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Theoretical and numerical investigation on the crush resistance of rhombic and kagome honeycombs
Composite Structures 96 (2013) 143–152
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Theoretical and numerical investigation on the crush resistance of rhombic and kagome honeycombs Xiong Zhang a,c,⇑, Hui Zhang b a
Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China School of Mechanical Engineering and Automation, Wuhan Textile University, Wuhan 430073, Hubei, PR China c Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Luoyu Road 1037, Wuhan 430074, PR China b
a r t i c l e
i n f o
Article history: Available online 9 October 2012 Keywords: Honeycomb Multi-cell Columns Out-of-plane properties Energy absorption
a b s t r a c t The out-of-plane crushing properties of rhombic and kagome honeycombs are studied in this paper by dividing the whole structure into basic angle elements: corner elements and X-shaped elements. Two theoretical models are presented to analyze the energy absorption mechanisms for inextensional mode of corner elements and one of collapse modes of X-shaped elements respectively. Expressions are derived to predict the mean crushing force of the angle elements. Numerical simulations are also carried out for angle elements under out-of-plane crushing by using nonlinear finite element code LS-DYNA. Crush resistance of angle elements with different geometric configurations including angle, width and thickness is analyzed. A comparison between theoretical predictions and numerical results shows that the theoretical models can effectively predict the mean crushing force of angle elements with a very good accuracy. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Honeycombs are widely applied in aerospace and transport engineering as core of sandwich or energy absorbers due to their excellent multifunctional properties. Up to now, the regular hexagonal honeycomb with two of the six cell walls having double thickness is the most extensively used commercial products although honeycombs with other sections such as square, triangle, rhomb and kagome can also be fabricated by various methods [1]. The out-of-plane crushing resistance of honeycombs is very much higher than their in-plane resistance and attracts the attention of many researchers. However, most of the researches are focused on the numerical and experimental aspects of the out-of-plane crushing properties of honeycombs [2–6] and there is little theoretical work being published in the open literature to study the crushing resistance of them with various sections. This is due to the complex nonlinear features and difficulties in energy dissipation analyses during the post-buckling and large plastic deformation behavior of honeycombs under crushing loading. Hexagonal and square honeycombs are the only two types of honeycomb structures being addressed theoretically [7–9]. The
⇑ Corresponding author at: Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China. Tel.: +86 27 87543538; fax: +86 27 87543501. E-mail address: [email protected] (X. Zhang). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.09.028
axial crushing stress of hexagonal honeycomb was firstly theoretically predicted by McFarland [7] and then studied by Wierzbicki [8]. The plateau stress of square honeycomb was predicted by Zhang et al. [9] with a simplified theoretical model based on Super Folding Element (SFE) method. Recently, thin-walled multi-cell metal extrusions were found to be highly efficient energy absorption structures and received extensive research interests [9–14]. Just like honeycombs, the same problem encountered is the lack of theoretical expressions for the crushing resistance of columns with various sections. In fact, honeycombs can also be considered as a kind of multi-cell columns with complex section and therefore any advance in the theory aspect can be employed for both type of structures. Although it is very challenging and difficult, there is a way out. No matter how complex the section of a multi-cell column is, it is constituted by a number of shells or plates connected with various angles and by different connection factors. A basic idea to obtain the crush resistance of a column with complex section under axial compression is to divide the column into a number of representative structural elements, determine the crush resistance of each element separately and finally sum up the contribution of every element [15]. However, to finish this process, first of all, we must know the crush resistance of each element. In this paper, the out-of-plane crushing resistances of rhombic and kagome honeycombs as shown in Fig. 1 are studied by using this basic concept. The constitutive structural elements of rhombic and kagome honeycombs are also given in Fig. 1c. There are two type of
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(a) Rhombic honeycomb
(b) Kagome honeycomb
(c) Angle elements
Fig. 1. Scheme of rhombic and kagome honeycomb and their constitutive angle elements.
constitutive angle elements: corner element and X-shaped element. The central angle of corner element can be acute, right or obtuse, which is determined by the configuration of honeycombs. The most familiar one of them is corner element with right angle which is constitutive elements of a square or rectangular column. A square or rectangular column can develop two types of deformation mode during axial loading: Type I: (quasi-)inextensional mode and Type II: extensional mode. Theoretical expressions for the mean crushing force of a right angle corner element deforming in these two modes can be found in literatures [16,17] for both static and dynamic loading. These expressions have been validated by lots of experiments and can be employed to give a rather good prediction for energy absorption of structures consisting of right angle corner elements. So far there are few works to study the crushing response of corner elements with acute or obtuse angle. Actually, the classic theoretical model of right angle corner elements that proposed by Wierzbicki and Abramowicz [16] can also be applied to acute and obtuse angle elements. However, recently it is found that this model may overestimate the influence of central angle on energy absorption. Based on this model, a 57.5% increase of mean force is predicted for inextensional mode when central angle is increased from 90° to 120°. By adopting finite element method, Zhang and Huh [18] investigated the influence of angle on the energy absorption of corner elements and X-shaped and Y-shape angle elements. They found that when inextensional mode is developed, the crushing resistance of corner elements increases gradually with central angle increasing from 30° to 90°, and then keeps almost invariant for obtuse angle less than 150°. Most recently, experimental investigations was carried out by Zhang and Zhang [19] to study the crush resistance of angle elements by axial compression of polygonal and rhombic columns. It was verified that the crushing resistance of angle elements almost kept constant for obtuse angle between 90° and 135°. Therefore, the existing models may not be applied to acute or obtuse angle elements. The numerical analysis of Zhang and Huh [18] showed that the angle u (see Fig. 1) had significant influence on energy absorption of X-shaped angle elements. The mean force is found to be increased by more than 50% when u increases from 30° to 90°. However, there is also no theoretical model to analyze this type of structural element except for u = 90°. Therefore, to study the out-of-plane crushing resistances of rhombic and kagome honeycombs, two analytical models have to be established to deal with corner element and X-shaped angle element respectively. This is the main task of the present work and since it’s quite challenging and difficult, some assumptions are adopted to simplify the analyses just as what was done in literatures [9,10].
2. Theoretical analysis To predict the crushing resistance of an element, theoretical model has to be established to take into consideration all the energy dissipation mechanisms during deformation of it. Both corner elements and X-shaped angle elements can deform in more than one collapse mode and therefore the analytical models have to be established according to these collapse modes. Just as right angle corner elements, the corner elements with arbitrary angle can generally deform in inextensional mode, extensional mode and mixed mode. Theoretically besides mixed mode, Xshaped angle elements can also develop four collapse modes as shown in Fig. 2. It’s impossible to establish all models in the present work and therefore the most commonly encountered collapse modes are selected to be analyzed here. According to Zhang and Huh [18], the acute corner elements tend to deform in inextensional mode and obtuse corner elements are inclined to develop extensional mode. However, pure extensional mode is developed only for very large angle (larger than 135° for instance). Therefore, inextensional mode of corner element will be analyzed here. As for X-shaped angle elements, numerical simulations in the later section show that mode s is the most familiar one when u – 90° and this mode will be investigated in the present work.
Fig. 2. Four possible collapse modes of X-shaped elements.
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2.1. Corner elements The classic kinematically admissible model of Wierzbicki and Abramowicz [16] can give a good prediction of the mean crushing force of right angle corner elements. In their model, the corner elements are divided into trapezoidal rigid surfaces and cylindrical, conical and toroidal deformable zones with moving hinge lines and energy dissipated in each region is analyzed. Since corner elements with acute and obtuse angle cannot be predicted well by this model, a simplified approach similar to Chen and Wierzbicki [10] are employed here. The basic folding element proposed will only consist of stationary hinge lines and triangular regions passed by travelling hinge lines. The mean crushing force can be derived by analyzed the energy dissipation in one fold wavelength 2H and the folding wavelength is the length of one complete fold. Different researchers adopted different definition in their theoretical model. In Fig. 3, two definitions of folding wavelength are denoted by H and H0 respectively and apparently H0 is two times of H. In the model of Wierzbicki [8], 2H is defined as the length of one fold while 2H0 is used instead for example in the models of Ohkubo et al. [20] and Song et al. [21]. In the present model, 2H is defined as folding wavelength. A representative corner element with panel width of B and height of 4H is shown in Fig. 4A. According to the model of Wierzbicki [8], the energy is dissipated by the bending of static hinge lines, the rolling of moving hinge lines and the membrane deformation during the formation of toroidal surface. As we know, if the rolling radius keeps constant during the move of a hinge line, the bending energy for the plastic hinge line to travel a distance will be
W rolling ¼ 2M 0 DS=r
ð1Þ
2
where M0 = r0 h /4 is the fully plastic bending moment per unit width, r0 and h represents the flow stress and wall thickness respectively. DS is the area passed by the hinge line and r is the rolling radius of curvature. At the same time, if the same area is formed by extensional or compressional deformation, the energy dissipated will be
W membrane ¼
Z
r0 hds ¼ 4M0 DS=h
ð2Þ
DS
It is interesting to find that the energy dissipated by rolling of plastic hinge line has the same form as the strain energy of membrane deformation if r is equal to h/2. During the axial compression of cor-
H' H
ner elements, the real rolling radius of curvature is quite complicated since it may vary with the thickness, central angle or other geometric parameters, and even vary with time during loading. On the other hand, the area controlled by membrane deformation is also hard to determine. It is reported by Wierzbicki [8] that the membrane energy is dissipated in the formation of toroidal surface. The calculation of this part of energy is also derived in their work and the extensional area is obtained as 4Hb/I1 (refer to the literature for definition of b, I1). However, there may still have other channels of membrane energy dissipation. Numerical analyses in the later section find that the thickening of panel wall in the corner region dissipates considerable amount of membrane energy although it is hard to quantify it. To simplify the analyses, in the following derivation, these two parts of energy dissipation as given by Eqs. (1) and (2) are integrated into one part as
W 0rolling ¼ 2M 0 DS=r 0
ð3Þ
And consequently, the energy dissipation is now constituted by two parts: bending energy of static hinge lines and the integrated rolling energy of moving hinge lines. The integrated rolling energy W 0rolling dissipated during one wavelength crushing will be calculated according to the areas that are passed by travelling hinge lines. A flattened angle element is shown in Fig. 4B, the shaded regions represent the areas that are passed by hinge lines during loading and this ideal model is extracted from the real deformation of an angle element as given in Fig. 4C. A paper model is also presented in Fig. 4D to illustrate the collapse of the ideal model of angle element. We can see that no extensional deformation will be necessary for the ideal model to develop a complete fold otherwise the paper will break during deformation. The difference of the ideal model and the real angle element consists in the bending curvature of panel along hinge lines. The bending curvature radius of the ideal model is zero while it is a value in the order of magnitude of panel thickness for a real structure. The mean crushing force of the element can now be derived by considering the energy equilibrium of the system in one folding wavelength 2H. The external work done by compression will be dissipated by plastic deformation in bending and rolling. That is
Pm 2H j ¼ W bending þ W 0rolling
where Pm denotes the mean crushing force over the collapse of a fold and j is the coefficient of effective crushing distance. The real crushing distance is actually less than 2H and according to Abramowicz and Jones [22], it is around 73% of the wavelength for a square tube deformed in inextensional mode. The value j is assumed as constant to be 73% for all central angle in the following derivation. The bending energy Wbending is relatively easy to obtain and the determination of it can adopt the same way as Wierzbicki and Abramowicz [16] since the definition of folding wavelength is identical. The length of the stationary hinge lines in a corner element is 2B and the bending angle should be p in one fold compression. Consequently, the energy dissipation in bending is 2pM0B. According to Wierzbicki and Abramowicz [16], it should be doubled due to the clamped boundary condition of the folding mechanism. Based on the plastic deformation process of the structure, a different explanation for the double of bending energy was proposed recently by Zhang and Zhang [19]. Nevertheless, it is certain that Wbending of the two-panel right angle element should be doubled and it will be calculated as
W bending ¼ 4pM 0 B
ð5Þ W 0rolling
Fig. 3. Definition of folding wavelength.
