Effective Moduli of a Continuous Fiber-Reinforced Lamina

A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 1 Effective Moduli of a Continuous Fiber-Reinforced Lamina

Review of Linear Constitutive Relations

General anisotropic material behavior is given by a relationship wherein all 6 tensor stress components are related to all 6 tensor strain components:

     xxC11 C 12 C 13 C 14 C 15 C 16  xx      yyC C C C C C  yy  21 22 23 24 25 26        zz C31 C 32 C 33 C 34 C 35 C 36    zz         yz C41 C 42 C 43 C 44 C 45 C 46   yz      zxC51 C 52 C 53 C 54 C 55 C 56   zx      C C C C C C xy 61 62 63 64 65 66    xy  or { } [C ]{  } A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 2 [C] is symmetric, but the matrix is full. The material constants in [C] are called the stiffness or elastic constants (or moduli).

Inverting the last relation gives

{ } [S ]{  }

[S] is usually called the compliance matrix. Note that [S ] [ C ]1.

The determination of the material constants for a general anisotropic material is extremely difficult since the material has mechanical properties (Young's modulus, Poisson's ratio, etc.) that vary with the direction in which they are measured, and all stresses are coupled with all strains. A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 3 Orthotropic Material (3-D) A material that has mechanical properties that can be associated with an orthogonal principal material coordinate system is called orthotropic. A typical example is a unidirectional composite lamina shown below: Orthotropic lamina with principal material (1,2,3) and non-principal (x,y,z) coordinates. Note that 1 is generally taken as the fiber direction, and 2 is transverse to the fiber but in the plane of a fiber layer. This unidirectional composite lamina has three mutually orthogonal planes of material property symmetry and is called an A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 4 orthotropic material. In the above figure, the 123 coordinates axes are referred to as the principal material coordinates since they are associated with the reinforcement directions. One can show that for specially orthotropic materials wherein the 123 axes are principal material directions, the compliance matrix has the form:

S11 S 12 S 13 0 0 0  S S S 0 0 0  21 22 23  S31 S 32 S 33 0 0 0  [S ]    0 0 0S44 0 0  0 0 0 0S55 0    0 0 0 0 0 S66  Note that shear stresses (and strains) are now uncoupled from normal stresses (and strains). Using the appropriate sequence of material uniaxial and shear tests (see Gibson, or any composite mechanics text), one can show that A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 5 the compliance terms can be written in terms of engineering material constants so that we have the following relation between engineering strains and stress. [Recall that engineering shear strain is twice the tensor shear strain, i.e., 12 2  12.]

11  1/E 1  21 / E 2   31 / E 3 0 0 0    11     /E 1/ E   / E 0 0 0     22   12 1 2 32 3   22  33    13/E 1   32 / E 2 1/ E 3 0 0 0    33        23 0 0 0 1/G 23 0 0    23  31 0 0 0 0 1/G 31 0    31       12  0 0 0 0 0 1/ G 12    12 

E1, E 2 , E 3 are Young's moduli in the 1, 2, 3 directions, ij   jj/  ii = Poisson's ratio for transverse normal strain in the j direction when a normal stress is applied in the i direction, and Gij are shear moduli in the i-j plane. A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 6

Since the compliance matrix [S] must be symmetric, we see that

  ij ji Ei E j

Hence, the strain-stress relation could also be written as:

11  1/E 1  12 / E 1   13 / E 1 0 0 0    11     /E 1/ E   / E 0 0 0     22   21 2 2 23 2   22  33    31/E 3   32 / E 3 1/ E 3 0 0 0    33        23 0 0 0 1/G 23 0 0    23  31 0 0 0 0 1/G 31 0    31       12  0 0 0 0 0 1/ G 12    12 

Note that for the specially orthotropic material, there are 9 engineering constants: E1, E 2 , E 3 , 12 ,  13 ,  23 , G 12 , G 13 , G 23. A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 7 Orthotropic Lamina in Plane Stress For a single laminae, the lamina is often assumed to be in a simple two-dimensional state of plane stress (in the 1-2 plane) such that 33  32   31  0. The lamina compliance relation simplifies to

