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A comment on the axial crush of metallic honeycombs by Wu and Jiang
ARTICLE IN PRESS
International Journal of Impact Engineering 28 (2003) 1143–1146
Letter to the Editor
A comment on the axial crush of metallic honeycombs by Wu and Jiang
Abstract In the paper published by Wu and Jiang (Int. J. Impact. Eng. 19(5/6) (1997) 439), the experimental results of quasi-static and impact tests on six types of honeycomb cellular structures have been compared with theoretical predictions. The values of some of the parameters used in the formulae in their paper, seem to be incorrect. In this paper the cause of inaccuracy and the correct values of these parameters have been presented. r 2003 Elsevier Ltd. All rights reserved.
1. Introduction Enboa Wu and Wu-Shung Jiang [1] have reported experimental results for six types of honeycomb cellular structures under quasi-static and impact loads in the axial direction. The specimens were all back-supported with a steel block and were loaded by blunt impactors whose cross-sectional areas were larger than those of specimens. They concluded that for preservation of better energy absorbing capacity per areal density, use of a honeycomb structure with a smaller cell size and core height, and made of a stronger material is suggested. Then they have compared experimental results with suitable analytical relations, but the values of half wavelength of progressive plastic buckling (H) and crush strength per areal density (Sc/r), related to analytical formulae [2], seem to be incorrect. 2. Theory [2] For determining the crushing strength of hexagonal cell structures subjected to axial loading, the method of energy in conjunction with a minimum principle in plasticity is used. The theory is first developed for an arbitrary angle between planes of angular element and then is specified for the 120 angle, appropriate for the hexagonal cell structures. The structure of honeycomb is regular and symmetric and thus can be assumed from one typical folding element consisting of two angle element joined together forming an angle of 120 (Fig. 1 of [2]). 0734-743X/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0734-743X(02)00107-0
ARTICLE IN PRESS 1144
Letter to the Editor / International Journal of Impact Engineering 28 (2003) 1143–1146
Fig. 1. Comparing of the crush strength per areal density among the experimental data and the theoretical predictions.
Fig. 2. Comparison of the measured and predicted values of progressive plastic buckling half-wavelength for types 1–6 specimens.
In view of the periodicity of the folding mechanism, the height of a representative folding element should be equal to the length of the local buckling wave, 2H (Fig. 2 of [2]). In order to determine the crushing behaviour of the hexagonal cell structures, it suffices to consider one such representative element positioned at the junction of three neighbouring cells. The work done on the element due to applied axial load is dissipated as plastic work on extension of some regions of element and movement of horizontal and inclined hinge lines. After calculating the total energy which is dissipated in the one angle element and multiplying this value by the number of elements of one hexagonal cell, one obtains Pm b D H ¼ 16:8 þ 3p þ 9:56 : ð1Þ h H b M0 It is reasonable that the free parameters of the collapse mode (H and b) would lead to the least possible value of the crushing force. Thus, these parameters can be determined from the optimality conditions: @Pm @Pm ¼ 0; ¼ 0: ð2Þ @H @b
ARTICLE IN PRESS Letter to the Editor / International Journal of Impact Engineering 28 (2003) 1143–1146
1145
The solution of the above set of equations takes the simple form H ¼ 0:821 ðhD2 Þ1=3 ;
ð3Þ
b ¼ 0:683 ðh2 DÞ1=3 :
ð4Þ
Substituting (3) and (4) back into (1) and using the definition of M0, we finally arrive at Pm ¼ 34:44 s0 ðh5 DÞ1=3 :
