Energy absorption properties of multi-layered corrugated paperboard in various ambient humidities

Materials and Design 32 (2011) 3476–3485

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Energy absorption properties of multi-layered corrugated paperboard in various ambient humidities Zhi-Wei Wang a,c,⇑, Yu-Ping E a,b,c a

Packaging Engineering Institute, Jinan University, 206 Qianshan Road, Zhuhai 519070, PR China College of Materials and Textiles, Zhejiang Sci-Tech University, Xiasha, Hangzhou 310018, PR China c Key Laboratory of Product Packaging and Logistics of Guangdong Higher Education Institutes, Jinan University, 206 Qianshan Road, Zhuhai 519070, PR China b

a r t i c l e

i n f o

Article history: Received 27 August 2010 Accepted 18 January 2011 Available online 18 February 2011 Keywords: B. Sandwich structures E. Environmental performance H. Material selection charts

a b s t r a c t This paper develops a mathematical model to depict the energy absorption properties of multi-layered corrugated paperboard (MLCP) in various ambient humidities. It is a piecewise function to model the energy absorptions corresponding to three deformation stages of MLCP (elastic stage, plateau stage and densification stage) separately. Simple formulas are derived for each stage which relating the energy absorption capacity of MLCP to the thickness-to-flute pitch ratio (t c =k) of corrugated-core cell, the mechanical properties of corrugated medium tested under a controlled atmosphere [23 °C and 50% relative humidity (RH)], and the RH. The theoretical energy absorption curves are then compared with experimental ones and good agreements are achieved for MLCP with wide range ratios of t c =k in various ambient humidities. Results of this research can be applied in the optimum design and material selection of cushioning packaging with multi-layered corrugated paperboard. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Multi-layered corrugated paperboard (MLCP, as shown in Fig. 1a) is a structure formed by gluing a specified number of paralleled single-wall corrugated paperboards, which are sandwich structures comprised of one layer of corrugated medium and two layers of linerboard (as illustrated in Fig. 1b). In recent years, MLCP has been gaining increasing attention as a replacement for polymeric materials in protective packaging of product. These benefited mainly from the environmentally friendly quality and the ability to absorb impact loading by converting it into plastic deformation energy. Therefore, the evaluation of cushioning and energy absorption property of MLCP is essential to engineering applications. Energy absorption diagram as an effective tool to characterise the energy absorption capacity of cushioning materials has been reported being successfully applied to plastic foam and aluminium alloy material [1–6]. Wang [7] then employed this tool to examine the energy absorption property of MLCP under an atmosphere of 20 °C in temperature and 65% in relative humidity (RH). In fact, MLCP is a paper-based packaging material and its energy absorption property may be influenced by the ambient humidity besides structural parameters. However, the existed literature in discussing the cushioning and energy absorption proper-

⇑ Corresponding author at: Packaging Engineering Institute, Jinan University, 206 Qianshan Road, Zhuhai 519070, PR China. Tel./fax: +86 756 8585200. E-mail address: [email protected] (Z.-W. Wang). 0261-3069/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2011.01.059

ties of MLCP were mostly focus on the structural factors [8–11], neglecting the impact of the relative humidity. Thus, this study aimed to investigate the effect of ambient humidity on the energy absorption characteristics of MLCP with various thickness-to-flute pitch ratios and to construct the relationship between the energy absorption of MLCP and ambient humidity, as well as the structural parameters thereof. For this purpose, the research work done by Wang and E [12] can serve as a reference for our study.

2. Modelling process 2.1. Simplified constitutive model of MLCP The energy absorption property of MLCP under flatwise compression (Fig. 1a, loading in T-direction) was investigated in this paper. A corrugated-core unit was shown in the enlarged drawing in Fig. 1a, where tc denotes the thickness of corrugated medium and k signifies the flute pitch, h and hc are the flute height and the thickness of single-wall corrugated board, respectively. The thickness of linerboard is negligible compared with the total thickness of corrugated board, thus hc approximately equals to h. The stress–strain curve of MLCP under flatwise compression exhibits four deformation stages (as can be seen in Fig. 2a), that is, elastic stage, first buckling stage, sub-buckling stage and densification stage. Elastic stage can be modelled by a perfect elastic material while the densification stage can be simplified as a vertical line. Compressive curve in first buckling stage and sub-buckling

