Solution Of Value Problem Research Articles

Discrete defect plasticity and implications for dissipation

Inelastic deformation of solids is almost always, if not always, associated with the evolution of discrete defects such as dislocations, vacancies, twins and shear transformation zones. The focus here is on discrete defects that can be modeled appropriately by continuum mechanics, but where the discreteness of the carriers of plastic deformation plays a significant role. The formulations are restricted to small deformation kinematics and the defects considered, dislocations and discrete shear transformation zones (STZs), are described by their linear elastic fields. In discrete defect plasticity both the stress–strain response and the partitioning between defect energy storage and defect dissipation are outcomes of an initial/boundary value problem solution. For such defects an explicit expression for the dissipation rate is presented and, because there is a length scale associated with the discrete defects, the stress–strain response and the evolution of the dissipation rate can be size dependent. Discrete dislocation plasticity modeling results are reviewed that illustrate the implications of defect dissipation evolution for friction, fracture, fatigue crack growth and thermal softening. Examples are also given of the consequences in constrained shear of three modes of the evolution of discrete defects for the size dependence of the stress–strain response and the associated dissipation. For a purely mechanical formulation the Clausius–Duhem inequality specializes to the requirement that the dissipation rate is non-negative. Requiring a non-negative dissipation rate for all points of a body and for all time imposes restrictions on kinetic relations for the evolution of discrete defects. In statistical mechanics, the Clausius–Duhem inequality can be violated for sufficiently small regions for a sufficiently short time. Explicit kinetic relations for discrete dislocation plasticity dissipation and for discrete STZ plasticity dissipation identify conditions that can lead to a negative dissipation rate. In continuum mechanics, at least in some circumstances, satisfaction of the Clausius–Duhem inequality can be regarded as a stability requirement. Nevertheless, simple one-dimensional continuum calculations illustrate that there can be a negative dissipation rate over a small region and for a short time period with overall stability maintained. Implications for discrete defect plasticity modeling are briefly discussed.

ВПЛИВ СТЕПЕНІ ДИСКРЕТИЗАЦІЇ ТЕХНІЧНОГО ОБ’ЄКТУ НА РЕЗУЛЬТАТИ ПРОГНОЗУ ЗА МГЕ

A significant difference between soils and homogeneous elastic bodies is that under action External loads residual deformations are always concomitant elastic, even at low loads. The sum of residual and elastic deformation is the total deformation of the soil base. The simultaneous presence in the soil of zones operating in both elastic and plastic zones requires the involvement of the theory of elasticity and plasticity to model its behavior [1-4]. It is known that the solution of the mixed problem of the theory of elasticity and the theory of soil plasticity brings the results of sedimentation calculations much closer to reality. The current trend towards automated calculation methods has dramatically changed the priorities towards the need to develop more reliable mathematical models of nonlinearly deformed soil massifs composed of layers with different properties. Urban planning and modern industry require the construction of responsible structures on increasingly complex engineering and geological conditions for which the rational type of foundations are piles. Widespread use of pile foundations requires the development of reliable methods for their calculation in order to obtain reliable design solutions. Therefore, the current stage of development of soil mechanics is characterized by an active transition to new computational models that more fully reflect the nonlinearity of deformation and rheological properties of soils and these issues remain an urgent problem today. The paper uses the numerical method of boundary elements, which emerged as a result of further theoretical development of a wide class of numerical methods, united under the common name of finite element theory. It is based on the existence of a fundamental solution of the boundary value problem, which corresponds to the source function given in the form of the Dirac delta function. The availability of a fundamental solution is very important from a practical point of view for the numerical implementation of the IHE task. A fundamental solution is a partial solution of the Laplace equation for a semi-infinite domain for a potential value of one given at some point. This type of solution is widely used in boundary value problems and is a Green's function or influence function. In the presence of a fundamental solution, finite elements are used to approximate the boundary of the domain, and the apparatus of classical integral equations is applied to the inner part of the domain/

Designing and Calculating the Nonlinear Elastic Characteristic of Longitudinal-Transverse Transducers of an Ultrasonic Medical Instrument Based on the Method of Successive Loadings.

