Bulletin of the JSME Vol.1, No.4, 2014 Mechanical

1Paper No.13-00086

© 2014 The Japan Society of Mechanical Engineers[DOI: 10.1299/mej.2014se0041]

Vol.1, No.4, 2014Bulletin of the JSME

Mechanical Engineering Journal

Numerical model for prediction of wrinkling behavior on a

thin-membrane structure

Shoko ARITA*, Takumi OKUMIYA* and Yasuyuki MIYAZAKI** * Department of Aerospace Engineering, Graduate School of Science and Technology, Nihon University

7-24-1 Narashinodai, Funabashi-shi, Chiba, 274-8501, Japan

E-mail: [email protected]

** Department of Aerospace Engineering, College of Science and Technology, Nihon University

7-24-1 Narashinodai, Funabashi-shi, Chiba, 274-8501, Japan

Abstract

One of the key aspects of developing gossamer space structures is the prediction of wrinkles and slacks in the

material. Wrinkles, which essentially refer to elastic buckling, have been analyzed numerically using finite

element methods (FEMs) with shell elements, but at a high computational cost. Therefore, membrane

elements, which ignore bending stiffness and consider only in-plane stress, have been employed to reduce the

computational cost. However, the compressive stiffness of the membrane cannot be ignored when predicting

wrinkle regions precisely in membrane structures. Some previous studies have employed membrane elements

considering small, constant non-zero values of compressive stiffness; these membrane elements can predict the

distribution of principal stress as the wrinkle regions. However, none of these traditional methods can

determine the value of compressive stiffness, and some parts of the principal stress distribution in slack areas

do not correspond to the actual phenomenon. Therefore, in order to determine compressive stiffness logically

and uniquely, we propose a new numerical calculation model, the modified-stiffness reduction model

(Mod-SRM), which is based on the stretchable elastic theory. Moreover, by comparison with the other FEM

models, we confirm that Mod-SRM represents the slack region more accurately than the traditional models.

Key words : Membrane, Wrinkle, Slack, Prediction, FEM

1. Introduction

1.1 Background

The design and development of gossamer space structures, such as large antennas, sun-shields, and solar sails, have

been quite active recently. Flexible, super-light, and highly storable structures, such as thin membranes, inflatable

structures, and tethers, have attracted increasing attention as materials for realizing such structures. Various deployment

experiments have been conducted in the past. For example, NASA launched the ECHO series of passive

communication test satellites in the 1960s and Spartan-207 (Fig. 1-1), which performed an Inflatable Antenna

Experiment (IAE) in the orbit around the earth, in 1996. Figure 1-2 shows behaviors during deployment and the shape

after the deployment of Spartan-207. As the figure shows, deployment of the three beams was not synchronized, was

subjected the beams to extreme bending, and caused severe vibration. This experiment showed that to ensure proper

deployment, certain technical problems, including the methods of storage and deployment control, must be addressed.

Received 4 June 2013

22

Arita ,Okumiya and Miyazaki, Mechanical Engineering Journal, Vol.1, No.4 (2014)


Fig. 1-1 Spartan-207 (Spartan Project, 1997)

Fig. 1-2 Behavior and shape of Spartan-207 (Spartan Project, 1997)

After Spartan-207, several studies examined the application of inflatable structures and membranes. Moreover, a

solar sail, which had been formulated as a propulsion structure for a deep space exploration craft, was launched in 2010

by JAXA/ISAS as a demonstrator for its later practical realization (Fig. 1-3, 1-4). The solar sail demonstrator, IKAROS,

has a square membrane 14 m wide on each side, and deploys by centrifugal force.

Fig. 1-3 Small solar power sail demonstrator IKAROS (Miyazaki, et al., 2011)

14m

28m

33



Fig. 1-4 Deployment of Membrane on IKAROS (Miyazaki, et al., 2011)

Gossamer members such as the membrane buckle easily under small compressive stresses, and thus wrinkles and

slacks occur. Because wrinkles and slacks affect the performance of the structures, it is necessary to predict their

behavior and characteristics in order to ensure deployment and operation of the space membrane structure. However, it

is difficult to understand the dynamic behavior, deformed shape, and characteristics of a large flexible structure by

performing ground tests. The deformation of these flexible structures is heavily influenced by gravity and air drag

because of their own lightness and flexibility. Therefore, it is difficult to conduct ground tests to simulate their behavior

in space, i.e., a microgravity and vacuum environment. For this reason, prediction by numerical analysis is necessary.

