## 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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