This paper presents a numerical method for studying the stress–strain state and obtaining the nonlinear elastic characteristics of longitudinal–transverse transducers. The authors propose a mathematical model that uses a direct numerical solution of the boundary value problem based on the plain curved rod equations in Matlab. The system’s stress–strain state and nonlinear elastic characteristic are obtained using the method of successive loadings based on the curved rod’s linearized equations. For most ultrasonic instruments, the operating frequency of ultrasonic vibrations is close to 20 kHz. On the other hand, the received own oscillation frequencies are close to the working range. Using the method of successive loadings in the mathematical complex Matlab, a numerical calculation of the stress–strain state of a flat, curved rod at large displacements has been carried out. The proposed model can be considered an initial approximation to the solution of the spatial problem of the longitudinal–torsional transducer.

About One Variational Problem, Leading to а Biharmonic Equation, and about the Approximate Solution of the Main Boundary Value Problem for this Equation

. Many important questions in the theory of elasticity lead to a variational problem associated with a biharmonic equation and to the corresponding boundary value problems for such an equation. The paper considers the main boundary value problem for the biharmonic equation in the unit circle. This problem leads, for example, to the study of plate deflections in the case of kinematic boundary conditions, when the displacements and their derivatives depend on the circular coordinate. The exact solution of the considered boundary value problem is known. The desired biharmonic function can be represented explicitly in the unit circle by means of the Poisson integral. An approximate solution of this problem is sometimes foundusing difference schemes. To do this, a grid with cells of small diameter is thrown onto the circle, and at each grid node all partial derivatives of the problem are replaced by their finite-difference relations. As a result, a system of linear algebraic equations arises for unknown approximate values of the biharmonic function, from which they are uniquely found. The disadvantage of this method is that the above system is not always easy to solve. In addition, we get the solution not at any point of the circle, but only at the nodes of the grid. For real calculations and numerical analysis of solutions to applied problems, the authors have constructed its unified analytical approximate representation on the basis of the known exact solution of the boundary value problem while using logarithms. The approximate formula has a simple form and can be easily implemented numerically. Uniform error estimates make it possible to perform calculations with a given accuracy. All coefficients of the quadrature formula for the Poisson integral are non-negative, which greatly simplifies the study of the approximate solution. An analysis of the quadrature sum for stability is carried out. An example of solving a boundary value problem is considered.

Solution of the Exterior Boundary Value Problem for the Helmholtz Equation Using Overlapping Domain Decomposition

Solution of the Exterior Boundary Value Problem for the Helmholtz Equation Using Overlapping Domain Decomposition

The Boundary Value Problem with Stationary Inhomogeneities for a Hyperbolic-Type Equation with a Fractional Derivative

The paper presents an analytical solution of a partial differential equation of hyperbolic-type, containing both second-order partial derivatives and fractional derivatives of order below the second. Examples of applying the solution of a boundary value problem with stationary inhomogeneities for a hyperbolic-type equation with a fractional derivative in modeling the behavior of polymer concrete under the action of loads are considered.

Existence of weak solutions to a general class of diffusive shallow medium type equations

Abstract In this article, we prove existence of weak solutions of a general class of diffusive shallow medium type equations. We truncate the coefficients from above and below. Then we prove existence to the problems associated with the truncated vector fields. At last, we show that the approximating solutions converge to the solution of the initial value problem.

A Solution of a Boundary Value Problem with Mixed Conditions for a Four-Dimensional Degenerate Elliptic Equation

The solvability issues of counterpart Holmgren’s boundary value problem with mixed conditions for a degenerate four-dimensional second-order Gellerstedt equation Hu≡ymzktluxx+xnzktluyy+xnymtluzz+xnymzkutt=0, m,n,k,l≡const>0, are studied in the finite domain R4+, where the values of normal derivatives are set on the piecewise smooth part of the boundary and the values of the desired function are set on the remaining part of the boundary. The main results of the work are the proof of the uniqueness of the considered problem solution by using an energy integral’s method and the construction of the solution of counterpart Holmgren’s boundary value problem in explicit form by means of Green’s function method, containing the hypergeometric Lauricella’s function FA4. Using the corresponding fundamental solution for the considered generalized Gellerstedt equation of elliptic type, we construct Green’s function. In addition, formulas of differentiation, some adjacent relations, decomposition formulas, and various properties of Lauricella’s hypergeometric functions were used to establish the main results for the aforementioned problem.

