Stability of a plate loaded by a heat source

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 31, No. 1, pp. 79–86, January, 1995.     

Stress-strain state of a plate unevenly heated through its thickness

Conclusion Using the method of differential trigonometric series, distinguished by its correctness and generality and the fact that it gives exact solutions for boundary-value problems, we obtained a new solution to the problem of thermal sresses in an unevenly heated plate, as well the problem of the deformation of a freely supported plate under the influence of moments uniformly distributed over its edges. The series entering into the deflections and bending and turning moments converge, and the solution coincides with well-known solutions.Nikolaev Institute of Shipbuiding. Translated from Prikladnaya Mekhanika, Vol. 17, No. 9, pp. 69–75, September, 1981.     

Thermoelastic state of an infinite plate weakened by a variable heat flow

International Applied Mechanics -     

Stress state of compound polygonal plate

The constructions made of bars and plates with holes, openings and bulges of various forms are widely used in modern industry. By loading these structural elements with different efforts, there appears concentration (accumulation) of stress whose values sometimes exceeds the admissible one. The durability of the given element is defined according to the quantity of these stresses. Since the failure of details and construction itself begins from the place where the stress concentration has the greatest value.

Therefore the exact determination of stress distribution in details (bars, plates, beams) is of great scientific and practical interest and is one of the important problems of the solid fracture.

Compound details (when the nucleus of different material is soldered to the hole) are often used to decrease the stress concentration.

In the present paper, we study a stress–strain state of polygonal plate weakened by a central elliptic hole with two linear cracks info which a rigid nucleus (elliptic cylinder with two linear bulges) of different material was put in (soldered) without preload.

The problem is solved by a complex variable functions theory stated in papers [Theory of Elasticity, Higher School, Moscow, 1976, p. 276; Plane Problem of Elasticity Theory of Plates with Holes, Cuts and Inclusions, Publishing House Highest School, Kiev, 1975, p. 228; Bidimensional Problem of Elasticity Theory, Stroyizdat, Moscow, 1991, p. 352; Science, Moscow (1996) 708; MSB AH USSR OTH 9 (1948) 1371].

Kolosov–Mushkelishvili complex potential (z) and ψ(z) satisfying the definite boundary conditions are sought in the form of sums of functional series.

After making several strict mathematical transformations, the problem is reduced to the solution of a system of linear algebraic equations with respect to the coefficients of expansions of functions (z) and ψ(z).

Determining the values of (z) and ψ(z), we can find the stress components σ_r, σ_θ and τ_rθ at any point of cross-section of the plate and nucleus on the basis of the known formulae. The obtained solution is illustrated by numerical example.

Changing the parameters A₁, m₁, e, A₂, and m₂ we can get the various contour plates.

For example, if we assume m₁=0, A₁=r, then the internal contour of L₁ becomes the circle of radius r with two rectilinear cracks (for the nucleus––a rectilinear bulges).

Further, if we assume a small semi-axis of the ellipse b₁ to be equal to zero (b₁=0), then a linear crack becomes the internal contour of L₁ (and the nucleus becomes the linear rigid inclusion made of other material). For m₂=0; A₂=R, the external contour L₂ turns into the circle of radius R.

The obtained method of solution may be applied and in other similar problems of elasticity theory; tension of compound polygonal plate, torsion and bending of compound prismatic beams, etc.     

Thermal stresses induced by a point heat source in a circular plate by quasi-static approach

The present paper deals with the determination of quasi-static thermal stresses due to an instantaneous point heat source of strength g_pi situated at certain circle along the radial direction of the circular plate and releasing its heat spontaneously at time t = τ. A circular plate is considered having arbitrary initial temperature and subjected to time dependent heat flux at the fixed circular boundary of r = b. The governing heat conduction equation is solved by using the integral transform method, and results are obtained in series form in terms of Bessel functions. The mathematical model has been constructed for copper material and the thermal stresses are discussed graphically.     

Convection regimes and heat transfer characteristics along a continuously moving heated vertical plate

The steady laminar flow and thermal characteristics of a continuously moving vertical sheet of extruded material are studied close to and far downstream from the extrusion slot. The velocity and temperature variations, obtained by a finite volume method, are used to map out the entire forced, mixed and natural convection regimes. The effects of the Prandtl number (Pr) and the buoyancy force parameter (B) on the friction and heat transfer coefficients are investigated. Comparisons with experimental measurements and solutions by others in the pure forced and pure natural convection regions are made. In the mixed convection region, the results are compared with available finite-difference solutions of the boundary layer equations showing excellent agreement. The region close to the extrusion slot is characterized as a non-similar forced-convection dominated region in which Nu_xRe_x^−1/2 drops sharply with increasing Richardson number (Ri_x). This is followed by a self-similar forced-convection dominated region in which Nu_xRe_x^−1/2 levels off with increasing Ri_x until the buoyancy effect sets in. The existence and extent of the latter region depend upon the value of B. A non-similar mixed convection region where increasing buoyancy effect enhances the heat transfer rate follows. Finally, this region is followed downstream by a self-similar natural-convection dominated region in which Nu_xRe_x^−1/2 approaches the pure natural convection asymptote at large Ri_x. Critical values of Ri_x to distinguish the various convection regimes are determined for different Pr and B.     

A modified fractional-order thermo-viscoelastic model and its application to a polymer micro-rod heated by a moving heat source

Classical thermo-viscoelastic models may be challenged to predict the precise thermo-mechanical behavior of viscoelastic materials without considering the memorydependent effect. Meanwhile, with the miniaturization of devices, the size-dependent effect on elastic deformation is becoming more and more important. To capture the memory-dependent effect and the size-dependent effect, the present study aims at developing a modified fractional-order thermo-viscoelastic coupling model at the microscale...