TMM4175 Lightweight Design TOC Next Prev
Lightweight design Mechanics of materials Composites Laminates Finite element analysis Case studies Miscellaneous
Case studies Lightweight drive shaft Lightweight column Lightweight panel FEA tool for flexural stiffness Crossbow Trebuchet Principal stresses and transformation Linear displacement field General transformation Transformation by substitution of axes Orientation of solid elements On transverse isotropy Fiber diameter distribution Micro-mechanical finite element models Elastic properties of woven plies Elastic properties of CSM Thin-walled pipes Composite drive shaft Elastic degradation of laminates Free egde effects of laminates Static FEA of a layered cantilever beam FEA of a sandwich beam Laminate optimization of in-plane load cases Compression testing of UD composites

On transverse isotropy

From Transformations, the transformed stiffness matrix for rotation about the z-axis is

\begin{equation} \mathbf{C{'}} = \mathbf{T}_{\sigma z}^{-1}\mathbf{C} \mathbf{T}_{\epsilon z} \end{equation}

The relevant functions are:

In [1]:
from compositelib import C3D, C3Dtz

Consider some orthotropic material with the engineering constants:

In [2]:
m1  = {'E1':50000, 'E2':20000, 'E3':10000,
       'v12':0.3,    'v13':0.4,  'v23':0.5, 'G12':5000,   'G13':4000, 'G23':3000 }

The stiffness matrix for the material is:

In [3]:
C=C3D(m1)

import numpy as np
print(np.array2string(C, precision=0, suppress_small=True, separator='  ', floatmode='maxprec') )

[[55875.  10217.   7024.      0.      0.      0.]
 [10217.  24725.   6999.      0.      0.      0.]
 [ 7024.   6999.  12312.      0.      0.      0.]
 [    0.      0.      0.   3000.      0.      0.]
 [    0.      0.      0.      0.   4000.      0.]
 [    0.      0.      0.      0.      0.   5000.]]

What if we take the average of all transformed stiffness matrices obtained by rotation about the z-axis from 0 to 359 degrees?

In [4]:
Cav=np.zeros((6,6))

count=0
for a in range(0,360): # not including 360
    Cav=Cav+C3Dtz(C,a)
    count=count+1
Cav=Cav/count
print(np.array2string(Cav, precision=0, suppress_small=True, separator='  ', floatmode='maxprec') )

[[35279.  15238.   7011.      0.      0.     -0.]
 [15238.  35279.   7011.      0.      0.      0.]
 [ 7011.   7011.  12312.      0.      0.      0.]
 [    0.      0.      0.   3500.     -0.      0.]
 [    0.      0.      0.     -0.   3500.      0.]
 [   -0.     -0.      0.      0.      0.  10021.]]

This stiffness matrix represents a material where the 1-2 plane is a plane of isotropy, since any random rotation of the stiffness matrix will result in the same matrix:

In [5]:
CavRandom=C3Dtz(Cav,12.34) # 12.34 represent a 'random' number here...

print(np.array2string(CavRandom, precision=0, suppress_small=True, separator='  ', floatmode='maxprec') )

[[35279.  15238.   7011.      0.      0.     -0.]
 [15238.  35279.   7011.      0.      0.      0.]
 [ 7011.   7011.  12312.      0.      0.      0.]
 [    0.      0.      0.   3500.      0.      0.]
 [    0.      0.      0.      0.   3500.      0.]
 [   -0.      0.      0.      0.      0.  10021.]]

Furthermore, we find that the following relation is true (why?):

\begin{equation} C_{66}= \frac{ (C_{11}-C_{12})}{2} \end{equation}
In [6]:
print(   (Cav[0,0]-Cav[0,1])/2   )

10020.753512132807
In [7]:
print(  Cav[5,5]   )

10020.753512132822

Try with angles from 0 to 360 in step of 10:

In [8]:
Cav=np.zeros((6,6))

count=0
for a in range(0,360,10): # 0, 10,...,350
    Cav=Cav+C3Dtz(C,a)
    count=count+1
Cav=Cav/count
print(np.array2string(Cav, precision=0, suppress_small=True, separator='  ', floatmode='maxprec') )

[[35279.  15238.   7011.      0.      0.      0.]
 [15238.  35279.   7011.      0.      0.      0.]
 [ 7011.   7011.  12312.      0.      0.     -0.]
 [    0.      0.      0.   3500.      0.      0.]
 [    0.      0.      0.     -0.   3500.      0.]
 [    0.     -0.     -0.      0.      0.  10021.]]

Step equal to 30?:

In [9]:
Cav=np.zeros((6,6))

count=0
for a in range(0,360,10): # 0, 10,...,350
    Cav=Cav+C3Dtz(C,a)
    count=count+1
Cav=Cav/count
print(np.array2string(Cav, precision=0, suppress_small=True, separator='  ', floatmode='maxprec') )

[[35279.  15238.   7011.      0.      0.      0.]
 [15238.  35279.   7011.      0.      0.      0.]
 [ 7011.   7011.  12312.      0.      0.     -0.]
 [    0.      0.      0.   3500.      0.      0.]
 [    0.      0.      0.     -0.   3500.      0.]
 [    0.     -0.     -0.      0.      0.  10021.]]

Just the angles 0, 45, 90 and 135:

In [10]:
Cav=np.zeros((6,6))

count=0
for a in (0,45,90,135):
    Cav=Cav+C3Dtz(C,a)
    count=count+1
Cav=Cav/count
print(np.array2string(Cav, precision=0, suppress_small=True, separator='  ', floatmode='maxprec') )

[[35279.  15238.   7011.      0.      0.      0.]
 [15238.  35279.   7011.      0.      0.      0.]
 [ 7011.   7011.  12312.      0.      0.      0.]
 [    0.      0.      0.   3500.     -0.      0.]
 [    0.      0.      0.      0.   3500.      0.]
 [    0.     -0.     -0.      0.      0.  10021.]]

And finally, the angles 0, 60 and 120:

In [11]:
Cav=np.zeros((6,6))

count=0
for a in (0,60,120):
    Cav=Cav+C3Dtz(C,a)
    count=count+1
Cav=Cav/count
print(np.array2string(Cav, precision=0, suppress_small=True, separator='  ', floatmode='maxprec') )

[[35279.  15238.   7011.      0.      0.      0.]
 [15238.  35279.   7011.      0.      0.     -0.]
 [ 7011.   7011.  12312.      0.      0.     -0.]
 [    0.      0.      0.   3500.      0.      0.]
 [    0.      0.      0.      0.   3500.      0.]
 [    0.      0.      0.      0.      0.  10021.]]
This case study serves as the background for an exercise on quazi-isotropic laminates. Therefore, no further comments.
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