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Mesomechanics of fatigue fracture for polycrystals with macroconcentrators
Theoretical and Applied Fracture Mechanics 30 (1998) 13±18
Mesomechanics of fatigue fracture for polycrystals with macroconcentrators V.S. Pleshanov *, V.V. Kibitkin, V.E. Panin Institute of Strength Physics and Materials Science, Siberian Branch of Russian Academy of Sciences, pr.Akademicheskii 2/1, Tomsk 634021, Russian Federation
Abstract Fatigue fracture of aluminum alloys is analyzed from analysis of time-spatial mesoscale substructures. The behavior involves many stages at the mesoscale and translational±rotational movement of the bulk structure element in addition to fragmentation. Mesofragments on a trajectory of the fatigue crack were also studied. Ó 1998 Elsevier Science Ltd All rights reserved. Keywords: Mesomechanics; Fatigue fracture; Stage; Crack; Mesovolume; Deformation; Mesofragmentation; Vector displacement ®eld
1. Introduction The process of fatigue fracture has been studied at the dierent scale levels [1±4]. At the microscale level, crystalline imperfections such as point defects, individual dislocations and ensembles, sliding bands in grains, etc. have been investigated including their interaction and evolution. At the macroscale level, the bulk properties of the solid are considered without taking the details of the inner structure into account. While progress has been made at both of these scale levels, the connection between the results at the micro- and macroscopic level for cyclic loading remains much to be desired. It is believed that prediction of fatigue crack initiation and propagation at the macroscopic level could not be obtained directly from the action of sliding and fracture at the microscopic level. The intermediate scale level of
* Corresponding author. Tel.: +7 382 2 286 892; fax: +7 382 2 259 576.
events has to be included. This is at the mesoscale level. Similar approach has been used in earlier works [5,6] where theoretical analyses were made to account for damage accumulation in individual blocks of material elements around a crack. The concept of energy density was applied to describe the fracture behavior for quasi-elastic and elastic± plastic material containing an initial defect. Consideration was given to the fracture process of individual elements near and away from the crack in addition to accounting for the change of fracture resistance of the material at the prospective site of crack growth. The approach of mesomechanics [7] to deformation and fracture entails multiscale level of selforganizing structural elements such that each scale level would be associated with the corresponding stress concentrators, say at the micro-, meso- and macroscale. They would be identi®ed by the character of deformation, element size and kinetics. Transition of deformation and fracture from one scale level to another would involve a change
0167-8442/98/$ ± see front matter Ó 1998 Elsevier Science Ltd All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 9 8 ) 0 0 0 3 9 - 1
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V.S. Pleshanov et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 13±18
of elastic energy dissipation. Defects generated at the mesoscale would be referred to as mesodefects which give rise to movement of the bulk structural elements or mesovolumes. Both translational and rotational modes would be involved. The material would acquire a mesosubstructure in the deformed state. This would ®ll in the gap between the microand macroscale level and provide a better understanding of material fracture under cyclic loading. 2. Experimental procedure and description Fatigue fracture of polycrystals at the mesoscale level is studied using a television-optics measurement system [7]. It includes a microscope, a scanning mechanism, a lighting unit, a TV camera, a block of specialized interfaces and a computer. To analyze the peculiarities of the mesoelement motion, the double exposure method was used to record the images on the surface. The cyclic loading was applied between two exposures. Computer processing of the images gave vector displacement ®elds on the surface. Then the distributions of the local components of plastic distortion can be computed from the longitudinal strain exy , shear strain exy and rotation xz . Experiments are made at 293°K for aluminum alloys with the chemical composition given in Table 1. The specimen dimensions are 75 ´ 10 ´ 0.8 mm3 with a central hole 2 mm in diameter as the macroscopic stress concentrator. A cyclic load is applied with amplitude 50 50 MPa and frequency of 1.5 Hz. The surface under examination is polished.