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Nomenclature WC

rpC epC rp0 Ep

e0c eDc rpD

E

0

E

EsC E0sC

energy absorption per unit volume stress at arbitrary point on stress–strain curve strain at arbitrary point on stress–strain curve average crushing stress in first buckling stage slope of the curve in sub-buckling stage strain at the end of elastic stage densification strain of MLCP stress corresponding to the densification strain elastic modulus of MLCP tested under flatwise compression in arbitrary humidity environment elastic modulus of MLCP tested under flatwise compression in an environment of 23 °C and 50% RH elastic modulus of corrugated medium tested in arbitrary humidity environment elastic modulus of corrugated medium tested in an environment of 23 °C and 50% RH

rysC r0ysC W C =E0sC rpC =E0sC tc k t c =k h hc nc

yield strength of corrugated medium tested in arbitrary humidity environment yield strength of corrugated medium tested in an environment of 23 °C and 50% RH normalised energy absorption per unit volume normalised stress thickness of corrugated medium flute pitch thickness-to-flute pitch ratio of corrugated-core cell flute height thickness of single-wall corrugated paperboard number of single-wall corrugated boards in a multi-layered corrugated paperboard

Fig. 1. Structural diagram of (a) Multi-layered corrugated paperboard and (b) Single-wall corrugated board.

stage is a wave-like uplift curve, but in the simplified material model, these two stages were collectively referred to as plateau stage. Although it was not a plateau in the strict sense, we borrowed this term in contrast to the deformation stages of paper

honeycomb [12]. Constitutive relation of this stage can be modelled by curvilinear equation analogy to that of plastic foam [7,11]. However, in order to facilitate the research, wave-like uplift curve in this stage was simplified as a line with the slope of Ep. The

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Fig. 2. Typical stress–strain curve of MLCP under flatwise compression. (a) Original stress–strain curve (I. Elastic stage; II. First buckling stage; III. Sub-buckling stage; IV. Densification stage); (b) Simplified material model.

simplified material model of MLCP was shown in Fig. 2b. Thus, the stress–strain relationship of MLCP can be depicted in a piecewise function

8  0 6 epC 6 e0c > < rpC ¼ E epC ; rpC ¼ rp0 þ Ep ðepC  e0c Þ; e0c 6 epC 6 eDc > : rpC P rp0 þ Ep ðeDc  e0c Þ; epC ¼ eDc

fied material model proposed here is feasible and accurate in predicting the energy absorption of MLCP. 2.2. Model of energy absorption curve

ð1Þ

where rpC and epC are the stress and strain at arbitrary point on stress–strain curve, respectively; rp0 denotes the average crushing stress of the first buckling; E⁄ signifies the elastic modulus of MLCP; e0c is the strain at the end of elastic stage, and eDc signifies the densification strain. Energy absorption per unit volume of MLCP can be obtained and compared by integration of the area under the simplified and the original stress–strain curve prior to the onset of densification (shaded part in the inset of Fig. 3). As illustrated in Fig. 3, the data lied closer to the unit-slope straight line, indicating that the simpli-

2.2.1. Energy absorption in elastic stage Energy absorption per unit volume of MLCP in this stage can be calculated by the following formula

WC ¼

Z

epC

rpC depC ¼

0

1 r2pC : 2 E

ð2Þ

Normalised by E0sC , the elastic modulus of corrugated medium tested under a controlled environment (23 °C, 50% RH), then we obtain

WC E0sC

1 E0sC ¼ 2 E

rpC E0sC

!2 ;

ð3Þ

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0.10

y=x

0.06

0.04

Stress σ pC /MPa

Experiments of energy absorption per unit volume WC /J/cm3

0.08

0.02

Ep

σp0

Energy absorption per unit volume WC

ε0c 0

0.02

0.04

εDc

Strain εpC

0.06

0.08

0.10 3

Predictions of energy absorption per unit volume WC /J/cm

Fig. 3. Comparison of the total energy absorption per unit volume between the simplified stress–strain relationships and the experimental ones.

