3d energy absorption diagram construction of paper honeycomb...

Research Article3D Energy Absorption Diagram Construction of PaperHoneycomb Sandwich Panel

Dongmei Wang 1 Ziyou Bai1 and Qianghua Liao2

1School of Media and Communication Shenzhen Polytechnic Shenzhen 518055 China2School of Mechanical and Electrical Engineering Shenzhen Polytechnic Shenzhen 518055 China

Correspondence should be addressed to Dongmei Wang 394794521qqcom

Received 12 September 2017 Revised 5 January 2018 Accepted 10 January 2018 Published 7 March 2018

Academic Editor Francesco Franco

Copyright copy 2018 DongmeiWang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Paper honeycomb sandwich panel is an environment-sensitive material Its cushioning property is closely related to its structuralfactors the temperature and humidity random shocks and vibration events in the logistics environment In order to visuallycharacterize the cushioning property of paper honeycomb sandwich panel in different logistics conditions the energy absorptionequation of per unit volume of paper honeycomb sandwich panel was constructed by piecewise function The three-dimensional(3D) energy absorption diagram of paper honeycomb sandwich panel was constructed by connecting the inflexion of energyabsorption curve It takes into account the temperature humidity strain rate and characteristics of the honeycomb structureOn the one hand this diagram breaks through the limitation of the static compression curve of paper honeycomb sandwich panelwhich depends on the test specimen and is applicable only to the standard condition On the other hand it breaks through thelimitation of the conventional 2D energy absorption diagram which has less information Elastic modulus was used to normalizethe plateau stress and energy absorption per unit volume This makes the 3D energy absorption diagram universal for differentmaterial sandwich panels It provides a new theoretical basis for packaging optimized design

1 Introduction

Paper honeycomb sandwich panel is an environmental-friendly low-cost good-mechanical propertymaterial whichis widely used as cushioning material for packaging for largemechanical and electrical products in recent years It hasbeen highly concerned by the packaging industry Paperhoneycomb sandwich panel has also been successfully usedin furniture and building industry as loading and fillingmaterial In short it has broad market prospects

Nowadays the research of the cushioning property ofpaper honeycomb sandwich panel is mostly focused on aspecific configuration of honeycomb Therefore the resultsare dependent on the test samples and are limited in thepractical application Xin and Guo [1] Cai and Lu [2] andZhang [3] studied the influence of temperature and relativehumidity on cushioning property and static compressiveproperty of paper honeycomb sandwich panel and foundthat as the relative humidity increased and the temperature

decreased the average critical stress plateau stress and thecushioning property of paper honeycomb sandwich paneldeclined Cui and Chen [4] studied the effects of differentlevels of precompression on the cushioning performanceof paper honeycomb sandwich panel The results showedthat with the increase in the precompression level the staticcompression performance static cushioning performanceand energy absorption of paper honeycomb sandwich panelreduced accordingly The above researches were based ontests and the simple qualitative relationship between thecushioning property of paper honeycomb sandwich paneland some parameters was obtained

Combined with the theory the cushioning property ofpaper honeycomb sandwich panel was studied from thestructure by some scholars Wang et al [5 6] assessedthe critical stress of paper honeycomb sandwich panel andobtained the assessment equation of critical stress Then theyestablished a static compression model of paper honeycombsandwich panel which is only suitable for the assessment

HindawiShock and VibrationVolume 2018 Article ID 4067062 6 pageshttpsdoiorg10115520184067062

2 Shock and Vibration

of the mechanical properties of paper honeycomb sandwichpanel under the static loading condition In practice cushioncurve is used to estimate the cushioning property of material[7] Based on the cushion curve Maiti et al [8] proposedthe energy absorption diagram which can be constructedaccording to the compressive stress-strain curve of materialsThe relationship between the stress and energy absorption ofcushion material can be characterized by energy absorptiondiagram In recent years some domestic scholars studiedthe characterization methods of energy absorption of paperhoneycomb sandwich panel actively Wang et al [9] andWang and Lu [10] constructed the energy absorption curve ofper unit volume paper honeycomb sandwich panel and thenthey linked the optimal inflexion point of energy absorptioncurve to build the energy absorption diagram of per unitvolume paper honeycomb sandwich panel with differentstructures Xiong et al [11] constructed the 3D image ofcorrugated paper among moisture content temperature andrelative humidity by experiments and studied the influenceof moisture content on elastic modulus of corrugated paperThe results showed that the moisture content decreaseswith temperature and increases with relative humidity Theelastic modulus decreases with moisture content Wang andE [12] and E and Wang [13] studied the influence of relativehumidity on the cushioning property of paper honeycombsandwich panel and fitted the relationship equation ofenergy absorption and various relative humidity The energyabsorption diagram of per unit volume paper honeycombsandwich panel which considered the relative humidity wasdrawn Wang et al [14] developed mathematical model inwhich the structure factors and temperature and relativehumidity are concerned through the cooperation of theoriesand experiments And the model is normalized by the elasticmodulus of themediumunder controlled condition thus thismodel can be used to predict the effect of various temperatureand relative humidity on the plateau stress of multilayercorrugated sandwich structure under flatwise compression

