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Abstract

At the request of the Author, the following article

Zhang, W, Hou, W, Hu, Ping and Ma, Z (2014) The Nonlinear Compressive Response and Deformation of an Auxetic Cellular Structure under In-Plane Loading Advances in Mechanical Engineering published 17 November 2014. doi: 10.1155/2014/214681has been retracted due to errors discovered by the authors.

On Page 3, the definition of component force in Equation 4 is incorrect.

(2) On Page 4, the definition of component force in Equation 11 is incorrect. Moreover this equation should not have parameterM(length of cell wall), because a mistake was made in the process of calculation.

Because of the above reasons, the conclusion obtained from the mechanical model is incorrect and should instead state that the Elastic Buckling and Plastic Collapse are both yield modes of this structure

(3) On Page 5, the FEA model used in this article is not appropriate, because the deformation of single cell is not homogeneous, which means that the geometrical non-linear effect on single cell model is greater. So in the actual whole structure we may not obtain the results that were described in Page 6 Paragraph 2.

(4) The data in figures 8 (page 6) and 9 (page 7) is incorrect and the values of effective Young’s modulus and plateau stress are much larger than reasonable values. The definition of effective stress is wrong in the FEA model, which means the effective stress should be calculated by the total width of cell instead of length of horizontal cell wall. For example, in Figure 8, the plateau stress can reach 140Mpa, this is not reasonable because there are many holes in this cellular structure, and its mechanical properties should be much lower than material properties of cell wall. The reasonable plateau stress should be around 2Mpa.

The authors takes responsibility for these errors and regret the publication of invalid results.

The nonlinear compressive response and deformation of an auxetic cellular structure that has periodic negative Poisson's ratio (NPR) cell were studied through theoretical and numerical method. The in-plane deformation and process of absorbing energy of this NPR cellular structure were discussed. The relative density of NPR cell was determined by geometric parameters and a theoretical approach to the strength of NPR cell was carried out to analyze the failure process under in-plane compression. Thus, to prove the effective of unit cell mechanical model, the size effect of this NPR cellular structure was discussed. The simulations of two different NPR cellular structures under in-plane compression were generated and their effective properties, strain-stress curves, and absorbed energy were analyzed. It can be concluded that the NPR cell that has lower relative density may have better capability of absorbing energy with enhanced Young's modulus. Finally the parametric analyses related to relative density were presented with NPR unit cell model and a set of quasistatic compressive experiments was carried out, which can be used to reveal the nonlinear properties of NPR cellular structure and to prove the effectiveness of theoretical method.

1. Introduction

The cellular structure plays an important role in most high performance racing cars, which attracts more and more scientists to study its topological structure and properties [1, 2]. Cellular structure is widely used to improve crashworthiness in the design of vehicle body because of lightweight and good energy absorption ability [1, 3, 4].

To work on properties of cellular structure, the most common method is to describe the effective mechanical properties by elastic constitutive of unit cell assuming that it is the continuous medium [5]. For cellular structure, the plateau stress is a very important indication to evaluate the performance of energy absorption. Gibson's research presented that the cellular structure which has excellent crashworthiness should have high, stable, and long-duration plateau stress [6]. Miller did some researches on the relationship between geometric parameters and plateau stress and evaluated the performance of energy absorption by limit compressive stress [7].

The traditional honeycomb structure has some disadvantages such as high peak stress, unstable plateau stress, and short duration. To improve these defects, a three dimensional negative Poisson's ratio (NPR) cellular structure was developed. Since the structure is made up of two dimensional cells, its performance is determined by in-plane properties of cell. Therefore, a lot of researchers worked on linear mechanical properties of this two-dimensional cell. It is worthwhile to note that the nonlinear properties are also important, so the failure mode and plateau stress are mainly discussed in this paper. Based on the analysis of nonlinear properties, the NPR cellular structure made up of these two-dimensional cells has negative effective Poisson's ratio and enhanced Young's modulus as well as the common advantages of traditional cellular structure. When the NPR cellular structure is under loading, the effective Poisson's ratio is less than zero, and the cell tends to be densified. Due to negative Poisson's ratio and enhanced Young's modulus, the NPR cell may reach higher limit compressive stress presenting higher plateau stress with lower density. To study the effect of geometric parameters on the plateau stress of cell, a mechanical analytical model and a finite element analysis (FEA) model were developed. The topological structure and deformation theory of cell were also included. Then a group of compressive experiments were carried out to prove the mechanical properties of the NPR cellular structure.

2. NPR Cellular Structure

NPR cellular structure is a three-dimensional cellular structure that includes negative Poisson's ratio cell, which, as shown in Figure 1, is a typical NPR cellular structure made up of an interconnected network of solid structure. The shape of two-dimensional NPR cell is shown in enlarged view of Figure 1. It is a two-dimensional structure, which can fill a plane area like honeycomb. More commonly, the cells can repeat in three directions of core structure to fill a space. Since the NPR cellular structure is only under loading parallel to symmetry axis of cell, the performance of NPR cellular structure is determined by in-plane properties of cell. Due to variable quantities of cells in every direction, the structure can present different mechanical properties in every direction to satisfy complex requirements of loading.

Figure 1:

NPR cellular structure and 2D representation of the 3D NPR cellular structure.

The side view of NPR cellular structure under in-plane compression is as shown in Figure 2. Under compression in y-direction, the long inclined cell wall deforms by stretching and bending and the short inclined cell wall deforms by compressing and bending. In the deformed diagram of Figure 2, the deformation of short inclined cell wall gives larger strain than that of long inclined cell wall in x-direction, which means that the overall strain of cell in x-direction is compressive strain. So the effective Poisson's ratio is negative. The strain-stress curve of NPR cellular structure is shown in Figure 3, which can be divided into three stages: elastic deformation, plateau, and densification [8]. In the elastic stage, the deformation occurs by bending of cell wall. Because of negative Poisson's ratio, the Young's modulus of NPR cellular structure can increase as the strain increases. And in the plateau stage, the strain-stress curve shows a long and stable period. The diagram of NPR cellular structure strain-stress curves are shown in Figure 3. The work per unit volume in deforming is the area under strain-stress curve up to the densification strain. Since very little energy is absorbed in the short elastic stage and the stress in the plateau stage is flat, the energy absorbed per unit volume can be simply defined as W = σ^•∊, where W is the energy absorbed per unit volume, σ^• is the plateau stress, and ∊ is the densification strain. In the following mechanical model, the plateau stress is the strength of cellular structure. So it can be defined as σ^• = σ_limit, where σ_limit is the stress when the NPR structure fails. In the figure, three curves indicate different effective densities of NPR cellular structures, which have the following relationships: ρ₁ > ρ₂ > ρ₃. Comparing ρ₁ with ρ₂, after the energy W is absorbed, the structure that has lower plateau stress reaches higher final stress. Comparing ρ₂ with ρ₃, the structure of ρ₃ has lower density, but it has higher plateau stress by negative Poisson's ratio and enhanced Young's modulus. It can be concluded from the above analysis that the cell should have lower final stress and higher plateau stress to improve the crashworthiness, and the NPR cellular structure that has lower density may have higher plateau stress.

Figure 2:

The side view of 3D NPR cellular structure.

Figure 3:

The stress-strain curves of different NPR cellular structures under in-plane compression.

3. NPR Cellular Structure Cell Mechanical Analytical Model

Based on the deformation analysis of NPR cellular structure, its mechanical analytical model under in-plane compression is shown in Figure 4. The properties of cellular structure depend on the relative density, which can be written as ρ*/ρ. A part of NPR cellular structure is chosen to calculate relative density, which is shown in the shadow area of Figure 4. So the relative density can be given by ρ*/ρ = A*/A, where A* is the area of cell wall in the shadow and A is the area of shadow. Through geometrical analysis, the relative density can be defined as ρ*/ρ = α(1 + β)(1 + K)L/(β + sin ⁡ φ)H, where α (thickness to length ratio) is defined as T_L = αL and T_M = αM and T_L and T_M are the thickness of long inclined cell wall and the thickness of short inclined cell wall, respectively, β (horizontal cell wall to inclined cell wall ratio) is defined as N = βL, L and M are the length of long inclined cell wall and length of short inclined cell wall, respectively, H is the height of cell, and N is the length of horizontal cell wall. Due to the requirements of NPR cellular structure, θ must be larger than φ, and M is determined by L, which can be written as K = M/L = sin ⁡ θsin ⁡ φ [9]. Therefore, the relative density is determined by α, β, and φ, and it can be concluded that α is proportional to relative density and β and φ are inversely proportional to relative density.

