- Email: [email protected]
On notch-strengthening and crack tip deformation in cellular metals
December 2002
Materials Letters 57 (2002) 532 – 536 www.elsevier.com/locate/matlet
On notch-strengthening and crack tip deformation in cellular metals E.W. Andrews, L.J. Gibson * Department of Materials Science and Engineering, Room 8-135, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Received 2 November 2001; accepted 28 March 2002
Abstract Using an image analysis technique, the deformation field in double-edged cracked specimens of an open-cell aluminum foam has been measured. The measured deformation field is consistent with a simple model in which failure occurs by tensile yield of the uncracked ligament and localized shearing (over a distance of about one cell) at the crack tip. An expression for notch strengthening based on this model provides good agreement with experimentally observed notch-strengthening results for both open- and closed-cell foams. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Foams; Mechanical properties; Fracture; Image analysis
1. Introduction A number of recent studies have reported results on the tensile strength of cellular metals with cracks [1– 4]. The results indicate that these materials typically show notch strengthening. The net section strength (tensile load at failure divided by remaining intact area in the cracked section) of a metal foam with a crack is greater than the tensile strength of the intact foam. The explanation for this notch strengthening is currently unclear. Successful application of these materials requires models to predict the failure conditions of cracked components, at the laboratory scale and in larger scale components. Better understanding of the material deformation mode allows models for predicting the failure of foams with cracks to be developed, and
*
Corresponding author. Tel.: +1-617-253-7107; fax: +1-617258-6275. E-mail address: [email protected] (L.J. Gibson).
allows laboratory results to be scaled up to engineering components. This letter reports preliminary measurements in which an image analysis technique was used to measure the in-plane deformation field around a crack in an open-cell aluminum foam. Motz and Pippan [2] previously used this technique to measure the deformation field around cracks in a closed-cell aluminum foam. This technique has also been applied to study the deformation of metal foams in uniaxial compression [5 –7] and compression fatigue [8], and the deformation field around holes in foams [9].
2. Experimental procedure A commercially available open-cell aluminum alloy (6101-T6) foam (trade name Duocel; ERG, Oakland, CA) with a nominal cell size of 40 pores per inch and a nominal relative density (foam density divided by the density of the solid the foam is made from) in the range 7 –8% was used in this study. The cells of the foam are
0167-577X/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 5 7 7 X ( 0 2 ) 0 0 8 2 4 - 8
E.W. Andrews, L.J. Gibson / Materials Letters 57 (2002) 532–536
533
deformation field was assessed from the image sequence using the Instron surface displacement analysis (SDA) software package (Instron; Canton, MA).
3. Results and discussion
Fig. 1. Schematics depicting (a) specimen geometry and (b) assumed failure mode.
elliptical in shape with major and minor axes of approximately 2.5 and 1.5 mm, respectively. The anisotropy in the cell shape gives rise to anisotropic mechanical properties. All the specimens were tested with the major axis of the cells parallel to the direction of the applied load. Dogbone-shaped specimens for tensile testing were prepared by the manufacturer using proprietary machining techniques. The length of the specimens was 220 mm, with a 120-mm gauge length. Within the gauge length, the cross-section had a width 2W of 20 mm and a thickness B of 20 mm. Doubleedged cracks were cut in the samples using a low-speed diamond blade to minimize damage to the specimen. The blade thickness was 0.35 mm, well below the cell size. The crack depth a varied from 2.6 to 7.4 mm. The geometry of the gauge section is shown schematically in Fig. 1a. The tests were run in displacement control, with the loading rate set to give a nominal strain rate of 104 s1. An extensometer with a gauge length of 25.4 mm was attached to the specimen to measure the local displacement around the cracked region of the gauge section. Each test was stopped at several points during the test, with the displacement held fixed, to allow an image of the specimen to be recorded. The images were taken using a 1K1K pixel array CCD camera (Pulnix; Sunnyvale, CA) with a telecentric lens. The in-plane
Fig. 2 shows the net section stress (peak tensile load divided by remaining intact area in the crack section) versus normalized crack depth (crack depth divided by specimen half-width) for tests on various aluminum foams [1 –3]. Alporas and Alcan are closed-cell foams while the ERG foam is open-cell. The notch-strengthening is evident. To explain the notch strengthening seen in their experiments, Andrews and Gibson [1] proposed a simple model, based on an analogy to flat punch indentation. The failure mode was assumed to be tensile failure of the uncracked ligament combined with localized shearing at the crack tips, shown schematically in Fig. 1b. The shearing is assumed to occur over a distance of approximately one-cell diameter. Based on this failure mode, the variation of the net * in terms of the cell size d, crack section strength, rpeak depth a, specimen half-width W and material plastic collapse strength, r*pl , was predicted to be: rpeak * d 1 : ¼1þ W ð1 a=W Þ r*pl
ð1Þ
To facilitate comparison with different materials and different specimen sizes, define rpeak * =r*pl +Dr, so
Fig. 2. Experimentally observed notch strengthening for metal foams.
