The quasistatics thermal stress analysis in coated half-space with mixed boundary heating conditions

Побудовано розв’язок плоскої квазістатичної задачі термопружності для півпростору з покриттям, на границі якого в смузі певної ширини задана температура, а зовні відбувається теплообмін за законом Ньютона. Розв’язок отримано із використанням інтегральних перетворень Лагерра й Фур’є та методу рядів Фур’є при розв’язуванні послідовностей парних інтегральних рівнянь. Наведено результати числового аналізу термонапруженого стану в півпросторі та покритті залежно від відносної товщини покриття та інтенсивності охолодження.Analysis of thermal stresses in bodies with coatings is important for many engineering researches. Taking into account the actual operating conditions of these structures frequently leads to mixed heating condition. The steady problem of thermoelasticity with mixed boundary conditions currently is sufficiently investigated. However, the corresponding transient problem, despite its relevance, is poorly understood. This is due to mathematical difficulties that arise in applying the integral Laplace transform. The authors of this paper developed a new effective method of constructing solutions of mixed boundary-value non-stationary problems. The half-space with a coating, on the surface of which on the band of 2d width the temperature distribution is given and outside of this area the heat transfer according to the Newton's law is performed, is analysed in the work. On the surface of separation of materials of half-space and coating the conditions of ideal thermomechanical contact are satisfied. The initial temperature of the coating and half-space is equal to zero. To the heat conductivity problem the Laguerre integral transformation in time variables and integral Fourier transformation in spatial variable are applied. As a result the triangular sequence of ordinary differential equations is obtained. The general solution of these sequences is obtained in the form of algebraic convolution. Taking into account the mixed boundary conditions results in dual integral equations. For solution of this problem the method of Newton's series is proposed. Taking advantage of this method the problem is reduced to the infinite system of algebraic equations, for which the convergence of reduction procedure is proved. The solution of thermoelasticity problem is built using resulting temperature field in the assumption, that the border of coating is free of load. The solution is obtained in the form of series in Laguerre polynomials. Calculations were carried for the half-space made of titanium alloy and ceramic coating

Reissner’s plate bilateral bending containing coaxial through-the-thickness slit and crack taking into account contact zone width of its faces

Досліджено напружено-деформований стан ізотропної пластини з прямолінійними співвісними наскрізними щілиною та тріщиною за двобічного згину розподіленими моментами на нескінченності. Береги дефектів до прикладання зовнішнього навантаження були вільними від нього, а під дією згинальних моментів на нескінченності береги тріщини прийшли у гладкий контакт уздовж області сталої ширини поблизу однієї з основ пластини. На основі методів теорії функцій комплексної змінної і комплексних потенціалів плоскої задачі теорії пружності та теорії згину пластин за Рейсснером розв’язок задачі зведено до системи сингулярних інтегральних рівнянь як на тріщині, так і на щілині, яку розв’язано числово за допомогою методу механічних квадратур. Побудовано графічні залежності для контактного зусилля між берегами тріщини, коефіцієнтів інтенсивності зусиль і моментів за різних параметрів задачі.Construction of the cracked plate bending problems solutions forms an important and actual in terms of theory development and practical applications scientific direction. They make it possible to determine stress and displacement distribution near the defects tips, as well as make recommendations for selecting optimal geometrical, physical and mechanical characteristics of plates depending on operating conditions in order to prevent destruction of engineering constructions. In this paper the stress-strain state of boundless isotropic plate with coaxial through-the-thickness slit and crack, the faces of which are free from the external loading is investigated. The plate is under the action of the uniformly distributed in a remote part bending moments, which vectors are parallel and perpendicular to the axe of the defects. It is assumed that under external loading the crack faces come in a smooth contact on all crack length along the two-dimensional area of constant width near the upper plate basis. As a result of the crack faces contact the solution of problem is presented in the form of two related problems solutions: the theory of elasticity plane problem and the problem of plates bending based on the equations of Reissner theory. On the basis of complex variable function theory methods and complex potentials the system of singular integral equations is obtained which is reduced by the mechanical quadratures method to the infinite system of linear algebraic equations. This system is solved numerically by the method of reduction using Gauss with a choice of main entry. The numerical analysis of problem at some parameters values is carried out and graphic dependences for contact force between the faces of crack, force and moment intensity factors are constructed. In particular cases known in the scientific literature results for Reissner’s plate bending problems with one crack considering the contact zone width of its faces, with two coaxial slits as well as for appropriate problems solved using equations of the classical theory of plates bending are obtained

