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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