534
E.W. Andrews, L.J. Gibson / Materials Letters 57 (2002) 532–536
the theoretical curve. We note that Motz and Pippan [2] tested the same material (Alporas) as Olurin et al. [3] but that their specimens were smaller than those of
Fig. 3. Normalized stress increase versus crack depth.
that Dr is the increase in section strength above the intact (uncracked) strength rpl *. Substituting and rearranging, we have that DrV 1 ¼ dV 1 a=W
ð2Þ
where DrV=Dr/rpl * and dV=d/W. Eq. (2) suggests that if the normalized stress increase, DrV/dV is plotted against crack depth a/W, the results for different materials and specimen sizes should all follow the same curve. Fig. 3 shows the data from Fig. 2 plotted in this way. It can be seen that the model is in good agreement with the experimentally observed notch strengthening trends for both open and closed-cell foams, except for the data of Motz and Pippan [2], which falls well below
Fig. 4. Net section stress versus strain. The test was paused and an image recorded for displacement analysis at the point shown.
Fig. 5. Images of the cracked region: (a) before loading and (b) at the point indicated in Fig. 4.
E.W. Andrews, L.J. Gibson / Materials Letters 57 (2002) 532–536
Olurin et al [3]: the width 2W and thickness B of the specimens of Motz and Pippan were 25 mm, while those of Olurin et al. were 60 mm. We suspect that if the specimens are too small and there are only a few cells spanning the uncracked ligament, our model is not appropriate and the strength of the samples will be lower than that predicted by Eq. (1). Their results may have been affected by the small specimen size relative to the cell size (3.5 mm for Alporas). The generally good agreement of the data with the equation is evidence for the validity of the assumed deformation mode. We now present measurements of the in-plane deformation field to provide further evidence for the validity of this model. The net section stress (load divided by remaining intact area in the cracked section) versus strain (recorded by the extensometer) is shown in Fig. 4 for a test at a crack size a/W=0.55. The test was paused and images were recorded at various points during loading. The arrow indicates the point at which the image used for the deformation analysis was recorded. The image is recorded at a point nearly at the peak load; the material is close to failure. After the peak load is reached, the load drops rapidly as many struts fail. Large strains are required before the specimen separates and the load goes to zero. The SDA software compares pairs of images and calculates the in-plane displacement components associated with the
535
deformation between the image pair. Fig. 5 shows the original (unloaded) image and a deformed image (at the point indicated in Fig. 4). The deformed image clearly shows that the crack has opened. Beyond that, visual inspection does not provide much information about the material deformation. The SDA software is used to determine the in-plane displacement field. The region of the image analyzed is shown by the white box. We analyzed only the lower half of the image due to the symmetry of the problem, and to avoid problems with the discontinuity in displacement across the cracks. We verified that the displacement fields were similar for the upper and lower regions of the image. Fig. 6 shows contour plots of the components of displacement in the x and y directions, u and v, as calculated by the software. The plots are superimposed on scale drawings of the cracked specimen, so the deformation field relative to the cracks may be seen. The displacement fields are not smooth and not exactly symmetric. This is likely due to the cellular nature of the material, or perhaps the surface imperfectly reflecting the bulk material deformation. The contour plot of u displacement shows roughly a transition from black to white from the left to right side of the specimen, a deformation gradient representing a lateral contraction (approximately 0.8% lateral strain). Irregularities in the field are present, likely due to the cellular structure. The contour plot of
Fig. 6. Contours of (a) u and (b) v displacement.