ð4Þ
The integrated rolling energy is given by Eq. (3) and the area of the triangular shaded region in Fig. 4B is equal to H2 tan(h/2) (when B P H tan(h/2)). That is
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B
θ /2
4H
2H B
A
B
C
D
Fig. 4. (A) A representative angle element; (B) triangular regions passed by travelling hinge lines; (C) deformed shape of an angle element with central angle 120°; and (D) paper model of angle element.
DS ¼ H2 tanðh=2Þ
ð6Þ
Therefore
0
The determination of r is very important and difficult. In extreme cases, when h equals to 0° and 180°, the angle element can be considered as a plain plate. Only bending deformation will occur during loading and therefore the rolling or membrane energy should be zero. Based on this consideration, the rolling radius is assumed to have the following expression: a2
r 0 ðhÞ ¼ a1 B h
1a2
ðtanðh=2Þ þ a3 = tanðh=2ÞÞ
ð7Þ
where a1, a2 and a3 are constant coefficients. Based on this function expression, the rolling radius r0 will be infinite when h equals to 0° and 180°. This seems reasonable since the membrane energy will vanish for h = 0° and 180°. When h = 180°, the area calculated by Eq. (6) will be infinite. But actually, the area passed by rolling hinge lines should be 2B H since Eq. (6) is effective only when B P H tan(h/2). Consequently, the membrane energy will still vanish for h = 180°. When B < H tan(h/2), the shaded region becomes a trapezoid and specially a rectangular for h = 180° rather than a triangle. The area of it can be easily derived. However, to simplify the analysis, Eq. (6) is employed to calculate the area of membrane deformation in the whole range of h. Substitute Eqs. (6) and (7) into Eq. (3) and then substitute Eqs. (3) and (5) into Eq. (4), we have
Pm 2H j ¼ 4pM 0 B þ
2M 0 H2 tanðh=2Þ r 0 ðhÞ
ð8Þ
The mean crushing force can be obtained as
j Pm M0
¼
2pB H tanðh=2Þ þ H r0 ðhÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pBr 0 ðhÞ= tanðh=2Þ
ð10Þ
ð11Þ
Substitute Eq. (11) into Eq. (9), the mean crushing force of the corner element would be
1 Pm ¼ r0 B0:5 h2 2j
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p tanðh=2Þ r0 ðhÞ
ð12Þ
The determination of parameters a1, a2 and a3 is very important and it can be determined according to the experimental results or theoretical results validated by applications. Based on experimental data, Magee and Thornton [23] reported that the best fitted exponent of h in the expression of mean force was 1.8 for square or rectangular tubes. Therefore, a2 can be set to 0.6. The numerical analysis carried out by Zhang and Huh [18] and experimental tests conducted by Zhang and Zhang [19] show that the increase of mean crushing force is smaller than 5% when central angle is increased from 90° to 120°. Substitute h = 90° and h = 120° into Eq. (12), it is found that a3 should be smaller than 0.16 to keep the increase of mean crushing force less than 5%. Here a3 is set to 0.06 corresponding to a 2% increase in mean force which is based on the experimental data of Zhang and Zhang [19]. Now the last parameter is a1 which can be determined according to the theoretical and experimental results of square tubes. According to Magee and Thornton [23], the mean crushing force of square columns can be obtained as 1:8
ð9Þ
The folding wavelength H can then be determined by the stationary condition of the mean crushing force
@Pm ¼0 @H
H¼
Pm ¼ 17ru C 0:2 h
ð13Þ
where C is the width of a square column and ru is the ultimate tensile stress. It’s found that ru should be replaced by the average value of yield stress ry and ultimate stress ru to give a good prediction of the experiment results of square columns [19]. A square column is constituted by four right corner element with B = C/2. Substitute h = 90° into Eq. (12) and equate it with a quarter of Eq. (13), a1
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can be determined. In Eq. (12), the flow stress r0 for material with power law hardening can be calculated by [24]
r0 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ry ru
ð14Þ
ð1 þ nÞ
where n is the power law exponent. The above process for the determination of parameters a1, a2 and a3 is based on the work of Magee and Thornton [23] and it can also be done in a different way. According to Wierzbicki and Abramowicz [16], the mean crushing force for square columns deforming in inextensional mode is 5=3
Pm ¼ 13:10r0 C 1=3 h
ð15Þ
Therefore, a2 can be set to 1/3. Still a3 is set to 0.06 and then a1 can be determined to be 0.163 by equating Eq. (12) and a quarter of Eq. (15) at h = 90°. For dynamic loading, dynamic amplification effects including strain rate effect and inertia effect should be considered in the expressions [25,26]. Then, we have
Pmd
k ¼ rd B0:5 h2 2j
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p tanðh=2Þ r 0 ðhÞ
ð16Þ
where k is the dynamic enhancement factor and rd is the dynamic flow stress. For strain rate sensitive material, the dynamic flow stress rd is always assessed by empirical Cowper–Symonds constitutive equation. The dynamic enhancement factor k is due to the inertia forces and an investigation of it was carried out for square columns by Langseth and Hopperstad [27]. 2.2. X-shaped elements The deformation behavior of X-shaped elements is much more complicated than corner elements. Due to symmetry of the structure, only elements with 0 < u 6 90° need to be analyzed. As shown in Fig. 2, four collapse modes can be developed for an X-shaped element. However, the possibility of these four modes to occur is not equal. For a crisscross element (u = 90°), all these four modes may be developed due to initial imperfections or triggers, while mode s is the most likely one to occur for X-shaped elements with small u. Consequently, a theoretical model is established here to predict the crush resistance of X-shaped elements deforming in mode s. The deformed shapes of an X-shaped element with u = 60° are shown in 5. Similar to three-panel elements [13], the deformation mechanisms of X-shaped elements during crushing are very complicated and considerable amount of energy is dissipated by the thickening of panels. The contour plots of thickness reduction of X-shaped elements are also shown in Fig. 5. We can see that in the intersection region, the thickening of shell is significant and most of elements in this region get a 20–40% increase in thickness. Other mechanisms such as bending of static hinge lines, rolling of moving hinge lines and extensional deformation are also presented during deformation. The feature line plot (top view) of deformed structure is given in Fig. 5B. It can be found that the intersection line of the element almost deforms along the symmetric plane of the element which equally divide the supplementary angle of u. Based on this feature, an folding model of X-shaped element deforming in this mode is presented in Fig. 6. Two panels of the element are stretched extensionally as denoted by blue1 lines and the other two crush each other as shown in red lines. Although the intersection line is apparently shorter than the other four lines
around after deformation, it’s assumed in the model that the length of intersection line is invariant during deformation and it will be 2H for one fold. In fact, the shortening of the intersection line reflects the thickening of the panels in that region from another aspect and this part of energy should not be omitted. For the two panels crushing each other, the energy dissipation mechanism is basically the rolling of travelling hinge lines and the energy can be derived by Eq. (1). The rolling radius of curvature in the equation is also hard to determine just as for corner elements. It varies with geometric parameters including angles and panel thickness and even varies with time during loading. By adopting the same simplification as corner elements, the energy dissipated by thickening of the panels is integrated into the rolling energy of the travelling hinge lines and the rolling radius of curvature is assumed a function of angle, panel width and thickness. Energy dissipation of the element during loading can now be classified into three parts: bending energy of static hinge lines, the membrane energy of panel stretching and the integrated rolling energy during the crushing of two panels. The bending energy Wbending of an X-shaped element can be calculated as
W bending ¼ 8pM 0 B
ð17Þ
The membrane energy of panel stretching is calculated by Eq. (2) and the area that is extended during loading is denoted by the triangular shaded region in Fig. 6B. The area of the shaded region is H2 tan(u/2) and it should be doubled for one fold wavelength. Therefore, the membrane energy is
W membrane ¼ 4M 0 DS=h ¼ 8M 0 H2 tanðu=2Þ=h
The integrated rolling energy W r olling is given by Eq. (3) and the area passed by moving hinge lines are given by the triangular shaded region in Fig. 6C whose area is also equal to 2H2 tan(u/2) for one fold wavelength. That is
W 0rolling ¼ 4M 0 H2 tanðu=2Þ=r 0 ðuÞ
ð19Þ
As mentioned above, the determination of r0 is quite difficult. It may vary with geometric parameters and even time. Based on this consideration, the rolling radius of curvature is assumed to have the following expression: 1b2
r0 ðuÞ ¼ b1 Bb2 h
sinðu=2Þ
ð20Þ
where b1 and b2 are constant coefficients. According to Eq. (20), the rolling radius r0 is a increasing function of angle u. This is reasonable since the rolling radius should increase when the angle u is increasing. As shown in Fig. 7, two panels are folded with different inclined angle u1 and u2 and u1 > u2. The folding process of the panels that crush each other is similar to the loading process of panels compressed by a rigid plane. Apparently, the rolling radius of panel folded with inclined angle u1 should be larger than that of panel folded with inclined angle u2. As for the determination of coefficients b1 and b2, there is relatively few experimental or theoretical results to be referenced compared to corner elements. Most of the experiments and theoretical analyses were conducted for hexagonal honeycombs. Therefore, two values are suggested here to be 1/8 and 0.5 for b1 and b2 respectively. Actually, a simplified model has been proposed to analyze the crisscross element by Zhang et al. [9]. However, only bending and membrane energy were considered there and the membrane energy is not related to the width B of the element. The suggested values of b1 and b2 will give approximately same predictions for crisscross elements. Substitute b1 and b2 into Eq. (20) and then Eq. (19). We have
W 0rolling ¼ 32M0 H2 B0:5 h 1 For interpretation of color in Fig. 6, the reader is referred to the web version of this article.