On the basis of a great deal of research on static com-pression scholars have further studied the energy-absorbingproperties of the paper honeycomb sandwich panel underdynamic impact E andWang [15] investigated the effect of thestrain rate on themechanical behavior and energy absorptionof paper honeycomb by experiments The experimentalresults show that the load-carrying capacity and the energyabsorption performance of paper honeycomb are insensitiveto the loading speed in low strain rate range (10minus4ndash10minus2 sminus1)Wang and Bai [16] found that there exists a certain strain rateeffect which leads to the difference of mechanical propertiesbetween dynamic and static condition A mechanical modelwas given to estimate plateau stress and yield stress at differentstrain rates based on experimental data by employing themethods ofCowperndashSymondsmodel and piecewise functionAnd then Bai et al [17] according to the mechanical modelbuilt the energy absorption diagram under different strainrates Moreover they have taken serious experiments and theresult showed a good agreement with theoretical models

From present documents all energy absorption prop-erties were researched on either relative humidity effector strain rate effect The energy absorption diagram of

material can be modeled by linking the optimal energyabsorption point under two dimensions However only asingle parameter was considered in the two-dimensionalenergy absorption diagramwhich limited its applicationThepurpose of this paper is to construct 3D diagram of paperhoneycomb sandwich panel which took structure strain rateand moisture content into account The diagram took therandom shock and variety of environment in the logisticscondition into account This will make the optimizationdesign of paper honeycomb sandwich panel more suitable forthe actual logistics environment

2 The Evaluation of Plateau Stress andCritical Stress Based on the Effect of StrainRate and Moisture Content

The influence of temperature and relative humidity on mois-ture content and cushioning property of paper honeycombsandwich panel was studied by reference [18 19] Then amodel of moisture content and the prediction equations ofcritical and plateau stress of paper honeycomb sandwichpanel are obtained and verified [6] We have the equationsas follows

119872= 205RH(1 minus 0005432RH) (1 minus 0005432RH + 332RH119879) 119879

(1)

where119872 is moisture content of paper honeycomb sandwichpanel RH is relative humidity 119879 is temperature ∘C

120590pk1198640119904 = 19298119890minus00769119872 (119905119897 )

3 (2)

120590pl1198640119904 = 00991119890minus00769119872 lowast (119905119897 )

53

+ 001067119890minus00769119872 (119905119897 )2

(3)

where 120590pk is critical stress of paper honeycomb sandwichpanel MPa 120590pl is plateau stress of paper honeycomb sand-wich panel MPa 1198640119904 is elastic modulus of corrugatingmedium under standard condition MPa 120590pk1198640119904 is nor-malized critical stress 120590pl1198640119904 is normalized plateau stress119890 = 27183 119905119897 is thickness-to-length ratio 119905 is thickness ofcorrugatingmediummm 119897 is length of honeycomb cell mm

Combined with the strain rate effect [16 17] studied themechanical behavior of paper honeycomb sandwich paneland obtained the prediction equations of dynamic critical andplateau stress The equations are as follows

1205901015840pk1198640119904 =18089 times 10minus4

(1 minus 120576119901119896) (1 + 028120576119901119896)2 +120590pk1198640119904 (4)

1205901015840pl1198640119904 =120590pl1198640119904 [1 + (

120576119861)1119902] (5)

Shock and Vibration 3

where 1205901015840pk is dynamic critical stress of paper honeycombsandwich panel MPa 1205901015840pl is dynamic plateau stress ofpaper honeycomb sandwich panel MPa 1205901015840pk1198640119904 is normal-ized dynamic critical stress 1205901015840pl1198640119904 is normalized dynamicplateau stress 120576pk is the strainwhen stress reaches the yieldingstress 120576 is strain rate sminus1 119861 119902 are parameters

Substituting (3) into (5) then the evaluation equation ofthe plateau stress of paper honeycomb sandwich panel formoisture content and strain rate can be represented as

1205901015840pl1198640119904 = [00991119890minus00769119872 lowast (119905119897 )

53

+ 001067119890minus00769119872 (119905119897 )2] [1 + ( 120576119861)

1119902] (6)

According to the static and dynamic impact tests 119861 and119902 can be obtained through curve fitting as follows [16]

119861 = 1528119905119897 + 209119902 = 05125 (7)

