Stress Plateau of Multilayered Corrugated Paperboard in Various Ambient Humidities

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1 PACKAGING TECHNOLOGY AND SCIENCE Packag. Techno. Sci. 212; 25: Pubished onine 18 August 211 in Wiey Onine Library (wieyonineibrary.com).971 Stress Pateau of Mutiayered Corrugated Paperboard in Various Ambient Humidities By Yu-Ping E 1,2,3 and Zhi-Wei Wang 2,3 * 1 Coege of Materias and Texties, Zhejiang Sci-Tech University, Xiasha, Hangzhou 3118, China 2 Packaging Engineering Institute, Jinan University, 26 Qianshan Road, Zhuhai, Guangdong 5197, China 3 Key Laboratory of Product Packaging and Logistics of Guangdong Higher Education Institutes, Jinan University, 26 Qianshan Road, Zhuhai, Guangdong 5197, China This paper presents a mathematica mode to predict the stress pateau of mutiayered corrugated paperboard under fatwise compression in various humidity environments. The mode reates the stress pateau to the thickness-to-fute pitch ratio of corrugated core ce, the yied stress of corrugated medium and the reative humidity in surrounding air. Mutiayered corrugated paperboards with a wide range of thicknessto-fute pitch ratios are investigated under severa eves of ambient humidities to expore the effect of reative humidity on the stress pateau of mutiayered corrugated paperboard. Comparison of the predictions and experiments is made, and a good correation is achieved corroborating the feasibiity and accuracy of the mode. The proposed method can be used for practica appication of the optimum design and materia seection of mutiayered corrugated paperboard. Copyright 211 John Wiey & Sons, Ltd. Received 25 August 21; Revised 29 May 211; Accepted 7 June 211 KEY WORDS: mutiayered corrugated paperboard; stress pateau; reative humidity INTRODUCTION Mutiayered corrugated paperboard (MLCP) is an inexpensive, recycabe and readiy avaiabe engineering materia that can be taiored to a variety of end uses. It has been gaining increasing attention as a repacement for poymeric materias in protective packaging of product. Fatwise compression (Figure 1a, oading in T direction) is the common oading condition for these uses. The resutant stress strain curve exhibits three deformation stages, that is initia stiffening stage, centra stage and densification stage (as shown in Figure 1b). A ong pateau with fuctuations appears in this curve, and the number of fuctuations has a correation with the number of ayers in the MLCP. This pateau ends MLCP a character to resist externa oads in transport process. Therefore, a theoretica mode to predict the stress pateau of MLCP poses significant meaning for its engineering appication. The centra stage in stress strain curve incudes the first bucking (the first fuctuation in Figure 1b) and the sub-bucking going with oca coapse. This stage is characterized by a wave-ike upift and can be simpified to be an obique ine (dashed ine in Figure 1b; we caed it stress pateau curve, which is the mean ine of this wave-ike upift). Its constitutive reation satisfies the foowing equation: s ¼ s P þ E p e e c e c e e Dc (1) * Correspondence to: Z.-W. Wang, Packaging Engineering Institute, Jinan University, 26 Qianshan Road, Zhuhai, Guangdong 5197, China. E-mai: wangzw@jnu.edu.cn Contract/grant sponsor: Project supported by the Nationa Natura Science Foundation of China, project supported by the Key Laboratory of Product Packaging and Logistics of Guangdong Higher Education Institutes, and project Y supported by Science Foundation of Zhejiang Sci-Tech University (ZSTU). Copyright 211 John Wiey & Sons, Ltd.

