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Table 4 Degrees of freedom and related functionals of the deformation function, for node 1 with \({\mathbf {x}}_0^1 = {\mathbf {x}}_0 \left( 0,0\right) \), and shape functions satisfying conditions () and ()

From: Introducing the Logarithmic finite element method: a geometrically exact planar Bernoulli beam element

Degree of freedom Functional of the deformation function Geometric meaning
\(u_{1,0,0}\) \( \mathrm{Re} \left( {{{\mathrm{\mathrm{Log}}}}{\mathinner { \frac{\partial {^{}}\mathbf {x}_\xi }{\partial {{{\mathbf {x}}_0}_\xi ^{}}} }} ({0, 0})}\right) \) Strain
\(u_{1,1,0}\) \(\mathrm{Im} \left( {{{\mathrm{\mathrm{Log}}}}{\mathinner { \frac{\partial {^{}}\mathbf {x}_\xi }{\partial {{{\mathbf {x}}_0}_\xi ^{}}} }} ({0, 0})}\right) \) Rotation
\(u_{1,0,1}\) \({\dot{\varepsilon }^\mathrm {mat}}({0, 0}) - {\dot{\varepsilon }_0^\mathrm {mat}}({0, 0})\) Material derivative of strain
\(u_{1,1,1}\) \({\kappa ^\mathrm {mat}}({0, 0}) - {\kappa _0^\mathrm {mat}} ({0, 0})\) Material curvature