Strain ellipse

Polar decomposition

Pluging LEFT polar decomposition $\boldsymbol{F} = \boldsymbol{V} \cdot \boldsymbol{R}$ to equation for LEFT Cauchy-Green deformation tensor $\boldsymbol{B}=\boldsymbol{F}\cdot\boldsymbol{F}^T$ results in: $$\boldsymbol{B}=\boldsymbol{V}^2$$

Pluging RIGHT polar decomposition $\boldsymbol{F} = \boldsymbol{R} \cdot \boldsymbol{U}$ to equation for RIGHT Cauchy-Green deformation tensor $\boldsymbol{C}=\boldsymbol{F}^T\cdot\boldsymbol{F}$ results in: $$\boldsymbol{C}=\boldsymbol{U}^2$$

Let's Python it

We also import our sg2lib library for deformation visualization. Some special function for polar decomposition, square root of matrix etc. could be found in SciPy linalg module, which is imported by sg2lib with alias la.



In [1]:

    
%pylab inline
from sg2lib import *









    



Populating the interactive namespace from numpy and matplotlib

Lets define deformation gradient matrix $\boldsymbol{F}$



In [2]:

    
F = array([[1, 1], [0, 1]])

Now we can calculate both LEFT and RIGHT Cauchy-Green deformation tensors $\boldsymbol{B}$ and $\boldsymbol{C}$



In [3]:

    
# calculate left Cauchy-Green tensor
B = dot(F, F.T)
# calculate right Cauchy-Green tensor
C = dot(F.T, F)

Than left stretch tensor $\boldsymbol{V}$ could be calculated as:



In [4]:

    
V = la.sqrtm(B)
print(V)
def_show(V)









    



[[ 1.34164079  0.4472136 ]
 [ 0.4472136   0.89442719]]

while right stretch tensor $\boldsymbol{U}$ could be calculated as:



In [5]:

    
U = la.sqrtm(C)
print(U)
def_show(U)









    



[[ 0.89442719  0.4472136 ]
 [ 0.4472136   1.34164079]]

Identical results can be calculated using SciPy polar decomposition function:



In [6]:

    
R, V = la.polar(F, 'left')
print(V)
def_show(V)









    



[[ 1.34164079  0.4472136 ]
 [ 0.4472136   0.89442719]]



In [7]:

    
R, U = la.polar(F, 'right')
print(U)
def_show(U)









    



[[ 0.89442719  0.4472136 ]
 [ 0.4472136   1.34164079]]

Properties of strain ellipse

The eigenvectors of $\boldsymbol{B}$ define orientation of principal axes of the strain ellipse/ellipsiod in deformed state. The eigenvalues are quadratic elogations along principal directions, i.e. the lengths of semi-axes of the strain ellipse or ellipsoid are the square roots of the corresponding eigenvalues of $\boldsymbol{B}$.

So in Python, to calculate orientation and axial ratio of strain ellipse for deformation gradient $\mathbf{F}$, we can use LEFT Cauchy-Green deformation tensor $\boldsymbol{B}$ and following code (Note the square root of axial ratio):



In [8]:

    
evals, evecs = eig(B)
# calculate axial ratio and orientation
ar = sqrt(max(evals) / min(evals))
x, y = evecs[:, evals.argmax()]
ori = degrees(arctan2(y, x))
print('Orientation: {}\nAxial ratio: {}'.format(ori, ar))









    



Orientation: 31.717474411461005
Axial ratio: 2.618033988749895

or we can use left stretch tensor $\boldsymbol{V}$ and following code:



In [9]:

    
evals, evecs = eig(V)
# calculate axial ratio and orientation
ar = max(evals) / min(evals)
x, y = evecs[:, evals.argmax()]
ori = degrees(arctan2(y, x))
print('Orientation: {}\nAxial ratio: {}'.format(ori, ar))









    



Orientation: 31.71747441146101
Axial ratio: 2.6180339887498953

Properties of reciprocal ellipse

The eigenvectors of $\boldsymbol{C}$ define orientation of principal axes of the reciprocal ellipse/ellipsoid in undeformed state. The eigenvalues are quadratic elongations along principal directions, i.e. the lengths of semi-axes of the reciprocal ellipse or ellipsoid are the square roots of the corresponding eigenvalues of $\boldsymbol{C}$.

To calculate orientation and axial ratio of reciprocal ellipse for deformation gradient $\mathbf{F}$, we can use RIGHT Cauchy-Green deformation tensor $\boldsymbol{C}$ and following code (Note the square root of axial ratio):



In [10]:

    
evals, evecs = eig(C)
# calculate axial ratio and orientation
ar = sqrt(max(evals) / min(evals))
x, y = evecs[:, evals.argmax()]
ori = degrees(arctan2(y, x))
print('Orientation: {}\nAxial ratio: {}'.format(ori, ar))









    



Orientation: -121.717474411461
Axial ratio: 2.618033988749895

or we can use right stretch tensor $\boldsymbol{U}$ and following code:



In [11]:

    
evals, evecs = eig(U)
# calculate axial ratio and orientation
ar = max(evals) / min(evals)
x, y = evecs[:, evals.argmax()]
ori = degrees(arctan2(y, x))
print('Orientation: {}\nAxial ratio: {}'.format(ori, ar))









    



Orientation: -121.717474411461
Axial ratio: 2.6180339887498953

Properties of simple shear

Following script will calculate evolution of orientation of finite strain ellipse major axis in simple shear in respect to $\gamma$



In [12]:

    
gammas = linspace(0, 10, 50)[1:]
thetas = []
for gamma in gammas:
    F = array([[1, gamma],[0, 1]])
    B = dot(F, F.T)
    e,v = eig(B)
    x, y = v[:, e.argmax()]
    thetas.append(degrees(arctan2(y, x)))
plot(gammas, thetas)
xlabel('gamma')
ylabel('theta');

and axial ration of finite strain ellipse in simple shear in respect to $\gamma$



In [13]:

    
gammas = linspace(0, 10, 1000)[1:]
ars = []
for gamma in gammas:
    F = array([[1, gamma],[0, 1]])
    B = dot(F, F.T)
    e,v = eig(B)
    ars.append(sqrt(max(e)/min(e)))
plot(gammas, ars)
xlabel('gamma')
ylabel('AR');



In [14]:

    
from IPython.core.display import HTML
def css_styling():
    styles = open("./css/sg2.css", "r").read()
    return HTML(styles)
css_styling()









    Out[14]: