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Prediction of residual stress-induced cracking by finite element analysis
scripts Metahgica et Materiah, Vol. 32, No. 3, pp. 359-364,1995 Copyright(B1994ElsevierScienceL,td Printedin the USA. All rightsreserved 0956-716X95$9.50t .OO
PREDICTION
OF RESIDUAL STRESS-INDI;CEI) BY FINITE ELEMENT ANAI,YSIS
CRACKING
S.Y. Kweon and S.K. Choi Dept. of Ceramic Science and Engineering Korea Advanced Institute of Science and Technology 373-l Kusung-dong, Yusung-gu, Taejon, Korea (Received July 8,1994) (Revised September 7,1994) Introduction With the development of new ceramic materials, including those for structural applications, there is an increasing demand to join ceramic components to metal structures. The main problem in joining ceramics to metals is the thermal expansion mismatch between the ceramic and the metal. Typically, metals have higher coefficients of thermal expansion (CTEs) than ceramics. The difference in the CTEs can lead to very high stress at the brazed region during cooling from the brazing temperature to room temperature. The high stress sometimes results in cracking of either the ceramic or the interface. The residual stresses may also degrade the mechanical strength of the brazed system. Therefore, the thermal stress problems should hc overcome to obtain a reliable joint between the ceramic and the metal. Several studies cl-51 have concentrated on either mcasunng or predicting the thermal stresses generatccl during cooling of the joint from brazing temperature as a function of joint materials and gcomctry. Th(, effect of the structure of the interface region on the toughness of a metal-to-ceramic joint has also received some attention [6,71. Some aspects of residual stress-induced cracking have been analyzed previously 14.81 but little effort has been made to present a comprehensive description of this phenomenon. The intent of the, present investigation is the prediction of the probable failure sites by using finite element analysis and the experimental characterization of cracking. An alumina-to-carbon steel joint was fabricated with a direct brazing technique. The residual stress distribution in alumina near interface is determined by finite element method (FEM) analysis and the analyzed results were testified by X-ray diffraction (XRD) measurements. In order to ohserve the predicted cracks in detail, the joining samples were sectioned, polished and characterized b!: scanning electroil microscopy(SEM1 with energy dispersive spectroscopy (EDS). Exlwrimental
Procedure
Pressureless-sintered polycrystalline alumina (with purity>98%1 and carbon steel ( -with O.Zwt%C) w#en used for the joining. The materials prepared in this study were cylindrical type and had a size of 14mm in diameter and 15mm in length. The surfaces to be joined were polished to mirror finish with l/4 urn diamond paste. The filler metal was 65wt?dAg-335wt%Cu-15wt%Ti $10~ and was prepared in a vacuum arc meltrng furnace. The joining was carried out under vacuum of -10 ‘Pa at 900eC for 40 minutes. The joining system was heated up to 780% at the rate of lZC/min followed by heating up to 900°C at 2.:i’C/min and cooled down to 20°C by furnace cooling. The interference between the components occurred as the joint was cooled down to 20°C‘ from brazing temperature. This interference results in permanent deformation of the hrazc and rcsiclual stresses in the metal. the braze, and the ceramic. The ANSYS 4.4A finite elcmcnt code was used to calculate the reslduai stress state in the joints. The thermally expanded geometry of each component at the \olidus temperatun (780%) of the filler metal was meshed by using 44nodccl rluadrikitcni1 asisymmctric solid elements. T’h<, configuration and dimensions used in FEM analysis arc shown in Fig.1. The material’s properties uscci in calculation are listed in Table 1 15,13,141. The material’s properties are functions of tcmpcrature. From i, simple elastic solution for thermal residual stresses in a layered system, the shear strcsscs developed at the, joint interfaces are dependent on the diffcrenccs in the products of Young’s modulus and CTE of the joinlllg materials l51. These differences are much less dependent on temperature than the individual matcriai’\ properties since Young’s moduli and CTEs change in an oplmsite way with temperature. Therefore, little I\ lost by considering only room temperature properties. The stress-strain response of the braze was assumed to be bilinear kinematic hardening based on the von Mises yield criterion which accounts for the Bauschingcr effect, where the elastic modulus, E,, is the slope in the elastic range, and the tangent modulus, E,, is th slope in the plastic range. However, the ceramic and the carbon steel wcrc assumed to hc only elastic.
