Journal of Space Engineering

Journal of SpaceEngineering

Vol.4, No.1, 2011

Mechanical Properties of Z-Fold Membraneunder Elasto-Plastic Deformation∗

Yasutaka SATOU∗∗ and Hiroshi FURUYA∗∗∗∗ Department of Built Environment, Tokyo Institute of Technology

4259–G3–6 Nagatsuta, Midori-ku, Yokohama 226–8502, Japan

E-mail: [email protected]

AbstractThis paper addresses a elasto-plastic behavior of creasing process for a z-fold mem-brane to examine the mechanical properties of the crease, which determine the foldingand deployment characteristics of a large membrane. To examine the elasto-plasticbehavior in terms of the layer pitch and the contact force for creasing the membrane,fold experiments are performed. The experimental results are evaluated numerically bydemonstrating elasto-plastic FEM analyses, and examined theoretically by introducinga mathematical model. In the FEM analyses, the precision is improved by investigat-ing numerical parameters; the penalty stiffness for the contact analyses, the numericaldamping in the equilibrium equation, and the size of the finite element mesh, whichare dominant parameters in non-linear FEM analyses. In the mathematical model, themechanics of the creasing process is formulated for elastic deformation. These resultsindicate that the loading process of the creasing properties is confirmed in terms ofthe contact force and the layer pitch. On the other hand, the further examinations arerequested for the unloading behavior to determine the mechanical properties of thecrease precisely.

Key words : FEM, Elastoplastic, Crease, Fold, Solar Sail

1. Introduction

Current interest in the use of large deployable space membranes has led to a number ofspace missions; solar sails, large aperture antennas, sunshields, solar power satellites, and et al.Since these structures are consists of thin and large membranes, the membrane is packed intothe rocket on the ground. In the course of the packaging, creases are generated, and the creasesaffect the folding and deployment of the membrane. For example, the packaging densityrelies on the contact force for creasing the membrane since the contact force determines thethickness of the crease and the layer pitch of the membrane. In the case of a solar powersail(1), the attachment area of solar cells is reduced depending on the region of the creaseto avoid the damages on the cells. In addition, the deployment dynamics is affected by thetensile behavior of the crease, and the flatness of the membrane after deployment depends onthe residual deformation of the crease.

Several researches were performed for the mechanical properties of the crease. For ex-ample, Okuizumi(2) expressed the crease behavior with a rotational spring correlated with thenatural crease angle to perform the deployment simulation of the solar sail using a spring-mass system. MacNeal et al.(3) derived the effective tensile modulus of a creased metallictape by assuming the crease as a plastic hinge. Furuya et al.(4) formulated the deploymentcharacteristics of a one-dimensional creased membrane, where the curved beam theory wasemployed to calculate the rotational stiffness of the crease. Gough et al.(5) introduced nonlin-ear anisotropic finite elements on the creased area to investigate the wrinkling behavior of acreased membrane using FEM. Papa et al.(6) represented the crease as the initial configurationobserved in the experiments to examine the stress distribution and the load-displacement rela-

∗Received 30 Mar., 2011 (No. 11-0228)[DOI: 10.1299/spacee.4.14]

Copyright c© 2011 by JSME

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Vol.4, No.1, 2011

tionships of the Miura-ori folding pattern. In our previous research works(7), folding analyseswere performed, where the creasing process was numerically simulated using FEM analy-ses to examine the fold properties of the membrane. However, few research works treat theelasto-plastic deformation of the crease.

Based on the above discussions, the elasto-plastic behavior of the crease is significantin the creasing process and in the deployment process. In the creasing process, the contactforce for creasing the membrane is one of the significant parameters because the contact forcedetermines the layer pitch of the membrane, which induces the plastic deformation in thecrease. In the deployment process, the tensile behavior of the creased membrane is importantto clarify the deployment dynamics. As the properties of the creased membrane depend on theplastic deformation in the creasing process, we focus on the elasto-plastic properties. Thus,the objective of the present paper is to identify the layer pitch which induces the plastic defor-mation of the membrane. We treat the crease in the z-fold membrane as one of the simple foldpatterns. At first, the effects of the elasto-plastic deformation on the mechanical propertiesof the crease are examined experimentally by investigating the relationships between the con-tact force and the layer pitch. Next, elasto-plastic FEM analyses are performed for the z-foldmembrane to evaluate the experimental results. Then, a mathematical model for linear elasticdeformation is formulated to examine the mechanics of the creasing process theoretically. Fi-nally, mechanical properties of the crease are discussed in terms of the contact force and thelayer pitch.

