Abstract—The methods (C-σm curve and Gm-σs curve) those

are adopted from the Chinese standard GB8166-87 and used forthe cushioning material design in voyage, architecture,automobile, transportation, and package fields have manyshortcomings. The novel cushioning defense technique-energyabsorption diagram method-were adapted to optimization designof cushioning materials in this paper. For this purpose, in thepaper the demand, functions and modules of the computer aideddesign system of cushioning materials were analyzed. Then thesystem was designed particularly, and theenergy-absorption-diagram production and optimizationmodules were mainly designed and realized. Finally somecommon examples on cushioning package design were computedin the final system, and the results from the final system and themethods in GB8166-87 were analyzed in contrast. The analysisand drop experiments on the designs according to theenergy-absorption-diagram method proved that it is morereliable and efficient for this method to be used in the cushioningmaterial optimization design than the methods in nationalstandard GB8166-87. (Postal Address: Mail Box No.806, Xi'anUniversity of Technology, Xi'an China; Post Code: 710048)

Index Terms—Algorithm; Cushioning-defense ComputerAided Design System; Cushioning Materials; Energy AbsorptionDiagrams; Optimization Design

I. INTRODUCTION

OST of the cushioning materials are cellular, light andgood energy absorbers, so they are extensively used in the

fields about voyage, architecture, transportation, automobileand packaging. The cushioning materials absorb all the impactenergy to defend the object under the level of its allowablestress. The traditional methods on the cushioning packagedesign of products adopted from the Chinese national standardGB8166-87 and the book [1] are cushioning coefficient(C)-maximum strain curve (σm) and maximum acceleration

Manuscript received July 10, 2006. The bureau of science and technology ofShaanxi province and its project GG200314 supported this work.

Sun Deqiang is with the Key Laboratory of Contemporary Design andIntegrated Manufacturing Technology of Ministry of Education NorthwesternPolytechnical University, Xi'an 710072 China. (phone: 86-29-8231-2803; fax:86-29-8322-0987; e-mail: wampire@126.com).

(Gm)-static strain (σs) curve methods. Both of them have manyshortcomings: 1) many dynamic or static experiments on thesame material must be done and the experimental data may benot consistent with the properties of the cushioning materials; 2)many experiments on different thickness of cushioningmaterials must be done, and the acquisition and collection ofdata are very difficult; 3) most of the cushioning materials arecellular and their properties depend on their relative densitiesso the optimization of the cushioning materials should focus ontheir densities and however in the traditional methods therelative densities of the cushioning materials are not taken intoaccount; 4) most of the cushioning materials are all viscoelasticand their stress-strain curves depend on the strain rate andtemperature, but the traditional methods don’t take them intoaccount; 5) The cushioning defense optimization design shouldbe based on abundant data of the cushioning materials andfocus on the proper basis material, relative density andthickness of the cushioning materials, but it is impossible forthe traditional methods to accomplish the optimization designlike this.

In 1984 Maiti et al. firstly established theenergy-absorption-diagram method in the reference [1] and in1997 Gibson thoroughly presented the algorithm in packagedesign based on this method in his book [2]. In China, themethods that are used by many scholars in their cushioningdefense CAD systems are still the traditional ones inGB8166-87 now [4,5,6]. The cushioning defense optimizationdesign is extensively used in the fields about voyage,architecture, transportation, automobile and packaging andinvolved in a great deal of computation and basic data, so itmust be done by computer. The software based onenergy-absorption-diagram method doesn’t appear in themarkets in China. For realization of this software, in this paperthe corresponding design system based onenergy-absorption-diagram method will be designed and thedemand analysis and realization techniques of this system willalso be finished. Finally the accomplished system carries outthe computations of two common examples of cushioningdesign.

Computer aided design system of cushioningmaterials

Sun Deqiang

Key Laboratory of Contemporary Design and Integrated Manufacturing Technology of Ministry of Education, NorthwesternPolytechnical University, Xi'an 710072 China

M

II. SYSTEM DEMAND ANALYSIS

A. System FunctionsThe system is developed based on the

energy-absorption-diagram method and also has basicfunctions of the cushioning design. It should have the functionsin the following.