2 188 Y.-P. E AND Z.-W. WANG a Load direction b I. Initia stiffening stage; II. centra stage; III. Densification stage I II III Stress E p T L P W c Strain Dc Figure 1. Structura diagram of (a) mutiayered corrugated paperboard and (b) typica stress strain curve of mutiayered corrugated paperboard under fatwise compression. where s and e are respectivey the stress and the strain at arbitrary point on the compressive curve in centra stage. s P is the average crushing stress of first bucking; it is the starting point of stress pateau curve. E p denotes the sope of stress pateau curve. e c signifies the maximum eastic strain, and e Dc is the densification strain. Wang 1,2 expored the constitutive reationship of MLCP in sub-bucking section by anaogy and comparison with eastic pastic foam, and the parameters in this reation were obtained by curve fitting of experimenta resuts. Therefore, it is an empirica reation that acks of theoretica anaysis. Research team from Victoria University has been activey invoved in research on the appication of mutiayered pre-compressed corrugated paperboard as cushioning materias for severa years, 3 6 but mechanica behaviours and cushioning properties of virgin corrugated paperboards differ widey with the pre-compressed corrugated paperboard. Their studies can serve as references for our study on mutiayered virgin corrugated paperboard. Mutiayered corrugated paperboard is a paper-based packaging materia, and its mechanica characteristics are sensitive to environmenta humidity in actua ogistics. Therefore, reative humidity (RH) in surrounding air is an important factor to affect the stress pateau of MLCP. However, itte information is avaiabe on this subject. Thus, the primary aim of this paper was to mode the reationship between the stress pateau of MLCP and the ambient humidity, as we as the structura parameters thereof. The mode was deveoped by anaysing the coapse mechanism and energy dissipation of corrugated core ces and the effect of RH on their mechanica behaviours. The accuracy of present mode was vaidated by comparing the predictions with the observations. MODELLING Unit ce of corrugated core In this paper, we focus on mutiayered virgin corrugated paperboard, which is formed by gueing a specified number of virgin corrugated paperboards (as shown in Figure 2a). Because of the periodic geometry of MLCP, a unit ce of corrugated core can be isoated as shown in Figure 2b. The fute profie can be simpified as a sinusoida 7 9 or cosinusoida 1 curve, a triange 11 or a trapezoida type, 12 and a tangent-arc-tangent shape. 13 However, fute peaks and vaeys wi fatten after gueing and converting process; thus, the actua fute profie is a simiar trapezoida type with the fattened peaks or vaeys connecting to the hypotenuse of the simiar trapezoid by arcs with a radius of r C as shown in Figure 2b, where h C and L C are respectivey the height and the ength of singe-wa corrugated board; signifies the fute pitch; the ength of the fattened peak and the hypotenuse are h and C, respectivey; θ C is the

3 STRESS PLATEAU OF MULTILAYERED CORRUGATED PAPERBOARD 189 a b Linerboard h Linerboard Peak Corrugated medium h C r C C C t C L C Fute pitch Vaey h/2 Figure 2. Structura diagram of (a) singe-wa corrugated board and (b) a unit ce of corrugated core. ange between the hypotenuse and the horizonta ine; and t C denotes the thickness of corrugated medium. Linerboard thickness is negigibe compared with the fute height; thus, the fute height is approximate to the thickness of singe-wa corrugated board. From the geometrica reationships shown in Figure 2b, we have ( h C ¼ C sin θ C þ 2r C ð1 cos θ C Þ 2 ¼ h þ 2r C sin θ C þ C cos θ C (2) Soving Equation (2), the foowing is obtained: 8 r C ¼ h C C sin θ C >< 21 ð cos θ C Þ ð h ¼ 2 C þ Þð1 cos θ C Þ 2h C sin θ C >: 21 ð cos θ C Þ (3) The assumptions for convenience of study are as foows. Linerboards are not incuded in the unit ce, and corrugated medium is uniform essentiay. The deformation pattern of the unit corrugated ce is ideaized as symmetrica. Its process is observed as foows: the ce was of unit ce enter the pastic deform stage after a momentary eastic deformation; four pastic hinge ines appear as the externa oad keeps increasing (Figure 3b), and they move in the direction shown by the arrows in Figure 3b unti the hypotenuse become vertica (Figure 3c); then the vertica ce was are crushed (Figure 3d); steadiy increasing externa force induces further squashing of the foded ce was and the apparition of crushing deformation for next ayer. According to Equation (1), two parameters are needed to determine the stress pateau of MLCP, i.e. the average crushing stress in first bucking and the sope of stress pateau curve. The former can be determined by anaysing the compressive coapse mechanism and energy dissipation of unit ce, whereas the atter is obtained by statistica anayses of experimenta data. Cacuation of energy dissipation The average crushing stress in first bucking is determined by energy method under the hypothesis that the work performed by the externa oad throughout the first bucking process is equa to the energy dissipated in form of pastic deformation, which incudes the energy dissipated by traveing pastic hinge and fexura bending of vertica wa. Energy dissipated by traveing pastic hinges. Haf of the corrugated core is taken into consideration as shown in Figure 4. Assume that the radius r C keeps constant when traveing pastic hinge moves to a new position by a distance of Δs. 14 During this process, arc AB is spread into ine AB, and ine BC bents into a circuar form firsty and then spread into ine B C, whereas ine CD bents into