536
E.W. Andrews, L.J. Gibson / Materials Letters 57 (2002) 532–536
stresses at the crack tips contribute to this multiaxial stress state.
4. Conclusions
Fig. 7. Contours of shear strain cxy.
v shows a semielliptical region of significant displacement extending from the uncracked ligament. The strong variation in vertical displacement along the xdirection near the crack tip (note the closely spaced contours near the crack tips) demonstrates that shearing is taking place near the crack tip. To further demonstrate this shearing, we show in Fig. 7 a contour plot of the engineering shear strain c as evaluated by the SDA software. Regions of high shear strains (around 4%) are visible at the crack tips. Images recorded during tests on other specimens with different crack sizes were analyzed and similar deformation fields were observed. Motz and Pippan [2] observed similar vertical displacement fields in their study of a cracks in a closed-cell aluminum foam, indicating that the deformation mode is similar for both the open and closed-cell foams. We assert that this measured inplane deformation field is consistent with the assumed failure mode shown in Fig. 1b. A key aspect of the model is localized shearing at the crack tips; this agrees well with the measured deformation field. Motz and Pippan [2] suggest that the generation of a multiaxial stress-state in the uncracked region is responsible for the notch strengthening. This may indeed contribute to the notch strengthening. The deformation field presented here suggests that shear
The in-plane displacement field of a cracked opencell foam sample subjected to tensile loading has been presented. The measured deformation field is consistent with a model in which failure occurs by tensile failure of the uncracked ligament combined with localized shearing at the crack tips. The results are similar to published results for closed-cell metal foams. This model predicts notch strengthening that is in good agreement with experimental results. Further details, such as the generation of normal stresses caused by constraint of the region ahead of the crack, need to be clarified. However, the results suggest that this simple model captures some of the features of tensile failure of cracked cellular metals and provides a starting point for further analysis.
Acknowledgements The financial support of the Office of Naval Research (contract N00014-96-1-1028) is gratefully acknowledged. Thanks are due to Ms. J.K. Chan for assistance with specimen preparation.
References [1] E.W. Andrews, L.J. Gibson, Scr. Mater. 44 (2001) 1005. [2] C. Motz, R. Pippan, Acta Mater. 49 (2001) 2463. [3] O.B. Olurin, N.A. Fleck, M.F. Ashby, Mater. Sci. Eng., A 291 (2000) 136. [4] A. Paul, T. Seshacharyulu, U. Ramamurty, Scr. Mater. 40 (1999) 809. [5] A.F. Bastawros, H. Bart-Smith, A.G. Evans, J. Mech. Phys. Solids 48 (2000) 301. [6] H. Bart-Smith, A.F. Bastawros, D.R. Mumm, A.G. Evans, D.J. Sypeck, H.N.G. Wadley, Acta Mater. 46 (1998) 3583. [7] K.Y.G. McCullough, N.A. Fleck, M.F. Ashby, Acta Mater. 47 (1999) 2323. [8] Y. Sugimura, A. Rabiei, A.G. Evans, A.M. Harte, N.A. Fleck, Mater. Sci. Eng., A 269 (1999) 38. [9] X. Badiche, S. Forest, T. Guibert, Y. Bienvenu, J.D. Bartout, P. Ienny, M. Croset, H. Bernet, Mater. Sci. Eng., A 289 (2000) 276.