1) Firstly it should have a indispensable function: tooptimize basis material, relative density and thickness ofcushioning materials according to their energy-absorptiondiagrams.

2) The basic data the system is based on are stress-straincurves, so the system should manage, maintain and produce allkinds of curves of the cushioning materials.

3) Under different conditions (with known or unknown basismaterial), the system should find the corresponding algorithmto optimize the cushioning materials for the object to beprotected.

4) The system should be able to accomplish structure designof the cushioning pads. The structure design includesfiguration and dimension confirmation of the cushioning padsand is accomplished according to area and thickness afteroptimization, configuration requirement of pads and thedimensions of the object to be protected. Finally the system alsomakes the production charts for the final pads.

5) The output and input of the design information must beconvenient. For different designs, the system not only inputs allrequired information conveniently, but also query, search anddisplay the results of every iteration step for every optimizationdesign.

B. System ModulesTo realize the functions above, the system should include the

following modules as shown figure 1.

1) Project Management Module. Each cushioning defenseoptimization design carried out in this system is called a projectand organized in the form of project. One project includesmany files, such as project management file, designinformation file and so on.

2) Data Management Module. In this module, the data aboutthe stress-strain curves of cushioning materials with differentrelative densities under different strain rates are managed andmaintained.

3) Optimization Module. The system can determine theoptimization kind according to the known design information.Different kind of optimization design has differentcorresponding algorithm, and the system can automaticallyfind the proper algorithm for one optimization design.

4) Structure Design Module. This module mainly finishesstructure design of cushioning pads mentioned above.

5) Output Module. This module mainly finishes functionsabout output and display of the results mentioned above.

Project management module mainly manages all kinds offiles; the functions of data management module can be realizedby database techniques; structure design module and outputmodule are related to graphics and image programming. Theseprogramming techniques are all practical. So the difficulty indeveloping this system lies in production of energy-absorptiondiagrams, algorithm of optimization design and design ofsystem database.

Knownparameters

Interface of the computer aided optimization design system of cushioningmaterials

Project managementmodule

Data managementmodule

Structure designmodule

Output module

Message Driven Functions

Optimized results

Structurecharts

Data for optimization

Structure designparameters

Optimization module

Outputcontrollingparameters

Figure 1 Constitution of the system modules

III. PRODUCTION OF ENERGY ABSORPTION DIGRAMS

A. Production Procedure of Energy Absorption Diagramsof the Cushioning MaterialsThe energy-absorption diagrams that the system is based on

are produced from stress-strain curves of cushioning materials.Most of cushioning materials are strain rate (or temperature)dependent and their stress-strain curves are related to strainrate (or temperature). Normalized energy per volume -normalized peak stain curves and their envelope are producedfrom stress-strain curves of cushioning materials as shown infigure 2.

Different stress-strain curves of the cushioning materialswith same basis material and different relative densities areshown in figure 2(a) when 1εε &&� and T=T1. When the stressincreases until pσ , the energy absorbed by one cushioningmaterial per volume is exactly W .

We canview W and

pσ asvariables,regard W asthe functionof pσ , and

normalizethem withmodulus ofelasticity

sE measured under theconditions

1εε &&� andT=T1.Then wewill get thefigure 2(b).From theshoulderpoint of thiscurve infigure 2(b)we can getthe maximum allowable stress of the protected object when thecushioning material with the corresponding density is used.Heavy line is the envelop of shoulder points of the cushioningmaterial with same basis material and different relativedensities that represents the relation between W and pσ of theoptimal cushioning material under the conditions 1εε &&� andT=T1.

In figure 2(c), at the same temperature and coordinationsystem all the envelops of the same cushioning materials underdifferent strain rates can be drawn. In figure 2(c), the fine linerepresents the envelop of shoulder points of the samecushioning material with different relative densities under

(a) Stress-strain curves of cushioning materialswith T=T1 and 1εε &&�

Strain

Stre

ss(M

Pa)

W

Relative densities increase.

