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Bold letters means matrix or vectors: A \boldsymbol A A
Bold letters with subscripts means the i i i j j j A \boldsymbol A A A i j \boldsymbol A_{ij} A ij
The k k k l l l i i i j j j A i j \boldsymbol A_{ij} A ij A \boldsymbol A A ( A i j ) k l (A_{ij})_{kl} ( A ij ) k l
0.1 Eq(14)
U ( ξ , η , ζ ) = ∑ i = 1 I 1 ∑ j = 1 J 1 ∑ k = 1 K 1 B i j k 1 P i ( ξ ) P j ( η ) P k ( ζ ) , V ( ξ , η , ζ ) = ∑ i = 1 I 2 ∑ j = 1 J 2 ∑ k = 1 K 3 B i j k 2 P i ( ξ ) P j ( η ) P k ( ζ ) , W ( ξ , η , ζ ) = ∑ i = 1 I 3 ∑ j = 1 J 3 ∑ k = 1 K 3 B i j k 3 P i ( ξ ) P j ( η ) P k ( ζ ) . \begin{array}{c}
U(\xi, \eta, \zeta)=\sum_{i=1}^{I_1} \sum_{j=1}^{J_1} \sum_{k=1}^{K_1} B^1_{i j k} P_{i}(\xi) P_{j}(\eta) P_{k}(\zeta), \\
V(\xi, \eta, \zeta)=\sum_{i=1}^{I_2} \sum_{j=1}^{J_2} \sum_{k=1}^{K_3} B^2_{i j k} P_{i}(\xi) P_{j}(\eta) P_{k}(\zeta), \\
W(\xi, \eta, \zeta)=\sum_{i=1}^{I_3} \sum_{j=1}^{J_3} \sum_{k=1}^{K_3} B^3_{i j k} P_{i}(\xi) P_{j}(\eta) P_{k}(\zeta).
\end{array} U ( ξ , η , ζ ) = ∑ i = 1 I 1 ∑ j = 1 J 1 ∑ k = 1 K 1 B ij k 1 P i ( ξ ) P j ( η ) P k ( ζ ) , V ( ξ , η , ζ ) = ∑ i = 1 I 2 ∑ j = 1 J 2 ∑ k = 1 K 3 B ij k 2 P i ( ξ ) P j ( η ) P k ( ζ ) , W ( ξ , η , ζ ) = ∑ i = 1 I 3 ∑ j = 1 J 3 ∑ k = 1 K 3 B ij k 3 P i ( ξ ) P j ( η ) P k ( ζ ) . We use the following relation ( i , j , k ) = ( 1 , 1 , 1 ) , ( 1 , 1 , 2 ) , … , ( 1 , 1 , K ) , ( 1 , 2 , 1 ) , … , ( 1 , 2 , K ) , … , ( 1 , J , K ) , … , ( I , J , K ) → p = 1 , 2 , 3 … , I J K (i,j,k)=(1,1,1),(1,1,2),\ldots, (1,1,K), (1,2,1), \ldots, (1,2,K), \ldots, (1,J,K), \ldots, (I,J,K) \rightarrow p = 1,2,3 \ldots, IJK ( i , j , k ) = ( 1 , 1 , 1 ) , ( 1 , 1 , 2 ) , … , ( 1 , 1 , K ) , ( 1 , 2 , 1 ) , … , ( 1 , 2 , K ) , … , ( 1 , J , K ) , … , ( I , J , K ) → p = 1 , 2 , 3 … , I J K ( i , j , k ) (i,j,k) ( i , j , k ) p p p B i j k m , m = 1 , 2 , 3 B^m_{i j k}, m=1,2,3 B ij k m , m = 1 , 2 , 3 A m \boldsymbol A_m A m p p p ( A m ) p = B i j k m (\boldsymbol A_m)_p=B^m_{ijk} ( A m ) p = B ij k m A = ( A 1 , A 2 , A 3 ) \boldsymbol A = (\boldsymbol A_1,\boldsymbol A_2,\boldsymbol A_3) A = ( A 1 , A 2 , A 3 )
U ( ξ , η , ζ ) = ∑ i = 1 I 1 ∑ j = 1 J 1 ∑ k = 1 K 1 ( A 1 ) p P i ( ξ ) P j ( η ) P k ( ζ ) , V ( ξ , η , ζ ) = ∑ i = 1 I 2 ∑ j = 1 J 2 ∑ k = 1 K 3 ( A 2 ) p P i ( ξ ) P j ( η ) P k ( ζ ) , W ( ξ , η , ζ ) = ∑ i = 1 I 3 ∑ j = 1 J 3 ∑ k = 1 K 3 ( A 3 ) p P i ( ξ ) P j ( η ) P k ( ζ ) . \begin{array}{c}
U(\xi, \eta, \zeta)=\sum_{i=1}^{I_1} \sum_{j=1}^{J_1} \sum_{k=1}^{K_1} (A_1)_p P_{i}(\xi) P_{j}(\eta) P_{k}(\zeta), \\
V(\xi, \eta, \zeta)=\sum_{i=1}^{I_2} \sum_{j=1}^{J_2} \sum_{k=1}^{K_3} (A_2)_p P_{i}(\xi) P_{j}(\eta) P_{k}(\zeta),\\
W(\xi, \eta, \zeta)=\sum_{i=1}^{I_3} \sum_{j=1}^{J_3} \sum_{k=1}^{K_3} (A_3)_p P_{i}(\xi) P_{j}(\eta) P_{k}(\zeta).