Nomenclaturepn : contact pressure on the contact sur-

faceg : penetration

kp : penalty stiffnessn : unit normal vector of the contact

surfaceλ : correction term which specify the

penetration toleranceFE : vector of external forceFI : vector of internal forcec : damping factor

M : artificial mass matrix calculatedwith unity density

v : vector of nodal velocitiesκ : curvature of creaseθ : deflection angleθ0 : deflection angle at s=a+bs : body fixed system

x, y, z : global coordinate systemE : Young’s modulus

I : moment of inertia of areaP : contact force by adjacent mem-

braneTs : in-plane force of body fixed systemQs : shear force of body fixed systemMs : bending moment of body fixed sys-

temTa+b : in-plane force at s=a+bQa+b : shear force at s=a+b

t : membrane thicknessu, w : displacement

u0, w0 : displacement of neutral lineRx,Rz : reaction force

h : layer pitchσs : stress along body fixed systemεs : strain along body fixed systemEi : definition of edgeS i : definition of surfacef1 : edge load of FEM analysesν : Poisson’s ratio

2. Fold experiments

The principal objectives of the fold experiments are to examine the configuration of thecrease, as well as to clarify the relationships between the contact force for creasing the mem-brane and the layer pitch.

2.1. Z-fold membraneFigure 1 indicates the cross-section of a z-fold membrane. We make the following three

assumptions for the cross-section. First, the contact force for creasing the membrane is domi-nant force; the membrane is creased by the contact force. Second, the membrane is uniformlycreased along the y-direction, and hence, the contact force is uniformly distributed. Third, the

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Vol.4, No.1, 2011

configuration of the cross section is repeating structure. When we introduce these assump-tions, it will be impossible to express the boundary condition around the end of the z-foldexactly. On the assumption of the repeating structure, the membrane in the selected area A inFig.1 is investigated, and h indicates the layer pitch of the z-fold membrane.

Fig. 1 Cross-section of z-fold membrane

2.2. Experimental setupIn the fold experiments, it is difficult to setup the boundary condition of the membrane in

the selected area A(Fig.1) because the location of the membrane on the left hand side in theselected area changes depending on the layer pitch. The fold experiments are thus performedfor the membrane in the selected area B.

The initial configuration of a membrane specimen is indicated in Fig.2. The membranespecimen is 101μm thick and PET. A broken line is the target crease line in the experiments.

Figure 3 indicates the side-view of the experimental setup. The membrane specimenis attached to a surface S A1 of a block A and a surface S B1 of a block B by double-sidedtape. The block A is placed on a level surface. On the block B, the contact force is appliedwith weights, where the block B is kept in a level position by walls. The friction betweenthe membrane and the blocks, and between the blocks and the walls are reduced by Naflonsheet(TOMBO-9001). The cross-section of the membrane is magnified by a microscope, andthe image is captured to measure the layer pitch with an image analysis software. In that case,the experimental error is ±0.23mm.

The contact force is loaded and unloaded with 2 cycles under the elastic deformation andunder the elasto-plastic deformation, respectively. On the second assumption in § 2.1, we treat2P as the contact force per unit length in the y-direction. When the contact force is appliedto the block B, the block B is supported by two contact points on the membrane as shown inFig.3. We assume the contact force is equally applied to each contact point, and thus, whenthe total load is 2P, the contact force P is applied to the crease.

Fig. 2 Initial configuration of membrane specimen

3. Elasto-plastic finite element analyses

Elasto-plastic FEM analyses have two main objectives. One is to confirm the elasto-plastic properties of the experimental results. Another is to evaluate the validity of the elasto-plastic FEM analyses for the creasing process.

3.1. Finite element model and creasing processIn the FEM analyses, on the assumption of the repeating structure for the cross-section

of the z-fold, we analyze the membrane model in the selected area A (Fig.1). To stabilize the

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Vol.4, No.1, 2011

Fig. 3 Side-view of experimental setup

FEM analyses numerically, the numerical creasing process is significant. Thus we introducea numerical creasing process proposed in our previous research(7).

