Quasi-static Thermoelasticity Research Articles

Development of model representations of thermal reaction viscoelastic bodies on the temperature field

Objectives. In recent decades, the relevance of research into the thermal response of solids to a temperature field has increased in connection with the creation of powerful energy emitters and their use in technological operations. There is a significant number of publications describing these processes using mathematical models of dynamic or quasi-static thermoelasticity, mainly for most technically important materials that obey Hooke’s law. However, at elevated temperatures and higher stress levels, the concept of an elastic body becomes insufficient: almost all materials exhibit more or less clearly the phenomenon of viscous flow. The real body begins to exhibit elastic and viscous properties and becomes viscoelastic. A rather complex problem arises: the development of dynamic (quasistatic) thermoviscoelasticity within the framework of the corresponding mathematical models of classical applied thermomechanics and mathematics. The purpose of the work is to consider the open problem of the theory of thermal shock in terms of a generalized model of thermoviscoelasticity under the conditions of classical Fourier phenomenology on the propagation of heat in solids. Three types of intense heating are considered: temperature, thermal, and medium heating. Intensive cooling modes can be equally considered. The task is posed: to develop model representations of dynamic (quasi-static) thermoviscoelasticity that allow accurate analytical solutions of the corresponding boundary value problems on their basis. This direction is practically absent in the scientific literature.Methods. Methods and theorems of operational calculus were used.Results. Model representations of the thermal response of viscoelastic bodies using the proposed new compatibility equation in displacements have been developed.Conclusions. New integro-differential relations are proposed based on linear rheological models for the Maxwell medium and the Kelvin medium, including both dynamic and quasi-static models for viscoelastic and elastic media, generalizing the results of previous studies. The proposed constitutive relations of the new form are applicable to describe the thermal response of quasi-elastic bodies of a canonical shape simultaneously in three coordinate systems with a system-defining parameter, which makes it possible to identify the influence of the topology of the region on the value of the corresponding temperature stresses.

Termostressed state of a three-layer rectangular plate under non-stationary convective heating conditions

The study considers a rectangular isotropic plate with a layered irregular structure. It is convectively non-stationarily heated by an external environment. The initial relationships of the non-stationary heat conduction and thermoelasticity problem are formulated using a five-mode mathematical model based on the shear deformation theory of thermoelasticity. Using the methods of Fourier and Laplace integral transforms, general solutions have been obtained for the non-stationary heat conduction problem and the quasi-static thermoelasticity problem for a hinge-supported plate along its edges. A numerical analysis of the temperature field, radial deflections, normal forces, bending moments, and normal stresses, depending on geometric parameters and the Bi criterion, has been performed for a three-layer plate. The materials of its layers are made of ceramics and metal. The temperature and mechanical parameters have been analyzed for the layering configuration of the plate: metal-ceramic-metal.

Constructing solutions for two-dimensional quasi-static problems of thermomechanics in terms of stresses for bodies with plane-parallel boundaries

A methodology to construct solutions for two-dimensional quasi-static thermomechanical problems for bodies with plane-parallel boundaries (2D-QS thermomechanical problems) is proposed. This approach begins with selecting equations for the plane quasi-static thermoelasticity problem in terms of stresses. The methodology approximates the distribution of non-zero stress tensor components through the body's thickness using cubic polynomials, with coefficients expressed in terms of integral characteristics of the stress tensor components over the thickness variable and their specified boundary values on the body’s lower and upper surfaces. Consequently, the original two-dimensional boundary problem is simplified to a one-dimensional boundary problem for the integral characteristics. For an infinite layer, solutions are found using the Fourier transform along the longitudinal coordinate, while for a strip plate, a finite integral transformation is applied along the transverse coordinate. General solutions for 2D-QS thermomechanical problems are formulated for the selected bodies under non-stationary volume forces and temperature fields. The resulting expressions for the stress tensor components are presented as convolutions of functions representing the boundary values on the bases (and end cross-sections for strip-plates) and functions describing homogeneous solutions to the one-dimensional boundary problems for the integral characteristics of the stress tensor components.

Approximation of Uncoupled Quasi-Static Thermoelasticity Solutions Based on Gaussians

A fast approximation method to three dimensional equations in quasi-static uncoupled thermoelasticity is proposed. We approximate the density via Gaussian approximating functions introduced in the method approximate approximations. In this way the action of the integral operators on such functions is presented in a simple analytical form. If the density has separated representation, the problem is reduced to the computation of one-dimensional integrals which admit efficient cubature procedures. The comparison of the numerical and exact solution shows that these formulas are accurate and provide the predicted approximation rate 2,4,6 and 8.