Some researchers have successfully introduced new methods of numerical analysis to the design of actual spacecraft

(Miyazaki, et al., 2011, Mori, et al., 2009), but their methods are still not entirely established for practical applications.

The wrinkles and slacks could impact the characteristics of flexible structures severely, so prediction of the amplitude

and area of wrinkles and slacks is important. This paper proposes a numerical method to predict them.

1.2 Previous Study

Bifurcation analysis or pseudobifurcation analysis with a shell element model is usually performed to calculate the

post-buckling deformation of a membrane. However, these methods have huge computational costs and need fine

meshes to obtain sufficiently accurate results. Because of the computational costs, it is not realistic to perform

simulations of large structures and dynamics with a shell element model. Hence, the deformation and motion analysis

of gossamer spacecrafts are often performed with membrane element models, such as for the deployment simulation of

IKAROS (Miyazaki, et al., 2011). A membrane element ignores bending stiffness and assumes a planar stress field, and

hence cannot represent three-dimensional (3D) deflections. This analysis method cannot, therefore, provide

characteristics such as height and wavelength of wrinkles.

A traditional membrane element model is based on the tension field theory and assumes zero compressive stiffness

in the wrinkle direction. However, Iwasa, et al. pointed out that the hypothesis does not hold exactly (Iwasa, et al,

2004). Therefore, Miyazaki, et al. proposed methods to represent the area and intensity of the wrinkles by giving some

small compressive stiffness in the wrinkle direction. One of the methods is the stiffness reduction model (SRM), which

is a model to give compressive stiffness as a ratio of the tensile stiffness (Miyazaki, 2006), and another is the maximum

compressive stress model (MCSM), which does not permit the compressive stress to exceed a certain value (Miyazaki

and Arita, 2008). However, the methods for determining the compressive stiffness are not established, and the

distribution of principal stress in slack areas is not valid (Inoue, 2009).

2. Purpose and Procedure

This study aims to solve the problems faced by traditional models by proposing a new numerical model based on

the stretchable elastic theory. In this paper, only the principal stress in the slack area is brought into focus. It is because

the problem which was pointed out in the traditional membrane element models is the problem of slack representation

(Inoue, 2009). The procedures followed in this study are detailed below.

Procedure 1:

First, we propose a new numerical calculation model, the modified-stiffness reduction model (Mod-SRM), which

employs the stretchable elastic theory, in order to determine the compressive stiffness logically and uniquely.

Procedure 2:

Next, we compare the results of Mod-SRM to those of the other models in order to verify if the proposed model

44



represents the slack region more accurately than the traditional models. We compare the distribution of the principal

stress of the proposed model with that of the traditional models and the shell element model, which represents the slack

region, in order to verify the proposed model. We do not evaluate the wrinkle area and taut area in this paper.

In this paper, Mod-SRM is proposed in Section 3 as Procedure 1. After that, comparison and verification of

Procedures 2 are given in Section 4.

3. Modified-Stiffness Reduction Model

3.1 Mathematical Model

In this section, a new mathematical model for a compressed membrane is proposed. This model determines the

compressive stiffness ratio uniquely, given the thickness of the membrane and length of the element, by introducing the

stretchable elastic theory into the element model. The compressive stiffness ratio is expressed as a function of strain.

When the shape of a wrinkle is seen as the deformation of an element, it must be first defined whether the

element’s constraints have rotational freedom or not. On the other hand, if the shape of a wrinkle is seen as the

deflection of a beam element, it must be defined whether the shape of the wrinkle is half a wavelength or one full

wavelength. Fig. 3-1 shows the shape of the wrinkle and its wavelength. The shape of the wrinkle of half a wavelength

and that of one full wavelength can be seen as the supported beams shown in Fig. 3-2, respectively. A simply supported

beam represents half a wavelength, while a rigidly supported beam represents one wavelength. This mathematical

model adopts the simply supported beam definition, and thus the total length of an element is / 2l shown in the left

figure of Fig. 3-2.