where E is the elastic modulus of MLCP in arbitrary RH. Defining the elastic modulus of MLCP tested under the controlled environment aforementioned as E0 , and then assuming the relative elastic modulus of MLCP (the elastic modulus ratio of MLCP tested in arbitrary atmosphere to that under the controlled atmosphere) changed with the ambient humidity as follows:

E

WC ¼ where

rp0 ðrpC  rp0 Þ 1 ðrpC  rp0 Þ2 Ep

¼ f1 ðRHÞ;

ð4Þ

where f1 (RH) is to be determined by experiments. The elastic modulus ratio of MLCP to corrugated medium varies proportionally to the cube of the thickness-to-flute pitch ratio [8], that is:

E0 E0sC

0

 k1

 3 tc ; k

ð5Þ

0

Where k1 is a constant. After substituting Eqs. (4) and (5) into Eq. (3), we have

WC E0sC

1 1 1  2 f1 ðRHÞ k01 ðtc =kÞ3

Let C 1C ¼ 12

WC E0sC

 C 1C

1 1 . f1 ðRHÞ k0 ðt c =kÞ3 1

rpC E0sC

rpC E0sC

!2 :

ð6Þ

Eq. (6) can be simplified as:

!2

:

ð7Þ

As can be seen from above equation, in elastic stage, the relationship between the normalised energy absorption per unit volume ðW C =E0sC Þ and the normalised stress ðrpC =E0sC Þ for MLCP with different t c =k ratios under various ambient humidities is a family of inclined lines with the same slop of 2 in logarithmic coordinates, but their intercepts depend upon the t c =k value of corrugated-core cell and the ambient humidity. 2.2.2. Energy absorption in plateau stage MLCP absorbs a predominant portion of kinetic energy in this stage. By neglecting the energy absorption in elastic stage, the energy absorption capacity can be approximately evaluated by

2

Ep

¼

1 2

r2pC  r2p0 Ep

ð8Þ

;

rp0 and Ep can be computed by following equations [13]

rP0 ¼ D1C rysC



E0

þ

 2   tc tc ¼ rysC g 1 ; k k

ð9Þ

Ep ¼ 2:2491rysC expð1:0448kÞ ¼ rysC g 2 ðkÞ; g 1 ðtkc Þ

ð10Þ

D1C ðtkc Þ2 ,

where ¼ and g 2 ðkÞ ¼ 2:2491 expð1:0448kÞ. D1C is a constant related to the configuration of the corrugated flute, and equals to 18.1923, 12.2437 and 11.3289 corresponding to the MLCP with A-flute, B-flute and C-flute, respectively; rysC signifies the yield strength of corrugated medium. Denoting r0ysC as the yield strength of corrugated medium tested under the controlled environment, then supposing the relative yield strength of corrugated medium (the yield strength ratios of corrugated medium measured in arbitrary environment to that under the controlled atmosphere) varied with the RH as follows:

rysC ¼ f ðRHÞ; r0ysC 2

ð11Þ

where f2 (RH) is an undetermined function. Substituting Eqs. (9)–(11) into Eq. (8) and then normalised by, the following is obtained:

WC E0sC

1 E0sC 1 1 ¼ 2 r0ysC f2 ðRHÞ g 2 ðkÞ E0

rpC E0sC

!2 

1 1 Let C 2C ¼ 12 r0sC f2 ðRHÞ , and C 3C ¼ 12 g 2 ðkÞ ysC can be rewritten as:

WC E0sC

¼ C 2C

rpC E0sC

1 r0ysC g 21 ðtc =kÞ f2 ðRHÞ: 2 E0sC g 2 ðkÞ r0ysC g 21 ðtc =kÞ E0sC

g 2 ðkÞ

ð12Þ

f2 ðRHÞ. Then, Eq. (12)

!2  C 3C :

ð13Þ

Therefore, in plateau stage, the energy absorption curve of MLCP with different tc =k ratios in various ambient humidities is a family of parabolas in linear coordinations, while a train of similar curves in logarithmic coordinates.

