Matrix generation

Init symbols for sympy



In [57]:

    
from sympy import *
from geom_util import *
from sympy.vector import CoordSys3D
N = CoordSys3D('N')
alpha1, alpha2, alpha3 = symbols("alpha_1 alpha_2 alpha_3", real = True, positive=True)
init_printing()

%matplotlib inline

%reload_ext autoreload
%autoreload 2
%aimport geom_util

Lame params



In [58]:

    
h1 = Function("H1")
h2 = Function("H2")
h3 = Function("H3")

H1 = h1(alpha1, alpha2, alpha3)
H2 = S(1)
H3 = h3(alpha1, alpha2, alpha3)

Metric tensor

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



In [59]:

    
G_up = getMetricTensorUpLame(H1, H2, H3)

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



In [60]:

    
G_down = getMetricTensorDownLame(H1, H2, H3)

Christoffel symbols



In [61]:

    
GK = getChristoffelSymbols2(G_up, G_down, (alpha1, alpha2, alpha3))

Gradient of vector

$ \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) = B \cdot D \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 [62]:

    
def row_index_to_i_j_grad(i_row):
    return i_row // 3, i_row % 3
        

B = zeros(9, 12)
B[0,1] = S(1)
B[1,2] = S(1)

B[2,3] = S(1)

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

B[6,9] = S(1)
B[7,10] = S(1)
B[8,11] = S(1)

for row_index in range(9):
    i,j=row_index_to_i_j_grad(row_index)
    B[row_index, 0] = -GK[i,j,0]
    B[row_index, 4] = -GK[i,j,1]
    B[row_index, 8] = -GK[i,j,2]

B









    Out[62]:





$$\left[\begin{array}{cccccccccccc}- \frac{\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 1 & 0 & 0 & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & \frac{\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0\\- \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 1 & 0 & 0 & 0 & 0 & - \frac{\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0\\- \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 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 & - \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0\\- \frac{\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 1 & 0\\\frac{\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & - \frac{\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 1\end{array}\right]$$

Strain tensor

$ \left( \begin{array}{c} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{12} \\ 2\varepsilon_{13} \\ 2\varepsilon_{23} \\ \end{array}

\right)

\left(E + E_{NL} \left( \nabla \vec{u} \right) \right) \cdot \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)$



In [63]:

    
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[63]:





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



In [64]:

    
def E_NonLinear(grad_u):
    N = 3

    du = zeros(N, N)

    #    print("===Deformations===")

    for i in range(N):
        for j in range(N):
            index = i*N+j
            du[j,i] = grad_u[index]

    #    print("========")

    a_values = S(1)/S(2) * du * G_up


    E_NL = zeros(6,9)
    E_NL[0,0] = a_values[0,0]
    E_NL[0,3] = a_values[0,1]
    E_NL[0,6] = a_values[0,2]

    E_NL[1,1] = a_values[1,0]
    E_NL[1,4] = a_values[1,1]
    E_NL[1,7] = a_values[1,2]

    E_NL[2,2] = a_values[2,0]
    E_NL[2,5] = a_values[2,1]
    E_NL[2,8] = a_values[2,2]

    E_NL[3,1] = 2*a_values[0,0]
    E_NL[3,4] = 2*a_values[0,1]
    E_NL[3,7] = 2*a_values[0,2]

    E_NL[4,0] = 2*a_values[2,0]
    E_NL[4,3] = 2*a_values[2,1]
    E_NL[4,6] = 2*a_values[2,2]

    E_NL[5,2] = 2*a_values[1,0]
    E_NL[5,5] = 2*a_values[1,1]
    E_NL[5,8] = 2*a_values[1,2]


    return E_NL


%aimport geom_util
#u=getUHat3D(alpha1, alpha2, alpha3)


u=getUHatU3Main(alpha1, alpha2, alpha3)

gradu=B*u


E_NL = E_NonLinear(gradu)*B

Physical coordinates

$u_i=u_{[i]} H_i$



In [65]:

    
P=zeros(12,12)
P[0,0]=H1
P[1,0]=(H1).diff(alpha1)
P[1,1]=H1
P[2,0]=(H1).diff(alpha2)
P[2,2]=H1
P[3,0]=(H1).diff(alpha3)
P[3,3]=H1

P[4,4]=H2
P[5,4]=(H2).diff(alpha1)
P[5,5]=H2
P[6,4]=(H2).diff(alpha2)
P[6,6]=H2
P[7,4]=(H2).diff(alpha3)
P[7,7]=H2

P[8,8]=H3
P[9,8]=(H3).diff(alpha1)
P[9,9]=H3
P[10,8]=(H3).diff(alpha2)
P[10,10]=H3
P[11,8]=(H3).diff(alpha3)
P[11,11]=H3
P=simplify(P)
P









    Out[65]:





$$\left[\begin{array}{cccccccccccc}\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} & 0 & 0 & \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\end{array}\right]$$



In [66]:

    
B_P = zeros(9,9)

for i in range(3):
    for j in range(3):
        ratio=1
        if (i==0):
            ratio = ratio*H1
        elif (i==1):
            ratio = ratio*H2
        elif (i==2):
            ratio = ratio*H3
            
        if (j==0):
            ratio = ratio*H1
        elif (j==1):
            ratio = ratio*H2
        elif (j==2):
            ratio = ratio*H3
            
        
        row_index = i*3+j
        
        B_P[row_index, row_index] = 1/ratio
        


Grad_U_P = simplify(B_P*B*P)
Grad_U_P









    Out[66]:





$$\left[\begin{array}{cccccccccccc}0 & \frac{1}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & \frac{\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & \frac{1}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & 0 & - \frac{\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0\\- \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & 0 & \frac{1}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 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 & \frac{1}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & - \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0\\- \frac{\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\frac{\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}\end{array}\right]$$



In [67]:

    
StrainL=simplify(E*Grad_U_P)
StrainL









    Out[67]:





$$\left[\begin{array}{cccccccccccc}0 & \frac{1}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & \frac{\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\frac{\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}\\- \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 1 & 0 & 0 & \frac{1}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & \frac{1}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & 0 & - \frac{\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & \frac{1}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{1}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & - \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 1 & 0\end{array}\right]$$



In [68]:

    
%aimport geom_util


u=getUHatU3Main(alpha1, alpha2, alpha3)

gradup=B_P*B*P*u


E_NLp = E_NonLinear(gradup)*B*P*u


simplify(E_NLp)









    Out[68]:





$$\left[\begin{matrix}\frac{1}{2 \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} + \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} - 2 \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{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 )} + \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} + \operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} + 2 \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}\right)\\\frac{\left(\frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}\\\frac{1}{2 \operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} + \operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} + 2 \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} + \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}\right)\\\frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)\\\frac{1}{\operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\left(- \left(\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)\\\frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{u_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{u_{2}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{u_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)\end{matrix}\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 [69]:

    
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[69]:





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



In [70]:

    
D_p_T = StrainL*T
simplify(D_p_T)









    Out[70]:





$$\left[\begin{matrix}0 & \frac{1}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{\alpha_{3}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & \frac{\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0\\0 & 0 & 0 & 0 & 0 & 0\\\frac{\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{\alpha_{3} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0\\- \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & - \frac{\alpha_{3} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0\\- \frac{\frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{- \alpha_{3} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & - \frac{\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & \frac{1}{\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}\\0 & 0 & 0 & 0 & - \frac{\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{\operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0\end{matrix}\right]$$



In [71]:

    
u = Function("u")
t = Function("theta")
w = Function("w")

u1=u(alpha1)+alpha3*t(alpha1)
u3=w(alpha1)

gu = zeros(12,1) 
gu[0] = u1
gu[1] = u1.diff(alpha1)
gu[3] = u1.diff(alpha3)


gu[8] = u3
gu[9] = u3.diff(alpha1)


gradup=Grad_U_P*gu

# o20=(K*u(alpha1)-w(alpha1).diff(alpha1)+t(alpha1))/2
# o21=K*t(alpha1)
# O=1/2*o20*o20+alpha3*o20*o21-alpha3*K/2*o20*o20
# O=expand(O)
# O=collect(O,alpha3)
# simplify(O)