Substituting (7) into (6) then (6) can be simplified asfollows1205901015840pl1198640119904 = [00991119890

minus00769119872 lowast (119905119897 )53 + 001067119890minus00769119872

lowast (119905119897 )2] [1 + ( 1205761528119905119897 + 209)

19512] (8)

Substituting 119905119897 = 003125 119905119897 = 00333 and 119905119897 = 00416into (8)

1205901015840pl11198640119904 = (30372119890minus00769119872 lowast 10minus4 + 10412119890minus00769119872

lowast 10minus5 ) [1 + ( 1205766825)19512]


lowast 10minus5) [1 + ( 1205767178)19512]


lowast 10minus5) [1 + ( 1205768446)19512]

(9)

The 3D diagram was drawn according to (9) as shown inFigure 1

It can be seen from Figure 1 under the same strain rateconditions with the increase of moisture content that thenormalized plateau stress exhibited a downward trend butthe decline is small In the samemoisture content conditionswith the increase of strain rate the stress rose exponentially

8

7

6

5

4

3

2

1100

80

4020

0 89

1011

1213

14

Moisture contentStrain rate

Nor

mal

ised

plat

eau

stres

s

times10minus4

60

003125t

l=

00333t

l=

00416t

l=

Figure 1 3D diagram of normalized plateau stress of paper honey-comb sandwich panel

Com

pres

sive s

tress

(M

Pa)

ＪＥ

l1

ＪＥ

Compressive strain ()D

ＪＦ

Figure 2 Theoretical stress-strain curve of paper honeycomb

3 The Evaluation of EnergyAbsorption Based on the Effect of StrainRate and Moisture Content

Figure 2 shows the theoretical stress-strain curve of paperhoneycomb sandwich panel The shaded area in Figure 2 isper unit volume energy absorption theoretical value Thestress-strain curve is divided into three stages the yield stagethe plateau stage and densification stage [6] Integrating 120590-120576we can obtain the per unit volume energy absorption value119882 It is an important factor tomeasure the energy absorptioncapacity of paper honeycomb sandwich panel

The cushioning property of paper honeycomb sandwichpanel is affected by various factors such as the impactfrom the transport process environmental temperature andhumidityThe influence of external factors on the cushioningproperty can be transformed into the influence of moisturecontent and strain rate Therefore the energy absorption


model is constructed in this paper with taking into accountthe honeycomb structure strain rate and moisture contentThis model aims to assess cushioning property of paperhoneycomb sandwich panel

Three stages of dynamic energy absorption equation ofpaper honeycomb sandwich panel were shown as follows [1720]

1198821198640119904 = (12059010158401198640119904 )2 119897radic34119905 0 le 120576 le 120576pk

1198821198640119904 = (1205901015840pk1198640119904 )

2 119897radic34119905 +1205901015840pl (120576 minus 120576pk)1198640119904 120576pk le 120576 le 120576119863

1198821198640119904 = (1205901015840pk1198640119904 )

2 119897radic34119905 +1205901015840pl (1 minus 395 (119905119897))1198640119904 120576 gt 120576119863

(10)

where119882 is the energy absorption per unit volume J1198821198640119904is the normalized energy absorption per unit volume 1205901015840 isdynamic stress at arbitrary point on stress-strain curve MPa120576 is the strain at arbitrary point on stress-strain curve 120576119863 isthe densification strain

Substituting (2) into (4) and then substituting (4) and (8)into (10) we can get equations as follows

1198821198640119904 = (12059010158401198640119904 )2 119897radic34119905 0 le 120576 le 120576pk

1198821198640119904 = [[18089 times 10minus4

(1 minus 120576pk) (1 + 028120576pk)2

+ 19298119890minus00769119872 (119905119897 )3]]2 119897radic34119905

+ [00991119890minus00769119872 (119905119897 )53

+ 001067119890minus00769119872 (119905119897 )2] [1

+ ( 1205761528119905119897 + 209)19512] (120576 minus 120576pk) 120576pk le 120576 le 120576119863

1198821198640119904 = [[18089 times 10minus4

(1 minus 120576pk) (1 + 028120576pk)2

+ 19298119890minus00769119872 (119905119897 )3]]2 119897radic34119905

+ [00991119890minus00769119872 (119905119897 )53

+ 001067119890minus00769119872 (119905119897 )2] [1

+ ( 1205761528119905119897 + 209)19512](1 minus 395 119905119897 ) 120576 gt 120576119863

(11)

Equations (11) are energy absorption model of paperhoneycomb sandwich panelThemoisture content and strainrate are taken into account

In the packaging optimization design of paper honey-comb sandwich panel the best point is the inflexion whichis the terminal point of plateau stage At this point namelywhen 120576 = 120576119863 the energy absorption value reached themaximumThe energy absorption value of that point consistsof two parts which are energy absorption value at the yieldstage and plateau stage The equation was shown as follows

