User:Egm6936.s09/Dimensional analysis

Write equations from the French thesis here.

Ref: Nordstrom et al., Influence of sheet dimensions on curl of paper, Journal of Pulp and Paper Science, Vol.24, No.1, pp.18-25, Jan 1998.

Eq.(2) in Nordstrom et al. (1998) can be written as follows:

\displaystyle \boldsymbol \epsilon =  \epsilon_{\alpha \beta} \boldsymbol e_\alpha \otimes \boldsymbol e_\beta  \,\,\,(1) $$ where $$\displaystyle \boldsymbol e_\alpha$$ is the cartesian basis vector along the coordinate axis $$\displaystyle x_\alpha$$, for $$\displaystyle \alpha = 1, 2, 3$$.

Using Kirhoff's assumptions and encountering effects of large deformation, components of Green Lagrange tensor are given as below:



\displaystyle \varepsilon _{\alpha \beta } = \frac{1}{2}\left( {u_{\alpha ,\beta }  + u_{\beta ,\alpha }  + w_{,\alpha } w_{,\beta } } \right) + z\kappa _{\alpha \beta }  = \varepsilon _{\alpha \beta }^M  + \varepsilon _{\alpha \beta }^B  \,\,\,(2) $$

We divided components of strain tensor into membrance strain and bending strain as:

\displaystyle \varepsilon _{\alpha \beta }^M = \frac{1}{2}\left( {u_{\alpha ,\beta }  + u_{\beta ,\alpha }  w_{,\alpha } w_{,\beta } } \right)\,\,\,(3) $$



\displaystyle \varepsilon _{\alpha \beta }^B = z\kappa _{\alpha \beta }   \,\,\,(4) $$ Thus

\displaystyle {\boldsymbol{\varepsilon = }}\varepsilon _{\alpha \beta } {\boldsymbol{e}}_\alpha   \otimes {\boldsymbol{e}}_\beta   = \frac{1}{2}\left( {u_{\alpha ,\beta }  + u_{\beta ,\alpha }  + w_{,\alpha } w_{,\beta } } \right){\boldsymbol{e}}_\alpha   \otimes {\boldsymbol{e}}_\beta   + z\kappa _{\alpha \beta } {\boldsymbol{e}}_\alpha   \otimes {\boldsymbol{e}}_\beta   = \varepsilon _{\alpha \beta }^M {\boldsymbol{e}}_\alpha   \otimes {\boldsymbol{e}}_\beta   + \varepsilon _{\alpha \beta }^B {\boldsymbol{e}}_\alpha   \otimes {\boldsymbol{e}}_\beta   = {\boldsymbol{\varepsilon }}^{\boldsymbol{M}}  + {\boldsymbol{\varepsilon }}^{\boldsymbol{B}}  = {\boldsymbol{\varepsilon }}^{\boldsymbol{M}}  + z{\boldsymbol{\kappa }}  \,\,\, (5) $$

with

\displaystyle {\boldsymbol{\varepsilon }}^{\boldsymbol{M}} = \varepsilon _{\alpha \beta }^M {\boldsymbol{e}}_\alpha   \otimes {\boldsymbol{e}}_\beta \,\,\, (6) $$



\displaystyle {\boldsymbol{\varepsilon }}^{\boldsymbol{B}} =  z{\boldsymbol{\kappa }}  = \varepsilon _{\alpha \beta }^B {\boldsymbol{e}}_\alpha   \otimes {\boldsymbol{e}}_\beta  \,\,\,(7) $$



\displaystyle {\boldsymbol{\kappa }} = \kappa _{\alpha \beta } {\boldsymbol{e}}_\alpha  \otimes {\boldsymbol{e}}_\beta   =  - w_{,\alpha \beta } {\boldsymbol{e}}_\alpha   \otimes {\boldsymbol{e}}_\beta    \,\,\,   (8) $$

where $${\boldsymbol{\varepsilon }}^{\boldsymbol{M}}$$,$${\boldsymbol{\varepsilon }}^{\boldsymbol{B}}$$, and $${\boldsymbol{\kappa}}$$ are membrance induced tensor, bending induced tensor, and curvature respectively.

