Cracks of higher modes in Cosserat continua

Rotational degrees of freedom in Cosserat continua give rise to higher fracture modes. Three new fracture modes correspond to the cracks that are surfaces of discontinuities in the corresponding components of independent Cosserat rotations. We develop a generalisation of J- integral that includes these additional degrees of freedom. The obtained path-independent integrals are used to develop a criterion of crack propagation for a special type of failure in layered materials with sliding layers. This fracture propagates as a progressive bending failure of layers – a “bending crack that is, a crack that can be represented as a distribution of discontinuities in the layer bending. This situation is analysed using a 2D Cosserat continuum model. Semi-infinite bending crack normal to layering is considered. The moment stress concentrates along the line that is a continuation of the crack and has a singularity of the power − 1/4. A model of process zone is proposed for the case when the breakage of layers in the process of bending crack propagation is caused by a crack (microcrack in our description) growing across the layer adjacent to the crack tip. This growth is unstable (in the moment-controlled loading), which results in a typical descending branch of moment stress – rotation discontinuity relationship and hence in emergence of a Barenblatt-type process zone at the tip of the bending crack.

Reconstruction of an acoustically sound-soft obstacle from one incident field and the far field pattern

The problem considered is that of determining the shape of a plane acoustically sound-soft obstacle from the knowledge of the far-field pattern for one time-harmonic incident field. An iterative procedure is proposed based on two boundary integrals representing the incident field and the far-field pattern, respectively. Numerical examples are included which show that the procedure gives accurate numerical approximations in relatively few iterations.

On the KAM - Theory Conditions for the Kirchhoff Top

* Partially supported by Grant MM523/95 with Ministry of Science and Technologies.

Limit Cycles of Perturbations of a Class of Quadratic Hamiltonian Vector Fields

We prove that in quadratic perturbations of generic Hamiltonian vector fields with two saddle points and one center there can appear at most two limit cycles. This bound is exact.

Perturbations of Systems Describing the Motion of a Particle in Central Fields

The present paper deals with the KAM-theory conditions for systems describing the motion of a particle in central field.

Numerical solution of parabolic Cauchy problems in planar corner domains

An iterative method for the parabolic Cauchy problem in planar domains having a finite number of corners is implemented based on boundary integral equations. At each iteration, mixed well-posed problems are solved for the same parabolic operator. The presence of corner points renders singularities of the solutions to these mixed problems, and this is handled with the use of weight functions together with, in the numerical implementation, mesh grading near the corners. The mixed problems are reformulated in terms of boundary integrals obtained via discretization of the time-derivative to obtain an elliptic system of partial differential equations. To numerically solve these integral equations a Nyström method with super-algebraic convergence order is employed. Numerical results are presented showing the feasibility of the proposed approach. © 2014 IMACS.

Operational Rules for a Mixed Operator of the Erdélyi-Kober Type

2000 Mathematics Subject Classification: 26A33 (main), 44A40, 44A35, 33E30, 45J05, 45D05

Fractional Calculus of the Generalized Wright Function

Mathematics Subject Classification: 26A33, 33C20.

Rough Maximal Oscillatory Singular Integral Operators

2000 Mathematics Subject Classification: Primary 42B20; Secondary 42B15, 42B25

On Multidimensional Analogue of Marchaud Formula for Fractional Riesz-Type Derivatives in Domains in R^n

2000 Mathematics Subject Classification: 26A33, 42B20

On Two Saigo’s Fractional Integral Operators in the Class of Univalent Functions

2000 Mathematics Subject Classification: Primary 26A33, 30C45; Secondary 33A35

On Integral Means for Fractional Calculus Operators of Multivalent Functions

2000 Mathematics Subject Classification: Primary 30C45, Secondary 26A33, 30C80

Some Fractional Extensions of the Temperature Field Problem in Oil Strata

This survey is devoted to some fractional extensions of the incomplete lumped formulation, the lumped formulation and the formulation of Lauwerier of the temperature field problem in oil strata. The method of integral transforms is used to solve the corresponding boundary value problems for the fractional heat equation. By using Caputo’s differintegration operator and the Laplace transform, new integral forms of the solutions are obtained. In each of the different cases the integrands are expressed in terms of a convolution of two special functions of Wright’s type.

Stochastic Solution of a KPP-Type Nonlinear Fractional Differential Equation

Mathematics Subject Classification: 26A33, 76M35, 82B31

Caputo-Type Modification of the Erdélyi-Kober Fractional Derivative

2000 Math. Subject Classification: 26A33; 33E12, 33E30, 44A15, 45J05