On the fully coupled quasi-static equations for the thermoelastic halfspace

Influence functions for fully coupled quasi-static thermoelasticity are presented. They can be used to calculate displacements, stresses as well as temperature distributions within a halfspace for arbitrarily shaped and time dependent heat sources or pressure distributions on the surface. To this end, the underlying equations are solved for a mechanically or thermally loaded rectangular element on the surface by potential theory. The solution procedure is described in detail. The obtained results are verified and may easily be transferred to the theory of poroelasticity where the underlying equations are structurally equal. Example calculations for a parabolic pressure field, heat flux and temperature field are performed and simulation results are discussed particularly in comparison to the uncoupled case. Estimations of the extent to which the Gough–Joule effect and its consequences can be neglected are given.

Stressed and strained state of layered cylindrical shell under local convective heating

The stress-strain state of a layered composite cylindrical shell under local heating by the environment due to convective heat exchange has been studied. The equation of the six-modal theory of thermoelasticity and the two-dimensional equation of thermal conductivity of inhomogeneous anisotropic shells are used for this purpose. The solution of the nonstationary heat transfer problem and the quasi-static thermoelasticity problem for a finite hinged orthogonally reinforced shell of symmetric structure is found by the methods of integral Fourier and Laplace transforms. Numerical results are given for the three-layer shell.

Uncoupled Quasistatic Problem of Thermoelasticity for a Two-Layer Hollow Thermally Sensitive Cylinder Under the Conditions of Convective Heat Exchange

We determine the unsteady temperature field depending on the radial coordinate and the thermoelastic state caused in a two-layer hollow thermally sensitive cylinder both by this field and the action of external force loads. The nonlinear boundary-value problem of heat conduction with discontinuous coefficients is reduced by the integro-interpolation method to the Cauchy problem for a system of ordinary differential equations, which is solved numerically. The stressed state is determined from the integral equations of the second kind and integral conditions obtained as a result of direct integration of the problem of quasistatic thermoelasticity in stresses. We study the influence of the temperature dependence of the thermal and mechanical characteristics of the layers of materials on the values and character of distributions of temperature and stresses caused by this dependence in a two-layer cylinder.

Influence of the Solid Shape on the Solution for the Uncoupled Quasi-Static Cyclic Thermoelasticity Problem in a Thermal Layer (Using Regular Solids as Example)

Analytical solutions are obtained for an uncoupled quasi-static cyclic thermoelasticity problem for regular solids: a half-space, a plate, a space with a cylindrical channel, a cylinder, a space with a spherical cavity, and a sphere. The solutions, which are trigonometric Fourier series of temporary variable, are transformed by changing the spatial variable into a form convenient for analysis of temperature oscillations and thermocyclic stresses in the thermal layer. For the solutions found, simple asymptotic dependences are established, suitable for large radii of curvature of the surface and body size. The influence of the solid shape on temperature oscillations and thermocyclic stresses in the thermal layer has been studied.

Generalized convolution quadrature based boundary element method for uncoupled thermoelasticity

Mechanical loads together with changing temperature conditions can be found in a wide variety of fields. Their effects on elastic media are reflected in the theory of thermoelasticity. For typical materials in engineering, very often a simplification of this coupled theory can be used, the so-called uncoupled quasistatic thermoelasticity. Therein, the effects of the deformations onto the temperature distribution is neglected and the mechanical inertia effects as well. The Boundary Element Method is used to solve numerically these equations in three dimensions. Since convolution integrals occur in this boundary element formulation, the Convolution Quadrature Method may be applied. However, very often in thermoelasticity the solution shows rapid changes and later on very small changes. Hence, a time discretisation with a variable time step size is preferable. Therefore, here, the so-called generalised Convolution Quadrature is applied, which allows for non-uniform time steps. Numerical results show that the proposed method works. The convergence behavior is, as expected, governed either by the time stepping method or the spatial discretisation, depending on which rate is smaller. Further, it is shown that for some problems the proposed use of the generalised Convolution Quadrature is the preferable.