359
360
RESIDUAL. STRESS
TABLE 1. Properties of the Components Used In Calculatmg Thermal Residual Stresses by Finite Element Analysis
Vol. 32, No. 3
the
It is important to estimate the residual stresses in the ceramic-to-metal joint for actual applications. The measurements after cutting were performed to verify the results of FEIM analysis. The residual stresses may redistribute because of cutting. But the distribution characteristics do not change, although the absolute values of the residual stresses at the same position decrease [91. The X-ray method has merits of being non-destructive and rapid. The irradiation area, however, has been too large to apply it in the joined interfaces. In this study a Cr I(, X-ray beam was collimated to lmm in diameter to be able to obtain the “real”, “not average”, stress values. Practical hints for X-ray residual stress determinations on ceramics and a summary of the work done in this field up to now are given in Ref. 1. X-ray elastic constants of alumina are also given in Ref. 1. Results
and Discussion
Two-dimensional axisymmetric analyses were performed to obtain a detailed residual stress distribution in the ceramic-to-metal joint. The result of the two dimensional stress analysis on 0 yy for the AlzOs-to-carbon steel joint is shown in Fig. 2. A peak tensile stress occurred at the ceramic near the interface on the edges. This is similar to the result reported by Suga and Elssner IlOl. Residual stresses at the ceramic along the center line in Fig.2 were measured by XRD technique (sin’4 method) [ll after cutting to verify the analytical result. The residual stress distributions measured are shown in Fig. 3 with FEM analysis data. It was concluded that the two-dimensional axisymmetric analysis performed shows good agreement with the XRD measurement qualitatively. So the FEM analysis method adopted in the present investigation is reliable for studying the residual stress distribution. In the case where there are cracks and tensile residual stresses at the ceramic near the interface, the strength of a joint drastically decreases [51. The formation of fine cracks at the interface between the twu different materials plays an important role in relieving the residual stress in the joint [Ill. Therefore, it is Important to understand the crack configuration in ceramic-to-metal joint. Selverian and Kang [121 reported that there was a close relationship between the spatial distributions of the extreme values of the maximum principal stress (ai) and the low I’, (the probability of survival) contours on the plots of survival probability. Bartlett and Evans L8J pointed out that the information about the imtiation of cracks can bc gained from the principal tensile stress, and crack trajectories are addressed by evaluation of the crack path along which the mode U stress intensity factor is zero. He also reported that cracks observed were 01: was perimeter crack, zone crack, edge crack, and branch crack. The maximum principal stress, considered to be an indicator of possible failure locations. Figure 4 shows the resultant maximum principal stress ( o ,) contours at 20°C calculated in this study for the AlzOs-to-carbon steel joint, and the directions of 0, drived by ANSYS 4.4A are shown in Fig.5. We can see the overall changes in the alumina and the brazing alloy region in Fig. 5(a). Figure 5(b) is the magnification of the region ‘A’ in (a). The symbol, ‘+‘, in Fig. 5 means that the direction is normal to sheet. Roughly speaking, there are three regions where maximum principal stress is higher than 200MPa (contour line ‘F’). These sites are interior of alumina. brazing alloy near interface, and alumina edge near interface. The information about the crack paths is also predicted by Fig. 5. The direction of Mode I crack initiation is normal to that of the maximum principal stress. These (Fig.4 and Fig.5) enable us to predict the continuous trajectories of cracks. Figure 6 is a schematic configuration of cracks forecasted in the ceramic component and the interface region. Inspection of the joint sectioned and polished with light microscopy revealed continuous perimeter cracks with no sign of internal crack. The shape of the perimeter crack (Fig. 7(a)) coincides with the path predicted in Fig. 5. The crack initiation did not occur at the ceramic-to-braze interface, but occurred in the ceramic at a distance about lmm from the ceramic-to-metal interface. Following initiation, the crack curved in a continuous trajecto?, turning direction to the interface in front of the compressive stress region before being arrested. Characterzation of the sample in detail by SEM showed the presence of additional cracks (branch cracks) having the feature shown in Fig. 7(b). The periodic branch cracks were observed in the reaction layer. In all bonds, reaction products were formed. In this system, two distinct reaction products were created as shown in Fig. 7(b). Data obtained by XRD, EDS. and TEM 1151 confirmed that the reaction products are Ti&u30 (thick part) and 6 -TiO (thin part). In order to obscrvc the cracks predicted at region C in Fig. 6, the fracture surface of a three point bending test sampll: was characterized by SEM. Large cracks were observed as shown in Fig. 7(c). Fracture strength of the samples which contain large perimeter crack was less than 5OMPa and that of the sintered alumina was about 400MPa. We concluded that those cracks were created not by the applied stresses during the three point bending test but by the stresses whose directions are normal to sheet. Their paths, thcrcfore, will be formed along the radial direction. WC
Vol. 32, No. 3
RESIDUAL STRESS
suggest that those cracks should be called ‘radial cracks’ and Evans [81 could not be observed in this system.