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2.2.3. Energy absorption in densification stage The energy absorbed in densification stage equals to that at the end of plateau stage, which can be predicted by substituting eDc into Eq. (13)

WC E0sC

¼ C 2C

rpD

!2  C 3C :

E0sC

ð14Þ

The stress corresponding to the densification strain () can be assessed by:

rpD ¼ rp0 þ Ep ðeDc  e0c Þ:

ð15Þ

Substituting Eq. (15) into Eq. (14), incorporating Eqs. (9) and (10), and ignoring the negligible t c =k small elastic strain e0c , we get

WC E0sC

 C 2C

r0ysC E0sC

!2

   2 tc þ g 2 ðkÞeDc  C 3C : f22 ðRHÞ g 1 k

ð16Þ

r0

2 2 2 tc Let C 4C ¼ C 2C ð EysC 0 Þ f2 ðRHÞðg 1 ð k Þ þ g 2 ðkÞeDc Þ  C 3C . Then, the normasC lised energy absorption per unit volume in densification stage can be represented as:

WC E0sC

 C 4C

ð17Þ

Thus, the energy absorption curve in densification stage is a family of horizontal lines; their levels are determined by the configuration of corrugated-core cell, the mechanical characteristics of corrugated medium and the RH. Combining Eqs. (7), (13), and (17), then we establish the energy absorption model for MLCP with any ratio in arbitrary ambient humidities:

8 rpC r rp 2 WC > 6 E0p0 > 0  C 1C ð 0 Þ ; > Es E0sC sC > EsC < rp0 rpC rpD rp 2 WC 0 ¼ C 2C ð 0 Þ  C 3C ; 0 6 0 6 0 EsC Es EsC EsC EsC > > > rpC rpD > : W0C  C 4C ; 0 P 0 E E E sC

sC

ð18Þ

sC

where

8 1 1 C 1C ¼ 12 f1 ðRHÞ > k01 ðtc =kÞ3 > > > > > E0 1 1 > > C 2C ¼ 12 r0sC f2 ðRHÞ > g 2 ðkÞ > ysC < 0 ysC E0sC

r

g 21 ðt c =kÞ

: C 3C ¼ 12 f ðRHÞ > g 2 ðkÞ 2 > > > > > r0 2 2 > 2 tc > C 4C ¼ C 2C ð EysC > 0 Þ f2 ðRHÞðg 1 ð k Þ þ g 2 ðkÞeDc Þ  C 3C > : sC

and flute valleys. Three common used types of corrugated-core, Aflute, B-flute and C-flute, were employed in this research, and they differed from each other in the flute pitch and the flute height (as shown in Fig. 1b). Multi-layered corrugated paperboard was obtained manually by gluing a designated number of identical single-wall corrugated boards. The inner-liner of single-wall corrugated board was pasted to the outer-liner of single-wall corrugated board next, and the flute peaks or valleys of each layer should keep on a straight line. The adhesive was PVA emulsion (white latex). Totally eight kinds of MLCP were used here, and the specifications of each kind of MLCP specimen were listed in Table 1. In the designation system of specimens, the number in middle of the symbol denotes the basis weight of corrugated medium (3, 5 and 7 stand for the corrugated medium with basis weight of 105, 120 and 145 g/m2, respectively), letter A on either side of these numbers signifies the linerboard with the basis weight of 150 g/m2, and letters A, B and C in parenthesis indicate the flute type. In order to investigate the influence of the tc =k ratio and the ambient humidity on the energy absorption properties of MLCP, the number of single-wall corrugated board for each MLCP specimen was designated as eight layers. Therefore, the total thicknesses of MLCPs with A-flute, B-flute and C-flute approximately equal to 40, 25 and 31 mm, correspondingly. 3.2. Experiment methods 3.2.1. Experiment equipments The test equipments involved were a temperature humidity programmable controller (GDJS-010, Zhongya Test Experiment Co. Ltd., Huaian, China) and two universal material testing machines with different full-scale load range (CMT 8502, used for testing the mechanical properties of corrugated medium, full-scale load range of 500 N, MTS Systems Co. Ltd., Shenzhen, China; and WDW-10C, used for testing the compression characteristics of MLCP, full-scale load range of 10 kN, Hualong Test Instruments Co. Ltd., Shanghai, China). 3.2.2. Experiment standards