stage involves the quasi-elastic crack growth at the microscopic scale level. The second and third stages correspond, respectively, to elastic±plastic crack growth at the mesoscale and unstable accelerated crack growth at the macroscale level. Identi®ed in this work are ®ve development stages of fracture of polycrystals at the mesoscale level. 3.1. Randomly distributed plastic shear At the early stage of 100 to 300 cycles of loading, the zones of plastic shear are randomly distributed and they are localized near the macroscopic concentrator such as a hole. This is illustrated by the variations of exy with x and y in Fig. 1. The size of the plastic shear zones changes with the amplitude of the cyclic load. It changes from 12 to 14 lm for low cycles for high cycles. As the material near the hole tends to harden plastically, the plastic shear zones disappear as the number of cycles increases into the range of 500 to 1000. 3.2. Quasi-growth of surface crack As the number of cycles reaches 40 to 70 ´ 103 , microtunnels and surface line cracks tend to form near the hole region on both sides of the specimen, The cracks are oriented normal to the direction of maximum tensile stress, a condition proposed in
3. Discussion of results Three stages of fatigue crack growth have been considered by conventional means [4]. The ®rst
Table 1 Chemical composition of aluminum alloy in wt% Cu
Mg
Mn
Si
Al
4.3
1.5
0.6
0.3
Remainder
Fig. 1. Shear strain exy distribution near hole for 100 cycles.
V.S. Pleshanov et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 13±18
[8]. They propagate at some depth inside the material [9] local to the microtunnels. Microtunnels generated in heterogeneous materials are sites of fatigue crack growth where the interconnecting ligaments would fracture owing to shear and rotation. As damage is accumulated ahead of the crack, breaking of the ligaments follows giving rise to jumps. The sharp stress concentration associated with the thin surface crack causes fracture in a brittle manner [10]. No signi®cant plastic shears were observed from the displacement ®elds in the regions of crack development. Independent development of surface cracks on both sides of the specimen is attributed to the lack of volume changes. 3.3. Elastic±plastic surface crack growth For load cycles of 55 to 85 ´ 103 , plastic deformation can be detected along the surface crack front. Movement of the mesovolume becomes selfconsistent. Formation of mesoscale fragmentations leads to the development of fatigue surface crack as shown in Fig. 2. The corresponding displacement vector ®eld in Fig. 3 shows several blocks or fragments with clearly de®ned borders. The boundaries of the mesoscale fragments de®ne the crack development path after additional cyclic
15
Fig. 2. Fatigue surface crack magni®ed 1000 times.
loadings. The jumping character of fracture continues to prevail. 3.4. Main crack Development of surface cracks is associated with increasing movement of the material volumes on the upper and bottom sides of the specimen. As the growth of the surface cracks becomes selfconsistent, they would unite and form the main crack at 80 to 110 ´ 103 cycles. The highly anisotropic character of the material in the region of the main crack can be seen in Fig. 4.
Fig. 3. Displacement ®eld after 57 ´ 103 cycles.
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V.S. Pleshanov et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 13±18
Fig. 4. Displacement ®eld in region of main fatigue cracks after 86 ´ 103 cycles.
In contrast to the conventional rupture-shear failure [11], motion of the main crack surface at the mesoscale level involves rupture-shear and rotation. Away from the crack tip region, the material undergoes extension in the direction of the load whereby normal rupture would dominate. Near the crack front, the material is fragmented such that the mesovolumes undergo translation
and rotation with highly localized shear strain exy and rotation xz . Their variations are shown graphically in Fig. 5. The key point of mesomechanics is the attribution of the rotational mode in the fracture process. Stress intensity at the main crack tip is lower than that of the surface crack. This is due to blunting of the main crack tip by plastic shear and rotation.
Fig. 5. Distribution of shear strain exy and rotation xz near main cracks tip after 86 ´ 103 cycles.
V.S. Pleshanov et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 13±18
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Fig. 6. Displacement ®eld at formation of plastic elongation zone in the region of ®nal termination after 88 ´ 103 cycles.
3.5. Plastic zone elongation and termination For cyclic load increment of 4 to 5 ´ 103 after formation of the main crack, cone-shaped plastic elongation appears near the crack tip [12] due to overload beyond yield. Fatigue fracture changes to plastic shear in the direction of maximum tangential stresses as illustrated in Fig. 6. Formation of the plastic elongation zone is similar to the mechanism of cyclic creep of polycrystals. The bands of localized plastic deformation are oriented at 45° to the load direction. These bands occur at the extremum points of local distortion. Final fracture of the specimen takes place after additional cyclic load increments of 500 to 1000. This is caused by the increase of rotation in the plastic elongation zone. The fatigue fracture surface is at 90° and ®nal termination is at 45° to the specimen plane.
