Three-dimensional Initial-boundary Value Problem Research Articles

A large deformation gradient theory for glassy polymers by means of micromorphic regularization

AbstractCold forming of polycarbonate films results in the formation of shear bands in the necking zone. The numerical results obtained from standard viscoplastic material models exhibit mesh size dependency, requiring mathematical regularization. For this purpose, we present in this work a large deformation gradient theory for a viscoplastic isotropic material model published before. We extend our model to a micromorphic model by introducing a new micromorphic variable as an additional degree of freedom along with its first gradient. This variable represents a microequivalent plastic strain. The relation between the macroequivalent plastic strain and the micromorphic variable is accomplished by a micromorphic coupling modulus. This coupling forces proximity between the macro- and microvariables, leading to the targeted regularization effect. The micromorphic model is implemented as a three-dimensional initial boundary value problem in an in-house finite element tool. The analysis is performed for both uniaxial and biaxial specimens. The provided numerical examples show the ability of our model to regularize shear bands within the specimens and address the issue of localization.

Параболічні крайові задачі математичної фізики в кусково-однорідному клиновидному циліндрично-круговому півпросторі з порожниною

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in piecewise homogeneous by the radial variable r wedge-shaped by the angular variable φ cylindrical-circular half-space with a cavity were constructed at first time by the method of classical integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices) in the proposed article. The cases of assigning on the verge of the wedge the boundary conditions of the 1st kind (Dirichlet) and the 2nd kind (Neumann) and their possible combinations (Dirichlet – Neumann, Neumann – Dirichlet) are considered. Finite integral Fourier transform by an angular variable, a finite integral Fourier transform on the Cartesian semiaxis (0; +∞) by an applicative variable z and a Weber hybrid integral transform type on the polar axis (R0; +∞) with n points of conjugation by a radial variable were used to construct solutions of investigated boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The consistent application of inverse integral transforms to the obtained solution in the space of images restores the solutions of the considered parabolic boundary value problems through their integral image in an explicit form in the space of the originals. At the same time, the main solutions to the problems were obtained in an explicit form.

Parabolic Boundary Value Problems in Unlimited Piecewise Homogeneous Wedge-Shaped Hollow Cylinder

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in unbounded by variable z piecewise-homogeneous by radially variable r wedge-shaped by an angularly variable φ hollow cylinder were constructed at first time by the method of classical integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices) in the proposed article. The cases of assigning on the verge of the wedge the boundary conditions of Dirichlet and Neumann and their possible combinations (Dirichlet — Neumann, Neumann — Dirichlet) are considered. Finite integral Fourier transform by an angular variable, a Fourier integral transform on a Cartesian axis by an applicative variable and a hybrid integral transform of the Hankel type of the second kind on a segment of the polar axis with n points of conjugation were used to construct classic solutions of investigated initial-boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The application of inverse integral transforms restores explicitly the solution of the considered problems through their integral image.

Concentrations Problem for Solutions to Compressible Navier–Stokes Equations

A three-dimensional initial-boundary value problem for the isentropic equations of the dynamics of a viscous gas is considered. The concentration phenomenon is that, for adiabatic exponent values $$\gamma \leqslant 3{\text{/}}2$$ , the finite energy can be concentrated on arbitrarily small sets. It is proved that, in the critical case γ = 3/2, the norm of the density of kinetic energy in the logarithmic Orlicz space is bounded by a constant that depends only on the initial and boundary data. This eliminates the possibility of the concentration phenomenon.

PARABOLIC BOUNDARY VALUE PROBLEMS IN A PIECEWISE HOMOGENEOUS WEDGE-SHAPED SOLID CYLINDER

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in piecewise homogeneous wedge-shaped solid cylinder were constructed at first time by the method of integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices). The cases of assigning on the verge of the wedge the boundary conditions of Dirichlet and Neumann and their possible combinations (Dirichlet – Neumann, Neumann – Dirichlet) are considered. Finite integral Fourier transform by an angular variable $\varphi \in (0; \varphi_0)$, a Fourier integral transform on the Cartesian segment $(-l_1;l_2)$ by an applicative variable $z$ and a hybrid integral transform of the Hankel type of the first kind on a segment $(0;R)$ of the polar axis with $n$ points of conjugation by an radial variable $r$ were used to construct solutions of investigated initial-boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The application of inverse integral transforms restores explicitly the solution of the considered problems through their integral image. The structure of the solution of the problem in the case of setting the Neumann boundary conditions on the wedge edges is analyzed. Exact analytical formulas for the components of the main solutions are written and the theorem on the existence of a single bounded classical solution of the problem is formulated. The obtained solutions are algorithmic in nature and can be used (using numerical methods) in solving applied problems.

