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Crack tip stress fields revealed by infrared photoelasticity in silicon crystals
Materials Science and Engineering A 387–389 (2004) 377–380
Crack tip stress fields revealed by infrared photoelasticity in silicon crystals Kenji Higashida∗ , Masaki Tanaka, Ena Matsunaga, Hironori Hayashi Department of Materials Science and Engineering, Faculty of Engineering, Kyushu University, 6-10-1 Higashi-ku, Fukuoka 812-8581, Japan Received 25 August 2003; received in revised form 9 December 2003
Abstract Stress fields around a crack tip in silicon crystals have been investigated by using infrared photoelasticity with the aim of clarifying the shielding effect due to crack tip dislocations on the steep increase of fracture toughness in the brittle-to-ductile transition (BDT). First, compact tension tests were carried out at room temperature to make in situ observation of elastic behavior of crack tip stress fields. The photoelastic images observed were in good agreement with those simulated for the usual elastic fields around the tip of a mode I crack. Next, to clarify the stress modification due to crack tip plasticity, three-point bending tests were also made by using notched specimens at high temperatures around 1000 K. After the high temperature test, in spite of the absence of the applied load, residual bright images were observed around the notch. Those images correspond to an internal stress due to dislocations multiplied around the notch, and they have an effect of shielding (accommodating) the stress concentration due to the applied load. The fracture toughness at room temperature was increased by the introduction of the residual stress. © 2004 Elsevier B.V. All rights reserved. Keywords: Brittle-to-ductile transition; Dislocations; Crack; Fracture toughness; Shielding
1. Introduction Dislocations introduced around a crack tip by an applied load contribute to a reduction in the local stress intensity factor by producing a back stress toward the crack tip. Such modification of stress state around a crack tip due to dislocations has been theoretically treated in terms of “crack tip shielding” [1–5]. Silicon crystals have attracted much attention as a model substance to investigate the fundamental mechanism of brittle-to-ductile transition (BDT). One of the most essential arguments for the BDT process is on the multiplication of crack tip dislocations and their shielding effect which has been considered to be the main cause of the steep increase of fracture toughness in the BDT [6–8]. However, observations or experimental analyses for such phenomena have been still limited [9–11], and further investigation is necessary to corroborate the shielding effect and its contribution to the increase of fracture toughness.
∗
Corresponding author. Tel.: +81 92 642 3660; fax: +81 92 642 3660. E-mail address: [email protected] (K. Higashida).
0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2004.05.058
In the present study, crack tip stress fields in silicon crystals have been investigated by using infrared photoelasticity and its image simulation. Particular emphasis is laid on the internal stress field generated by dislocations introduced at high temperatures. On the basis of these observations, the effect of crack tip shielding and its contribution to the BDT is discussed.
2. Experimental procedures Silicon single crystals of p-type wafers commercially available were employed. Compact tension (CT) specimens were cut out from {1 1 1} silicon wafers (Fig. 1(a)). Tensile tests of the CT specimens were made at room temperature to observe elastic field around a crack tip. Three-point bending tests of notched specimens were also made at high temperatures around 1000 K. The orientation of those specimens for high temperature tests is illustrated in Fig. 1(b). Silicon crystals are transparent for infrared light whose wavelength is longer than 1.2 m. Photoelastic observations were made using crossed Nicols under the respective conditions of β = 0 and β = /4, where β is the angle between
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Here, I0 is the intensity of incident polarized light, δ is the phase difference between the two transmitted lights due to birefringence, φ is the angle between the crack plane and the principal direction of stress state. The phase difference δ is expressed as 1/2 2 σxy 2π 2 δ= qd + 4σxy , (2) λ σyy − σxx
Fig. 1. (a) CT specimen, (b) specimen for three-point bending tests.