Fig. 3-1 Shape of the wrinkle and the model

Fig. 3-2 Model shape of supported beam

An infinitesimal line element dx subjected to a load P , as shown in Fig. 3-3, is represented as

d

d d (1 )d

d

s

x

y s

z

x e (1)

where ds is the length of the infinitesimal line element before deformation, e is unit vector parallel to the dx , and

s is the axial strain, which is represented as below using Young’s modulus E and the cross-sectional area A :

s

T

EA (2)

s is x axis value of the position of the infinitesimal line element before deformation, and x is the distance from the

edge point on the left side of the element as shown in Fig. 3-3. The ranges of the values s and x are defined as Eq.

(31). The axial force T acting on the beam is expressed as

cosT P (3)

l/2

l

P

l/2 P

l P P

Simply supported beam Rigidly supported beam

55



Fig. 3-3 Parameters

Thus, the following relation is obtained:

cosd

(1 ) (1 ) sind

0

s ss

xe (4)

where is the angle between the tangent of the infinitesimal line element and the x axis with counterclockwise

direction. Using this definition, Eq. (2) is rewritten as

coss

P

EA (5)

As shown in Fig.3-4, at the right side of the infinitesimal element, the internal force vector P is written as

0

0

P

P (6)

Using the moment of inertia of area I , the bending moment vector M is written as

0

0

EI

M (7)

where superscript means the first derivative with respect to s.

As shown in Fig.3-4 the balance of the moments yields

d dM x P 0 (8a)

Dividing both sides by ds gives

M x P 0 (8b)

Fig. 3-4 Infinitesimal element

Substituting Eq. (4), Eq. (6), and Eq. (7) into Eq. (8b), the following relation is obtained:

cos1 sin 0

PEI P

EA

(9)

where superscript means second derivative with respect to s.

P P 0

dx

0

0M

ds

z

x

y

Before deformation After deformation

/ 2l

dM M

M

e

P

P

s

x

dM

P dx

z

x

yP

66



Multiplying both sides of Eq. (9) with ,

2

sin sin 2 02

PEI P

EA (10)

Finally, the following equation is obtained by integrating Eq. (10):

221

( ) cos cos 2 .2 4

PEI P const

EA (11)

By considering 0 at 0 , the following equation is obtained:

2 22

0 0

1( ) cos cos2 cos cos2

2 4 4

P PEI P P

EA EA (12)

which can be rewritten as

20 0

2( ) (cos cos ) 1 (cos cos )

2

P P

EI EA (13)

Then, Eq. (13) can be written as below with non-dimensional parameters and :

22

0 02( ) 2 (cos cos ) 1 (cos cos )

2l (14)

P EA (15)

where

2

2 2

2

,P EI

EI EAl

l

(16)

0 holds for the beam shown in Fig. 3-2. Therefore, the following relation holds:

0 0

d

d2 (cos cos ) 1 (cos cos )

2

s l (17)

Let us introduce a parameter t defined as Eq. (18):

01sin d sin d

2

Ct t (18)

where

0 0cosC (19)

0 0

0

1 1cos cos ( )

2 2

: 0

: 0

C Ct C t

t

(20)

Then, sin 0 holds on the left of the beam in Fig. 3-2. Using this condition, Eq. (18) can be rewritten as below:

0 0

2

d 1 sin 1 sin

d 2 sin 2 1

C t C t

t C (21)

The following equation is obtained from Eq. (20):

77



2

0

1 (1 )(1 )

1(1 cos )(1 )

2

C C C

Ct C

(22)

Using Eq. (22), Eq. (21) can be rewritten as

0

0

0

d 1 sin

d 2 1(1 cos )(1 )

2

1 sin

2 (1 cos )(1 )

C t

t Ct C

C t

t C

(23)

The following equation is obtained from Eq. (20):

00

1(1 cos )

2

CC C t (24)

Using Eq. (24), Eq. (17) can be rewritten as

00

d

d 12 (1 cos ) 1 ( )

2 2

s l

Ct C C

(25)

Using Eq. (23) and Eq. (25), the following relation is obtained:

0

00

0

d d d

d d d

1 sin

21 (1 cos )(1 )2 (1 cos ) 1 ( )

2 2

1

2(1 ) 1 ( )

2

s s

t t

l C t

C t Ct C C

l

C C C

(26)

In this formulation, the fact that sin 0t holds on the left side of the beam is used.

