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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
##### Keywords:
string theory; energy functional
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