ð18Þ
0
0:5
= cosðu=2Þ
ð21Þ
The energy equilibrium equation of the system in one folding wavelength is
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Fig. 5. Deformed shapes and thickness reduction (%) plots of a X-shaped elements with u = 60° (A) top view of deformed shape; (B) top view of feature lines; (C) general view; (D) left view; and (E) front view.
B
A
C
Fig. 6. Folding model of panels in an X-shaped element deforming in mode s.
According to the stationary condition of mean crushing force,
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u u tanðu=2Þ 4 t þ 0:5 0:5 H ¼ pB= h B h cosðu=2Þ
ð25Þ
and the mean crushing force is
Pm ¼
2
j
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r0 B0:5 h1:5 p tanðu=2Þ þ 4p secðu=2Þðh=BÞ0:5
ð26Þ
For dynamic loading, we have
Pmd ¼
2k
j
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rd B0:5 h1:5 p tanðu=2Þ þ 4p secðu=2Þðh=BÞ0:5
ð27Þ
3. Numerical analysis and discussion
Fig. 7. Analogy between folding of panels that crush each other and compression of panels with rigid plane.
Pm 2H j ¼ W bending þ W membrane þ W 0rolling
ð22Þ
Substitute Eqs. (17), (18), (21) into Eq. (22). We have
Pm 2H j ¼ 8pM 0 B þ
8M 0 H2 tanðu=2Þ 32M 0 H2 þ 0:5 0:5 h B h cosðu=2Þ
ð23Þ
The expression can be rearranged as
j Pm M0
¼
! 4pB tanðu=2Þ 4 þ þ 0:5 0:5 4H H h B h cosðu=2Þ
ð24Þ
To validate the proposed theoretical models for corner elements and X-shaped elements, numerical analyses are carried out for these elements with different geometric configurations including angle, width and thickness. The theoretical models are proposed for inextensional mode of corner elements and collapse mode s of X-shaped elements. Therefore, initial imperfections or triggers are introduced here to induce the occurrence of corresponding deformation mode. The finite element models employed here are same as those adopted in literature [18] including the material and structural properties. The material is aluminum alloy AA6060 T4 and the mechanical properties of it are listed here: Young’s modulus E = 68.2 GPa, initial yield stress ry = 80 MPa, the ultimate stress ru = 173 MPa, Poisson’s ratio m = 0.3 and the power law exponent n = 0.23. The engineering stress–strain curve was also presented in literature [18]. The finite element models of
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X. Zhang, H. Zhang / Composite Structures 96 (2013) 143–152
els. The structures is modeled with Belytschko–Tsay 4-node shell element with five integration points through the thickness and one integration point in the element plane. The characteristic size of mesh is selected to be 1 or 2 mm based on the width B of the elements. The material AA6060 T4 is modeled with material model#123 in LS-DYNA [28]. Two types of contacts are employed in the numerical analysis. Automatic single surface contact is used to account for the contact of panels of angle elements while automatic node-to-surface contact is applied between the angle elements and the rigid plane. A coulomb friction coefficient of 0.15 is employed in all contact. 3.1. Validation of theoretical model for corner elements
A
B
Fig. 8. Finite element models for angle elements: (a) corner element and (b) Xshaped element.
corner element and X-shaped element are shown in Fig. 8. The form of the triggers introduced are also given in the figure. The structural elements are loaded by a rigid plane with a constant velocity of 10 m/s in the axial direction. Clamped boundary conditions are applied at the bottom of the elements and symmetric boundary condition is used in the unconnected end of the pan-
In the study of Zhang and Huh [10], numerical simulations were carried out for corner elements with different angles and widths. The central angle analyzed by them is between 30° and 150° while the width of panel ranges from 20 mm to 40 mm. One problem in their analyses is that the mean force of angle elements is overestimated because the length of the elements L = 180 mm is too short to obtain stable value of mean force. In the present work, the corner elements are analyzed again with the length of L = 300 mm and the triggers are introduced with a depth of 0.2 mm at 18 mm below the top end. The mean forces of the corner elements obtained by numerical analyses are listed in Table 1. Not all elements deform in inextensional mode during loading and the
Table 1 Comparison of theoretical and numerical results for Pm of corner elements. Test
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 a b
h (°)
30 45 60 75 90 105 120 135 150 30 45 60 75 90 105 120 135 150 30 45 60 75 90 105 120 135 150 90 90 90 90 90 90 90 90 90 90 90 90
B (mm)
20 20 20 20 20 20 20 20 20 30 30 30 30 30 30 30 30 30 40 40 40 40 40 40 40 40 40 20 20 20 20 30 30 30 30 40 40 40 40
h (mm)
1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 0.6 1.2 1.8 2.4 0.6 1.2 1.8 2.4 0.6 1.2 1.8 2.4
Angle elements with extensional mode. Angle elements with non-compact deformation mode.
Numerical Pm (N)
1389.0 1653.5 1721.0 1780.9 1810.3 1832.0 1838.6 1895.1 1626.2a 1548.2 1771.4 1848.3 1836.2 1882.9 1966.4 1961.2 1993.8 1954.4 1778.2 1978.9 2075.1 2117.8 2081.6 2096.4 2188.3 2201.0 2197.4 513.0 1810.3 4062.6 7431.5 628.0b 1882.9 4321.7 7482.1 773.3b 2081.6 4638.2 7786.9
Eq. (28)
Eq. (29)
Pm (N)
Error (%)
Pm (N)
Error (%)
1422.8 1659.3 1774.6 1836.4 1872.4 1894.6 1908.8 1917.9 1923.6 1543.0 1799.5 1924.5 1991.6 2030.6 2054.6 2070.0 2079.9 2086.1 1634.4 1906.1 2038.5 2109.5 2150.8 2176.3 2192.6 2203.1 2209.6 537.7 1872.4 3884.7 6520.0 583.1 2030.6 4212.9 7070.8 617.7 2150.8 4462.4 7489.6
2.4 0.4 3.1 3.1 3.4 3.4 3.8 1.2 18.3 0.3 1.6 4.1 8.5 7.8 4.5 5.5 4.3 6.7 8.1 3.7 1.8 0.4 3.3 3.8 0.2 0.1 0.6 4.8 3.4 4.4 12.3 7.1 7.8 2.5 5.5 20.1 3.3 3.8 3.8
1468.5 1712.6 1831.6 1895.4 1932.5 1955.4 1970.0 1979.5 1985.4 1681.0 1960.4 2096.7 2169.7 2212.2 2238.4 2255.1 2265.9 2272.7 1850.2 2157.7 2307.7 2388.1 2434.8 2463.7 2482.1 2494.0 2501.4 608.7 1932.5 3798.5 6135.4 696.8 2212.2 4348.2 7023.2 766.9 2434.8 4785.8 7730.1
5.7 3.6 6.4 6.4 6.8 6.7 7.1 4.5 22.1 8.6 10.7 13.4 18.2 17.5 13.8 15.0 13.7 16.3 4.0 9.0 11.2 12.8 17.0 17.5 13.4 13.3 13.8 18.7 6.8 6.5 17.4 11.0 17.5 0.6 6.1 0.8 17.0 3.2 0.7
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X. Zhang, H. Zhang / Composite Structures 96 (2013) 143–152
mean crushing force of elements developing extensional mode are marked with (). Based on the theoretical results of Magee and Thornton [23] and Wierzbicki and Abramowicz [16] for right corner elements, two groups of the parameters a1, a2 and a3 can be determined for the rolling radius in Eq. (7). According to Eq. (14), the flow stress r0 for AA6060 T4 is calculated to be 106 MPa and a1 can be determined to be 0.82 by equating Eq. (12) with a quarter of Eq. (13) at h = 90°. Therefore, a1, a2 and a3 are set to 0.82, 0.6 and 0.06 respectively based on the results of Magee and Thornton [23]. In addition, a1, a2 and a3 has been determined to be 0.163, 1/3 and 0.06 respectively according to the results of Wierzbicki and Abramowicz [16]. Therefore, the mean crushing force can be predicted by
PmdM
or
k ¼ rd B0:2 h1:8 2j PmdW ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p tanðh=2Þ 0:82ðtanðh=2Þ þ 0:06= tanðh=2ÞÞ
k rd B1=3 h5=3 2j sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p tanðh=2Þ 0:163ðtanðh=2Þ þ 0:06= tanðh=2ÞÞ
ð28Þ
simulations are conducted for right angle elements with thickness ranging from 0.6 to 2.4 mm. As shown in Table 1, the theoretical predictions by Eq. (28) are still in good agreement with numerical results. The error is generally below 8% except for element #31 and #36. These two elements have the minimum and maximum C/h (=2B/h) ratio in all elements. The C/h ratio is 16.7 and 133.3 for element #31 and #36 respectively. By observing the collapse mode of elements, it is found that all the elements develop inextensional mode except that the elements #32 and #36 develop a type of non-compact deformation mode as described by Reid et al. [29]. The deformed shapes of element #32 and #36 are given in Fig. 9 and an experimental photo of non-compact deformation mode is also presented. The non-compact deformation mode is reported to occur when the C/h ratio is high [29]. The C/h ratio of element #32 and #36 is 100 and 133.3. Apparently, it is reasonable for these elements to develop non-compact deformation mode, which leads to the big error between analytical and numerical results. As for element #31, the reason for the error of 12.3% is not quite clear, but the low C/h ratio is believed to be one of the reasons. For Eq. (29), it gives a relatively poor prediction when compared to Eq. (28). However, the error is lower than 20%.
