General Biomimetic Turing Machine
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The General theory for BTM
We consier a morphing slender kirgami ribbon for conformable deformation. The movement equation of the principal curvatrue frame can be written as:
\[\partial_s\left( \begin{array}{c} \mathbf{d}_1\\ \mathbf{d}_2\\ \mathbf{n}\\ \end{array} \right)=\left( \begin{matrix} 0& -(\theta'+\phi')& (\kappa_2-\kappa_1)\sin\theta\cos\theta\\ \theta'+\phi'& 0& -\kappa_1\sin^2\theta-\kappa_2\cos^2\theta\\ (\kappa_1-\kappa_2)\sin\theta\cos\theta& \kappa_1\sin^2\theta+\kappa_2\cos^2\theta& 0\\ \end{matrix} \right)\left( \begin{array}{c} \mathbf{d}_1\\ \mathbf{d}_2\\ \mathbf{n}\\ \end{array} \right)\]The movement equation of the twisted Frenet frame can be written as:
\[\partial_s\left( \begin{array}{c} \mathbf{B}_1\\ \mathbf{T}\\ \mathbf{N}_1\\ \end{array} \right)=\left( \begin{matrix} 0& -\kappa\sin\varphi& \varphi'-\tau\\ \kappa\sin\varphi& 0& \kappa\cos\varphi\\ \tau-\varphi'& -\kappa\cos\varphi& 0\\ \end{matrix} \right)\left( \begin{array}{c} \mathbf{B}_1\\ \mathbf{T}\\ \mathbf{N}_1\\ \end{array} \right)\]We can also write the movement of the ribbon as material strain:
\[\partial_s\left( \begin{array}{c} \mathbf{d}_1\\ \mathbf{d}_2\\ \mathbf{d}_3\\ \end{array} \right)=\left( \begin{matrix} 0& m& m_1\\ -m& 0& m_2\\ -m_1& -m_2& 0\\ \end{matrix} \right)\left( \begin{array}{c} \mathbf{d}_1\\ \mathbf{d}_2\\ \mathbf{d}_3\\ \end{array} \right)\]Finally we have the equation for inverse design:
\[\begin{cases} (\kappa_2-\kappa_1)\sin\theta\cos\theta=m_1\\ \kappa_1\sin^2\theta+\kappa_2\cos^2\theta=m_2\\ \phi'=-m-\theta' \end{cases}\]When $\kappa_2=-\kappa_1$, we have:
\[\begin{cases} \phi'=-m-\theta' \\ \tan2\theta=-\frac{m_1}{m_2}\\ \kappa_1=\frac{m_2}{\cos2\theta} \end{cases}\]