Evaluation of Apparent Elastic Moduli for Mulite ...

Proceedings of 50th Congress of ISTAM (An International Meet) IIT-Kharagpur, December 14-17, 2005

Evaluation of Apparent Elastic Moduli for Mulite/Alumina Ceramic Matrix Composite using Finite Element Method 1

2

M. Sudheer , R. Shankara Reddy , K. S. Shivakumar Aradhya

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1

PG Student, Dept. of Mechanical Engg., P.E.S.C.E., Mandya 571401, Karnataka State 2 PhD Scholar, Dept. of Mechanical Engg., U.V.C.E., Bangalore 560001 3 Associate Director, Gas Turbine Research Estt., C. V. Raman Nagar, Bangalore 560093

ABSTRACT This paper presents the results of investigations carried out to develop a methodology to estimate the elastic moduli along non-principal material directions (Apparent Elastic Moduli) using Finite Element Techniques. This method has been extended for computation of elastic moduli for Mulite/Alumina Ceramic Matrix Composite system. A parametric study has been carried out to study the effect of fiber orientation on elastic moduli. The results obtained are in good agreement with the analytical solutions. Key Words: Apparent Engineering Constants, Mulite/Alumina Ceramic Matrix Composite, Non-principal Material Directions, Finite Element Method.

INTRODUCTION Composite materials are gradually gaining increasing importance as structural materials in the present day engineering design and development activity due to their attractive mechanical and thermal properties such as high strength-to-weight ratio, high stiffness-to-weight ratio, thermal shock and corrosion resistances etc.[1]. Aircrafts and spacecrafts are typical weight-sensitive structures in which composite materials are costeffective and finding extensive applications. Ceramic Matrix Composites (CMCs) are an important class of composites, which provide as alternate substitutes for conventional engineering materials when specific mechanical properties necessary for high ambient temperature applications are desirable [2]. Apart from possessing high strength to weight ratio and high stiffness to weight ratio, they offer elevated temperature stability, low thermal conductivity etc. The mechanical and thermal properties of composites along material principal directions are generally available in literature or from the manufacturer. However, the properties along non-principal material directions, which are invariably required in stress calculations during design validation process, are not readily available. One has to use closed form equations, which give approximate values of these engineering constants. This paper presents the results of investigations carried out to develop a methodology to estimate the elastic moduli along non-principal material directions (Apparent Elastic Moduli) using Finite Element Techniques. This method has been extended for computation of elastic moduli for Mulite/Alumina Ceramic Matrix Composite system. The front-end commercial software ANSYS is adopted in the present study. The FEM formulation is carried out using plane stress modeling approach with PLANE42 element. The results thus obtained are compared with the analytically calculated values.

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CLOSED FORM SOLUTIONS FOR APPARENT ELASTIC MODULI [ 3 ] The analytical solutions for apparent engineering constants for an orthotropic lamina that is stressed in non-principal material direction are given by (Fig. 1) :

1 = EX 1 Ey

 1 2 12  1 1  sin2  cos 2   cos 4     sin4  E1 E1  E2  G12 1 sin4  E1

=

   xy = E x  12  E1 yx Ey



 xy

 1 212   G E1  12

 sin

4





(1)

 1  sin2  cos 2   cos 4  E 2 

cos 4 



 1 1 1    E E G 2 12  1

(2)

   sin2  cos 2    

(yx can be calculated from the general relation)

Ex

 2 412 2 1 1 = 2     G xy E2 E1 G12  E1

(3)

(4)



 2 1  sin  cos 2   sin 4   cos 4  G12 



(5)

For the details of notations used in the above equations one is referred to the nomenclature presented at the end of the paper. E1, E2, 12, and G12 values can be determined from Rule of Mixtures.

PREDICTION OF APPARENT ELASTIC MODULI USING 2-D FEM APPROACH The details of evaluating the apparent elastic moduli E x, Ey, xy, yx and Gxy are discussed in the following sections. Evaluation of Apparent Young’s Modulus (Ex) and Poisson’s Ratio (xy)

Geometry and Dimensions of the Test coupon Figure 2 shows geometry and dimensions of the FEM test coupon used. The test coupon is a square lamina of edge length 200 μm. The size of test coupon is taken on a micro-mechanical scale. Thickness of the test coupon is taken as 20 μm which is equal to the size of the fiber. Fiber volume fraction considered in the current study is 0.4. The fiber angle is varied from 0º to 90º in steps of 15º.