Existence and uniqueness results for a fractional differential equations with nonlocal boundary conditions

In this paper, we establish sufficient conditions for the existence and uniqueness of solution of a boundary value problem of differential equations of fractional order involving the nonlocal boundary condition.

Решение задачи о раскрытии метаном трещин в предельно напряженной зоне пласта

The conditions were studied concerning the cracks opening by in-situ methane in the formation marginal zone during its transition to the maximum stressed state. The calculations are based on the joint use of the fundamental methods of deformable solids mechanics, statics of the flowing medium, and the concept of academician S.A. Khristianovich on the distribution of pressure of intra-layer methane during the movement of the mine face at a constant speed. The stress field of a coal-rock mass containing a seam with a weak interlayer and a working through it, was constructed within the framework of an elastic-plastic problem of the stressed state of a coal-rock mass. It is reduced to the integral equation of the second external boundary value problem of elasticity theory and solved by the boundary element method. The stress distribution in the extremely stressed zone of the formation is constructed by the method of characteristics known in the theory of differential equations. In the model of the ultimate stress state of a structurally homogeneous formation, the limiting zones are formed from the edge itself. In the formation with an unstable interlayer, the interlayer passes to the limiting state first, and at some distance from the edge, the formation itself. After that, the formation and the intermediate layer are deformed as integral whole. The cracks in the formation, which are its slip lines, open when the pressure of in-situ methane exceeds the stresses normal to the contour of the slip line. These stresses are related to the values of the principal stresses found from the solution of the boundary value problem. From this condition, the dimensions of the formation edge zone with free methane surfaces are determined.

On a type-2 fuzzy approach to solution of second-order initial value problem

On a type-2 fuzzy approach to solution of second-order initial value problem

РЕЗОНАНСНІ ВЛАСТИВОСТІ ШАРУВАТОГО ҐРУНТОВОГО МАСИВУ, ЩО МІСТИТЬ ПРОШАРОК З КОЛИВНИМИ ВКЛЮЧЕННЯМИ

The paper is devoted to modeling of the seismic reaction of an inhomogeneous soil massif consisting of three layers that can incorporate oscillating inclusions. The description of the behavior of soil strata is carried out on the basis of the continual approach, within which the dynamics of the medium is determined by a system of equations of motion and equations of state, where the elastic or visco-elastic properties of the medium are indicated. Classical linear models are quite simple and well verified experimentally, but due to the complex structure of natural geomedia, these models cannot always adequately describe wave processes in them. This requires improvement of models in such a way as to take into account the internal structure of materials. In this paper, to describe the shear dynamics of an inhomogeneous soil massif, the equations of motion for a continuous medium in the form of mutually penetrating continua are used, one of which coincides with the Kelvin-Voigt carrier medium, and the other one is described by a set of noninteracting partial oscillators. Various inclusions, cracks, or cavities filled with gases and/or liquids can act as oscillators. The equations of motion are supplemented by the compatibility conditions at the boundaries of the layers. The set of layers is considered, among which only one contains oscillating inclusions. The one-dimensional boundary value problem with a free surface and the harmonic law of deformation of massif's rigid bedrock is solved. Using the exact solution of the boundary value problem, the amplification factor of strains for massifs with characteristics close to natural is calculated. Moreover, the iterative procedure, which makes it possible to write down the final formula for the amplification factor in the case of a layered medium with layers of more than 3 is developed. It is shown that the incorporation of inclusions in one of the layers leads to the appearance of an additional resonance frequency, a shift of resonances to the low-frequency region, the appearance of zones with significant damping of resonance peaks. The results obtained make it possible to improve the computational methods for determining the quantitative parameters of seismic hazard when work on seismic micro zoning of construction and operational sites in the seismic regions of Ukraine is carried out.

