The remarkable nature of radially symmetric equilibrium states of aeolotropic nonlinearly elastic bodies

This paper treats the radially symmetric equilibrium states of aeolotropic nonlinearly elastic solid cylinders and balls under constant normal forces on their boundaries. It is shown that the aeolotropy gives rise to solutions describing both intact and cavitating states, which exhibit an array of remarkable new phenomena, not suggested by the solutions for isotropic bodies. E.g., it is shown that there are materials having a critical pressure such that for applied pressures on the boundary below the critical value, the normal pressures at the center of the body are zero and for applied pressures above the critical value, the normal pressures at the center are infinite. There are also materials for which there is no equilibrium state with center intact when the boundary is subjected to uniform tension. It is also shown that the equilibrium states treated here are the only radially symmetric equilibrium states. Thus the strange phenomena discovered here must be present in such stable equilibrium states.     

On formation of singularities in one-dimensional nonlinear thermoelasticity

Dedicated to Bernard D. Coleman on the occasion of his 60^th birthday     

On the Gibbs conditions of stable equilibrium,convexity and the second-order variations of thermodynamic potentials in nonlinear thermoelasticity

The Gibbs conditions of stable thermodynamic equilibrium are formulated for nonlinear thermoelastic materials, based on the constrained minimization of four fundamental thermodynamic potentials. Sufficient conditions for incremental stability are stated in each case. The previously unexplored connections between the second-order variations of thermodynamic potentials are used to establish the convexity or concavity properties of all thermodynamic potentials in relation to each other, and to derive the relationships between the specific heats at constant stress and deformation, and between the isentropic and isothermal elastic moduli and compliances. The comparison with the derivation based on the classical thermodynamic approach is also given.     

Parametric class of solutions of thermoelasticity equations for transversally isotropic bodies

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 24, No. 7, pp. 52–57, July, 1988.     

Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity

The global existence of smooth solutions to the equations of nonlinear thermoelasticity is shown for a one-dimensional homogeneous reference configuration. Dirichlet boundary conditions are studied and the asymptotic behaviour of the solutions as t is described.Dedicated to Professor Rolf Leis and to Proffessor Mutsuhide Matsumura on the occasion of their sixtieth birthdays in 1991     

Geometric phases in the motion of rigid bodies

Archive for Rational Mechanics and Analysis - In this paper we make some differential-geometric observations on the kinematics of convex surfaces rolling along a fixed plane in ?3, and on the...     

Instability and non-existence in the nonlinear theory of thermoelasticity without energy dissipation

Received June 7, 2000 / Published online April 18, 2001     

Limiting equilibrium in problems of nonlinear filtration

The article indicates a method for determining the field of the pressure of a non-Newtonian liquid, existing in a state of limiting equilibrium in a porous medium. This kind of question arises in the solution of problems in the theory and practice of the exploitation of petroleum deposits containing petroleum with an initial pressure gradient.     

Radially and axially symmetric motions of a class of transversely isotropic compressible hyperelastic cylindrical tubes

Modeling and control of diabetes mellitus (DM) are difficult due to the highly nonlinear attitude, time-delay effects, the impulse kind input signals and the lack of continuously available blood glucose (BG) level to be regulated. Regarding the mentioned problems, identification of DM model is crucial. Furthermore, due to the lack of information about the internal states (which cannot be measured in everyday life) and because the BG level is not available in every moment over time, adaptive robust control design method regardless exact model dependency would successfully handle these unfavorable effects without simplifications. The recently developed nonlinear robust fixed point transformation (RFPT)-based controller design method requires only a roughly approximate model in order to realize the controller structure. Moreover, parallel simulated approximate models—in order to provide additional internal information—can be used with the method. In this paper, the usability of the novel RFPT-based technique is demonstrated on the physiological problem of diabetes.     

A state space formalism and perturbation method for generalized thermoelasticity of anisotropic bodies

A state space formalism for generalized anisotropic thermoelasticity accounting for thermomechanical coupling and thermal relaxation is developed, which includes the classical thermoelasticity as a special case. By properly grouping the field variables using matrix notations, the basic equations of thermoelasticity are formulated into a state equation and an output equation in terms of the state vector. To obtain the solution for a specific problem it suffices to solve the state equation under the prescribed conditions. For weak thermomechanical coupling an asymptotic solution can be obtained by using the method of perturbation with multiple scales. Propagation of plane harmonic thermoelastic waves in an anisotropic medium is studied within the context.     

Stability of heterogeneous equilibrium in a system with liquid phases

Gibbs's method is used to study the equilibrium and stability in a heterogeneous system consisting of single-component liquid phases. The coefficient of surface tension is assumed to be a constant independent of the state of the phases. An expression is obtained for the second variation of the corresponding functional, and this expression can be used to analyze the stability of nucleating centers. It is shown that in the absence of external force fields and surface tension forces the equilibrium state of a closed thermodynamic system consisting of single-component liquid phases satisfying the classical Gibbs inequalities is always stable.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 88–94, March–April, 1982.     

Shear deformable composite plates with nonlinear curvatures: modeling and nonlinear vibrations of symmetric laminates

Moving from a general plate theory, a modified general shear deformable laminated plate theory (MGFSDT) exhibiting nonlinear curvatures but still allowing for some worth features of linear curvature models (von Karman) is formulated. Starting from MGFSDT partial differential equations, a minimal discretized model (Duffing equation) for symmetric cross-ply laminates nonlinear vibrations, whose coefficients account for shear deformability and nonlinear curvatures, is obtained via the Galerkin procedure. The variable features of such a Duffing model as obtainable via alternative kinematic assumptions at the continuum level are highlighted. Through the comparison of a number of underlying models in different technical situations, information on the influence of shear deformability on system nonlinear response and on the influence of nonlinear curvatures are obtained. Frequency–response curves through a multiple scale analysis are presented for different continuous models, kinds of material, mode numbers and boundary conditions.     

Determination of the ultimate states of elastoplastic bodies

The equations of quasistatic deformation of elastoplastic bodies are considered in a geometrical linear formulation. After discretization of the equations with respect to spatial variables by the finite-element method, the problem of determining equilibrium onfigurations reduces to integration of a system of nonlinear ordinary differential equations. In the ultimate state of a body of an ideal elastoplastic material, the matrix of the system degenerates and the problem becomes singular. A regularization algorithm for determining solutions of the problems for the ultimate states of bodies is proposed. Numerical solutions of test problems of determining the ultimate loads and equilibrium configurations for ideal elastoplastic bodies confirm the reliability of the regularization algorithm proposed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 196–204, September–October, 2000.