StrainNL = E_NonLinear(gradup)*gradup
simplify(StrainNL)









    Out[71]:





$$\left[\begin{matrix}\frac{1}{2 \operatorname{H_{1}}^{4}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{4}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right)^{2} \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{4}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} + \left(\left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{d}{d \alpha_{1}} w{\left (\alpha_{1} \right )}\right)^{2} \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(\left(\alpha_{3} \frac{d}{d \alpha_{1}} \theta{\left (\alpha_{1} \right )} + \frac{d}{d \alpha_{1}} u{\left (\alpha_{1} \right )}\right) \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)\\0\\\frac{\left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right)^{2} \left(\frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{4}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} + \frac{1}{2 \operatorname{H_{1}}^{4}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \theta{\left (\alpha_{1} \right )} - w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2} + \frac{w^{2}{\left (\alpha_{1} \right )} \left(\frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)^{2}}{2 \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}\\0\\\frac{1}{\operatorname{H_{1}}^{4}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{4}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} \left(- \left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) \left(\left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{d}{d \alpha_{1}} w{\left (\alpha_{1} \right )}\right) \operatorname{H_{1}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(\alpha_{3} \theta{\left (\alpha_{1} \right )} + u{\left (\alpha_{1} \right )}\right) \operatorname{H_{1}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}^{3}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(\left(\alpha_{3} \frac{d}{d \alpha_{1}} \theta{\left (\alpha_{1} \right )} + \frac{d}{d \alpha_{1}} u{\left (\alpha_{1} \right )}\right) \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \left(\operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \theta{\left (\alpha_{1} \right )} - w{\left (\alpha_{1} \right )} \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right) \operatorname{H_{3}}^{2}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}\right)\\0\end{matrix}\right]$$

Square theory

$u^1 \left( \alpha_1, \alpha_2, \alpha_3 \right)=u_{10}\left( \alpha_1 \right)p_0\left( \alpha_3 \right)+u_{11}\left( \alpha_1 \right)p_1\left( \alpha_3 \right)+u_{12}\left( \alpha_1 \right)p_2\left( \alpha_3 \right) $

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

$u^3 \left( \alpha_1, \alpha_2, \alpha_3 \right)=u_{30}\left( \alpha_1 \right)p_0\left( \alpha_3 \right)+u_{31}\left( \alpha_1 \right)p_1\left( \alpha_3 \right)+u_{32}\left( \alpha_1 \right)p_2\left( \alpha_3 \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) = L \cdot \left( \begin{array}{c} u_{10} \\ \frac { \partial u_{10} } { \partial \alpha_1} \\ u_{11} \\ \frac { \partial u_{11} } { \partial \alpha_1} \\ u_{12} \\ \frac { \partial u_{12} } { \partial \alpha_1} \\ u_{30} \\ \frac { \partial u_{30} } { \partial \alpha_1} \\ u_{31} \\ \frac { \partial u_{31} } { \partial \alpha_1} \\ u_{32} \\ \frac { \partial u_{32} } { \partial \alpha_1} \\ \end{array} \right) $



In [72]:

    
L=zeros(12,12)
h=Symbol('h')
p0=1/2-alpha3/h
p1=1/2+alpha3/h
p2=1-(2*alpha3/h)**2

L[0,0]=p0
L[0,2]=p1
L[0,4]=p2

L[1,1]=p0
L[1,3]=p1
L[1,5]=p2

L[3,0]=p0.diff(alpha3)
L[3,2]=p1.diff(alpha3)
L[3,4]=p2.diff(alpha3)

L[8,6]=p0
L[8,8]=p1
L[8,10]=p2

L[9,7]=p0
L[9,9]=p1
L[9,11]=p2

L[11,6]=p0.diff(alpha3)
L[11,8]=p1.diff(alpha3)
L[11,10]=p2.diff(alpha3)

