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Corrugated Shells

Init symbols for sympy





In [1]:



    


from sympy import *
from sympy.vector import CoordSys3D
N = CoordSys3D('N')
x1, x2, x3 = symbols("x_1 x_2 x_3")
alpha1, alpha2, alpha3 = symbols("alpha_1 alpha_2 alpha_3")
R, L, ga, gv = symbols("R L g_a g_v")
init_printing()

Corrugated cylindrical coordinates





In [2]:



    


a1 = pi / 2 + (L / 2 - alpha1)/R

x = (R + alpha3 + ga * cos(gv * a1)) * cos(a1)
y = alpha2
z = (R + alpha3 + ga * cos(gv * a1)) * sin(a1)

r = x*N.i + y*N.j + z*N.k

Base Vectors $\vec{R}_1, \vec{R}_2, \vec{R}_3$





In [3]:



    


R1=r.diff(alpha1)
R2=r.diff(alpha2)
R3=r.diff(alpha3)



    











In [4]:



    


trigsimp(R1)



    


















    
Out[4]:







$$(- \frac{g_{a} g_{v}}{R} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{i}_{N}} + (\frac{g_{a} g_{v}}{R} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{k}_{N}}$$
















In [5]:



    


R2



    


















    
Out[5]:







$$\mathbf{\hat{j}_{N}}$$
















In [6]:



    


R3



    


















    
Out[6]:







$$(- \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{i}_{N}} + (\cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{k}_{N}}$$

Base Vectors $\vec{R}^1, \vec{R}^2, \vec{R}^3$





In [7]:



    


eps=trigsimp(R1.dot(R2.cross(R3)))
R_1=simplify(trigsimp(R2.cross(R3)/eps))
R_2=simplify(trigsimp(R3.cross(R1)/eps))
R_3=simplify(trigsimp(R1.cross(R2)/eps))



    











In [8]:



    


R_1



    


















    
Out[8]:







$$(\frac{R \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}}{R + \alpha_{3} + g_{a} \cos{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )}})\mathbf{\hat{i}_{N}} + (\frac{R \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}}{R + \alpha_{3} + g_{a} \cos{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )}})\mathbf{\hat{k}_{N}}$$
















In [9]:



    


R_2



    


















    
Out[9]:







$$\mathbf{\hat{j}_{N}}$$
















In [10]:



    


R_3



    


















    
Out[10]:







$$(- \frac{1}{R + \alpha_{3} + g_{a} \cos{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )}} \left(g_{a} g_{v} \sin{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \left(R + \alpha_{3} + g_{a} \cos{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right))\mathbf{\hat{i}_{N}} + (\frac{1}{R + \alpha_{3} + g_{a} \cos{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )}} \left(- g_{a} g_{v} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )} + \left(R + \alpha_{3} + g_{a} \cos{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right))\mathbf{\hat{k}_{N}}$$

Jacobi matrix:

$ A = \left( \begin{array}{ccc} \frac{\partial x_1}{\partial \alpha_1} & \frac{\partial x_1}{\partial \alpha_2} & \frac{\partial x_1}{\partial \alpha_3} \\ \frac{\partial x_2}{\partial \alpha_1} & \frac{\partial x_2}{\partial \alpha_2} & \frac{\partial x_3}{\partial \alpha_3} \\ \frac{\partial x_3}{\partial \alpha_1} & \frac{\partial x_3}{\partial \alpha_2} & \frac{\partial x_3}{\partial \alpha_3} \\ \end{array} \right)$

$ \left[ \begin{array}{ccc} \vec{R}_1 & \vec{R}_2 & \vec{R}_3 \end{array} \right] = \left[ \begin{array}{ccc} \vec{e}_1 & \vec{e}_2 & \vec{e}_3 \end{array} \right] \cdot \left( \begin{array}{ccc} \frac{\partial x_1}{\partial \alpha_1} & \frac{\partial x_1}{\partial \alpha_2} & \frac{\partial x_1}{\partial \alpha_3} \\ \frac{\partial x_2}{\partial \alpha_1} & \frac{\partial x_2}{\partial \alpha_2} & \frac{\partial x_3}{\partial \alpha_3} \\ \frac{\partial x_3}{\partial \alpha_1} & \frac{\partial x_3}{\partial \alpha_2} & \frac{\partial x_3}{\partial \alpha_3} \\ \end{array} \right) = \left[ \begin{array}{ccc} \vec{e}_1 & \vec{e}_2 & \vec{e}_3 \end{array} \right] \cdot A$

$ \left[ \begin{array}{ccc} \vec{e}_1 & \vec{e}_2 & \vec{e}_3 \end{array} \right] =\left[ \begin{array}{ccc} \vec{R}_1 & \vec{R}_2 & \vec{R}_3 \end{array} \right] \cdot A^{-1}$





In [11]:



    


dx1da1=R1.dot(N.i)
dx1da2=R2.dot(N.i)
dx1da3=R3.dot(N.i)

dx2da1=R1.dot(N.j)
dx2da2=R2.dot(N.j)
dx2da3=R3.dot(N.j)

dx3da1=R1.dot(N.k)
dx3da2=R2.dot(N.k)
dx3da3=R3.dot(N.k)

A=Matrix([[dx1da1, dx1da2, dx1da3], [dx2da1, dx2da2, dx2da3], [dx3da1, dx3da2, dx3da3]])
simplify(A)



    


















    
Out[11]:







$$\left[\begin{matrix}\frac{1}{R} \left(- g_{a} g_{v} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )} + \left(R + \alpha_{3} + g_{a} \cos{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) & 0 & - \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\\0 & 1 & 0\\\frac{1}{R} \left(g_{a} g_{v} \sin{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \left(R + \alpha_{3} + g_{a} \cos{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) & 0 & \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\end{matrix}\right]$$
















In [29]:



    


A_inv = A**-1
trigsimp(A_inv[0,0])



    


















    
Out[29]:







$$\frac{R \left(2 \cos{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} + 2 + \frac{2 \alpha_{3}}{R} \cos{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{2 \alpha_{3}}{R} - \frac{g_{a} g_{v}}{R} \cos{\left (\frac{L g_{v}}{2 R} - \frac{L}{R} + \frac{\pi g_{v}}{2} - \frac{\alpha_{1} g_{v}}{R} + \frac{2 \alpha_{1}}{R} \right )} + \frac{g_{a} g_{v}}{R} \cos{\left (\frac{L g_{v}}{2 R} + \frac{L}{R} + \frac{\pi g_{v}}{2} - \frac{\alpha_{1} g_{v}}{R} - \frac{2 \alpha_{1}}{R} \right )} + \frac{2 g_{a}}{R} \cos{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )} + \frac{g_{a}}{R} \cos{\left (\frac{L g_{v}}{2 R} - \frac{L}{R} + \frac{\pi g_{v}}{2} - \frac{\alpha_{1} g_{v}}{R} + \frac{2 \alpha_{1}}{R} \right )} + \frac{g_{a}}{R} \cos{\left (\frac{L g_{v}}{2 R} + \frac{L}{R} + \frac{\pi g_{v}}{2} - \frac{\alpha_{1} g_{v}}{R} - \frac{2 \alpha_{1}}{R} \right )}\right)}{4 \left(- \frac{g_{a} g_{v}}{R} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}\right)}$$
















In [13]:



    


trigsimp(A.det())



    


















    
Out[13]:







$$1 + \frac{\alpha_{3}}{R} + \frac{g_{a}}{R} \cos{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}$$

Metric tensor

${\displaystyle \hat{G}=\sum_{i,j} g^{ij}\vec{R}_i\vec{R}_j}$





In [14]:



    


g11=R1.dot(R1)
g12=R1.dot(R2)
g13=R1.dot(R3)

g21=R2.dot(R1)
g22=R2.dot(R2)
g23=R2.dot(R3)

g31=R3.dot(R1)
g32=R3.dot(R2)
g33=R3.dot(R3)

G=Matrix([[g11, g12, g13],[g21, g22, g23], [g31, g32, g33]])
G=trigsimp(G)
G



    


