Laminated cross-ply cylindrical shell due to transient heating

The stress-deformed state of an antisymmetric cross-ply laminated circular closed cylindrical shell subjected to initial local heating in initially is investigated. For the study, the mathematical model of the shear theory of inhomogeneous shells of the Tymoshenko type is used. The two-dimensional heat equation is deduced under the linear distribution of temperature over the thickness. By methods of Fourier and Laplace integral transformations the solution of the non-stationary heat conduction problem and the quasi-static thermoelasticity problem for a finite simply supported circular cylindrical shell is obtained. Numerical results are presented for four-layer orthotropic composite. Cite as: U. V. Zhydyk, “Laminated cross-ply cylindrical shell due to transient heating,” Prykl. Probl. Mekh. Mat. , Issue 17, 113–120 (2019) (in Ukrainian), https://doi.org/10.15407/apmm2019.17.113-120

Analysis of the thermoelastic state of the inhomogeneous through the thickness isotropic cylindrical shell

The stress-deformed state of a functionally graded isotropic circular closed cylindrical shell subjected to local heating of the outer surface of the environment by means of convective heat transfer is investigated. For the study, the mathematical model of the shear theory of inhomogeneous shells of the Tymoshenko type is used. The two-dimensional heat equation is deduced under the linear distribution of temperature over the thickness. By methods of Fourier and Laplace integral transformations the solution of the non-stationary heat conduction problem and the quasi-static thermoelasticity problem for a finite simply supported circle cylindrical shell is obtained. Numerical results are presented for metal–ceramic composites. Cite as: U. V. Zhydyk, “Analysis of the thermoelastic state of the inhomogeneous through the thickness isotropic cylindrical shell,” Prykl. Probl. Mekh. Mat. , Issue 16, 119–125 (2018) (in Ukrainian), http://doi.org/10.15407/apmm2018.16.119-125

Analysis of the Pump Strength to Extend its Lifetime

The paper deals with the estimation of the residual strength of the body of water jet pump SN-10 /50K type operating in the beyond design lifetime in the line of NPP unit sprinkler pumps. The results of theoretical studies of its stressed-strained state are presented taking into account change in the geometry of body parts, which was observed after completion of the design lifetime. Static strength assessment was carried out for the main operating modes of the pump operation (normal operating conditions and hydraulic tests), as well as for conditions of an emergency situation. Corresponding researches were carried out in the framework of numerical computer simulation on the basis of the finite element method using up-to-date software complexes. 3D finite element models have been developed that take into account actual geometry of the pump components and the forecast of its possible change for a period of extended lifetime. The change in the design geometry is taken into account based on the extrapolation of the data of thickness measurement of the pump body walls obtained during the long service period. Based on the built finite-element models, the tasks of thermal conductivity and thermoelasticity have been consistently solved. The phenomenon of thermal shock on body parts was simulated that allow assessing residual pump strength in case of an emergency. Corresponding simulation was carried out by solving the problems of nonstationary thermal conductivity and the related problem of quasi-static thermoelasticity. Such an approach made it possible to determine the distribution of the temperature field over time under thermal shock, and the distribution of the stressed-strained state parameters of the pump at certain time moments.

Solution of the non-axisymmetric quasistatic thermoelasticity problem for multilayer cylinder with identical Lamé coefficients

За допомогою побудованих функцій Гріна квазістатичної задачі термопружності визначено неосесиметричний термопружний стан багатошарового необмеженого порожнистого циліндра з однаковими коефіцієнтами Ламе шарів за дії внутрішніх та поверхневих джерел тепла і нерівномірного розподілу початкової температури. Досліджено термопружний стан двошарового циліндра, зумовлений нормально розподіленим на зовнішній поверхні циліндра потоком тепла, який рухається по твірній циліндра.

Development of the numerical model for evaluating the temperature field and thermal stresses in structural elements of aircrafts

An approximate method for estimating the thermal stresses of the aircraft key components of the simple geometric shape (the edges of the hull and wings, the nose fairing) has been developed. The mathematical model of such estimates is based on the solution of the quasi-static thermoelasticity problem. The solution is evaluated in the area with curvilinear boundaries, and the shape of these boundaries changes under the influence of thermal and mechanical loads. Thus the computational domain is transformed to an area where the regular Cartesian (structured) grid can be introduced. The initial validation and verification of the developed numerical methodology was carried out. Numerical modeling of temperature fields and thermal stresses in the simplest components of aircraft structures (cylinder blunted over the sphere and the shell) is performed.

Thermoelasticity problem for a multilayer coating/half-space assembly with undercoat crack subjected to convective thermal loading

ABSTRACTThe transient thermal stress crack problem for a half-space with a multilayer coating under thermal surface loading containing an undercoat crack, perpendicular to the interface, is considered. The problem is solved using the principle of superposition and uncoupled quasi-static thermoelasticity. Transient temperature distribution and corresponding thermal stresses for the uncracked multilayer assembly are obtained in a closed analytical form using the model with generalized thermal boundary conditions of heat exchange of a half-space with ambient media via the coating. The crack problem is formulated as a perturbation mixed boundary value problem, in which the crack surface loading should be equal and opposite to the thermal stresses obtained for the uncracked medium, and is reduced to a singular integral equation and solved numerically. Numerical computations are performed for the analysis of influence of the coating upon thermal stresses and thermal stress intensity factor.