L









    Out[72]:





$$\left[\begin{array}{cccccccccccc}- \frac{\alpha_{3}}{h} + 0.5 & 0 & \frac{\alpha_{3}}{h} + 0.5 & 0 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & - \frac{\alpha_{3}}{h} + 0.5 & 0 & \frac{\alpha_{3}}{h} + 0.5 & 0 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{1}{h} & 0 & \frac{1}{h} & 0 & - \frac{8 \alpha_{3}}{h^{2}} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - \frac{\alpha_{3}}{h} + 0.5 & 0 & \frac{\alpha_{3}}{h} + 0.5 & 0 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{\alpha_{3}}{h} + 0.5 & 0 & \frac{\alpha_{3}}{h} + 0.5 & 0 & - \frac{4 \alpha_{3}^{2}}{h^{2}} + 1\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & - \frac{1}{h} & 0 & \frac{1}{h} & 0 & - \frac{8 \alpha_{3}}{h^{2}} & 0\end{array}\right]$$



In [73]:

    
D_p_L = StrainL*L
simplify(D_p_L)









    Out[73]:





$$\left[\begin{array}{cccccccccccc}0 & \frac{- \alpha_{3} + 0.5 h}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{\alpha_{3} + 0.5 h}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{- 4 \alpha_{3}^{2} + h^{2}}{h^{2} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & - \frac{\left(\alpha_{3} - 0.5 h\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{\left(\alpha_{3} + 0.5 h\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{\left(- 4 \alpha_{3}^{2} + h^{2}\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h^{2} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\- \frac{\left(\alpha_{3} - 0.5 h\right) \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{\left(\alpha_{3} + 0.5 h\right) \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{\left(- 4 \alpha_{3}^{2} + h^{2}\right) \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h^{2} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & - \frac{1}{h \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{1}{h \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & - \frac{8 \alpha_{3}}{h^{2} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0\\\frac{\left(\alpha_{3} - 0.5 h\right) \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & - \frac{\left(\alpha_{3} + 0.5 h\right) \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{\left(4 \alpha_{3}^{2} - h^{2}\right) \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h^{2} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\frac{\left(\alpha_{3} - 0.5 h\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} - \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{- \left(\alpha_{3} + 0.5 h\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{- 8 \alpha_{3} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} + \left(4 \alpha_{3}^{2} - h^{2}\right) \frac{\partial}{\partial \alpha_{3}} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h^{2} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{\left(\alpha_{3} - 0.5 h\right) \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & \frac{- \alpha_{3} + 0.5 h}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & - \frac{\left(\alpha_{3} + 0.5 h\right) \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & \frac{\alpha_{3} + 0.5 h}{h \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & \frac{\left(4 \alpha_{3}^{2} - h^{2}\right) \frac{\partial}{\partial \alpha_{1}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h^{2} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & \frac{- 4 \alpha_{3}^{2} + h^{2}}{h^{2} \operatorname{H_{1}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}\\0 & 0 & 0 & 0 & 0 & 0 & \frac{\left(\alpha_{3} - 0.5 h\right) \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & - \frac{\left(\alpha_{3} + 0.5 h\right) \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0 & \frac{\left(4 \alpha_{3}^{2} - h^{2}\right) \frac{\partial}{\partial \alpha_{2}} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}}{h^{2} \operatorname{H_{3}}{\left (\alpha_{1},\alpha_{2},\alpha_{3} \right )}} & 0\end{array}\right]$$



In [ ]:

    
h = 0.5
exp=(0.5-alpha3/h)*(1-(2*alpha3/h)**2)#/(1+alpha3*0.8)
p02=integrate(exp, (alpha3, -h/2, h/2))
integral = expand(simplify(p02))
integral

Mass matrix



In [84]:

    
rho=Symbol('rho')
B_h=zeros(3,12)
B_h[0,0]=1
B_h[1,4]=1
B_h[2,8]=1

M=simplify(rho*P.T*B_h.T*G_up*B_h*P)
M









    Out[84]:





$$\left[\begin{array}{cccccccccccc}\rho & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & \rho & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \rho & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$$