ð5Þ
In the above equations h, D, s0H, b and Pm are the wall thickness and cell edge (Fig. 2 on page 160 and also the second line on page 169 of Wierzbicki’s paper [2]) of honeycomb, the yield strength of honeycomb material, half wavelength of progressive plastic buckling, minor radius of curvature of extended surface and mean crushing load of hexagonal structure, respectively. 3. Discussion 3.1. Correction for unit of density, r In Table 1 of [1] (honeycomb specimens specifications), the unit of density (r) is given in kg/mm3. By referring to Handbook of composites [3], it can easily be seen that the correct unit for density (r) in Table 1 should be kg/m3. 3.2. Correction for Sc =r In Section 3.4 of paper [1], the crush strength per areal density (Sc/r) of six types of honeycombs are compared with calculated results using Eq. (7.8) in [2] (Fig. 9 in [1]). The values for Wierzbicki model in this figure seem to be incorrect due to taking the value of the cell size (minor diameter of cells), S, as the value of the cell edge, D. The crushing strength of honeycomb is determined from Pm ; ð6Þ Sc ¼ A Table 1 Specimens properties and their half wavelength of progressive plastic buckling (H) and crush strength per areal density (Sc/r) Type no. Cell size (mm) s0 (MPa) r(kg/m3) Sc/r (experimental) (106 N-mm/kg) Sc/r (Analytical) [1] (106 N-mm/kg) Sc/r (Analytical) [4] (106 N-mm/kg) H (experimental) (mm) [1] H (analytical) (mm) [1] H (analytical) (mm) [4]
1
2
3
4
5
6
4.763 255 49.656 22.07 16.61 13.83 0.916 0.683 0.474
4.763 255 49.656 22.37 16.61 13.83 — — —
3.175 255 72.081 31.20 22.51 18.74 0.665 0.521 0.361
3.175 255 72.081 34.72 22.51 18.74 0.665 0.521 0.361
3.175 345 72.081 37.64 30.44 25.35 0.669 0.521 0.361
3.175 345 72.081 39.04 30.44 25.35 0.701 0.521 0.361
ARTICLE IN PRESS Letter to the Editor / International Journal of Impact Engineering 28 (2003) 1143–1146
1146
where A is the total area occupied by one cell (or the surface which undergoes the applied load). By substituting Eq. (5) in Eq. (6) the crushing strength can be obtained Sc ¼ 16:55 s0 ðh=sÞ5=3 :
ð7Þ
Wu and Jiang [1] assumed that D in Eq. (5) is the cell size which is not right because D is the cell edge and therefore the accurate values of (Sc/r) for six types of specimens will be equal to those shown in Table 1. Fig. (1) compares the measured [1] and calculated [1] and corrected [4] values of the crush strength per areal density of six types of specimens. 3.3. Correction for H In Section 3.6 of paper [1], the quasi-static test results for half wavelength of progressive plastic buckling (H) of specimens have compared with analytical formula which is derived by Wierzbicki [2] [Eq. (3)]. In Eq. (3), h and D are the wall thickness and the edge size of cells, respectively, while the authors give D as the cell size (minor diameter of cells) and on the basis of this assumption Fig. (10) of paper [1] is plotted and the authors [1] have concluded that the predicted values for H are E80% of the values of the experimental data. If the correct values of D are used, the theoretical values will be about 53% of the experimental data as shown in Table 1. After applying the correct value of D in the equations, the real values of (H) and (Sc/r), may be obtained and compared to experimental results (Figs. 1 and 2). 4. Conclusion After applying the correct value of D, the comparison shows [4] that the difference between the test results and the predicted values is greater than those have been brought in [1]. References [1] [2] [3] [4]
Wu E, Wu-Shung Jiang. Axial crush of metallic honeycombs. Int J Impact Eng 1997; 19(5/6): 439–456. Wierzbicki T. Crushing analysis of metal honeycombs. Int J Impact Eng 1983;1(2):157–74. Peters ST. Handbook of composites. London: Chapman & Hall, 1998 p. 265. Alavinia A, Liaghat GH. Internal report, Tarbiat Modarres University, 2001.