Figure 4 indicates the numerical creasing process. A broken line indicates the initialconfiguration of the membrane. As shown in the figure, a strip membrane is used to reducethe numerical cost. The body fixed system of the membrane is indicated with s. There aretwo steps to demonstrate the FEM analyses. In step 1, an edge load f1 is applied to the edgeEx1 to bend the membrane. Since the contact force is assumed to be the dominant forcein the creasing process, the value of the edge load is sufficiently small comparing with thecontact force. On the assumption of the repeating structure, a moment applied to the edgeEx1 is 0Nmm and the edge Ex0 is clamped. In step 2, the contact force is applied with arigid surface. Since the contact area between the membrane and the rigid surface is unknown,the boundary condition of the displacement of the rigid surface is applied, and we obtain thecontact force. The friction between the membrane and the block is neglected in the FEMbecause the friction is reduced by the Naflon sheets in the experiments. On the assumption ofthe repeating structure, the edge Ex1 is h/2 away from y − z surface.

Fig. 4 Creasing process for FEM analyses

3.2. Material propertiesThe tensile tests for the membrane specimen of the fold experiments are performed to

evaluate the experimental results quantitatively by the elasto-plastic FEM analyses. In thetensile test, the membrane thickness, the gauge length, and the membrane width are 100μm,250mm, and 14mm, respectively. Figure 5 indicates the stress-strain relationships obtainedby the tensile test. The broken lines represent the loading and the unloading processes. In thecourse of the tensile test, the plastic strain is occurred when the stress is 65MPa. Using thestress-strain relationships, we obtain the Young’s modulus; 4.8 ± 0.3GPa.

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Vol.4, No.1, 2011

Fig. 5 Stress-strain relationship of membrane specimen

3.3. Setup for FEM analysesThe FEM analyses are demonstrated with ABAQUS/Standard(8), a commercial software.

As the creasing process of the membrane induces geometrical nonlinearity, the analysis pro-cedure is geometrical nonlinear and static, where the implicit integration scheme is employed.To carry out the analysis procedure, General Static and NLgeom option are applied. The non-linear stress-strain relationships of the membrane, which are obtained by the tensile test, areexpressed by piecewise linear function, where the compressive stress-strain relationships areassumed to be equivalent to the tensile one. In the course of the creasing process, when thelayer pitch is small, the shear deformation around the crease will not be negligible. Hence,S4R shell elements are used, which employs the thick shell theory when the shell thickness islarge.

Contact analyses are performed between the membrane and the rigid surface. In thisresearch, Augmented Lagrange Method is applied for the contact analyses. The contact defi-nition in the Augmented Lagrange Method is as follows.

pn = kpg · n + λ (1)

where, pn, g, kp, n, and λ represent contact pressure on the contact surface, penetration,penalty stiffness, unit normal vector of the contact surface, and correction term which specifythe penetration tolerance. As shown in Eq.(1), the contact force depends on the penalty stiff-ness. The relationships between the contact force and the penalty stiffness are investigatedas shown in Fig.6. As shown in the figure, the contact force becomes a constant value whenthe penalty stiffness is more than 1.00E+4 MPa/mm. Although the precision of the contactanalyses is higher as the penalty stiffness is increased, the numerical cost is also increased.Thus, the penalty stiffness is set to be 1.00E+4MPa/mm.

Numerical damping is added to the global equilibrium equations as shown in Eq.(2) inorder to stabilize the analyses of the non-linear geometrical problem numerically.

FE − FI − cMv = 0 (2)

where, FE, FI, c, M, and v represent vector of external force, vector of internal force, dampingfactor, artificial mass matrix calculated with unity density, and vector of nodal velocities.When we introduce the numerical damping, the dissipated energy by the numerical dampinghas to be sufficiently small comparing with the internal energy to improve the precision of theFEM results. The relationships between the damping factor and the internal energy are thusinvestigated in Fig.7. The results indicate that the internal energy is asymptotically close to avalue, and thus, the damping factor is requested to be less than 1.00E-6.

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Vol.4, No.1, 2011

Figure 8 indicates the finite element mesh. The edges Ex0 and Ex1 are correspond to theedges shown in Fig.4. The mesh size also affects the FEM results, and hence, the relationshipsbetween the mesh size around the crease and the contact force are examined as shown in Fig.9.The results indicate that the contact force is asymptotically close to a value when the mesh sizeis smaller than 0.250mm. Although higher precision is expected as finer mesh size is used,the numerical cost also becomes higher. Hence, the mesh size is selected to be 0.250mm.