1198821198640119904 = (1205901015840pk1198640119904 )

2 119897radic34119905 +1205901015840pl (1 minus 395 (119905119897))1198640119904 (12)

The energy absorption value of yield stage is much lessthan that of plateau stage Hence (12) can be simplified as

1198821198640119904 asymp1205901015840pl (1 minus 395 (119905119897))1198640119904 (13)

Substituting (8) into (13) then we can get

1198821198640119904 = [00991119890minus00769119872 (119905119897 )

53

+ 001067119890minus00769119872 (119905119897 )2] [1

+ ( 1205761528119905119897 + 209)19512](1 minus 395 119905119897 )

(14)

Equation (14) is the prediction equation of normalizedenergy absorption per unit volume of paper honeycombsandwich panel which the moisture content and strain ratewere concerned with

Substituting 119905119897 = 003125 119905119897 = 00333 and 119905119897 =00416 into (14) the energy absorption of paper honeycombsandwich panel can be expressed as

11988211198640119904 = (26622119890minus00769119872 lowast 10minus4 + 91267119890minus00769119872

lowast 10minus6) [1 + ( 1205766825)19512]


lowast 10minus5) [1 + ( 1205767178)19512]


lowast 10minus5) [1 + ( 1205768446)19512]

(15)

The 3D diagram was drawn in accordance with (15) asshown in Figure 3


Table 1 The comparison of experimental and predictive energy absorption

1198821198640119904 predictiveExperimental (10minus4) Strain rate 120576 (sminus1)Thickness-to-length 119905119897 10 50 100003125 1306014112 1963820013 393504123500333 1437114532 2101821429 409374012500416 1999019128 2676024257 4704745675

6

5

4

3

2

1

0100

8060

4020

0 89

1011

1213

14


Nor

mal

ised

ener

gy ab

sorp

tion

per u

nit v

olum

e

times10minus4

003125t

l=

00333t

l=

00416t

l=

Figure 3 3D diagram of normalized energy absorption of paperhoneycomb sandwich panel

Figure 3 indicates that in the same moisture contentand strain rate conditions the normalized maximum energyabsorption value rises with the increase of 119905119897 At a certainstrain rate the normalized maximum energy absorptionvalue exhibits a downward trendwith the increase ofmoisturecontent but the decline range is small In the same mois-ture content conditions the normalized maximum energyabsorption value increaseswith the increase of strain rateThehigher the strain rate is the more significant the magnitudeis

4 Experimental Verification

In the case of moisture content being 10 (14) can beused to predict the energy absorption of three kinds ofpaper honeycomb sandwich panel In order to verify theaccuracy of the above equations three kinds of sampleswere tested and their energy absorption were obtained Theexperimental details and information about geometry andmaterial characteristics are showed in [17] And then thecomparison between the theoretically predictive values andexperimental values was shown in Table 1

Statistically the coefficient of determination 1198772 is usedin the context of statistical models and is to predict futureoutcomes on the basis of other related information It pro-vides a measure of how well future outcomes are likely tobe predicted by the model 1198772 ranges from 0 to 1 An 1198772

5

4

3

2

1

00 1 2 3 4 5

Predictive value (10minus4)

Expe

rimen

tal v

alue

(10minus4)

R2 = 09551

Figure 4 The relationship between experimental and predictedenergy absorption

of 1 indicates that the model shows perfect consistency withthe experimental data In Figure 4 1198772 is 09551 which means(14) can be better used to forecast the value of the energyabsorption of paper honeycomb sandwich panel

5 Conclusion

In this paper based on theory and experiment the energyabsorption equation of per unit volume of paper honeycombsandwich panel was constructed by piecewise function Andthen the inflexion of energy absorption curve was connectedto construct 3D diagram of energy absorption of paperhoneycomb sandwich panel The diagram can be expressedas the theoretical formula It takes the temperature humiditystrain rate and characteristics of honeycomb structure intoaccount On the one hand this diagram breaks through thelimitation of the static compression curve of paper honey-comb sandwich panel which depends on the test specimenand is applicable only to standard condition on the otherhand it breaks through the limitation of the conventional 2Denergy absorption diagram which has less information

Elastic modulus was used to normalize the plateau stressand energy absorption per unit volume which make the 3Denergy absorption diagram universal for different materialsandwich panels The optimization of paper honeycombsandwich panel and corrugated sandwich panel can be real-ized by using 3D energy absorption diagram which providesa new theoretical basis for optimum packaging design In


the future once you know the elastic modulus of base paperwith the help of 3D energy absorption diagram mechanicalproperties and energy absorption properties of paper honey-comb sandwich plate under different temperature humidityand compression state can been predicted without makingsamples to test

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Contractgrant sponsor is National Natural Science Founda-tion of China Project (51675349)

References

[1] C L Xin and Y F Guo ldquoExperimental research and analysison static cushioning properties of honeycomb paperboardrdquoPackaging Engineering vol 29 no 1 pp 56ndash58 2008

[2] Y Cai and L X Lu ldquoThe influence of humidity on the out-of-plane strength of paper honeycombrdquoMechanical and ElectricalInformation vol 17 pp 41ndash43 2004

[3] Z K Zhang ldquoExperiment research on the effects of relativehumidity on the static compression properties of paper hon-eycomb panelrdquo Packaging Engineering vol 29 no 9 pp 12-132008

[4] Y Cui and L Chen ldquoThe effect of pre-compression on theenergy absorption of honeycomb cardboardrdquo Packaging Engi-neering vol 38 no 9 pp 141ndash145 2017

[5] D M Wang and Z W Wang ldquoOut-of-plane compressiveproperties of hexagonal paper honeycombsrdquo Chinese Journal ofMechanical Engineering vol 20 no 2 pp 115ndash119 2007

[6] D M Wang and Q H Liao ldquoResearch on the mechanicalperformancemodeling of honeycomb paperboard under quasi-static compressionrdquo Packaging Engineering vol 27 no 4 pp129ndash132 2006

[7] M A Sek M Minett V Rouillard and B Bruscella ldquoA newmethod for the determination of cushion curvesrdquo PackagingTechnology and Science vol 13 no 6 pp 249ndash255 2000

[8] S K Maiti L J Gibson and M F Ashby ldquoDeformation andenergy absorption diagrams for cellular solidsrdquo Acta Metallur-gica et Materialia vol 32 no 11 pp 1963ndash1975 1984

[9] D M Wang Z W Wang and Q H Liao ldquoEnergy absorptiondiagrams of paper honeycomb sandwich structuresrdquo PackagingTechnology and Science vol 22 no 2 pp 63ndash67 2009

[10] J Wang and L X Lu ldquoEnergy-absorption diagrams of honey-combpaperboards under static compression in different relativehumidityrdquo Packaging Engineering vol 1 pp 5ndash7 2011

[11] W Xiong J Wang and D M Wang ldquoInfluence of temperatureand relative humidity on elastic modulus of corrugated paperrdquoPackaging Engineering vol 32 no 21 pp 45ndash48 2011

[12] Z W Wang and Y P E ldquoMathematical modelling of energyabsorption property for paper honeycomb in various ambienthumiditiesrdquo Materials and Corrosion vol 31 no 9 pp 4321ndash4328 2010

[13] Y P E and Z W Wang ldquoEffect of relative humidity on energyabsorption properties of honeycomb paperboardsrdquo PackagingTechnology and Science vol 23 no 8 pp 471ndash483 2010

[14] D M Wang H X Gong and Z Y Bai ldquoEffect investigationof relative humidity and temperature on multi-layer corrugatedsandwich structuresrdquo Journal of Sandwich Structures and Mate-rials vol 15 no 2 pp 156ndash167 2013

[15] Y P E and Z W Wang ldquoEnergy-absorbing properties ofpaper honeycombs under low and intermediate strain ratesrdquoPackaging Technology and Science vol 25 no 3 pp 173ndash1852012

[16] D M Wang and Z Y Bai ldquoMechanical property of paperhoneycomb structure under dynamic compressionrdquo Materialsand Corrosion vol 77 pp 59ndash64 2015

[17] Z Y Bai D M Wang and Z F Xu ldquoModel creation of strainratendashdependent energy absorption for paper honeycomb sand-wich structurerdquo Journal of Sandwich Structures and Materialsvol 17 no 4 pp 359ndash375 2015

[18] D M Wang J Wang and Q H Liao ldquoInvestigation ofmechanical property for paper honeycomb sandwich compositeunder different temperature and relative humidityrdquo Journal ofReinforced Plastics and Composites vol 32 no 13 pp 987ndash9972013

[19] W Xiong Energy absorptionmodel research of paper honeycombsandwich structure based on temperature and relative humidity[Master thesis] Zhejiang Sci-Tech University China 2011

[20] Z F Xu Energy absorption model research of paper honeycombsandwich structure based on strain rate [Master thesis] XirsquoanUniversity of Technology China 2011

