One-dimensional Boundary Value Problem Research Articles

Frictional heating of the brake disc with essential nonlinearity thermal barrier coating

A mathematical model to investigate temperature field distributions in a brake disc with thermal barrier coating (TBC) was proposed. For this purpose, a one-dimensional boundary-value heat conduction problem was formulated for the coating (TBC) - substrate (disc) system. It was assumed that the materials of both the coating and the substrate are thermally sensitive and characterized by essential thermal nonlinearity. The surface of the coating was heated by a heat flux with intensity proportional to the friction power during single braking. Thermal contact between the coating and substrate is perfect. The solution to the problem was obtained by means of an integro-interpolation variant of the method of lines and the DIFSUB package for numerical solving of systems of ordinary differential equations. Quasi-static thermal stresses in the strip with taking into account change in temperature mechanical properties, were determined, too. Numerical analysis was conducted for a coating made from Yttria-Stabilized Zirconium ZrO2–3Y2O3 (YSZ) deposited on an AISI 1040 steel disc.

Mathematical model for the electrochemical impedance response of a continuous glucose monitor

A mathematical model is developed for the impedance response of immobilized glucose oxidase electrochemical biosensors. The coupling between the homogeneous reactions and heterogeneous reactions considered in the model included anomerization between α-d-glucose and β-d-glucose and four reversible enzymatic catalytic reactions transforming β-d-glucose and oxygen into gluconic acid and hydrogen peroxide. The electroactive hydrogen peroxide was considered to be reversibly oxidized or reduced at the electrode. The electrochemical system was modeled mathematically as a one-dimensional boundary-value problem and solved by use of Newman's BAND algorithm. The corresponding impedance was calculated for each specified frequency. The resulted limiting current, reaction profiles, and impedance response provide insights into the influence of system parameters such as interstitial glucose concentration and enzymatic rate constants. This model has a potential application in predicting sensor design and diagnosing sensor failure mechanisms.

The use of two-point Taylor expansions in singular one-dimensional boundary value problems I

We consider the second-order linear differential equation (x+1)y″+f(x)y′+g(x)y=h(x) in the interval (−1,1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet–Neumann). The functions f(x), g(x) and h(x) are analytic in a Cassini disk Dr with foci at x=±1 containing the interval [−1,1]. Then, the end point of the interval x=−1 may be a regular singular point of the differential equation. The two-point Taylor expansion of the solution y(x) at the end points ±1 is used to study the space of analytic solutions in Dr of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the analytic solutions when they exist.

Non-Conventional Thermodynamics and Models of Gradient Elasticity.

We consider material bodies exhibiting a response function for free energy, which depends on both the strain and its gradient. Toupin–Mindlin’s gradient elasticity is characterized by Cauchy stress tensors, which are given by space-like Euler–Lagrange derivative of the free energy with respect to the strain. The present paper aims at developing a first version of gradient elasticity of non-Toupin–Mindlin’s type, i.e., a theory employing Cauchy stress tensors, which are not necessarily expressed as Euler–Lagrange derivatives. This is accomplished in the framework of non-conventional thermodynamics. A one-dimensional boundary value problem is solved in detail in order to illustrate the differences of the present theory with Toupin–Mindlin’s gradient elasticity theory.

Modelling load transfer and mixed-mode fracture of ductile adhesive composite joints

A new general computational framework for the stress and effective strength analysis of ductile adhesive composite joints is proposed. Composite adherends are modelled using First Order Shear Deformation Theory accounting for bending-extension coupling and an asymptotic approach for the bondline within deformation theory of plasticity is employed. The governing one-dimensional boundary value problem is solved using an efficient finite-difference scheme in Matlab. Based on the proposed stress solution, different failure criteria motivated either by strength criteria or fracture mechanics are implemented and benchmarked against experimental data for an effective joint strength assessment.A comparison to numerical finite element analyses demonstrates the model's ability to render the adhesive stress field accurately. The impact of different stress-strain curve approximations and effective stress quantities within deformation theory of plasticity are studied and discussed. Finally, effective joint strength predictions are compared to experimental data from literature. The paper concludes with a discussion on distinct failure criteria for elastic-plastic adhesive joints and recommendations for reliable accurate joint strength predictions.

One-dimensional nonlinear boundary value problems with variable exponent

In this paper, a class of nonlinear differential boundary value problems with variable exponent is investigated. The existence of at least one non-zero solution is established, without assuming on the nonlinear term any condition either at zero or at infinity. The approach is developed within the framework of the Orlicz-Sobolev spaces with variable exponent and it is based on a local minimum theorem for differentiable functions.

Thermal Stresses Due to Frictional Heating with Time-Dependent Specific Power of Friction

Abstract In this paper influence of temporal profile of the specific friction power (i.e. the product of the coefficient of friction, sliding velocity and contact pressure) on thermal stresses in a friction element during braking was investigated. Spatio-temporal distributions of thermal stresses were analytically determined for a subsurface layer of the friction element, based on the model of thermal bending of a thick plate with unfixed edges (Timoshenko and Goodier, 1970). To conduct calculations, the fields of dimensionless temperature were used. These fields were received in the article (Topczewska, 2017) as solutions to a one-dimensional boundary-value problem of heat conduction for a semi-space heated on its outer surface by fictional heat flux with three, different time profiles of the friction power.