Fig. 6 Effects of penalty stiffness on contact force Fig. 7 Effects of damping factor on internal energy

Fig. 8 Finite element mesh

Fig. 9 Effects of mesh size on contact force

3.4. Crease configurationThe crease configurations, which are obtained by the experiments and the FEM analyses,

are indicated in Fig.10. As shown in these figures, in the region between the contact area tothe blocks and the origin point of the x−z coordinates, the configurations of the FEM analysesare good agreement with that of the experiments. On the other hand, when the z coordinateis greater than the contact area, there are differences between the experimental results and theFEM results, as obviously shown in Fig.10c and d; the configurations of the experiments areasymmetric about the y − z plane. We discuss the asymmetric configuration using Fig.11 and12, which is the overview of the crease configuration. In the loading process, the asymmetricconfiguration is obtained when the layer pitch is smaller than 1.8mm (Fig.11a to Fig.11b).On the other hand, in the unloading process, the asymmetric configuration is not obtainedwhen the layer pitch is larger than 1.0mm (Fig.12a to Fig.12b). This results indicate thatthe asymmetry occurs when the layer pitch is small. When the layer pitch is small, in-planecompressive stress is applied to the membrane due to the friction between the membrane andthe block. As the results, the in-plane compressive stress induces the asymmetric configurationas the buckling behavior.

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Vol.4, No.1, 2011

Fig. 10 Configuration of crease

Fig. 11 Overview of crease configuration in loading process

Fig. 12 Overview of crease configuration in unloading process

4. Mathematical model for linear elastic deformation

In this section, a mathematical model for linear elastic deformation is formulated to eval-uate the experimental and FEM results theoretically in the next section, as well as to examinethe mechanics of the creasing process.

Figure 13 indicates the mathematical model of the z-fold membrane on the assumptionof the repeating structure. In the model, the contact force P is applied by a concentrated force.

Figure 14 shows the beam element of the mathematical model. The displacement u andw are derived as,

u = u0 − tsinθ, w = w0 − t(1 − cosθ) (3)

We assume that the elongation along the neutral line of the membrane is negligible, and thefollowing equations are derived as,

sinθ =dw0

ds, cosθ = 1 +

du0

ds(4)

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Vol.4, No.1, 2011

Fig. 13 Mathematical model of z-fold membrane

Fig. 14 Beam element

The strain along the s axis in the membrane is described as,

εs =

√(1 +

duds

)2

+

(dwds

)2

− 1 (5)

Substituting Eq.(3) to Eq.(5), and using Eq.(4), the following relationship is obtained as,

εs = −tdθds

(6)

The following equations for the in-plane force, shear force, and moment are also derived as,

dds

(Tscosθ) − dds

(Qssinθ) = 0 (7)

dds

(Tssinθ) − dds

(Qscosθ) = 0 (8)

dMs

ds= −Qs (9)

Integrating Eqs.(7) and (8), the following equations are derived as,

Tscosθ − Qssinθ + c1 = 0 (10)

Tssinθ + Qscosθ + c2 = 0 (11)

where, c1 and c2 are integration constants. Equilibrium equations in x-direction and z-directionare derived using the reaction forces in x-direction Rx and z-direction Rz, as,

Rx|s=0 − P + Ta+bcosθ0 − Qa+bsinθ0 = 0 (12)

Rz|s=0 + Ta+bsinθ0 + Qa+bcosθ0 = 0 (13)

4.1. In-plane force and shear force(a) 0 ≤ s ≤ a

The repeating structure gives following boundary condition.

θ|s=0 = 0 (14)

Substituting Eq.(14) into Eqs.(10) and (11), the following relationships are derived as,

T0 + c1 = 0 (15)

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Vol.4, No.1, 2011

Q0 + c2 = 0 (16)

Using Eqs.(12), (13), (15), and (16), the integration constants c1 and c2 are determined, andthus, the in-plane force and the shear force are derived using Eqs.(10) and (11) as,

Ts = −Pcosθ + Ta+bcos(θ − θ0) + Qa+bsin(θ − θ0) (17)

Qs = Psinθ − Ta+bsin(θ − θ0) + Qa+bcos(θ − θ0) (18)

(b) a ≤ s ≤ a + bAs the deflection angle θ is π/2 at s = a, Eq.(10) and (11) are,

−Qa + c1 = 0 (19)

Ta + c2 = 0 (20)

At s = a, the following relationships are obtained.