Figure 4:

The geometrical parameters NPR cell and its mechanical analytical model under in-plane compression.

3.1. Linear Elastic Deformation

Base on the conclusion of previous research [9], the effective Young's modulus E^• and Poisson's ratio ν^• can be defined as follows:

\begin{matrix} E^{•} = \frac{σ}{ε}, \\ ν^{•} = - \frac{ε_{y}}{ε_{x}}, \end{matrix}

(1)

where σ is the stress applied on the cell wall, ∊ is the effective strain given by stress, ∊_y is the effective strain in y-direction, and ∊_x is the effective strain in x-direction. In the research, we can conclude that the effective Young's modulus may increase under loading and the effective Poisson's ratio is negative.

3.2. Elastic Buckling

When a NPR cell is loaded in x-direction, the cell deforms by cell wall bending and compression. In Figure 4, the force P can be divided into two parts. The cell wall tends to bend by the force perpendicular to cell wall, which means the cell can fail by plastic hinges. At the same time the cell wall tends to compress by the force parallel to cell wall, which means the cell wall can fail by elastic buckling. So, in the process of compression, the reason why plateau occurs is that the cell wall fails by elastic buckling and plastic collapse. In the following analysis, the mechanical model is divided into two parts: compression model and bending model. The plateau stress can be determined by minimum limit stress of two models.

In the compressive mechanical model of cell, the long cell wall and short cell wall are under compression and stretching, respectively. And the long cell wall can be assumed as a beam that is constrained on one side and loaded on the other side. When the load exceeds the Euler critical load, the elastic buckling of cell occurs [10]. The Euler critical load of cell is given by:

P_{crit} = \frac{n^{2} π^{2} E_{s} I}{L^{2}},

(2)

where n is a coefficient that defines the type of constraint, which is between 0.5 and 2. When the constrained side can rotate freely, n is equal to 0.5. When the constrained side is fully fixed, n is equal to 2. E_s is Young's modulus of cell material. The second moment of inertia of the cell wall is defined as I = b(αL)³/12, where L is the length of long cell wall and b is the width of cell. Based on the geometrical analysis of cell, the component force of long cell wall in the direction of its axis is given by:

P_{y} = σ β L b \cos ϕ .

(3)

When P_y = P_crit, the elastic buckling occurs. The limit elastic buckling stress of cell is given by:

σ_{e l} = \frac{n^{2} π^{2} α^{3} E_{s}}{12 β \cos ϕ} .

(4)

3.3. Plastic Collapse

When the cell wall of NPR cellular structure tends to bend, the bending moment created by force that is perpendicular to cell wall may reach the fully plastic moment, giving two plastic hinges of each cell wall that are shown in Figure 4. This gives strain-stress curve a plateau at the limit stress. By setting the long cell wall and short cell wall as object, respectively, their loads under stress σ are given by:

\begin{matrix} P_{L} = σ β L b, \\ P_{M} = σ β L b . \end{matrix}

(5)

Therefore, the plastic works given by four plastic hinges 1, 2, 3, and 4 during a plastic rotation ϕ are given by:

\begin{matrix} W_{L} = 2 * \frac{1}{2} φ σ β L^{2} b \sin ϕ, \\ W_{M} = 2 * \frac{1}{2} φ σ β L^{2} b \sin θ . \end{matrix}

(6)

The plastic works given by plastic moment of cell are given by:

\begin{matrix} W_{P L} = 2 * \frac{1}{4} σ_{y s} b {t_{L}}^{2}, \\ W_{P M} = 2 * \frac{1}{4} σ_{y s} b {t_{M}}^{2}, \end{matrix}

(7)

where σ_ys is the yield strength.

