Differential Geometry for Shell Kinematics

  1. Differential Forms
    1. First Fundamental Form
    2. Second Fundamental Form
    3. Third Fundamental Form
    4. The Metric Tensor as a function of the fundamental forms
  2. Shell Kinematics

Differential Forms

First Fundamental Form

The first fundamental form is calculated using

aαβ=⟨aα,aβ⟩ a_{\alpha\beta}=\langle a_\alpha,a_\beta \rangle

Second Fundamental Form

Third Fundamental Form

The Metric Tensor as a function of the fundamental forms

gα=aα−ξ3bαλaα g_\alpha=a_\alpha-\xi^3 b^\lambda_\alpha a_\alpha g3=∂φ∂ξ3=a3 g_3=\frac{\partial \varphi}{\partial \xi^3}=a_3 gαβ=⟨(aα−ξ3bαλaλ),(aβ−ξ3bβμaμ)⟩=⟨aα,aβ⟩−ξ3bαλ⟨aλ,aβ⟩−ξ3bβμ⟨aμ,aα⟩+(ξ3)2bαλbβμ⟨aλ,aμ⟩=aαβ−ξ3bαλaλβ−ξ3bβμaμα+(ξ3)2bαλbβμaλμ=aαβ−ξ3bβα−ξ3bαβ+(ξ3)2bαλbλβ=aαβ−2ξ3bαβ+(ξ3)2cαβ\begin{aligned} g_{\alpha \beta}&= \langle (a_\alpha-\xi^3 b^\lambda_\alpha a_\lambda),(a_\beta-\xi^3 b^\mu_\beta a_\mu) \rangle \\ &=\langle a_\alpha,a_\beta \rangle-\xi^3 b^\lambda_\alpha \langle a_\lambda,a_\beta \rangle -\xi^3 b^\mu_\beta\langle a_\mu,a_\alpha \rangle+(\xi^3)^2 b^\lambda_\alpha b^\mu_\beta\langle a_\lambda,a_\mu \rangle\\ &=a_{\alpha\beta}-\xi^3 b^\lambda_\alpha a_{\lambda\beta}-\xi^3 b^\mu_\beta a_{\mu\alpha} +(\xi^3)^2 b^\lambda_\alpha b^\mu_\beta a_{\lambda\mu} \\ &=a_{\alpha\beta}-\xi^3 b_{\beta\alpha}-\xi^3 b_{\alpha\beta} +(\xi^3)^2 b^\lambda_\alpha b_{\lambda\beta} \\ &=a_{\alpha\beta}-2\xi^3 b_{\alpha\beta}+(\xi^3)^2 c_{\alpha\beta} \end{aligned} gα3=⟨gα,g3⟩=0 g_{\alpha 3}=\langle g_\alpha,g_3 \rangle=0 g33=⟨g3,g3⟩=1 g_{33}=\langle g_3,g_3 \rangle=1