    
Out[14]:







$$\left[\begin{matrix}\left(- \frac{g_{a} g_{v}}{R} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} + \left(\frac{g_{a} g_{v}}{R} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} & 0 & \frac{g_{a} g_{v}}{R} \sin{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}\\0 & 1 & 0\\\frac{g_{a} g_{v}}{R} \sin{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )} & 0 & 1\end{matrix}\right]$$

${\displaystyle \hat{G}=\sum_{i,j} g_{ij}\vec{R}^i\vec{R}^j}$





In [15]:



    


g_11=R_1.dot(R_1)
g_12=R_1.dot(R_2)
g_13=R_1.dot(R_3)

g_21=R_2.dot(R_1)
g_22=R_2.dot(R_2)
g_23=R_2.dot(R_3)

g_31=R_3.dot(R_1)
g_32=R_3.dot(R_2)
g_33=R_3.dot(R_3)

G_con=Matrix([[g_11, g_12, g_13],[g_21, g_22, g_23], [g_31, g_32, g_33]])
G_con=trigsimp(G_con)
G_con



    


















    
Out[15]:







$$\left[\begin{matrix}\frac{R^{2}}{\left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}\right)^{2}} & 0 & - \frac{R g_{a} g_{v} \sin{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}}{\left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}\right)^{2}}\\0 & 1 & 0\\- \frac{R g_{a} g_{v} \sin{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}}{\left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}\right)^{2}} & 0 & \frac{1}{\left(R + \alpha_{3} + g_{a} \cos{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )}\right)^{2}} \left(- g_{a} g_{v} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )} + \left(R + \alpha_{3} + g_{a} \cos{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} + \frac{1}{\left(R + \alpha_{3} + g_{a} \cos{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )}\right)^{2}} \left(g_{a} g_{v} \sin{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \left(R + \alpha_{3} + g_{a} \cos{\left (\frac{g_{v}}{2 R} \left(L + \pi R - 2 \alpha_{1}\right) \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2}\end{matrix}\right]$$
















In [16]:



    


G_inv = G**-1
G_inv



    


















    
Out[16]:







$$\left[\begin{matrix}\frac{1}{\left(- \frac{g_{a} g_{v}}{R} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} + \left(\frac{g_{a} g_{v}}{R} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} - \frac{g_{a}^{2} g_{v}^{2}}{R^{2}} \sin^{2}{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}} & 0 & - \frac{g_{a} g_{v} \sin{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}}{R \left(\left(- \frac{g_{a} g_{v}}{R} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} + \left(\frac{g_{a} g_{v}}{R} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} - \frac{g_{a}^{2} g_{v}^{2}}{R^{2}} \sin^{2}{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}\right)}\\0 & 1 & 0\\- \frac{g_{a} g_{v} \sin{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}}{R \left(\left(- \frac{g_{a} g_{v}}{R} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} + \left(\frac{g_{a} g_{v}}{R} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} - \frac{g_{a}^{2} g_{v}^{2}}{R^{2}} \sin^{2}{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}\right)} & 0 & \frac{\left(- \frac{g_{a} g_{v}}{R} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} + \left(\frac{g_{a} g_{v}}{R} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2}}{\left(- \frac{g_{a} g_{v}}{R} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} + \left(\frac{g_{a} g_{v}}{R} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{1}{R} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)^{2} - \frac{g_{a}^{2} g_{v}^{2}}{R^{2}} \sin^{2}{\left (g_{v} \left(\frac{L}{2 R} + \frac{\pi}{2} - \frac{\alpha_{1}}{R}\right) \right )}}\end{matrix}\right]$$

Derivatives of vectors

Derivative of base vectors





In [17]:



    


dR1dalpha1 = trigsimp(R1.diff(alpha1))
dR1dalpha1



    


















    
Out[17]:







$$(\frac{g_{a} g_{v}^{2}}{R^{2}} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} + \frac{2 g_{a}}{R^{2}} g_{v} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + \frac{1}{R^{2}} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{i}_{N}} + (- \frac{g_{a} g_{v}^{2}}{R^{2}} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} + \frac{2 g_{a}}{R^{2}} g_{v} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \sin{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )} - \frac{1}{R^{2}} \left(R + \alpha_{3} + g_{a} \cos{\left (g_{v} \left(\frac{\pi}{2} + \frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right)\right) \right )}\right) \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{k}_{N}}$$

$ \frac { d\vec{R_1} } { d\alpha_1} = -\frac {1}{R} \left( 1+\frac{\alpha_3}{R} \right) \vec{R_3} $





In [18]:



    


dR1dalpha2 = trigsimp(R1.diff(alpha2))
dR1dalpha2



    


















    
Out[18]:







$$\mathbf{\hat{0}}$$
















In [19]:



    


dR1dalpha3 = trigsimp(R1.diff(alpha3))
dR1dalpha3



    


















    
Out[19]:







$$(\frac{1}{R} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{i}_{N}} + (\frac{1}{R} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{k}_{N}}$$

$ \frac { d\vec{R_1} } { d\alpha_3} = \frac {1}{R} \frac {1}{1+\frac{\alpha_3}{R}} \vec{R_1} $





In [20]:



    


dR2dalpha1 = trigsimp(R2.diff(alpha1))
dR2dalpha1



    


















    
Out[20]:







$$\mathbf{\hat{0}}$$
















In [21]:



    


dR2dalpha2 = trigsimp(R2.diff(alpha2))
dR2dalpha2



    


















    
Out[21]:







$$\mathbf{\hat{0}}$$
















In [22]:



    


dR2dalpha3 = trigsimp(R2.diff(alpha3))
dR2dalpha3



    


















    
Out[22]:







$$\mathbf{\hat{0}}$$
















In [23]:



    


dR3dalpha1 = trigsimp(R3.diff(alpha1))
dR3dalpha1



    


















    
Out[23]:







$$(\frac{1}{R} \cos{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{i}_{N}} + (\frac{1}{R} \sin{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )})\mathbf{\hat{k}_{N}}$$

$ \frac { d\vec{R_3} } { d\alpha_1} = \frac {1}{R} \frac {1}{1+\frac{\alpha_3}{R}} \vec{R_1} $





In [24]:



    


dR3dalpha2 = trigsimp(R3.diff(alpha2))
dR3dalpha2



    


















    
Out[24]:







$$\mathbf{\hat{0}}$$
















In [25]:



    


dR3dalpha3 = trigsimp(R3.diff(alpha3))
dR3dalpha3



    


















    
Out[25]:







$$\mathbf{\hat{0}}$$

$ \frac { d\vec{R_3} } { d\alpha_3} = \vec{0} $

Derivative of vectors

$ \vec{u} = u^1 \vec{R_1} + u^2\vec{R_2} + u^3\vec{R_3} $

$ \frac { d\vec{u} } { d\alpha_1} = \frac { d(u^1\vec{R_1}) } { d\alpha_1} + \frac { d(u^2\vec{R_2}) } { d\alpha_1}+ \frac { d(u^3\vec{R_3}) } { d\alpha_1} = \frac { du^1 } { d\alpha_1} \vec{R_1} + u^1 \frac { d\vec{R_1} } { d\alpha_1} + \frac { du^2 } { d\alpha_1} \vec{R_2} + u^2 \frac { d\vec{R_2} } { d\alpha_1} + \frac { du^3 } { d\alpha_1} \vec{R_3} + u^3 \frac { d\vec{R_3} } { d\alpha_1} = \frac { du^1 } { d\alpha_1} \vec{R_1} - u^1 \frac {1}{R} \left( 1+\frac{\alpha_3}{R} \right) \vec{R_3} + \frac { du^2 } { d\alpha_1} \vec{R_2}+ \frac { du^3 } { d\alpha_1} \vec{R_3} + u^3 \frac {1}{R} \frac {1}{1+\frac{\alpha_3}{R}} \vec{R_1}$

