modeling of multi-layer composite material pipes under internal pressure

7/31/2019 MODELING OF MULTI-LAYER COMPOSITE MATERIAL PIPES UNDER INTERNAL PRESSURE

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(Manuscript No: I12725-01)

May 21, 2012/Accepted: June 10, 2012

1

MODELING OF MULTI-LAYER

COMPOSITE MATERIAL PIPES UNDER

INTERNAL PRESSURE

Serhii Meckaelovech Vereshaka

Sumy State University

Sumy, Ukraine.

Tel: +380 542 334058, Fax: +380 542 334058

(Email: [email protected])

Emad Toma Karash

Sumy State University

Sumy, Ukraine.

Tel: +380 542 334058, Fax: +380 542 334058

(Email: [email protected])

Abstract - The analysis of the methods of definition of elastic characteristics of separate layers reinforced byhigh-module fibers on the macro level is given. In the case, when the composite itself represents a set of layers

with different directions of reinforce, the technique of the given elastic characteristics definition and componentof matrixes ofrigidity of all layers package as a whole is offered. Comparison of theoretical and experimental

results, and also the analysis of data, given in the already known publications, confirms the correctness of theoffered technique.

Introduction

Designing constructions of composite materials, one meets a great number of possible versions and schemes for

reinforcing. Therefore, a theoretical problem to determine optimal deformation and strength properties of such

materials in a combination with a minimum cost of experiment seems to be urgent. In a composite material witha regular structure, as a rule, repeated elements are present in the form of single-directed layers. Neglecting

structure non-uniformity at a micro-level of every layer, one could find efficient characteristics of individuallayers at a macro-level. In this case, a material deformation model would have a quasi-uniform structure, which

be composed of single-directed layers with various angles of arrangement. Analysis of different approaches [1

5] to a calculation of elastic characteristics of a composite material demonstrated that a correct evaluation of an

effect of reinforcement arrangement schemes on physical and mechanical characteristics of a material could bederived solving boundary-value problems of an elasticity theory for a multiply-connected region. However, such

calculation cannot exclude errors conditioned by deviation of a real material structure from its idealized model

and is associated with a laborious numerical analysis.

A principle of summation of the repeated elemental layers serves a basis for an approximated calculation ofelastic characteristics of composite materials. The elastic characteristics of an elemental layer, as a rule, can be

determined in two stages. First, one should find the characteristics of a reduced matrix by averaging elastic

properties of fibers of an orthogonal-reinforced material layer. It is assumed that material components (a fiber

and a matrix) are isotropic, linearly elastic, and work jointly at all deformation stages. In addition, the followingassumptions were accepted: the stresses, which were perpendicular to fibers, when a normal load was acting

along the fibers, were not taken into account; lateral strains occurring under tension-compression of everycomponent were proportional to its volume content in a matrix; consideration of stress concentration at a fiber-

matrix interface should be excluded. At the second stage, calculation of the layer characteristics was performedon the basis of elastic properties of fibers and the modified matrix.


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International Journal Of Structronics & Mechatronics

2

Theoretical bases of calculation of elastic characteristics of multilayer materials

For orthotropic material calculated according to the elastic properties of high-modulus fiber-reinforced layer

have the form [7]:

1. ;1

)1)(1()(

1

)(

3

)(

1)(

1

)(

1 Mk

kk

B

kkEEE

;)1)(1)(1(

)1)(1(2)(

3

)(

1

)(

3

)(

1)(

2 M

B

kk

kkK E

vE

;)1)(1(

)1)(1(2)(

1

)(

3

)(

1)(

3

)(

3 M

B

k

kk

B

kkE

vEE

;.)1)(1)(1(

)1)(1(2)(

3

)(

3

)(

1

)(

1

)(

3)(

12

B

M

B

kkk

kK

Bk

E

E

v

vv

;)1(;)1( )(3)(

3)(

23)(

3)(

3)(

13 Mkk

Bk

Bkk

Bk vvvvvv

;)1)(1(

)1(;

)1)(1(

)1()(

1

)(

3

)(

3)(

23)(

3

)(

1

)(

1)(

12 Mkk

k

k

Mkk

k

kGGGG

,)1)(1(

)1)(1()(

3

)(

1

)(

1

)(

1)(

13 Mkk

kk

kGG

Where the subscript "" refers to the reinforcement (fiber), "" refers to the binder (matrix);)(

3

)(

1,

kk

- the

relative volume content reinforcement layer in the direction of axes (1) and (3) (see Fig. 1). Shear modulus of

the fiber and matrix are determined by the dependencies

2. .)1(2

;)1(2

M

M

M

B

B

B

V

EG

V

EG

Where: ,

- Poisson's ratio coefficient of reinforcement)(

1

k

, which characterizes the relative volume

content of fibers, can be determined by the formula:

3. ,4

)( )()(

2)()(

1

k

Bk

k

Bki

h

d

Where:)(k

h thickness of the reinforced layer,)(k

Bd fiber diameter; )(Bi frequency of reinforcement )(3

determined using empirical relationships and, as typically, changes in the interval)(

1

)(

3 )15,005,0(

.

