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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.

MSC:

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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