Moreover, Eq. (26) can be defined as

d( )

d 2

s lf t

t (27)

where

0

1( )

(1 ) 1 ( )2

f t

C C C

(28)

By integrating Eq. (27), the following equation is obtained:

0( )d

2f t t (29)

Meanwhile, the following relation is derived from Eq. (4) and Eq. (5):

d(1 )cos

ds

x

s (30)

where

: 0 / 2

: 0 (1 ) / 2

s l

x l (31)

In this equation, is the equivalent in-plane compressive strain, defined as shown in Fig. 3-5. *E means Young's

modulus after buckling, and denotes the ratio of stiffness in the compressive direction to that in the tensile

88



direction.

Fig. 3-5 parameters

Thus, Eq. (30) can be rewritten as below, using Eq. (27):

d(1 ) ( )

d 2

: 0

: 0 (1 ) / 2

x lC C f t

t

t

x l

(32)

By integrating Eq. (32) about t , the following equation is obtained:

0

(1 )(1 ) ( )d

2 2

l lC C f t t (33)

Finally, the following relation is obtained:

0(1 ) ( )d (1 )

2C C f t t (34)

That is to say, given the equivalent plane compressive strain and material constant , the relation between the

non-dimensional compressive load and strain can be obtained when the load and angle 0 are determined

under the following two conditions:

0

0

( )d2

(1 ) ( )d (1 )2

f t t

C C f t t

(35)

The procedure to give and get and 0 using the simultaneous equations in Eq. (35) is complex. Therefore, one

gives 0 to get by Newton’s method under the condition of Eq. (36), and then gets by Eq. (37):

0( )d

2f t t (36)

0

0

(1 ) ( )d1

( )d

C C f t t

f t t

(37)

In this method, the relation between and is obtained by increasing the value of 0 gradually from 0. The

obtained relation is used for the FEM analysis after it is curve-fitted to a cubic equation. This theory is introduced by

modifying the stiffness matrix in the process of actual simulation. After direction and value of the principal stress is

obtained along the procedure of the calculation of the traditional two-dimensional membrane element, it is judged

whether it is compressive or tensile depending on the sign of the value. If it is compressive stress, the stiffness matrix is

modified using the cubic equation obtained from Eq. (37). The detailed process is explained in the paper by Miyazaki

(2006).

3.2 Comparison with Previous Models

At first, two previous models of membrane element for prediction of wrinkling are explained in 3.2.1 and 3.2.2. In

addition to that, Fig.3-6 shows the differences of them. The features of the Mod-SRM are explained in 3.3.3.

3.2.1 Stiffness reduction model (SRM) (Miyazaki, 2006)

l / 2P P

P PE* = αE

P P

ε l / 2

99



The SRM differs from the conventional model, which assumes a complete tension field. In this model, a small

positive value is given for stiffness in the wrinkle direction. The value of the compressive stiffness ratio 0 or 1

denotes the ratio of stiffness in the compressive direction to that in the tensile direction. The stiffness matrix is

modified depending on the stress state of an element as below:

taut: 0 2

1

11

EΓ Γ (38)

wrinkle: 0 0

2

0011

EΓ (39)

slack: 0 0 1

2

0 1 10 11

EΓ (40)

where Γ : Modified stiffness matrix E : Young’s modulus

: Poisson’s ratio

0 1, : Compressive stiffness ratio of the minor and major principal stress directions, respectively

The stress states are defined according to the states of principal stress. The taut element means the element where

both direction of the principal stress are tensile stress. The wrinkle element means the element where the major

principal stress is tensile and the minor principal stress is compressive. The slack element means the element where

both direction of the principal stress are compressive stress.