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of the mechanical properties of paper honeycomb sandwichpanel under the static loading condition In practice cushioncurve is used to estimate the cushioning property of material[7] Based on the cushion curve Maiti et al [8] proposedthe energy absorption diagram which can be constructedaccording to the compressive stress-strain curve of materialsThe relationship between the stress and energy absorption ofcushion material can be characterized by energy absorptiondiagram In recent years some domestic scholars studiedthe characterization methods of energy absorption of paperhoneycomb sandwich panel actively Wang et al [9] andWang and Lu [10] constructed the energy absorption curve ofper unit volume paper honeycomb sandwich panel and thenthey linked the optimal inflexion point of energy absorptioncurve to build the energy absorption diagram of per unitvolume paper honeycomb sandwich panel with differentstructures Xiong et al [11] constructed the 3D image ofcorrugated paper among moisture content temperature andrelative humidity by experiments and studied the influenceof moisture content on elastic modulus of corrugated paperThe results showed that the moisture content decreaseswith temperature and increases with relative humidity Theelastic modulus decreases with moisture content Wang andE [12] and E and Wang [13] studied the influence of relativehumidity on the cushioning property of paper honeycombsandwich panel and fitted the relationship equation ofenergy absorption and various relative humidity The energyabsorption diagram of per unit volume paper honeycombsandwich panel which considered the relative humidity wasdrawn Wang et al [14] developed mathematical model inwhich the structure factors and temperature and relativehumidity are concerned through the cooperation of theoriesand experiments And the model is normalized by the elasticmodulus of themediumunder controlled condition thus thismodel can be used to predict the effect of various temperatureand relative humidity on the plateau stress of multilayercorrugated sandwich structure under flatwise compression

On the basis of a great deal of research on static com-pression scholars have further studied the energy-absorbingproperties of the paper honeycomb sandwich panel underdynamic impact E andWang [15] investigated the effect of thestrain rate on themechanical behavior and energy absorptionof paper honeycomb by experiments The experimentalresults show that the load-carrying capacity and the energyabsorption performance of paper honeycomb are insensitiveto the loading speed in low strain rate range (10minus4ndash10minus2 sminus1)Wang and Bai [16] found that there exists a certain strain rateeffect which leads to the difference of mechanical propertiesbetween dynamic and static condition A mechanical modelwas given to estimate plateau stress and yield stress at differentstrain rates based on experimental data by employing themethods ofCowperndashSymondsmodel and piecewise functionAnd then Bai et al [17] according to the mechanical modelbuilt the energy absorption diagram under different strainrates Moreover they have taken serious experiments and theresult showed a good agreement with theoretical models

From present documents all energy absorption prop-erties were researched on either relative humidity effector strain rate effect The energy absorption diagram of

material can be modeled by linking the optimal energyabsorption point under two dimensions However only asingle parameter was considered in the two-dimensionalenergy absorption diagramwhich limited its applicationThepurpose of this paper is to construct 3D diagram of paperhoneycomb sandwich panel which took structure strain rateand moisture content into account The diagram took therandom shock and variety of environment in the logisticscondition into account This will make the optimizationdesign of paper honeycomb sandwich panel more suitable forthe actual logistics environment

2 The Evaluation of Plateau Stress andCritical Stress Based on the Effect of StrainRate and Moisture Content

The influence of temperature and relative humidity on mois-ture content and cushioning property of paper honeycombsandwich panel was studied by reference [18 19] Then amodel of moisture content and the prediction equations ofcritical and plateau stress of paper honeycomb sandwichpanel are obtained and verified [6] We have the equationsas follows

119872= 205RH(1 minus 0005432RH) (1 minus 0005432RH + 332RH119879) 119879

(1)

where119872 is moisture content of paper honeycomb sandwichpanel RH is relative humidity 119879 is temperature ∘C

120590pk1198640119904 = 19298119890minus00769119872 (119905119897 )

3 (2)

120590pl1198640119904 = 00991119890minus00769119872 lowast (119905119897 )

53

+ 001067119890minus00769119872 (119905119897 )2

(3)

where 120590pk is critical stress of paper honeycomb sandwichpanel MPa 120590pl is plateau stress of paper honeycomb sand-wich panel MPa 1198640119904 is elastic modulus of corrugatingmedium under standard condition MPa 120590pk1198640119904 is nor-malized critical stress 120590pl1198640119904 is normalized plateau stress119890 = 27183 119905119897 is thickness-to-length ratio 119905 is thickness ofcorrugatingmediummm 119897 is length of honeycomb cell mm

Combined with the strain rate effect [16 17] studied themechanical behavior of paper honeycomb sandwich paneland obtained the prediction equations of dynamic critical andplateau stress The equations are as follows

1205901015840pk1198640119904 =18089 times 10minus4

(1 minus 120576119901119896) (1 + 028120576119901119896)2 +120590pk1198640119904 (4)

1205901015840pl1198640119904 =120590pl1198640119904 [1 + (

120576119861)1119902] (5)




1205901015840pl1198640119904 = [00991119890minus00769119872 lowast (119905119897 )

53

+ 001067119890minus00769119872 (119905119897 )2] [1 + ( 120576119861)

1119902] (6)


119861 = 1528119905119897 + 209119902 = 05125 (7)



lowast (119905119897 )2] [1 + ( 1205761528119905119897 + 209)