ð29Þ
Before the application of Eqs. (28) and (29), dynamic flow stress rd of the material and the dynamic enhancement factor k should be determined. Since AA6060 T4 is strain rate insensitive material, dynamic flow stress rd can be replaced by r0 and the dynamic enhancement factor k can be set to 1.2 for corner elements as proposed by Langseth and Hopperstad [27]. By applying Eqs. (28) and (29), the theoretical predictions of mean forces for corner elements are given in Table 1. It can be found that the theoretical results predicted by Eq. (28) show very good agreement with the numerical results while Eq. (29) overestimates the mean crushing force. For Eq. (28), although the central angle h varies from 30° to 150° and width B ranges from 20 mm to 40 mm, the error is generally smaller than 5%. The maximum error is 18.3%, which comes from the element #9 with B = 20 and h = 150°. This is reasonable since extensional mode is developed for this element and the theoretical model is developed for inextensional mode in the present work. For the elements developing inextensional mode, the maximum error is 8.5%. For Eq. (29), the error is generally between 5% and 20%. The overestimation of Eq. (29) to some extent demonstrate that the exponent of h in the expression of mean force should be 1.8 to get better predictions for corner elements. To verify the effectiveness of Eqs. (28) and (29) for corner elements with different wall thickness, a group of supplementary
3.2. Validation of theoretical model for X-shaped elements Numerical analyses are carried out for X-shaped elements with width B ranging from 10 mm to 20 mm and thickness h between 0.3 mm and 1.2 mm. Since the fold wavelength 2H of X-shaped elements is very much lower than corner elements, the length of the structures is selected to be L = 180 mm which is enough for X-shaped elements to obtain stable value of mean force. The angle u of the elements analyzed here is between 30° and 90° with an interval of 15°. Very small angle u may lead to some numerical problem in contact. The triggers are introduced with a depth of 0.1 mm and the position of the triggers below the top end is adjusted according to the width B and thickness h of the elements since the differences for the fold wavelengths of X-shaped elements with different geometric parameters are much big here. The mean forces of the X-shaped elements obtained by numerical analyses are listed in Table 2. The numerical simulations show that not all X-shaped elements analyzed deform in collapse mode s although triggers are introduced to induce its occurrence. Almost all elements with angle u 6 75° develop collapse mode s, while the collapse modes of crisscross elements are various. This is reasonable and can be expected since the initial trigger only takes effect in the first fold and crisscross elements will switch easily to other mode if the chance of all collapse modes is equal. In addition, an interesting
Fig. 9. Non-compact deformation mode: (A) element #32 (C/t = 100); (B) element #36 (C/t 133.3); and (C) experimental test (C/t 100) (Reid et al. [29]).
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X. Zhang, H. Zhang / Composite Structures 96 (2013) 143–152 Table 2 Comparison of theoretical and numerical results for Pm of X-shaped elements. Test
Angle u(°)
B (mm)
h (mm)
Numerical Pm (N)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
30 45 60 75 90 30 45 60 75 90 30 45 60 75 90 30 45 60 75 90 30 45 60 75 90 30 45 60 75 90 30 45 60 75 90 30 45 60 75 90 30 45 60 75 90
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
0.3 0.3 0.3 0.3 0.3 0.6 0.6 0.6 0.6 0.6 0.9 0.9 0.9 0.9 0.9 0.3 0.3 0.3 0.3 0.3 0.6 0.6 0.6 0.6 0.6 0.9 0.9 0.9 0.9 0.9 0.6 0.6 0.6 0.6 0.6 0.9 0.9 0.9 0.9 0.9 1.2 1.2 1.2 1.2 1.2
298.5 359.4 403.2 443.1 475.2 899.1 1128.2 1259.1 1382.3 1682.0 1843.0 2337.4 2537.6 2798.2 3129.9 348.7 421.9 468.5 505.6 532.3 1057.8 1271.0 1424.7 1522.1 1640.9 2106.0 2578.6 2990.7 3291.9 3318.5 1170.7 1423.6 1579.7 1726.5 1847.0 2358.2 2842.3 3147.8 3403.1 3698.7 3777.5 4590.2 5154.6 5607.2 6451.3
Front view
Side view
Front view
A
Eq. (27) (j = 0.73)
Eq. (27) (Numerical j)
Pm (N)
Error (%)
j
Pm (N)
Error (%)
318.6 346.3 376.7 411.1 451.6 1028.0 1102.4 1186.7 1284.6 1402.2 2049.6 2182.8 2336.6 2518.2 2738.9 363.2 398.3 436.1 478.3 527.4 1164.1 1258.0 1362.2 1481.6 1623.2 2313.0 2480.5 2670.0 2890.3 3154.9 1274.3 1385.2 1506.7 1644.4 1806.4 2525.3 2722.7 2943.2 3196.6 3498.3 4112.0 4409.7 4746.7 5138.4 5608.7
6.7 3.7 6.6 7.2 5.0 14.3 2.3 5.8 7.1 16.6 11.2 6.6 7.9 10.0 12.5 4.1 5.6 6.9 5.4 0.9 10.0 1.0 4.4 2.7 1.1 9.8 3.8 10.7 12.2 4.9 8.8 2.7 4.6 4.8 2.2 7.1 4.2 6.5 6.1 5.4 8.9 3.9 7.9 8.4 13.1
0.79 0.75 0.72 0.72 0.69 0.77 0.75 0.72 0.69 0.61 0.74 0.72 0.70 0.69 0.59 0.80 0.76 0.73 0.73 0.71 0.79 0.75 0.74 0.72 0.72 0.76 0.75 0.71 0.69 0.67 0.79 0.77 0.73 0.74 0.71 0.77 0.76 0.73 0.69 0.70 0.75 0.73 0.72 0.70 0.68
294.8 338.1 380.7 418.8 480.9 976.0 1071.5 1195.8 1367.9 1681.1 2021.9 2226.9 2421.4 2647.1 3382.5 331.4 384.8 435.4 477.6 538.9 1075.7 1231.7 1351.9 1497.5 1650.8 2218.4 2428.7 2732.4 3057.9 3454.6 1177.5 1318.9 1513.6 1617.3 1854.4 2401.1 2611.4 2947.6 3371.0 3671.6 3984.6 4409.7 4797.8 5392.8 6040.8
1.2 5.9 5.6 5.5 1.2 8.5 5.0 5.0 1.0 0.1 9.7 4.7 4.6 5.4 8.1 5.0 8.8 7.1 5.5 1.2 1.7 3.1 5.1 1.6 0.6 5.3 5.8 8.6 7.1 4.1 0.6 7.4 4.2 6.3 0.4 1.8 8.1 6.4 0.9 0.7 5.5 3.9 6.9 3.8 6.4
Side view
B
Front view
Side view
C
Fig. 10. Deformed shapes of X-shaped elements: (A) element #23; (B) element #24; and (C) element #25.
feature is observed for deformed shapes of X-shaped elements with collapse mode s. In Fig. 10, the deformed shapes of three elements #23–#25 with angle u = 60°, 75° and 90° are presented. It
can be found that the front view of deformed panels of element #23 is almost symmetric, however, this symmetry is broken with u increasing. For element #24, the deformed lobes of the two
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X. Zhang, H. Zhang / Composite Structures 96 (2013) 143–152
panels that crush each other are slightly staggered and the extent of stagger is increased for element #25. This phenomenon reflects the increase in possibility of X-shaped elements to switch to other modes with u increasing. Here, we still classify this mode as collapse mode s despite the occurrence of the stagger. The theoretical expression for mean crushing force of X-shaped elements is given in Eq. (27). The dynamic flow stress rd of the material is the same as r0 and is equal to 106 MPa. However, the coefficient of effective crushing distance j and the dynamic enhancement factor k is not determined yet. The coefficient of effective crushing distance j is assigned to be 0.73 as an average value of X-shaped elements Since there is no experimental or theoretical results available. Another method to determine the coefficient j is to extract the effective crushing distance from numerical results. As for dynamic enhancement factor k, the same value 1.2 as corner elements is adopted here. Both the theoretical predictions with j = 0.73 and j extracted from numerical results are presented in Table 2. It can be found that Eq. (27) with j extracted from numerical simulations gives a relatively much better prediction of the mean crushing force. The error is generally below 10%. The results of Eq. (27) with j = 0.73 also compare very well with the numerical results and the error is generally below 15%. It should be mentioned that the collapse modes of crisscross elements are various, that is, the proposed theoretical model may give a reasonable prediction for mean force of crisscross elements without considering the possible collapse mode. This may be due to the fact that the mean forces of different collapse modes are close. It can be concluded that the theoretical model can give a good prediction for the crush resistance of X-shaped elements. 4. Conclusion Theoretical and numerical analyses on the crush resistance of rhombic and kagome honeycombs are conducted in this paper. The energy absorption characteristics of constitutive angle elements of the honeycombs, that is corner elements and X-shaped elements, are theoretically modeled and analyzed. Since the angle elements may deform in various collapse modes, theoretical models are only presented for the most frequently occurring collapse modes of these elements which are inextensional mode of corner elements and collapse mode s of X-shaped elements. Finite element analyses are carried out for angle elements with different geometric configurations including angle, width and thickness. A comparison between theoretical predictions and numerical results shows that the proposed theoretical models can give a very good prediction for the mean crushing force of angle elements. Experimental studies on the crush resistance of honeycombs and angle elements with different configurations are required and of great importance in future research. Acknowledgements The present work was supported by National Natural Science Foundation of China (No. 11002060) and the Fundamental Research Funds for the Central Universities, HUST (No. 2010QN002).