Hooke's Law describes relation of stress and strain has the following form:

\displaystyle {\boldsymbol{\sigma = C}}:{\boldsymbol{\varepsilon }} \,\,\, (9) $$

where stress tensor

\displaystyle \sigma _{\alpha \beta } {\boldsymbol{e}}_\alpha  \otimes {\boldsymbol{e}}_\beta \,\,\,(10) $$

and stiffness tensor

\displaystyle {\boldsymbol{C}} = C_{\alpha \beta \gamma \delta } {\boldsymbol{e}}_\alpha  \otimes {\boldsymbol{e}}_\beta   \otimes {\boldsymbol{e}}_\gamma   \otimes {\boldsymbol{e}}_\beta \,\,\,(11) $$

or stress-strain relation written in component forms is:

\displaystyle \sigma _{\alpha \beta } = C_{\alpha \beta \gamma \delta } \varepsilon _{\gamma \delta } \,\,\,(12) $$

The membrance forces are:

\displaystyle {\boldsymbol{N}} = \int {{\boldsymbol{\sigma }}dz{\boldsymbol{ = }}\int {{\boldsymbol{C}}:{\boldsymbol{\varepsilon }}dz = } } N_{\alpha \beta } {\boldsymbol{e}}_\alpha  \otimes {\boldsymbol{e}}_\beta \,\,\ (13) $$

or

\displaystyle {\boldsymbol{N}} = \int {{\boldsymbol{C}}:\left( {{\boldsymbol{\varepsilon }}^{\boldsymbol{M}} + {\boldsymbol{\varepsilon }}^{\boldsymbol{B}} } \right)dz}  = \int {{\boldsymbol{C}}:{\boldsymbol{\varepsilon }}^{\boldsymbol{M}} dz}  + \int {{\boldsymbol{C}}:{\boldsymbol{\varepsilon }}^{\boldsymbol{B}} dz = } \left( {\int {{\boldsymbol{C}}dz} } \right):{\boldsymbol{\varepsilon }}^{\boldsymbol{M}}  + \left( {\int {{\boldsymbol{C}}zdz} } \right):{\boldsymbol{\kappa }} = {\boldsymbol{A}}:{\boldsymbol{\varepsilon }}^{\boldsymbol{M}}  + {\boldsymbol{B}}:{\boldsymbol{\kappa }} \,\,\, (14) $$

where

\displaystyle {\boldsymbol{A}} = \left( {\int {{\boldsymbol{C}}dz} } \right) = A_{\alpha \beta \gamma \delta } {\boldsymbol{e}}_\alpha  \otimes {\boldsymbol{e}}_\beta   \otimes {\boldsymbol{e}}_\gamma   \otimes {\boldsymbol{e}}_\beta   = \left( {\int {C_{\alpha \beta \gamma \delta } dz} } \right){\boldsymbol{e}}_\alpha   \otimes {\boldsymbol{e}}_\beta   \otimes {\boldsymbol{e}}_\gamma   \otimes {\boldsymbol{e}}_\beta \,\,\,(15) $$



\displaystyle {\boldsymbol{B}} = \left( {\int {{\boldsymbol{C}}zdz} } \right) = B_{\alpha \beta \gamma \delta } {\boldsymbol{e}}_\alpha  \otimes {\boldsymbol{e}}_\beta   \otimes {\boldsymbol{e}}_\gamma   \otimes {\boldsymbol{e}}_\beta   = \left( {\int {C_{\alpha \beta \gamma \delta } zdz} } \right){\boldsymbol{e}}_\alpha   \otimes {\boldsymbol{e}}_\beta   \otimes {\boldsymbol{e}}_\gamma   \otimes {\boldsymbol{e}}_\beta \,\,\,(16) $$

Thus,

\displaystyle N_{\alpha \beta } = A_{\alpha \beta \gamma \delta } \varepsilon _{\gamma \delta }^M  + B_{\alpha \beta \gamma \delta } \kappa _{\alpha \beta } \,\,\,(17) $$

Similarly, bending moments are:

\displaystyle {\boldsymbol{M}} = \int {{\boldsymbol{\sigma }}zdz{\boldsymbol{ = }}\int {{\boldsymbol{C}}:{\boldsymbol{\varepsilon }}zdz = } } M_{\alpha \beta } {\boldsymbol{e}}_\alpha  \otimes {\boldsymbol{e}}_\beta \,\,\,(18) $$ or