When W = W_P, the fully plastic hinges occur. The limit stresses of long cell wall and short cell wall can be written as follows:

\begin{matrix} σ_{L} = \frac{σ_{y s} α^{2}}{2 β \sin ϕ}, \\ σ_{M} = \frac{σ_{y s} α^{2} \sin ϕ}{2 β \sin^{2} θ} . \end{matrix}

(8)

Since sin ⁡ φ < sin ⁡ θ and σ_L > σ_M, the limit stress of cell is determined by short cell wall. The plastic collapse limit stress can be written as follows:

σ_{p l} = \frac{σ_{y s} α^{2} M^{2}}{2 β L^{2} \sin ϕ} .

(9)

Comparing (4) with (9), elastic buckling can occur before plastic collapse under some conditions. When elastic buckling occurs before plastic collapse, the critical value can be determined by:

σ_{e l} = σ_{p l} .

(10)

Combining (4), (9), and (10) gives us the following:

α = \frac{6 \cos ϕ \sin ϕ σ_{y s}}{n^{2} π^{2} \sin^{2} θ E_{s}} .

(11)

For common NPR cell, φ = 30°, θ = 60°, and n = 1, (11) can be written as follows:

α \approx \frac{σ_{y s}}{E_{s}} .

(12)

For most materials, σ_ys/E_s is between 10⁻³ and 10⁻². Elastic buckling precedes plastic collapse, only when thickness to length ratio α is below 10⁻³. But most thickness to length ratio of NPR cell is between 0.05 and 0.3, so the NPR cell fails by bending of cell wall, which means the plastic collapse is the only failure mode of NPR cellular structure. The values of plateau stress given by FEA method and theoretical method are shown in Figure 5. The limit stress given by compressive model is larger than that given by bending model, so the plateau stress of NPR cell is only determined by plastic collapse. Comparing the results of FEA with that of plastic collapse in this figure, when the cell angle φ is smaller than 25°, the results of theoretical method shows good agreement with FEA method. Meanwhile, the effect of shear stress is ignored in the theoretical method, so the results of FEA method are smaller than those of theoretical method.

Figure 5:

The comparison of FEA and theoretical results with different cell angle φ.

Based on the analysis of plastic collapse, the limit stress of cell is inversely proportional to horizontal cell wall to inclined cell wall ratio and cell angle and proportional to thickness to length ratio.

4. NPR Cellular Structure Simulation and Analysis

By developing an FEA model, the effects of geometric parameters on plateau stress of NPR cellular structure cell were carried out. The FEA model of unit cell is shown in Figure 6. For the constants of L = 30 mm, M = 16 mm, and b = 2 mm, φ was taken between 10° and 35°, α was taken between 0.05 and 0.3, and β was taken between 0.1 and 0.6. The mesh size was determined by convergence requirements. The material in the simulation is reinforced PA66 whose Young's modulus is 2.2 GPa, yield stress is 150 MPa, and Poisson's ratio is 0.33. By fixing vertical displacement of bottom horizontal cell well, applying the distributed load on the top horizontal cell wall, and enforcing the periodic boundary conditions in x-direction, the FEA model of NPR cellular structure cell was carried out. Based on the FEA model, an application was developed to build parametric FEA models automatically, which can be used to study on the relationships between geometric parameters and plateau stress.

Figure 6:

FEA model of NPR cell under in-plane compression.

Beside the cell shape, material, and relative density, the properties of NPR structure also depend on the size effect, which means the ratio of NPR sample size to NPR cell size needs to be considered. Figure 7 shows the plateau stresses of NPR sample with different numbers of NPR cells. The plateau stress increases as the number of NPR cell increases, and its value tends to be flat when the number of NPR cell reaches seven. So if the number of NPR cell is less than seven, the size effect cannot be ignored in the design of cellular structure. In the following simulation, the FEA model of unit cell was only used to study the relationship between geometric parameters and plateau stress. So the results of simulation can be used as reference.

Figure 7:

The plateau stress plotted against the number of NPR cells with given geometrical parameters that α = 0.3, φ = 20^∘, and β = 0.1.