References [1] Gibson LJ, Ashby MF. Cellular solids: structure and properties. Cambridge: Cambridge University Press; 1997. [2] Yamashita M, Gotoh M. Impact behavior of honeycomb structures with various cell specifications – numerical simulation and experiment. Int J Impact Eng 2005;32(1–4):618–30. [3] Xue Z, Hutchinson JW. Crush dynamics of square honeycomb sandwich cores. Int J Numer Meth Eng 2006;65(13):2221–45. [4] Deqiang S, Weihong Z, Yanbin W. Mean out-of-plane dynamic plateau stresses of hexagonal honeycomb cores under impact loadings. Compos Struct 2010;92(11):2609–21. [5] Aktay L, Johnson AF, Kröplin B-H. Numerical modelling of honeycomb core crush behaviour. Eng Fract Mech 2008;75(9):2616–30. [6] Xu S, Beynon JH, Ruan D, Lu G. Experimental study of the out-of-plane dynamic compression of hexagonal honeycombs. Compos Struct 2012;94(8):2326–36. [7] McFarland RK. Hexagonal cell structures under post-buckling axial load. Aiaa J 1963;1(6):1380–5. [8] Wierzbicki T. Crushing analysis of metal honeycombs. Int J Impact Eng 1983;1(2):157–74. [9] Zhang X, Cheng G, Zhang H. Theoretical prediction and numerical simulation of multi-cell square thin-walled structures. Thin Wall Struct 2006;44(11): 1185–1191. [10] Chen W, Wierzbicki T. Relative merits of single-cell, multi-cell and foam-filled thin-walled structures in energy absorption. Thin Wall Struct 2001;39(4): 287–306. [11] Kim HS. New extruded multi-cell aluminum profile for maximum crash energy absorption and weight efficiency. Thin Wall Struct 2002;40(4):311–27. [12] Hou S, Li Q, Long S, Yang X, Li W. Multiobjective optimization of multi-cell sections for the crashworthiness design. Int J Impact Eng 2008;35(11): 1355–1367. [13] Zhang X, Zhang H. Numerical and theoretical studies on energy absorption of three-panel angle elements. Int J Impact Eng 2012;46:23–40. [14] Najafi A, Rais-Rohani M. Mechanics of axial plastic collapse in multi-cell, multi-corner crush tubes. Thin Wall Struct 2011;49(1):1–12. [15] Abramowicz W. Macro element method on crashworthiness of vehicles. In: Ambrosio J, editor. Crashworthiness – energy management and occupant protection. NewYork: Springer Wien; 2001. p. 291–320. [16] Wierzbicki T, Abramowicz W. On the crushing mechanics of thin-walled structures. J Appl Mech-T ASME 1983;50(4a):727–34. [17] Abramowicz W, Jones N. Dynamic progressive buckling of circular and square tubes. Int J Impact Eng 1986;4(4):243–70. [18] Zhang X, Huh H. Crushing analysis of polygonal columns and angle elements. Int J Impact Eng 2010;37(4):441–51. [19] Zhang X, Zhang H. Experimental and numerical investigation on crush resistance of polygonal columns and angle elements. Thin-wall Struct 2012;57:25–36. [20] Ohkubo Y, Akamatsu T, Shirasawa K. Mean crushing strength of closed-hat section members SAE Prepr; 1974 [740040]. [21] Song HW, Fan ZJ, Yu G, Wang QC, Tobota A. Partition energy absorption of axially crushed aluminum foam-filled hat sections. Int J Solids Struct 2005;42(9–10):2575–600. [22] Abramowicz W, Jones N. Dynamic axial crushing of square tubes. Int J Impact Eng 1984;2(2):179–208. [23] Magee CL, Thornton PH. Design considerations in energy absorption by structural collapse. Trans SAE 1978;87:2041–55. [24] Santosa SP, Wierzbicki T, Hanssen AG, Langseth M. Experimental and numerical studies of foam-filled sections. Int J Impact Eng 2000;24(5):509–34. [25] Calladine CR, English RW. Strain-rate and inertia effects in the collapse of two types of energy-absorbing structure. Int J Mech Sci 1984;26(11–12):689–701. [26] Tam LL, Calladine CR. Inertia and strain-rate effects in a simple plate-structure under impact loading. Int J Impact Eng 1991;11(3):349–77. [27] Langseth M, Hopperstad OS. Static and dynamic axial crushing of square thinwalled aluminium extrusions. Int J Impact Eng 1996;18(7–8):949–68. [28] Hallquist J. LS-DYNA User’s Manual. LS-DYNA 970 ed. Livermore Software Technology Corporation; 2003. [29] Reid SR, Reddy TY, Gray MD. Static and dynamic axial crushing of foam-filled sheet metal tubes. Int J Mech Sci 1986;28(5):295–322.
Contents lists available at SciVerse ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Theoretical and numerical investigation on the crush resistance of rhombic and kagome honeycombs Xiong Zhang a,c,⇑, Hui Zhang b a
Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China School of Mechanical Engineering and Automation, Wuhan Textile University, Wuhan 430073, Hubei, PR China c Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, Luoyu Road 1037, Wuhan 430074, PR China b
a r t i c l e
i n f o
Article history: Available online 9 October 2012 Keywords: Honeycomb Multi-cell Columns Out-of-plane properties Energy absorption
a b s t r a c t The out-of-plane crushing properties of rhombic and kagome honeycombs are studied in this paper by dividing the whole structure into basic angle elements: corner elements and X-shaped elements. Two theoretical models are presented to analyze the energy absorption mechanisms for inextensional mode of corner elements and one of collapse modes of X-shaped elements respectively. Expressions are derived to predict the mean crushing force of the angle elements. Numerical simulations are also carried out for angle elements under out-of-plane crushing by using nonlinear finite element code LS-DYNA. Crush resistance of angle elements with different geometric configurations including angle, width and thickness is analyzed. A comparison between theoretical predictions and numerical results shows that the theoretical models can effectively predict the mean crushing force of angle elements with a very good accuracy. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Honeycombs are widely applied in aerospace and transport engineering as core of sandwich or energy absorbers due to their excellent multifunctional properties. Up to now, the regular hexagonal honeycomb with two of the six cell walls having double thickness is the most extensively used commercial products although honeycombs with other sections such as square, triangle, rhomb and kagome can also be fabricated by various methods [1]. The out-of-plane crushing resistance of honeycombs is very much higher than their in-plane resistance and attracts the attention of many researchers. However, most of the researches are focused on the numerical and experimental aspects of the out-of-plane crushing properties of honeycombs [2–6] and there is little theoretical work being published in the open literature to study the crushing resistance of them with various sections. This is due to the complex nonlinear features and difficulties in energy dissipation analyses during the post-buckling and large plastic deformation behavior of honeycombs under crushing loading. Hexagonal and square honeycombs are the only two types of honeycomb structures being addressed theoretically [7–9]. The
⇑ Corresponding author at: Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China. Tel.: +86 27 87543538; fax: +86 27 87543501. E-mail address: [email protected] (X. Zhang). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.09.028
axial crushing stress of hexagonal honeycomb was firstly theoretically predicted by McFarland [7] and then studied by Wierzbicki [8]. The plateau stress of square honeycomb was predicted by Zhang et al. [9] with a simplified theoretical model based on Super Folding Element (SFE) method. Recently, thin-walled multi-cell metal extrusions were found to be highly efficient energy absorption structures and received extensive research interests [9–14]. Just like honeycombs, the same problem encountered is the lack of theoretical expressions for the crushing resistance of columns with various sections. In fact, honeycombs can also be considered as a kind of multi-cell columns with complex section and therefore any advance in the theory aspect can be employed for both type of structures. Although it is very challenging and difficult, there is a way out. No matter how complex the section of a multi-cell column is, it is constituted by a number of shells or plates connected with various angles and by different connection factors. A basic idea to obtain the crush resistance of a column with complex section under axial compression is to divide the column into a number of representative structural elements, determine the crush resistance of each element separately and finally sum up the contribution of every element [15]. However, to finish this process, first of all, we must know the crush resistance of each element. In this paper, the out-of-plane crushing resistances of rhombic and kagome honeycombs as shown in Fig. 1 are studied by using this basic concept. The constitutive structural elements of rhombic and kagome honeycombs are also given in Fig. 1c. There are two type of
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X. Zhang, H. Zhang / Composite Structures 96 (2013) 143–152
(a) Rhombic honeycomb
(b) Kagome honeycomb
(c) Angle elements
Fig. 1. Scheme of rhombic and kagome honeycomb and their constitutive angle elements.
constitutive angle elements: corner element and X-shaped element. The central angle of corner element can be acute, right or obtuse, which is determined by the configuration of honeycombs. The most familiar one of them is corner element with right angle which is constitutive elements of a square or rectangular column. A square or rectangular column can develop two types of deformation mode during axial loading: Type I: (quasi-)inextensional mode and Type II: extensional mode. Theoretical expressions for the mean crushing force of a right angle corner element deforming in these two modes can be found in literatures [16,17] for both static and dynamic loading. These expressions have been validated by lots of experiments and can be employed to give a rather good prediction for energy absorption of structures consisting of right angle corner elements. So far there are few works to study the crushing response of corner elements with acute or obtuse angle. Actually, the classic theoretical model of right angle corner elements that proposed by Wierzbicki and Abramowicz [16] can also be applied to acute and obtuse angle elements. However, recently it is found that this model may overestimate the influence of central angle on energy absorption. Based on this model, a 57.5% increase of mean force is predicted for inextensional mode when central angle is increased from 90° to 120°. By adopting finite element method, Zhang and Huh [18] investigated the influence of angle on the energy absorption of corner elements and X-shaped and Y-shape angle elements. They found that when inextensional mode is developed, the crushing resistance of corner elements increases gradually with central angle increasing from 30° to 90°, and then keeps almost invariant for obtuse angle less than 150°. Most recently, experimental investigations was carried out by Zhang and Zhang [19] to study the crush resistance of angle elements by axial compression of polygonal and rhombic columns. It was verified that the crushing resistance of angle elements almost kept constant for obtuse angle between 90° and 135°. Therefore, the existing models may not be applied to acute or obtuse angle elements. The numerical analysis of Zhang and Huh [18] showed that the angle u (see Fig. 1) had significant influence on energy absorption of X-shaped angle elements. The mean force is found to be increased by more than 50% when u increases from 30° to 90°. However, there is also no theoretical model to analyze this type of structural element except for u = 90°. Therefore, to study the out-of-plane crushing resistances of rhombic and kagome honeycombs, two analytical models have to be established to deal with corner element and X-shaped angle element respectively. This is the main task of the present work and since it’s quite challenging and difficult, some assumptions are adopted to simplify the analyses just as what was done in literatures [9,10].
2. Theoretical analysis To predict the crushing resistance of an element, theoretical model has to be established to take into consideration all the energy dissipation mechanisms during deformation of it. Both corner elements and X-shaped angle elements can deform in more than one collapse mode and therefore the analytical models have to be established according to these collapse modes. Just as right angle corner elements, the corner elements with arbitrary angle can generally deform in inextensional mode, extensional mode and mixed mode. Theoretically besides mixed mode, Xshaped angle elements can also develop four collapse modes as shown in Fig. 2. It’s impossible to establish all models in the present work and therefore the most commonly encountered collapse modes are selected to be analyzed here. According to Zhang and Huh [18], the acute corner elements tend to deform in inextensional mode and obtuse corner elements are inclined to develop extensional mode. However, pure extensional mode is developed only for very large angle (larger than 135° for instance). Therefore, inextensional mode of corner element will be analyzed here. As for X-shaped angle elements, numerical simulations in the later section show that mode s is the most familiar one when u – 90° and this mode will be investigated in the present work.
Fig. 2. Four possible collapse modes of X-shaped elements.