4 19 Y.-P. E AND Z.-W. WANG a b Pastic hinge ine c d Figure 3. Symmetrica deformation process of unit corrugated core ce. (a) Initia situation; (b) fattening of circuar arc; (c) the hypotenuse turn to be straight turn to be straight; (d) fod on vertica ce wa. 2 h/2 Δs r C h/2 C h C 2H C D C h/2 A ( A ) r C Δs C B B C r D C Figure 4. Traveing pastic hinges on fute peak and vaey (dashed ine stand for the fina configuration). an arc C D with radius of r C. Thus, energy dissipated by this traveing pastic hinge can be computed using the foowing equation: E 1C ¼ AB L C L C L C M C þ 2BC M C þ CD M C r C r C r C 2M C L C r C Δs (4)

5 STRESS PLATEAU OF MULTILAYERED CORRUGATED PAPERBOARD 191 where M C signifies the fuy pastic bending moment per unit ength. By considering pane strain condition, 14 M C ¼ ffiffi 3 sysc t 2 C =6, s ysc denotes the yied stress of corrugated medium, and L C is the ength p of singe-wa corrugated board and aso the ength of the traveing pastic hinge ine. The traveing pastic hinges on each end of the hypotenuse move in the direction shown by arrows in Figure 3b, thus shorten the ength of two hypotenuses, and they become vertica eventuay. Length of arcs on both ends of the hypotenuse are negigibe compared with the ength of the hypotenuse (Figure 4); therefore, the foowing geometrica reationships can be obtained: 8 >< C 2 ¼ H C þ Δs >: 2 ¼ h þ 2Δs where H C is haf ength of the vertica wa in Figure 4. Combining Equations (3) and (5), the foowing is obtained: (5) 8 Δs ¼ 1 h C sin θ C C >< 2 1 cos θ C 2 >: H C ¼ C 1 h C sin θ C 2 1 cos θ C (6) By substituting Equations (3) and (6) into Equation (4), the tota energy dissipated by four traveing pastic hinge ines can be represented as h C sin θ C C ð1 cos θ C Þ E 11 ¼ 4E 1C 8M C L C (7) h C C sin θ C Energy dissipated by fexura bending of vertica ce wa. Two hypotenuses of the simiar trapezoid become vertica and then crushed because the externa oad exceeds their utimate bearing capacities. Assume that the materia of ce wa is rigid-perfecty pastic and no stretch on the midsurface of ce wa occurs. Furthermore, assume that the vertica ce wa is foded ongitudinay, which fits reasonaby we with experimenta observation. Such an axia-symmetric foding needs three pastic hinges (the top hinge 1, the midde 2 and the bottom 3) as iustrated in Figure 5. The genera expression for the energy dissipation rate due to the fexura bending of ce wa is E : 2C ¼ X3 i¼1 M C jθ : icj ic (8) where θ : ic and ic are the rate of anguar motion and ength of hinge i, respectivey. According to Figure 5, we have θ : 1C ¼ a: 1 1C ¼ L C θ : 2C ¼ a: 1 þ a: 3 2C ¼ L C (9) θ : 3C ¼ a: 3 3C ¼ L C Because the foding is symmetric, thus a 1 ¼ a 3 ; a : 1 ¼ a: 3 (1)