Then $ \frac { d\vec{u} } { d\alpha_1} = \left( \frac { du^1 } { d\alpha_1} + u^3 \frac {1}{R} \frac {1}{1+\frac{\alpha_3}{R}} \right) \vec{R_1} + \frac { du^2 } { d\alpha_1} \vec{R_2} + \left( \frac { du^3 } { d\alpha_1} - u^1 \frac {1}{R} \left( 1+\frac{\alpha_3}{R} \right) \right) \vec{R_3}$

$ \frac { d\vec{u} } { d\alpha_2} = \frac { d(u^1\vec{R_1}) } { d\alpha_2} + \frac { d(u^2\vec{R_2}) } { d\alpha_2}+ \frac { d(u^3\vec{R_3}) } { d\alpha_2} = \frac { du^1 } { d\alpha_2} \vec{R_1} + \frac { du^2 } { d\alpha_2} \vec{R_2} + \frac { du^3 } { d\alpha_2} \vec{R_3} $

Then $ \frac { d\vec{u} } { d\alpha_2} = \frac { du^1 } { d\alpha_2} \vec{R_1} + \frac { du^2 } { d\alpha_2} \vec{R_2} + \frac { du^3 } { d\alpha_2} \vec{R_3} $

$ \frac { d\vec{u} } { d\alpha_3} = \frac { d(u^1\vec{R_1}) } { d\alpha_3} + \frac { d(u^2\vec{R_2}) } { d\alpha_3}+ \frac { d(u^3\vec{R_3}) } { d\alpha_3} = \frac { du^1 } { d\alpha_3} \vec{R_1} + u^1 \frac { d\vec{R_1} } { d\alpha_3} + \frac { du^2 } { d\alpha_3} \vec{R_2} + u^2 \frac { d\vec{R_2} } { d\alpha_3} + \frac { du^3 } { d\alpha_3} \vec{R_3} + u^3 \frac { d\vec{R_3} } { d\alpha_3} = \frac { du^1 } { d\alpha_3} \vec{R_1} + u^1 \frac {1}{R} \frac {1}{1+\frac{\alpha_3}{R}} \vec{R_1} + \frac { du^2 } { d\alpha_3} \vec{R_2}+ \frac { du^3 } { d\alpha_3} \vec{R_3} $

Then $ \frac { d\vec{u} } { d\alpha_3} = \left( \frac { du^1 } { d\alpha_3} + u^1 \frac {1}{R} \frac {1}{1+\frac{\alpha_3}{R}} \right) \vec{R_1} + \frac { du^2 } { d\alpha_3} \vec{R_2}+ \frac { du^3 } { d\alpha_3} \vec{R_3}$

Gradient of vector

$\nabla_1 u^1 = \frac { \partial u^1 } { \partial \alpha_1} + u^3 \frac {1}{R} \frac {1}{1+\frac{\alpha_3}{R}}$

$\nabla_1 u^2 = \frac { \partial u^2 } { \partial \alpha_1} $

$\nabla_1 u^3 = \frac { \partial u^3 } { \partial \alpha_1} - u^1 \frac {1}{R} \left( 1+\frac{\alpha_3}{R} \right) $

$\nabla_2 u^1 = \frac { \partial u^1 } { \partial \alpha_2}$

$\nabla_2 u^2 = \frac { \partial u^2 } { \partial \alpha_2}$

$\nabla_2 u^3 = \frac { \partial u^3 } { \partial \alpha_2}$

$\nabla_3 u^1 = \frac { \partial u^1 } { \partial \alpha_3} + u^1 \frac {1}{R} \frac {1}{1+\frac{\alpha_3}{R}}$

$\nabla_3 u^2 = \frac { \partial u^2 } { \partial \alpha_3} $

$\nabla_3 u^3 = \frac { \partial u^3 } { \partial \alpha_3}$

$ \nabla \vec{u} = \left( \begin{array}{ccc} \nabla_1 u^1 & \nabla_1 u^2 & \nabla_1 u^3 \\ \nabla_2 u^1 & \nabla_2 u^2 & \nabla_2 u^3 \\ \nabla_3 u^1 & \nabla_3 u^2 & \nabla_3 u^3 \\ \end{array} \right)$





In [37]:



    


u1=Function('u^1')
u2=Function('u^2')
u3=Function('u^3')
q=Function('q') # q(alpha3) = 1+alpha3/R
K = Symbol('K') # K = 1/R

u1_nabla1 = u1(alpha1, alpha2, alpha3).diff(alpha1) + u3(alpha1, alpha2, alpha3) * K / q(alpha3)
u2_nabla1 = u2(alpha1, alpha2, alpha3).diff(alpha1)
u3_nabla1 = u3(alpha1, alpha2, alpha3).diff(alpha1) - u1(alpha1, alpha2, alpha3) * K * q(alpha3)

u1_nabla2 = u1(alpha1, alpha2, alpha3).diff(alpha2)
u2_nabla2 = u2(alpha1, alpha2, alpha3).diff(alpha2)
u3_nabla2 = u3(alpha1, alpha2, alpha3).diff(alpha2)

u1_nabla3 = u1(alpha1, alpha2, alpha3).diff(alpha3) + u1(alpha1, alpha2, alpha3) * K / q(alpha3)
u2_nabla3 = u2(alpha1, alpha2, alpha3).diff(alpha3)
u3_nabla3 = u3(alpha1, alpha2, alpha3).diff(alpha3)
# $\nabla_2 u^2 = \frac { \partial u^2 } { \partial \alpha_2}$

grad_u = Matrix([[u1_nabla1, u2_nabla1, u3_nabla1],[u1_nabla2, u2_nabla2, u3_nabla2], [u1_nabla3, u2_nabla3, u3_nabla3]])
grad_u



    


















    
Out[37]:







$$\left[\begin{matrix}\frac{K \operatorname{u^{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{q{\left (\alpha_{3} \right )}} + \frac{\partial}{\partial \alpha_{1}} \operatorname{u^{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{\partial}{\partial \alpha_{1}} \operatorname{u^{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & - K q{\left (\alpha_{3} \right )} \operatorname{u^{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\partial}{\partial \alpha_{1}} \operatorname{u^{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\\\frac{\partial}{\partial \alpha_{2}} \operatorname{u^{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{\partial}{\partial \alpha_{2}} \operatorname{u^{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{\partial}{\partial \alpha_{2}} \operatorname{u^{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\\\frac{K \operatorname{u^{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{q{\left (\alpha_{3} \right )}} + \frac{\partial}{\partial \alpha_{3}} \operatorname{u^{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{\partial}{\partial \alpha_{3}} \operatorname{u^{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{\partial}{\partial \alpha_{3}} \operatorname{u^{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\end{matrix}\right]$$
















In [38]:



    


G_s = Matrix([[q(alpha3)**2, 0, 0],[0, 1, 0], [0, 0, 1]])
grad_u_down=grad_u*G_s
expand(simplify(grad_u_down))



    


















    
Out[38]:







$$\left[\begin{matrix}K q{\left (\alpha_{3} \right )} \operatorname{u^{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + q^{2}{\left (\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{u^{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{\partial}{\partial \alpha_{1}} \operatorname{u^{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & - K q{\left (\alpha_{3} \right )} \operatorname{u^{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \frac{\partial}{\partial \alpha_{1}} \operatorname{u^{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\\q^{2}{\left (\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{u^{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{\partial}{\partial \alpha_{2}} \operatorname{u^{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{\partial}{\partial \alpha_{2}} \operatorname{u^{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\\K q{\left (\alpha_{3} \right )} \operatorname{u^{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + q^{2}{\left (\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{u^{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{\partial}{\partial \alpha_{3}} \operatorname{u^{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \frac{\partial}{\partial \alpha_{3}} \operatorname{u^{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\end{matrix}\right]$$

$ \left( \begin{array}{c} \nabla_1 u_1 \\ \nabla_2 u_1 \\ \nabla_3 u_1 \\ \nabla_1 u_2 \\ \nabla_2 u_2 \\ \nabla_3 u_2 \\ \nabla_1 u_3 \\ \nabla_2 u_3 \\ \nabla_3 u_3 \\ \end{array}

\right)