Material Properties Properties of isotropic mulite and isotropic alumina at room temperature are given in Table 1. Table 1. Material Properties of Mulite/Alumina Composite system at room temperature [4]. Properties

Matrix (Mulite)

Fiber (Alumina)

Young’s Modulus (GPa)

215

380

Poisson’s ratio

0.26

0.24

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FEM Modeling Details Finite element mesh is generated for the test coupon using 4 noded quadrilateral elements (PLANE42). To improve the accuracy of the results, a finer mesh density is used which was arrived through a convergence study. A typical FEM mesh for fiber volume fraction 0.4 and fiber angle 75º is shown in the Fig. 3. Following is the summary of the FEM mesh. Type of element

: PLANE42

Number of nodes: 568 Number of d.o.f.’s: 1136

Number of elements: 526

Geometric and Load Boundary Conditions The details of boundary conditions considered in the present case are as follows (Fig. 4):  The nodes lying on the Y–axis (edge OB) are constrained from moving in the X-direction. (Ux=0). The mid-node on the same edge is constrained from moving in both X and Y directions (U x and Uy = 0).  Nodes lying on right edge AC of the model are coupled to move together in X-direction.  Nodes lying on the top edge BC and bottom edge OA of the model are coupled separately to move together in Y-direction.  A tensile stress 100X10 N/μm is applied along the right edge AC of the model. -6

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RESULTS AND DISCUSSIONS From FEM analysis peak values of displacements U x and Uy along X and Y directions are recorded for different fiber orientations. By using values of U x and Uy corresponding apparent elastic moduli are computed from appropriate material constitutive relations. Figure 5 shows the variation of apparent Young’s modulus E x as a function of fiber angle θ. The apparent Young’s modulus gradually reduces from E1 (281GPa) to E2 (260.19 GPa) in a non-linear way. Figure also shows the comparison of variation of apparent Young’s modulus with the analytical solution [Eq.(1)]. FEM solutions are in close agreement with analytical values. In Fig. 6 is shown the variation of apparent Poisson’s ratio xy as a function of fiber angle θ. Apparent Poisson’s ratio xy varies from 12 (0.252) to 21 (0.2333) in a non-linear way. However, unlike for the apparent Young’s modulus, peak value of apparent Poisson’s ratio xy occurs at θ = 45º. Figure also shows comparison of FEM predicted values with that obtained using analytical solution [Eq. (2)]. FEM results are in close agreement with analytical values.

Evaluation of Apparent Young’s Modulus (Ey) and Apparent Poisson’s Ratio (yx) The analysis in this case has been carried out using same FEM test coupon shown in Fig.2 and FEM model shown in Fig.3. Material Properties are presented in Table 1.

Geometric and Load Boundary Conditions The details of boundary conditions considered in the present case are as follows (Fig. 7):

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 The nodes lying on X-axis (edge OA) are constrained from moving in the Y-direction

(Uy=0). The

mid-node on the same edge is constrained from moving in both X and Y directions (U x and Uy = 0).  Nodes lying on left edge OB and right edge AC of the model are coupled separately to move together in X-direction.  Nodes lying on the top edge BC are coupled to move together in Y-direction.  A tensile stress of 100X10 N/μm is applied on the top edge BC of the model. -6

2

Results and Discussions Figure 8 shows the variation of apparent Young’s modulus E y as a function of fiber angle θ. The apparent Young’s modulus Ey gradually increases from E2 (260.19 GPa) to E1 (281GPa) in a non-linear way. It is observed that behavior of Ey is mirror symmetric to Ex. Figure also shows the comparison of variation of apparent Young’s modulus Ey with the analytical solution [Eq.(3)]. Obviously in this case also FEM solutions are in close agreement with analytical values. In Fig. 9 is shown the variation of apparent Poisson’s ratio yx as a function of fiber angle θ. Apparent Poisson’s ratio yx varies from 21 (0.2333) to 12 (0.252) in a nonlinear way. Variation of yx is mirror symmetric to xy . Figure also shows comparison of FEM predicted values with that obtained using analytical solution [Eq. (4)]. Obviously in this case also FEM results are in close agreement with analytical values.

Evaluation of Apparent Shear Modulus (Gxy)

Geometry and Dimensions of the Test coupon (Fig. 10) Test coupon is of square shape with side equal to 4a units. Thickness of the test coupon is taken as 20 μm. The value of ‘a’ is calculated as

100

2



 100 2 . The actual test section is embedded inside the FEM test

coupon as shown in Fig. 10. It has a square shape with side 200 μm. Its edges are at 45º with respect to the edges of the main test coupon. Material surrounding the test section is having the equivalent properties (E 1 and 12) determined from rule of mixtures which is a function of fiber volume fraction. A biaxial loading on the test coupon will introduce a pure shear stress on the test section (Fig. 12). In the Fig. 10 fiber orientation angle is 75º.