4. Conclusion Fatigue fracture is identi®ed as a multistage process at the mesoscale level. Each stage corresponds to certain type of translation and rotational motion of the volume elements. The fracture
process starts with mesofragmentation and localized rotation near the crack front. Fatigue fracture trajectory can be de®ned as the noncrystallographic boundaries of the mesofragments.
References [1] H. Liebowitz (Ed.), Fracture, vol. 1±7, Academic Press, New York, 1968. [2] V.S. Ivanova, S.E. Gurevich, I.M. Kopjev et al., Fatigue and Brittleness of Metallic Materials, Nauko, Moskva, 1968. [3] V.T. Troshenko, Fatigue and Inelasticity of Metals, Kiev, 1971. [4] V.S. Ivanova, A.S. Balankin, A.A. Bunin et al., Synergetics and Fractal of Metal Science, Nauka, Moscow, 1994. [5] G.C. Sih, B. Macdonald, Fracture mechanics applied to engineering problems: Strain energy density fracture criterion, J. Eng. Fract. Mech. 6 (1974) 361±386. [6] G.C. Sih, Mechanics and physic of energy density theory, Theor. Appl. Fract. Mech. 4 (1985) 157±173. [7] V.E. Panin, V.E. Egorushkin, P.V. Makarov et al., Physical Mesomechanics and computer Aided Design of Materials, T. 1, Nauka, Novosibirsk, 1995. [8] F. Erodgan, G.C. Sih, On the crack extension of plates under plane loading and transverse shear, J. Basic Eng. 85 (1963) 519±525. [9] V.S. Ivanova, A.A. Shanjavskii, Quantitative Fractography. Fatigue Fracture Ustalostnoe Metallurgy, Cheljabinsk, 1988.
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V.S. Pleshanov et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 13±18
[10] R.E. Peterson, Stress concertration Factors, Wiley, New York, 1974. [11] B. Tomkins, Role of mechanics in corrosion fatigue, Metal Sci. 13 (1979) 387±395.
[12] V.A. Vainshtok, A.J. Krasovskii, G.N. Nadezzhdin et al., Application of stereoscopic fractography for analysing development of crack resistance, Problemi Prochnosti 11 (1978) 101±108.
Mesomechanics of fatigue fracture for polycrystals with macroconcentrators V.S. Pleshanov *, V.V. Kibitkin, V.E. Panin Institute of Strength Physics and Materials Science, Siberian Branch of Russian Academy of Sciences, pr.Akademicheskii 2/1, Tomsk 634021, Russian Federation
Abstract Fatigue fracture of aluminum alloys is analyzed from analysis of time-spatial mesoscale substructures. The behavior involves many stages at the mesoscale and translational±rotational movement of the bulk structure element in addition to fragmentation. Mesofragments on a trajectory of the fatigue crack were also studied. Ó 1998 Elsevier Science Ltd All rights reserved. Keywords: Mesomechanics; Fatigue fracture; Stage; Crack; Mesovolume; Deformation; Mesofragmentation; Vector displacement ®eld
1. Introduction The process of fatigue fracture has been studied at the dierent scale levels [1±4]. At the microscale level, crystalline imperfections such as point defects, individual dislocations and ensembles, sliding bands in grains, etc. have been investigated including their interaction and evolution. At the macroscale level, the bulk properties of the solid are considered without taking the details of the inner structure into account. While progress has been made at both of these scale levels, the connection between the results at the micro- and macroscopic level for cyclic loading remains much to be desired. It is believed that prediction of fatigue crack initiation and propagation at the macroscopic level could not be obtained directly from the action of sliding and fracture at the microscopic level. The intermediate scale level of
* Corresponding author. Tel.: +7 382 2 286 892; fax: +7 382 2 259 576.