10

0.030.10.21εε &&�

T=T1

(b) Envelop after normalization

Normalized peak strain

Nor

mal

ized

ener

gype

rvol

ume Envelop with 1εε &&�

For different relativedensities

10-3

10-4

10-

5

0.01

0.03

0.1

0.2

10-4 10-3 10-2

(c) Envelops after normalization for different strain rates

0.01

0.10.2

10-5

10-4 10-3 10-2

For different relativedensities

Nor

mal

ized

ener

gype

rvol

ume

Normalized peak strain

Computing the impact velocity mUv 2=

and strain rate11 2 tmU=ε&

Y

N

Calculating F=ma and σp=F/A,and setting first thickness 1t

Figure 4 Optimization design flow of cushioning materialswith known basis material

From database searching the relative density andabsorbed energy of the cushioning material

according to the conditions above; calculatingnew thickness ( )WAUt =2

Inputting known parameters: modulus ofelasticity Es of cushioning material; mass m,

maximum acceleration a, bearing area A, impactkinetic energy U of the object to be protected

Start

≤− 12 tt error?21 tt =

End

some strain rate and temperature. While the heavy line infigure 2(c) represents the envelop of shoulder points of thecushioning material with same relative density under differentstrain rates.

Abundant data of the energy-absorption diagrams can begotten according to the procedure above and they are the basisfor the following optimization design of the cushioningmaterials.

B. Production of Energy Absorption Diagrams in theSystem

To get the energy-absorption diagrams in this system, weshould create the tables of the cushioning materials and theirstress-strain curves in the system database. The former includesthe information about the properties of the cushioningmaterials, and the latter contains the data those are coordinatesof the key points on the stress-strain curves. According to thesekey points interpolation or fitting techniques can help to maketheir stress-strain curves and meanwhile integral of thesecurves makes the final energy-absorption diagrams. Thecommon techniques are Lagrange interpolation, spline fittingand B-spline curve methods.

IV. OPTIMIZATION DESIGN ALGORITHM OF CUSHIONINGMATERIALS

The optimization designs of the cushioning materials are oftwo kinds. One is with known basis material that optimizes therelative density and thickness of the cushioning materials, andthe other is with unknown basis material that optimizes therelative density and basis material.

A. Cushioning Optimization Design Algorithm with theKnown Basis MaterialThe basis material of the cushioning material is known, and

its modulus of elasticity sE is constant. The mass m, bearingarea A, impact kinetic energy U and maximum allowableacceleration of the object to be protected a are given. Theoptimization algorithm is in the following as shown in figure 3.

1) Calculating the maximum force F=ma the protectedobject is subject to. So the peak stress AFp =σ and normalized

peak stress level is sp Eσ .

2) Setting the first thickness of cushioning material as 1t ,and the impact velocity is mUv 2= . So the initial strain rateis 11 2 tmU=ε& .

3) From the energy-absorption diagrams, we can get thevalue of sEW according to the values of the normalized peakstress, and then get the energy per volume W absorbed by thecushioning material. The new thickness is ( )WAUt =2 .

4) The new strain rate is 22 2 tmU=ε& . Then the newenergy per volume W absorbed by cushioning material can becalculated by the formula in step 3).

Iteration will go on until the absolute value of 12 tt − is equalto and less than the error initialized. The thickness at this timeis the final and optimal and the optimal relative density is alsogotten from its energy-absorption diagrams.

This algorithm above is realized in the system as shown infigure 4. To realize this algorithm, we must do one key thingthat the system can search the relative density and absorbedenergy per volume of the cushioning materials under theconditions of some strain rate and peak stress pσ .

Normalized peak stress

Nor

mal

ized

ener

gype

rvol

ume

10-1

10-4

10-5

Figure3 Optimization problem with given materials

10-2

10-3

Strain rate1ε& when

thickness is1t

Strain rate2ε& when

thickness is2t

10-4 10-3 10-2 10-1

Two valuesof

sEW

Maximumallowable stress

0.05

0.1

0.3

Normalized peak stress

Nor

mal

ized

ener

gype

rvol

ume

10-1

10-4

10-5

Figure 5 The optimization problem without given material

10-2

10-3

ε&

10-4 10-3 10-2 10-1

First valueof

sEW0.05

0.1

0.3Desired value of

sEW

First value of

sp Eσ

Desiredvalue of

sp Eσ

Dashed line with slope k

Point A

Point B

B. Cushioning Optimization Design Algorithm with theUnknown Basis MaterialThe thickness of the cushioning material t and the mass m,

bearing area A, impact kinetic energy U and maximumallowable acceleration of the object to be protected a are given.The optimization algorithm is in the following as shown infigure 5.