The geometry of the reinforced layer is shown in (Fig. 1.a). All quantities with index (K) refer to the K- layer

shell.

Elasticity relationships for an orthotropic unidirectional reinforced layer in its axis of symmetry, taking into

account physical and technical constants (1) - (3) in matrix form are as follows:

4. )()()()()()( , kkkkkk ba


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Serhii Meckaelovech Vereshaka, Emad Toma Karash

3

Tzkzkzkzkzkzk

k

Tkkkkkk

k

],,,,,[

,],,,,,[

)(

21

)(

31

)(

32

)(

33

)(

22

)(

11)(

)(

21

)(

31

)(

32

)(

33

)(

22

)(

11)(

Column matrix of stress and strain layer in the direction of axes of symmetry1 , 2 (Fig. 1, b);

)(

66

)(

55

)(

44

)(

33

)(

32

)(

31

)(

23

)(

22

)(

21

)(

13

)(

12

)(

11

/

)(

)(

66

)(

55

)(

44

)(

33

)(

32

)(

31

)(

23

)(

22

)(

21

)(

13

)(

12

)(

11

/

)(

00000

00000

00000

000

000

000

,

00000

00000

00000

000

000

000

k

k

k

kkk

kkk

kkk

k

k

k

k

kkk

kkk

kkk

k

b

b

b

bbb

bbb

bbb

b

a

a

a

aaa

aaa

aaa

a

Matrix stiffness and pliability K - rank orthotropic layer in the direction of symmetry axes 1 , 2 respectively.

a)

b)

Figure 1: Unidirectional fiber reinforcement scheme

Solving two systems of equations (4) with respect to stiffness)k(

ija , we can find the following relations:

5. ,, 12133311)(2212233322)(11 kkkkkkkkkk bbbabbba ,, 133122313)(121

2

122211

)(

33

k

kkkkk

k

kkkkbbbbabbba

,, 123111312)(23113222312)(13 kkkkkkkkkkkk bbbbabbbba

kkkkkkkkk

kbbbbbbbbb 133221312312332211

,

233211331221312213

kkkkkkkkkbbbbbbbbb

kk

k

k

k

k

ab

ab

a

66

)(

66

55

)(

55

44

)(

441,1,1


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4

Compliance coefficients kijb can be written by technical constants (1):

,,,,

1

1

1221

3

3113

2

2112

1

11 k

k

k

k

k

k

k

k

k

Eb

Eb

Eb

Eb

,,,,12

2332

1

1331

3

3223

2

22 k

k

k

k

kk

k

kk

Eb

Eb

Eb

Eb

kk

k

kk

k

Gb

Gb

Gb

Eb

12

66

13

55

23

44

3

33

1,

1,

1,

1

.

If the thin-walled element consists of unidirectional reinforced layers, the axis of local coordinate systems thatdo not coincide with the axes of the global coordinate system, that is, for example, in cross-reinforced shells, it

is possible to vary the material properties due to the angle of reinforcement

Let

- the angle between the axes of symmetry k layer shell )(1

k

, )(2

k

and the directions of coordinate

lines

)(

1

k

,

)(

2

k

, .. corner reinforcement. It is known that in the rotated axes (

)(

1

k

,

)(

2

k

, z) reinforced layerhas anisotropic properties and has one plane of elastic symmetry. Then becoming fair ratio of elasticity:

6. )()()( kkk a Where

7.