\left( \begin{array}{c} \left( 1+\frac{\alpha_2}{R} \right)^2 \frac { \partial u^1 } { \partial \alpha_1} + u^3 \frac {\left( 1+\frac{\alpha_3}{R} \right)}{R} \\ \left( 1+\frac{\alpha_2}{R} \right)^2 \frac { \partial u^1 } { \partial \alpha_2} \\ \left( 1+\frac{\alpha_3}{R} \right)^2 \frac { \partial u^1 } { \partial \alpha_3} + u^1 \frac {\left( 1+\frac{\alpha_3}{R} \right)}{R} \\ \frac { \partial u^2 } { \partial \alpha_1} \\ \frac { \partial u^2 } { \partial \alpha_2} \\ \frac { \partial u^2 } { \partial \alpha_3} \\ \frac { \partial u^3 } { \partial \alpha_1} - u^1 \frac {\left( 1+\frac{\alpha_3}{R} \right)}{R} \\ \frac { \partial u^3 } { \partial \alpha_2} \\ \frac { \partial u^3 } { \partial \alpha_3} \\ \end{array} \right) $

$ \left( \begin{array}{c} \nabla_1 u_1 \\ \nabla_2 u_1 \\ \nabla_3 u_1 \\ \nabla_1 u_2 \\ \nabla_2 u_2 \\ \nabla_3 u_2 \\ \nabla_1 u_3 \\ \nabla_2 u_3 \\ \nabla_3 u_3 \\ \end{array}

\right)

B \cdot \left( \begin{array}{c} u^1 \\ \frac { \partial u^1 } { \partial \alpha_1} \\ \frac { \partial u^1 } { \partial \alpha_2} \\ \frac { \partial u^1 } { \partial \alpha_3} \\ u^2 \\ \frac { \partial u^2 } { \partial \alpha_1} \\ \frac { \partial u^2 } { \partial \alpha_2} \\ \frac { \partial u^2 } { \partial \alpha_3} \\ u^3 \\ \frac { \partial u^3 } { \partial \alpha_1} \\ \frac { \partial u^3 } { \partial \alpha_2} \\ \frac { \partial u^3 } { \partial \alpha_3} \\ \end{array} \right) $





In [39]:



    


B = zeros(9, 12)
B[0,1] = (1+alpha3/R)**2
B[0,8] = (1+alpha3/R)/R

B[1,2] = (1+alpha3/R)**2

B[2,0] = (1+alpha3/R)/R
B[2,3] = (1+alpha3/R)**2

B[3,5] = S(1)
B[4,6] = S(1)
B[5,7] = S(1)

B[6,9] = S(1)
B[6,0] = -(1+alpha3/R)/R
B[7,10] = S(1)
B[8,11] = S(1)

B



    


















    
Out[39]:







$$\left[\begin{array}{cccccccccccc}0 & \left(1 + \frac{\alpha_{3}}{R}\right)^{2} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{R} \left(1 + \frac{\alpha_{3}}{R}\right) & 0 & 0 & 0\\0 & 0 & \left(1 + \frac{\alpha_{3}}{R}\right)^{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{1}{R} \left(1 + \frac{\alpha_{3}}{R}\right) & 0 & 0 & \left(1 + \frac{\alpha_{3}}{R}\right)^{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\frac{1}{R} \left(-1 - \frac{\alpha_{3}}{R}\right) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]$$

Deformations tensor





In [40]:



    


E=zeros(6,9)
E[0,0]=1
E[1,4]=1
E[2,8]=1
E[3,1]=1
E[3,3]=1
E[4,2]=1
E[4,6]=1
E[5,5]=1
E[5,7]=1
E



    


















    
Out[40]:







$$\left[\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0\end{matrix}\right]$$
















In [41]:



    


Q=E*B
Q=simplify(Q)
Q



    


















    
Out[41]:







$$\left[\begin{array}{cccccccccccc}0 & \frac{1}{R^{2}} \left(R + \alpha_{3}\right)^{2} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{R^{2}} \left(R + \alpha_{3}\right) & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & \frac{1}{R^{2}} \left(R + \alpha_{3}\right)^{2} & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & \frac{1}{R^{2}} \left(R + \alpha_{3}\right)^{2} & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0\end{array}\right]$$

Tymoshenko theory

$u^1 \left( \alpha_1, \alpha_2, \alpha_3 \right)=u\left( \alpha_1 \right)+\alpha_3\gamma \left( \alpha_1 \right) $

$u^2 \left( \alpha_1, \alpha_2, \alpha_3 \right)=0 $

$u^3 \left( \alpha_1, \alpha_2, \alpha_3 \right)=w\left( \alpha_1 \right) $

$ \left( \begin{array}{c} u^1 \\ \frac { \partial u^1 } { \partial \alpha_1} \\ \frac { \partial u^1 } { \partial \alpha_2} \\ \frac { \partial u^1 } { \partial \alpha_3} \\ u^2 \\ \frac { \partial u^2 } { \partial \alpha_1} \\ \frac { \partial u^2 } { \partial \alpha_2} \\ \frac { \partial u^2 } { \partial \alpha_3} \\ u^3 \\ \frac { \partial u^3 } { \partial \alpha_1} \\ \frac { \partial u^3 } { \partial \alpha_2} \\ \frac { \partial u^3 } { \partial \alpha_3} \\ \end{array} \right) = T \cdot \left( \begin{array}{c} u \\ \frac { \partial u } { \partial \alpha_1} \\ \gamma \\ \frac { \partial \gamma } { \partial \alpha_1} \\ w \\ \frac { \partial w } { \partial \alpha_1} \\ \end{array} \right) $





In [42]:



    


T=zeros(12,6)
T[0,0]=1
T[0,2]=alpha3
T[1,1]=1
T[1,3]=alpha3
T[3,2]=1

T[8,4]=1
T[9,5]=1
T



    


















    
Out[42]:







$$\left[\begin{matrix}1 & 0 & \alpha_{3} & 0 & 0 & 0\\0 & 1 & 0 & \alpha_{3} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\end{matrix}\right]$$
















In [43]:



    


Q=E*B*T
Q=simplify(Q)
Q



    


















    
Out[43]:







$$\left[\begin{matrix}0 & \frac{1}{R^{2}} \left(R + \alpha_{3}\right)^{2} & 0 & \frac{\alpha_{3}}{R^{2}} \left(R + \alpha_{3}\right)^{2} & \frac{1}{R^{2}} \left(R + \alpha_{3}\right) & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & \frac{1}{R^{2}} \left(R + \alpha_{3}\right)^{2} & 0 & 0 & 1\\0 & 0 & 0 & 0 & 0 & 0\end{matrix}\right]$$

Elasticity tensor(stiffness tensor)

General form





In [44]:



    


from sympy import MutableDenseNDimArray
C_x = MutableDenseNDimArray.zeros(3, 3, 3, 3)

for i in range(3):
    for j in range(3):        
        for k in range(3):
            for l in range(3):
                elem_index = 'C^{{{}{}{}{}}}'.format(i+1, j+1, k+1, l+1)
                el = Symbol(elem_index)
                C_x[i,j,k,l] = el
                
C_x



    


