The relationships between geometric parameters and plateau stress of cell are shown in Figure 8. In these figures, the plateau stress is defined as the fact that its value is equal to the stress when the slope of strain-stress curve is zero. In Figure 8(a), the horizontal cell wall to inclined cell wall ratio β is taken as 0.1 and three curves in this figure indicate different thickness to length ratios α, which are taken as 0.1, 0.2, and 0.3. As the cell angle φ decreases, the plateau stress of cell increases. It is worthwhile to note that the cell with smaller cell angle has higher increasing rate of plateau stress. In Figure 8(b), the horizontal cell wall to inclined cell wall ratio β is taken as 0.1. The three curves indicate different cell angles, which are taken as 10°, 20°, and 30°. As the thickness to length ratio α increases, the plateau stress increases. When α is larger than 0.1, the relationship between α and plateau stress is almost linear. In Figure 8(c), the same conclusion can be presented: plateau stress is inversely proportional to horizontal cell wall to inclined cell wall ratio β. And the cell with smaller β has higher decreasing rate of plateau stress. Since the relative density of cell is inversely proportional to β and proportional to φ and α. The following conclusion can be presented: the plateau stress of NPR cellular structure is proportional to relative density.

Figure 8:

The plateau stress plotted against geometric parameters of NPR cell, where φ is cell angle, α is thickness to length ratio, and β is horizontal cell wall to inclined cell wall ratio.

The cell stress, effective Young's modulus, and energy absorbed per unit volume with respect to strain are shown in Figure 9. In these figures, the parameters of structure A are taken as α = 0.1, φ = 15°, and β = 0.1, and the parameters of structure B are taken as α = 0.2, φ = 27°, and β = 0.1, so the relative density of structure A is smaller than that of structure B. In Figure 9(a), when the strain is larger than 0.06, the two structures fail almost at the same strain. In the flat stage of Figure 9(a), it can be concluded that the plateau stress of structure A is larger than that of structure B. In the process of compression, as the cell angle φ decreases, plastic hinges are more difficult to appear based on the mechanical analysis. So the NPR cellular structure with lower relative density may present higher strength considering geometrical nonlinearity. The area between strain-stress curve and x-coordinate axis is the energy that is absorbed per unit volume, which is shown in Figure 9(b). When the strain is larger than 0.03, the energy absorbed per unit volume of structure A exceeds that of structure B, and its increasing rate is larger than that of structure B. In Figure 9(c), the effective Young's modulus E^• is calculated by E^• = Δσ/Δ∊, where Δσ is the increment of stress and Δ∊ is the increment of strain in Figure 9(a). It is worth noting that the equation above only works in the elastic stage, so the results in the plateau stage are only a reference. When the strain reaches 0.7, the value of effective Young's modulus is negative and very close to zero, which means the plastic collapse occurs. The initial effective Young's modulus (stiffness of cell) of structure A is lower than that of structure B. As the strain increases, the increasing rate and final value of effective Young's modulus of structure A is higher than that of structure B.

Figure 9:

The cell stress, effective Young's modulus, and energy per unit volume plotted against effective strain of cell, where the parameters of structure A and structure B are taken as α = 0.1, φ = 15°, and β = 0.1 and α = 0.2, φ = 27°, and β = 0.1.

5. In-Plane Compressive Experiments

A group of in-plane compressive experiments were carried out to prove analysis above. The samples in these experiments are made of reinforced PA66, and its material properties are defined in the simulation. These experiments were divided into two sets. In the first set, the three samples that are 2D single cell sample, 2D 3 * 3 sample, and 3D 2 * 3 * 6 sample all have same cell but the number of cells is different. The 2D samples were used to prove size effect. When the number of cells in one direction is relatively large, the 3D sample was used to avoid overall elastic buckling. For given parameters φ = 20°, α = 0.3, β = 0.1, b = 2 mm, L = 30 mm, and M = 16 mm, the samples and hydraulic experimental bench are shown in Figure 10. To simulate the process of quasistatic compression, the velocity of loading was set to 1 mm/min. In the second set, there are four samples that are single cell samples. These samples have different parameters, their thickness to length ratio α is taken as 0.1, 0.15, 0.2, and 0.25, and other parameters are same.

Figure 10:

In-plane compressive experiment and 2D 3 * 3 and 3D 2 * 3 * 6 NPR cellular structure samples.

