considering that this material is strain rate sensitive, the numerical models were calibrated with the modulus of elasticity measured in tension tests at the observed strain rates. Numerical evaluations were performed with the finite element method in conjunction with the volume energy density criterion. An improvement in the evaluation of the crack propagation angle was observed. In order to complete the analysis, the crack initiation angle was also evaluated with the strain energy density factor S, considering the mechanical properties of PMMA, as evaluated at the observed strain rates, and the stress intensity factors k1 and k2. Results are in agreement with those observed experimentally. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Crack propagation stability is an engineering problem that has attracted attention for many years. Much work has been done since the pioneer*
Corresponding author. E-mail address: [email protected] (L.H. Herna´ndezGo´mez).
ing work in [1]. It considered the crack initiation direction for an angled crack which was predicted with the maximum circumferential stress criterion. This work inspired the investigation of several additional factors that are involved in the fracture process. Crack extension stability was studied using the eigenfunction series expansion [2,3]. This approach was also considered in [4] to study the influence of transversely applied stress by
0167-8442/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2004.06.008
54
L.H. Herna´ndez-Go´mez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53–61
including the second term of the series obtained in [2,3]. This was extended for the case of tension and bending [5,6]. Regarding crack path extension, the work in [7] investigated effects caused by the applied load direction, the curvature crack radius and the complete stress and/or energy field. Later, crack growth instability was evaluated numerically [8,9] and experimentally [10]. All this work is based on the single edge notch specimen under biaxial loading. Recently, crack initiation and propagation in beams with edge crack under mixed loading was analysed [11]. This was done by using an energetic approach based on the S-theory [12]. The aforementioned works have shown that there are many factors involved. Some of them are local while others are global. These different factors have to be separated in order to establish their individual contribution to the fracture process. This complicates further when mixed mode loading conditions are developed. In all these cases, quasi-static loading conditions were assumed. However, when the time of load application varies, strain rate conditions are developed at the crack tip. In materials, which are rate sensitive, mechanical properties change with the loading rate. In other words, fracture energy decreases with increasing strain rate. This has been observed by others [13,14]. As a consequence, the evaluation of the crack initiation angle must take into account these factors. Under these considerations, the question being: how will the crack initiation angle be affected under mixed mode loading? This situation is analysed in this work following an experimental–numerical approach. The results are then compared with those obtained by means of the strain energy density factor S [15].
2. Experimental analysis The experimental work has been divided in two parts. The first part evaluates the PMMA modulus of elasticity at different strain rates and the second part evaluates the crack initiation angle. The objective is to first evaluate the mechanical properties under tensile load at the observed strain rates at the crack tip when crack initiation takes place.
Table 1 PMMA modulus of elasticity at different strain rates Strain rate (s1)
Modulus of elasticity (GPa)
Ref.
0.00001 0.0001 0.008 0.065 0.290 20 35
2.6 2.9 2.9 3.4 3.9 3.8 4.3
[16] [16] This work This work This work [17] [17]
This was done with an 8502 Instron machine with several loading rates. Appropriate tension specimens were tested. Table 1 summarises this data. For the comparison of this data, it has to be borne in mind, atleast, two points, namely (1) the compliance of the testing machine and (2) the batch variation characteristics of the raw material. These factors were minimized as much as possible by making direct readings of the specimen deformation on the gage length of the specimen with a strain gage. Besides, all the specimens were produced from the same sheet. For the analysis of crack propagation direction, PMMA plates with central angled cracks were loaded under tension, along its major axis of symmetry. The crack angle with respect to the horizontal plane was set to be 0°, 30°, 45° and 60°. Fig. 1 shows the dimensions of the specimens. In all these cases, the strain field was evaluated with a strain gage located at the crack tip neighbourhood. Specimens were tested with a 8502 Instron machine with the following loading rates: 300 and 3000 mm/min and the strain field variation was recorded with a SYSTEM-6000-6100 of Vishay of Measurement Group, Inc. Reproducibility of the results was checked, by testing 10 specimens with the same geometrical conditions. It is important to keep in mind that, it is difficult to establish the strain rate at the very crack tip. So, an average evaluation was obtained with a 3 mm strain gage located as close as possible to the very crack tip. The case of 0° is well known and the results were used to validate the experimental and numerical results. A typical strain rate history is shown in Fig. 2. This corresponds to a specimen tested at 3000 mm/min with an horizontal crack. As it can
L.H. Herna´ndez-Go´mez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53–61
55
Fig. 1. Dimension of the cracked specimens and strain gage location. In all cases the thickness is 6.35 mm. (a) 0°, (b) 30°, (c) 45° and (d) 60°.
be seen, the strain rate varies as the loading process takes place. Crack initiation is well defined with a curve peak.