19512] (8)



lowast 10minus5 ) [1 + ( 1205766825)19512]


lowast 10minus5) [1 + ( 1205767178)19512]


lowast 10minus5) [1 + ( 1205768446)19512]

(9)



8

7

6

5

4

3

2

1100

80

4020

0 89

1011

1213

14


Nor

mal

ised

plat

eau

stres

s

times10minus4

60

003125t

l=

00333t

l=

00416t

l=


Com

pres

sive s

tress

(M

Pa)

ＪＥ

l1

ＪＥ


ＪＦ








1198821198640119904 = (12059010158401198640119904 )2 119897radic34119905 0 le 120576 le 120576pk

1198821198640119904 = (1205901015840pk1198640119904 )


1198821198640119904 = (1205901015840pk1198640119904 )

2 119897radic34119905 +1205901015840pl (1 minus 395 (119905119897))1198640119904 120576 gt 120576119863

(10)



1198821198640119904 = (12059010158401198640119904 )2 119897radic34119905 0 le 120576 le 120576pk

1198821198640119904 = [[18089 times 10minus4

(1 minus 120576pk) (1 + 028120576pk)2

+ 19298119890minus00769119872 (119905119897 )3]]2 119897radic34119905

+ [00991119890minus00769119872 (119905119897 )53

+ 001067119890minus00769119872 (119905119897 )2] [1

+ ( 1205761528119905119897 + 209)19512] (120576 minus 120576pk) 120576pk le 120576 le 120576119863

1198821198640119904 = [[18089 times 10minus4

(1 minus 120576pk) (1 + 028120576pk)2

+ 19298119890minus00769119872 (119905119897 )3]]2 119897radic34119905

+ [00991119890minus00769119872 (119905119897 )53

+ 001067119890minus00769119872 (119905119897 )2] [1

+ ( 1205761528119905119897 + 209)19512](1 minus 395 119905119897 ) 120576 gt 120576119863

(11)



1198821198640119904 = (1205901015840pk1198640119904 )

2 119897radic34119905 +1205901015840pl (1 minus 395 (119905119897))1198640119904 (12)


1198821198640119904 asymp1205901015840pl (1 minus 395 (119905119897))1198640119904 (13)


1198821198640119904 = [00991119890minus00769119872 (119905119897 )

53

+ 001067119890minus00769119872 (119905119897 )2] [1

+ ( 1205761528119905119897 + 209)19512](1 minus 395 119905119897 )

(14)




lowast 10minus6) [1 + ( 1205766825)19512]


lowast 10minus5) [1 + ( 1205767178)19512]


lowast 10minus5) [1 + ( 1205768446)19512]

(15)





6

5

4

3

2

1

0100

8060

4020

0 89

1011

1213

14


Nor

mal

ised

ener

gy ab

sorp

tion

per u

nit v

olum

e

times10minus4

003125t

l=

00333t

l=

00416t

l=






5

4

3

2

1

00 1 2 3 4 5


Expe

rimen

tal v

alue

(10minus4)

R2 = 09551



5 Conclusion







Acknowledgments


References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





1205901015840pl1198640119904 = [00991119890minus00769119872 lowast (119905119897 )

53

+ 001067119890minus00769119872 (119905119897 )2] [1 + ( 120576119861)

1119902] (6)


119861 = 1528119905119897 + 209119902 = 05125 (7)



lowast (119905119897 )2] [1 + ( 1205761528119905119897 + 209)

19512] (8)



lowast 10minus5 ) [1 + ( 1205766825)19512]


lowast 10minus5) [1 + ( 1205767178)19512]


lowast 10minus5) [1 + ( 1205768446)19512]

(9)



8

7

6

5

4

3

2

1100

80

4020

0 89

1011

1213

14


Nor

mal

ised

plat

eau

stres

s

times10minus4

60

003125t

l=

00333t

l=

00416t

l=


Com

pres

sive s

tress

(M

Pa)

ＪＥ

l1

ＪＥ


ＪＦ








1198821198640119904 = (12059010158401198640119904 )2 119897radic34119905 0 le 120576 le 120576pk

1198821198640119904 = (1205901015840pk1198640119904 )


1198821198640119904 = (1205901015840pk1198640119904 )

2 119897radic34119905 +1205901015840pl (1 minus 395 (119905119897))1198640119904 120576 gt 120576119863

(10)



1198821198640119904 = (12059010158401198640119904 )2 119897radic34119905 0 le 120576 le 120576pk

1198821198640119904 = [[18089 times 10minus4

(1 minus 120576pk) (1 + 028120576pk)2

+ 19298119890minus00769119872 (119905119897 )3]]2 119897radic34119905

+ [00991119890minus00769119872 (119905119897 )53

+ 001067119890minus00769119872 (119905119897 )2] [1

+ ( 1205761528119905119897 + 209)19512] (120576 minus 120576pk) 120576pk le 120576 le 120576119863