)(

66

)(

63

)(

62

)(

61

)(

55

)(

54

)(45)(44

)(

36

)(

33

)(

32

)(

31

)(

26

)(

23

)(

22

)(

21

)(

16

)(

13

)(

12

)(

11

)(

00

0000

0000

00

00

00

kkkk

kk

kk

kkkk

kkkk

kkkk

k

aaaa

aa

aa

aaaa

aaaa

aaaa

a

Matrix of stiffness coefficients k rank of an anisotropic layer in the direction of the major coordinate lines

;, 21

Tkkkkkk

k

Tkkkkkk

k

],,,,,[

,],,,,,[

)(12

)(13

)(23

)(33

)(22

)(11)(

)(12

)(13

)(23

)(33

)(22

)(11)(

Column matrix of stress and strain layer in the direction of the major coordinate lines 21,

. Matrix

coefficients are expressed through the coefficients of the matrix )(ka

with dependencies:

,sincossin22cos 4)(2222)(66)(124)(11)(11 kkkkkkkkk aaaaa

,coscossin22sin 42222661241122 kkkkkkkkk aaaaa

,cossin22 12226612221112 kkkkkkkk aaaaaa

,sincos,2

23

2

13133333 k

k

k

kkkkaaaaa

,cossin

2

23

2

1323 k

k

k

kkaaa

,sincos

2

55

2

4444 k

k

k

kkaaa ,cossin554445 kkkkk aaa


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5

,cossin

255

24455 k

k

k

kkaaa

,cossin132336 kkkkk aaa

,cossin22 66226612221166 kkkkkkkk aaaaaa

,cossinsincos2

cossin22

6612

2

11

2

22

16 kk

kk

kk

k

k

k

k

k

aa

aaa

,cossin

sincos2

sincos

226612

211222

26

kk

kkkk

kkkk

k

aa

aaa

For further presentation of the material system of equations (6)(7) conveniently represented as:

8. 3)(3)(

3

)()()()(,

kkkkkkaa

Where in equation (8) introduced the following notation:

9.,],[,],,,[

,],[,],,,[

)(

13

)(

23

3

)(

)(

12

)(

33

)(

22

)(

11)(

)(

13

)(

23

3

)(

)(

12

)(

33

)(

22

)(

11)(

Tkk

k

Tkkkk

k

Tkk

k

Tkkkk

k

)(

55

)(

54

)(

45

)(

44

3)(

)(

66

)(

63

)(

62

)(

61

)(

36

)(

33

)(

32

)(

31

)(

26

)(

23

)(

22

)(

21

)(

16

)(

13

)(

12

)(

11

)( ,

kk

kk

k

kkkk

kkkk

kkkk

kkkk

k

aa

aaa

aaaa

aaaa

aaaa

aaaa

a

In the case where the composite is a set of n different oriented layers of unidirectional material contained elasticcharacteristics of the package layers are the obviouswith relations:

10. 33

3,

aa

Where

n

kk

k

ijij haa 1)(

)(

,

hhh

kk /)()(

relative thickness of the k-rank layer.

The elastic constants of multilayer stack in tension can be obtained by transforming the system of equations (10)

to mean:

11..0

,0

,0

,

1266336322621161

1236333322321131

1226332322221121

121633132212111111

aaaa

aaaa

aaaa

aaaa


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6

Substituting

11

11

1

E in the first equation of the system of equations (11) and pre-expressing strain

123322,, with assistance

11 from the remaining (3-x) equations (11), it is easy to find the value

1E :

12.11

1det

aE

Where 11

minor element11

aof the matrix

a . Similarly, there are other values of the constants:

13.33

3

22

2

det,

det

aE

aE

Young modulus;

14.

55

2454423

44

2455513

44

12)(,)(,det

a

aaGa

aaGaG

Shear modulus;

15.22

23

23

11

13

13

11

12

12 ,,M

M

M

M

M

M

Poisson's ratios. The remaining three values of Poisson's ratio323121

,, re using the well-known relations

16. )3,2,1,( jiEE ijijij Here, the first index of the Poisson coefficient indicated a direction of load application; the second one

demonstrated a direction of the lateral deformation, which was induced by this force.

On the basis of the proposed algorithm, using an applied packet of PC MATHCAD 14 programs, we obtained

numerical values of elastic characteristics of the reinforced material. A carbon-plastic [8, 9], which was

composed of 16 layers with a codeS]0/45/0/45/0[

22

, and a glass-plastic with a longitudinal-

transversal scheme of arrangement of 19 single layersSS]0/)90/0[(

were used as an example for a

calculation of elastic characteristics of a cross-reinforced material.