Two-Dimensional Mixed Problem of Thermoelasticity for a Semistrip

We study the stress state of a semistrip with conditions of smooth contact imposed on one of its lateral surfaces and coupling conditions given on the other surface under the action of normal loading and temperature on the end face. The original problem is reduced to a one-dimensional vector boundary-value problem by the method of integral transformations. By using the tools of matrix differential calculus and Green matrix function, we reduce the obtained problem to a singular integral equation for the derivative of displacements on the end face of the semistrip and solve this equation by the method of orthogonal polynomials. We perform the numerical analysis of fields of displacements and stresses inside the semistrip and establish the zones of tensile stresses on the lateral face of the semistrip and the conditions of their formation.

Using Asymptotic Analysis for Developing a One-Dimensional Substance Transport Model in the Case of Spatial Heterogeneity

We study a solution with an internal transition layer of a one-dimensional boundary value problem for the stationary reaction–advection–diffusion differential equation that arises in mathematical modeling of transport phenomena in the surface layer of the atmosphere in the case of non-uniform vegetation on the assumption of space isotropy along one of the horizontal axes and neutral atmospheric stratification. The parameters of the model at which a boundary value problem has a stable stationary solution with an internal transition layer localized near the boundary between different vegetation types are provided. The existence of such a solution and its local Lyapunov stability and uniqueness are proven. The results can be used for developing multidimensional substance transfer models in the case of a spatial heterogeneity.

Influence of the Friction Power on Temperature in the Process of Braking

For three experimental dependences used to describe the changes in the specific power of friction as a function of the braking time, we construct the exact solutions of the corresponding thermal problems of friction. For this purpose, we use the Duhamel formula and the well-known analytic solution of a one-dimensional boundary-value problem of heat conduction for two semiinfinite bodies and a constant power of friction on the contact surface. We illustrate the application of obtained solutions to modeling the process of friction heating of a pad–disk tribosystem. We also study the influence of the chosen time profiles of friction power on the temperature distribution in a cast-iron (ChNMKh) disk and in a pad made of FM–16L retinax.

Exact analytical solution for a pie-shaped wedge thick plate under oscillating load

A finite elastic wedge-shaped thick plate is considered. One of the faces is based on a rigid base, and the other is exposed to a dynamic oscillating load through an absolutely rigid overlay. On the side faces, conditions of sliding contact are fulfilled; at its end, the stresses are given. The solution is based on a special linear transformation of Lame’s equations and application of the vector integral transformations method. The proposed approach leads to a one-dimensional vector boundary value problem for which an exact solution is constructed. The analysis of the eigenfrequency value distribution and an estimation of the edge resonance frequency are done. An analogous problem is solved for the case when at the lower wedge plate’s face the sliding conditions are given. In order to establish the possibility of separation of the lower base, a comparison of the obtained values of the stress on the bottom plate with the stresses arising in the analogous formulation of the static problem for wedge plate considering its own weight is worked out.

Continuity of the Solutions of One-Dimensional Boundary-Value Problems in Hölder Spaces with Respect to the Parameter

We introduce the most general class of linear boundary-value problems for systems of ordinary differential equations of order r ≥ 2 whose solutions belong to a complex Holder space C n+r,α ([a, b]), where n ∈ ℤ +, 0 < α ≤ 1 and [a, b] ⊂ ℝ. We establish sufficient conditions under which the solutions of these problems continuously depend on the parameter in the Holder space C n + r , α ([a, b]).

Positive solutions for one-dimensional third-order p-Laplacian boundary value problems

In this paper, we give improved results on the existence of positive solutions for the following one-dimensional p-Laplacian equation with nonlinear boundary conditions: \t\t\t{(ϕp(y″))′+b(t)g(t,y(t))=0,0<t<1,λ1ϕp(y(0))−β1ϕp(y′(0))=0,λ2ϕp(y(1))+β2ϕp(y′(1))=0,y″(0)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} (\\phi_{p} ( y'' )) ' + b ( t ) g ( t, y ( t ) ) = 0, \\quad 0 < t < 1, \\\\ \\lambda_{1}\\phi_{p} ( y ( 0 ) ) - \\beta_{1} \\phi_{p} ( y' ( 0 ) ) = 0, \\\\ \\lambda_{2}\\phi_{p} ( y ( 1 ) ) + \\beta_{2} \\phi_{p} ( y' ( 1 ) ) = 0,\\qquad y'' ( 0 ) = 0, \\end{cases} $$\\end{document} where phi_{p} ( s ) = | s | ^{ p-2 } s, p >1 . Constructing an available integral operator and combining fixed point index theory, we establish some optimal criteria for the existence of bounded positive solutions. The interesting point of the results is that the term b ( t ) may be singular at t=0 and/or t=1. Moreover, the nonlinear term g(t, y) is also allowed to have singularity at y=0. In particular, our results extend and unify some known results.