Investigation of surface structure, electrokinetic and stability properties of highly dispersed Ho₂O₃-Yb₂O₃/SiO₂ nanocomposites

A series of highly dispersed Ho2O3–Yb2O3/SiO2 nanocomposites was synthesized using a liquid-phase method and examined using Fourier transform infrared (FTIR) spectroscopy, X-ray diffraction (XRD), nitrogen adsorption–desorption, transmission electron microscopy (TEM), Scanning Electron Microscopy (SEM), and photon correlation spectroscopy (PCS). X-ray fluorescence spectrometry (XRF) confirmed a similar amount of weight percentage of Ho, Yb and Si oxides in the prepared samples. Samples HoYbSi1 (Ho2O3:Yb2O3:SiO2 = 0.5:10:89.5, wt. %), HoYbSi2 (Ho2O3:Yb2O3:SiO2 = 1:10:89, wt. %) and HoYbSi3 (Ho2O3:Yb2O3:SiO2 = 2:10:88, wt. %) calcined at 550 °C are amorphous. TEM and SEM analysis confirm a sphere-like morphology with a quite homogeneous size and shape. As compared with the initial silica, the agglomerated particles of nanocomposites in the aqueous medium are in the range from 200 to 850 nm according to PCS data. The effect of anionic polyacrylic acid (PAA) adsorption on fumed silica (SiO2) and Ho2O3–Yb2O3/SiO2 nanocomposite surfaces on suspension stability was studied. The turbidymetry method was used to monitor the initial silica and triple nanooxides suspensions stability as a function of time

Modelowanie stanu naprężenia dwuwymiarowego ośrodka sprężystegozawierającego doskonałe i niedoskonałe kontaktujące cienkie inkluzje oraz nakładki

This study considers modelling of two-dimensional stress state of solids containing thin elastic inclusions. In modeling the coupling principle for continua of different dimension is utilized. Basing on the model of inclusion under the perfect contact three other models of imperfect contact are developed. The simplest one is a model of thin inclusion, which is completely delaminated at certain segments. Two other models take into account a smooth contact between inclusion and a solid, and also a contact with friction. The developed models are easy to introduce into the used boundary element approach. The model of inclusion, completely debonded at one face, is also used in modeling of solids with thin elastic overlays or stringers.W pracy przy użyciu zasady sprzężenia kontinuów różnowymiarowych rozważane są sposoby matematycznego modelowania zagadnienia płaskiego dla ośrodków sprężystych zawierających cienkie inkluzje. Zasadniczy model matematyczny dla cienkiej inkluzji połączonej w doskonały sposób z tarczą uwzględnia możliwość sprężystego odkształcenia w kierunkach: poprzecznym i wzdłużnym względem osi inkluzji oraz efekty jej zginania. Zostały sprecyzowane trzy szczególne modele niedoskonałego kontaktu inkluzji. Najprostszym z nich jest model cienkiego wtrącenia, które wzdłuż pewnych segmentów jest odseparowane od macierzy. Dwa inne modele uwzględniają gładki kontakt inkluzji z ciałem oraz kontakt z tarciem. Opracowane modele łatwo łączy się ze schematem metody elementów brzegowych. Model cienkiej inkluzji zupełnie odseparowanej wzdłuż jednej strony nadaje się również do badania ośrodków z cienkim wzmocnieniem powierzchniowym (nakładką)

Boundary element analysis of anisotropic thermomagnetoelectroelastic solids with 3D shell-like inclusions