361
from now on. Zone cracks
reported
by Bartlett
The result of two-dimensional axisymmetric analysis on 6 yy by FEM shows good qualitative agreement with that of the XRD stress measurement. Therefore, FEM adopted in this study is reliable for studying the residual stress distribution. The maximum principal stress, d I, is a good indicator of possible failure sites in the joint during the cooling process. With the contribution of FEM analysis, three types of cracks were observed: perimeter crack, branch crack, and radial crack. References ____ 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
B. Eigenmann, B.Scholtes, and E. Macherauch, Mater. Sci. & Eng., A118, l(1989) O.T. Iancu and D. Munz, J. Am. Ceram. Sot., 73(5), 1144(1990) K. Suganuma, T. Okamoto, and K. Kamachi, J. Mater. Sci., 22, 270X1987) M.Y. He and A.G. Evans, Acta Metall. Mater., 39(7), 1587(1991) A. Levy, J. Am. Ceram. Sot., 74(g), 21410991) R.E. Loehman, A.P. Tomsia, J.A. Pask, and S.M. Johnson, J. Am. Ceram. Sot., 73(3), 55X1990) W.C. Lee and C.S. Kang. J. of the Korean Inst. of Metals, 29(4). 313(1991) A. Bartlett and A.G. Evans, Acta Metall. Mater., 39(7), 1579(1991) H. Kobayashi, Y. Arai, H. Nakamura, and T. Sato. Mater. Sci & Brig.. R143. 91(1991) T. Suga and G. Elssner, MRS Int’l Mtg. on Adv. Matls., 8, 99(1989) W.A. Zdaniewski, J.C. Conway, Jr., and H.P. Kirchner, J. Am. Ceram. SIX.. 70(Z). 110(19X7) J.H. Selverian and S.H. Kang, Am. Ceram. Sot. Bull., 71(10), 1511(1992) Ii. Misuhara, E. Huebel, and T. Oyama, Am. Ceram. Sot. Bull., 68(g),, 1591(19X9) WC. Campbell-Heselwood, “Smithells Metals Reference Book,” 6th et&ion. edited by EA. Brandes, 14-23, Butterworths, London, 1983 15. SK. Choi and S.Y. Kweon, “Characterization of Interfacial Microstructure In Active Filler Metal/ Alumina Brazement,” J. Am. Ceram. Sot., to be submitted
X (mm) FIG. 1. Finite element model with boundary conditions for two-dimensional axisymmetric analysis.
E
B.A.
FIG. 2. A contour plot Indicating of the (i ,.” stress.
\
MIN.
the distribution
362
RESIDUAL STRESS
Vol. 32, No. 3
FIG. 3. Comparison of the result calculated by FEM with the values measured by XRD method along the center line shown in Fig.2.
I
1 G
ALO,
Max.
FIG. 4. Distribution of the maximum principal stress, 0 I, in an alumina-m-carbon steel joint subjected to thermai load at 20-C
363
Vol. 32, No. 3
(a)
(b) (a) overall FIG. 5. Directions of the maximum principal stress, o I changes of the directions in the joint, and (b) the magnificatioq of the region ‘A’ in (a). Bold line in (b) is the ‘perimeter crack expected in a real system.
RESIDUAL STRESS
364
Vol. 32, No. 3
I
center 11‘le FIG. 6. Schematic drawing of cracking patterms predicted in the joint with the assistance of the FEM analysx. There are three probable cracks: perimeter crack. branch crack, and radial crack.