1198821198640119904 = [[18089 times 10minus4

(1 minus 120576pk) (1 + 028120576pk)2

+ 19298119890minus00769119872 (119905119897 )3]]2 119897radic34119905

+ [00991119890minus00769119872 (119905119897 )53

+ 001067119890minus00769119872 (119905119897 )2] [1

+ ( 1205761528119905119897 + 209)19512](1 minus 395 119905119897 ) 120576 gt 120576119863

(11)



1198821198640119904 = (1205901015840pk1198640119904 )

2 119897radic34119905 +1205901015840pl (1 minus 395 (119905119897))1198640119904 (12)


1198821198640119904 asymp1205901015840pl (1 minus 395 (119905119897))1198640119904 (13)


1198821198640119904 = [00991119890minus00769119872 (119905119897 )

53

+ 001067119890minus00769119872 (119905119897 )2] [1

+ ( 1205761528119905119897 + 209)19512](1 minus 395 119905119897 )

(14)




lowast 10minus6) [1 + ( 1205766825)19512]


lowast 10minus5) [1 + ( 1205767178)19512]


lowast 10minus5) [1 + ( 1205768446)19512]

(15)





6

5

4

3

2

1

0100

8060

4020

0 89

1011

1213

14


Nor

mal

ised

ener

gy ab

sorp

tion

per u

nit v

olum

e

times10minus4

003125t

l=

00333t

l=

00416t

l=






5

4

3

2

1

00 1 2 3 4 5


Expe

rimen

tal v

alue

(10minus4)

R2 = 09551



5 Conclusion







Acknowledgments


References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





1198821198640119904 = (12059010158401198640119904 )2 119897radic34119905 0 le 120576 le 120576pk

1198821198640119904 = (1205901015840pk1198640119904 )


1198821198640119904 = (1205901015840pk1198640119904 )

2 119897radic34119905 +1205901015840pl (1 minus 395 (119905119897))1198640119904 120576 gt 120576119863

(10)



1198821198640119904 = (12059010158401198640119904 )2 119897radic34119905 0 le 120576 le 120576pk

1198821198640119904 = [[18089 times 10minus4

(1 minus 120576pk) (1 + 028120576pk)2

+ 19298119890minus00769119872 (119905119897 )3]]2 119897radic34119905

+ [00991119890minus00769119872 (119905119897 )53

+ 001067119890minus00769119872 (119905119897 )2] [1

+ ( 1205761528119905119897 + 209)19512] (120576 minus 120576pk) 120576pk le 120576 le 120576119863

1198821198640119904 = [[18089 times 10minus4

(1 minus 120576pk) (1 + 028120576pk)2

+ 19298119890minus00769119872 (119905119897 )3]]2 119897radic34119905

+ [00991119890minus00769119872 (119905119897 )53

+ 001067119890minus00769119872 (119905119897 )2] [1

+ ( 1205761528119905119897 + 209)19512](1 minus 395 119905119897 ) 120576 gt 120576119863

(11)



1198821198640119904 = (1205901015840pk1198640119904 )

2 119897radic34119905 +1205901015840pl (1 minus 395 (119905119897))1198640119904 (12)


1198821198640119904 asymp1205901015840pl (1 minus 395 (119905119897))1198640119904 (13)


1198821198640119904 = [00991119890minus00769119872 (119905119897 )

53

+ 001067119890minus00769119872 (119905119897 )2] [1

+ ( 1205761528119905119897 + 209)19512](1 minus 395 119905119897 )

(14)




lowast 10minus6) [1 + ( 1205766825)19512]


lowast 10minus5) [1 + ( 1205767178)19512]


lowast 10minus5) [1 + ( 1205768446)19512]

(15)





6

5

4

3

2

1

0100

8060

4020

0 89

1011

1213

14


Nor

mal

ised

ener

gy ab

sorp

tion

per u

nit v

olum

e

times10minus4

003125t

l=

00333t

l=

00416t

l=






5

4

3

2

1

00 1 2 3 4 5


Expe

rimen

tal v

alue

(10minus4)

R2 = 09551



5 Conclusion







Acknowledgments


References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





6

5

4

3

2

1

0100

8060

4020

0 89

1011

1213

14


Nor

mal

ised

ener

gy ab

sorp

tion

per u

nit v

olum

e

times10minus4

003125t

l=

00333t

l=

00416t

l=






5

4

3

2

1

00 1 2 3 4 5


Expe

rimen

tal v

alue

(10minus4)

R2 = 09551



5 Conclusion







Acknowledgments


References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






Acknowledgments


References























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


3d energy absorption diagram construction of paper honeycomb...

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