1) Calculating the energy per volume absorbed bycushioning materials ( )AtUW = and the peak stress

Amap =σ .2) Setting a proper value for sE and search the point of

intersection between the horizontal dashed line of the firstsEW and the vertical dashed line of sp Eσ A.

3) From the point A and down the dashed line withslope k ( pW σ ) which is through point A, the W , pσ areconstant and sE is variable.

4) Computing the strain rate according to the formulatmU2=ε& . The point of intersection of the dashed line and

the envelope line with εε &= B is gotten.

5) The value of sEW and sp Eσ on the point B are all

gotten and the value of sE is also gotten according to theknown value of W and pσ .

To realize this algorithm, we must do another key thing thatthe system can search the absorbed energy per volume W of thecushioning materials with some relative density under somepeak stress level pσ .

V. DESIGN OF SYSTEM DATABASE AND TOTAL SYSTEM FLOW

The system database is produced by the relational databasemanagement system (RDBMS) SQL Server that is based onstructural querying language (SQL) and multithreadingtechniques. This RDBMS uses two-dimensional tables,keywords and relation charts to realize the purpose of savingdata efficiently and avoiding reiteration. The main tables of thesystem database are ones about performance and stress-straincurves of the cushioning materials, and others aboutinformation and computation results of projects. These tablesare performance table of cushioning materials, stress-straintable of cushioning materials, table of project information,table of project information and so on. There is a one-to-manyconstraint relation between performance table of cushioningmaterials and stress-strain table of cushioning materials. Thereis also a one-to-many constraint relation between performancetable of cushioning materials and stress-strain table ofcushioning materials. Now the total system flow is shown infigure 6.

VI. EXAMPLES

A. Optimization Design Example with the Known BasisMaterialThe mass of the product to be packaged m is 0.5kg, bearing

area A=0.01m2, impact velocity v=4.5m/s and maximumallowable acceleration a=10g. So energy to be absorbed is

522 == mvU J, the maximum force 50== maF N, allowablepeak stress 5== AFpσ kN/m2, and the equivalent height is

h=1m. Polyurethane foams are used as cushioning materialsand is modulus of elasticity is 50=sE MN/ m2. Normalizedpeak stress is 410−=sp Eσ . With the initial thickness 0.9m, thecompleted system works out the final and proper thickness

0.15m and relative density 0.01 after several iterations. Theresults are shown in the left column of table 1.

Table 1 Comparison of results designed according toenergy-absorption-diagram, cushioning coefficient-maximum stress andmaximum acceleration-static stress methods (with the known material)

Proj

ectm

anag

emen

tmod

ule

Opt

imiz

atio

nm

odul

e

Finalresults

Performance table ofcushioning materials

Stress-strain table of cushioning materials

Inputtingthe knownparameters

Requireddata

Relational database

Figure 6 Total flow chart of system

Data managementmodule

Structure designmodule

Output module

Data Management

Outputcontrolparameters

Designparameters

Designresults

Designrequiremen

Table of projectinformation Table of project

information

Syst

emso

ftwar

e

According to the methods in national standard GB8122-87and reference [1], results for polyurethane foams with relativedensity 0.01 calculated with the cushioningcoefficient-maximum stress and maximum acceleration-staticstrain methods are shown in the middle and right columns ofthe table 1.

B. Optimization Design Example with the unknown BasisMaterialThe mass of the product to be protected m is 2.5kg, bearing

area A=0.025m2, drop height is h=1m, the thickness ofcushioning material t is 0.2m and maximum allowableacceleration a=10g. The completed system works out that themodulus of elasticity of cushioning material is 28 MN/m2 and itis the modulus of elasticity of polyamide. The proper relativedensity is 0.1. The results are shown in the left column of table2.