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2.1. Corner elements The classic kinematically admissible model of Wierzbicki and Abramowicz [16] can give a good prediction of the mean crushing force of right angle corner elements. In their model, the corner elements are divided into trapezoidal rigid surfaces and cylindrical, conical and toroidal deformable zones with moving hinge lines and energy dissipated in each region is analyzed. Since corner elements with acute and obtuse angle cannot be predicted well by this model, a simplified approach similar to Chen and Wierzbicki [10] are employed here. The basic folding element proposed will only consist of stationary hinge lines and triangular regions passed by travelling hinge lines. The mean crushing force can be derived by analyzed the energy dissipation in one fold wavelength 2H and the folding wavelength is the length of one complete fold. Different researchers adopted different definition in their theoretical model. In Fig. 3, two definitions of folding wavelength are denoted by H and H0 respectively and apparently H0 is two times of H. In the model of Wierzbicki [8], 2H is defined as the length of one fold while 2H0 is used instead for example in the models of Ohkubo et al. [20] and Song et al. [21]. In the present model, 2H is defined as folding wavelength. A representative corner element with panel width of B and height of 4H is shown in Fig. 4A. According to the model of Wierzbicki [8], the energy is dissipated by the bending of static hinge lines, the rolling of moving hinge lines and the membrane deformation during the formation of toroidal surface. As we know, if the rolling radius keeps constant during the move of a hinge line, the bending energy for the plastic hinge line to travel a distance will be
W rolling ¼ 2M 0 DS=r
ð1Þ
2
where M0 = r0 h /4 is the fully plastic bending moment per unit width, r0 and h represents the flow stress and wall thickness respectively. DS is the area passed by the hinge line and r is the rolling radius of curvature. At the same time, if the same area is formed by extensional or compressional deformation, the energy dissipated will be
W membrane ¼
Z
r0 hds ¼ 4M0 DS=h
ð2Þ
DS
It is interesting to find that the energy dissipated by rolling of plastic hinge line has the same form as the strain energy of membrane deformation if r is equal to h/2. During the axial compression of cor-
H' H
ner elements, the real rolling radius of curvature is quite complicated since it may vary with the thickness, central angle or other geometric parameters, and even vary with time during loading. On the other hand, the area controlled by membrane deformation is also hard to determine. It is reported by Wierzbicki [8] that the membrane energy is dissipated in the formation of toroidal surface. The calculation of this part of energy is also derived in their work and the extensional area is obtained as 4Hb/I1 (refer to the literature for definition of b, I1). However, there may still have other channels of membrane energy dissipation. Numerical analyses in the later section find that the thickening of panel wall in the corner region dissipates considerable amount of membrane energy although it is hard to quantify it. To simplify the analyses, in the following derivation, these two parts of energy dissipation as given by Eqs. (1) and (2) are integrated into one part as
W 0rolling ¼ 2M 0 DS=r 0
ð3Þ
And consequently, the energy dissipation is now constituted by two parts: bending energy of static hinge lines and the integrated rolling energy of moving hinge lines. The integrated rolling energy W 0rolling dissipated during one wavelength crushing will be calculated according to the areas that are passed by travelling hinge lines. A flattened angle element is shown in Fig. 4B, the shaded regions represent the areas that are passed by hinge lines during loading and this ideal model is extracted from the real deformation of an angle element as given in Fig. 4C. A paper model is also presented in Fig. 4D to illustrate the collapse of the ideal model of angle element. We can see that no extensional deformation will be necessary for the ideal model to develop a complete fold otherwise the paper will break during deformation. The difference of the ideal model and the real angle element consists in the bending curvature of panel along hinge lines. The bending curvature radius of the ideal model is zero while it is a value in the order of magnitude of panel thickness for a real structure. The mean crushing force of the element can now be derived by considering the energy equilibrium of the system in one folding wavelength 2H. The external work done by compression will be dissipated by plastic deformation in bending and rolling. That is
Pm 2H j ¼ W bending þ W 0rolling
where Pm denotes the mean crushing force over the collapse of a fold and j is the coefficient of effective crushing distance. The real crushing distance is actually less than 2H and according to Abramowicz and Jones [22], it is around 73% of the wavelength for a square tube deformed in inextensional mode. The value j is assumed as constant to be 73% for all central angle in the following derivation. The bending energy Wbending is relatively easy to obtain and the determination of it can adopt the same way as Wierzbicki and Abramowicz [16] since the definition of folding wavelength is identical. The length of the stationary hinge lines in a corner element is 2B and the bending angle should be p in one fold compression. Consequently, the energy dissipation in bending is 2pM0B. According to Wierzbicki and Abramowicz [16], it should be doubled due to the clamped boundary condition of the folding mechanism. Based on the plastic deformation process of the structure, a different explanation for the double of bending energy was proposed recently by Zhang and Zhang [19]. Nevertheless, it is certain that Wbending of the two-panel right angle element should be doubled and it will be calculated as
W bending ¼ 4pM 0 B
ð5Þ W 0rolling
Fig. 3. Definition of folding wavelength.
ð4Þ
The integrated rolling energy is given by Eq. (3) and the area of the triangular shaded region in Fig. 4B is equal to H2 tan(h/2) (when B P H tan(h/2)). That is
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B
θ /2
4H
2H B
A
B
C
D
Fig. 4. (A) A representative angle element; (B) triangular regions passed by travelling hinge lines; (C) deformed shape of an angle element with central angle 120°; and (D) paper model of angle element.
DS ¼ H2 tanðh=2Þ
ð6Þ
Therefore
0
The determination of r is very important and difficult. In extreme cases, when h equals to 0° and 180°, the angle element can be considered as a plain plate. Only bending deformation will occur during loading and therefore the rolling or membrane energy should be zero. Based on this consideration, the rolling radius is assumed to have the following expression: a2
r 0 ðhÞ ¼ a1 B h
1a2
ðtanðh=2Þ þ a3 = tanðh=2ÞÞ
ð7Þ
where a1, a2 and a3 are constant coefficients. Based on this function expression, the rolling radius r0 will be infinite when h equals to 0° and 180°. This seems reasonable since the membrane energy will vanish for h = 0° and 180°. When h = 180°, the area calculated by Eq. (6) will be infinite. But actually, the area passed by rolling hinge lines should be 2B H since Eq. (6) is effective only when B P H tan(h/2). Consequently, the membrane energy will still vanish for h = 180°. When B < H tan(h/2), the shaded region becomes a trapezoid and specially a rectangular for h = 180° rather than a triangle. The area of it can be easily derived. However, to simplify the analysis, Eq. (6) is employed to calculate the area of membrane deformation in the whole range of h. Substitute Eqs. (6) and (7) into Eq. (3) and then substitute Eqs. (3) and (5) into Eq. (4), we have
Pm 2H j ¼ 4pM 0 B þ
2M 0 H2 tanðh=2Þ r 0 ðhÞ
ð8Þ
The mean crushing force can be obtained as
j Pm M0
¼
2pB H tanðh=2Þ þ H r0 ðhÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pBr 0 ðhÞ= tanðh=2Þ
ð10Þ
ð11Þ
Substitute Eq. (11) into Eq. (9), the mean crushing force of the corner element would be
1 Pm ¼ r0 B0:5 h2 2j
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p tanðh=2Þ r0 ðhÞ
ð12Þ
The determination of parameters a1, a2 and a3 is very important and it can be determined according to the experimental results or theoretical results validated by applications. Based on experimental data, Magee and Thornton [23] reported that the best fitted exponent of h in the expression of mean force was 1.8 for square or rectangular tubes. Therefore, a2 can be set to 0.6. The numerical analysis carried out by Zhang and Huh [18] and experimental tests conducted by Zhang and Zhang [19] show that the increase of mean crushing force is smaller than 5% when central angle is increased from 90° to 120°. Substitute h = 90° and h = 120° into Eq. (12), it is found that a3 should be smaller than 0.16 to keep the increase of mean crushing force less than 5%. Here a3 is set to 0.06 corresponding to a 2% increase in mean force which is based on the experimental data of Zhang and Zhang [19]. Now the last parameter is a1 which can be determined according to the theoretical and experimental results of square tubes. According to Magee and Thornton [23], the mean crushing force of square columns can be obtained as 1:8
ð9Þ
The folding wavelength H can then be determined by the stationary condition of the mean crushing force
@Pm ¼0 @H
H¼
Pm ¼ 17ru C 0:2 h
ð13Þ
where C is the width of a square column and ru is the ultimate tensile stress. It’s found that ru should be replaced by the average value of yield stress ry and ultimate stress ru to give a good prediction of the experiment results of square columns [19]. A square column is constituted by four right corner element with B = C/2. Substitute h = 90° into Eq. (12) and equate it with a quarter of Eq. (13), a1
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can be determined. In Eq. (12), the flow stress r0 for material with power law hardening can be calculated by [24]
r0 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ry ru
ð14Þ
ð1 þ nÞ
where n is the power law exponent. The above process for the determination of parameters a1, a2 and a3 is based on the work of Magee and Thornton [23] and it can also be done in a different way. According to Wierzbicki and Abramowicz [16], the mean crushing force for square columns deforming in inextensional mode is 5=3
Pm ¼ 13:10r0 C 1=3 h
ð15Þ
Therefore, a2 can be set to 1/3. Still a3 is set to 0.06 and then a1 can be determined to be 0.163 by equating Eq. (12) and a quarter of Eq. (15) at h = 90°. For dynamic loading, dynamic amplification effects including strain rate effect and inertia effect should be considered in the expressions [25,26]. Then, we have
Pmd
k ¼ rd B0:5 h2 2j
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p tanðh=2Þ r 0 ðhÞ
ð16Þ
where k is the dynamic enhancement factor and rd is the dynamic flow stress. For strain rate sensitive material, the dynamic flow stress rd is always assessed by empirical Cowper–Symonds constitutive equation. The dynamic enhancement factor k is due to the inertia forces and an investigation of it was carried out for square columns by Langseth and Hopperstad [27]. 2.2. X-shaped elements The deformation behavior of X-shaped elements is much more complicated than corner elements. Due to symmetry of the structure, only elements with 0 < u 6 90° need to be analyzed. As shown in Fig. 2, four collapse modes can be developed for an X-shaped element. However, the possibility of these four modes to occur is not equal. For a crisscross element (u = 90°), all these four modes may be developed due to initial imperfections or triggers, while mode s is the most likely one to occur for X-shaped elements with small u. Consequently, a theoretical model is established here to predict the crush resistance of X-shaped elements deforming in mode s. The deformed shapes of an X-shaped element with u = 60° are shown in 5. Similar to three-panel elements [13], the deformation mechanisms of X-shaped elements during crushing are very complicated and considerable amount of energy is dissipated by the thickening of panels. The contour plots of thickness reduction of X-shaped elements are also shown in Fig. 5. We can see that in the intersection region, the thickening of shell is significant and most of elements in this region get a 20–40% increase in thickness. Other mechanisms such as bending of static hinge lines, rolling of moving hinge lines and extensional deformation are also presented during deformation. The feature line plot (top view) of deformed structure is given in Fig. 5B. It can be found that the intersection line of the element almost deforms along the symmetric plane of the element which equally divide the supplementary angle of u. Based on this feature, an folding model of X-shaped element deforming in this mode is presented in Fig. 6. Two panels of the element are stretched extensionally as denoted by blue1 lines and the other two crush each other as shown in red lines. Although the intersection line is apparently shorter than the other four lines
around after deformation, it’s assumed in the model that the length of intersection line is invariant during deformation and it will be 2H for one fold. In fact, the shortening of the intersection line reflects the thickening of the panels in that region from another aspect and this part of energy should not be omitted. For the two panels crushing each other, the energy dissipation mechanism is basically the rolling of travelling hinge lines and the energy can be derived by Eq. (1). The rolling radius of curvature in the equation is also hard to determine just as for corner elements. It varies with geometric parameters including angles and panel thickness and even varies with time during loading. By adopting the same simplification as corner elements, the energy dissipated by thickening of the panels is integrated into the rolling energy of the travelling hinge lines and the rolling radius of curvature is assumed a function of angle, panel width and thickness. Energy dissipation of the element during loading can now be classified into three parts: bending energy of static hinge lines, the membrane energy of panel stretching and the integrated rolling energy during the crushing of two panels. The bending energy Wbending of an X-shaped element can be calculated as
W bending ¼ 8pM 0 B
ð17Þ
The membrane energy of panel stretching is calculated by Eq. (2) and the area that is extended during loading is denoted by the triangular shaded region in Fig. 6B. The area of the shaded region is H2 tan(u/2) and it should be doubled for one fold wavelength. Therefore, the membrane energy is
W membrane ¼ 4M 0 DS=h ¼ 8M 0 H2 tanðu=2Þ=h
The integrated rolling energy W r olling is given by Eq. (3) and the area passed by moving hinge lines are given by the triangular shaded region in Fig. 6C whose area is also equal to 2H2 tan(u/2) for one fold wavelength. That is
W 0rolling ¼ 4M 0 H2 tanðu=2Þ=r 0 ðuÞ
ð19Þ
As mentioned above, the determination of r0 is quite difficult. It may vary with geometric parameters and even time. Based on this consideration, the rolling radius of curvature is assumed to have the following expression: 1b2
r0 ðuÞ ¼ b1 Bb2 h
sinðu=2Þ
ð20Þ
where b1 and b2 are constant coefficients. According to Eq. (20), the rolling radius r0 is a increasing function of angle u. This is reasonable since the rolling radius should increase when the angle u is increasing. As shown in Fig. 7, two panels are folded with different inclined angle u1 and u2 and u1 > u2. The folding process of the panels that crush each other is similar to the loading process of panels compressed by a rigid plane. Apparently, the rolling radius of panel folded with inclined angle u1 should be larger than that of panel folded with inclined angle u2. As for the determination of coefficients b1 and b2, there is relatively few experimental or theoretical results to be referenced compared to corner elements. Most of the experiments and theoretical analyses were conducted for hexagonal honeycombs. Therefore, two values are suggested here to be 1/8 and 0.5 for b1 and b2 respectively. Actually, a simplified model has been proposed to analyze the crisscross element by Zhang et al. [9]. However, only bending and membrane energy were considered there and the membrane energy is not related to the width B of the element. The suggested values of b1 and b2 will give approximately same predictions for crisscross elements. Substitute b1 and b2 into Eq. (20) and then Eq. (19). We have
W 0rolling ¼ 32M0 H2 B0:5 h 1 For interpretation of color in Fig. 6, the reader is referred to the web version of this article.