Materials Letters 57 (2002) 532 – 536 www.elsevier.com/locate/matlet
On notch-strengthening and crack tip deformation in cellular metals E.W. Andrews, L.J. Gibson * Department of Materials Science and Engineering, Room 8-135, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Received 2 November 2001; accepted 28 March 2002
Abstract Using an image analysis technique, the deformation field in double-edged cracked specimens of an open-cell aluminum foam has been measured. The measured deformation field is consistent with a simple model in which failure occurs by tensile yield of the uncracked ligament and localized shearing (over a distance of about one cell) at the crack tip. An expression for notch strengthening based on this model provides good agreement with experimentally observed notch-strengthening results for both open- and closed-cell foams. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Foams; Mechanical properties; Fracture; Image analysis
1. Introduction A number of recent studies have reported results on the tensile strength of cellular metals with cracks [1– 4]. The results indicate that these materials typically show notch strengthening. The net section strength (tensile load at failure divided by remaining intact area in the cracked section) of a metal foam with a crack is greater than the tensile strength of the intact foam. The explanation for this notch strengthening is currently unclear. Successful application of these materials requires models to predict the failure conditions of cracked components, at the laboratory scale and in larger scale components. Better understanding of the material deformation mode allows models for predicting the failure of foams with cracks to be developed, and
*
Corresponding author. Tel.: +1-617-253-7107; fax: +1-617258-6275. E-mail address: [email protected] (L.J. Gibson).
allows laboratory results to be scaled up to engineering components. This letter reports preliminary measurements in which an image analysis technique was used to measure the in-plane deformation field around a crack in an open-cell aluminum foam. Motz and Pippan [2] previously used this technique to measure the deformation field around cracks in a closed-cell aluminum foam. This technique has also been applied to study the deformation of metal foams in uniaxial compression [5 –7] and compression fatigue [8], and the deformation field around holes in foams [9].
2. Experimental procedure A commercially available open-cell aluminum alloy (6101-T6) foam (trade name Duocel; ERG, Oakland, CA) with a nominal cell size of 40 pores per inch and a nominal relative density (foam density divided by the density of the solid the foam is made from) in the range 7 –8% was used in this study. The cells of the foam are
0167-577X/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 5 7 7 X ( 0 2 ) 0 0 8 2 4 - 8
E.W. Andrews, L.J. Gibson / Materials Letters 57 (2002) 532–536
533
deformation field was assessed from the image sequence using the Instron surface displacement analysis (SDA) software package (Instron; Canton, MA).
3. Results and discussion
Fig. 1. Schematics depicting (a) specimen geometry and (b) assumed failure mode.
elliptical in shape with major and minor axes of approximately 2.5 and 1.5 mm, respectively. The anisotropy in the cell shape gives rise to anisotropic mechanical properties. All the specimens were tested with the major axis of the cells parallel to the direction of the applied load. Dogbone-shaped specimens for tensile testing were prepared by the manufacturer using proprietary machining techniques. The length of the specimens was 220 mm, with a 120-mm gauge length. Within the gauge length, the cross-section had a width 2W of 20 mm and a thickness B of 20 mm. Doubleedged cracks were cut in the samples using a low-speed diamond blade to minimize damage to the specimen. The blade thickness was 0.35 mm, well below the cell size. The crack depth a varied from 2.6 to 7.4 mm. The geometry of the gauge section is shown schematically in Fig. 1a. The tests were run in displacement control, with the loading rate set to give a nominal strain rate of 104 s1. An extensometer with a gauge length of 25.4 mm was attached to the specimen to measure the local displacement around the cracked region of the gauge section. Each test was stopped at several points during the test, with the displacement held fixed, to allow an image of the specimen to be recorded. The images were taken using a 1K1K pixel array CCD camera (Pulnix; Sunnyvale, CA) with a telecentric lens. The in-plane
Fig. 2 shows the net section stress (peak tensile load divided by remaining intact area in the crack section) versus normalized crack depth (crack depth divided by specimen half-width) for tests on various aluminum foams [1 –3]. Alporas and Alcan are closed-cell foams while the ERG foam is open-cell. The notch-strengthening is evident. To explain the notch strengthening seen in their experiments, Andrews and Gibson [1] proposed a simple model, based on an analogy to flat punch indentation. The failure mode was assumed to be tensile failure of the uncracked ligament combined with localized shearing at the crack tips, shown schematically in Fig. 1b. The shearing is assumed to occur over a distance of approximately one-cell diameter. Based on this failure mode, the variation of the net * in terms of the cell size d, crack section strength, rpeak depth a, specimen half-width W and material plastic collapse strength, r*pl , was predicted to be: rpeak * d 1 : ¼1þ W ð1 a=W Þ r*pl
ð1Þ
To facilitate comparison with different materials and different specimen sizes, define rpeak * =r*pl +Dr, so
Fig. 2. Experimentally observed notch strengthening for metal foams.