events has to be included. This is at the mesoscale level. Similar approach has been used in earlier works [5,6] where theoretical analyses were made to account for damage accumulation in individual blocks of material elements around a crack. The concept of energy density was applied to describe the fracture behavior for quasi-elastic and elastic± plastic material containing an initial defect. Consideration was given to the fracture process of individual elements near and away from the crack in addition to accounting for the change of fracture resistance of the material at the prospective site of crack growth. The approach of mesomechanics [7] to deformation and fracture entails multiscale level of selforganizing structural elements such that each scale level would be associated with the corresponding stress concentrators, say at the micro-, meso- and macroscale. They would be identi®ed by the character of deformation, element size and kinetics. Transition of deformation and fracture from one scale level to another would involve a change
0167-8442/98/$ ± see front matter Ó 1998 Elsevier Science Ltd All rights reserved. PII: S 0 1 6 7 - 8 4 4 2 ( 9 8 ) 0 0 0 3 9 - 1
14
V.S. Pleshanov et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 13±18
of elastic energy dissipation. Defects generated at the mesoscale would be referred to as mesodefects which give rise to movement of the bulk structural elements or mesovolumes. Both translational and rotational modes would be involved. The material would acquire a mesosubstructure in the deformed state. This would ®ll in the gap between the microand macroscale level and provide a better understanding of material fracture under cyclic loading. 2. Experimental procedure and description Fatigue fracture of polycrystals at the mesoscale level is studied using a television-optics measurement system [7]. It includes a microscope, a scanning mechanism, a lighting unit, a TV camera, a block of specialized interfaces and a computer. To analyze the peculiarities of the mesoelement motion, the double exposure method was used to record the images on the surface. The cyclic loading was applied between two exposures. Computer processing of the images gave vector displacement ®elds on the surface. Then the distributions of the local components of plastic distortion can be computed from the longitudinal strain exy , shear strain exy and rotation xz . Experiments are made at 293°K for aluminum alloys with the chemical composition given in Table 1. The specimen dimensions are 75 ´ 10 ´ 0.8 mm3 with a central hole 2 mm in diameter as the macroscopic stress concentrator. A cyclic load is applied with amplitude 50 50 MPa and frequency of 1.5 Hz. The surface under examination is polished.
stage involves the quasi-elastic crack growth at the microscopic scale level. The second and third stages correspond, respectively, to elastic±plastic crack growth at the mesoscale and unstable accelerated crack growth at the macroscale level. Identi®ed in this work are ®ve development stages of fracture of polycrystals at the mesoscale level. 3.1. Randomly distributed plastic shear At the early stage of 100 to 300 cycles of loading, the zones of plastic shear are randomly distributed and they are localized near the macroscopic concentrator such as a hole. This is illustrated by the variations of exy with x and y in Fig. 1. The size of the plastic shear zones changes with the amplitude of the cyclic load. It changes from 12 to 14 lm for low cycles for high cycles. As the material near the hole tends to harden plastically, the plastic shear zones disappear as the number of cycles increases into the range of 500 to 1000. 3.2. Quasi-growth of surface crack As the number of cycles reaches 40 to 70 ´ 103 , microtunnels and surface line cracks tend to form near the hole region on both sides of the specimen, The cracks are oriented normal to the direction of maximum tensile stress, a condition proposed in
3. Discussion of results Three stages of fatigue crack growth have been considered by conventional means [4]. The ®rst
Table 1 Chemical composition of aluminum alloy in wt% Cu
Mg
Mn
Si
Al
4.3
1.5
0.6
0.3
Remainder
Fig. 1. Shear strain exy distribution near hole for 100 cycles.
V.S. Pleshanov et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 13±18
[8]. They propagate at some depth inside the material [9] local to the microtunnels. Microtunnels generated in heterogeneous materials are sites of fatigue crack growth where the interconnecting ligaments would fracture owing to shear and rotation. As damage is accumulated ahead of the crack, breaking of the ligaments follows giving rise to jumps. The sharp stress concentration associated with the thin surface crack causes fracture in a brittle manner [10]. No signi®cant plastic shears were observed from the displacement ®elds in the regions of crack development. Independent development of surface cracks on both sides of the specimen is attributed to the lack of volume changes. 3.3. Elastic±plastic surface crack growth For load cycles of 55 to 85 ´ 103 , plastic deformation can be detected along the surface crack front. Movement of the mesovolume becomes selfconsistent. Formation of mesoscale fragmentations leads to the development of fatigue surface crack as shown in Fig. 2. The corresponding displacement vector ®eld in Fig. 3 shows several blocks or fragments with clearly de®ned borders. The boundaries of the mesoscale fragments de®ne the crack development path after additional cyclic
15
Fig. 2. Fatigue surface crack magni®ed 1000 times.
loadings. The jumping character of fracture continues to prevail. 3.4. Main crack Development of surface cracks is associated with increasing movement of the material volumes on the upper and bottom sides of the specimen. As the growth of the surface cracks becomes selfconsistent, they would unite and form the main crack at 80 to 110 ´ 103 cycles. The highly anisotropic character of the material in the region of the main crack can be seen in Fig. 4.