Properties of component compositions:

Carbon-Plastic according to the certificate data, an elastic modulus

E , a shear modulus

G , and a Poisson

coefficient

for the carbon fiber LU-03 were equal to 235000P, 90400P and 0,4, respectively. The

mechanical characteristics of the carbon-plastic binder (copolymer of an epoxytriphenol and

anilineformadehyde epoxy) were the following: ,3500 PE ,1320 PG

43.0

. In

every mono-layer of 0.25 mm thickness, a volume, which was occupied by fibers, was 55% of the total volume.Glass-Plastic an epoxy polymer 5-211B having the following parameters

4.0,1500,4200

PGMPE : was employed as a matrix of the glass plastic. A

tissue of with a satin structure T-10-80 was employed as a reinforcing element. The tissue thickness was

0.25mm. Its density base density was 36 filaments/cm and 20 filaments/cm for the bundle. The tissue wasfabricated from weaved aluminum-boron-silicate fibers 6-2611( glass). The fiber diameter was 6 x 10-


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7

3mm. The fiber mechanical characteristics were ,31000,74800 PGPE 2.0

. An

amount of fibers in one stream was 800. As a result of calculations, which were performed by the authors [8, 9],it was demonstrated that the elastic modulus of the filament was equal to 74506MPa, the shear modulus and the

Poisson coefficient of the filament was assumed as it was for the fibers.

Technical constants of elasticity of the considered multilayer composites obtained on the basis of the

relationships (1) - (16) are summarized in Table 1.

Table 1 Elastic characteristic of carbon plastic and fiberglass.

In this case, the glass plastic(fiberglass) was considered to represent a traversal isotropic material, which was

composed of 19 single-directed bases corresponding to experimentally determined value of the elastic modulus

E11

. A relative volume content of the layer reinforcement towards the axis 3 direction was assumed to be

)(

1

)(

305.0

kk . A comparison of results presented in Table 1 with those obtained by the authors of [8, 9]

confirmed that the technique, which was employed to determine averaged technical parameters of the multi-

layered composite, was correct. Physical and mechanical characteristics for the transversal shear and reduction

2313332313,,,, EGG were ruled out.

Stressed state of multi-layered cylinders under action of internal pressure

In [10], a solution of a problem of a stress-deformed state of an anisotropic cylindrical shell of a finite length

induced by an action of internal and external hydrostatic pressure for the case of state without moment was

proposed to be solved similar to a solution of Lame problem for a thick-wall isotropic cylinder. Under an action

of only internal pressure p, expressions for solution of lame problem for a thick-wall isotropic cylinder. Underan action of only internal pressure P, expressions for normal pressure tangents for a cylindrical coordinate

system (Fig. 2) should be written as:

17. ,)(1)(11

12

2

12

2

1

1

kkkkr

r

r

rpr

18. ,)(1)(1k1

12

2

12

2

1

1

kkkk

rr

rrp

19. ,)(1)(1k1

12

2

12

2

1

1

kkkk

r

r

rrp

20. ,0,)(1)(1

12

2

12

2

1

1

rz

kk

k

k

k

k

zr

rg

rgpr

Wherezr

,, normal stresses respectively in the radial, circumferential and longitudinal directions;

rzz , shear stresses in the circumferential and radial directions; 21 , rr inner and outer radii of the cylinder;


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8

21rr coordinate of the cylinder; )6,...,2,1,(b ji

ijcoefficients compliance matrix reinforced

material and that are associated with coefficients stiffness matrices (10) relations

b =1

)(a

,

3b =1

3)(a

.

In addition, in (17) - (20) introduces additional notation:

21. ;;;66

2616

66

2616

22

33

kg

kgk kk

.)6,...,2,1,(33

33

ij jib

bbb

ji

ij

Dependences (17) to (21) were derived taking into account an assumption that an elastic symmetry, which was

perpendicular to a normal to a middle cylinder surface, was present in every point of the cylinder. In this case ofanisotropy, under an action of a normal pressure, the thick-wall cylinder will not only change the curvature radii

of transversal cross-sections but also change the initial length and be whirled.

Employing formulas (17) to (21), to derive the values of maximal and minimal stresses taking place at points of

the internal and external surfaces of the considered cylinder seems to be not complicated (1

r ,2

r ).

We should like to note that calculating the stressed state of anisotropic cylindrical shells employing the formulas

(17) to (21), theoretical result for thin-walled cylinders could be derived with a practically assumed accuracy(

8,0/21rr ) . In this case, the indicated dependences do not allow one to determine changes of the cylinder

stressed state induced by a presence of material inter-phase structure defects and an influence of conditions of

ends fixation.

Figure 2: Design scheme of the shell in a cylindrical coordinate system

On the basis of the presented calculation models and techniques, which were developed for calculation of such

class of problems, the stressed state of the carbon plastic cylinder, which had the internal surface radius

mmr 096.01

, was studied. The shell was fabricated by reeling up a single-directed glass strip. As a whole,such cylindrical shell was composed of 16 single-directed layers. The reeling angle of every layer was

determined using a code of material structure. Totally, 10 versions of the shell reinforcing were considered(Table.2).


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9

Table 2 The structure of the laminate cylinder.