3. Numerical analysis The numerical work has been divided in three parts: (1) Model development, (2) Model calibration and (3) Evaluation of crack initiation direction. More specifically, FRANC code [18] was used for this purpose. A finite element model with 4300 nodes was generated. This mesh has rectangular elements of eight nodes and eight quarter point elements were used at the crack tip. The whole plate was simulated because there no sym-
metry prevails when angled cracks are introduced. Crack initiation angle was evaluated with the energy density criterion. In this case, KI and KII are required for these calculations. In order to validate FRANC for this purpose, these parameters were evaluated under quasi-static loading with the same geometry of the cracked specimens. They were loaded quasi-statically in the same range. The numerical results were compared with those obtained with photoelasticity, using polycarbonate specimens. Convergence between the experimental and numerical results was obtained. This suggests that FRANC is adequate for such calculations. In a second step, the model calibration was done. Its purpose was to obtain the numerical reproduction of the strain rate recorded by the
56
L.H. Herna´ndez-Go´mez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53–61
Fig. 2. Strain rate at the crack tip neighbourhood of a specimen with an horizontal crack, and tested with a loading rate of 3000 mm/min.
strain gages bonded near the crack tip. All the calculations were performed with the same loading rate that was applied by the testing machine. Besides, in order to make a comparison, another finite element analysis was done with the typical modulus of elasticity under quasi-static loading (2.9 GPa) commonly reported in the open literature, as in [16]. In Fig. 3, it is shown the typical
divergence between these results. This situation was improved when the value of the modulus of elasticity at the observed strain rate, during the loading process, was introduced in the calculations. As shown in Fig. 4, both results are in agreement. In the third step, the fracture analysis of the centred angled crack plate, loaded under tension,
Fig. 3. Divergence between the numerical evaluation of the strain rate under quasi-static conditions and the experimental results.
L.H. Herna´ndez-Go´mez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53–61
57
Fig. 4. Converge between numerical and experimental evaluation of the strain rate at the crack tip neighbourhood, (a) 300 mm/min, (b) 3000 mm/min.
was done. KI and KII were evaluated. In order to make a comparison, this evaluation was made with the mechanical properties at the observed strain rates and under quasi-static loading. In the last case, a modulus of elasticity, equals to 2.9 GPa was also taken into account. Crack initiation angle was evaluated in accordance with the energy criterion with the following equation: 2K I K II h ¼ arc tan 2 ð1Þ K I þ K 2II The results are summarised in Table 2.
4. Analysis with the strain energy density concept Another way to evaluate the crack initiation angle is with the fracture theory based on the field strength of the local strain energy density proposed in [15]. In this case, the energy release rate is not required and mixed mode crack extension problems may be treated. The fundamental parameter of this theory, the strain energy density factor S, is direction sensitive and it is evaluated in two dimensions with the following relation:
L.H. Herna´ndez-Go´mez et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 53–61
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Table 2 Crack initiation angle evaluated with the energy criterion under quasi-static and dynamic loading Crack angle (°)
Load speed (mm/min)
KI quasi-static (MPa m1/2)
KI observed strain rate (MPa m1/2)
KII quasi-static (MPa m1/2)
KII observed strain rate (MPa m1/2)
Crack initiation angle quasi-static (°)
Crack initiation angle at observed strain rate (°)
0 0 30 30 45 45 60 60
300 3000 300 3000 300 3000 300 3000
1.811 1.811 1.209 1.209 0.8429 0.8429 0.2079 0.2079
1.699 1.634 1.240 1.190 0.842 0.840 0.2093 0.2069
0.005411 0.005411 0.4140 0.4140 0.8202 0.8202 1.2935 1.2935
0.005081 0.005363 0.3909 0.3929 0.8437 0.8444 1.3900 1.4912
0.342 0.342 31.50 31.50 44.98 44.99 62.39 62.39
0.342 0.376 29.83 30.76 44.99 45.00 61.40 60.22
S ¼ a11 k 21 þ 2a12 k 1 k 2 þ a22 k 22
ð2Þ
where k1 and k2 are the stress intensity factors under loading mode I and II, respectively. These parameters are related to the energy release rate, in generalized plane stress, there results GI ¼ ðpk 21 Þ=E and GII ¼ ðpk 22 Þ=E. The coefficients aij (i, j = 1, 2) are given by 1 ½ð3 4m cos hÞð1 þ cos hÞ 16l 1 2 sin h½cos h ð1 2mÞ a12 ¼ 16l 1 ½4ð1 mÞð1 cos hÞ a22 ¼ 16l þ ð1 þ cos hÞð3 cos h 1Þ
a11 ¼
ð3Þ Fig. 5. Parameters involved in the calculation of the crack initiation angle.
where h is the polar angle, which varies around the crack tip, m is the PoissonÕs ratio and l is the shear modulus of elasticity. Fig. 5 shows the parameters involved in the crack initiation angle evaluation at the crack tip neighbourhood. Once S is established, crack initiation will take place in a radial direction h, from the crack tip, along which the strain energy density is minimum. Hence the crack initiation angle determined from
ulus is strain rate dependent, therefore it was calculated from the modulus of elasticity observed during the experimental tests. The results obtained are shown in Table 3 and they are compared with those obtained with the maximum circumferential stress criterion. In all the cases reported in Table 3, the load speed was 3000 mm/min. These are the situations, which are more sensitive to the strain rate.