    
Out[44]:







$$\left[\begin{matrix}\left[\begin{matrix}C^{1111} & C^{1112} & C^{1113}\\C^{1121} & C^{1122} & C^{1123}\\C^{1131} & C^{1132} & C^{1133}\end{matrix}\right] & \left[\begin{matrix}C^{1211} & C^{1212} & C^{1213}\\C^{1221} & C^{1222} & C^{1223}\\C^{1231} & C^{1232} & C^{1233}\end{matrix}\right] & \left[\begin{matrix}C^{1311} & C^{1312} & C^{1313}\\C^{1321} & C^{1322} & C^{1323}\\C^{1331} & C^{1332} & C^{1333}\end{matrix}\right]\\\left[\begin{matrix}C^{2111} & C^{2112} & C^{2113}\\C^{2121} & C^{2122} & C^{2123}\\C^{2131} & C^{2132} & C^{2133}\end{matrix}\right] & \left[\begin{matrix}C^{2211} & C^{2212} & C^{2213}\\C^{2221} & C^{2222} & C^{2223}\\C^{2231} & C^{2232} & C^{2233}\end{matrix}\right] & \left[\begin{matrix}C^{2311} & C^{2312} & C^{2313}\\C^{2321} & C^{2322} & C^{2323}\\C^{2331} & C^{2332} & C^{2333}\end{matrix}\right]\\\left[\begin{matrix}C^{3111} & C^{3112} & C^{3113}\\C^{3121} & C^{3122} & C^{3123}\\C^{3131} & C^{3132} & C^{3133}\end{matrix}\right] & \left[\begin{matrix}C^{3211} & C^{3212} & C^{3213}\\C^{3221} & C^{3222} & C^{3223}\\C^{3231} & C^{3232} & C^{3233}\end{matrix}\right] & \left[\begin{matrix}C^{3311} & C^{3312} & C^{3313}\\C^{3321} & C^{3322} & C^{3323}\\C^{3331} & C^{3332} & C^{3333}\end{matrix}\right]\end{matrix}\right]$$

Include symmetry





In [45]:



    


C_x_symmetry = MutableDenseNDimArray.zeros(3, 3, 3, 3)

def getCIndecies(index):
    if (index == 0):
        return 0, 0
    elif (index == 1):
        return 1, 1
    elif (index == 2):
        return 2, 2
    elif (index == 3):
        return 0, 1
    elif (index == 4):
        return 0, 2
    elif (index == 5):
        return 1, 2
    
for s in range(6):
    for t in range(s, 6):
        i,j = getCIndecies(s)
        k,l = getCIndecies(t)
        elem_index = 'C^{{{}{}{}{}}}'.format(i+1, j+1, k+1, l+1)
        el = Symbol(elem_index)
        C_x_symmetry[i,j,k,l] = el
        C_x_symmetry[i,j,l,k] = el
        C_x_symmetry[j,i,k,l] = el
        C_x_symmetry[j,i,l,k] = el
        C_x_symmetry[k,l,i,j] = el
        C_x_symmetry[k,l,j,i] = el
        C_x_symmetry[l,k,i,j] = el
        C_x_symmetry[l,k,j,i] = el

                
C_x_symmetry



    


















    
Out[45]:







$$\left[\begin{matrix}\left[\begin{matrix}C^{1111} & C^{1112} & C^{1113}\\C^{1112} & C^{1122} & C^{1123}\\C^{1113} & C^{1123} & C^{1133}\end{matrix}\right] & \left[\begin{matrix}C^{1112} & C^{1212} & C^{1213}\\C^{1212} & C^{2212} & C^{1223}\\C^{1213} & C^{1223} & C^{3312}\end{matrix}\right] & \left[\begin{matrix}C^{1113} & C^{1213} & C^{1313}\\C^{1213} & C^{2213} & C^{1323}\\C^{1313} & C^{1323} & C^{3313}\end{matrix}\right]\\\left[\begin{matrix}C^{1112} & C^{1212} & C^{1213}\\C^{1212} & C^{2212} & C^{1223}\\C^{1213} & C^{1223} & C^{3312}\end{matrix}\right] & \left[\begin{matrix}C^{1122} & C^{2212} & C^{2213}\\C^{2212} & C^{2222} & C^{2223}\\C^{2213} & C^{2223} & C^{2233}\end{matrix}\right] & \left[\begin{matrix}C^{1123} & C^{1223} & C^{1323}\\C^{1223} & C^{2223} & C^{2323}\\C^{1323} & C^{2323} & C^{3323}\end{matrix}\right]\\\left[\begin{matrix}C^{1113} & C^{1213} & C^{1313}\\C^{1213} & C^{2213} & C^{1323}\\C^{1313} & C^{1323} & C^{3313}\end{matrix}\right] & \left[\begin{matrix}C^{1123} & C^{1223} & C^{1323}\\C^{1223} & C^{2223} & C^{2323}\\C^{1323} & C^{2323} & C^{3323}\end{matrix}\right] & \left[\begin{matrix}C^{1133} & C^{3312} & C^{3313}\\C^{3312} & C^{2233} & C^{3323}\\C^{3313} & C^{3323} & C^{3333}\end{matrix}\right]\end{matrix}\right]$$

Isotropic material





In [46]:



    


C_isotropic = MutableDenseNDimArray.zeros(3, 3, 3, 3)

C_isotropic_matrix = zeros(6)

mu = Symbol('mu')
la = Symbol('lambda')

for s in range(6):
    for t in range(s, 6):
        if (s < 3 and t < 3):
            if(t != s):
                C_isotropic_matrix[s,t] = la
                C_isotropic_matrix[t,s] = la
            else:
                C_isotropic_matrix[s,t] = 2*mu+la
                C_isotropic_matrix[t,s] = 2*mu+la
        elif (s == t):
            C_isotropic_matrix[s,t] = mu
            C_isotropic_matrix[t,s] = mu
            
for s in range(6):
    for t in range(s, 6):
        i,j = getCIndecies(s)
        k,l = getCIndecies(t)
        el = C_isotropic_matrix[s, t]
        C_isotropic[i,j,k,l] = el
        C_isotropic[i,j,l,k] = el
        C_isotropic[j,i,k,l] = el
        C_isotropic[j,i,l,k] = el
        C_isotropic[k,l,i,j] = el
        C_isotropic[k,l,j,i] = el
        C_isotropic[l,k,i,j] = el
        C_isotropic[l,k,j,i] = el

                
C_isotropic



    


















    
Out[46]:







$$\left[\begin{matrix}\left[\begin{matrix}\lambda + 2 \mu & 0 & 0\\0 & \lambda & 0\\0 & 0 & \lambda\end{matrix}\right] & \left[\begin{matrix}0 & \mu & 0\\\mu & 0 & 0\\0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & \mu\\0 & 0 & 0\\\mu & 0 & 0\end{matrix}\right]\\\left[\begin{matrix}0 & \mu & 0\\\mu & 0 & 0\\0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}\lambda & 0 & 0\\0 & \lambda + 2 \mu & 0\\0 & 0 & \lambda\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0\\0 & 0 & \mu\\0 & \mu & 0\end{matrix}\right]\\\left[\begin{matrix}0 & 0 & \mu\\0 & 0 & 0\\\mu & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0\\0 & 0 & \mu\\0 & \mu & 0\end{matrix}\right] & \left[\begin{matrix}\lambda & 0 & 0\\0 & \lambda & 0\\0 & 0 & \lambda + 2 \mu\end{matrix}\right]\end{matrix}\right]$$
















In [47]:



    


def getCalpha(C, A, q, p, s, t):
    res = S(0)
    for i in range(3):
        for j in range(3):        
            for k in range(3):
                for l in range(3):
                    res += C[i,j,k,l]*A[q,i]*A[p,j]*A[s,k]*A[t,l]
    return simplify(trigsimp(res))
                    


C_isotropic_alpha = MutableDenseNDimArray.zeros(3, 3, 3, 3)

for i in range(3):
    for j in range(3):        
        for k in range(3):
            for l in range(3):
                c = getCalpha(C_isotropic, A_inv, i, j, k, l)
                C_isotropic_alpha[i,j,k,l] = c

C_isotropic_alpha[0,0,0,0]



    


















    
Out[47]:







$$\frac{R^{4} \left(\lambda + 2 \mu\right)}{\left(R + \alpha_{3}\right)^{4}}$$
















In [48]:



    


C_isotropic_matrix_alpha = zeros(6)

for s in range(6):
    for t in range(6):
        i,j = getCIndecies(s)
        k,l = getCIndecies(t)
        C_isotropic_matrix_alpha[s,t] = C_isotropic_alpha[i,j,k,l]
        
C_isotropic_matrix_alpha



    


