6 192 Y.-P. E AND Z.-W. WANG 2 Δs h/2 1 r C 1 H C h C 2 3 H C 3 h/2 Δs Figure 5. Foding on the vertica ce wa. where a 1 and a 3 are anguar rotation of hinges 1 and 3, respectivey; a : 1 and a: 3 are the rates of anguar motion of these two hinges. The tota energy dissipation rate due to the fexura bending of the ce wa is E : 22 ¼ 2E: 2C ¼ 2M CL C ja : 1 jþj a: 1 þ a: : 1 jþja 1 j ¼ 8MC L C a : 1 (11) Then the energy dissipated by the fexure bending of hinge ines is obtained by integrating Equation (11) with respect to a in the imits [, p/2]. E 22 ¼ 4pM C L C (12) Average crushing stress in first bucking Assuming that the work performed by externa force P is equa to the deformation energy of a unit corrugated ce, then P h C ¼ E 11 þ E 22 (13) Substituting Equations (7) and (12) into Equation (13), the foowing is derived: P ¼ 4 p ffiffi! h 3 C C sinθ C þ cosθ C 1 3 h C þ p 2 t C s ysc L C (14) C sinθ C 2 h C The reationship between the externa oad and the average crushing stress is given by P ¼ s P L C (15) Therefore, average crushing stress in first bucking can be represented as foows: s P ¼ 4 p ffiffiffi 3 3 s ysc h C p 2 þ h C= C sinθ C þ cosθ C 1 h C = C sinθ C tc 2 (16)

7 STRESS PLATEAU OF MULTILAYERED CORRUGATED PAPERBOARD 193 Let D 1C ¼ ffiffi 4 p 3 3 h C h p 2 þ h C= C sinθ C þ cosθ C 1 h C = C sinθ C i, then Equation (16) can be simpified into the form t 2 C s P ¼ D 1C s ysc (17) The constant D 1C in Equation (17) is reated to the configuration of corrugated fute. Once the futetype is seected, the ange θ C (determined by the profie of corrugator, and θ C =55 in this case), the ratio of fute pitch to its height (/h C ) and the ratio of fute height to the ength of the hypotenuse (h C / C ) are determined and given in Tabe 1. Then the constant D 1C corresponding to each fute-type can be computed. Therefore, Equation (17) can be further subdivided into three equations corresponding to three common used fute-types. t 2 C A-fute s Ρ ¼ 18:19s ysc t 2 C B-fute s Ρ ¼ 12:24s ysc (18) t 2 C C-fute s Ρ ¼ 11:33s ysc According to Equation (18), the average crushing stress depends on the fute type, the thickness-tofute pitch ratio of corrugated core ce and the yied stress of ce wa materias. If the corrugated medium and the fute type are specified, the average crushing stress can be evauated convenienty. EXPERIMENTS Corrugated medium and mutiayered corrugated paperboard Both corrugated medium and singe-wa corrugated paperboard were suppied by Jackson Packaging Co., Ltd (Dongguan, China). The basis weight of corrugated medium were 15, 12 and 145g/m 2, with the thickness.15,.19 and.22mm, respectivey. Singe-wa corrugated boards were composed of a corrugated core (made from corrugated medium mentioned previousy) between two iners of Kraft sheets with basis weight of 15g/m 2. The iners and medium were gued together by starch adhesives aong the outsides of the peaks and vaeys. Three types of common corrugated futes, A-type, B-type and C-type, were empoyed in this research. They differed from each other in the fute pitch and fute height. Geometric dimensions of specimen are shown in Tabe 1. In designation system of specimens isted in Tabe 1, the number in the midde of symbo denotes the basis weight of corrugated medium (3, 5 and 7 stand for corrugated medium with basis weight of 15, 12 and 145g/m 2, respectivey), etter A on either side of number signifies iners with the basis weight of 15g/m 2, and etters A, B and C in parenthesis indicate the fute type. Mutiayered corrugated paperboard was obtained by gueing a designated number of identica singe-wa corrugated paperboards of the same size stacked one on top of the other. The machine directions of the iners were kept parae, as were the fute directions. There were two kinds of mutiayered corrugated paperboards in this research. One was the MLCP with same number of ayers (eight ayers in this research) but different thickness-to-fute pitch ratios; another was the MLCP with the same thickness (4mm in our research) but different number of ayers (8, 13 and 12 corresponding to A-fute, B-fute and C-fute, respectivey). Yied stress of corrugated medium Corrugated medium was tested to determine its yied stress under different environmenta conditions. The tests were conducted under a constant dispacement veocity of 1.5mm/min with the China Nationa Standard GB/T for reference. Five repicate tests were conducted. A specimens

