[PDF] - Analytical Model for Computing Thermal Expansion Coefficients and PDF Document

SLIDE 1

18TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS

1 Introduction Unit cell models for evaluating the coefficients of thermal expansion (CTE’s) of unidirectional lamina have been extensively studied before. In his book, Hyer [1] by using the modified rule-of-mixture (ROM) approach presented an expression for

1



and

2

 , the longitudinal and transverse coefficients

f thermal expression for lamina, respectively.

Shapery [2] used energy method to determine

1



and

2



f unidirectional composite. Rosen and

Hashin [3] considered the composite as two- phase medium through the thermoelasticity approach,

btained an expression for

1

 and

2



.Tendon and

Chatterjee [4] focused experimental study on transversely CTE and Ishikawa, et. al [5] developed an analytical model for both

1



and

2



and conducted an experimental study. In the previous mentioned models, the expressions of

1

 are similar

to the one obtained by “ROM” model but not

2

 . In

the rule-of-mixture (ROM) model,

1



gives a function of the volume fraction of the fiber and matrix as well of their modulus. However, the expression of

2

 was found to be a function of

1

 ,

too. It should be noted that all of the models have not

included the effect of CTE’s due to fiber configuration in lamina. Moreover, no thermal induced stresses for fiber and matrix were

investigated. The purpose of this study is to develop

a closed-form solution capable of evaluating the CTE’s as well as stresses on each constituent in a lamina. 2 Analytical Model of Coefficients of Thermal Expansion 2.1 Analytical Approach In the past work, a unit cell model was divided into two regions, fiber and matrix, respectively according to their volume fraction. In so doing, the configuration of the fiber in the unit cell is ignored. The unit cell and its dimensions considered in this study are shown in Fig. 1. The cell contains an elliptical cross-section of fiber. The width and the thickness of the unit cell are represented by W and h and the semi major and minor axes of the elliptical fiber are designated by a and b, respectively. The unit cell of lamina cross-section is spliced into infinitesimal layers as also shown in Fig. 1. Each layer constitutes of various percentages of fiber and matrix constituents. This approach was successfully taken in development of stiffness model for the rod reinforced laminate by Wang and Chan [6].

2.2 Constitutive Equations for Sub Layer

In order to obtain the thermal induced load, the stiffness of the unit cell is needed. In the following derivation, the subscripts, f and m refer to the fiber and matrix constituents and the superscripts, 1 and 2 refer to the sub layer properties along the fiber and transverse to the fiber directions, respectively. The super subscript T is designated as the total properties including the mechanical and thermal components. The fiber was considered as orthotropic and the matrix as isotropic. Perfect bonding between the fiber and the matrix and no void presence in the layer are assumed. With these assumptions, the following relations can be established:

Analytical Model for Computing Thermal Expansion Coefficients and Thermal Stress in Unidirectional Lamina

N. Srisuk and W.S. Chan*

Department of mechanical and Aerospace Engineering University of Texas at Arlington, Arlington, Texas, U.S.A.

* Corresponding author (Chan@uta.edu)

Keywords: Coefficients of Thermal Expansion; Thermal Stress; Composites; Unit Cell Model; Elliptical Fiber Configuration

SLIDE 2

(1) (2) (3) The fiber volume fraction, Vf in each sub layer can be written from the geometry of the fiber as

2 2

, , 2 2 2 1 [ , ]

r i

h h z a a V w b z z a a W W a                           

(4) where wi is the width of the fiber part of the i-th sub layer. Substituting the stress/strain relationship for each constituent, the reduced stiffness matrix of the sub layer can be written as

      

2 2 12 12 11 11 11 11 22 22 22 f f m f f m f m m f f f

V V Q Q Q Q Q V Q Q Q V Q        

   

12 22 12 22 12 22 12 22 22 22 f m m f f m f m f f f

Q Q Q Q V Q Q Q Q Q V Q     

 

22 22 22 22 22 22 f m m f f f

Q Q Q Q Q V Q   

(5)

 

66 66 66 66 66 66 f m m f f f

Q Q Q Q Q V Q    The matrix of Q is defined as the same expression as given in most of the composite text such as in Ref. 1. For each sub layer, the stress and strain relationships are given as

1 2

1 1 11 12 2 22 2 12 66 12

f f

T f f f f T f f f f f f

T Q Q Q T sym Q                                              

1 2

1 11 12 2 22 12 66 12

m m

T m m m m T m m m m m m

T Q Q Q T sym Q                                               (6)

1 2

1 1 11 12 2 22 2 12 66 12

f

T T

T Q Q Q T sym Q                                              

It is also noted that thermal strain for the shear component is zero.