534
E.W. Andrews, L.J. Gibson / Materials Letters 57 (2002) 532–536
the theoretical curve. We note that Motz and Pippan [2] tested the same material (Alporas) as Olurin et al. [3] but that their specimens were smaller than those of
Fig. 3. Normalized stress increase versus crack depth.
that Dr is the increase in section strength above the intact (uncracked) strength rpl *. Substituting and rearranging, we have that DrV 1 ¼ dV 1 a=W
ð2Þ
where DrV=Dr/rpl * and dV=d/W. Eq. (2) suggests that if the normalized stress increase, DrV/dV is plotted against crack depth a/W, the results for different materials and specimen sizes should all follow the same curve. Fig. 3 shows the data from Fig. 2 plotted in this way. It can be seen that the model is in good agreement with the experimentally observed notch strengthening trends for both open and closed-cell foams, except for the data of Motz and Pippan [2], which falls well below
Fig. 4. Net section stress versus strain. The test was paused and an image recorded for displacement analysis at the point shown.
Fig. 5. Images of the cracked region: (a) before loading and (b) at the point indicated in Fig. 4.
E.W. Andrews, L.J. Gibson / Materials Letters 57 (2002) 532–536
Olurin et al [3]: the width 2W and thickness B of the specimens of Motz and Pippan were 25 mm, while those of Olurin et al. were 60 mm. We suspect that if the specimens are too small and there are only a few cells spanning the uncracked ligament, our model is not appropriate and the strength of the samples will be lower than that predicted by Eq. (1). Their results may have been affected by the small specimen size relative to the cell size (3.5 mm for Alporas). The generally good agreement of the data with the equation is evidence for the validity of the assumed deformation mode. We now present measurements of the in-plane deformation field to provide further evidence for the validity of this model. The net section stress (load divided by remaining intact area in the cracked section) versus strain (recorded by the extensometer) is shown in Fig. 4 for a test at a crack size a/W=0.55. The test was paused and images were recorded at various points during loading. The arrow indicates the point at which the image used for the deformation analysis was recorded. The image is recorded at a point nearly at the peak load; the material is close to failure. After the peak load is reached, the load drops rapidly as many struts fail. Large strains are required before the specimen separates and the load goes to zero. The SDA software compares pairs of images and calculates the in-plane displacement components associated with the
535
deformation between the image pair. Fig. 5 shows the original (unloaded) image and a deformed image (at the point indicated in Fig. 4). The deformed image clearly shows that the crack has opened. Beyond that, visual inspection does not provide much information about the material deformation. The SDA software is used to determine the in-plane displacement field. The region of the image analyzed is shown by the white box. We analyzed only the lower half of the image due to the symmetry of the problem, and to avoid problems with the discontinuity in displacement across the cracks. We verified that the displacement fields were similar for the upper and lower regions of the image. Fig. 6 shows contour plots of the components of displacement in the x and y directions, u and v, as calculated by the software. The plots are superimposed on scale drawings of the cracked specimen, so the deformation field relative to the cracks may be seen. The displacement fields are not smooth and not exactly symmetric. This is likely due to the cellular nature of the material, or perhaps the surface imperfectly reflecting the bulk material deformation. The contour plot of u displacement shows roughly a transition from black to white from the left to right side of the specimen, a deformation gradient representing a lateral contraction (approximately 0.8% lateral strain). Irregularities in the field are present, likely due to the cellular structure. The contour plot of
Fig. 6. Contours of (a) u and (b) v displacement.