ð19Þ

Above two equations connect the energy absorption capacity of MLCP with the thickness-to-flute pitch ratio of corrugated-core cell, the mechanical characteristics of corrugated medium and the RH. Therefore, we can predict the energy absorption capacity for a given MLCP applied in known humidity environments. 3. Experiment materials and method 3.1. Materials Both corrugated medium and single-wall corrugated board were supplied by the professional manufacturing industry of Jackson Packaging Co. Ltd. (Dongguan, China). The corrugated medium was made from recycled fibre, and grammages were 105, 120 and 145 g/m2 (the corresponding thickness are, respectively, 0.15, 0.19 and 0.22 mm). Single-wall corrugated board was composed of two layers of linerboard with 150 g/m2 Kraft paper and one layer of fluting made from corrugated mediums aforesaid, and they were glued together by starch adhesives along the outsides of the peaks

ISO 187-1990 Paper, board and pulps: standard atmosphere for conditioning and testing and procedure for monitoring the atmosphere and conditioning of samples [14]; GB/T 1040.1-2006 Plastic-determination of tensile propertiespart 1: general principles [15]; GB/T 8168–2008 Testing method of static compression for package cushioning materials [16]; GB/T 1453–2005 Test method for flatwise compression properties of sandwich constructions or cores [17].

Table 1 Specifications and characteristics of MLCP specimens (nc = 8). Specimen

Basic weight of linerboard (g/m2)

Basis weight of corrugated medium (g/ m2)

Thickness of corrugated medium tc (mm)

Flute type

Thickness of MLCP T (mm)

A3A(A) A3A(B) A3A(C) A5A(A) A5A(B) A5A(C) A7A(B) A7A(C)

150 150 150 150 150 150 150 150

105 105 105 120 120 120 145 145

0.15 0.15 0.15 0.19 0.19 0.19 0.22 0.22

A B C A B C B C

40 25 31 40 25 31 25 31

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3.2.3. Experiment conditions All specimens were pre-conditioned in the temperature humidity programmable controller to reach their own equilibrium moisture content in each experimental RH, i.e., 40%, 50%, 65%, 75%, 85% and 95%, at fixed temperature of 23 °C for 48 h prior to tests. The corrugated medium was tested to determine its elastic modulus and yield strength under different environmental conditions. All specimens were cut into the same area of 12.7  152 mm2, with a gauge length of 65 mm. The long edge of specimen was paralleled to the machine direction (MD) of corrugated medium. The tests were conducted under a constant displacement velocity of 1 ± 0.5 mm/min, and every test was conducted in six repeats. Linear regression of stress–strain curve between strains of 0.05 and 0.25 was defined as the elastic modu-

lus of corrugated medium, and the stress corresponding to strain of 0.3 was taken as the yield strength, as recommended by China National Standard GB/T 1040.1-2006. MLCP was tested under quasi-static compression under different environmental conditions to determine their energy absorption characteristics as response to ambient humidities. The uniaxial compression characteristics of MLCP may be affected by the dimension effect. Some researchers have studied the dimension effect in metal foam and metal honeycomb [18–20], the results advised that the boundary-layer effects will vanish when the specimens larger than seven cell sizes. Thus for corrugated-core cell, the side length of specimen should not less than 60 mm. By considering the dimension of the compression apparatus we employed in this study, the dimension of specimen was

1.2 105g/m2 120g/m2 145g/m2

1.0

0.8

0.6

0.4

0.2

0

40

50

60

70

80

90

100

Fig. 4. Relationship between the relative yield strength of corrugated medium and the ambient humidities.

1.2 Experiments Fitting curve 1.0

0.8

0.6

0.4

0.2

0

40

50

60

70

80

90

100

Fig. 5. Relationship between the relative elastic modulus of MLCP and the ambient humidities.