Fig. 2. Coordinates associated with the crack. β defines the direction of the polarized light vector.
the incident polarized light vector and the crack plane (see Fig. 2). The intensity of transmitted light of polarized light microscope, IT , is described by the following equation [12]. δ IT = I0 sin2 (1) sin2 (2β + 2Φ). 2
where q is the photoelastic constant, d is the specimen thickness, and λ is the wave length. In this equation, the stress components σ xx , σ yy , and σ xy are defined by the coordinate shown in Fig. 2. The angle φ in Eq. (1) is expressed as 2σxy 1 −1 Φ= tan . (3) 2 (σyy − σxx ) From Eqs. (1) and (3), it is understood that the intensity of transmitted light IT increases with increasing the value of {σ xy /(σ yy − σ xx )} under the condition of β = 0. Therefore, this condition is favorable to observe the shear stress component σ xy . On the other hand, the condition of β = /4 was employed to observe the stress component σ yy normal to the crack plane such as tensile or compressive stress field. Because, under this condition, the intensity IT becomes large when the value of {σ xy /(σ yy − σ xx )} approaches to zero.
3. Results and discussion Fig. 3(a)–(c) shows photoelastic images obtained at room temperature under the condition of β = 0, /8, and /4, respectively. Applied stress intensity factor is around
Fig. 3. Observed (a–c) and computed (d–f) photoelastic images of mode I crack tip field in a Si crystal. (a) and (d) are obtained under the condition of β = 0, (b) and (e) are under β = /8, (c) and (f) are under β = /4.
K. Higashida et al. / Materials Science and Engineering A 387–389 (2004) 377–380
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0.3 MPa m1/2 . Image simulation was also made assuming that the images observed are formed only by the applied KI field. The computed images are shown in Fig. 3(d) (β = 0), (e) (β = /8), and (f) (/4), which are in good agreement with the observed images. In these figures, simple white broad images are visible around the crack tip, and these images became larger with increasing the external tensile load. When the external load was removed, they disappeared and no residual images were seen around the crack tip. At room temperature, crack extension in silicon crystals occurs in perfectly brittle manner, and it was also confirmed by TEM that no dislocations were introduced during crack extension at room temperature [13]. Thus, the images observed in Fig. 3 correspond to the simple crack tip field of mode I, where no plastic deformation occurs. In order to investigate the shielding effect on the increase of fracture toughness, three-point bending tests of notched specimens were performed at high temperatures, where the increase of fracture toughness is enough to cause the BDT
of silicon crystals. Fig. 4(a) shows a photoelastic image obtained from a specimen deformed at the temperature around 1000 K. Here, β was set to be /4, and external stress was removed after the high temperature test. In spite of the absence of the external stress, a white image is seen in front of the notch, indicating that the image observed corresponds to the internal stress field due to dislocations introduced during the high temperature test. Compared with the image shown in Fig. 3(c) (β = /4), note that the image observed in Fig. 4(a) is not symmetrical with respect to the notch. Fig. 4(b) shows the simulated image assuming a dislocation structure illustrated in Fig. 4(c), where the edge dislocations are arranged on the {1 1 1} slip plane ahead of the crack tip. Based on the results of actual observations by TEM [13], the spacing between adjacent dislocations in the array was set to be 1/3 m. Some characteristics observed in Fig. 4(a) is well simulated in Fig. 4(b), which indicates that the dislocation structure assumed in Fig. 4(c) is not invalid, although it may be much simplified compared with the actual dislocation structure. For further understanding of the stress modification due to the plastic deformation, Fig. 5(a)–(c) exhibits the contour maps of the stress fields of σ yy , σ xx , and σ xy , respectively (see the coordinates in Fig. 4(c)). They were calculated using the model in Fig. 4(c), and the photoelastic image in Fig. 4(b) was obtained from these stresses. In these figures, particular attention should be paid on the contour map of σ yy in Fig. 5(a), which illustrates the stress component normal to the crack plane. As is seen in Fig. 5(a), a compressive stress field (shaded area) appears in front of the crack tip. Such compressive stress field is theoretically possible to restrain the tensile stress concentration when an external tensile load is applied. This effect of stress modification by dislocations was termed “crack tip shielding”.
Fig. 4. (a) Photoelastic image (β = /4) observed around a notch in the specimen deformed at the temperature around 1000 K. No stress is applied. (b) Photoelatic image (β = /4) computed by using the model shown in (c).