Table 2 Comparison of results designed according toenergy-absorption-diagram, cushioning coefficient-maximum stress and

maximum acceleration-static stress methods (with the unknown material)

For comparison, results for polyamide foams with relativedensity 0.1 calculated with the cushioningcoefficient-maximum stress and maximum acceleration-staticstrain methods are shown in the middle and right columns ofthe table 2.

The results of the two examples above show that thedimensions of the cushioning materials gotten byenergy-absorption-diagram method is less than thosecalculated by the traditional methods, which demonstrates thatthe energy-absorption-diagram method fully makes use of theenergy absorption characteristic of the cushioning materials. Inthese examples the energy-absorption-diagram method alsotake the relative densities and strain rate (or temperature) of thecushioning materials into account to optimize them, and thetraditional methods don’t realize this purpose.

VII. CONCLUSION

The following tasks are finished in this paper.1) The optimization design system of cushioning materials is

designed and realized.2) Production principle of the energy-absorption diagrams is

represented.3) Optimization algorithms of cushioning materials are

designed and realized.4) The final examples proved that the

energy-absorption-diagram method not only overcomes theshortcomings of the traditional methods but also takes therelative density and strain rate into account, which make theoptimization more reasonable and efficient.

The key problem we must resolve to use theenergy-absorption-diagram method to optimize cushioningmaterials is that abundant stress-strain data of cushioningmaterials with different relative densities under different strainrates must be gotten through experiments.

Methods Results

Energy-absorption-diagrammethod

For the first iterationThe initial thickness is t =0.9m, initial strain rate

is 11 9 −== stvε& , normalized energy per volume is

5.5=sEW , and absorbed energy per volume is

W=2750J/m3.For the second iteration

The revised thickness 24.0=t m, strain rateis 1

1 8.18 −== stvε& � 5101.7 −×=sEW and

W=3540J/m3

For the third iterationFinal results: thickness is 0.15m and relative density is0.01.

Cushioningcoefficient-maximum stressmethod

Maximum stress is == Aammσ 5kN/m2 and the

minimum cushioning coefficient on the C-σm curves ofpolyurethane foams is 2.1. The final thickness is0.21m.

Maximumacceleration-static strain method

Static strain is == Amgtσ 500N/m2.

The point of intersection of lines with a andtσ is

between 0.2 and 0.3 on the G-σs curves ofpolyurethane foams. The final thickness is0.23m.

Methods Results

Energy-absorption-diagrammethod

Energy absorbed by cushioning materials 25== mghU J

Absorbed energy per volume ( ) 4105×== AtUW J/m3

Maximum allowable force of product 2500== maF NAllowable peak strain of product 510== AFpσ N/m2

Impact velocity 5.42 == ghv m/s

The initial sE is 100 MN/m2, the initial 4105 −×=sEW and

310−=sp Eσ �The strain rate stv 225==ε& , and the final

sEW andsp Eσ are 1.8�10-3 and 3.7�10-3 respectively.

The optimal 28=sE MN/m2 and the optimal relative density

is 0.1.Cushioningcoefficient-maximum stressmethod

Maximum stress is10kN/m2 and the minimum cushioningcoefficient on the C-σm curves of polyurethane foams is 24.4.The final thickness is 0.24m.

Maximumacceleration-static strainmethod

Static strain is 103N/m2. The point of intersection of lines with a100g and 500 is between 0.2 and 0.3 on the G-σs curves ofpolyurethane foams. The final thickness is 0.25m.

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[3] Lorna J. Gibson, Michanel F. Ashby. Cellular solids: structure andproperties, Oxford, England, Pergamon Press, 1997, ch.8, pp. 252-343.

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[7] Weaire, D. and Fortes, M. A. “Stress and strain in liquid and solid foams”.Adv. Pys., vol. 43: 685-738, 1994.

[8] Suh and K. W. Skochdopole, R. E. “ Foamed plastics”. Encylopedia ofPolymer Science, vol. 2, pp. 82, 1980.

[9] Weaire, D. and Fortes, M. A. “Stress and strain in liquid and solid foams”,Adv. Pys., vol. 43, pp. 685-738, 1994.