\displaystyle {\boldsymbol{M}} = \int {{\boldsymbol{C}}:\left( {{\boldsymbol{\varepsilon }}^{\boldsymbol{M}} + {\boldsymbol{\varepsilon }}^{\boldsymbol{B}} } \right)zdz}  = \int {{\boldsymbol{C}}:{\boldsymbol{\varepsilon }}^{\boldsymbol{M}} zdz}  + \int {{\boldsymbol{C}}:{\boldsymbol{\varepsilon }}^{\boldsymbol{B}} zdz = } \left( {\int {{\boldsymbol{C}}zdz} } \right):{\boldsymbol{\varepsilon }}^{\boldsymbol{M}}  + \left( {\int {{\boldsymbol{C}}z^2 dz} } \right):{\boldsymbol{\kappa }} = {\boldsymbol{B}}:{\boldsymbol{\varepsilon }}^{\boldsymbol{M}}  + {\boldsymbol{D}}:{\boldsymbol{\kappa }} \,\,\,(19) $$

where

\displaystyle {\boldsymbol{D}} = D_{\alpha \beta \gamma \delta } {\boldsymbol{e}}_\alpha  \otimes {\boldsymbol{e}}_\beta   \otimes {\boldsymbol{e}}_\gamma   \otimes {\boldsymbol{e}}_\beta   = \left( {\int {C_{\alpha \beta \gamma \delta } z^2 dz} } \right){\boldsymbol{e}}_\alpha   \otimes {\boldsymbol{e}}_\beta   \otimes {\boldsymbol{e}}_\gamma   \otimes {\boldsymbol{e}}_\beta \,\,\, (20) $$

whence

\displaystyle M_{\alpha \beta } = B_{\alpha \beta \gamma \delta } \varepsilon _{\gamma \delta }^M  + D_{\alpha \beta \gamma \delta } \kappa _{\alpha \beta } \,\,\, (21) $$

Combining equations (14) and (19) into matrix form:

\displaystyle \left[ \begin{array}{l} {\boldsymbol{N}} \\ {\boldsymbol{M}} \\ \end{array} \right] = \left[ {\begin{array}{*{20}c} {\boldsymbol{A}} & {\boldsymbol{B}} \\ {\boldsymbol{B}} & {\boldsymbol{D}} \\ \end{array}} \right]:\left[ \begin{array}{l} {\boldsymbol{\varepsilon }}^{\boldsymbol{M}} \\ {\boldsymbol{\kappa }} \\ \end{array} \right] \,\,\, (22) $$

$$   \begin{align} G   & = x_1^{g_1} x_2^{g_2} \cdots x_q^{g_q} F ( \Pi_{q+1} \Pi_{q+2} \cdots \Pi_{p}) \\ (G) & =   L^{\rho} M^{m} T^{t} \Theta^{\theta} \\ (X_i) & =   L^{l_i} M^{m_i} T^{t_i} \Theta^{\theta_i} \end{align} $$

$$\begin{array}{|c|c|c|c|c|c|} & L & M & T & \Theta & \\ \hline G & l & m & t & \theta & \\ x_1 & l_1 & m_1 & t_1 & \theta_1 & e_1 \\ x_2 & l_2 & m_2 & t_2 & \theta_2 & e_2 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ x_p & l_p & m_p & t_p & \theta_p & e_p \\ \end{array}$$

$$   \begin{align} G_n & =   x_1^{e_1} x_2^{e_2} \cdots x_p^{e_p} \\ l   & = l_i\,e_i \\ m   & = m_i\,e_i \\ t   & = t_i\,e_i \\ \theta & =   \theta_i\,e_i \\ l_j\,e_j & =  l - (l_k\,e_k) \\ m_j\,e_j & =  m - (m_k\,e_k) \\ t_j\,e_j & =   t- (t_k\,e_k) \\ \theta_j\,e_j & =  \theta - (\theta_k\,e_k) \\ e_j & =  f_j (l,m,t,\theta,e_5,e_6, \cdots,e_p) = g_j (l,m,t,\theta) + h_j (e_5, \cdots,e_p) \\ \end{align} $$

$$  \begin{align} G_n & = x_1^{g_1+h_1} x_2^{g_2+h_2} x_3^{g_3+h_3} x_4^{g_4+h_4} x_5^{e_5}x_6^{e_6} \cdots x_p^{e_p} \\ & = x_1^{g_1} x_2^{g_2} \cdots x_4^{g_4} \left \{ x_1^{h_1 (e_5, e_6, \cdots, e_p)}, x_2^{h_2} x_3^{h_3} x_4^{h_4} x_5^{e_5} \cdots x_p^{e_p} \right \} \\ & = x_1^{g_1} x_2^{g_2} x_3^{g_3} x_4^{g_4} \left \lbrace (x_1 x_2 x_3 x_4 x_5)^{e_5} (x_1 x_2 x_3 x_4 x_6)^{e_6} \cdots (x_1 x_2 x_3 x_4)^{e_p} \right \rbrace \end{align} $$