Qa = −P + Rx|s=0 (21)

Ta = −Rz|s=0 (22)

The integration constants c1 and c2 are determined by Eqs.(19)-(22), and the in-plane forceand the shear force are,

Ts = Ta+bcos(θ − θ0) + Qa+bsin(θ − θ0) (23)

Qs = −Ta+bsin(θ − θ0) + Qa+bcos(θ − θ0) (24)

4.2. Equilibrium equationThe bending moment is calculated using Eq.(6) as,

Ms = −∫σstdA = −E

∫εstdA = EI

dθds

(25)

The shear force is calculated from Eqs.(9) and (25) as,

Qs = −dMs

ds= −EI

d2θ

ds2(26)

By using Eqs.(7), (8), and (26), the governing equation is described by the following differen-tial equation;

−EId3θ

ds3+ Ts

dθds= 0 (27)

Substituting Eqs.(17) and (23) to Eq.(27), the following equations are obtained as,(a) 0 ≤ s ≤ a

EId3θ

ds3+ {Pcosθ − Ta+bcos(θ − θ0) − Qa+bsin(θ − θ0)}dθ

ds= 0 (28)

(b) a ≤ s ≤ a + b

EId3θ

ds3+ {−Ta+bcos(θ − θ0) − Qa+bsin(θ − θ0)}dθ

ds= 0 (29)

On the assumption of the repeating structure, the following boundary condition is obtained.

dθds|s=a+b = 0 (30)

Integrating Eq.(29) and using Eq.(30), the equilibrium equation for a ≤ s ≤ a + b is obtainedas Eq. (31).

EI2

(dθds

)2 + Ta+b{cos(θ − θ0) − 1} + Qa+bsin(θ − θ0) (31)

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Vol.4, No.1, 2011

Considering the continuity at θ = π/2, the equilibrium equation for 0 ≤ s ≤ a is derived usingEqs.(28) and (31) as,

EI2

(dθds

)2 − Pcosθ + Ta+b{cos(θ − θ0) − 1} + Qa+bsin(θ − θ0) (32)

Also, the curvature is obtained by using Eq.(32) as,

κ =dθds=

√2

EI

√Pcosθ + Ta+b{1 − cos(θ − θ0)} − Qa+bsin(θ − θ0) (33)

4.3. Relationships between contact force and layer pitchTo obtain the configuration of the crease, the equation of the displacement of the neutral

line Eq.(4) is integrated as,

u0 |s=s −u0 |s=0=∫ s

0(cosθ − 1)ds, w0 |s=s −w0 |s=0=

∫ s

0sinθds (34)

The boundary conditions of u0 and w0 in s = 0 are

u0 |s=0= 0, w0 |s=0= 0 (35)

Using Eqs.(34), (35) and (33), the displacement of the neutral line is obtained as,

u0 =

√EI2

∫ θ0

(cosθ − 1)dθ√Pcosθ + Ta+b{1 − cos(θ − θ0)} − Qa+bsin(θ − θ0)

(36)

w0 =

√EI2

∫ θ0

sinθdθ√Pcosθ + Ta+b{1 − cos(θ − θ0)} − Qa+bsin(θ − θ0)

(37)

As shown in Fig.13, the relationship between the layer pitch h and u0 is,

h =∫ a

0ds + u0 |s=a=

∫ π/20

dsdθ

dθ + u0 |s=a (38)

Using Eqs. (33), (36), and (38), the layer pitch h is derived as,

h =

√EI2

∫ π/20

cosθdθ√Pcosθ + Ta+b{1 − cos(θ − θ0)} − Qa+bsin(θ − θ0)

(39)

Note that Eq.(39) indicates the layer pitch is proportional to the 1/2 power of the bendingstiffness EI.

5. Results and discussion

Mechanics of the creasing process is discussed in terms of the contact force and thelayer pitch by the experiments, the FEM analyses, and the mathematical model to examinethe effects of the elasto-plastic behavior on the mechanical properties of the crease in the z-fold membrane. The relationships between the contact force and the layer pitch are indicatedin Fig.15. The experimental data are illustrated with dots. The black-solid line indicatesthe results of the elasto-plastic FEM analyses. The green-long broken line, the green-middlebroken line, and the green-short broken line represent the numerical examples of Eq.(39),where the Young’s moduli are 4.8GPa, 5.1GPa and 4.5GPa, respectively. In the mathematicalmodel, Eq.(39) is integrated with respect to θ, where the variable s is changed to the variableθ. Also, Ta+b is assumed to be 0.00N/mm as the friction between the membrane and the blockis reduced by the Naflon sheet in the experiments, and Qa+b is also 0.00N/mm because the thinmembrane is used in the experiments. In that case, the variable θ0 is not included in Eq.(39).The other numerical parameters of the FEM and the mathematical model are indicated inTable 1.