11  S 11 S 120    11       S S 0   22  21 22   22      12  0 0 S 66    12  or 11  1/E 1  12 / E 1 0    11        /E 1/ E 0   22  21 2 2   22      12  0 0 1/G 12    12 

Hence, there are only 5 non-zero compliances (only are 4 are independent since [S] is symmetric) and 4 independent material constants (E1, E 2 , 12 , G 12). A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 8

The last equation can be inverted to obtain the lamina stiffness relation, but is written in terms of tensor strains as:

11  Q 11 Q 120    11       Q Q 0   22  21 22   22      12  0 0 2Q 66    12  12 / 2  or 11    11      22  [Q ]   22      12    12  12 / 2  where the Qij are components of the lamina stiffness matrix and are given by: A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 9 S E Q 22  1 11 2 S11 S 22 S 12 112  21 S E Q 11  2 22 2 S11 S 22 S 12 112  21 S E  E Q12  12 2  21 1  Q 122 21 S11 S 22 S 12 112  21 1   12  21 1 Q66  G 12 S66 A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 10 Some typical values of orthotropic lamina engineering constants:

Material E1( Msi ) E2 ( Msi ) G12 ( Msi ) 12 v f Scotchply 1002 5.6 1.2 0.6 0.26 0.45 E-glass/eposy Kevlar 49/934 11.0 0.8 0.33 0.34 0.65 Aramid/epoxy AS/3501 20.0 1.3 1.0 0.3 0.65 Graphite/epoxy Boron/5505 29.6 2.68 0.81 0.23 0.5 Boron/epoxy v f = Volume fraction = ratio of volume of fibers to total volume of composite A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 11 Transformation of Material Properties (1-2 to x-y) (or the Generally Orthotropic Lamina)

In order to analyze laminates having multiple laminae with fibers in different directions, it is necessary to determine material properties in an arbitrary x-y coordinate system in terms of material properties in the 1-2 principal material directions. This is a simple transformation similar to stress transformation done in ENGR 214 (from which Mohr's Circle is obtained).

Consider a lamina that has the principal 1 material axes at angle  to the x axis (+ counterclockwise) as shown below. We can transform forces from x-y to 1-2 coordinates using the simple relationship: F  c s  Fx   F x  1   T *          F2  s c  Fy   F y  ccos , s  sin  A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 12 1,2 is the local (material) coordinate system

x,y is the global coordinate system

The stress transforms as a second order tensor, i.e.,

xx  xy  T 11  12     T*   T * yx  yy  21  22  A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 13 This can be expanded to give the familiar set of equations seen in ENGR 214 and/or AERO 306, i.e.,

2 2 xx c 11  s  22  2 sc  12 2 2 yy s 11  c  22  2 sc  12 2 2 xy sc 11  sc  22 ( c  s )  12 or, in matrix notation as

2 2   cos sin  2sin  cos    xx   11    2 2   yy  sin  cos  2sin  cos     22   2 2     sin cos  sin  cos  cos   sin  12  xy   

The strain transforms the same as stress if we use tensor strain (engineering shear strain is not a tensor quantity). Recall that A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 14 engineering shear strain is twice the tensor shear strain, i.e., 12 2  12.