    
Out[48]:







$$\left[\begin{matrix}\frac{R^{4} \left(\lambda + 2 \mu\right)}{\left(R + \alpha_{3}\right)^{4}} & \frac{R^{2} \lambda}{\left(R + \alpha_{3}\right)^{2}} & \frac{R^{2} \lambda}{\left(R + \alpha_{3}\right)^{2}} & 0 & 0 & 0\\\frac{R^{2} \lambda}{\left(R + \alpha_{3}\right)^{2}} & \lambda + 2 \mu & \lambda & 0 & 0 & 0\\\frac{R^{2} \lambda}{\left(R + \alpha_{3}\right)^{2}} & \lambda & \lambda + 2 \mu & 0 & 0 & 0\\0 & 0 & 0 & \frac{R^{2} \mu}{\left(R + \alpha_{3}\right)^{2}} & 0 & 0\\0 & 0 & 0 & 0 & \frac{R^{2} \mu}{\left(R + \alpha_{3}\right)^{2}} & 0\\0 & 0 & 0 & 0 & 0 & \mu\end{matrix}\right]$$

Orthotropic material





In [49]:



    


C_orthotropic = MutableDenseNDimArray.zeros(3, 3, 3, 3)

C_orthotropic_matrix = zeros(6)

for s in range(6):
    for t in range(s, 6):
        elem_index = 'C^{{{}{}}}'.format(s+1, t+1)
        el = Symbol(elem_index)
        if ((s < 3 and t < 3) or t == s):
            C_orthotropic_matrix[s,t] = el
            C_orthotropic_matrix[t,s] = el
            
for s in range(6):
    for t in range(s, 6):
        i,j = getCIndecies(s)
        k,l = getCIndecies(t)
        el = C_orthotropic_matrix[s, t]
        C_orthotropic[i,j,k,l] = el
        C_orthotropic[i,j,l,k] = el
        C_orthotropic[j,i,k,l] = el
        C_orthotropic[j,i,l,k] = el
        C_orthotropic[k,l,i,j] = el
        C_orthotropic[k,l,j,i] = el
        C_orthotropic[l,k,i,j] = el
        C_orthotropic[l,k,j,i] = el

                
C_orthotropic



    


















    
Out[49]:







$$\left[\begin{matrix}\left[\begin{matrix}C^{11} & 0 & 0\\0 & C^{12} & 0\\0 & 0 & C^{13}\end{matrix}\right] & \left[\begin{matrix}0 & C^{44} & 0\\C^{44} & 0 & 0\\0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & C^{55}\\0 & 0 & 0\\C^{55} & 0 & 0\end{matrix}\right]\\\left[\begin{matrix}0 & C^{44} & 0\\C^{44} & 0 & 0\\0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}C^{12} & 0 & 0\\0 & C^{22} & 0\\0 & 0 & C^{23}\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0\\0 & 0 & C^{66}\\0 & C^{66} & 0\end{matrix}\right]\\\left[\begin{matrix}0 & 0 & C^{55}\\0 & 0 & 0\\C^{55} & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0\\0 & 0 & C^{66}\\0 & C^{66} & 0\end{matrix}\right] & \left[\begin{matrix}C^{13} & 0 & 0\\0 & C^{23} & 0\\0 & 0 & C^{33}\end{matrix}\right]\end{matrix}\right]$$

Orthotropic material in shell coordinates





In [50]:



    


def getCalpha(C, A, q, p, s, t):
    res = S(0)
    for i in range(3):
        for j in range(3):        
            for k in range(3):
                for l in range(3):
                    res += C[i,j,k,l]*A[q,i]*A[p,j]*A[s,k]*A[t,l]
    return simplify(trigsimp(res))
                    


C_orthotropic_alpha = MutableDenseNDimArray.zeros(3, 3, 3, 3)

for i in range(3):
    for j in range(3):        
        for k in range(3):
            for l in range(3):
                c = getCalpha(C_orthotropic, A_inv, i, j, k, l)
                C_orthotropic_alpha[i,j,k,l] = c

C_orthotropic_alpha[0,0,0,0]



    


















    
Out[50]:







$$\frac{R^{4}}{\left(R + \alpha_{3}\right)^{4}} \left(C^{11} \cos^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + 2 C^{13} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{33} \sin^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + 4 C^{55} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right)$$
















In [51]:



    


C_orthotropic_matrix_alpha = zeros(6)

for s in range(6):
    for t in range(6):
        i,j = getCIndecies(s)
        k,l = getCIndecies(t)
        C_orthotropic_matrix_alpha[s,t] = C_orthotropic_alpha[i,j,k,l]
        
C_orthotropic_matrix_alpha



    


















    
Out[51]:







$$\left[\begin{matrix}\frac{R^{4}}{\left(R + \alpha_{3}\right)^{4}} \left(C^{11} \cos^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + 2 C^{13} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{33} \sin^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + 4 C^{55} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) & \frac{R^{2}}{\left(R + \alpha_{3}\right)^{2}} \left(C^{12} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{23} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) & \frac{R^{2}}{\left(R + \alpha_{3}\right)^{2}} \left(C^{11} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{13} \sin^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{13} \cos^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{33} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} - 4 C^{55} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) & 0 & \frac{R^{3}}{8 \left(R + \alpha_{3}\right)^{3}} \left(- 2 C^{11} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} - C^{11} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + 2 C^{13} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + 2 C^{33} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} - C^{33} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + 4 C^{55} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )}\right) & 0\\\frac{R^{2}}{\left(R + \alpha_{3}\right)^{2}} \left(C^{12} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{23} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) & C^{22} & C^{12} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{23} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & 0 & - \frac{R \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )}}{2 R + 2 \alpha_{3}} \left(C^{12} - C^{23}\right) & 0\\\frac{R^{2}}{\left(R + \alpha_{3}\right)^{2}} \left(C^{11} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{13} \sin^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{13} \cos^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{33} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} - 4 C^{55} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) & C^{12} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{23} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & C^{11} \sin^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + 2 C^{13} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{33} \cos^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + 4 C^{55} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & 0 & \frac{R}{8 \left(R + \alpha_{3}\right)} \left(- 2 C^{11} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} + C^{11} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} - 2 C^{13} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + 2 C^{33} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} + C^{33} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} - 4 C^{55} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )}\right) & 0\\0 & 0 & 0 & \frac{R^{2}}{\left(R + \alpha_{3}\right)^{2}} \left(C^{44} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{66} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) & 0 & - \frac{R \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )}}{2 R + 2 \alpha_{3}} \left(C^{44} - C^{66}\right)\\\frac{R^{3}}{8 \left(R + \alpha_{3}\right)^{3}} \left(- 2 C^{11} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} - C^{11} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + 2 C^{13} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + 2 C^{33} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} - C^{33} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + 4 C^{55} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )}\right) & - \frac{R \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )}}{2 R + 2 \alpha_{3}} \left(C^{12} - C^{23}\right) & \frac{R}{8 \left(R + \alpha_{3}\right)} \left(- 2 C^{11} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} + C^{11} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} - 2 C^{13} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + 2 C^{33} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} + C^{33} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} - 4 C^{55} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )}\right) & 0 & \frac{R^{2}}{\left(R + \alpha_{3}\right)^{2}} \left(C^{11} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} - 2 C^{13} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{33} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{55} \sin^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} - 2 C^{55} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{55} \cos^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\right) & 0\\0 & 0 & 0 & - \frac{R \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )}}{2 R + 2 \alpha_{3}} \left(C^{44} - C^{66}\right) & 0 & C^{44} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{66} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\end{matrix}\right]$$

Physical coordinates

$u^1=\frac{u_{[1]}}{1+\frac{\alpha_3}{R}}$

$\frac{\partial u^1} {\partial \alpha_3}=\frac{1}{1+\frac{\alpha_3}{R}} \frac{\partial u_{[1]}} {\partial \alpha_3} + u_{[1]} \frac{\partial} {\partial \alpha_3} \left( \frac{1}{1+\frac{\alpha_3}{R}} \right) = =\frac{1}{1+\frac{\alpha_3}{R}} \frac{\partial u_{[1]}} {\partial \alpha_3} - u_{[1]} \frac{1}{R \left( 1+\frac{\alpha_3}{R} \right)^2} $