3.2.2 Maximum compressive stress model (MCSM) (Miyazaki and Arita 2008)

During compression, the experiment shows that after buckling, the compressive stiffness becomes extremely

small. However, before buckling, the relation between the stress and strain stays linear. In the SRM model, the stress–

strain relation stays linear permanently, and SRM does not provide for the concept of buckling. SRM differs from the

real phenomenon in this respect.

The maximum compressive stress model addresses to these shortcomings. In this model, a membrane cannot bear

more than certain constant compressive stress. When the critical compressive strain 0cr is surpassed, the state of

the membrane is determined to be undergoing wrinkling if its principal stress holds the following condition:

0 0 1 02( )

1cr

EE (41)

where

0 1, : Maximum and minimum principal strain, respectively

After wrinkling, the principal stress maintains the following values:

0 0

1 1 0( )

cr

cr

E

E (42)

If 1 0crE , the principal stress maintains 0 1 0crE .

Fig. 3-6 Difference of models SRM, MCSM (left), Mod-SRM (right)

3.2.3 Features of Mod-SRM

SRM enables the representation of compressive deformation in a flexible structure, but it is not perfect in that it

differs from the real phenomenon after buckling, and the way to give the compressive stiffness ratio is to attempt a

calculation and seek a suitable value. MCSM attempts at solving the SRM problems, and succeeds in improving the

representation of the phenomenon after buckling. However, it still does not have an accurate theoretical base in that the

Ma x imu m Co mp re s sive

Stre s s Mo de l cr

S tiffn e s s Re d u c tio n Mo d e l

0cr

( )f Before buckling

2( )

1

Ef

After buckling3 2( )f a b c d

Buckling point

1010



stress has a constant value after the critical compressive stress. Mod-SRM exhibits better theoretical base for its

mathematical model, and can determine parameters uniquely when the shape of a model is determined. It solves,

therefore, the two problems encountered by the previous models.

4. Comparison between Numerical Results

In this section, we compare the results of shell element methods, SRM, MCSM, and Mod-SRM.

4.1 Membrane Model

Table 4-1 and Fig. 4-1 shows the description of the membrane model used for the simulation. The simulation using

the models of shell element, SRM, MCSM, and Mod-SRM are carried out under the same conditions as Table 4-1 and

Fig. 4-1.

Table 4-1 Description of the membrane model

4.2 Comparison of results

According to Procedure 2 (Section 2), we compare the distribution of principal stress of the shell element,

traditional membrane element models, and Mod-SRM to evaluate Mod-SRM. We compared the results at δu=0.5mm,

1.0mm, 1.5mm. The evaluation is based on the assumption that the shell element gives correct results.

The results are shown in Fig. 4-2 to Fig. 4-16 and in Table 4-2 to Table 4-5. Figure 4-2, Fig. 4-3, and Fig. 4-4 show

the distribution of major and minor principal stress vectors of the four models. The vectors are described for each mesh.

The direction of each line segment means the direction of the principal stress, and the length of each line segment

means the largeness of the principal stress. Red line segment means tensile stress, and blue line segment means

compressive stress. Figure 4-5 explains the description of line segments of each mesh in Fig. 4-2, Fig. 4-3, and Fig. 4-4.

Figure 4-6, Fig. 4-7, and Fig. 4-8 discern the slack elements based on Fig. 4-2, Fig. 4-3, and Fig. 4-4. Moreover, Fig.

4-9, Fig. 4-10, Fig. 4-11, Fig. 4-12, Fig. 4-13, Fig. 4-14, Table 4-2, Table 4-3, and Table 4-4 treat the value of the minor

principal stress of each mesh based on Fig. 4-2, Fig. 4-3, and Fig. 4-4.

Figure 4-6, Fig. 4-7, and Fig. 4-8 show the distribution of slack elements. The green area is the slack area. We

compared the minor principal stress value of the slack area shown in Fig. 4-6, Fig. 4-7, and Fig. 4-8 for the four

models. In this paper, 0 is the minor principal stress and 1 is the major principal stress where 0 1 . As

shown in Fig. 4-5, in the slack region, where both 0 and 1 have minus signs, 0 is the minor principal stress if

0 1 . The compressive stress in the slack area of the shell element shows almost zero because the shell element

allows out-of-plane deformation. On the other hand, the compressive stress in the slack area of the membrane element

tends to be large value because the membrane element dose not allows out-of-plane deformation. Therefore, we

evaluate by the minor principal stress, which absolute value is larger than the major principal stress in the slack area.