Fig. 1 Geometry and Dimension of Unit Cell Model
Fig. 1 Geometry and Dimension of Unit Cell Model

SLIDE 3

3 PAPER TITLE

2.3 CTE’s for Sub Layer Substituting equation 1through 3 into 6, we obtain α1 and α2 of each sub layer as expressed below:

12

22 1 12 2 1 2 11 22

Q C Q C Q Q Q    

12

11 2 12 1 2 2 11 22

Q C Q C Q Q Q    

(7) Where C1 and C2 are given as

     

12 12 12 1 22 2 12 22 1 22 22 22

1

f f m f f f f f m m m m m f f f

V V Q Q Q Q Q Q C Q Q V Q            

  



11 1 12 2 11 12 11 12 f f f f f m m m m m m m m

V Q Q Q Q Q Q            

     

22 12 1 22 2 12 22 2 22 22 22

1

f F f f f f m m m m m f f f

V Q Q Q Q Q C Q Q V Q           

12 1 22 2 f f f f

Q Q    

(8)

2.4 Thermal Induced Load of Unit Cell

The thermal resultant in-plane forces,

Th

N     and

moment,

Th

M     of lamina due to the change of

temperature can be obtained by integrating the constituent equation through the thickness of cross- sectional area with no external force. In this study, the fiber is assumed to be placed at its center of the unit cell. Hence, the thermal induced moment is

zero. With aid of Equation 7, the thermal induced

load can be simplified as

1 2 2 2 Th x h Th h y Th xy

N C N T C dz N



                      



(9) It should be noted that C1and C2 are a function of Vf which is dependent of z. The detailed integration was carried out in a closed form expression shown in

Ref. [7].

2.5 Thermal Expansion Coefficients of Lamina

The CTE’s can be obtained from mid-plane strains,

1

 and

2

 as shown below:

1 1

T    

and

2 2

T    

(10) Where

1

 and

2

 can be obtained from the thermal

induced load as shown.

1 11 12 Th Th x x

a N a N   

2 12 22 Th Th x x

a N a N   

(11) The explicit expression of the lamina compliance matrix, [a] was derived in Ref. 7.

2.5 Thermal Stress of Fiber and Matrix of Lamina

For any given point of z, the thermal stress can be

btained from equation (6). In doing so,

1 T

 and

2 T



should be replaced by

1

 and

2

 .

3 Present Model Validation By Finite

Element Method

A three- dimensional finite element model using ANSY 11.0 program was developed. E-glass fiber with circular cross-section and Epoxy 3501-6 matrix were used as an example to demonstrate the validity

f the analytical solution. The material constants are

taken from Ref. [8] as listed below: E1f= E2f = E3f =10.5x106 psi; G12f= G23f = G13f =4.3x106 psi ν 12f= ν 23f = ν 13f =0.23; α1f= α 2f = α 3f =2.8x10-6 in/in/0F Em= 0.62x106 psi; Gm= 0.24x106 psi; ν m= 0.35; Solid95 of 20-node solid element with three degrees

f freedom in each node was used. The dimension in

length and cross-section of the model used was 0.05 in x 0.005 in x 0.005 in for

1

 and 0.0005 in x 0.005

in x 0.005 in for

2

 . The present results compared

with the data given in Ref. [8] was in difference of 0.37% for

1

 and 3 % for

2

 .

SLIDE 4

4 Parametric Studies The model and materials used in the previous section were used for this study. The fiber volume fraction ranging from 0 to 78% and the ration of fiber major and minor axes from 0.70 to 1.42 were considered.