Thermostressed state of a pad–disk tribosystem during single braking

The temperature and stress distributions in a disk under single braking have been studied. The finite-element method (FEM) has been used to obtain a numerical solution to the heat-conduction boundary problem for a steadily decelerated rotating disk heated in a local area of the working surface by a frictional heat flow. Given the temperature field, the finite-element method has been used to solve the corresponding quasistatic thermoelasticity boundary problem. The elastostatic boundary problem has been numerically solved separately for a disk loaded with distributed normal and shear stresses in the contact area. The complete field of stresses in the disk has been determined by adding up elastic and thermoelastic components. Numerical analysis has been performed for a cerametallic (FMC-11) pad—cast-iron (ChNMKh) disk friction couple.

ISS In Different Norms For 1-D Parabolic Pdes With Boundary Disturbances

For one-dimensional parabolic partial differential equations with disturbances at both boundaries and distributed disturbances we provide input-to-state stability (ISS) estimates in various norms. Due to the lack of an ISS Lyapunov functional for boundary disturbances, the proof methodology uses (i) an eigenfunction expansion of the solution, and (ii) a finite-difference scheme. The ISS estimate for the sup-norm leads to a refinement of the well-known maximum principle for the heat equation. Finally, the obtained results are applied to quasi-static thermoelasticity models that involve nonlocal boundary conditions. Small-gain conditions that guarantee the global exponential stability of the zero solution for such models are derived.

Quasi-static thermal stresses due to a concentrated moving heat source in a thin plate

Temperature and thermal stress distributions in a two-dimensional infinite thin plate subjected to a moving heat source with variable power and velocity are obtained by solving quasi-static thermoelasticity equations analytically with the aid of a thermoelastic displacement potential. The results show good agreement with experimental data for a stationary source with constant power and with a steady-state analytical solution in the open literature. It is shown that the quasi-static solution can predict changes of the thermal stress field during the movement of the heat source, and can give the effect of changes of power and velocity of the heat source on the thermal stress field during its movement.

Thermal Stressed State of a Disk in the Process of Multiple Braking

On the basis of the finite-element method, we propose a mathematical model of frictional heating of a disk in the process of multiple braking. We obtain the numerical solution of the axisymmetric initial boundary-value problem of heat conduction and the boundary-value problem of quasistatic thermoelasticity for a disk periodically heated on the friction surface by a heat flow whose intensity is proportional to the specific power of friction. The numerical analyses of temperature and temperature stresses in the process of multiple braking are performed for a disk made of cast iron and a cermet pad.

Second-order two-scale finite element algorithm for dynamic thermo–mechanical coupling problem in symmetric structure

The new second-order two-scale (SOTS) finite element algorithm is developed for the dynamic thermo-mechanical coupling problems in axisymmetric and spherical symmetric structures made of composite materials. The axisymmetric structure considered is periodic in both radial and axial directions and homogeneous in circumferential direction. The spherical symmetric structure is only periodic in radial direction. The dynamic thermo-mechanical coupling model is presented and the equivalent compact form is derived. Then, the cell problems, effective material coefficients and the homogenized thermo-mechanical coupling problem are obtained successively by the second-order asymptotic expansion of the temperature increment and displacement. The homogenized material obtained is manifested with the anisotropic property in the circumferential direction. The explicit expressions of the homogenized coefficients in the plane axisymmetric and spherical symmetric cases are given and both the derivation of the analytical solutions of the cell functions and the quasi-static thermoelasticity problems are discussed. Based on the SOTS method, the corresponding finite-element procedure is presented and the unconditionally stable implicit algorithm is established. Some numerical examples are solved and the mutual interaction between the temperature and displacement field is studied under the condition of structural vibration. The computational results demonstrate that the second-order asymptotic analysis finite-element algorithm is feasible and effective in simulating and predicting the dynamic thermo-mechanical behaviors of the composite materials with small periodic configurations in axisymmetric and spherical symmetric structures. This may provide a vital computational tool for analyzing composite material internal temperature distribution and structural deformation induced by the dynamic thermo-mechanical coupling response under strong aerothermodynamic environment. Second-order two-scale (SOTS) expansions for dynamic thermo-mechanical coupling problems in symmetric structure are obtained.SOTS finite element algorithm is proposed and unconditional stable implicit scheme is established.Anisotropic material is obtained by homogenization with enhanced strength and minor thermal expansion in circumferential direction.The mutual interaction and simultaneous vibration of the temperature and displacement field are simulated.A new computational tool is presented for analyzing thermo-mechanical response of composites under strong aerothermodynamic environment.