Sensitivities of the damping factor, the penalty stiffness, and the mesh size to the FEMresults are examined as shown in Fig.15. The orange-broken line indicates the FEM resultwhich uses the default value of the damping factor; 2.00E-4. Although the orange-broken line

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Vol.4, No.1, 2011

is agreement with the black-solid line as shown in Fig.15, we cannot obtain the asymptoticvalue of the internal energy in Fig.7. The blue-broken line and the purple-broken line indicatethe results which employ the smaller penalty stiffness 1.00E+0MPa/mm and the larger meshsize 0.500mm, respectively. These values calculate the contact force about 10% larger than theasymptotic values as shown in Fig.6 and Fig.9. These broken lines in Fig.15 become separatedfrom the black-solid line when the layer pitch is smaller than 1.1mm. Thus, we have to selectthe proper parameters to obtain the precise FEM results.

We discuss the relationships between the contact force and the layer pitch in Fig.15 bydivided into three regions; region A, B, and C. As shown in the figure, the layer pitch 4.1mmdetermines the bounds of the region A and B. When the layer pitch is 4.1mm, in the FEManalyses, the maximum stress in the crease is calculated to be 65MPa, which is the yield stressobtained by the tensile test. In the case of the smaller layer pitch than 4.1mm, the maximumstress in the crease is increased. Thus, the deformation of the region A is treated as elasticity,and the region B represents the elasto-plastic properties in the loading process. Also, theregion C shows the unloading process of the contact force from the layer pitch 0.72mm.

For the region A in Fig.15, the experimental and FEM results are in agreement qualita-tively with the results of the mathematical model. Hence, the mechanical properties of theregion A can be represented by the linear elastic theory.

For the region B in Fig.15, the experimental results are expressed by the elasto-plasticFEM analyses. By using the elasto-plastic properties obtained by the tensile test, the resultsof the FEM analyses are in good agreement with that of the experiments.

For the region C in Fig.15, the unloading process of the contact force is indicated. Asshown in the figure, the gradient of the FEM analyses(−1.7N/mm2) is larger than that ofthe experiments(−2.2N/mm2). These results indicate that the bending stiffness in the FEManalysis becomes larger than that of the experiments as the layer pitch is increased. Thus, inthe unloading behavior, the bending stiffness is treated too large in the FEM analyses.

In Fig.11b and 12a, we indicated the asymmetric configuration of the cross-section. Theeffects of the asymmetric configuration on the difference between the experimental resultsand the FEM ones are discussed. In the experiments, although the asymmetric configurationis obtained in the loading process for the region B, the difference between the experimentalresults and the FEM ones is less than the experimental error(± 0.23mm). Hence, in the regionB, the significant differences are not induced regardless of the presence of the asymmetricconfiguration. On the other hand, in the region C, the difference is larger than the experimentalerror, and the difference exists almost all the results in the region C. However, the asymmetricconfiguration is not obtained when the layer pitch is larger than 1.0mm. Thus, the differencein the region C is dominantly caused by the elasto-plastic material model of the unloadingprocess in the FEM, and hence, the effects of the asymmetric configuration do not induce thesignificant difference.

Based on the above discussions, the experimental results are agreement with the resultsof FEM and the mathematical model in the loading process of creasing the membrane. On theother hand, the unloading behavior has to be examined precisely to determine the mechanicalproperties of the crease.

6. Conclusions

Mechanical properties of a creasing process for a z-fold membrane were discussed exper-imentally, numerically, and theoretically to examine the elasto-plastic behavior of the crease.For the creased z-fold membrane, the cross-section was assumed to be a repeating structure.In the elasto-plastic FEM analyses, proper numerical parameters were selected to improvethe precision of the numerical results. The experimental results of the creasing process wereagreement with the results of the elasto-plastic FEM analyses and the results of the math-ematical model in the loading process. In the unloading process, there was the qualitativedifference between the results of the FEM and the experiments, and thus, the effects of the

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Vol.4, No.1, 2011

Fig. 15 Contact force vs layer pitch

Table 1 Numerical parameters(a) FEM analyses

membrane thickness, t 101μmedge load, f1 1.00E-5 N/mmPoisson’s ratio, ν 0.300penalty stiffness, kp 1.00E+4MPa/mmdamping factor, c 1.00E-6mesh size 0.250mm

(b) Mathematical modelmembrane thickness, t 101μmin-plane force, Ta+b 0.00 N/mmshear force, Qa+b 0.00 N/mm

bending stiffness on the unloading behavior had to be examined in detail.

References

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Vol.4, No.1, 2011

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