The strain-stress curves of 2D 3 * 3 sample developed by FEA and experiment are shown in Figure 11. When the NPR cellular structure sample fails, the strain-stress curves have a plateau. Table 1 shows the calculation result of 2D 3 * 3 sample. Figure 12 shows FEA and experimental results with variable cell quantity and geometric parameters. It is worthwhile to note that the results of experiments are smaller than that of FEA. The reason why there is a discrepancy between FEA and experiment is that there are some manufacturing defects in the samples, but the agreement is acceptable.

Table 1:

The plateau stress of 3*3 NPR cellular structure sample.

	FEA	Experiment

The first peak value MPa	105.28	95.34
The average of first circle MPa	101.76	92.38
The second peak value MPa	115.97	108.45
The average of second circle MPa	109.12	98.32
Effective Poisson's ratio	−12.34	−11.97

Figure 11:

In-plane compressive stress-strain curve of 3 * 3 NPR cellular structure sample.

Figure 12:

The in plane plateau stress with a range of cell numbers and thickness to length ratios.

6. Conclusion

The analysis of failure process of NPR cellular structure is helpful to understand the theory of energy absorption. The limit stress of elastic buckling and plastic collapse are determined by theoretical approach. From the equations, it can be concluded that the elastic buckling occurs only when the thickness to length ratio α is below 10⁻³. So for common NPR cellular structure, its failure mode is plastic collapse.

Based on the analysis of theoretical model, the plateau stress of NPR cellular structure is very important for energy absorption under compression. By developing a parametric FEA model, the relationship between topological structure and plateau stress was carried out: the plateau stress is inversely proportional to horizontal cell wall to inclined cell wall ratio β and proportional to thickness to length ratio α and cell angle φ.

In general, lower weight and higher plateau stress is main target in the design of energy absorbing structure. Due to negative Poisson's ratio and enhanced Young's modulus, the NPR cellular structure can achieve this merit, which provides theoretical significance on improving performance of cellular structure.

Conflict of Interests

The authors declare that there is no conflict of interests regarding to the publication of this paper.

Footnotes

Acknowledgment

The authors wish to thank MKP Structural Design Associates, Ann Arbor, USA, for providing necessary financial assistance and experimental equipment.

References

Zarei

H. R.

and Kroger

, “Optimization of the foam-filled aluminum tubes for crush box application,” Thin-Walled Structures, vol. 46, no. 2, pp. 214–221, 2008.

Hou

Long

Yang

, and Li

, “Crashworthiness design for foam filled thin-wall structures,” Materials and Design, vol. 30, no. 6, pp. 2024–2032, 2009.

Buitrago

B. L.

Santiuste

Sánchez-Sáez

Barbero

, and Navarro

, “Modelling of composite sandwich structures with honeycomb core subjected to high-velocity impact,” Composite Structures, vol. 92, no. 9, pp. 2090–2096, 2010.

Cricri

Perrella

, and Calì

, “Honeycomb failure processes under in-plane loading,” Composites B: Engineering, vol. 45, no. 1, pp. 1079–1090, 2013.

Burlayenko

V. N.

and Sadowski

, “Effective elastic properties of foam-filled honeycomb cores of sandwich panels,” Composite Structures, vol. 92, no. 12, pp. 2890–2900, 2010.

Gibson

L. J.

and Ashby

M. F.

, Cellular Solids-Structure and Properties, Cambridge University Press, Cambridge, Mass, USA, 1997.

Miller

Smith

C. W.

, and Evans

K. E.

, “Honeycomb cores with enhanced buckling strength,” Composite Structures, vol. 93, no. 3, pp. 1072–1077, 2011.

Masters

I. G.

and Evans

K. E.

, “Models for the elastic deformation of honeycombs,” Composite Structures, vol. 35, no. 4, pp. 403–422, 1996.

Zhang

, and Hu

, “Mechanical properties of a cellular vehicle body structure with negative Poisson's ratio and enhanced strength,” Journal of Reinforced Plastics and Composites, vol. 33, no. 4, pp. 342–349, 2014.

10.

Chen

and Pugno

N. M.

, “In-plane elastic buckling of hierarchical honeycomb materials,” European Journal of Mechanics: A/Solids, vol. 34, pp. 120–129, 2012.