FALL 2013 - AEE 569
HOMEWORK #3
(30 Points)
(DUE 13.11.2013)
1- The engineering constants of a graphite-epoxy unidirectional ply are given as:
E1= 148 GPa
G12= 4.55 GPa
E2= 9.65 GPa
v21=0.3
v23=0.6
We assume that UD ply is modeled as a transversely isotropic material such that material behaves
as isotropic material in the 2-3 plane (1: fiber direction, 2: transverse direction to the fiber, 3:
thickness direction). So, we can write G23=E2/2(1+v23) which is the equivalent relation for
G=E/2(1+v) for isotropic materials.
a) Construct the 3D compliance matrix S for the tranversely isotropic UD ply utilizing only
the independent elastic constants. Express the compliance matrix utilizing symbols of the
elastic constants such as E1, E2 etc.
b) Obtain the 3D compliance matrix numerically utilizing the values of the elastic constants
given above.
c) Find the eigen values of the compliance matrix that you have determined in part b.
Comment if the specified set of elastic constants are valid or not.
d) Check all the restrictions imposed on the elastic constants. Determine if any of the
restrictions are violated or not.
2- Consider a symmetric laminate with orthotropic layers. A representative volume which may
represent the thickness of the entire laminate contains N plies of various materials and fiber
orientations. The effective constitutive equation of the representative volume of the laminate is
given by:
 xx  C11

 
 yy  C12
 zz  C
13
   
 yz   0
 xz   0

 
 xy  C16
C12 C13
0
0
C16   xx 
 
C22 C23
0
0 C26   yy 
C23 C33
0
0 C36   zz 
 
0
0 C44 C45
0   yz 

0
0 C45 C55
0   xz 
 
C26 C36
0
0 C66   xy 
where σi and ϵi and 𝛾𝑖𝑗 are the effective stresses and strains , and 𝐶𝑖𝑗 are the effective elastic
constants. Effective stresses and strains are defined as:
1
𝑘
𝑘
𝜎𝑖𝑗 = ℎ ∑𝑁
𝑘=1 𝜎𝑖𝑗 ℎ
1
𝑘
𝑘
𝜖𝑖𝑗 = ℎ ∑𝑁
𝑘=1 𝜖𝑖𝑗 ℎ
1
𝑘
𝑘
𝛾𝑖𝑗 = ℎ ∑𝑁
𝑘=1 𝛾𝑖𝑗 ℎ
ij=xx,yy,zz
ij=yz,xz,xy
where h is the total thickness of the laminate and the superscript k indicates the kth layer.
Thus, hk refers to the thickness of the individual layer.
We will assume that distribution of interlaminar normal stresses and in-plane strains are
constant through the thickness of the laminate.
𝜎𝑧𝑧 = 𝜎𝑘𝑧𝑧
That is :
and
𝜖𝑖𝑗 = 𝜖𝑘𝑖𝑗
𝛾𝑖𝑗 = 𝛾𝑘𝑖𝑗
where ij=xx, yy
We want to estimate the effective elastic coefficients 𝐶𝑖𝑗 of the laminate in terms elastic
coefficients of the individual layers. Effective elastic coefficients can be obtained by applying
some particular deformation modes and utilizing the distribution of interlaminar normal stresses
and in-plane strains given above together with the stress-strain relations of each lamina.
Apply the following deformation modes to the representative volume of the laminate and
determine the effective elastic coefficient 𝐶11 in terms of the elastic coefficients of the individual
layers and effective elastic coefficients 𝐶13 and 𝐶33 which are assumed to be known in terms of
elastic coefficients of the individual layers.
Deformation modes: Apply pure effective strain 𝜖𝑥𝑥
free.
and let 𝜖𝑦𝑦 = 𝛾𝑥𝑦 =0 but keep 𝜖𝑧𝑧
1
Hint: Write the expression for 𝜎𝑥𝑥 in terms of effective strains and utilize 𝜎𝑥𝑥 = ∑𝑁
𝜎 𝑘 ℎ𝑘
ℎ 𝑘=1 𝑥𝑥
and also 𝜎𝑧𝑧 =0. You can use hk/h=tk as the volume fraction of the kth lamina. You can show
this for a laminate with two layers and then generalize to N layers to make the writing easier for
you.