8 194 Y.-P. E AND Z.-W. WANG Tabe 1. Characteristics of experiment sampes. Specimen Basis weight of inerboard (g/m 2 ) Basis weight of corrugated medium (g/m 2 ) Thickness of corrugated medium tc (mm) Fute type Fute height hc (mm) Fute pitch λ(mm) Fute pitch-toheight fute ratio /hc Fute heightto hypotenuse ength ratio hc/c Thicknessto-fute pitch ratio tc/ A3A(A) A A3A(B) B A3A(C) C A5A(A) A A5A(B) B A5A(C) C A7A(B) B A7A(C) C

9 STRESS PLATEAU OF MULTILAYERED CORRUGATED PAPERBOARD 195 were cut into the same area of 12.7mm152mm, with gauge ength of 65mm, and the ong edge of specimen was parae to the machine direction of corrugated medium. The ce wa thickness was tested according to the China Nationa Standard GB/T A specimens were preconditioned under each experimenta RH (respectivey at 4%, 5%, 65%, 75%, 85% and 95% RH) and at the temperature of 23 C for 48h prior to tests. The instruments used for the experiments incuded a universa materia testing machine (CMT 852, fu-scae oad range of 5N, MTS Systems Co., Ltd, Shenzhen, China) and a temperature humidity programmabe controer (GDJS-1, Zhongya Test Experiment Co., Ltd, Huaian, China). Fatwise compression of mutiayered corrugated paperboard Mutiayered corrugated paperboards were tested under quasi-static compression in different RH to determine their average crushing stress and stress pateau as response to RH. A specimens were preconditioned under each environmenta RH (respectivey at 4%, 5%, 65%, 75%, 85% and 95% RH) and at the temperature of 23 C for 48h prior to tests. The dimension of specimens was 1mm 1mm. The oad was appied vertica to the axis of the corrugated fute with a constant rate of 12 3mm/min as recommended by the China Nationa Standard GB/T Each compression test was repicated five times. The oad-defection reationships of MLCP with different thickness-to-fute pitch ratios were tested on a universa materia testing machine (WDW-1C, fu-scae oad range of 1kN, Huaong test equipment Co., Ltd, Shanghai, China) equipped with continuous data acquisition. RESULTS AND DISCUSSION Comparison of theoretica vaues cacuated from Equation (18) and experimenta resuts of average crushing stress for MLCPs with a wide range of t C / ratios under a controed atmosphere (23 C, 5% RH) is shown in Figure 6. The predictions and experiments ie coser to the unit-sope straight ine, indicating that the present mode provides an accurate prediction of the average crushing stress of first bucking. It is cear from Equation (18) that for mutiayered corrugated paperboards having the same fute type, the normaized average crushing stress (s P /s ysc ) is a function of thickness-to-fute pitch ratio but irreevant to the seection of corrugated medium. Therefore, the normaized average crushing stress versus thickness-to-fute pitch ratio curves are potted and superposed with experiment data (Figure 7). As can be seen from Figure 7, the average crushing stress increases with the increasing of the vaue Experimenta data of the average crushing stress σ P /MPa A3A(A) A5A(A) A3A(B) A5A(B) A7A(B) A3A(C) A5A(C) A7A(C) y=x Predictions of the average crushing stress σ P /MPa Figure 6. Comparison of the observations and predictions of the average crushing stress for mutiayered corrugated paperboard with different thickness-to-fute pitch ratios (23 C, 5% RH).