oS ¼0 oh
5. Discussion of the results
ð4Þ
In this work, the crack inclination angle is taken into account in the calculations by means of the values of the SIF k1 and k2, because their values are function of the orientation of the crack plane. These parameters were calculated numerically with the finite element method. Besides, the shear mod-
The tension tests have shown that PMMA becomes more rigid as the strain rate increases. This is reflected with the increment of the modulus of elasticity as the strain rate growths. Comparing KI and KII evaluations at the observed strain rates
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Table 3 Crack initiation angle evaluated with the strain energy density factor S and the maximum circumferential stress criterion Crack inclination angle
0° 30° 45° 60°
Crack initiation angle This work
Strain energy density concept
Maximum circumferential stress
0.376° 30.76° 45° 60.22°
0° 31.2° 46.5° 62.7°
0° 26.346° 45.207° 62.618°
with those made under quasi-static conditions (Table 2) a divergence of results is found. In fact, the biggest difference of KI values is for angles lesser than 45°. This is the range in which KI is a dominant factor in the fracture process. On the other hand, the upper divergence of KII lies on the range of angles greater than 45°. This is the case in which this parameter plays an important role in crack direction. In the case of the specimen with an horizontal crack, KII should be zero. Nonetheless, the numer-
ical results show that this value is small. This can be explained by the fact that the whole plate was modelled and some nodes may not lay on the crack plane, especially with those that are used in the fracture parameters calculation. Regarding the crack initiation angle, it is nearly 0°. This was expected. In spite of these differences, it can be considered that the numerical evaluations are in line with reality. With respect to the other cases, the specimens have been made in such a way that the crack runs along the horizontal plane, because
Fig. 6. Fracture surface of a 60° cracked specimen tested at 3 mm/min.
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Fig. 7. Fracture surface of a 60° cracked specimen tested at 3000 mm/min.
it is perpendicular to the applied tension load. This situation is observed when all the calculations are made with the parameters obtained with the calibrated model at the observed strain rate. When quasi-static conditions are applied it can be seen there is some divergence. The crack surfaces were microscopically analysed. Typical results are shown in Figs. 6 and 7, in which the real image is magnified 35 times. In both cases, the initial crack front is on the top of each picture. This is the limit between the mark left by tool that completed the crack front, and the crack surface. In the last area, different sort of marks are depicted. In the case of the specimen tested at low strain rate (Fig. 6), the crack surface close to the crack front has many ribs. On the other hand, the crack loaded with the biggest strain rate (Fig. 7), has a surface which is ‘‘clean’’. This reflects that the latter has used less energy. As the crack propagates, there is a bright region and at a far distance from the crack front the surface becomes rough. Nonetheless, in the case where a
lower strain rate was applied (Fig. 6), there are deeper marks. In other words, as the strain rate increases, the crack surface is smoother. It is important to mention that this evaluation reflects the fact that less fracture energy is required as the strain rate increases. To complete the discussion, the results with the highest strain rate were compared with those obtained with the strain energy density factor S and the maximum circumferential stress. As it can be seen, the results of this work and those obtained with the S factor are in agreement. Besides, the case of the horizontal crack (0°), the crack initiation angle is 0°, as it was expected. In accordance with the authorÕs opinion, two points have to be kept in mind when this approach is followed. Namely, (1) the material mechanical properties at the observed strain rates are easily introduced in the calculations by means of the coefficients a11, a12 and a22 and (2) the SIFÕs have to be evaluated as accurate as possible. These should lead to an exact evaluation of the crack angle initiation under
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mixed loading mode, as the material is strain rate dependent. Finally, the work in [15] has shown that the predictions based on the maximum stress criterion do deviate from those obtained by the strain energy density criterion. The results reported in Table 3 are in agreement with this statement.
6. Conclusions The accurate evaluation of the stress intensity factors of a crack under mixed mode loading is relevant in order to establish the crack initiation angle. However, when the strain rate increases at the crack tip, a quasi-static evaluation may no be valid. Under this scheme, the crack direction angle of propagation calculated varies in relation with the one observed in quasi-static conditions. This is confirmed with the values of fracture parameters, such as KI and KII. The calculations are improved when the changes of mechanical properties with strain rate are taken into account. This situation is also valid when the strain energy density factor S is applied. The crack initiation angle was calculated along the radial direction on which the strain energy density is a minimum. The obtained results are in agreement with those observed experimentally. Also, this confirms the fact that the stationary value of Smin can be used as an intrinsic material parameter, and from this a mixed mode fracture criterion can be stated. Besides, the expected divergence with maximum circumferential stress was also observed.
Acknowledgments The grant 34950-U awarded by Consejo Nacional de Ciencia y Tecnologı´a and the support given to this project by COFAA and CGEPI of Instituto Polite´cnico Nacional are grateful acknowledged. Also, the authors thank Mr. Ca´ndido Zamora for the final numerical calculations.
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