\end{array} U ( ξ , η , ζ ) = ∑ i = 1 I 1 ∑ j = 1 J 1 ∑ k = 1 K 1 ( A 1 ) p P i ( ξ ) P j ( η ) P k ( ζ ) , V ( ξ , η , ζ ) = ∑ i = 1 I 2 ∑ j = 1 J 2 ∑ k = 1 K 3 ( A 2 ) p P i ( ξ ) P j ( η ) P k ( ζ ) , W ( ξ , η , ζ ) = ∑ i = 1 I 3 ∑ j = 1 J 3 ∑ k = 1 K 3 ( A 3 ) p P i ( ξ ) P j ( η ) P k ( ζ ) . ##Eq 16
Π = V max − T max → L = V max − T max \Pi=V_{\max }-T_{\max } \rightarrow L=V_{\max }-T_{\max } Π = V m a x − T m a x → L = V m a x − T m a x 0.2 Eq 17
∂ L ∂ B i j k 1 = 0 , ∂ L ∂ B i j k 2 = 0 , ∂ L ∂ B i j k 3 = 0. \frac{\partial L}{\partial B^1_{i j k}}=0, \frac{\partial L}{\partial B^2_{i j k}}=0, \frac{\partial L}{\partial B^3_{i j k}}=0 . ∂ B ij k 1 ∂ L = 0 , ∂ B ij k 2 ∂ L = 0 , ∂ B ij k 3 ∂ L = 0. By using the index transform, we have
∂ L ∂ ( A 1 ) p = 0 , ∂ L ∂ ( A 2 ) p = 0 , ∂ L ∂ ( A 3 ) p = 0. \frac{\partial L}{\partial (A_1)_p}=0, \frac{\partial L}{\partial (A_2)_p}=0, \frac{\partial L}{\partial (A_3)_p}=0 . ∂ ( A 1 ) p ∂ L = 0 , ∂ ( A 2 ) p ∂ L = 0 , ∂ ( A 3 ) p ∂ L = 0. 0.3 Eq21
[ Π 11 Π 12 Π 13 Π 21 Π 22 Π 23 Π 31 Π 32 Π 33 ] [ A 1 A 2 A 3 ] = [ 0 0 0 ] . \left[\begin{array}{ll}
\boldsymbol \Pi_{11} & \boldsymbol \Pi_{12} & \boldsymbol \Pi_{13} \\
\boldsymbol \Pi_{21} & \boldsymbol \Pi_{22} & \boldsymbol \Pi_{23}\\
\boldsymbol \Pi_{31} & \boldsymbol \Pi_{32} & \boldsymbol \Pi_{33}
\end{array}\right]
\left[\begin{array}{l}
{\boldsymbol A_1} \\
{\boldsymbol A_2} \\
{\boldsymbol A_3}
\end{array}\right]
=\left[\begin{array}{c}
{\boldsymbol 0} \\
{\boldsymbol 0} \\
{\boldsymbol 0}
\end{array}\right] . Π 11 Π 21 Π 31 Π 12 Π 22 Π 32 Π 13 Π 23 Π 33 A 1 A 2 A 3 = 0 0 0 . where
( Π i j ) p q = ∂ 2 L ∂ ( A i ) p ∂ ( A j ) q (\Pi_{ij})_{pq} = \frac{\partial^2 L}{\partial (A_i)_p \partial (A_j)_q} ( Π ij ) pq = ∂ ( A i ) p ∂ ( A j ) q ∂ 2 L In fact,
Π = K − ω 2 M \boldsymbol \Pi= \boldsymbol K -\omega^2 \boldsymbol M Π = K − ω 2 M and
( K i j ) p q = ∂ 2 V m a x ∂ ( A i ) p ∂ ( A j ) q (K_{ij})_{pq} = \frac{\partial^2 V_{\rm max}}{\partial (A_i)_p \partial (A_j)_q} ( K ij ) pq = ∂ ( A i ) p ∂ ( A j ) q ∂ 2 V max ω 2 ( M i j ) p q = ∂ 2 T m a x ∂ ( A i ) p ∂ ( A j ) q \omega^2(M_{ij})_{pq} = \frac{\partial^2 T_{\rm max}}{\partial (A_i)_p \partial (A_j)_q} ω 2 ( M ij ) pq = ∂ ( A i ) p ∂ ( A j ) q ∂ 2 T max Π 11 = ∂ L ∂ ( A 1 ) p = E c 2 λ ( 1 + v ) ∫ − 1 1 ∫ − 1 1 ∫ − 1 1 { v 1 − 2 v ( ε ˉ ξ ξ + ε ˉ η η + ε ˉ ζ ζ ) λ P l ( ξ ) ∂ P m ( η ) ∂ ( η ) P n ( ζ ) + λ ε ˉ η η P l ( ξ ) ∂ P m ( η ) ∂ ( η ) P n ( ζ ) + 1 2 [ ε ˉ ξ η ∂ P l ( ξ ) ∂ ( ξ ) P m ( η ) P n ( ζ ) + λ γ ε ˉ η ζ P l ( ξ ) P m ( η ) ∂ P n ( ζ ) ∂ ( ζ ) ] } d ξ d η d ζ + E ˉ c 2 λ ( 1 + v ˉ ) ∫ − 1 1 ∫ − 1 1 ∫ 1 1 + 2 c ˉ c { v ˉ 1 − 2 v ˉ ( ε ˉ ξ ξ + ε ˉ η η + ε ˉ ζ ζ ) ⋅ λ P l ( ξ ) ∂ P m ( η ) ∂ ( η ) P n ( ζ ) + λ ε ˉ η η P l ( ξ ) ∂ P m ( η ) ∂ ( η ) P n ( ζ ) + 1 2 [ ε ˉ ξ η ∂ P l ( ξ ) ∂ ( ξ ) P m ( η ) P n ( ζ ) + λ γ ε ˉ η ζ P l ( ξ ) P m ( η ) ∂ P n ( ζ ) ∂ ( ζ ) ] } d ξ d η d ζ − { ρ 8 a b c ω 2 ∫ − 1 1 ∫ − 1 1 ∫ − 1 1 V [ P l ( ξ ) P m ( η ) P n ( ζ ) ] d ξ d η d ζ + ρ ˉ 8 a b c ω 2 ∫ − 1 1 ∫ − 1 1 ∫ 1 1 + 2 c ˉ c V [ P l ( ξ ) P m ( η ) P n ( ζ ) ] d ξ d η d ζ } \begin{aligned}
\Pi_{11}=\frac{\partial L}{\partial (A_1)_p}=& \frac{E c}{2 \lambda(1+v)} \int_{-1}^{1} \int_{-1}^{1} \int_{-1}^{1}\left\{\frac { v } { 1 - 2 v } \left(\bar{\varepsilon}_{\xi \xi}+\bar{\varepsilon}_{\eta \eta}\right.\right.\\
&\left.+\bar{\varepsilon}_{\zeta \zeta}\right) \lambda P_{l}(\xi) \frac{\partial P_{m}(\eta)}{\partial(\eta)} P_{n}(\zeta)+\lambda \bar{\varepsilon}_{\eta \eta} P_{l}(\xi) \frac{\partial P_{m}(\eta)}{\partial(\eta)} P_{n}(\zeta) \\
&+\frac{1}{2}\left[\bar{\varepsilon}_{\xi \eta} \frac{\partial P_{l}(\xi)}{\partial(\xi)} P_{m}(\eta) P_{n}(\zeta)\right.\\
&\left.\left.+\frac{\lambda}{\gamma} \bar{\varepsilon}_{\eta \zeta} P_{l}(\xi) P_{m}(\eta) \frac{\partial P_{n}(\zeta)}{\partial(\zeta)}\right]\right\} d \xi d \eta d \zeta \\