10 196 Y.-P. E AND Z.-W. WANG Normaised average crushing stress σ P /σ ys C Experiments of A-fute Experiments of B-fute Experiments of C-fute Prediction cure of A-fute Prediction cure of B-fute Prediction cure of C-fute Thickness-to fute pitch ratio t c.4.5 Figure 7. Curves of normaized average stress versus thickness-to-fute pitch ratio for mutiayered corrugated paperboard. of t C / ratio, and predicting curves can successfuy anticipate the average crushing stress of corrugated ce with various t C / ratios. The wave-ike upift curve in centra stage can be simpified as an obique ine with the sope of E p as mentioned previousy. Anaysing the sope E p for MLCPs with different fute types, different corrugated medium and different number of ayers (Figure 8), it appears that the sope decreases with increasing fute pitch, but the reationship between the sope and the thickness of corrugated medium and the number of ayers tends to have no reguarity. The sope E p normaized by the yied stress of corrugated medium varies with the fute pitch by a certain rue shown in Figure 8c. By fitting the experimenta data with the exponentia aw of attenuation, the foowing is obtained: E p ¼ 2:25s ysc expð 1:5Þ R 2 ¼ :95 (19) Substituting Equations (19) and (17) into Equation (1), the foowing is obtained: t 2 C s ¼ D 1C s ysc þ 2:25sysC expð 1:5Þ e e c (2) According to Equation (18), the constant D 1C equas respectivey to 18.19, and corresponding to A-fute, B-fute and C-fute. Thus, the stress pateau curve of MLCP is determined by the combination of yied stress of corrugated medium, ratio t C / and the fute configuration. By foowing these given parameters and ignoring the reative sma eastic strain, stress pateau curves of MLCPs with different thickness-to-fute pitch ratios can be potted theoreticay. Figure 9 shows the comparison between the theoretica and experimenta stress pateau curves of MLCPs with different thickness-to-fute pitch ratios under the controed atmosphere (23 C, 5% RH). It can be seen from Figure 9 that in genera, the theoretica curves fit we with the mean ines of experimenta wave-ike curves, and the predicting curves are sighty higher than experimenta ones. The inherent hygroscopic properties associated with paper-based materias and water degradation of starch adhesive make MLCP very sensitive to ambient humidity, especiay for the yied stress of corrugated medium. By taking the yied stress of corrugated medium tested under the controed atmospheres as reference and denoting it with s ysc, the reative yied stress (RYS) is defined as a ratio of the yied stress of corrugated medium tested under arbitrary environment to that under the controed atmosphere. The RYSs of corrugated medium were potted against the RH in Figure 1. As it can be seen from this figure, the RYS versus RH curves of corrugated mediums with basis weight of 15, 12 and 145g/m 2 were in substantia agreement; the empirica reationship can be obtained by fitting these experiments.

11 STRESS PLATEAU OF MULTILAYERED CORRUGATED PAPERBOARD 197 Sope of stress pateau curve Ep a 15g/m 2 ( t c =.15mm) 12g/m 2 ( t c =.19mm) 145g/m 2 ( t c =.22mm) Sope of stress pateau curve Ep b B-fute Fute type C-fute n c =8 n c =13 n c =12 Normaised sope of stress pateau curve E p /σ ysc c A3A(B) A5A(B) A7A(B) A3A(C) A5A(C) A7A(C) specimen type Fute pitch /mm 15 g/m 2 12 g/m g/m Figure 8. Infuence factors of the sope of stress pateau curve (23 C, 5% RH). (a) Thickness of corrugated medium; (b) the number of ayers and (c) fute pitch. RYS ¼ s ysc s ¼ 1:1 ysc 1 þ exp RH 91:11 R 2 ¼ :99 (21) 8:98 Substituting Equation (21) into Equation (17) shapes s P ¼ D 1 1C s ysc t 2 C 1 þ exp RH (22) 8:98 1:15

