Finite element model of RV

FiberMatrix

5.4 Finite element model of RV

Chapter 5:Design of laminated composite plates .... composite plies

119

Chapter 5:Design of laminated composite plates .... composite plies

120

Substituting Eq. (5.16) in Eq. (5.14), the strain vector ({ }ε ) at any point within the RV can be expressed as follows,

^{{ }}

^

^

 

^{L d}

 

^

T

0 0 0

[ ] 0 0 0

0 0 0

x z y

L y z x

z y x

   

   

 

  

 

     

 

  

 

    

 

(5.17)

The constitutive relations for fiber and matrix phases within the RV can be written as, {



^s} [ C^s]{ },



^s s1, 2

(5.18) where, the superscript s denotes fiber or matrix phase according to its value as 1 or 2, respectively; the matrix ([C^s]) is the stiffness matrix for fiber or matrix phase. The first variation of strain energy (δU ) of RV can be expressed as,

2

T 1

{ } { }

  





 

V_s ^{s s s} s

U dV (5.19) where,



is an operator for first variation; V_s is the volume of fiber (s1) or matrix (s2) phase.

Using Eqs. (5.17)-(5.18) in Eq. (5.19), the first variation of strain energy (



U ) can be written as,

2

T T

1

{ } [ ] [ ][ ]{ }

 





 

V_s ^{s s}

s

U d L C L d dV (5.20) For deriving FE model of RV, its volume is discretized using 27-node isoparametric hexahedral element. The displacement vector ({ }d ) for i^th node of an element can be expressed as,

 

 ^{{ } ,} 1, 2,3..., 27

i i i i

d u v w i

(5.21)

Chapter 5:Design of laminated composite plates .... composite plies

121

The displacement vector at any point within a typical element can be written as,

 

d   ^{ }N

 

d^e

(5.22) where, [ ]N is the shape function matrix and {d^e} is the elemental nodal displacement vector. Using Eq. (5.22) in Eq. (5.20), the simplified expression for the first variation of strain energy (δU^e) of a typical element can be written as,

^U^{e }



^d^e



^K^e

 

d^e

(5.23) where,

s^e

     

e s e

V s

K N^ L ^ C L N dV

   

 



 ^    ^

(5.23a) where, V_s^e is the elemental volume within fiber (s = 1) or matrix (s = 2) phase volume.

Upon assembling the elemental equations (Eq. (5.23)) into the global space, the global expression for the first variation of strain energy can be obtained as,

^

^{U }

  ^

^{X }

^{ }

^K

 

^X (5.24) where, [ ]K is the global stiffness matrix; { }X is the global nodal displacement vector.

The specified nodal displacements over the boundary surfaces of RV could be incorporated in the expression of strain energy (Eq. (5.24)) following a simple procedure as demonstrated by Cook et al. (2003). For a specified nodal displacement (say, 

Xj

element of { }X ), the first variation of the corresponding element ( 

Xj) of the global nodal displacement vector ({ }X ) is equal to zero (  

Xj

δ 0). So, the j^th row of [ ]K is to be deleted and a column of [ ]K with the same index ( j) (say, ^{fj^}) is also to be removed for constituting the displacement load vector as follows,

^{U }



^Xr



^

 ^{ }

^Kr

 

^{Xr { }fj Xj}



(5.25)

Chapter 5:Design of laminated composite plates .... composite plies

122

In Eq. (5.25), [K_r] and ^{X^_r^} are the resulting global stiffness matrix and global nodal displacement vector after imposition of specified nodal displacement ( 

Xj). Equation (5.25) can also be written in generalized form when a number of nodal displacements (N_d) are specified as,

^

^{U }

 ^

^Xr



^

 ^{ }

^Kr

 

^{Xr { }P}



(5.26) where,

¹

{ } { }



 



^{Nd j j}

j

P f X

(5.26a) According to the principle of minimum potential energy (



U 0), the following governing equations of equilibrium can be obtained as,

[ ]{ } { }

r r

K X P (5.27) For the specified displacement boundary conditions (Table 5.1) over the boundary surfaces of the RV, the corresponding displacement field can be obtained by the solution of Eq. (5.27). Utilizing this displacement field, the volume-average strain vectors of RV, layers and fiber phase of a layer can be computed using Eqs. (5.13a-b).