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Table 2 Densification strain of MLCP in different humidities (23 °C, nc = 8). Specimen

Thickness-to-flute pitch ratio t c =k

A3A(A) A3A(C) A5A(A) A5A(C) A3A(B) A7A(C) A5A(B) A7A(B)

0.0185 0.0228 0.0235 0.0288 0.0291 0.0334 0.0369 0.0427

Densification strain eDc 40% RH

50% RH

65% RH

75% RH

85% RH

95% RH

Average

0.7451 0.7760 0.7782 0.7545 0.7297 0.7673 0.7342 0.6945

0.7470 0.7729 0.7505 0.7414 0.7325 0.7670 0.7312 0.7101

0.7489 0.7791 0.7633 0.7653 0.7228 0.7526 0.7312 0.7115

0.7470 0.7808 0.7559 0.7560 0.7279 0.6769 0.7580 0.6619

0.7470 0.7765 0.7687 0.7530 0.7297 0.7525 0.7445 0.7015

0.7434 0.7689 0.7654 0.7725 0.7414 0.7269 0.7360 0.7099

0.7464 0.7757 0.7637 0.7578 0.7307 0.7155 0.7392 0.6983

settled to be 100  100 mm2 finally. The compressive load was applied with a constant displacement velocity of 12 ± 3 mm/ min. Every compression test was effectively conducted in five repeats, and each measurement was completed within 5 min after

the specimen being removed from the temperature humidity programmable controller. Linear regression of stress vs. strain data located in the elastic stage was defined as the elastic modulus of MLCP.

Fig. 6. Theoretical energy absorption curves of (a) MLCP with the same thickness-to-flute pitch ratio under various relative humidities; and (b) MLCP with different thickness-to-flute pitch ratios under a controlled atmosphere (23 °C, 50%RH).

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against the RH in Fig. 4, and the empirical expression of the function f2 (RH) was acquired by curve fitting,

4. Results and discussion 4.1. Mechanical characteristics of corrugated medium and MLCP

f2 ðRHÞ ¼ According to the experimental results, the elastic modulus and the yield strength of corrugated medium tested under the controlled environment were [13]

105 g=m2 : E0sC ¼ 2:943 GPa;

r0ysC ¼ 8:171 MPa; 120 g=m : ¼ 1:904 GPa; r0ysC ¼ 5:731MPa; 145 g=m2 : E0sC ¼ 2:582 GPa; r0ysC ¼ 7:349 MPa: 2

3483

E0sC

ð20Þ

The yield strength of corrugated medium in arbitrary environment was measured by experiments [13]. Then, the relative yield strength of corrugated medium was computed by using Eq. (11) and plotted

1:001 1 þ expðRH92:038 Þ 11:206

ð21Þ

The elastic modulus of MLCP in arbitrary and the controlled environment aforementioned were tested by experiments [13]. Then, the relative elastic modulus of MLCP was computed by using Eq. (4) and plotted against the RH in Fig. 5. By applying curve fitting to these data, the expression of the function f1 (RH) was obtained:

f1 ðRHÞ ¼

0:982  : 1 þ exp RH86:888 8:058

ð22Þ

According to Eq. (5), the elastic modulus ratio of MLCP to corrugated medium varies proportionally to the cube of the thickness-to-flute

Fig. 7. Energy absorption diagram of MLCP based on the effect of relative humidity (solid line stand for the contour with the same value of tc =k while dashed line represent the contour with constant level of RH).

Fig. 8. Comparison of the energy absorption curves of MLCP between the theoretical one and the experimental one (t c =k = 0.0334, 23 °C, 50%RH).

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pitch ratio [8]. Thus, a linear approximation dependency between the ratio of E0 to E0sC and the cube of t c =k value was obtained by experiments [13],

E

0

E0sC

 3 tc ¼ 2:908 : k

ð23Þ

The densification strain is defined as the onset of densification and taken to be the turning point from sub-buckling stage to densification stage. The densification strains of MLCP in various relative humidities were listed in Table 2. As can be seen from these data, the densification strain was inert to relative humidity, thus the experiment results under six levels of RH can be averaged and taken as the densification strain of this kind of MLCP. The average densification strains of MLCP with different configurations were inversely related to t c =k except for two small fluctuations, and their relationship can be represented as:

t k

eDc ¼ 0:801  2:139 c :