12 198 Y.-P. E AND Z.-W. WANG.6.5 Experiments Prediction.6.5 Experiments Prediction Experiments Prediction.6.5 Experiments Prediction Experiments Prediction.6.5 Experiments Prediction Experiments Prediction.6.5 Experiments Prediction

13 STRESS PLATEAU OF MULTILAYERED CORRUGATED PAPERBOARD 199 Reative yied stress RYS/% g/m 2 12g/m 2 145g/m Reative humidity RH/% Figure 1. Reationship between reative yied stress of corrugated medium and reative humidities in surrounding air. where D 1C ¼ 1:1D 1C. This constant equas respectivey to 18.37, and corresponding to A-fute, B-fute and C-fute. Substituting Equation (21) into Equation (19), the foowing is obtained: 1 E p ¼ 2:27s ysc 1 þ exp RH expð 1:5Þ (23) 8:98 1:15 Substituting Equations (22) and (23) into Equation (1), the prediction equation of stress pateau for MLCPs with different ratios of t C / under various ambient humidities is acquired: 1 s ¼ s ysc t 2 C 1 þ exp RH D 1C þ 2:27 expð 1:5Þ e e c 8:98 1:15 (24) According to Equation (24), the stress pateau function of MLCP can be computed with the aid of known parameters of t C / ratio and the RH. Stress pateau of MLCP depends primariy on the average crushing stress in first bucking and the sope of stress pateau curve. However, the atter is reativey sma in most cases, and then the stress pateau curve of MLCP can be roughy considered as a horizonta ine. Therefore, the prediction of the average crushing stress in first bucking in various ambient humidities is a vita concern for evauating the energy absorption property of MLCP in design practices. The experimenta resuts of the normaized average crushing stresses of eight types of MLCPs under six eves of RH testing conditions are obtained and isted in Tabe 2. The experiments suggest that the normaized average crushing stress increases with the increasing of t C / vaue and decreases as the RH increases. Comparison between the experiments and the theoretica predictions cacuated from Equation (24) is iustrated in Figure 11. In genera, the points ie cose to the unit-sope straight ine, indicating that the theory provides a good prediction of the average crushing stress in various environmenta conditions. Figure 9. Comparison of the experimenta and predicting stress pateau curves of mutiayered corrugated paperboard (23 C, 5% RH). (a) A3A (A); (b) A3A (B); (c) A3A (C); (d) A5A (A); (e) A5A (B); (f) A5A (C); (g) A7A (B) and (h) A7A(C).

14 2 Y.-P. E AND Z.-W. WANG Tabe 2. Normaized average crushing stress of mutiayered corrugated paperboard in various ambient humidities. Specimen Thicknessto-fute pitch ratio t C / Normaized average crushing stress s p /s ysc 4% RH 5% RH 65% RH 75% RH 85% RH 95% RH A3A(A) A3A(C) A5A(A) A5A(C) A3A(B) A7A(C) A5A(B) A7A(B) Predictions of the normaised average crushing stress σ P /σ ysc y=x Experimenta data of the normaised average crushing stress σ P /σ ysc Figure 11. Comparison of the observations and predictions of the normaized average crushing stress for mutiayered corrugated paperboard in various ambient humidities. CONCLUSIONS Theoretica mode is estabished to predict the stress pateau of MLCPs with different thickness-tofute pitch ratios under various ambient humidities. The mode is a simpe formua reating the stress pateau to the thickness-to-fute pitch ratio of corrugated core ce, the yied stress of corrugated medium, the fute pitch and the ambient humidity. Therefore, the stress pateau of MLCP in actua ogistica environments can be evauated with the aid of corrugated configuration, ce wa materia and the RH. Predictions of stress pateau for MLCP with various vaues of t C / in different environmenta humidities are found in good agreement with the observations, indicating that the mode deveoped here coud be used in practica appication. Resuts of this research can be expected to predict the energy absorption properties of MLCPs with different configurations in actua ogistica environments and provide usefu information for the optimum design and the materia seection of cushion packaging. ACKNOWLEDGEMENTS This work is supported by the Nationa Natura Science Foundation of China (Project 57751), the Key Laboratory of Product Packaging and Logistics of Guangdong Higher Education Institutes, and the Science Foundation of Zhejiang Sci-Tech University (ZSTU; project under Grant No Y).