FIG. 7. SEM micrographs of the cracks ohscrv~~d in a jolnr system: (a) 13erimetcr crack, (1,) branch crack. (c) radial crack,
PREDICTION
OF RESIDUAL STRESS-INDI;CEI) BY FINITE ELEMENT ANAI,YSIS
CRACKING
S.Y. Kweon and S.K. Choi Dept. of Ceramic Science and Engineering Korea Advanced Institute of Science and Technology 373-l Kusung-dong, Yusung-gu, Taejon, Korea (Received July 8,1994) (Revised September 7,1994) Introduction With the development of new ceramic materials, including those for structural applications, there is an increasing demand to join ceramic components to metal structures. The main problem in joining ceramics to metals is the thermal expansion mismatch between the ceramic and the metal. Typically, metals have higher coefficients of thermal expansion (CTEs) than ceramics. The difference in the CTEs can lead to very high stress at the brazed region during cooling from the brazing temperature to room temperature. The high stress sometimes results in cracking of either the ceramic or the interface. The residual stresses may also degrade the mechanical strength of the brazed system. Therefore, the thermal stress problems should hc overcome to obtain a reliable joint between the ceramic and the metal. Several studies cl-51 have concentrated on either mcasunng or predicting the thermal stresses generatccl during cooling of the joint from brazing temperature as a function of joint materials and gcomctry. Th(, effect of the structure of the interface region on the toughness of a metal-to-ceramic joint has also received some attention [6,71. Some aspects of residual stress-induced cracking have been analyzed previously 14.81 but little effort has been made to present a comprehensive description of this phenomenon. The intent of the, present investigation is the prediction of the probable failure sites by using finite element analysis and the experimental characterization of cracking. An alumina-to-carbon steel joint was fabricated with a direct brazing technique. The residual stress distribution in alumina near interface is determined by finite element method (FEM) analysis and the analyzed results were testified by X-ray diffraction (XRD) measurements. In order to ohserve the predicted cracks in detail, the joining samples were sectioned, polished and characterized b!: scanning electroil microscopy(SEM1 with energy dispersive spectroscopy (EDS). Exlwrimental
Procedure
Pressureless-sintered polycrystalline alumina (with purity>98%1 and carbon steel ( -with O.Zwt%C) w#en used for the joining. The materials prepared in this study were cylindrical type and had a size of 14mm in diameter and 15mm in length. The surfaces to be joined were polished to mirror finish with l/4 urn diamond paste. The filler metal was 65wt?dAg-335wt%Cu-15wt%Ti $10~ and was prepared in a vacuum arc meltrng furnace. The joining was carried out under vacuum of -10 ‘Pa at 900eC for 40 minutes. The joining system was heated up to 780% at the rate of lZC/min followed by heating up to 900°C at 2.:i’C/min and cooled down to 20°C by furnace cooling. The interference between the components occurred as the joint was cooled down to 20°C‘ from brazing temperature. This interference results in permanent deformation of the hrazc and rcsiclual stresses in the metal. the braze, and the ceramic. The ANSYS 4.4A finite elcmcnt code was used to calculate the reslduai stress state in the joints. The thermally expanded geometry of each component at the \olidus temperatun (780%) of the filler metal was meshed by using 44nodccl rluadrikitcni1 asisymmctric solid elements. T’h<, configuration and dimensions used in FEM analysis arc shown in Fig.1. The material’s properties uscci in calculation are listed in Table 1 15,13,141. The material’s properties are functions of tcmpcrature. From i, simple elastic solution for thermal residual stresses in a layered system, the shear strcsscs developed at the, joint interfaces are dependent on the diffcrenccs in the products of Young’s modulus and CTE of the joinlllg materials l51. These differences are much less dependent on temperature than the individual matcriai’\ properties since Young’s moduli and CTEs change in an oplmsite way with temperature. Therefore, little I\ lost by considering only room temperature properties. The stress-strain response of the braze was assumed to be bilinear kinematic hardening based on the von Mises yield criterion which accounts for the Bauschingcr effect, where the elastic modulus, E,, is the slope in the elastic range, and the tangent modulus, E,, is th slope in the plastic range. However, the ceramic and the carbon steel wcrc assumed to hc only elastic.