FEM Modeling Details Finite element mesh is generated for the test coupon using PLANE42 elements. To improve the accuracy of the results, a finer mesh density is used which was arrived through a convergence study. Figure 11 shows a typical FEM mesh for fiber angle of 75º. Following is the summary of the FEM mesh.

Type of element

: PLANE42

Number of nodes : 1118 Number of d.o.f.’s : 2236

Number of elements: 1066

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Geometric and Load Boundary Conditions The details of boundary conditions considered in the present case are as follows (Fig. 12):  The mid-node lying on the right edge AC and left edge OB of the test coupon is restricted from moving in the Y – direction. (Uy = 0)  The mid-node lying on top edge BC and bottom edge OA of the test coupon is restricted from moving in the X- direction. (Ux = 0)  Nodes lying on right edge AC and left edge OB of the model are coupled separately to move together in X-direction.  Nodes lying on top edge BC and bottom edge OA of the model are coupled separately to move together in Y-direction.  A tensile stress of 100X10 N/μm is applied on the right edge AC and left edge OB of the test -6

2

coupon.  A compressive stress 100X10 N/μm is applied on the top edge BC and bottom edge OA of the test -6

2

coupon.

Results and Discussions Figure 13 shows the variation of apparent Shear modulus Gxy as a function of fiber angle θ. The Shear modulus Gxy reaches its maximum value at θ = 45º. This indicates that efficiency of the composite system to resist the shear forces is maximum when the fibers are oriented at an angle θ = 45º. At θ = 45º, Gxy = 110.15 GPa where as G12 = 107.52 GPa. Behavior of Gxy is mirror symmetric about θ = 45º. Figure also shows comparison of FEM predicted values with that obtained using analytical solution [Eq. (5)]. FEM results are in close agreement with analytical values. CONCLUSIONS 1.

Apparent Young’s Modulus Ex decreases with increase in the fiber orientation angle θ. The variation of E x with respect to fiber angle θ is not very significant, due to the fact that difference between E 1 and E2 is small. FEM values are in close agreement with analytical solutions. Maximum deviation being 0.61% at θ = 90º in case of Ex. Behavior of Ey is mirror symmetric to Ex and maximum deviation 0.87% at θ = 15º in case of Ey.

2.

Apparent Poisson’s ratio xy varies in a non-linear way with respect to fiber angle θ. Peak value of Poisson’s ratio occurs at 45º fiber angle. FEM values are in close agreement with analytical solutions, maximum deviation is 0.18% at θ = 75º. Variation of yx is mirror symmetric to xy . Maximum deviation is 0.71% at θ = 15º in case of yx.

3.

Variation of Apparent Shear modulus Gxy with fiber angle θ is non-linear with a maximum value occurring at 45º fiber angle. FEM values are slightly higher compared to corresponding analytical values. The maximum deviations being 3.68% both at θ = 0º and 90º.

4.

From the above conclusions, one can observe that the FEM predicted Elastic Moduli values are in close agreement with corresponding analytically estimated solutions. The maximum error is well within 4%. 198

NOMENCLATURE Ef, Em : Young’s moduli of fiber and matrix respectively (GPa) f, m : Poisson’s ratios of fiber and matrix respectively Gf, Gm : Shear moduli of fiber and matrix respectively (GPa) Vf, Vm : Volume fractions of fiber and matrix respectively E1, E2 : Longitudinal and transverse moduli respectively of the composite (GPa) 12, 21 : Major and minor Poisson’s ratios respectively of the composite G12 : Shear modulus of the composite (GPa) Ex, Ey : Apparent Young’s moduli of the composite (GPa) xy, yx : Apparent Poisson’s ratios of the composite Gxy : Apparent Shear modulus of the composite (GPa) θ : Fiber orientation angle (Degrees) d.o.f.’s: Degrees of freedom REFERENCES [1]

Isaac M. Daniel and Ori Ishai, Engineering Mechanics of Composite Materials, Oxford University Press, New York, 1994.

[2]

Chawla K. K., Ceramic Matrix Composites, Chapman & Hall, London, 1993.

[3]

Jones, R.M., Mechanics of Composite Materials. McGraw-Hill, New York, 1975.

[4]

Shankara Reddy R., Shivakumar Aradhya K. S., Proc., 49 Rourkela, pp. 223-231.

[5]

ANSYS(R) Help System, Release 5.4, UP19970828, 1997.

th

Congress of ISTAM. Dec 2004, NIT

Y 1 2 θ

σ

σ X

Fig.1 Unidirectional lamina stressed in non-principal material direction (XY)

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200

201

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