4.1 Effect of Fiber Volume Fraction on

1



and

2



Figures 2 and 3 display the variation of

1

 and

2



, respectively. The figures also include the results

btained

from the FEM and the Schapery’s equation [2]. In the Shapery’s model,

1

 and

2

 are given as

1 1 1 1 1 1 f f f m m m f f m m

E V E V E V E V      

and

 

1 2 2 12 12 1 2

1 1

f f f f m m m f

V V                      

(12) As indicated,

1

 has an excellent agreement among

all of the methods for the entire fiber volume

fraction. However, for

2



.the results from the present method gives closer to the FEM results. The results are also shown that for the small fiber volume fraction, all of the results are in a better agreement.

4.2 Effect of Fiber Configuration on

1

 and

2



In this study, the fiber volume fraction is fixed at 55%. The variation of the major to minor axis ratio, a/b ranges from <1, =1 (circular) and >1 are

considered. As illustrated in Fig. 4,

1

 is a fairly

constant and

2

 significantly increases as the ratio

increases.

Fig. 2

1

 vs. Fiber volume fraction

Fig. 3

2

 vs. Fiber volume fraction

Fig. 4

1

 and

2

 vs. a/b ratio

SLIDE 5

18TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS

5 Conclusions A micromechanics model with an elliptical fiber cross-section was developed to calculate both the longitudinal and transverse thermal expansion coefficients and the fiber and matrix stresses of lamina. An ANSY finite element model was used to validate the results obtained from the present method. Excellent agreement between the present and FEM results was

btained

for the coefficient

f

thermal expansion along the longitudinal direction. For the coefficient of thermal expansion along the transverse direction, the present result give better closed to FEM compared to the published results.

Effect of CTE’s due to the fiber volume fraction ( ) and the fiber configuration (axial ratio, a/b of elliptical cross-section) were studied. The results of the longitudinal CTE obtained by the present method are in excellent agreement with the results obtained from finite element (FEA) and the Rule-of-Mixture (ROM) methods for all of ’s . However, the present results of the CTE along the transverse direction show a significant difference from the ROM results but close to the finite element results. The difference in the results is also dependent on the fiber volume fraction. In the fiber configuration study, the results of the longitudinal CTE for both elliptical and circular cross-sections of the fiber indicate no difference. However, the results in the transverse direction of CTE are significantly different from each other. It is found that the CTE along the transverse direction increases as the

increases. The study shows that the

fiber shape is significant effect in the transverse CTE but not in the longitudinal CTE.

References

1. Hyer, M.W., Stress Analysis of Fiber-Reinforced

Composite Materials, McGraw-Hill Company, 1997.

2. Shapery, R.A., “Thermal Expansion Coefficients
f Composite Materials based on Energy

Principles,” Journal of Composite Materials, Vol. 2(3), pp.380-404, 1968.

3. Rosen, B.W. and Hashin, Z.,”Effective Thermal

Expansion Coefficients and Specific Heats of Composite Materials,” International Journal of Engineering Science, Vol. 8, pp157-183, 1070.

4. Tandon, G.P. and Chatterjee, A.,”The Transverse

Coefficient

f

Thermal Expansion

f

a Unidirectional Composite,” Journal of Material Science, Vol. 26, pp2759-2764, 1991.

5. Ishikawa,T., Koyama, K. and Kobayashi, S.,

“Thermal Expansion Coefficients

f

Unidirectional Composites,” Journal

f

Composite Materails, Vol. 12, pp153-168, 1978.

6. Wang, J.S. and Chan, W. S. “Effects of Defects
n the Buckling load of Rodpack Laminates”,

Journal of American helicopter Society, Vol. 45,

pp. 216-221, 2000.
7. SrisuK, N., A Micromechanics Model of Thermal

Expansion Coefficient in Fiber Reinforced Composites, Master Thesis, Department of mechanical and Aerospace Engineering, University of Texas at Arlington, Dec. 2010.

8. Daniel, I.M. and Ishai, O., Engineering