$$  \begin{align} G_n = x_1^{g_1(lmt)} x_2^{g_2(lmt)} x_3^{g_3(lmt)} x_4^{g_4(lmt)} \pi_5 (x_1 x_2 x_3 x_4 x_5) \cdots \pi_p (x_1 x_2 x_3 x_4 x_p) \end{align} $$

$$  \begin{align} G_n = x_1^{g_1} x_2^{g_2} x_3^{g_3} x_4^{g_4} \end{align}

$$ $$  \begin{align} \pi_5 = L^{o} M^{o} T^{o} \Theta^{o} \end{align} $$

$$  \begin{align} G   = \sum_{n} G_n \end{align} $$

$$  \begin{align} F  = \sum_{n} (\pi_5 \pi_6 \cdots \pi_p)_n \end{align} $$

$$  \begin{align} G_n = x_1^{g_1} x_2^{g_2} x_3^{g_3} x_4^{g_4} F(\pi_5 \pi_6 \cdots \pi_p) \end{align} $$

$$\begin{array}{|c|c|c|c|c|} & L & M & T & \\ \hline G & l& m & t & \\ \rho & -3 & +1 & 0 & e_1 \\ u & +1 & 0 & -1 & e_2 \\ L & +1 & 0 & 0 & e_3 \\ \mu & -1 & +1 & -1 & e_4 \\ g & +1 & 0 & -2 & e_5 \\ \sigma & 0 & 1 & -2 & e_6 \\ \end{array}$$

$$  \begin{align} l   & = - 3 e_1 + e_2 + e_3 - e_4 + e_5 \\ m  & = e_1 + e_4 + e_6 \\ t   & = - e_2 - e_4 - 2 e_5 - 2 e_6 \\ e_1 & = m - e_4 - e_6 \\ e_2 & = - t - e_4 - 2 e_5 - 2 e_6 \\ e_3 & = l + 3 m + t - e_4 + e_5 - e_6 \\ \end{align} $$

$$  \begin{align} G  & = \rho^{m - e_4 - e_6} u^{- t - e_4 - 2 e_5 - 2 e_6} L^{l + 3 m + t - e_4 + e_5 - e_6} \mu^{e_4} g^{e_5} \sigma^{e_6} \\ & = \rho^{m } u^{- t} L^{l + 3 m + t} \left ( \frac{\mu}{u L \rho} \right )^{e_4} \left ( \frac{gL}{u^{2}} \right )^{e_5} \left ( \frac{\sigma}{\rho u^{2} L} \right )^{e_6} \end{align} $$

$$  \begin{align} R  & = \rho \frac{u^{2}}{2} S F (Re, Fr, Wb) \\ Re  & = \frac{uL\rho}{\mu}, \quad Fr = \frac{u}{\sqrt{gL}} \quad and \quad Wb = \frac{\rho u^{2} L}{\sigma} \end{align} $$

$$  \begin{align} k_d & = \frac{d_2}{d_1} = const \\ u_2 & = k_u\,u_1 \\ t_2 & = k_t\,t_1 \\ u_2 & = \frac{d(d_2)}{d(t_2)} \\ k_u\,u_1 & = \frac{k_d}{k_t} \frac{d(d_1)}{d(t_1)} \\ u_1 & = \frac{d(d_1)}{d(L_1)} \\ k_u & = \frac{k_d}{k_t} \\ k_Y & = \frac{k_u}{k_t} = \frac{k_d}{k_t^{2}} = \frac{k_u^{2}}{k_d} \\ k_F & = k_m\,k_Y = k_G = k_p = k_\nu \\ \end{align} $$