ПОСТРОЕНИЕ БАЗИСА ДЛЯ ОДНОМЕРНЫХ КРАЕВЫХ ЗАДАЧ В СИСТЕМАХ СИМВОЛЬНЫХ ВЫЧИСЛЕНИЙ

ПОСТРОЕНИЕ БАЗИСА ДЛЯ ОДНОМЕРНЫХ КРАЕВЫХ ЗАДАЧ В СИСТЕМАХ СИМВОЛЬНЫХ ВЫЧИСЛЕНИЙ

Continuity of the Solutions of One-Dimensional Boundary-Value Problems with Respect to the Parameter in the Slobodetskii Spaces

For a system of linear ordinary differential equations of the first order, we study the broadest class of inhomogeneous boundary-value problems whose solutions belong to the Slobodetskii space W + 1 ((a, b),ℂ m ) with m ∈ ℕ, s > 0, and p ∈ (1,∞). We prove a theorem on the Fredholm property of these problems. We also establish conditions under which the problems are uniquely solvable in the Slobodetskii space and their solutions are continuous in this space with respect to the parameter.

Finite-Layer Method: Evaluation of Stresses and the Modal Components of Energy Release Rate on the Midplane of Edge-Cracked Composite Specimens

On the basis of the finite-layer method, a technique for calculating the two-dimensional stress-strain state and interlaminar stresses in edge-cracked specimens in tests on determination of the Mode I, II, and III fracture toughnesses of composites is proposed. Taking into account the symmetry and antisymmetry of specimens, the calculation is reduced to a boundary-value problem for a system of partial differential equations describing the deformation of one of the layers. Using the method of straight lines, the two-dimensional problem is reduced to a one-dimensional boundary-value problem, which is solved by the stable Godunov–Grigorenko numerical method. Calculations for specimens subjected to three types of the tests are performed, and the distributions of normal and tangential stresses on the specimen midplane and the distributions of modal components of the fracture toughnesses GI, GII, and GIII along the delamination front line are presented. Comparisons with results of the one-dimensional solution and with known results obtained by the virtual crack closure technique are presented.

Propagation of converging spherical deformation waves in a heteromodular elastic medium

The unsteady one-dimensional boundary-value problem of shock deformation of a medium bounded by a sphere is solved. The propagation of converging deformation wave fronts in an elastic material with different tensile and compressive strengths is studied. A boundary condition is obtained that provides the formation of a converging spherical shock wave with constant velocity. The impact conditions on the boundary of the heteromodular sphere are determined that can lead to the formation of a transition zone (a spherical layer of constant density) between the compression and tension regions.

Stochastic Model of Heat Conduction with Stochastic Boundary Conditions

One-dimensional stochastic models of heat conduction with different types of boundary conditions have been developed. Boundary-value problems for mean temperature and temperature field dispersion in the region of heat conduction have been formulated. A generalization of the indicated one-dimensional boundary-value problems to the case of a large number of measurements in invariant form is given.

Buckling behavior of a radially loaded corrugated orthotropic thin-elliptic cylindrical shell on an elastic foundation

This paper is aimed to investigate how the corrugation parameters and the Winkler foundation affect the buckling behavior of isotropic and orthotropic thin-elliptic cylindrical shells with cosine-shaped meridian subjected to radial loads. The buckling three-dimensional equations of the shell are amended by including the Winkler foundation modulus based on the Flügge thin shell theory and Fourier's approach is used to deform the displacement fields as trigonometric functions in the longitudinal direction of shell. Using the transfer matrix of the shell, the governing equations of buckling can be written in a matrix differential equation of variable coefficients as a one-dimensional boundary-value problem that is solved numerically as an initial-value problem by the Romberg integration approach. The proposed model is adopted to determine the basic loads and the corresponding buckling deformations for the symmetrical and antisymmetrical modes of buckling. The sensitivity of the buckling behavior and bending deformations to the corrugation parameters, Winkler foundation moduli, ellipticity and orthotropy of the shell structure is studied for different type-modes of buckling. The obtained results indicate that this design model allows us to find out how the local properties of the shell and its stiffness in existence of an elastic foundation are related and would serve as benchmarks for future works in this important area.

B-splines in the Analog Equation Method for the generalized beam analysis including warping effects

In this paper, the Analog Equation Method (AEM), a boundary element based method, is employed for the analysis of homogenous beams of arbitrary cross section (thin- or thick-walled) taking into account nonuniform warping and shear deformation effects (shear lag due to both flexure and torsion), considering a B-spline approximation for the fictitious loads of a substitute problem. The fictitious loads are established using a BEM-based technique and the solution of the original problem is obtained from the integral representation of the solution of the substitute problem. The beam is subjected to the combined action of arbitrarily distributed or concentrated axial and transverse loading, as well as to bending, twisting and warping moments. Its edges are subjected to the most general boundary conditions, including also elastic support. Nonuniform warping distributions are taken into account by employing four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions, which are obtained by solving corresponding boundary value problems, formulated exploiting the longitudinal local equilibrium equation. Ten one-dimensional boundary value problems are described by second-order differential equations. Integrating B-splines in the AEM technique, the computational cost is efficiently reduced while the results are highly accurate.