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Derivation of the nonlinear bending-torsion theory for inextensible rods by \(\Gamma\)-convergence. (English) Zbl 1053.74027
Let \[ E^{(h)}(v)=\int_{\Omega_h}W(\nabla v(z))\,dz \] be the functional of elastic energy of a deformation \(v\in W^{1,2}(\Omega_h;\mathbb{R}^3)\), where \(\Omega_h=(0,L)\times hS\) and \(S\) is an open subset of \(\mathbb{R}^2\). After rescaling, one gets \(\frac{1}{h^2} E^{(h)}(v) = I^{(h)}(y)\) with \(y\in W^{1,2}(\Omega_h;\mathbb{R}^3)\), \(\Omega_h=(0,L)\times S\). As their main result, the authors identify, under certain assumptions, the \(\Gamma\)-limit [cf. G. Dal Maso, An introduction to \(\Gamma\)-convergence. Progress in Nonlinear Differential Equations and their Applications 8, Basel: Birkhäuser (1993; Zbl 0816.49001)] of the sequence of functionals \(\{\frac{1}{h^2} I^{(h)}\}\) with respect to weak (and strong) topology of \(W^{1,2}\). Their result implies that the nonlinear bending-torsion theory for inextensible rods is the \(\Gamma\)-limit of three-dimensional nonlinear elasticity. Similar results have been achieved with like methods in recent papers by O. Pantz [Arch. Ration. Mech. Anal. 167, No. 3, 179–209 (2003; Zbl 1030.74031)].

MSC:
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74G65 Energy minimization in equilibrium problems in solid mechanics
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
49J45 Methods involving semicontinuity and convergence; relaxation
74B20 Nonlinear elasticity
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