The results of the comparison about the minor principal stress value are shown in Fig. 4-9, Fig. 4-10, Fig. 4-11, Fig.

4-12, Fig. 4-13, Fig. 4-14, Table 4-2, Table 4-3, and Table 4-4. Figure 4-15 explains the numbering of mesh and the plot

shape in Fig. 4-9, Fig. 4-10, Fig. 4-11, Fig. 4-12, Fig. 4-13, and Fig. 4-14. Each plot point in these figures represents the

stress for each element. The horizontal axis does not represent the x axis but the element numbers. Each plot point for

the minor principal stress is located in ascending order of the element number from the left to the right. That is to say,

the location of the plot point of each element with a larger number is shifted a little to the right direction for the

convenience of notation as explained by Fig.4-16. The value of the minor principal stress is the value at the center point

of each mesh. Figure 4-9 and Fig. 4-10 shows the minor principal stress at δu=0.5mm. The range of vertical axis of Fig.

4-9 is from -160MPa to 20MPa where the minor principal stress of all slack elements can be seen, and that of Fig. 4-10

Material polyimide

Length　 L[mm] 300

Height　 H[mm] 100

Thickness　 T[μm] 12.5

Young's modulus　E [GPa] 3.0

Poisson ratio 　 ν[-] 0.3

Density ρ[kg/m3] 1420

Number of division for length direction [-] 150

Number of division for height direction [-] 50

δu [mm] 0.5, 1.0, 1.5

δu

L

H

Free

side

Free

side

Constraint side

Constraint side Fig. 4-1 Overview of the membrane model

1111



is from -10MPa to 1MPa which make the comparison at the small stress value easier. Figure 4-11 and Fig. 4-12 shows

the minor principal stress at δu=1.0mm. For the same reason, the range of vertical axis of Fig. 4-11 is from -260MPa to

20MPa, and that of Fig. 4-12 is from -10MPa to 1MPa. Figure 4-13 and Fig. 4-14 shows the minor principal stress at

δu=1.5mm. The range of vertical axis of Fig. 4-13 is from -320MPa to 20MPa, and that of Fig. 4-14 is from -10MPa to

1MPa. The region of the horizontal axis of Fig. 4-9, Fig. 4-10, Fig. 4-11, Fig. 4-12, Fig. 4-13, and Fig. 4-14 is

determined so that it includes all the slack area of the shell element model and other membrane models. The only

elements which are slack in any of membrane element models are contained in each figure. However, the minor

principal stress of the shell element is plotted even if it is wrinkle or taut in order to evaluate the membrane element

models. The stress gap in Table 4-2, Table 4-3 and Table 4-4 is defined to be the difference between the minor

principal stress of the membrane element and the corresponding minor principal stress of the shell element. The area

for the calculation of the stress gap of each membrane element model is restricted to the slack area of each membrane

element model.

Figure 4-9, Fig. 4-11, and Fig. 4-13 show that severely large values of the principal stress are seen in the slack area

of MCSM. Figure 4-6, Fig. 4-7, and Fig 4-8 show that among the slack regions of the three membrane models the slack

region of SRM is not represented properly. The maximum gap of the minor principal stress between the shell element

and SRM is small (Table 4-2, Table 4-3, and Table 4-4) but each stress value is not close to the result of the shell

element (Fig. 4-10, Fig. 4-12, and Fig. 4-14). Table 4-2, Table 4-3, and Table 4-4 shows the average of the stress gap of

Mod-SRM is the smallest. Therefore, we can conclude that the representation of the slack area is improved by

Mod-SRM from the traditional membrane element models.

Finally, we show the calculation time of each model in Table 4-5. Mod-SRM takes enough less time than the shell

element.