A three-dimensional soil–water coupled FE analysis of hollow cylinder test concerning non-uniform deformation

The hollow cylinder shear test is somewhat controversial due to its non-uniform stresses and strains, and has been analyzed by simple theoretical methods and two-dimensional FE calculations. In this paper, the hollow cylinder test under strain control was carried out numerically by treating the specimen as a three-dimensional initial-boundary value problem considering the inertial forces. At first, besides the known nonuniform strain, the non-uniformities of excess pore water pressure and overconsolidation ratio have been shown to benefit from a soil–water coupled analysis that employs the SYS Cam-clay model. Then, the influence of the specimen geometries, including wall thicknesses, heights and outer diameters on the non-uniformity was investigated sequentially. A new method for evaluating non-uniformity was proposed, which is suitable for the three-dimensional analysis. The response under a uniform deformation field, which is indicated by “the perfect path”, was presented to draw a comparison with the apparent behaviors, with non-uniformities taken into consideration. It should be noted that there is a critical height to prevent failure at the specimen ends according to the apparent behavior. Finally, the torque-controlled experiment indicated that 4 ribs could not transfer the torque reliably while 6 or 8 ribs were feasible.

Dynamical two-dimensional models of solid-fluid interaction

The present paper is devoted to the construction and investigation of two-dimensional hierarchical models for solid-fluid interaction. Applying the variational approach, the three-dimensional initial-boundary value problem is reduced to a sequence of two-dimensional problems and the existence and uniqueness of their solutions in suitable functional spaces is proved. The convergence of the sequence of vector-functions of three space variables to the solution of the original problem is proved and under additional regularity conditions the rate of approximation is estimated.

On Dynamical Three-Dimensional Fluid-Solid Interaction Problem

Abstract The present paper is devoted to the investigation of one dynamical three-dimensional mathematical model of the fluid-solid interaction. The variational formulation of the corresponding initial boundary problem is considered and a problem for abstract second order evolution equation is formulated, which is a generalization of the three-dimensional initial boundary value problem. For the stated abstract problem the existence and uniqueness of solution, and the energy equality are proved, which yield the corresponding result for the dynamical three-dimensional problem of fluid-solid interaction.

Electron Kinetics in Nonisothermal Plasmas with Spatially Two-Dimensional Structures

A method for the study of the steady-state nonlocal electron kinetics in radially and axially inhomogeneous cylindrical plasmas is presented. The method is based on the solution of the relevant space-dependent kinetic equation for the electron velocity distribution using the two-term expansion. A three-dimensional initial boundary value problem for the isotropic distribution component with the total energy as the evolution variable of the kinetic problem has to be treated. The technique is applied to analyze the electron kinetics in a cylindrical DC column plasma under impact of a space-independent axial field and a radially increasing radial field. Particularly, the spatial relaxation of the electron gas in axial direction in response to a disturbance of the axial homogeneity is investigated. A detailed analysis of the spatially resolved energy balance is given. Considerable modification of the results with respect to those obtained by earlier studies of axially homogeneous DC column plasmas, as well as of relaxation processes in one spatial dimension, has been found.

On an initial boundary-value problem for the equation of magnetohydrodynamics with the hall and ion-slip effects

This paper is concerned with the three-dimensional initial boundary-value problem for the equations of magnetohydrodynamics with additional nonlinear terms stemming from a more general relationship between the electric field and the current density. The problem governs the motion of a viscous incompressible conducting liquid in a bounded container with an ideal conducting surface. The existence of a solution which is close to a certain basic solution is proved. The solution is found in the anosotropic Sobolev spaces W p 2,1 with p>5/2. The proof relies on the theory of general parabolic initial boundary-value problems. Bibliography: 16 titles.

Three-dimensional initial boundary-value problem of the convection of a viscous weakly compressible fluid. II. Uniqueness and stability of generalized solutions

We continue our study of the three-dimensional initial boundary-value problem of the convection of a viscous thermally inhomogeneous weakly compressible fluid which fills a cavity in a solid body. We prove theorems on the uniqueness of the generalized solution of this problem and its continuity with respect to initial conditions and perturbations. We obtain estimates of exponential type for the decay of solutions (in the mean) for large time.

Three-dimensional initial boundary-value problem of the convection of a viscous weakly compressible fluid. I. Global solvability

By using asymptotic methods, we study the three-dimensional initial boundary-value problem of the convection of a viscous thermally inhomogeneous weakly compressible fluid which fills a cavity in a solid body. A theorem about the global solvability of this problem (with respect to time) is proved. For solving this problem, we suggest a convergent iteration process of a special form.