$$  \begin{align} & E_1: \quad \rho_1 \frac{\partial{u_{1i}}}{\partial{t}} + \rho_1 u_{1j} \frac{\partial{u_{1j}}}{\partial{x_{1j}}} = - \frac{\partial{p1}}{\partial{x_{1i}}} + \rho_i g_{1i} + \frac{\partial{\tau_{1ij}}}{\partial{x_{1j}}} \\ & (g_{11} = 0, \quad g_{12} = 0, \quad g_{13} = -g) \\ & \underset{Initial}{\underset{\uparrow}{I_t + I_s}} = \underset{Pression}{\underset{\uparrow}{P}} + \underset{Gravity}{\underset{\uparrow}{G}} + \underset{Force}{\underset{\uparrow}{F}} \quad I_t = \rho \frac{\partial{u_1}}{\partial{t}}, \quad P = - \frac{\partial{p1}}{\partial{x_1i}} \\ & E_2: \quad k_{\rho} \frac{k_u}{k_t} I_t + k_{\rho} \frac{k_u^{2}}{k_d} I_s = \frac{k_p}{k_d} P + k_{\rho} k_{g} G + \frac{k_u k_\mu}{k_d^{2}} F \\ & \tau_{ij} = \lambda \frac{\partial{u_k}}{\partial{x_k}} \delta_{ij} + \mu \left ( \frac{\partial{u_i}}{\partial{x_j}} + \frac{\partial{u_j}}{\partial{x_i}} \right ) \\ \end{align} $$

$$  \begin{align} 1^{o}) I_t\,-\,I_s\,:  \frac{k_\rho k_u}{k_t} & = \frac{k_\rho k_u^{2}}{k_d} \\   k_u & = \frac{k_d}{k_t} \\   2^{o}) I_s\,-\,P\,: \frac{k_\rho k_u^{2}}{k_d} & = \frac{k_p}{k_d} \\ \frac{k_u^{2} k_\rho}{k_p} & = 1 \\ \end{align} $$

$$  \begin{align} \frac{u_1^{2} \rho_1}{P_1} & = \frac{u_2^{2} \rho_2}{P_2} \\ a^{2} & = \gamma \frac{p}{\rho} \\ k_a^{2} & = k_\gamma \frac{k_p}{k_\rho} \\ \frac{k_\gamma k_u^{2}}{k_a} & = 1 \\ \frac{u_1}{a_1} & = \frac{u_2}{a_2} \\ \end{align} $$

$$  \begin{align} 3^{o}) I_s\,-\,G\,:   k_\rho \frac{k_u^{2}}{k_d} & = k_\rho k_g  \\   \frac{k_u^{2}}{k_g k_d} & = 1 \\   4^{o})  I_s\,-\,F\,: k_\rho \frac{k_u^{2}}{k_d} & = \frac{k_u k_\mu}{k_d^{2}} \\ \frac{k_u k_\rho k_d}{k_\mu} & = 1 \\ \end{align} $$

$$  \begin{align} F_r & = \frac{u}{\sqrt{gD}} \,, \, R_e = \frac{\rho u d}{\mu} = \frac{u d}{\nu} \\ \rho \frac{dh}{dt} & = \frac{dp}{dt} + \frac{\partial}{\partial{x_i}} k \frac{\partial{T}}{\partial{x_i}} + \phi \\ \phi & = \lambda \left ( \frac{\partial{u_k}}{\partial{x_k}} \right )^{2} + \frac{u}{2} \left (  \frac{\partial{u_i}}{\partial{x_j}} + \frac{\partial{u_j}}{\partial{x_i}} \right )^{2} \\ E  & = P\, + \, T\, + \, D \\ \end{align} $$

$$  \begin{align} & 1^{o}) \quad E\,-\,P\,:  \quad k_\gamma = 1 \quad \gamma = \frac{C_p}{C_v} \\   & 2^{o}) \quad E\,-\,T\,: \quad \frac{k_\rho k_{C_p} k_u k_d}{k_k} = 1 \,, \, Pe = \frac{\rho C_p VD}{k} = \frac{C_p \mu}{k} \,. \, \frac{\rho VD}{\mu} = Pr \,. \, Re \\ & 3^{o}) \quad E\,-\,D\,:  \quad \frac{k_\rho k_{C_p} k_T k_d}{k_\mu k_u} = 1 \,, \, E_k = \frac{C_p T}{u^{2}} \\   \end{align} $$

$$  \begin{align} W_D & = \frac{\rho u^{2} d}{\sigma} \\ k_u & = \sqrt{k_d} \\ k_\mu & = k_u \,. \, k_\rho \,. \, k_d \\ k_d & = \left ( \frac{k_\mu}{k_\rho} \right )^{\frac{2}{3}} \\ \end{align} $$