5.5 Finite element formulation for graded laminated composite plate

The main objective in the design of present graded fiber-reinforced composite lamina is to utilize it for reducing the mismatch of material properties and stresses at the inter- surfaces of plies in laminated composite structures. In order to substantiate this intent, a bending analysis of laminated composite plates comprised of presently designed graded composite plies is performed. Since a higher side-to-thickness ratio for laminated composite plates is considered in the present analysis, geometrically nonlinear bending analysis of the plates is carried out using FE procedure.

Chapter 5:Design of laminated composite plates .... composite plies

123

Figure 5.3 shows a laminated composite plate. The length, width and height of the plate are denoted by a, b and h, respectively. Unless otherwise mentioned, the plies are of uniform thickness. The middle plane of the plate is considered as the reference plane and a corner of this plane is the origin of the reference coordinate system (oxyz) (Fig. 5.3). Since a thin laminated plate is considered, the kinematics of deformation of the overall plate is defined according to the first order shear deformation theory (FSDT) as given in Eq. 2.8 with a shear correction factor as 5/6 (Reddy 2003) Equation 2.8 can be rewritten as,

0 x

u( x, y , z,t ) u (x, y ,t ) z (x, y ,t ) 



,

0 y

v(x, y , z,t ) v ( x, y ,t ) z ( x,y ,t) 



_,

w( x,y , z,t ) w ( x, y ,t ) 0 (2.8)

a h b

h/2

x Y

Z

Fig. 5.3 Schematic diagram of a laminated graded composite plate.

According to this displacement field (FSDT) and von Karman nonlinear strain- displacement relations, the strain vectors and its first variation can be written as,

Chapter 5:Design of laminated composite plates .... composite plies

124

{ }_b ([L^L_b] [ L_N]z L[ _]){ }d , { } [ ]{ }_s  L_s d

{_b}([L^L_b] [ L^d_N]z L[ _]){d}, {_s} [ ]{ L_s d}

(5.28) where,

0 0 0 0

0 0 0 0

0 0 0

 

 

 

  

  

  

  

 

 

 

L b

x

L y

y x

[ ] ,

0 0 1 0 0

2

0 0 1 0 0

2

0 0 0 0

N

w x x L w

y y w

x y

   

 

   

   

  

   

   

 

   

[ ]

,

0 0 0 0

0 0 0 0 0 0 0



  

 

  

 

  

  

  

 

 

 

 

x

L y

y x

[ ] ,

0 0 0 0

0 0 0 0

0 0 0 0

d N

w x x L w

y y

w w

x y y x

   

 

   

   

  

   

     

  

   

 

[ ] ,

0 0 0 1

0 0 1 0

s

L y

x

  

  

 





 

  

 

[ ]

(5.28a)

Chapter 5:Design of laminated composite plates .... composite plies

125

In Eq. (5.28),

 

^d is the generalized displacement vector as given in Eq. (2.9) and repeated here as follows,

 

d [u₀ v₀ w₀ _x _y]^T (2.9) Since the material properties of graded composite plies vary along the thickness

direction (z), their stiffness coefficients can be expressed as the functions of thickness- coordinate (z). Accordingly, the constitutive relations for a graded composite ply can be written as,

_b^k  C_b^k z _b^k { } [ ( )]{ } ,

{ } [ ( )]{ }_s^k  C z_s^k _s^k , k1, 2, 3,…….N_p where,

11 12 16

12 22 26

16 26 66

( ) ( ) ( )

[ ( )] ( ) ( ) ( )

( ) ( ) ( )

k k k

k k k k

b

k k k

c z c z c z

C z c z c z c z

c z c z c z

 

 

  

 

 

,

44 45

45 55

( ) ( ) [ ( )]

( ) ( )

k k

k

s k k

c z c z C z c z c z

 

  

 

(5.29) where, the superscript k denotes a particular ply within the laminate.N_p is the total number of plies.[C_b^k( )] and z [ ( )]C zs^k are the transformed stiffness matrices of k^th-ply with respect to the reference coordinate system (xyz).