On One Solution of the Boundary Value Problem for a Pseudohyperbolic Equation of Fourth Order

On One Solution of the Boundary Value Problem for a Pseudohyperbolic Equation of Fourth Order

On Solution of First Order Initial Value Problems using Laplace Transform in Fuzzy Environment

This study aimed at solving a nonhomogeneous linear first order initial value problem by means of Laplace transform method in fuzzy environment. The conditions for a fuzzy function to be H−differentiable and gH−differentiability are well established. Finally, example is constructed to test the applicability or otherwise of the established results.

Adam-Bash forth-Multan Predictor Corrector Method for Solving First Order Initial value problem and Its Error Analysis

This paper presents fourth order Adams predictor corrector numerical scheme for solving initial value problem. First, the solution domain is discretized. Then the derivatives in the given initial value problem are replaced by finite difference approximations and the numerical scheme that provides algebraic systems of difference equations is developed. The starting points are obtained by using fourth order Runge-Kutta method and then applying the present method to finding the solution of Initial value problem. To validate the applicability of the method, two model examples are solved for different values of mesh size. The stability and convergence of the present method have been investigated. The numerical results are presented by tables and graphs. The present method helps us to get good results of the solution for small value of mesh size h. The proposed method approximates the exact solution very well. Moreover, the present method improves the findings of some existing numerical methods reported in the literature.

Well-posed results for nonlocal biparabolic equation with linear and nonlinear source terms

In this paper, we consider the biparabolic problem under nonlocal conditions with both linear and nonlinear source terms. We derive the regularity property of the mild solution for the linear source term while we apply the Banach fixed-point theorem to study the existence and uniqueness of the mild solution for the nonlinear source term. In both cases, we show that the mild solution of our problem converges to the solution of an initial value problem as the parameter epsilon tends to zero. The novelty in our study can be considered as one of the first results on biparabolic equations with nonlocal conditions.

Метод сеток приближенного решения начально-краевых задач для обобщенных уравнений конвекции-диффузии

В прямоугольной области исследуются начально-краевые задачи для одномерных по пространству обобщенных уравнений конвекции-диффузии с оператором Бесселя и дробными производными в смысле Римана~--- Лиувилля и Капуто порядка $\alpha$ (0<$\alpha$<1) и с граничные условия первого и третьего рода. Уравнение конвекции-диффузии дробного порядка с оператором Бесселя возникает при переходе от трехмерного уравнения конвекции-диффузии дробного порядка к цилиндрическим (сферическим) координатам, в случае, когда решение $u=u(r)$ не зависит ни от~$z$, ни от~$\varphi$. Для численного решения рассматриваемых задач строятся монотонные разностные схемы второго порядка точности по параметрам сетки, аппроксимирующие эти задачи на равномерных сетках. С~помощью метода энергетических неравенств для решения начально-краевых задач получены априорные оценки в дифференциальной и разностной трактовках при предположении существования регулярного решения исходной дифференциальной задачи. Из полученных априорных оценок следуют единственность и устойчивость решения по правой части и начальным данным, а также в~силу линейности разностных задач сходимость решения соответствующей разностной задачи к~решению исходной дифференциальной задачи со скоростью $O(h^2+\tau^2)$.

Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation

In this work, an explicit solution of the initial-boundary value problem for a multidimensional time-fractional differential equation is constructed. The possibility of obtaining this equation from an integro-differential wave equation with a Mittag–Leffler–type memory kernel is shown. An explicit solution to the problem under consideration is obtained using the Laplace and Fourier transforms, the properties of the Fox H-functions and the convolution theorem.

О решении в явном виде краевой задачи Неймана для дифференциального уравнения Бауэра в круговых областях

О решении в явном виде краевой задачи Неймана для дифференциального уравнения Бауэра в круговых областях

Noether property and approximate solution of the Riemann boundary value problem on closed curves

Noether property and approximate solution of the Riemann boundary value problem on closed curves