O pewnym ustalonym zagadnieniu przestrzennym termosprężystości

A steady-state three-dimensional thermo-elastic problem

The object of this paper is to determine the temperature field T and the stress components (σ_ij) in an elastic semi-space, assuming that the temperature in the region Γ of the plane bounding the semi-space is known and equal to T₀(ξ, η), the region outside Γ being kept at T = 0.

The problem is solved by means of the Green's function. The temperature field T* is determined and the stress components (σ_ij^*), in the case where T = 0 over the whole plane z = 0 except an infinitely small region dΓ where T = T₀. Using integral expressions (1.1) and (1.2), T and σ_ij are determined for any temperature distribution T₀(ξ, η) inside the region Γ. In the second part of the paper, the Green's functions T* and Φ* and the Galerkin's function φ are determined, using the Fourier transformation. The Green's functions obtained for the stress (σ_ij^*) are characterized by the fact that the three components σ_zz^*, σ_zx^*, and σ_zy^* are zero.

In the third part of the paper, consideration is given to the particular case of T₀ = const over a rectangle whose sides are 2a and 2b, and that of T₀ = const over an infinite strip of width 2a. In the latter case, only the stress σ_yy^* differs from zero.