On infinitesimal deformation of thermoelastic membrane. (Russian) Zbl 0931.74043

A thermoelastic membrane is considered as a surface \(\Sigma\) in three-dimensional Euclidean space. For an infinitesimal deformation of the membrane determined by a vector field \(\vec u\), the deformation tensor \(\gamma_{ij} = {1 \over 2}(\nabla_i u_j + \nabla_j u_i)\) has the form \( \gamma_{ij}=\alpha \tau g_{ij} + {1 \over E}((1+\kappa)E_{ij} - \kappa \theta g_{ij}) \), where \(E_{ij}\) is the strain tensor, \(\theta=g^{ij}E_{ij}\), \(E\) is the elasticity modulus, and \(\kappa\) is Poisson’s ratio. The problem is to find \(\vec u\) if \(E_{ij}\) and \(\kappa\) are given. In the paper under review, this problem is solved in the case when \(\Sigma\) is a torus. The integrability conditions are found, and the components of \(\vec u\) are written out in an explicit form.


74K15 Membranes
74F05 Thermal effects in solid mechanics
53Z05 Applications of differential geometry to physics
53A05 Surfaces in Euclidean and related spaces
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