G.H. Liaghat Mechanical Engineering Department, Tarbiat Modarres University, P.O. Box 14155-4838, Tehran, Iran A. Alavinia Mechanical Engineering Department, Bu-Ali Sina University, Hamedan, Iran E-mail address: [email protected]
International Journal of Impact Engineering 28 (2003) 1143–1146
Letter to the Editor
A comment on the axial crush of metallic honeycombs by Wu and Jiang
Abstract In the paper published by Wu and Jiang (Int. J. Impact. Eng. 19(5/6) (1997) 439), the experimental results of quasi-static and impact tests on six types of honeycomb cellular structures have been compared with theoretical predictions. The values of some of the parameters used in the formulae in their paper, seem to be incorrect. In this paper the cause of inaccuracy and the correct values of these parameters have been presented. r 2003 Elsevier Ltd. All rights reserved.
1. Introduction Enboa Wu and Wu-Shung Jiang [1] have reported experimental results for six types of honeycomb cellular structures under quasi-static and impact loads in the axial direction. The specimens were all back-supported with a steel block and were loaded by blunt impactors whose cross-sectional areas were larger than those of specimens. They concluded that for preservation of better energy absorbing capacity per areal density, use of a honeycomb structure with a smaller cell size and core height, and made of a stronger material is suggested. Then they have compared experimental results with suitable analytical relations, but the values of half wavelength of progressive plastic buckling (H) and crush strength per areal density (Sc/r), related to analytical formulae [2], seem to be incorrect. 2. Theory [2] For determining the crushing strength of hexagonal cell structures subjected to axial loading, the method of energy in conjunction with a minimum principle in plasticity is used. The theory is first developed for an arbitrary angle between planes of angular element and then is specified for the 120 angle, appropriate for the hexagonal cell structures. The structure of honeycomb is regular and symmetric and thus can be assumed from one typical folding element consisting of two angle element joined together forming an angle of 120 (Fig. 1 of [2]). 0734-743X/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0734-743X(02)00107-0
ARTICLE IN PRESS 1144
Letter to the Editor / International Journal of Impact Engineering 28 (2003) 1143–1146
Fig. 1. Comparing of the crush strength per areal density among the experimental data and the theoretical predictions.
Fig. 2. Comparison of the measured and predicted values of progressive plastic buckling half-wavelength for types 1–6 specimens.
In view of the periodicity of the folding mechanism, the height of a representative folding element should be equal to the length of the local buckling wave, 2H (Fig. 2 of [2]). In order to determine the crushing behaviour of the hexagonal cell structures, it suffices to consider one such representative element positioned at the junction of three neighbouring cells. The work done on the element due to applied axial load is dissipated as plastic work on extension of some regions of element and movement of horizontal and inclined hinge lines. After calculating the total energy which is dissipated in the one angle element and multiplying this value by the number of elements of one hexagonal cell, one obtains Pm b D H ¼ 16:8 þ 3p þ 9:56 : ð1Þ h H b M0 It is reasonable that the free parameters of the collapse mode (H and b) would lead to the least possible value of the crushing force. Thus, these parameters can be determined from the optimality conditions: @Pm @Pm ¼ 0; ¼ 0: ð2Þ @H @b
ARTICLE IN PRESS Letter to the Editor / International Journal of Impact Engineering 28 (2003) 1143–1146
1145
The solution of the above set of equations takes the simple form H ¼ 0:821 ðhD2 Þ1=3 ;
ð3Þ
b ¼ 0:683 ðh2 DÞ1=3 :
ð4Þ
Substituting (3) and (4) back into (1) and using the definition of M0, we finally arrive at Pm ¼ 34:44 s0 ðh5 DÞ1=3 :
ð5Þ