ð24Þ

4.2. Energy absorption diagram of MLCP Two steps were involved to acquire the energy absorption curves for a given MLCP. Firstly, parameters of C1C, C2C, C3C and C4C in Eq. (19) have to be computed with the help of the known expressions of f1 (RH) and f2 (RH) and the known values of tc =k, as well as the RH; and then substituting the calculated C1C, C2C, C3C and C4C into Eq. (18), the relationship between the normalised energy absorption in each stage and the normalised stress can be determined. Secondly, the normalised energy absorption per unit volume is plotted against the normalised stress for a range of thickness-to-flute pitch ratios in logarithmic coordinates by using MATLABÒ software (version 7.1 R14, The Math Works Inc., USA), and then the energy absorption curves were constructed. Energy absorption curves obtained from the theoretical model aforesaid were plotted to illustrate the energy absorption properties of MLCP with the specified thickness-to-flute pitch ratio of 0.0334 under different surrounding environments (respectively at

a

b

Fig. 9. Energy absorption curves obtained from experiments. (a) MLCP with the same thickness-to-flute pitch ratio under various relative humidities; (b) MLCP with different thickness-to-flute pitch ratios under a controlled atmosphere (23 °C, 50% RH).

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40%, 50%, 65%, 75%, 85% and 95% in RH with the temperature fixed at 23 °C) in Fig. 6a, and to illustrate the energy absorption properties of MLCP with different values of tc =k under the controlled environment in Fig. 6b. Inflexions of these curves signify the optimum points for energy absorption properties of MLCP. As illustrated in Fig. 6a, the inflexions of energy absorption curves for MLCP with the same t c =k ratio shift to the lower-left corner with the increasing of RH, suggesting a weakening of load carrying property and energy absorption capacity of MLCP. Fig. 6b indicates that the inflexions of energy absorption curves for MLCP used under the same relative humidity of 50% move to the higher-right corner as the increasing of the t c =k ratio. The energy absorption diagram of MLCP illustrated in Fig. 7 is constructed by connecting the inflexions of energy absorption curves with the same t c =k value and the inflexions of energy absorption curves having constant RH level respectively. As illustrated in this figure, energy absorption properties of MLCP increase proportionally to the t c =k ratios on the contours with constant RH levels, while inversely proportional to the ambient humidities on the contours with the same t c =k ratios. Eight kinds of MLCPs (thickness-to-flute pitch ratios are 0.0185, 0.0228, 0.0235, 0.0288, 0.0291, 0.0334, 0.0369 and 0.0427, respectively) were compressed under six levels of relative humidity (40%, 50%, 65%, 75%, 85% and 95% in RH, respectively, with the temperature fixed at 23 °C). Therefore, eight contours with the same value of and six contours having constant RH level were obtained from experiments. However, in order to make the energy absorption diagram of MLCP clear and distinguishable, only eight contours were shown in Fig. 7 to represent energy absorption properties of MLCPs with four t c =k ratios of 0.0185, 0.0235, 0.0291 and 0.0334 under four levels of RH (i.e., 40%, 65%, 85% and 95%). By using these contours, the optimal points of energy absorption for MLCP with arbitrary value of t c =k in prescribed humidity environment can be acquired by means of interpolation. 4.3. Verification of model Take the MLCP with the thickness-to-flute pitch ratio of 0.0334 for example, theoretical and experimental energy absorption curves were plotted and compared in Fig. 8. It can be seen that the theoretical energy absorption curve correlated closely with the experimental one particularly in plateau stage, indicating the energy absorption model established here can well simulate the energy absorption performance of MLCP. The theoretical curve located at the upper left of the experimental one in elastic stage, and lowered than the experimental one in densification stage. These differences were mainly caused by the simplification of material model of MLCP. Energy absorption curves of MLCP with different t c =k ratios tested under various ambient humidities (40%, 50%, 65%, 75%, 85% and 95%, respectively) were plotted in Fig. 9. These curves were converted from the original stress–strain curves by integrating technique using the MATLABÒ program, and then compared with the theoretical energy absorption curves in Fig. 6. Results show that the theoretical model can reflect the influence of relative humidity and thickness-to-flute pitch ratio on the energy absorption capacity of MLCP precisely. Furthermore, both theoretical and experimental energy absorption curves have proved that the energy absorption properties of MLCP decrease with the increasing of RH, and increase with the increasing of the thickness-to flute pitch ratio. 5. Conclusions A mathematical model was developed to predict the energy absorption properties of MLCPs with different thickness-to-flute