536
E.W. Andrews, L.J. Gibson / Materials Letters 57 (2002) 532–536
stresses at the crack tips contribute to this multiaxial stress state.
4. Conclusions
Fig. 7. Contours of shear strain cxy.
v shows a semielliptical region of significant displacement extending from the uncracked ligament. The strong variation in vertical displacement along the xdirection near the crack tip (note the closely spaced contours near the crack tips) demonstrates that shearing is taking place near the crack tip. To further demonstrate this shearing, we show in Fig. 7 a contour plot of the engineering shear strain c as evaluated by the SDA software. Regions of high shear strains (around 4%) are visible at the crack tips. Images recorded during tests on other specimens with different crack sizes were analyzed and similar deformation fields were observed. Motz and Pippan [2] observed similar vertical displacement fields in their study of a cracks in a closed-cell aluminum foam, indicating that the deformation mode is similar for both the open and closed-cell foams. We assert that this measured inplane deformation field is consistent with the assumed failure mode shown in Fig. 1b. A key aspect of the model is localized shearing at the crack tips; this agrees well with the measured deformation field. Motz and Pippan [2] suggest that the generation of a multiaxial stress-state in the uncracked region is responsible for the notch strengthening. This may indeed contribute to the notch strengthening. The deformation field presented here suggests that shear
The in-plane displacement field of a cracked opencell foam sample subjected to tensile loading has been presented. The measured deformation field is consistent with a model in which failure occurs by tensile failure of the uncracked ligament combined with localized shearing at the crack tips. The results are similar to published results for closed-cell metal foams. This model predicts notch strengthening that is in good agreement with experimental results. Further details, such as the generation of normal stresses caused by constraint of the region ahead of the crack, need to be clarified. However, the results suggest that this simple model captures some of the features of tensile failure of cracked cellular metals and provides a starting point for further analysis.
Acknowledgements The financial support of the Office of Naval Research (contract N00014-96-1-1028) is gratefully acknowledged. Thanks are due to Ms. J.K. Chan for assistance with specimen preparation.
References [1] E.W. Andrews, L.J. Gibson, Scr. Mater. 44 (2001) 1005. [2] C. Motz, R. Pippan, Acta Mater. 49 (2001) 2463. [3] O.B. Olurin, N.A. Fleck, M.F. Ashby, Mater. Sci. Eng., A 291 (2000) 136. [4] A. Paul, T. Seshacharyulu, U. Ramamurty, Scr. Mater. 40 (1999) 809. [5] A.F. Bastawros, H. Bart-Smith, A.G. Evans, J. Mech. Phys. Solids 48 (2000) 301. [6] H. Bart-Smith, A.F. Bastawros, D.R. Mumm, A.G. Evans, D.J. Sypeck, H.N.G. Wadley, Acta Mater. 46 (1998) 3583. [7] K.Y.G. McCullough, N.A. Fleck, M.F. Ashby, Acta Mater. 47 (1999) 2323. [8] Y. Sugimura, A. Rabiei, A.G. Evans, A.M. Harte, N.A. Fleck, Mater. Sci. Eng., A 269 (1999) 38. [9] X. Badiche, S. Forest, T. Guibert, Y. Bienvenu, J.D. Bartout, P. Ienny, M. Croset, H. Bernet, Mater. Sci. Eng., A 289 (2000) 276.