359
360
RESIDUAL. STRESS
TABLE 1. Properties of the Components Used In Calculatmg Thermal Residual Stresses by Finite Element Analysis
Vol. 32, No. 3
the
It is important to estimate the residual stresses in the ceramic-to-metal joint for actual applications. The measurements after cutting were performed to verify the results of FEIM analysis. The residual stresses may redistribute because of cutting. But the distribution characteristics do not change, although the absolute values of the residual stresses at the same position decrease [91. The X-ray method has merits of being non-destructive and rapid. The irradiation area, however, has been too large to apply it in the joined interfaces. In this study a Cr I(, X-ray beam was collimated to lmm in diameter to be able to obtain the “real”, “not average”, stress values. Practical hints for X-ray residual stress determinations on ceramics and a summary of the work done in this field up to now are given in Ref. 1. X-ray elastic constants of alumina are also given in Ref. 1. Results
and Discussion
Two-dimensional axisymmetric analyses were performed to obtain a detailed residual stress distribution in the ceramic-to-metal joint. The result of the two dimensional stress analysis on 0 yy for the AlzOs-to-carbon steel joint is shown in Fig. 2. A peak tensile stress occurred at the ceramic near the interface on the edges. This is similar to the result reported by Suga and Elssner IlOl. Residual stresses at the ceramic along the center line in Fig.2 were measured by XRD technique (sin’4 method) [ll after cutting to verify the analytical result. The residual stress distributions measured are shown in Fig. 3 with FEM analysis data. It was concluded that the two-dimensional axisymmetric analysis performed shows good agreement with the XRD measurement qualitatively. So the FEM analysis method adopted in the present investigation is reliable for studying the residual stress distribution. In the case where there are cracks and tensile residual stresses at the ceramic near the interface, the strength of a joint drastically decreases [51. The formation of fine cracks at the interface between the twu different materials plays an important role in relieving the residual stress in the joint [Ill. Therefore, it is Important to understand the crack configuration in ceramic-to-metal joint. Selverian and Kang [121 reported that there was a close relationship between the spatial distributions of the extreme values of the maximum principal stress (ai) and the low I’, (the probability of survival) contours on the plots of survival probability. Bartlett and Evans L8J pointed out that the information about the imtiation of cracks can bc gained from the principal tensile stress, and crack trajectories are addressed by evaluation of the crack path along which the mode U stress intensity factor is zero. He also reported that cracks observed were 01: was perimeter crack, zone crack, edge crack, and branch crack. The maximum principal stress, considered to be an indicator of possible failure locations. Figure 4 shows the resultant maximum principal stress ( o ,) contours at 20°C calculated in this study for the AlzOs-to-carbon steel joint, and the directions of 0, drived by ANSYS 4.4A are shown in Fig.5. We can see the overall changes in the alumina and the brazing alloy region in Fig. 5(a). Figure 5(b) is the magnification of the region ‘A’ in (a). The symbol, ‘+‘, in Fig. 5 means that the direction is normal to sheet. Roughly speaking, there are three regions where maximum principal stress is higher than 200MPa (contour line ‘F’). These sites are interior of alumina. brazing alloy near interface, and alumina edge near interface. The information about the crack paths is also predicted by Fig. 5. The direction of Mode I crack initiation is normal to that of the maximum principal stress. These (Fig.4 and Fig.5) enable us to predict the continuous trajectories of cracks. Figure 6 is a schematic configuration of cracks forecasted in the ceramic component and the interface region. Inspection of the joint sectioned and polished with light microscopy revealed continuous perimeter cracks with no sign of internal crack. The shape of the perimeter crack (Fig. 7(a)) coincides with the path predicted in Fig. 5. The crack initiation did not occur at the ceramic-to-braze interface, but occurred in the ceramic at a distance about lmm from the ceramic-to-metal interface. Following initiation, the crack curved in a continuous trajecto?, turning direction to the interface in front of the compressive stress region before being arrested. Characterzation of the sample in detail by SEM showed the presence of additional cracks (branch cracks) having the feature shown in Fig. 7(b). The periodic branch cracks were observed in the reaction layer. In all bonds, reaction products were formed. In this system, two distinct reaction products were created as shown in Fig. 7(b). Data obtained by XRD, EDS. and TEM 1151 confirmed that the reaction products are Ti&u30 (thick part) and 6 -TiO (thin part). In order to obscrvc the cracks predicted at region C in Fig. 6, the fracture surface of a three point bending test sampll: was characterized by SEM. Large cracks were observed as shown in Fig. 7(c). Fracture strength of the samples which contain large perimeter crack was less than 5OMPa and that of the sintered alumina was about 400MPa. We concluded that those cracks were created not by the applied stresses during the three point bending test but by the stresses whose directions are normal to sheet. Their paths, thcrcfore, will be formed along the radial direction. WC
Vol. 32, No. 3
RESIDUAL STRESS
suggest that those cracks should be called ‘radial cracks’ and Evans [81 could not be observed in this system.
