On radii of absorbing sets for Kuramoto-Sivashinsky equation

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Minimum principles,convexity, and thermodynamics in viscoelasticity

A linear viscoelastic solid is considered along with the complete set of thermodynamic restrictions on the relaxation function. It is shown that such reslrictions imply the validity of a dissipativity condition, so far regarded as unrelated to the second law. Next it is remarked that the thermodynamic restrictions imply the convexity of a commonly used bilinear functional and the stationarity only if the class of displacement field is appropriate. Then it is proved that the Laplace transform of the solution to the mixed problem gives the strict minimum of a bilinear functional and vice versa. Finally, a bilinear functional with a weight function is considered and it is shown that the solution to the mixed problem gives the strict minimum and vice versa.     

On the relation between polyconvexity and rank-one convexity in nonlinear elasticity

For elastic stored energy functions we examine some of the implications of polyconvexity and rank-one convexity, as well as the connection between the two conditions in the presence of certain additional restrictions.     

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.     

On linking n-dimensional anisotropic and isotropic Green’s functions for infinite space,half-space,bimaterial, and multilayer for conduction

We establish exact mathematical links between the n-dimensional anisotropic and isotropic Green’s functions for diffusion phenomena for an infinite space, a half-space, a bimaterial and a multilayered space. The purpose of this work is not to attempt to present a solution procedure, but to focus on the general conditions and situations in which the anisotropic physical problems can be directly linked with the Green’s functions of a similar configuration with isotropic constituents. We show that, for Green’s functions of an infinite and a half-space and for all two-dimensional configurations, the exact correspondences between the anisotropic and isotropic ones can always be established without any regard to the constituent conductivities or any other information. And thus knowing the isotropic Green’s functions will readily provide explicit expressions for anisotropic Green’s functions upon back transformation. For three- and higher-dimensional bimaterials and layered spaces, the correspondence can also be found but the constituent conductivities need to satisfy further algebraic constraints. When these constraints are fully satisfied, then the anisotropic Green’s functions can also be obtained from those of the isotropic ones, or at least in principle.     

Integral convexity argument for plasticity function and monotonicity of iteration process for elasto-plastic problems

The variational approach for boundary value problems related to non-linear materials is presented. The approach is based on the introduced integral convexity argument and monotone operator theory. This permits to obtain an existence and uniqueness of the solution within the range of Kachanov's theory, under natural and general conditions. The introduced argument allows to prove monotonicity of the sequence of potentials on the sequence of iterations. As a result monotonicity of the iteration process for elastoplastic problems is obtained. Theoretical results are illustrated by computational experiments.     

On some nonlinear nonisotropic quasi-self-similar functions

Nonlinear nonisotropic quasi-self-similar functions are important because of their relation with many physical phenomena such as fully developed turbulence or diffusion limited aggregates. Such functions are superpositions of “similar" structures at different scales, reminiscent of some modelization of turbulence. In this paper we continue to study such a class of functions. We extend our results in [Ben Mabrouk. A., Far East J. Dynam. Syst. 7(1), 23–63 (2005)] to some nonlinear cases and where some separation condition is not satisfied. Mathematics subject classification (2000): 42C40, 28A80, 76M55     

On the invariance of time-dependent Hamilton＇s functions

The general framework of Poincaré's formalism is used to establish the connection between conservation laws and invariance properties of Hamilton's function under infinitesimal transformations when these laws and the Hamiltonian are time-dependent. An example illustrative of the theory is also considered. The English text was polished by Yunming Chen     

Homogenization of nonlinearly elastic materials,microscopic bifurcation and macroscopic loss of rank-one convexity

Communicated by R. V. Kohn     

轴对称非协调元分片检验检验函数的结构

导出了轴对称非协调元分片检验中检验函数的形式,其位移函数是一种不完全的线性函数。从而给出了轴对称非协调调元分析检验更准确,完全的描述。