ð18Þ
0
0:5
= cosðu=2Þ
ð21Þ
The energy equilibrium equation of the system in one folding wavelength is
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Fig. 5. Deformed shapes and thickness reduction (%) plots of a X-shaped elements with u = 60° (A) top view of deformed shape; (B) top view of feature lines; (C) general view; (D) left view; and (E) front view.
B
A
C
Fig. 6. Folding model of panels in an X-shaped element deforming in mode s.
According to the stationary condition of mean crushing force,
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u u tanðu=2Þ 4 t þ 0:5 0:5 H ¼ pB= h B h cosðu=2Þ
ð25Þ
and the mean crushing force is
Pm ¼
2
j
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r0 B0:5 h1:5 p tanðu=2Þ þ 4p secðu=2Þðh=BÞ0:5
ð26Þ
For dynamic loading, we have
Pmd ¼
2k
j
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rd B0:5 h1:5 p tanðu=2Þ þ 4p secðu=2Þðh=BÞ0:5
ð27Þ
3. Numerical analysis and discussion
Fig. 7. Analogy between folding of panels that crush each other and compression of panels with rigid plane.
Pm 2H j ¼ W bending þ W membrane þ W 0rolling
ð22Þ
Substitute Eqs. (17), (18), (21) into Eq. (22). We have
Pm 2H j ¼ 8pM 0 B þ
8M 0 H2 tanðu=2Þ 32M 0 H2 þ 0:5 0:5 h B h cosðu=2Þ
ð23Þ
The expression can be rearranged as
j Pm M0
¼
! 4pB tanðu=2Þ 4 þ þ 0:5 0:5 4H H h B h cosðu=2Þ
ð24Þ
To validate the proposed theoretical models for corner elements and X-shaped elements, numerical analyses are carried out for these elements with different geometric configurations including angle, width and thickness. The theoretical models are proposed for inextensional mode of corner elements and collapse mode s of X-shaped elements. Therefore, initial imperfections or triggers are introduced here to induce the occurrence of corresponding deformation mode. The finite element models employed here are same as those adopted in literature [18] including the material and structural properties. The material is aluminum alloy AA6060 T4 and the mechanical properties of it are listed here: Young’s modulus E = 68.2 GPa, initial yield stress ry = 80 MPa, the ultimate stress ru = 173 MPa, Poisson’s ratio m = 0.3 and the power law exponent n = 0.23. The engineering stress–strain curve was also presented in literature [18]. The finite element models of
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X. Zhang, H. Zhang / Composite Structures 96 (2013) 143–152
els. The structures is modeled with Belytschko–Tsay 4-node shell element with five integration points through the thickness and one integration point in the element plane. The characteristic size of mesh is selected to be 1 or 2 mm based on the width B of the elements. The material AA6060 T4 is modeled with material model#123 in LS-DYNA [28]. Two types of contacts are employed in the numerical analysis. Automatic single surface contact is used to account for the contact of panels of angle elements while automatic node-to-surface contact is applied between the angle elements and the rigid plane. A coulomb friction coefficient of 0.15 is employed in all contact. 3.1. Validation of theoretical model for corner elements
A
B
Fig. 8. Finite element models for angle elements: (a) corner element and (b) Xshaped element.
corner element and X-shaped element are shown in Fig. 8. The form of the triggers introduced are also given in the figure. The structural elements are loaded by a rigid plane with a constant velocity of 10 m/s in the axial direction. Clamped boundary conditions are applied at the bottom of the elements and symmetric boundary condition is used in the unconnected end of the pan-
In the study of Zhang and Huh [10], numerical simulations were carried out for corner elements with different angles and widths. The central angle analyzed by them is between 30° and 150° while the width of panel ranges from 20 mm to 40 mm. One problem in their analyses is that the mean force of angle elements is overestimated because the length of the elements L = 180 mm is too short to obtain stable value of mean force. In the present work, the corner elements are analyzed again with the length of L = 300 mm and the triggers are introduced with a depth of 0.2 mm at 18 mm below the top end. The mean forces of the corner elements obtained by numerical analyses are listed in Table 1. Not all elements deform in inextensional mode during loading and the
Table 1 Comparison of theoretical and numerical results for Pm of corner elements. Test
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 a b
h (°)
30 45 60 75 90 105 120 135 150 30 45 60 75 90 105 120 135 150 30 45 60 75 90 105 120 135 150 90 90 90 90 90 90 90 90 90 90 90 90
B (mm)
20 20 20 20 20 20 20 20 20 30 30 30 30 30 30 30 30 30 40 40 40 40 40 40 40 40 40 20 20 20 20 30 30 30 30 40 40 40 40
h (mm)
1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 0.6 1.2 1.8 2.4 0.6 1.2 1.8 2.4 0.6 1.2 1.8 2.4
Angle elements with extensional mode. Angle elements with non-compact deformation mode.
Numerical Pm (N)
1389.0 1653.5 1721.0 1780.9 1810.3 1832.0 1838.6 1895.1 1626.2a 1548.2 1771.4 1848.3 1836.2 1882.9 1966.4 1961.2 1993.8 1954.4 1778.2 1978.9 2075.1 2117.8 2081.6 2096.4 2188.3 2201.0 2197.4 513.0 1810.3 4062.6 7431.5 628.0b 1882.9 4321.7 7482.1 773.3b 2081.6 4638.2 7786.9
Eq. (28)
Eq. (29)
Pm (N)
Error (%)
Pm (N)
Error (%)
1422.8 1659.3 1774.6 1836.4 1872.4 1894.6 1908.8 1917.9 1923.6 1543.0 1799.5 1924.5 1991.6 2030.6 2054.6 2070.0 2079.9 2086.1 1634.4 1906.1 2038.5 2109.5 2150.8 2176.3 2192.6 2203.1 2209.6 537.7 1872.4 3884.7 6520.0 583.1 2030.6 4212.9 7070.8 617.7 2150.8 4462.4 7489.6
2.4 0.4 3.1 3.1 3.4 3.4 3.8 1.2 18.3 0.3 1.6 4.1 8.5 7.8 4.5 5.5 4.3 6.7 8.1 3.7 1.8 0.4 3.3 3.8 0.2 0.1 0.6 4.8 3.4 4.4 12.3 7.1 7.8 2.5 5.5 20.1 3.3 3.8 3.8
1468.5 1712.6 1831.6 1895.4 1932.5 1955.4 1970.0 1979.5 1985.4 1681.0 1960.4 2096.7 2169.7 2212.2 2238.4 2255.1 2265.9 2272.7 1850.2 2157.7 2307.7 2388.1 2434.8 2463.7 2482.1 2494.0 2501.4 608.7 1932.5 3798.5 6135.4 696.8 2212.2 4348.2 7023.2 766.9 2434.8 4785.8 7730.1
5.7 3.6 6.4 6.4 6.8 6.7 7.1 4.5 22.1 8.6 10.7 13.4 18.2 17.5 13.8 15.0 13.7 16.3 4.0 9.0 11.2 12.8 17.0 17.5 13.4 13.3 13.8 18.7 6.8 6.5 17.4 11.0 17.5 0.6 6.1 0.8 17.0 3.2 0.7
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mean crushing force of elements developing extensional mode are marked with (). Based on the theoretical results of Magee and Thornton [23] and Wierzbicki and Abramowicz [16] for right corner elements, two groups of the parameters a1, a2 and a3 can be determined for the rolling radius in Eq. (7). According to Eq. (14), the flow stress r0 for AA6060 T4 is calculated to be 106 MPa and a1 can be determined to be 0.82 by equating Eq. (12) with a quarter of Eq. (13) at h = 90°. Therefore, a1, a2 and a3 are set to 0.82, 0.6 and 0.06 respectively based on the results of Magee and Thornton [23]. In addition, a1, a2 and a3 has been determined to be 0.163, 1/3 and 0.06 respectively according to the results of Wierzbicki and Abramowicz [16]. Therefore, the mean crushing force can be predicted by
PmdM
or
k ¼ rd B0:2 h1:8 2j PmdW ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p tanðh=2Þ 0:82ðtanðh=2Þ þ 0:06= tanðh=2ÞÞ
k rd B1=3 h5=3 2j sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p tanðh=2Þ 0:163ðtanðh=2Þ þ 0:06= tanðh=2ÞÞ
ð28Þ
simulations are conducted for right angle elements with thickness ranging from 0.6 to 2.4 mm. As shown in Table 1, the theoretical predictions by Eq. (28) are still in good agreement with numerical results. The error is generally below 8% except for element #31 and #36. These two elements have the minimum and maximum C/h (=2B/h) ratio in all elements. The C/h ratio is 16.7 and 133.3 for element #31 and #36 respectively. By observing the collapse mode of elements, it is found that all the elements develop inextensional mode except that the elements #32 and #36 develop a type of non-compact deformation mode as described by Reid et al. [29]. The deformed shapes of element #32 and #36 are given in Fig. 9 and an experimental photo of non-compact deformation mode is also presented. The non-compact deformation mode is reported to occur when the C/h ratio is high [29]. The C/h ratio of element #32 and #36 is 100 and 133.3. Apparently, it is reasonable for these elements to develop non-compact deformation mode, which leads to the big error between analytical and numerical results. As for element #31, the reason for the error of 12.3% is not quite clear, but the low C/h ratio is believed to be one of the reasons. For Eq. (29), it gives a relatively poor prediction when compared to Eq. (28). However, the error is lower than 20%.