The paper presents novel boundary element technique for analysis of anisotropic thermomagnetoelectroelastic solids containing cracks and thin shell-like soft inclusions. Dual boundary integral equations of heat conduction and thermomagnetoelectroelasticity are derived, which do not contain volume integrals in the absence of distributed body heat and extended body forces. Models of 3D soft thermomagnetoelectroelastic thin inclusions are adopted. The issues on the boundary element solution of obtained equations are discussed. The efficient techniques for numerical evaluation of kernels and singular and hypersingular integrals are discussed. Nonlinear polynomial mappings are adopted for smoothing the integrand at the inclusion’s front, which is advantageous for accurate evaluation of field intensity factors. Special shape functions are introduced, which account for a square-root singularity of extended stress and heat flux at the inclusion’s front. Numerical example is presented

Self-regular stress integral equations for the axisymmetric elasticity

The stress hypersingular integral equations of axisymmetric elasticity are considered. The singular and hypersingular integrals are regularized using the imposition of auxiliary polynomial solution, and self-regular integral equations are obtained for bounded and unbounded domains. The stress-BEM formulation is considered basing on the proposed equations. Considered numerical examples show high efficiency of the proposed approach. New problem for inclusion in finite cylinder is considered

Bending of Reissner's plate containing cracks with the account of their faces contact zone width

W pracy zbadano zagadnienie zginania nieograniczonej izotropowej płyty obciążonej jednorodnie rozłożonymi momentami gnącymi w nieskończoności. Płyta jest osłabiona trzema równoległymi szczelinami, których brzegi są wolne od obciążeń zewnętrznych. Zakłada się, że brzegi szczelin w górnej swojej części kontaktują się wzdłuż całej długości, a strefa kontaktu ma stałą szerokość dla każdej szczeliny. Rozwiązanie zagadnienia jest superpozycją rozwiązań dwóch zagadnień: tarczy płaskiej oraz zagadnienia zginania płyty typu Reissnera. Wykorzystując metody teorii funkcji zmiennych zespolonych i potencjałów zespolonych, otrzymano układ osobliwych równań całkowych rozwiązywanych numerycznie metodą mecha- nicznych kwadratur. Przedstawiono wyniki analizy numerycznej przykładów dla jednej, dwóch i trzech szczelin.This paper considers the bending of unbounded isotropic plate loaded at infinity with uniformly distributed bending moments. The plate is weakened with three collinear cracks with traction-free faces. It is assumed that the crack faces are in a smooth contact on the top face of a plate along their length. The contact region for each crack has a constant height. Due to the contact of crack faces the solution of the problem is obtained as a superposition of two ones: plane stress problem and problem of Reissner plate bending. Basing on the complex variable method the system of singular integral equations is ob- tained. It is solved numerically using the mechanical qudrature technique. The analysis of numerical data is provided

Bending of Orthotropic Plate Containing a Crack Parallel to the Median Plane

This paper considers cylindrical bending of the plate containing a crack parallel to plate's faces. The analytical model of the problem is obtained using the improved theory of plates bending, which considers transverse deformation of the plate. Received analytical results are compared with the numerical data of the boundary element approach, which is modified to suit the considered contact problem. The results of analytical and numerical techniques are in a good agreement both for the isotropic and anisotropic plates

Doubly Periodic Sets of Thin Branched Inclusions in the Elastic Medium: Stress Concentration and Effective Properties

This paper considers the doubly periodic problem of elasticity for anisotropic solids containing regular sets of thin branched inclusions. A coupling principle for continua of different dimension is utilized for modeling of thin inhomogeneities and the boundary element technique is adopted for numerical solution of the problem. The branches of the inclusion can interact both inside the representative volume element and at the interface of neighbor representative elements. A particular example of the elastic medium reinforced by a doubly periodic set of I-beams is considered. Stress intensity and stress concentration inside and outside thin inclusions are determined. The dependence of the effective mechanical properties of the reinforced composite material on the volume fraction of the filament and its rigidity is obtained