Fig. 5. Contour maps of the stress fields of σ yy (a), σ xx (b) and σ xy (c) calculated using the model as shown in Fig. 4(c). Shaded areas in the figures indicate negative value, i.e., compressive stress field in case of σ yy or σ xx .
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indicate that the images observed are corresponding to the internal stress field which shields (accommodates) the applied stress concentration. Finally, influence of the internal compressive stress field on the fracture toughness was examined by comparing apparent fracture toughness KIF of notched specimens with and without high temperature deformation. When the specimen was not subjected to high temperature deformation, the value of KIF at room temperature was about 1.7 MPa m1/2 . On the other hand, the KIF value of a specimen pre-deformed at 973 K was increased to 3.1 MPa m1/2 in spite that the fracture toughness test was carried out at room temperature. The increase of the KIF value must be attributed to the internal stress field introduced by the high temperature deformation. Thus, the effect of crack tip shielding due to dislocations actually operates and increases the fracture toughness of silicon crystals.
4. Conclusion Infrared photoelasticity was employed to visualize a crack tip stress field in silicon crystals. Computer simulation of the images was also made to analyze the stress field. The effect of crack tip shielding due to dislocations was verified by those observations, and it is considered to be essential in the increase of fracture toughness in the brittle-to-ductile transition of silicon crystals. In order to make more quantitative analysis for the shielding effect, it is necessary to correlate the intensity of transmitted light with the value of local stress intensity factors at the crack tip, which is one of the remaining problems to be solved in the further investigation.
Fig. 6. (a) and (b) are photoelastic images (β = /4) observed around the tip of notch; (c) and (d) are simulated images for (a) and (b), respectively. In (a) and (c), no stresses are applied to the specimens, i.e., the images are due to only internal stresses. In (b) and (d), mode I external tensile loads are applied to the specimens.
In order to confirm that the observed imaged in Fig. 4 corresponds to such compressive stress, we examined the change of the photoelastic image when external tensile loads were applied to the specimen. If the internal stress is compressive, it should be canceled by applying external tensile loads. Fig. 6(a) shows a photoelastic image (β = /4) without tensile load, i.e., the image due to only the internal stress field. Fig. 6(b) shows a image under tensile loading, where white broad image observed in the left side of Fig. 6(a) is reduced. Fig. 6(c) and (d) shows the images simulated for those without and with applied load, respectively. Also in the simulated image in Fig. 6(d), broad image in the left side of Fig. 6(c) is reduced by the external tensile load. These
References [1] R. Thomson, Physics of fracture, in: H. Seitz, D. Turnbull (Eds.), Solid State Physics, vol. 39, Academic Press, New York, USA, 1986, pp. 39–53. [2] S. Majumdar, S.J. Burns, Acta Matall. 29 (1981) 579. [3] I.-H. Lin, R. Thomson, Acta Metall. 34 (1986) 187. [4] K. Higashida, N. Narita, K. Matsunaga, R. Onodera, Phys. Stat. Sol. (a) 149 (1995) 429. [5] K. Higashida, N. Narita, Mater. Trans. 42 (2001) 33. [6] C.St. John, Phil. Mag. 32 (1975) 1193. [7] P.B. Hirsch, S.G. Roberts, J. Samuels, Proc. R. Soc. Lond. A 41 (1989) 25. [8] A. George, G. Michot, Mater. Sci. Eng. A164 (1993) 118. [9] K. Higashida, N. Narita, R. Onodera, S. Minato, S. Okazaki, Mater. Sci. Eng. A237 (1997) 72. [10] K. Higashida, T. Kawamura, T. Morikawa, Y. Miura, N. Narita, R. Onodera, Mater. Sci. Eng. A319–321 (2001) 683. [11] K. Higashida, N. Narita, M. Tanaka, T. Morikawa, Y. Miura, R. Onodera, Philos. Mag. A 82 (2002) 3263. [12] R. Bullough, Phys. Rev. 110 (1958) 620. [13] M. Tanaka, K. Higashida, T. Kishikawa, T. Morikawa, Mater. Trans. 43 (2002) 2169.