ð29Þ
Before the application of Eqs. (28) and (29), dynamic flow stress rd of the material and the dynamic enhancement factor k should be determined. Since AA6060 T4 is strain rate insensitive material, dynamic flow stress rd can be replaced by r0 and the dynamic enhancement factor k can be set to 1.2 for corner elements as proposed by Langseth and Hopperstad [27]. By applying Eqs. (28) and (29), the theoretical predictions of mean forces for corner elements are given in Table 1. It can be found that the theoretical results predicted by Eq. (28) show very good agreement with the numerical results while Eq. (29) overestimates the mean crushing force. For Eq. (28), although the central angle h varies from 30° to 150° and width B ranges from 20 mm to 40 mm, the error is generally smaller than 5%. The maximum error is 18.3%, which comes from the element #9 with B = 20 and h = 150°. This is reasonable since extensional mode is developed for this element and the theoretical model is developed for inextensional mode in the present work. For the elements developing inextensional mode, the maximum error is 8.5%. For Eq. (29), the error is generally between 5% and 20%. The overestimation of Eq. (29) to some extent demonstrate that the exponent of h in the expression of mean force should be 1.8 to get better predictions for corner elements. To verify the effectiveness of Eqs. (28) and (29) for corner elements with different wall thickness, a group of supplementary
3.2. Validation of theoretical model for X-shaped elements Numerical analyses are carried out for X-shaped elements with width B ranging from 10 mm to 20 mm and thickness h between 0.3 mm and 1.2 mm. Since the fold wavelength 2H of X-shaped elements is very much lower than corner elements, the length of the structures is selected to be L = 180 mm which is enough for X-shaped elements to obtain stable value of mean force. The angle u of the elements analyzed here is between 30° and 90° with an interval of 15°. Very small angle u may lead to some numerical problem in contact. The triggers are introduced with a depth of 0.1 mm and the position of the triggers below the top end is adjusted according to the width B and thickness h of the elements since the differences for the fold wavelengths of X-shaped elements with different geometric parameters are much big here. The mean forces of the X-shaped elements obtained by numerical analyses are listed in Table 2. The numerical simulations show that not all X-shaped elements analyzed deform in collapse mode s although triggers are introduced to induce its occurrence. Almost all elements with angle u 6 75° develop collapse mode s, while the collapse modes of crisscross elements are various. This is reasonable and can be expected since the initial trigger only takes effect in the first fold and crisscross elements will switch easily to other mode if the chance of all collapse modes is equal. In addition, an interesting
Fig. 9. Non-compact deformation mode: (A) element #32 (C/t = 100); (B) element #36 (C/t 133.3); and (C) experimental test (C/t 100) (Reid et al. [29]).
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X. Zhang, H. Zhang / Composite Structures 96 (2013) 143–152 Table 2 Comparison of theoretical and numerical results for Pm of X-shaped elements. Test
Angle u(°)
B (mm)
h (mm)
Numerical Pm (N)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
30 45 60 75 90 30 45 60 75 90 30 45 60 75 90 30 45 60 75 90 30 45 60 75 90 30 45 60 75 90 30 45 60 75 90 30 45 60 75 90 30 45 60 75 90
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
0.3 0.3 0.3 0.3 0.3 0.6 0.6 0.6 0.6 0.6 0.9 0.9 0.9 0.9 0.9 0.3 0.3 0.3 0.3 0.3 0.6 0.6 0.6 0.6 0.6 0.9 0.9 0.9 0.9 0.9 0.6 0.6 0.6 0.6 0.6 0.9 0.9 0.9 0.9 0.9 1.2 1.2 1.2 1.2 1.2
298.5 359.4 403.2 443.1 475.2 899.1 1128.2 1259.1 1382.3 1682.0 1843.0 2337.4 2537.6 2798.2 3129.9 348.7 421.9 468.5 505.6 532.3 1057.8 1271.0 1424.7 1522.1 1640.9 2106.0 2578.6 2990.7 3291.9 3318.5 1170.7 1423.6 1579.7 1726.5 1847.0 2358.2 2842.3 3147.8 3403.1 3698.7 3777.5 4590.2 5154.6 5607.2 6451.3
Front view
Side view
Front view
A
Eq. (27) (j = 0.73)
Eq. (27) (Numerical j)
Pm (N)
Error (%)
j
Pm (N)
Error (%)
318.6 346.3 376.7 411.1 451.6 1028.0 1102.4 1186.7 1284.6 1402.2 2049.6 2182.8 2336.6 2518.2 2738.9 363.2 398.3 436.1 478.3 527.4 1164.1 1258.0 1362.2 1481.6 1623.2 2313.0 2480.5 2670.0 2890.3 3154.9 1274.3 1385.2 1506.7 1644.4 1806.4 2525.3 2722.7 2943.2 3196.6 3498.3 4112.0 4409.7 4746.7 5138.4 5608.7
6.7 3.7 6.6 7.2 5.0 14.3 2.3 5.8 7.1 16.6 11.2 6.6 7.9 10.0 12.5 4.1 5.6 6.9 5.4 0.9 10.0 1.0 4.4 2.7 1.1 9.8 3.8 10.7 12.2 4.9 8.8 2.7 4.6 4.8 2.2 7.1 4.2 6.5 6.1 5.4 8.9 3.9 7.9 8.4 13.1
0.79 0.75 0.72 0.72 0.69 0.77 0.75 0.72 0.69 0.61 0.74 0.72 0.70 0.69 0.59 0.80 0.76 0.73 0.73 0.71 0.79 0.75 0.74 0.72 0.72 0.76 0.75 0.71 0.69 0.67 0.79 0.77 0.73 0.74 0.71 0.77 0.76 0.73 0.69 0.70 0.75 0.73 0.72 0.70 0.68
294.8 338.1 380.7 418.8 480.9 976.0 1071.5 1195.8 1367.9 1681.1 2021.9 2226.9 2421.4 2647.1 3382.5 331.4 384.8 435.4 477.6 538.9 1075.7 1231.7 1351.9 1497.5 1650.8 2218.4 2428.7 2732.4 3057.9 3454.6 1177.5 1318.9 1513.6 1617.3 1854.4 2401.1 2611.4 2947.6 3371.0 3671.6 3984.6 4409.7 4797.8 5392.8 6040.8
1.2 5.9 5.6 5.5 1.2 8.5 5.0 5.0 1.0 0.1 9.7 4.7 4.6 5.4 8.1 5.0 8.8 7.1 5.5 1.2 1.7 3.1 5.1 1.6 0.6 5.3 5.8 8.6 7.1 4.1 0.6 7.4 4.2 6.3 0.4 1.8 8.1 6.4 0.9 0.7 5.5 3.9 6.9 3.8 6.4
Side view
B
Front view
Side view
C
Fig. 10. Deformed shapes of X-shaped elements: (A) element #23; (B) element #24; and (C) element #25.
feature is observed for deformed shapes of X-shaped elements with collapse mode s. In Fig. 10, the deformed shapes of three elements #23–#25 with angle u = 60°, 75° and 90° are presented. It
can be found that the front view of deformed panels of element #23 is almost symmetric, however, this symmetry is broken with u increasing. For element #24, the deformed lobes of the two
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X. Zhang, H. Zhang / Composite Structures 96 (2013) 143–152
panels that crush each other are slightly staggered and the extent of stagger is increased for element #25. This phenomenon reflects the increase in possibility of X-shaped elements to switch to other modes with u increasing. Here, we still classify this mode as collapse mode s despite the occurrence of the stagger. The theoretical expression for mean crushing force of X-shaped elements is given in Eq. (27). The dynamic flow stress rd of the material is the same as r0 and is equal to 106 MPa. However, the coefficient of effective crushing distance j and the dynamic enhancement factor k is not determined yet. The coefficient of effective crushing distance j is assigned to be 0.73 as an average value of X-shaped elements Since there is no experimental or theoretical results available. Another method to determine the coefficient j is to extract the effective crushing distance from numerical results. As for dynamic enhancement factor k, the same value 1.2 as corner elements is adopted here. Both the theoretical predictions with j = 0.73 and j extracted from numerical results are presented in Table 2. It can be found that Eq. (27) with j extracted from numerical simulations gives a relatively much better prediction of the mean crushing force. The error is generally below 10%. The results of Eq. (27) with j = 0.73 also compare very well with the numerical results and the error is generally below 15%. It should be mentioned that the collapse modes of crisscross elements are various, that is, the proposed theoretical model may give a reasonable prediction for mean force of crisscross elements without considering the possible collapse mode. This may be due to the fact that the mean forces of different collapse modes are close. It can be concluded that the theoretical model can give a good prediction for the crush resistance of X-shaped elements. 4. Conclusion Theoretical and numerical analyses on the crush resistance of rhombic and kagome honeycombs are conducted in this paper. The energy absorption characteristics of constitutive angle elements of the honeycombs, that is corner elements and X-shaped elements, are theoretically modeled and analyzed. Since the angle elements may deform in various collapse modes, theoretical models are only presented for the most frequently occurring collapse modes of these elements which are inextensional mode of corner elements and collapse mode s of X-shaped elements. Finite element analyses are carried out for angle elements with different geometric configurations including angle, width and thickness. A comparison between theoretical predictions and numerical results shows that the proposed theoretical models can give a very good prediction for the mean crushing force of angle elements. Experimental studies on the crush resistance of honeycombs and angle elements with different configurations are required and of great importance in future research. Acknowledgements The present work was supported by National Natural Science Foundation of China (No. 11002060) and the Fundamental Research Funds for the Central Universities, HUST (No. 2010QN002).
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