361
from now on. Zone cracks
reported
by Bartlett
The result of two-dimensional axisymmetric analysis on 6 yy by FEM shows good qualitative agreement with that of the XRD stress measurement. Therefore, FEM adopted in this study is reliable for studying the residual stress distribution. The maximum principal stress, d I, is a good indicator of possible failure sites in the joint during the cooling process. With the contribution of FEM analysis, three types of cracks were observed: perimeter crack, branch crack, and radial crack. References ____ 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
B. Eigenmann, B.Scholtes, and E. Macherauch, Mater. Sci. & Eng., A118, l(1989) O.T. Iancu and D. Munz, J. Am. Ceram. Sot., 73(5), 1144(1990) K. Suganuma, T. Okamoto, and K. Kamachi, J. Mater. Sci., 22, 270X1987) M.Y. He and A.G. Evans, Acta Metall. Mater., 39(7), 1587(1991) A. Levy, J. Am. Ceram. Sot., 74(g), 21410991) R.E. Loehman, A.P. Tomsia, J.A. Pask, and S.M. Johnson, J. Am. Ceram. Sot., 73(3), 55X1990) W.C. Lee and C.S. Kang. J. of the Korean Inst. of Metals, 29(4). 313(1991) A. Bartlett and A.G. Evans, Acta Metall. Mater., 39(7), 1579(1991) H. Kobayashi, Y. Arai, H. Nakamura, and T. Sato. Mater. Sci & Brig.. R143. 91(1991) T. Suga and G. Elssner, MRS Int’l Mtg. on Adv. Matls., 8, 99(1989) W.A. Zdaniewski, J.C. Conway, Jr., and H.P. Kirchner, J. Am. Ceram. SIX.. 70(Z). 110(19X7) J.H. Selverian and S.H. Kang, Am. Ceram. Sot. Bull., 71(10), 1511(1992) Ii. Misuhara, E. Huebel, and T. Oyama, Am. Ceram. Sot. Bull., 68(g),, 1591(19X9) WC. Campbell-Heselwood, “Smithells Metals Reference Book,” 6th et&ion. edited by EA. Brandes, 14-23, Butterworths, London, 1983 15. SK. Choi and S.Y. Kweon, “Characterization of Interfacial Microstructure In Active Filler Metal/ Alumina Brazement,” J. Am. Ceram. Sot., to be submitted
X (mm) FIG. 1. Finite element model with boundary conditions for two-dimensional axisymmetric analysis.
E
B.A.
FIG. 2. A contour plot Indicating of the (i ,.” stress.
\
MIN.
the distribution
362
RESIDUAL STRESS
Vol. 32, No. 3
FIG. 3. Comparison of the result calculated by FEM with the values measured by XRD method along the center line shown in Fig.2.
I
1 G
ALO,
Max.
FIG. 4. Distribution of the maximum principal stress, 0 I, in an alumina-m-carbon steel joint subjected to thermai load at 20-C
363
Vol. 32, No. 3
(a)
(b) (a) overall FIG. 5. Directions of the maximum principal stress, o I changes of the directions in the joint, and (b) the magnificatioq of the region ‘A’ in (a). Bold line in (b) is the ‘perimeter crack expected in a real system.
RESIDUAL STRESS
364
Vol. 32, No. 3
I
center 11‘le FIG. 6. Schematic drawing of cracking patterms predicted in the joint with the assistance of the FEM analysx. There are three probable cracks: perimeter crack. branch crack, and radial crack.
FIG. 7. SEM micrographs of the cracks ohscrv~~d in a jolnr system: (a) 13erimetcr crack, (1,) branch crack. (c) radial crack,