Thickness is mm25.0 . The other characteristics of the monolayer components are carbon earlier.

For any given structure of carbon fiber were found physical and mechanical characteristics as a compositematerial with one plane of elastic symmetry (Table 3). In this Cartesian coordinate system (Fig. 1) is replaced by

a cylindrical (Fig. 2).

Table 3 The elastic constants of carbon fiber.

Values of normal and tangential stresses at points of the internal and external cylinder surface under internal

pressure intensity MPq 25 are presented in Table 4. Analysis of results demonstrated that a changed code

did not practically influence the values of normal stresses in a circle direction . In this case, an essential

change of stress valuesz of a transversal shear z and normal axial stresses z took place.

Studies, which were performed for a stressed state of multi-layered shell having

S]06/45/0/45/0[

22

code, when the layer thickness was successively increased till a given value

(Fig. 35), are of interest. As a whole, the shell thickness was determined by an expression 12rh r . The

value2r shown in Fig. 3 5, a value of the shell internal surface did not change and was equal to

mmr 096.01 . Elastic constants for the presented set of layers did not depend on the shell thickness.


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10

Table 4 Stress state of thickness cylinder mm8h

Figure 3: Relation between stress and thickness of the shell

Figure 4: Relation between stressz

and thickness of the shell

Figure 5: Relation between stressz

and thickness of the shell


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11

We should like to note that the increased shell thickness did not practically change a difference of normal

stresses

in the circle direction at points of the internal and external surfaces. So, for example, if the shell

thickness was mm3hr-r12

, this difference was 20.4 MPa, if it was h = 8 mm, the difference was 21.7

MPa. Analyzing dependences of z and z stresses on the shell thickness, one could notice that in

anisotropic shells, when r/h>20, essentially high stress valuesz

andz

arose. Such stresses could be a

reason for destruction of the binder in a considered reinforced material. In this case, conditions for an ideal

contact between the layers, which were considered in a continuous-structure theory of anisotropic plates and

shells turned out to be essentially violated.

Conclusion

The paper proposed a method for determining the elastic constants of anisotropic material, which consists of a

set of reinforced layers. Reviewed ten variants of multilayer anisotropic hollow cylinders with different structurereinforcement. For each variant of reinforcement to determine the stress state of the cylinder under the action of

internal pressure. It is shown that with increasing thickness of the membrane stress of transverse shear andnormal stresses in the longitudinal direction decreases.

Summary

The analysis of separate layers reinforced by high-module fibers on the macro level elastic properties definitionmethods is offered. In the case when a composite is a set of layers with different re-enforcement directions,

methodology of determination the reduced elastic properties over all layers in package in tote is offered. Thedeflected mode of multi-layered hollow cylinder with the different variants of re-enforcement of its separate

layers under internal pressure is analyzed.

References

1. Wild, P. M. & Vickers, G. W. (1997). Analysis of Filament-Wound Cylindrical Shells under CombinedCentrifugal, Pressure and Axial Loading, Composites: Part A, 28A, 47-55.

2. Sonnen M., Laval C. & Seifert A. (2004). Computerized Calculation of Composite Laminates andStructures: Theory and Reality, Material S.A.

3. Tsai, S. W. & Roy A. K. (1988). Design of Thick Composite Cylinders, Journal of Pressure VesselTechnology..

4. Sayman, O. (2005). Analysis of Multi-Layered Composite Cylinders Under Hygrothermal Loading,Composites Part A, 1-11.

5. Roy, A. K., Massard, T. N.(1992). A Design Study of Multilayered Composite Spherical Pressure Vessels,Journal of Reinforced Plastic and Composites, VII, 479-493

6. Xia, M., Takayanagi, H. & Kemmochi, K. (2001). Analysis of Multi-Layered Filament- Wound Composite7. Parnas, L. & Katrc, N. (2002). Design of Fiber-Reinforced Composite Pressure Vessels under Various

Loading Conditions, Composite Structures, 58, 83-95.

8. Xia, M., Takayanagi, H. & Kemmochi, K. (2002). Bending Behaviour of FilamentWound Fiber-ReinforcedSandwich Pipes, Composite Structures, 56, 201-210.

9. Mirza, S., Bryan, A., Noori, M. (2001). Fiber-Reinforced Composite Cylindrical Vessel with Lugs,Composite Structures, 53, 143-151.


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10. Chang, R. R. (2000). Experimental and Theoretical Analyses of First-Ply Failure of Laminated, CompositePressure Vessels, Composite Structures, 49, 237-243.

modeling of multi-layer composite material pipes under internal pressure

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