In the above equations h, D, s0H, b and Pm are the wall thickness and cell edge (Fig. 2 on page 160 and also the second line on page 169 of Wierzbicki’s paper [2]) of honeycomb, the yield strength of honeycomb material, half wavelength of progressive plastic buckling, minor radius of curvature of extended surface and mean crushing load of hexagonal structure, respectively. 3. Discussion 3.1. Correction for unit of density, r In Table 1 of [1] (honeycomb specimens specifications), the unit of density (r) is given in kg/mm3. By referring to Handbook of composites [3], it can easily be seen that the correct unit for density (r) in Table 1 should be kg/m3. 3.2. Correction for Sc =r In Section 3.4 of paper [1], the crush strength per areal density (Sc/r) of six types of honeycombs are compared with calculated results using Eq. (7.8) in [2] (Fig. 9 in [1]). The values for Wierzbicki model in this figure seem to be incorrect due to taking the value of the cell size (minor diameter of cells), S, as the value of the cell edge, D. The crushing strength of honeycomb is determined from Pm ; ð6Þ Sc ¼ A Table 1 Specimens properties and their half wavelength of progressive plastic buckling (H) and crush strength per areal density (Sc/r) Type no. Cell size (mm) s0 (MPa) r(kg/m3) Sc/r (experimental) (106 N-mm/kg) Sc/r (Analytical) [1] (106 N-mm/kg) Sc/r (Analytical) [4] (106 N-mm/kg) H (experimental) (mm) [1] H (analytical) (mm) [1] H (analytical) (mm) [4]
1
2
3
4
5
6
4.763 255 49.656 22.07 16.61 13.83 0.916 0.683 0.474
4.763 255 49.656 22.37 16.61 13.83 — — —
3.175 255 72.081 31.20 22.51 18.74 0.665 0.521 0.361
3.175 255 72.081 34.72 22.51 18.74 0.665 0.521 0.361
3.175 345 72.081 37.64 30.44 25.35 0.669 0.521 0.361
3.175 345 72.081 39.04 30.44 25.35 0.701 0.521 0.361
ARTICLE IN PRESS Letter to the Editor / International Journal of Impact Engineering 28 (2003) 1143–1146
1146
where A is the total area occupied by one cell (or the surface which undergoes the applied load). By substituting Eq. (5) in Eq. (6) the crushing strength can be obtained Sc ¼ 16:55 s0 ðh=sÞ5=3 :
ð7Þ
Wu and Jiang [1] assumed that D in Eq. (5) is the cell size which is not right because D is the cell edge and therefore the accurate values of (Sc/r) for six types of specimens will be equal to those shown in Table 1. Fig. (1) compares the measured [1] and calculated [1] and corrected [4] values of the crush strength per areal density of six types of specimens. 3.3. Correction for H In Section 3.6 of paper [1], the quasi-static test results for half wavelength of progressive plastic buckling (H) of specimens have compared with analytical formula which is derived by Wierzbicki [2] [Eq. (3)]. In Eq. (3), h and D are the wall thickness and the edge size of cells, respectively, while the authors give D as the cell size (minor diameter of cells) and on the basis of this assumption Fig. (10) of paper [1] is plotted and the authors [1] have concluded that the predicted values for H are E80% of the values of the experimental data. If the correct values of D are used, the theoretical values will be about 53% of the experimental data as shown in Table 1. After applying the correct value of D in the equations, the real values of (H) and (Sc/r), may be obtained and compared to experimental results (Figs. 1 and 2). 4. Conclusion After applying the correct value of D, the comparison shows [4] that the difference between the test results and the predicted values is greater than those have been brought in [1]. References [1] [2] [3] [4]
Wu E, Wu-Shung Jiang. Axial crush of metallic honeycombs. Int J Impact Eng 1997; 19(5/6): 439–456. Wierzbicki T. Crushing analysis of metal honeycombs. Int J Impact Eng 1983;1(2):157–74. Peters ST. Handbook of composites. London: Chapman & Hall, 1998 p. 265. Alavinia A, Liaghat GH. Internal report, Tarbiat Modarres University, 2001.
G.H. Liaghat Mechanical Engineering Department, Tarbiat Modarres University, P.O. Box 14155-4838, Tehran, Iran A. Alavinia Mechanical Engineering Department, Bu-Ali Sina University, Hamedan, Iran E-mail address: [email protected]