On the theory of thermoelastic bending of reinforced shells and plates

The author examines an approach to the construction of a theory of thermoelastic bending of arbitrarily reinforced shells and plates that takes into account the actual structure, the deformation characteristic, and the actual thermomechanical properties of the elements which make up the composite. The final equations are obtained with and without allowance for transverse shear strains. The problem of the thermoelastic bending of a thin, arbitrarily reinforced rectangular plate hinged at the edges, when transverse shear strains can be neglected, is considered as an example.Institute of Hydrodynamics, Siberian Division, Academy of Sciences of the USSR, Novosibirsk. Translated from Mekhanika Polimerov, No. 5, pp. 861–873, September–October, 1972.     

Fundamental solution for thermoelastic bending of orthotropic spherical shells

A fundamental solution is constructed for the thermoelastic bending of thin gently curved orthotropic spherical shells. Newtonian heat exchange with the surrounding medium is taken into account. Numerical results are given for the case of symmetric heat exchange. Donetsk State University. Translated from Teoreticheskaya i Prikladnya Mekhnika, No. 30, pp. 126–132, 1999.     

On the dynamics of phase transitions

The dynamics of phase transitions is described in ordinary thermodynamics i.e. the bodies are considered to be homogeneous. Stability properties of phase transitions are investigated by Lyapunov's method.     

On the classification of phase transitions

A new classification of phase transitions is presented, based on an exact mathematical model of phase, on rigorous definitions and results regarding phase transitions which are classified to be of zeroth, second and first order. Then the relation between the known classifications due to Ehrenfest and Tisza, respectively, becomes clear.     

On the convergence of stable phase transitions

We consider the local behavior of critical points of the functional as ε → 0. Here, W is a double-well potential and U is a regular domain in ℝⁿ, n ≥ 2. Assuming that {u^ε}_ε>0 is stable for n = 2 and locally energy-minimizing for n = 3, we show that the level sets of solutions converge in an average sense to a stationary (n − 1)-rectifiable varifold. Our study is based on estimates derived from the second variation formula and is entirely local. © 1998 John Wiley & Sons, Inc.     

Variational problem describing martensitic-austenitic phase transitions with zero surface tension

For a martensitic-austenitic phase medium described by the classical Hooke law, a variational problem on phase transitions is considered. Solutions of this problem are given in explicit form. Bibliography: 12 titles Dedicated to O.A. Ladyzhenskaya on the occasion of her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997, pp. 231–243 Translated by V.G. Karazeeva     

Optimization problems of the stress-deformed state of thermoelastic shells

We discuss general questions about the choice of optimization criteria for the stressed state of thermoelastic shells taking account of the functional measures of the space-time variability of its characteristics. The efficiency of the proposed approach is illustrated using the example of the solution of a problem for an axisymmetrically heated cylindrical shell.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 11–18.     

On phase transitions of elastic continua

Summary An elastic body undergoing a phase transition is modelled as a mixture of the two separate phases. Classical results for liquid-gas transitions are obtained from the relevant Clausius-Duhem inequality and their counterparts derived for a particular class of solid-solid transitions.

---

Résumé Un corps élastique en changement de phase est représenté par un mélange des deux phases. On retrouve les résultats classiques pour les changements de phase liquide-gazeuse à partir de l'inégalité de Clausius-Duhem et obtient les résultats correspondants pour une certaine classe de changements de phase solide-solide.

     

On Julia sets concerning phase transitions

The sets of the points corresponding to the phase transitions of the Potts model on the diamond hierarchical lattice for antiferromagnetic coupling are studied. These sets are the Julia sets of a family of rational mappings. It is shown that they may be disconnected sets. Furthermore, the topological structures of these sets are described completely.     

On exact expressions of the bending tensor in the nonlinear theory of thin shells

Some equivalent exact expressions of the bending tensor in the nonlinear theory of thin shells are reviewed. It is noted that the bending tensor, proposed by Shen et al. (2010) [X.Q. Shen, K.T. Li, Y. Ming “The modified model of Koiter’s type for the nonlinearly elastic shells”, Appl. Math. Model. 34 (2010) 3527-3535] as a third-degree polynomial of displacements, is an approximate expression, not the exact one. Then integrability of the fourth kinematic boundary condition, associated with two different but equivalent exact expressions of the bending tensor, is briefly discussed. Finally, a few modified definitions of the bending tensor proposed in the literature are recalled. Within the first-approximation theory they all lead to energetically equivalent models of elastic shells.     

The direct method in time-dependent bending of thermoelastic plates

The initial-boundary value problems with Dirichlet and Neumann boundary conditions arising in the theory of bending of thermoelastic plates with transverse shear deformation are reduced to time-dependent boundary integral equations by means of the Somigliana representation formulas. The solvability of these equations is then investigated in Sobolev-type spaces.     

Boundary integral equations in time-dependent bending of thermoelastic plates

The initial-boundary value problems with the Dirichlet and Neumann boundary conditions arising in the theory of bending of thermoelastic plates with transverse shear deformation are reduced to time-dependent boundary integral equations by means of layer potentials. The solvability of these equations is then investigated in Sobolev-type spaces.     

Finite bending of plates into cylindrical shells

Summary The use of finite components of strain and a linear stress-strain relation shows that a plate can be bent into a cylindrical shell whose section is given by rⁿ cos nθ=aⁿ, where n=(3−2σ)⁻¹, σ being the Poisson’s ratio. The case (σ=1/2) and the one given by r^2/5 cos2/5 θ=a^2/5, (σ=1/4) are discussed in detail. To Antonio Signorini on his 70^th birth day.     

On phase transitions for subshifts of finite type

It has recently been shown that a strongly irreducible subshift of finite type in two or more dimensions may have more than one measure of maximal entropy. In this paper we obtain some results on when (i.e. for what kinds of subshifts of finite type) this happens, and when it does not. In particular, we show that the parameter of a certain subshift of finite type introduced by Burton and Steif has a critical value, below which we have a unique measure of maximal entropy, and above which we have non-uniqueness.     

On a linear model of mechanical phase transitions

Existence and infinite speed of propagation are shown for the equation $$h_n \left( {x,t} \right) = \frac{1}{{\alpha \pi }}\int_{ - \infty }^\infty {\frac{{Ah_{xxx} \left( {\xi ,t} \right) - Bh_{xt} \left( {\xi ,t} \right)}}{{\left( {x - \xi } \right)}}d\xi } , - \infty< x< \infty ,0< t< T,$$ which is the linearization of a model equation for the mechanical phase transition phenomenon of melting-freezing waves. In addition, from the formal solution of the equation one can compute a decay rate.     

A fundamental solution of the equations of thermoelastic bending of orthotropic plates

We construct a fundamental solution of the equations of thermoelastic bending of orthotropic plates under symmetric heat transfer and a linear heart distribution over the thickness. On the lateral surfaces convective heat transfer with a surrounding medium occurs in accordance with Newton's law. The effect of the orthotropy parameters and heat transfer on the magnitude of the bending moment and torque is studied numerically. Two figures. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 87–92.     

Modeling of thermoelastic processes in heterogeneous anisotropic shells with initial deformations

A refined mathematical model of the dynamic problem of coupled thermoelasticity of heterogeneous anisotropic shells taking into account the anisotropy of thermomechanical properties of the material both in the median surface and in the transversal direction is developed. The model includes initial deformations and is based on the assumption that the components of the vector of displacements and temperature are linearly distributed over the thickness. The reliability of the proposed model is evaluated by comparing the solutions obtained on its basis with the corresponding solutions obtained by using the theory of elasticity.