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pitch ratios under various ambient humidities simulating the actual logistics environment. The model was normalised by the elastic modulus of corrugated medium tested under the controlled atmosphere, which makes the diagram universal to MLCP made of different base materials. A correlation between the theoretical and experimental energy absorption curves indicated that the theoretical model can be used to evaluate the energy absorption properties of MLCP in various ambient humidities. Both theoretical and experimental energy absorption curves show that the energy absorption properties of MLCP decrease with the increasing of RH, and increase with the increasing of the thickness-to-flute pitch ratio. The model presented here can facilitate the construction of energy absorption curves of MLCP; therefore, we can quickly seek the optimal energy absorption points of MLCPs with different thickness-to-flute pitch ratios in various humidity conditions. Acknowledgements The authors acknowledge the financial support of this research by Project 50775100 supported by National Natural Science Foundation of China. References [1] Maiti SK, Gibson LJ, Ashby MF. Deformation and energy absorption diagrams for cellular solids. Acta Metal 1984;32(11):1964–75. [2] Prakash O, Sang H, Embury JD. Structure and properties of AL-SiC foam. Mat Sci Eng A – Struct 1995;199(2):195–203. [3] Gibson LJ, Ashby MF. Cellular solids: structure and properties. second ed. Cambridge: Cambridge University Press; 1997. [4] Avalle M, Belingardi G, Montanini R. Characterization of polymeric structural foams under compressive impact loading by means of energy-absorption diagram. Int J Impact 2001;25(5):455–72. [5] Lu GX, Yu TX. Energy absorption of structures and materials. Woodhead Publishing Ltd. and CRC Press LLC; 2003. [6] Wang ZH. Studies on the dynamic mechanical properties and energy absorption of aluminum alloy foams. Doctoral dissertation of Taiyuan University of Technology. Taiyuan, China; 2005 [in Chinese]. [7] Wang DM. Energy absorption diagram of multi-layer corrugated boards. J Wuhan Univ Technol (Mater Sci Ed) 2010;25(2):58–61. [8] Wang DM. Cushioning property and characteristic studies on honeycomb paperboards and corrugated paperboards. Doctoral Dissertation of Jiangnan University. Wuxi; 2007 [in Chinese]. [9] Pan XZ. Study on the structural properties of corrugated cardboard composites. Master’s Dissertation of Nanjing Forestry University. Nanjing; 2007 [in Chinese]. [10] Naganathan P, He J, Kirkpatrick J. The effect of compression of enclosed air on the cushioning properties of corrugated fiberboard. Packag Technol Sci 1999;12(2):81–91. [11] Minett M. A study of air flow effects on the cushioning characteristics of multi layered pre-compressed fiberboard. Doctoral dissertation of Victoria University. Melbourne, Australia; 2006. [12] Wang Z-W, E YP. Mathematical modelling of energy absorption property for paper honeycomb in various ambient humidities. Mater Des 2010;31(9):4321–8. [13] E YP. Influence of relative humidity and strain rate on the energy absorption properties of paper-based cushioning materials. Doctoral Dissertation of Jiangnan University. Wuxi; 2010 [in Chinese]. [14] ISO 187-1990 paper, board and pulps: standard atmosphere for conditioning and testing and procedure for monitoring the atmosphere and conditioning of samples. International Organization for Standardization; 1990. [15] GB/T 1040.1-2006 Plastic-determination of tensile properties-part 1: general principles. Beijing: Standards Press of China; 2006 [in Chinese]. [16] GB/T 8168-2008 Testing method of static compression for package cushioning materials. Beijing: Standards Press of China; 2008 (in Chinese). [17] GB/T 1453-2005 Test method for flatwise compression properties of sandwich constructions or cores. Beijing: Standards Press of China; 2005 [in Chinese]. [18] Andrews EW, Gioux G, Onck PR, et al. Size effects in ductile cellular solids. Part II: experimental results. Int J Mech Sci 2001;43(3):701–13. [19] Hayes AM, Wang AJ, Dempsey BM, et al. Mechanics of linear cellular alloys. Mech Mater 2004;36(8):691–713. [20] Onck PR. Scale effects in cellular metals. MRS Bull 2003;28(4):279–83.