Crack tip stress fields revealed by infrared photoelasticity in silicon crystals Kenji Higashida∗ , Masaki Tanaka, Ena Matsunaga, Hironori Hayashi Department of Materials Science and Engineering, Faculty of Engineering, Kyushu University, 6-10-1 Higashi-ku, Fukuoka 812-8581, Japan Received 25 August 2003; received in revised form 9 December 2003
Abstract Stress fields around a crack tip in silicon crystals have been investigated by using infrared photoelasticity with the aim of clarifying the shielding effect due to crack tip dislocations on the steep increase of fracture toughness in the brittle-to-ductile transition (BDT). First, compact tension tests were carried out at room temperature to make in situ observation of elastic behavior of crack tip stress fields. The photoelastic images observed were in good agreement with those simulated for the usual elastic fields around the tip of a mode I crack. Next, to clarify the stress modification due to crack tip plasticity, three-point bending tests were also made by using notched specimens at high temperatures around 1000 K. After the high temperature test, in spite of the absence of the applied load, residual bright images were observed around the notch. Those images correspond to an internal stress due to dislocations multiplied around the notch, and they have an effect of shielding (accommodating) the stress concentration due to the applied load. The fracture toughness at room temperature was increased by the introduction of the residual stress. © 2004 Elsevier B.V. All rights reserved. Keywords: Brittle-to-ductile transition; Dislocations; Crack; Fracture toughness; Shielding
1. Introduction Dislocations introduced around a crack tip by an applied load contribute to a reduction in the local stress intensity factor by producing a back stress toward the crack tip. Such modification of stress state around a crack tip due to dislocations has been theoretically treated in terms of “crack tip shielding” [1–5]. Silicon crystals have attracted much attention as a model substance to investigate the fundamental mechanism of brittle-to-ductile transition (BDT). One of the most essential arguments for the BDT process is on the multiplication of crack tip dislocations and their shielding effect which has been considered to be the main cause of the steep increase of fracture toughness in the BDT [6–8]. However, observations or experimental analyses for such phenomena have been still limited [9–11], and further investigation is necessary to corroborate the shielding effect and its contribution to the increase of fracture toughness.
∗
Corresponding author. Tel.: +81 92 642 3660; fax: +81 92 642 3660. E-mail address: [email protected] (K. Higashida).
0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2004.05.058
In the present study, crack tip stress fields in silicon crystals have been investigated by using infrared photoelasticity and its image simulation. Particular emphasis is laid on the internal stress field generated by dislocations introduced at high temperatures. On the basis of these observations, the effect of crack tip shielding and its contribution to the BDT is discussed.
2. Experimental procedures Silicon single crystals of p-type wafers commercially available were employed. Compact tension (CT) specimens were cut out from {1 1 1} silicon wafers (Fig. 1(a)). Tensile tests of the CT specimens were made at room temperature to observe elastic field around a crack tip. Three-point bending tests of notched specimens were also made at high temperatures around 1000 K. The orientation of those specimens for high temperature tests is illustrated in Fig. 1(b). Silicon crystals are transparent for infrared light whose wavelength is longer than 1.2 m. Photoelastic observations were made using crossed Nicols under the respective conditions of β = 0 and β = /4, where β is the angle between
378
K. Higashida et al. / Materials Science and Engineering A 387–389 (2004) 377–380
Here, I0 is the intensity of incident polarized light, δ is the phase difference between the two transmitted lights due to birefringence, φ is the angle between the crack plane and the principal direction of stress state. The phase difference δ is expressed as 1/2 2 σxy 2π 2 δ= qd + 4σxy , (2) λ σyy − σxx
Fig. 1. (a) CT specimen, (b) specimen for three-point bending tests.