&+\frac{\bar{E} c}{2 \lambda(1+\bar{v})} \int_{-1}^{1} \int_{-1}^{1} \int_{1}^{1+\frac{2 \bar{c}}{c}}\left\{\frac{\bar{v}}{1-2 \bar{v}}\left(\bar{\varepsilon}_{\xi \xi}+\bar{\varepsilon}_{\eta \eta}+\bar{\varepsilon}_{\zeta \zeta}\right)\right.\\
& \cdot \lambda P_{l}(\xi) \frac{\partial P_{m}(\eta)}{\partial(\eta)} P_{n}(\zeta)+\lambda \bar{\varepsilon}_{\eta \eta} P_{l}(\xi) \frac{\partial P_{m}(\eta)}{\partial(\eta)} P_{n}(\zeta) \\
&+\frac{1}{2}\left[\bar{\varepsilon}_{\xi \eta} \frac{\partial P_{l}(\xi)}{\partial(\xi)} P_{m}(\eta) P_{n}(\zeta)\right.\\
&\left.\left.+\frac{\lambda}{\gamma} \bar{\varepsilon}_{\eta \zeta} P_{l}(\xi) P_{m}(\eta) \frac{\partial P_{n}(\zeta)}{\partial(\zeta)}\right]\right\} d \xi d \eta d \zeta \\
&-\left\{\frac{\rho}{8} a b c \omega^{2} \int_{-1}^{1} \int_{-1}^{1} \int_{-1}^{1} V\left[P_{l}(\xi) P_{m}(\eta) P_{n}(\zeta)\right] d \xi d \eta d \zeta\right.\\
&\left.+\frac{\bar{\rho}}{8} a b c \omega^{2} \int_{-1}^{1} \int_{-1}^{1} \int_{1}^{1+\frac{2 \bar{c}}{c}} V\left[P_{l}(\xi) P_{m}(\eta) P_{n}(\zeta)\right] d \xi d \eta d \zeta\right\}
\end{aligned} Π 11 = ∂ ( A 1 ) p ∂ L = 2 λ ( 1 + v ) E c ∫ − 1 1 ∫ − 1 1 ∫ − 1 1 { 1 − 2 v v ( ε ˉ ξ ξ + ε ˉ η η + ε ˉ ζ ζ ) λ P l ( ξ ) ∂ ( η ) ∂ P m ( η ) P n ( ζ ) + λ ε ˉ η η P l ( ξ ) ∂ ( η ) ∂ P m ( η ) P n ( ζ ) + 2 1 [ ε ˉ ξ η ∂ ( ξ ) ∂ P l ( ξ ) P m ( η ) P n ( ζ ) + γ λ ε ˉ η ζ P l ( ξ ) P m ( η ) ∂ ( ζ ) ∂ P n ( ζ ) ] } d ξ d η d ζ + 2 λ ( 1 + v ˉ ) E ˉ c ∫ − 1 1 ∫ − 1 1 ∫ 1 1 + c 2 c ˉ { 1 − 2 v ˉ v ˉ ( ε ˉ ξ ξ + ε ˉ η η + ε ˉ ζ ζ ) ⋅ λ P l ( ξ ) ∂ ( η ) ∂ P m ( η ) P n ( ζ ) + λ ε ˉ η η P l ( ξ ) ∂ ( η ) ∂ P m ( η ) P n ( ζ ) + 2 1 [ ε ˉ ξ η ∂ ( ξ ) ∂ P l ( ξ ) P m ( η ) P n ( ζ ) + γ λ ε ˉ η ζ P l ( ξ ) P m ( η ) ∂ ( ζ ) ∂ P n ( ζ ) ] } d ξ d η d ζ − { 8 ρ ab c ω 2 ∫ − 1 1 ∫ − 1 1 ∫ − 1 1 V [ P l ( ξ ) P m ( η ) P n ( ζ ) ] d ξ d η d ζ + 8 ρ ˉ ab c ω 2 ∫ − 1 1 ∫ − 1 1 ∫ 1 1 + c 2 c ˉ V [ P l ( ξ ) P m ( η ) P n ( ζ ) ] d ξ d η d ζ }