15 STRESS PLATEAU OF MULTILAYERED CORRUGATED PAPERBOARD 21 Nomencature P s P s e e c e Dc M C s ysc s ysc E p h C L C Externa oad Average crushing stress in first bucking Stress at arbitrary point on the compressive curve in centra section Strain at arbitrary point on the compressive curve in centra section Maximum eastic strain Densification strain Fu pastic bending moment per unit ength Yied stress of corrugated medium under a controed atmosphere (23 C, 5% RH) Yied stress of corrugated medium under arbitrary ambient humidities Sope of the stress pateau curve Height of singe-wa corrugated board Length of singe-wa corrugated board Fute pitch C Length of the hypotenuse of a simiar trapezoid θ C Ange between the hypotenuse and the horizonta ine t C Thickness of corrugated medium h Length of fattened peak r C Radius of the arc connecting the fattened peak and the hypotenuse H C Haf ength of the fod Δs Move distance of traveing pastic hinge ine θ : ic Rate of anguar motion of hinge i ic Length of hinge i E 11 Energy dissipated by traveing pastic hinges E 22 Energy dissipated by fexura bending of vertica wa n C Number of ayers in the mutiayered corrugated paperboard a 1,a 3 Anguar rotation of hinge 1 and 3 a : 1 ; a: 3 Rate of anguar motion of hinge 1 and 3 t C / Thickness-to-fute pitch ratio /h C Fute pitch-to-fute height ratio h C / C Fute height-to-hypotenuse ength ratio s P /s ysc Normaized average crushing stress in first bucking RH Reative humidity in surrounding air RYS Reative yied stress of corrugated medium REFERENCES 1. Wang DM. Energy absorption diagram of muti-ayer corrugated boards. Journa of Wuhan University of Technoogy- Materias Science Edition 21; 25(2): Wang DM. PhD Dissertation of Jiangnan University. Cushioning property and characteristics studies on honeycomb paperboards and corrugated paperboards. Wuxi, China 27 (in Chinese). 3. Minett M, Sek M. Behaviour of corrugated fiberboard as a cushioning materia. Proceedings of the 11th IAPRI Word Conference on Packaging, Singapore, Naganathan P, He J, Kirkpatrick J. The effect of compression of encosed air on the cushioning properties of corrugated fiberboard. Packaging Technoogy and Science 1999; 12(2): Minett M, Sek M. Pre-contact effect on acceeration puses during cushion testing of protective packaging. Proceedings of the 14th IAPRI Word Conference on Packaging, Stockhom, Sweden, Minett M. PhD Dissertation of Victoria University. A study of air fow effects on the cushioning characteristics of muti ayered pre-compressed fiberboard. Mebourne, Austraia Aboura Z, Tabi N, Aaoui S, Benzeggagh ML. Eastic behavior of corrugated cardboard: experiments and modeing. Composite Structures 24; 63(1): Anders H, Matti R. Large strain easto-pastic mode of paper and corrugated board. Internationa Journa of Soids and Structures 28; 45(11 12): Haj-Ai R, Choi J, Wei B-S, Popi R, Schaepe M. Refined noninear finite eement modes for corrugated fiberboards. Composite Structures 29; 87(4): Isaksson P, Krusper A, Gradin PA. Shear correction factors for corrugated core structures. Composite Structures 27; 8(1):

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