For an applied uniformly distributed transverse mechanical load of intensity, p, the first variation of the total potential energy (Tp) of the laminated composite plate can be written as,

¹

 

0 0 1



^

    



 

    

 







  

^

p k

k

a b N

h k k k k

p h b b s s

k

T { } {^T } { } {^T } dz ( w p dydx) (5.30) Substituting Eqs. (5.28)-(5.29) in Eq. (5.30), the following expression for T_p can be obtained,

Chapter 5:Design of laminated composite plates .... composite plies

126

T T T T

T T T T

0 0

{ } ([ ] [ ] ) [ ]([ ] [ ]) [ ] [ ]([ ] [ ]) ([ ] [ ] )[ ][ ] [ ] [ ][ ] [ ] [ ][ ] { }

d

a b bL N b bL N bL N

p d

bL N s s s

d L L A L L L B L L

T dydx

L L B L L D L L A L d w p



  







     

 

      

 



 

(5.31) In Eq. (5.31), the rigidity matrices ([A_b],[A_s],[ ]B ,[D]) are given by,

1

1









p k k

N h k

b h b

k

A C z dz

[ ] [ ( )] ,

1

1









p k k

N h k

s h s

k

A C z dz

[ ] [ ( )] ,

1

1









p k k

N h k

h b k

B C z zdz

[ ] [ ( )] ,

1 2

1









p k k

N h k

h b k

D C z z dz

[ ] [ ( )]

(5.32)

where, hk and hk1 are the thickness (z)-coordinates of the bottom and top surfaces of kth ply, respectively. For deriving FE model of the plate, the xy-plane of the plate is discreatized using 9-node isoparametric quadrilateral element. Correspondingly, the generalized displacement vector (Eq. (2.9)) within a typical element can be written according to Eq. (2.21) ({ }d [ ]{N d^e}). Employing the principle of minimum potential energy (Tp⁰) after the substitution of Eq. (2.21) in Eq. (5.31), the equilibrium equations for a typical element can be obtained as,



^[KL^{e] [} KN^{e] { } {}



d^{e } P^e}

(5.33) where,

e e e

L Lb Ls

K  K  K [ ] [ ] [ ] ,

 

e

e

Ls A s s s e

K 



B ^T A B dA

[ ] [ ] [ ][ ] ,

 

e

e L L L

Lb A b b b b e

K 



B ^T A B  B B_  B_ ^T B B  D B_ dA

[ ] [ ] [ ][ ] [ ][ ] [ ] [ ][ ] [ ][ ] ,

Chapter 5:Design of laminated composite plates .... composite plies

127



^{ }







e   

e L d L

N A b b N N b b e

K B ^T A B ^T B B B ^T A B B B dA

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ][ ] ,

^{{ }}



^{[ ] [T 0 0 0 0]T}



e

e A e

P 



N p dA

(5.33a) In Eq. (5.33a), A_e is the area of a typical element in the xy-plane; [K_L^e] and [K_N^e ] are the linear and nonlinear counterparts of the elemental stiffness matrix; [K^e_Lb] and [K_Ls^e ] are the bending and shear counterparts of the linear elemental stiffness matrix ([K_L^e]); {P } is the elemental nodal load vector. The forms of different strain-displacement matrices appearing in Eq. (5.33a) are similar to those given in Eq. (2.23) where the operator matrices ([ ]L^L_b , [L_N] , [ ], L_s [L_] , [L^d_N]) are defined in Eq. (5.28a). In the present formulation, since the bending ([K_Lb^e ]) and shear ([K_Ls^e ]) counterparts of the linear stiffness matrix ([KL^e] ) are separately formulated, the selective integration can be implemented in a straight forward manner. Assembling the elemental equations (Eq.

(5.33)) in the global space, the global nonlinear FE equations of equilibrium of the laminated composite plate under the uniformly distributed transverse mechanical load can be obtained as,



^[KL^{] [} KN^{] { } { }}



X ^ P

(5.34) where,[K_L] and [K_N] are the linear and nonlinear counterparts of the global stiffness matrix; { }P is the global nodal load vector; { }X is the global nodal displacement vector.