Deformation the Lagrangian description

• Deformation map \(\varphi:\mathbb{R}^3\to\mathbb{R}^3\) such that \(p=\varphi(P)\)
• • \(P\) position in the reference undeformed shape
  • \(p\) position in the deformed configuration.

• Deformation Gradient \(F\)

• • - \(\displaystyle F(P)=\frac{\partial \varphi}{\partial P}(P)=\frac{\partial p}{\partial P}\in\mathbb{R}^{3\times 3}\)
  • - Characterizes the local deformation associated to \(\varphi\)
  • • Position \(P+\mathrm{d}P\) is mapped into \(\displaystyle \varphi(P+\mathrm{d}P)\simeq p+\frac{\partial \varphi}{\partial P}\,\mathrm{d}P\)
  • - \(\displaystyle F(P)=\left(\begin{array}{ccc}\frac{\partial \varphi_x}{\partial X} & \frac{\partial \varphi_x}{\partial Y} & \frac{\partial \varphi_x}{\partial Z} \\ \frac{\partial \varphi_y}{\partial X} & \frac{\partial \varphi_y}{\partial Y} & \frac{\partial \varphi_y}{\partial Z} \\ \frac{\partial \varphi_z}{\partial X} & \frac{\partial \varphi_z}{\partial Y} & \frac{\partial \varphi_z}{\partial Z} \end{array}\right)\)