2 2 xx c 11  s  22  2 sc  12 2 2 yy s 11  c  22  2 sc  12 2 2 xy sc 11  sc  22 ( c  s )  12

This last relation can be written in matrix notation as

2 2   cos sin  2sin  cos    xx   11    2 2   yy  sin  cos  2sin  cos    22   2 2      / 2 sin cos  sin  cos  cos   sin  12  12 / 2  xy xy   

Note that the square matrices in and are identical. Define the transformation matrix [T] as A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 15

cos2 sin 2  2sin  cos     [T ] sin2 cos 2   2sin  cos     sin cos  sin  cos  cos2   sin 2   

Note that [T] is not the square matrix in and , but is similar. Inverting this matrix, we obtain

cos2 sin 2  2sin  cos     [T ]1  sin 2 cos 2  2sin  cos     sin cos  sin  cos  cos2   sin 2   

Comparing equation and , we see that can be written in terms of [T ]1: A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 16   xx  11    1   yy  T    22       xy  12  Likewise for the strain,

  xx  11    1   yy  T   22      / 2  xy  xy / 2  12 12 

Note that equation can be inverted to obtain:

  11   xx     22  T    yy    / 2    12 12  xy  xy / 2  A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 17 Now we are ready to transform the compliance from material (1,2) directions to global (x,y) directions. First substitute equation into equation to obtain:

  11    11   xx       22 [Q ]   22   [ Q ] T    yy       / 2    12   12 12  xy  xy / 2 

Now substitute equation into equation to obtain

    xx 11   xx  1   1   yy T  22    T [ Q ] T    yy        xy 12    xy  xy / 2 

So we have now have the stress-strain relation in x-y directions but written in terms of the stiffness in 1-2 material directions: A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 18     xx  xx  1   yy  T [ Q ] T    yy      xy    xy  xy / 2 

The triple matrix product must then be the transformed lamina stiffness matrix in x-y global directions. Hence, we define the transformed lamina stiffness matrix in x-y global directions by:

[Q ]  T1 [ Q ] T  and becomes

      xx  xxQ11 Q 12 Q 16   xx        yy [Q ]   yy    Q21 Q 22 Q 26    yy     Q Q Q    xy    xy  xy/ 2 61 62 66    xy   xy / 2 

Carrying out the matrix multiplication gives: A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 19 4 2 2 4 Q11 Q 11cos  2( Q 12  2 Q 66 )sin  cos   Q 22 sin 

2 2 4 4 Q12( Q 11  Q 22  4 Q 66 )sin cos   Q 12 (sin   cos  )

4 2 2 4 Q22 Q 11sin  2( Q 12  2 Q 66 )sin  cos   Q 22 cos 

3 3 Q16( Q 11  Q 12  2 Q 66 )sin cos   ( Q 12  Q 22  2 Q 66 )sin  cos  3 3 Q26( Q 11  Q 12  2 Q 66 )sin cos   ( Q 12  Q 22  2 Q 66 )sin  cos  2 2 4 4 Q66( Q 11  Q 22  2 Q 12  2 Q 66 )sin cos   Q 66 (sin   cos  )

Note that the stiffness matrix [Q ] now looks like an anisotropic material since the 3x3 has nine non-zero terms. However, the material is still orthotropic because the stiffness matrix can be expressed in terms of 4 independent lamina stiffness terms ( Q11, Q 12 , Q 22 , Q 66).

The compliance matrix in can be similarly written. From , A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 20

11    11      22  [S ]   22      12    12  Transform to global direction (similarly to that done for Q) to obtain:       xx  xx  xx  T     yy T [ S ] T   yy   [ S ]   yy        xy  xy/ 2    xy    xy  where 4 2 2 4 S11 S 11cos  (2 S 12  S 66 )sin  cos   S 22 sin 

2 2 4 4 S12( S 11  S 22  S 66 )sin cos   S 12 (sin   cos  )

4 2 2 4 S22 S 11sin  (2 S 12  S 66 )sin  cos   S 22 cos  A08 - Effective Moduli of a Continuous Fiber-Reinforced Lamina 21 3 3 S16(2 S 11  2 S 12  S 66 )sin cos   (2 S 22  2 S 12  S 66 )sin  cos  3 3 S26(2 S 11  2 S 12  S 66 )sin cos   (2 S 22  2 S 12  S 66 )sin  cos  2 2 4 4 S662(2 S 11  S 22  4 S 12  S 66 )sin cos   S 66 (sin   cos  )