Shell Kinematics

U(ξ1,ξ2,ξ3)=u(ξ1,ξ2)+ξ3θλ(ξ1,ξ2)aλ(ξ1,ξ2) U(\xi^1,\xi^2,\xi^3)=u(\xi^1,\xi^2)+\xi^3 \theta_\lambda(\xi^1,\xi^2)a^\lambda(\xi^1,\xi^2) eij=12(⟨gi,U,j⟩+⟨gj,U,i⟩) e_{ij}=\frac 1 2 \left ( \langle g_i,U_{,j} \rangle+\langle g_j,U_{,i} \rangle \right ) ∂u∂ξα=∂∂ξα(uλaλ+u3a3)=uλ∣αaλ+bαλuλa3+u3,αa3+u3a3,α=uλ∣αaλ−bλαu3aλ+(u3,α+bαλuλ)a3=(uλ∣α−bλαu3)aλ+(u3,α+bαλuλ)a3\begin{aligned} \frac{\partial u}{\partial\xi^\alpha}&= \frac{\partial}{\partial\xi^\alpha}(u_\lambda a^\lambda+u_3a_3) \\ &=u_{\lambda|\alpha}a^\lambda+b^\lambda_\alpha u_\lambda a_3+u_{3,\alpha}a_3+u_3 a_{3,\alpha}\\ &=u_{\lambda|\alpha}a^\lambda-b_{\lambda\alpha}u_3 a^\lambda +(u_{3,\alpha}+b^\lambda_\alpha u_\lambda)a_3 \\ &=(u_{\lambda|\alpha}-b_{\lambda\alpha}u_3)a^\lambda+(u_{3,\alpha}+b^\lambda_\alpha u_\lambda)a_3 \end{aligned} ∂∂ξα(θλaλ)=θλ∣αaλ+bαλθλa3 \frac{\partial}{\partial \xi^\alpha} (\theta_\lambda a^\lambda) =\theta_{\lambda|\alpha} a^\lambda+b^\lambda_\alpha \theta_\lambda a_3 ∂U∂ξα=∂u∂ξα+ξ3∂(θλaλ)∂ξα=(uλ∣α−bλαu3+ξ3θλ∣α)aλ+(u3,α+bαλuλ+ξ3bαλθλ)a3\begin{aligned} \frac{\partial U}{\partial \xi^\alpha} &= \frac{\partial u}{\partial \xi^\alpha} +\xi^3 \frac{\partial (\theta_\lambda a^\lambda)}{\partial \xi^\alpha} \\ &=(u_{\lambda|\alpha}-b_{\lambda\alpha}u_3+\xi^3 \theta_{\lambda|\alpha})a^\lambda +(u_{3,\alpha}+b^\lambda_\alpha u_\lambda+\xi^3 b^\lambda_\alpha \theta_\lambda)a_3 \end{aligned} ∂U∂ξ3=θλaλ \frac{\partial U}{\partial \xi^3}=\theta_\lambda a^\lambda ⟨gβ,U,α⟩=⟨(aβ−ξ3bβμaμ),(uλ∣α−bλαu3+ξ3θλ∣α)aλ+(u3,α+bαλuλ+ξ3bαλθλ)a3⟩=(uλ∣α−bλαu3+ξ3θλ∣α)δβλ−(uλ∣α−bλαu3+ξ3θλ∣α)ξ3bβμδμλ=uβ∣α−bβαu3+ξ3θβ∣α−(uμ∣α−bμαu3+ξ3θμ∣α)ξ3bβμ=uβ∣α−bβαu3+ξ3θβ∣α−ξ3bβμuμ∣α+ξ3bβμbμαu3−(ξ3)2bβμθμ∣α=uβ∣α−bβαu3+ξ3(θβ∣α−bβμuμ∣α+cαβu3)−(ξ3)2bβμθμ∣α\begin{aligned} \langle g_\beta,U_{,\alpha} \rangle&= \langle (a_\beta-\xi^3 b^\mu_\beta a_\mu),(u_{\lambda|\alpha}-b_{\lambda\alpha}u_3 +\xi^3 \theta_{\lambda|\alpha})a^\lambda+(u_{3,\alpha}+b^\lambda_\alpha u_\lambda +\xi^3 b^\lambda_\alpha \theta_\lambda)a_3 \rangle \\ &=(u_{\lambda|\alpha}-b_{\lambda\alpha}u_3+\xi^3 \theta_{\lambda|\alpha})\delta_{\beta\lambda} -(u_{\lambda|\alpha}-b_{\lambda\alpha}u_3+\xi^3 \theta_{\lambda|\alpha})\xi^3 b^\mu_\beta \delta_{\mu\lambda} \\ &=u_{\beta|\alpha}-b_{\beta\alpha}u_3+\xi^3 \theta_{\beta|\alpha} -(u_{\mu|\alpha}-b_{\mu\alpha}u_3+\xi^3 \theta_{\mu|\alpha}) \xi^3 b^\mu_\beta \\ &=u_{\beta|\alpha}-b_{\beta\alpha}u_3+\xi^3 \theta_{\beta|\alpha} -\xi^3 b^\mu_\beta u_{\mu|\alpha}+\xi^3 b^\mu_\beta b_{\mu\alpha} u_3 -(\xi^3)^2 b^\mu_\beta \theta_{\mu|\alpha} \\ &=u_{\beta|\alpha}-b_{\beta\alpha}u_3 +\xi^3 (\theta_{\beta|\alpha}-b^\mu_\beta u_{\mu|\alpha}+c_{\alpha\beta}u_3) -(\xi^3)^2 b^\mu_\beta \theta_{\mu|\alpha} \end{aligned} ⟨g3,U,α⟩=⟨a3,U,α⟩=u3,α+bαλuλ+ξ3bαλθλ\begin{aligned} \langle g_3,U_{,\alpha} \rangle=\langle a_3,U_{,\alpha} \rangle=u_{3,\alpha}+b^\lambda_\alpha u_\lambda +\xi^3 b^\lambda_\alpha \theta_\lambda \end{aligned} ⟨gα,U,3⟩=⟨(aα−ξ3bαλaλ),θμaμ⟩=θμδαμ−ξ3bαλδλμθμ=θα−ξ3bαλθλ\begin{aligned} \langle g_\alpha,U_{,3} \rangle &= \langle (a_\alpha-\xi^3 b^\lambda_\alpha a_\lambda),\theta_\mu a^\mu \rangle \\ &=\theta_\mu \delta_{\alpha\mu}-\xi^3 b^\lambda_\alpha \delta_{\lambda\mu} \theta_\mu =\theta_\alpha-\xi^3 b^\lambda_\alpha \theta_\lambda \end{aligned}
CC BY-SA 4.0 Miguel Bustamante. Last modified: August 08, 2023. Website built with Franklin.jl and the Julia programming language.