Fig. 4-2 Distribution of principal stress at δu=0.5mm


shell element

nt

SRM

MCSM Mod-SRM

shell element

nt

SRM

MCSM Mod-SRM

1212




Fig. 4-5 Description of line segments of each mesh

Fig. 4-6 Distribution of slack elements at δu=0.5mm


shell element

nt

Mod-SRM MCSM

SRM

taut wrinkle slack

minor major minor major minor major

shell element

nt

SRM

MCSM Mod-SRM

shell element

nt

SRM

MCSM Mod-SRM

1313




Fig. 4-9 Minor stress of slack elements at δu=0.5mm (larger scale)

Fig. 4-10 Minor stress of slack elements at δu=0.5mm (smaller scale)

shell element

nt

SRM

MCSM Mod-SRM

-160

-140

-120

-100

-80

-60

-40

-20

0

20

22

~5

0

77

~1

00

13

0~

15

0

18

2~

20

02

33

~2

50

28

3~

29

93

35

~3

50

38

6~

39

94

38

~4

50

49

0~

50

05

41

~5

50

59

3~

60

06

45

~6

50

69

7~

70

07

50

80

08

50

90

0

Str

ess

[MP

a]

Element number [-]

66

01

66

51

67

01

67

51

68

01

~6

80

46

85

1~

68

56

69

01

~6

90

86

95

1~

69

60

70

01

~7

011

70

51

~7

06

37

10

1~

711

47

15

1~

71

66

72

01

~7

21

7

72

51

~7

26

8

73

01

~7

31

9

73

51

~7

37

1

74

01

~7

42

4

74

51

~7

47

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

22~

50

77~

100

130~

150

182~

200

233~

250

283~

299

335~

350

386~

399

438~

450

490~

500

541~

550

593~

600

645~

650

697~

700

75

08

00

85

09

00

Str

ess

[MP

a]

Element number [-]

6601

6651

6701

6751

68

01

~6

80

46

85

1~

68

56

69

01

~6

90

86

95

1~

69

60

70

01

~7

011

70

51

~7

06

37

10

1~

711

47

15

1~

71

66

72

01

~7

21

7

72

51

~7

26

8

73

01

~7

31

9

73

51

~7

37

1

74

01

~7

42

4

74

51

~7

47

9

1414






-260-240-220-200-180-160-140-120-100

-80-60-40-20

020

17

~5

0

73

~1

00

12

8~

15

0

18

1~

20

0

23

2~

25

02

82

~2

99

33

4~

35

03

84

~3

99

43

7~

45

04

88

~5

00

54

0~

55

05

91

~6

00

64

3~

65

06

95

~7

00

74

7~

75

0800

850

900

Str

ess

[MP

a]

Element number [-]

66

01

66

51

67

01

6751~

6754

6801~

6806

685

1~

685

86901~

6910

6951~

6961

7001~

7013

7051~

7064

7101~

7116

7151~

7167

720

1~

721

8725

1~

726

9

7301~

7320

7351~

7373

7401~

7428

7451~

7484

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

17~

50

73~

100

128~

150

181

~200

232~

250

282~

299

334~

350

384~

399

437~

450

488~

500

540~

550

591~

600

643~

650

695~

700

747~

750

80

08

50

90

0

Str

ess

[MP

a]

Element number [-]

66

01

66

51

67

01

6751~

6754

6801~

6806

6851~

6858

6901~

6910

6951~

6961

7001~

7013

7051~

7064

7101~

7116

7151~

7167

7201~

7218

7251~

7269

7301~

7320

7351~

7373

7401~

7428

7451~

7484

-320-300-280-260-240-220-200-180-160-140-120-100-80-60-40-20

020

16~

50

72~

100

127~

150

180~

200

231~

250

282~

299

334~

350

384~

399

436~

450

488~

500

539~

550

591~

600

643~

650

694~

700

746~

750

799~

800

850

Str

ess

[MP

a]

Element number [-]

66

51

6701~

6702

6751~

6755

6801~

6807

6851~

6858

6901~

6910

6951~

6962

7001~

7013

7051~

7065

7101~

7116

7151~

7167

7201~

7218

7251~

7270

7301~

7321

7351~

7374

7401~

7429

7451~

7485

1515




Table 4-2 Gap of minor principal stress in slack area at δu=0.5mm



Fig. 4-15 Element number and plot shape in the figures of minor stress of slack elements