In [52]:



    


P=eye(12,12)
P[0,0]=1/(1+alpha3/R)
P[1,1]=1/(1+alpha3/R)
P[2,2]=1/(1+alpha3/R)
P[3,0]=-1/(R*(1+alpha3/R)**2)
P[3,3]=1/(1+alpha3/R)
P



    


















    
Out[52]:







$$\left[\begin{array}{cccccccccccc}\frac{1}{1 + \frac{\alpha_{3}}{R}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & \frac{1}{1 + \frac{\alpha_{3}}{R}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & \frac{1}{1 + \frac{\alpha_{3}}{R}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{1}{R \left(1 + \frac{\alpha_{3}}{R}\right)^{2}} & 0 & 0 & \frac{1}{1 + \frac{\alpha_{3}}{R}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right]$$
















In [53]:



    


Def=simplify(E*B*P)
Def



    


















    
Out[53]:







$$\left[\begin{array}{cccccccccccc}0 & \frac{1}{R} \left(R + \alpha_{3}\right) & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{R^{2}} \left(R + \alpha_{3}\right) & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & \frac{1}{R} \left(R + \alpha_{3}\right) & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{1}{R} & 0 & 0 & \frac{1}{R} \left(R + \alpha_{3}\right) & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0\end{array}\right]$$
















In [54]:



    


rows, cols = Def.shape
D_p=zeros(rows, cols)
q = 1+alpha3/R
for i in range(rows):
    ratio = 1
    if (i==0):
        ratio = q*q
    elif (i==3 or i == 4):
        ratio = q
    
    for j in range(cols):
        D_p[i,j] = Def[i,j] / ratio

D_p = simplify(D_p)
D_p



    


















    
Out[54]:







$$\left[\begin{array}{cccccccccccc}0 & \frac{R}{R + \alpha_{3}} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{R + \alpha_{3}} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\0 & 0 & 1 & 0 & 0 & \frac{R}{R + \alpha_{3}} & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{1}{R + \alpha_{3}} & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \frac{R}{R + \alpha_{3}} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0\end{array}\right]$$

Stiffness tensor





In [55]:



    


C_isotropic_alpha_p = MutableDenseNDimArray.zeros(3, 3, 3, 3)
q=1+alpha3/R
for i in range(3):
    for j in range(3):        
        for k in range(3):
            for l in range(3):
                fact = 1
                if (i==0):
                    fact = fact*q
                if (j==0):
                    fact = fact*q
                if (k==0):
                    fact = fact*q
                if (l==0):
                    fact = fact*q
                C_isotropic_alpha_p[i,j,k,l] = simplify(C_isotropic_alpha[i,j,k,l]*fact)
            
C_isotropic_matrix_alpha_p = zeros(6)

for s in range(6):
    for t in range(6):
        i,j = getCIndecies(s)
        k,l = getCIndecies(t)
        C_isotropic_matrix_alpha_p[s,t] = C_isotropic_alpha_p[i,j,k,l]
        
C_isotropic_matrix_alpha_p



    


















    
Out[55]:







$$\left[\begin{matrix}\lambda + 2 \mu & \lambda & \lambda & 0 & 0 & 0\\\lambda & \lambda + 2 \mu & \lambda & 0 & 0 & 0\\\lambda & \lambda & \lambda + 2 \mu & 0 & 0 & 0\\0 & 0 & 0 & \mu & 0 & 0\\0 & 0 & 0 & 0 & \mu & 0\\0 & 0 & 0 & 0 & 0 & \mu\end{matrix}\right]$$
















In [56]:



    


C_orthotropic_alpha_p = MutableDenseNDimArray.zeros(3, 3, 3, 3)
q=1+alpha3/R
for i in range(3):
    for j in range(3):        
        for k in range(3):
            for l in range(3):
                fact = 1
                if (i==0):
                    fact = fact*q
                if (j==0):
                    fact = fact*q
                if (k==0):
                    fact = fact*q
                if (l==0):
                    fact = fact*q
                C_orthotropic_alpha_p[i,j,k,l] = simplify(C_orthotropic_alpha[i,j,k,l]*fact)
            
C_orthotropic_matrix_alpha_p = zeros(6)

for s in range(6):
    for t in range(6):
        i,j = getCIndecies(s)
        k,l = getCIndecies(t)
        C_orthotropic_matrix_alpha_p[s,t] = C_orthotropic_alpha_p[i,j,k,l]
        
C_orthotropic_matrix_alpha_p



    


















    
Out[56]:







$$\left[\begin{matrix}C^{11} \cos^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + 2 C^{13} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{33} \sin^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + 4 C^{55} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & C^{12} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{23} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & C^{11} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{13} \sin^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{13} \cos^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{33} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} - 4 C^{55} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & 0 & - \frac{C^{11}}{4} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} - \frac{C^{11}}{8} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{C^{13}}{4} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{C^{33}}{4} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} - \frac{C^{33}}{8} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{C^{55}}{2} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} & 0\\C^{12} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{23} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & C^{22} & C^{12} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{23} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & 0 & \frac{1}{2} \left(- C^{12} + C^{23}\right) \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} & 0\\C^{11} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{13} \sin^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{13} \cos^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{33} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} - 4 C^{55} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & C^{12} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{23} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & C^{11} \sin^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + 2 C^{13} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{33} \cos^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + 4 C^{55} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & 0 & - \frac{C^{11}}{4} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{C^{11}}{8} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} - \frac{C^{13}}{4} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{C^{33}}{4} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{C^{33}}{8} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} - \frac{C^{55}}{2} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} & 0\\0 & 0 & 0 & C^{44} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{66} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & 0 & \frac{1}{2} \left(- C^{44} + C^{66}\right) \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )}\\- \frac{C^{11}}{4} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} - \frac{C^{11}}{8} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{C^{13}}{4} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{C^{33}}{4} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} - \frac{C^{33}}{8} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{C^{55}}{2} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} & \frac{1}{2} \left(- C^{12} + C^{23}\right) \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} & - \frac{C^{11}}{4} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{C^{11}}{8} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} - \frac{C^{13}}{4} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{C^{33}}{4} \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} + \frac{C^{33}}{8} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} - \frac{C^{55}}{2} \sin{\left (\frac{2}{R} \left(L - 2 \alpha_{1}\right) \right )} & 0 & C^{11} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} - 2 C^{13} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{33} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{55} \sin^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} - 2 C^{55} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{55} \cos^{4}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} & 0\\0 & 0 & 0 & \frac{1}{2} \left(- C^{44} + C^{66}\right) \sin{\left (\frac{1}{R} \left(L - 2 \alpha_{1}\right) \right )} & 0 & C^{44} \sin^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )} + C^{66} \cos^{2}{\left (\frac{1}{R} \left(\frac{L}{2} - \alpha_{1}\right) \right )}\end{matrix}\right]$$

Tymoshenko





In [84]:



    


D_p_T = D_p*T
K = Symbol('K')
D_p_T = D_p_T.subs(R, 1/K)
simplify(D_p_T)



    


















    
Out[84]:







$$\left[\begin{matrix}0 & \frac{1}{K \alpha_{3} + 1} & 0 & \frac{\alpha_{3}}{K \alpha_{3} + 1} & \frac{K}{K \alpha_{3} + 1} & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\- \frac{K}{K \alpha_{3} + 1} & 0 & \frac{1}{K \alpha_{3} + 1} & 0 & 0 & \frac{1}{K \alpha_{3} + 1}\\0 & 0 & 0 & 0 & 0 & 0\end{matrix}\right]$$

Square of segment

$A=\frac {\theta}{2} \left( R + h_2 \right)^2-\frac {\theta}{2} \left( R + h_1 \right)^2$





In [89]:



    


theta, h1, h2=symbols('theta h_1 h_2')
square_geom=theta/2*(R+h2)**2-theta/2*(R+h1)**2
expand(simplify(square_geom))



    


















    
Out[89]:







$$- R h_{1} \theta + R h_{2} \theta - \frac{h_{1}^{2} \theta}{2} + \frac{h_{2}^{2} \theta}{2}$$