Fig. 2. Coordinates associated with the crack. β defines the direction of the polarized light vector.
the incident polarized light vector and the crack plane (see Fig. 2). The intensity of transmitted light of polarized light microscope, IT , is described by the following equation [12]. δ IT = I0 sin2 (1) sin2 (2β + 2Φ). 2
where q is the photoelastic constant, d is the specimen thickness, and λ is the wave length. In this equation, the stress components σ xx , σ yy , and σ xy are defined by the coordinate shown in Fig. 2. The angle φ in Eq. (1) is expressed as 2σxy 1 −1 Φ= tan . (3) 2 (σyy − σxx ) From Eqs. (1) and (3), it is understood that the intensity of transmitted light IT increases with increasing the value of {σ xy /(σ yy − σ xx )} under the condition of β = 0. Therefore, this condition is favorable to observe the shear stress component σ xy . On the other hand, the condition of β = /4 was employed to observe the stress component σ yy normal to the crack plane such as tensile or compressive stress field. Because, under this condition, the intensity IT becomes large when the value of {σ xy /(σ yy − σ xx )} approaches to zero.
3. Results and discussion Fig. 3(a)–(c) shows photoelastic images obtained at room temperature under the condition of β = 0, /8, and /4, respectively. Applied stress intensity factor is around
Fig. 3. Observed (a–c) and computed (d–f) photoelastic images of mode I crack tip field in a Si crystal. (a) and (d) are obtained under the condition of β = 0, (b) and (e) are under β = /8, (c) and (f) are under β = /4.
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0.3 MPa m1/2 . Image simulation was also made assuming that the images observed are formed only by the applied KI field. The computed images are shown in Fig. 3(d) (β = 0), (e) (β = /8), and (f) (/4), which are in good agreement with the observed images. In these figures, simple white broad images are visible around the crack tip, and these images became larger with increasing the external tensile load. When the external load was removed, they disappeared and no residual images were seen around the crack tip. At room temperature, crack extension in silicon crystals occurs in perfectly brittle manner, and it was also confirmed by TEM that no dislocations were introduced during crack extension at room temperature [13]. Thus, the images observed in Fig. 3 correspond to the simple crack tip field of mode I, where no plastic deformation occurs. In order to investigate the shielding effect on the increase of fracture toughness, three-point bending tests of notched specimens were performed at high temperatures, where the increase of fracture toughness is enough to cause the BDT
of silicon crystals. Fig. 4(a) shows a photoelastic image obtained from a specimen deformed at the temperature around 1000 K. Here, β was set to be /4, and external stress was removed after the high temperature test. In spite of the absence of the external stress, a white image is seen in front of the notch, indicating that the image observed corresponds to the internal stress field due to dislocations introduced during the high temperature test. Compared with the image shown in Fig. 3(c) (β = /4), note that the image observed in Fig. 4(a) is not symmetrical with respect to the notch. Fig. 4(b) shows the simulated image assuming a dislocation structure illustrated in Fig. 4(c), where the edge dislocations are arranged on the {1 1 1} slip plane ahead of the crack tip. Based on the results of actual observations by TEM [13], the spacing between adjacent dislocations in the array was set to be 1/3 m. Some characteristics observed in Fig. 4(a) is well simulated in Fig. 4(b), which indicates that the dislocation structure assumed in Fig. 4(c) is not invalid, although it may be much simplified compared with the actual dislocation structure. For further understanding of the stress modification due to the plastic deformation, Fig. 5(a)–(c) exhibits the contour maps of the stress fields of σ yy , σ xx , and σ xy , respectively (see the coordinates in Fig. 4(c)). They were calculated using the model in Fig. 4(c), and the photoelastic image in Fig. 4(b) was obtained from these stresses. In these figures, particular attention should be paid on the contour map of σ yy in Fig. 5(a), which illustrates the stress component normal to the crack plane. As is seen in Fig. 5(a), a compressive stress field (shaded area) appears in front of the crack tip. Such compressive stress field is theoretically possible to restrain the tensile stress concentration when an external tensile load is applied. This effect of stress modification by dislocations was termed “crack tip shielding”.
Fig. 4. (a) Photoelastic image (β = /4) observed around a notch in the specimen deformed at the temperature around 1000 K. No stress is applied. (b) Photoelatic image (β = /4) computed by using the model shown in (c).
