Materials Science, Vol. 42, No. 2, 2006 STRESSED STATE OF A THERMOSENSITIVE PLATE IN A CENTRAL-SYMMETRIC TEMPERATURE FIELD R. M. Kushnir and V. S. Popovych UDC 539.3 The problem of thermoelasticity for a thin thermosensitive plate placed in a central-symmetric temperature field is reduced by the perturbation method to a recurrent sequence of boundary-val- ue problems for differential equations with constant coefficients. On this basis, we obtain solu- tions for the cases of load-free circular washer, infinite plate with circular hole, circular disk, and infinite plate. The stress–strain state of an infinite plate containing a circular hole is investigated. Thin plates are fairly extensively used as structural elements of contemporary equipment operating in media with high or low temperatures and significant temperature drops both in space and in the course of time. Under the indicated conditions, the engineering practice imposes elevated requirements to the accuracy of determina- tion of the stress–strain state of thin plates. This, can be attained by improving the corresponding mathematical models in which, in particular, it is necessary to take into account the temperature dependence (thermosensitiv- ity) of the characteristics of materials. In [1], the equation of thermoelasticity is deduced for thin thermosensi- tive plates free of loads by using the Kirchhoff – Love hypothesis under the assumption that Poisson’s ratio is constant. Similarly, in [2], the thermoelastic state of a thermosensitive plate is modeled for the case where the lateral surfaces of the plate are uniformly loaded by a constant pressure. Some special problems for thin thermo- sensitive plates were studied in [3–9]. In the present work, we construct the general solution of the problem of thermoelasticity for a thin load-free plate placed in a central-symmetric temperature field under the assumption that all characteristics of the plate are temperature dependent. Statement of the Problem Consider a thin plate whose mechanical (shear modulus G, Poisson’s ratio ν, and the coefficient of linear thermal expansion α t ) and thermal (heat-conduction coefficient λ t and volumetric heat capacity c υ ) charac- teristics are functions of temperature. The plate is placed in a central-symmetric nonstationary temperature field t ( r, τ ) , where r is a radial coordinate and τ is time. The initial temperature of the plate at which it is strain- free is constant and equal to t p ( t ( r, 0 ) = t p ). In view of the symmetry of the problem, only the radial compo- nent of displacements u ( r, τ ) and two components of the stress tensor are nonzero in a polar coordinate system r, ϕ [1, 10], namely, σ rr = 2 1 Gt t u r t u r t () - () ∂ ∂ + () - ()       ν ν Φ and σ ϕϕ = 2 1 Gt t t u r u r t () - () () ∂ ∂ + - ()       ν ν Φ . (1) Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, Lviv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 2, pp. 5–12, March–April, 2006. Original article submitted October 21, 2005. 1068–820X/06/4202–0145 © 2006 Springer Science+Business Media, Inc. 145