${\displaystyle A=\int_{0}^{L}\int_{h_1}^{h_2} \left( 1+\frac{\alpha_3}{R} \right) d \alpha_1 d \alpha_3}, L=R \theta$





In [88]:



    


square_int=integrate(integrate(1+alpha3/R, (alpha3, h1, h2)), (alpha1, 0, theta*R))
expand(simplify(square_int))



    


















    
Out[88]:







$$- R h_{1} \theta + R h_{2} \theta - \frac{h_{1}^{2} \theta}{2} + \frac{h_{2}^{2} \theta}{2}$$

Virtual work

Isotropic material physical coordinates





In [58]:



    


simplify(D_p.T*C_isotropic_matrix_alpha_p*D_p)



    


















    
Out[58]:







$$\left[\begin{array}{cccccccccccc}\frac{\mu}{\left(R + \alpha_{3}\right)^{2}} & 0 & 0 & - \frac{\mu}{R + \alpha_{3}} & 0 & 0 & 0 & 0 & 0 & - \frac{R \mu}{\left(R + \alpha_{3}\right)^{2}} & 0 & 0\\0 & \frac{R^{2} \left(\lambda + 2 \mu\right)}{\left(R + \alpha_{3}\right)^{2}} & 0 & 0 & 0 & 0 & \frac{R \lambda}{R + \alpha_{3}} & 0 & \frac{R \left(\lambda + 2 \mu\right)}{\left(R + \alpha_{3}\right)^{2}} & 0 & 0 & \frac{R \lambda}{R + \alpha_{3}}\\0 & 0 & \mu & 0 & 0 & \frac{R \mu}{R + \alpha_{3}} & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{\mu}{R + \alpha_{3}} & 0 & 0 & \mu & 0 & 0 & 0 & 0 & 0 & \frac{R \mu}{R + \alpha_{3}} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & \frac{R \mu}{R + \alpha_{3}} & 0 & 0 & \frac{R^{2} \mu}{\left(R + \alpha_{3}\right)^{2}} & 0 & 0 & 0 & 0 & 0 & 0\\0 & \frac{R \lambda}{R + \alpha_{3}} & 0 & 0 & 0 & 0 & \lambda + 2 \mu & 0 & \frac{\lambda}{R + \alpha_{3}} & 0 & 0 & \lambda\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \mu & 0 & 0 & \mu & 0\\0 & \frac{R \left(\lambda + 2 \mu\right)}{\left(R + \alpha_{3}\right)^{2}} & 0 & 0 & 0 & 0 & \frac{\lambda}{R + \alpha_{3}} & 0 & \frac{\lambda + 2 \mu}{\left(R + \alpha_{3}\right)^{2}} & 0 & 0 & \frac{\lambda}{R + \alpha_{3}}\\- \frac{R \mu}{\left(R + \alpha_{3}\right)^{2}} & 0 & 0 & \frac{R \mu}{R + \alpha_{3}} & 0 & 0 & 0 & 0 & 0 & \frac{R^{2} \mu}{\left(R + \alpha_{3}\right)^{2}} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \mu & 0 & 0 & \mu & 0\\0 & \frac{R \lambda}{R + \alpha_{3}} & 0 & 0 & 0 & 0 & \lambda & 0 & \frac{\lambda}{R + \alpha_{3}} & 0 & 0 & \lambda + 2 \mu\end{array}\right]$$

Isotropic material physical coordinates - Tymoshenko





In [95]:



    


W = simplify(D_p_T.T*C_isotropic_matrix_alpha_p*D_p_T*(1+alpha3*K)**2)
W



    


















    
Out[95]:







$$\left[\begin{matrix}K^{2} \mu & 0 & - K \mu & 0 & 0 & - K \mu\\0 & \lambda + 2 \mu & 0 & \alpha_{3} \left(\lambda + 2 \mu\right) & K \left(\lambda + 2 \mu\right) & 0\\- K \mu & 0 & \mu & 0 & 0 & \mu\\0 & \alpha_{3} \left(\lambda + 2 \mu\right) & 0 & \alpha_{3}^{2} \left(\lambda + 2 \mu\right) & K \alpha_{3} \left(\lambda + 2 \mu\right) & 0\\0 & K \left(\lambda + 2 \mu\right) & 0 & K \alpha_{3} \left(\lambda + 2 \mu\right) & K^{2} \left(\lambda + 2 \mu\right) & 0\\- K \mu & 0 & \mu & 0 & 0 & \mu\end{matrix}\right]$$
















In [96]:



    


h=Symbol('h')
E=Symbol('E')
v=Symbol('nu')
W_a3 = integrate(W, (alpha3, -h/2, h/2))
W_a3 = simplify(W_a3)
W_a3.subs(la, E*v/((1+v)*(1-2*v))).subs(mu, E/((1+v)*2))



    


















    
Out[96]:







$$\left[\begin{matrix}\frac{E K^{2} h}{2 \nu + 2} & 0 & - \frac{E K h}{2 \nu + 2} & 0 & 0 & - \frac{E K h}{2 \nu + 2}\\0 & h \left(\frac{E \nu}{\left(- 2 \nu + 1\right) \left(\nu + 1\right)} + \frac{2 E}{2 \nu + 2}\right) & 0 & 0 & K h \left(\frac{E \nu}{\left(- 2 \nu + 1\right) \left(\nu + 1\right)} + \frac{2 E}{2 \nu + 2}\right) & 0\\- \frac{E K h}{2 \nu + 2} & 0 & \frac{E h}{2 \nu + 2} & 0 & 0 & \frac{E h}{2 \nu + 2}\\0 & 0 & 0 & \frac{h^{3}}{12} \left(\frac{E \nu}{\left(- 2 \nu + 1\right) \left(\nu + 1\right)} + \frac{2 E}{2 \nu + 2}\right) & 0 & 0\\0 & K h \left(\frac{E \nu}{\left(- 2 \nu + 1\right) \left(\nu + 1\right)} + \frac{2 E}{2 \nu + 2}\right) & 0 & 0 & K^{2} h \left(\frac{E \nu}{\left(- 2 \nu + 1\right) \left(\nu + 1\right)} + \frac{2 E}{2 \nu + 2}\right) & 0\\- \frac{E K h}{2 \nu + 2} & 0 & \frac{E h}{2 \nu + 2} & 0 & 0 & \frac{E h}{2 \nu + 2}\end{matrix}\right]$$
















In [91]:



    


A_M = zeros(3)
A_M[0,0] = E*h/(1-v**2)
A_M[1,1] = 5*E*h/(12*(1+v))
A_M[2,2] = E*h**3/(12*(1-v**2))
Q_M = zeros(3,6)
Q_M[0,1] = 1
Q_M[0,4] = K
Q_M[1,0] = -K
Q_M[1,2] = 1
Q_M[1,5] = 1
Q_M[2,3] = 1

W_M=Q_M.T*A_M*Q_M
W_M



    


















    
Out[91]:







$$\left[\begin{matrix}\frac{5 E K^{2} h}{12 \nu + 12} & 0 & - \frac{5 E K h}{12 \nu + 12} & 0 & 0 & - \frac{5 E K h}{12 \nu + 12}\\0 & \frac{E h}{- \nu^{2} + 1} & 0 & 0 & \frac{E K h}{- \nu^{2} + 1} & 0\\- \frac{5 E K h}{12 \nu + 12} & 0 & \frac{5 E h}{12 \nu + 12} & 0 & 0 & \frac{5 E h}{12 \nu + 12}\\0 & 0 & 0 & \frac{E h^{3}}{- 12 \nu^{2} + 12} & 0 & 0\\0 & \frac{E K h}{- \nu^{2} + 1} & 0 & 0 & \frac{E K^{2} h}{- \nu^{2} + 1} & 0\\- \frac{5 E K h}{12 \nu + 12} & 0 & \frac{5 E h}{12 \nu + 12} & 0 & 0 & \frac{5 E h}{12 \nu + 12}\end{matrix}\right]$$

Content source: tarashor/vibrations

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