Fig. 5. Contour maps of the stress fields of σ yy (a), σ xx (b) and σ xy (c) calculated using the model as shown in Fig. 4(c). Shaded areas in the figures indicate negative value, i.e., compressive stress field in case of σ yy or σ xx .
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indicate that the images observed are corresponding to the internal stress field which shields (accommodates) the applied stress concentration. Finally, influence of the internal compressive stress field on the fracture toughness was examined by comparing apparent fracture toughness KIF of notched specimens with and without high temperature deformation. When the specimen was not subjected to high temperature deformation, the value of KIF at room temperature was about 1.7 MPa m1/2 . On the other hand, the KIF value of a specimen pre-deformed at 973 K was increased to 3.1 MPa m1/2 in spite that the fracture toughness test was carried out at room temperature. The increase of the KIF value must be attributed to the internal stress field introduced by the high temperature deformation. Thus, the effect of crack tip shielding due to dislocations actually operates and increases the fracture toughness of silicon crystals.
4. Conclusion Infrared photoelasticity was employed to visualize a crack tip stress field in silicon crystals. Computer simulation of the images was also made to analyze the stress field. The effect of crack tip shielding due to dislocations was verified by those observations, and it is considered to be essential in the increase of fracture toughness in the brittle-to-ductile transition of silicon crystals. In order to make more quantitative analysis for the shielding effect, it is necessary to correlate the intensity of transmitted light with the value of local stress intensity factors at the crack tip, which is one of the remaining problems to be solved in the further investigation.
Fig. 6. (a) and (b) are photoelastic images (β = /4) observed around the tip of notch; (c) and (d) are simulated images for (a) and (b), respectively. In (a) and (c), no stresses are applied to the specimens, i.e., the images are due to only internal stresses. In (b) and (d), mode I external tensile loads are applied to the specimens.
In order to confirm that the observed imaged in Fig. 4 corresponds to such compressive stress, we examined the change of the photoelastic image when external tensile loads were applied to the specimen. If the internal stress is compressive, it should be canceled by applying external tensile loads. Fig. 6(a) shows a photoelastic image (β = /4) without tensile load, i.e., the image due to only the internal stress field. Fig. 6(b) shows a image under tensile loading, where white broad image observed in the left side of Fig. 6(a) is reduced. Fig. 6(c) and (d) shows the images simulated for those without and with applied load, respectively. Also in the simulated image in Fig. 6(d), broad image in the left side of Fig. 6(c) is reduced by the external tensile load. These
References [1] R. Thomson, Physics of fracture, in: H. Seitz, D. Turnbull (Eds.), Solid State Physics, vol. 39, Academic Press, New York, USA, 1986, pp. 39–53. [2] S. Majumdar, S.J. Burns, Acta Matall. 29 (1981) 579. [3] I.-H. Lin, R. Thomson, Acta Metall. 34 (1986) 187. [4] K. Higashida, N. Narita, K. Matsunaga, R. Onodera, Phys. Stat. Sol. (a) 149 (1995) 429. [5] K. Higashida, N. Narita, Mater. Trans. 42 (2001) 33. [6] C.St. John, Phil. Mag. 32 (1975) 1193. [7] P.B. Hirsch, S.G. Roberts, J. Samuels, Proc. R. Soc. Lond. A 41 (1989) 25. [8] A. George, G. Michot, Mater. Sci. Eng. A164 (1993) 118. [9] K. Higashida, N. Narita, R. Onodera, S. Minato, S. Okazaki, Mater. Sci. Eng. A237 (1997) 72. [10] K. Higashida, T. Kawamura, T. Morikawa, Y. Miura, N. Narita, R. Onodera, Mater. Sci. Eng. A319–321 (2001) 683. [11] K. Higashida, N. Narita, M. Tanaka, T. Morikawa, Y. Miura, R. Onodera, Philos. Mag. A 82 (2002) 3263. [12] R. Bullough, Phys. Rev. 110 (1958) 620. [13] M. Tanaka, K. Higashida, T. Kishikawa, T. Morikawa, Mater. Trans. 43 (2002) 2169.