Fig. 3. Displacement ®eld after 57 ´ 103 cycles.
16
V.S. Pleshanov et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 13±18
Fig. 4. Displacement ®eld in region of main fatigue cracks after 86 ´ 103 cycles.
In contrast to the conventional rupture-shear failure [11], motion of the main crack surface at the mesoscale level involves rupture-shear and rotation. Away from the crack tip region, the material undergoes extension in the direction of the load whereby normal rupture would dominate. Near the crack front, the material is fragmented such that the mesovolumes undergo translation
and rotation with highly localized shear strain exy and rotation xz . Their variations are shown graphically in Fig. 5. The key point of mesomechanics is the attribution of the rotational mode in the fracture process. Stress intensity at the main crack tip is lower than that of the surface crack. This is due to blunting of the main crack tip by plastic shear and rotation.
Fig. 5. Distribution of shear strain exy and rotation xz near main cracks tip after 86 ´ 103 cycles.
V.S. Pleshanov et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 13±18
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Fig. 6. Displacement ®eld at formation of plastic elongation zone in the region of ®nal termination after 88 ´ 103 cycles.
3.5. Plastic zone elongation and termination For cyclic load increment of 4 to 5 ´ 103 after formation of the main crack, cone-shaped plastic elongation appears near the crack tip [12] due to overload beyond yield. Fatigue fracture changes to plastic shear in the direction of maximum tangential stresses as illustrated in Fig. 6. Formation of the plastic elongation zone is similar to the mechanism of cyclic creep of polycrystals. The bands of localized plastic deformation are oriented at 45° to the load direction. These bands occur at the extremum points of local distortion. Final fracture of the specimen takes place after additional cyclic load increments of 500 to 1000. This is caused by the increase of rotation in the plastic elongation zone. The fatigue fracture surface is at 90° and ®nal termination is at 45° to the specimen plane.
4. Conclusion Fatigue fracture is identi®ed as a multistage process at the mesoscale level. Each stage corresponds to certain type of translation and rotational motion of the volume elements. The fracture
process starts with mesofragmentation and localized rotation near the crack front. Fatigue fracture trajectory can be de®ned as the noncrystallographic boundaries of the mesofragments.
References [1] H. Liebowitz (Ed.), Fracture, vol. 1±7, Academic Press, New York, 1968. [2] V.S. Ivanova, S.E. Gurevich, I.M. Kopjev et al., Fatigue and Brittleness of Metallic Materials, Nauko, Moskva, 1968. [3] V.T. Troshenko, Fatigue and Inelasticity of Metals, Kiev, 1971. [4] V.S. Ivanova, A.S. Balankin, A.A. Bunin et al., Synergetics and Fractal of Metal Science, Nauka, Moscow, 1994. [5] G.C. Sih, B. Macdonald, Fracture mechanics applied to engineering problems: Strain energy density fracture criterion, J. Eng. Fract. Mech. 6 (1974) 361±386. [6] G.C. Sih, Mechanics and physic of energy density theory, Theor. Appl. Fract. Mech. 4 (1985) 157±173. [7] V.E. Panin, V.E. Egorushkin, P.V. Makarov et al., Physical Mesomechanics and computer Aided Design of Materials, T. 1, Nauka, Novosibirsk, 1995. [8] F. Erodgan, G.C. Sih, On the crack extension of plates under plane loading and transverse shear, J. Basic Eng. 85 (1963) 519±525. [9] V.S. Ivanova, A.A. Shanjavskii, Quantitative Fractography. Fatigue Fracture Ustalostnoe Metallurgy, Cheljabinsk, 1988.
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V.S. Pleshanov et al. / Theoretical and Applied Fracture Mechanics 30 (1998) 13±18
[10] R.E. Peterson, Stress concertration Factors, Wiley, New York, 1974. [11] B. Tomkins, Role of mechanics in corrosion fatigue, Metal Sci. 13 (1979) 387±395.
[12] V.A. Vainshtok, A.J. Krasovskii, G.N. Nadezzhdin et al., Application of stereoscopic fractography for analysing development of crack resistance, Problemi Prochnosti 11 (1978) 101±108.