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

116~

50

72~

100

127~

150

180

~200

231~

250

282~

299

334~

350

384~

399

436~

450

488

~500

539~

550

591~

600

643~

650

694~

700

746~

750

799~

800

85

0

Str

ess

[MP

a]

Element number [-]

66

51

67

01

~6

70

26

75

1~

67

55

68

01

~6

80

76

85

1~

68

58

69

01

~6

91

06

95

1~

69

62

70

01

~7

01

37

05

1~

70

65

71

01

~7

11

67

15

1~

71

67

72

01

~7

21

87

25

1~

72

70

73

01

~7

32

1

73

51

~7

37

4

74

01

~7

42

9

74

51

~7

48

5

Maximum value of stress gap

between shell element [MPa]

Average of the

stress gap [MPa]

Element number of maximum

stress gap [-]

SRM 10.67 4.76 50, 7451

MCSM 150.27 34.51 50, 7451

Mod-SRM 38.28 2.63 50, 7451



Average of the

stress gap [MPa]


stress gap [-]

SRM 23.51 10.22 50, 7451

MCSM 249.97 30.16 50, 7451

Mod-SRM 52.12 3.70 50, 7451



Average of the

stress gap [MPa]


stress gap [-]

SRM 36.24 15.84 50, 7451

MCSM 312.51 30.17 50, 7451

Mod-SRM 62.45 4.83 50, 7451

50 100 7500

・・・

・・・

・・・

2 52 7452

1 51 7451

・・・

・・・

・・・

L

H

1616



Fig. 4-16 Example of magnified view (22~50 of horizontal axis in Fig. 4-7)

Table 4-5 Calculation time

5. Conclusions

The following facts were obtained from this study:

1. We proposed Mod-SRM, which can directly, logically, and uniquely determine parameters where the shape of a

model is determined.

2. It is found that within the range of parameters studied in this paper, Mod-SRM can represent the distribution of the

minor principal stress in the slack area more accurately than traditional membrane element models.

References Inoue, S., Predicrion methods of wrinkling in thin-membrane, 27th International Symposium on Space Technology and

Science (2009), 2009-c-20s. Iwasa, T., Natori, M.C. and Higuchi, K., Evaluation of tension field theory for wrinkling analysis with respect to the

post-buckling study, Journal of Applied Mechanics, Vol.71 (2004), pp.532-540. Miyazaki, Y. and Arita, K., Simplified model of membrane for wrinkle analysis of gossamer structure, 49th

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference (2008), AIAA2008-2134, 7-10.

Miyazaki, Y., Shirasawa, Y., Mori, O., Sawada, H., Okuizumi, N., Sakamoto, H., Matunaga, S., Furuya, H. and Natori, M., Conserving finite element dynamics of gossamer structure and its application to spinning solar sail ‘IKAROS’, 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (2011) , AIAA-2011-2181.

Miyazaki, Y., User’s manual of nonlinear elasto-dynamic analysis code NEDA Ver. 1.7 (2008). Miyazaki, Y., Wrinkle/slack model and finite element dynamics of membrane, International Journal for Numerical

Methods in Engineering, Vol.66, No.7 (2006), pp.1179-1209. Mori, O., Sawada, H., Funase, R., Morimoto, M., Endo, T., Yamamoto, T., Tsuda, Y., Kawakatsu, Y. and Kawaguchi, J.,

First solar power sail demonstration by IKAROS, 27th International Symposium on Space Technology and Science (2009) , 2009-o-4-07v.

Okumiya, T., Inoue, S. and Miyazaki, Y., A study on verification of mathematical model of wrinkle of thin membrane, 20th Space Engineering Conference (2012), C4.

Spartan Project, Preliminary mission report: Spartan 207 / Inflatable Antenna Experiment Flown on STS-77, NASA Goddard Space Center, Greenbelt, MD (1997).

Wong, Y.W. and Pellegrino S., Computation of wrinkle amplitudes in thin membrane, 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (2002), AIAA2002-1369, 22-25.

-10-9-8-7-6-5-4-3-2-101

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Str

ess

[MP

a]

Element number [-]

shell SRM MCSM Mod-SRM

calculation time [min] 201 11 10 14

Documents

Bulletin of the JSME Vol.1, No.4, 2014 Mechanical