3D thermoelastic analysis of rotating disks having arbitrary profile based on a variable kinematic 1D finite element method

A variable kinematic 1D finite element (FE) method is presented for 3D thermoelastic analysis of rotating disks with variable thickness. The principle of minimum potential energy is used to derive general governing equations of the disks subjected to body forces, surface forces, concentrated forces, and thermal loads. To solve the equations, the 1D Carrera unified formulation (CUF), which enables to go beyond the kinematic assumptions of classical beam theories, is employed. Based on the 1D CUF, the disk is considered as a beam, which can be discretized into a finite number of 1D elements along its axis. The displacement field over the beam’s cross section is approximated by Lagrange expansions. This methodology leads to an FE formulation that is invariant with respect to the order of expansions used over the cross sections, and thus the 3D problem reduces to a 1D problem. The effect of the cross section discretization on displacement and stress fields is investigated. Results obtained from this method are in good agreement with the reference analytical and finite difference solutions. The proposed innovative method can be very effective in the thermoelastic analysis of rotating disks.     

Contact Zone Approach for an Electrically Impermeable Crack in Piezoelectric Materials Under Thermal-Mechanical Loading

A partial contact zone model is developed for the stress and electric displacement fields due to the obstruction of a uniform heat flux by an electrically impermeable crack in piezoelectric materials. Green's function method is used to reduce the problem to a set of singular integral equations that are solved in closed form. When the crack is assumed to be traction free, the crack opening displacement is found to be negative over one-half of the crack unless a sufficiently large far field tensile stress is superposed. The problem is reformulated assuming a contact zone at one crack tip. The extent of this zone, the stress and electric displacement intensity factors at each crack tip are obtained as functions of the applied mechanical stress and heat flux.     

A Novel Boundary-Integral Based Finite Element Method for 2D and 3D Thermo-Elasticity Problems

In this article a new hybrid boundary integral-based (HBI) finite element method (FEM) is presented for analyzing two-dimensional (2D) and three-dimensional (3D) thermoelastic problems with arbitrary distribution of body force and temperature changes. The method of particular solution is used to decompose the displacement field into homogeneous part and particular part. The homogeneous solution is obtained by using the HBI-FEM with fundamental solutions, yet the particular solution related to the body force and temperature change is approximated by radial basis function (RBF). The detailed formulation for both 2D and 3D HBI-FEM for thermoelastic problems are given, and two different approaches for treating the inhomogenous terms are presented and compared. Five numerical examples are presented to demonstrate the accuracy and performance of the proposed method. When compared with the existing analytical solutions or ABAQUS results, it is found that the proposed method works well for thermoelastic problems and also when using a very coarse mesh, results with satisfactory accuracy can be obtained.     

Green's function for infinite thin plate with elliptic hole under bending heat source and interaction between elliptic hole and crack under uniform bending heat flux

Green's function is derived for the bending problem of an infinite thin plate with an elliptic hole under a bending heat source. Then the interaction problem between an elliptic hole and a crack in a thin plate under uniform bending heat flux is analyzed. First, the complex variable method is developed for the thermoelastic problem of bending. Then an exact solution in explicit form is derived for the Green's function by using the complex variable method. Distributions of temperature moment, heat flux moments, bending moments along the hole edge are shown in figures. For solving the interaction problem, a solution for an infinite thin plate with an adiabatic elliptic hole under uniform bending heat flux, and two Green's functions of the plate under a bending heat source couple and a bending dislocation are given. The interaction problem then reduces into singular integral equations using the Green's functions and the principle of superposition. After the equations are solved numerically, the moment intensity factors at crack tips are presented in the figures.     

Thermal Residual Stresses in One-Directional Functionally Graded Plates Subjected to In-Plane Heat Flux

This study carries out the transient thermal residual stress analyses of functionally graded clamped plates for different in-plane material compositions and in-plane heat fluxes. The heat conduction and Navier equations representing the two-dimensional thermoelastic problem were discretized using the finite-difference method, and the set of linear equations were solved using the pseudo singular value method. Both in-plane temperature distributions and the heat transfer period were affected considerably by the compositional gradient. The type of in-plane heat flux had a minor effect on the temperature profile, but on the heat transfer period. The high stress levels appeared in the ceramic-rich regions. The normal and equivalent stresses exhibited a sharp change in the plates with ceramic-rich as well as metal-rich compositions, and the concentrated on a narrow ceramic layer. A smooth stress variation was achieved through the graded region with a balanced composition of ceramic and metal-phases, and the stress discontinuities disappeared. The in-plane shear stress was negligible. The equivalent stress exhibited a linear temporal variation for both constant and sinusoidal heat fluxes, but a nonlinear variation for the exponential heat flux. In case the heat flux is applied along the metal edge (metal-to-ceramic plate) instead of the ceramic edge, the displacement and stress components exhibited similar distributions to those of a ceramic-to-metal plate but in the opposite direction. As a result, the distribution of in-plane material composition affects only normal stress distributions, whereas the peak stress levels occur in the ceramic-rich regions. Since the normal stresses concentrate along a narrow ceramic layer for ceramic-rich or metal-rich compositions, a balanced in-plane material composition distribution of ceramic and metal would be useful to avoid probable local ceramic fracture or damage.     

NUMERICAL SOLUTIONS OF THERMOELASTIC WAVE PROBLEMS BY THE METHOD OF CHARACTERISTICS

This article is concerned with the numerical treatment of thermal and thermal stress waves in thermoelastic solids. To keep the numerical treatment general, the development of the formulation is based on the generalized theory of thermoelasticity. A number of thermoelastic wave problems, which involve one or two space variables, are treated, in a uniform manner, by a system of first-order partial differential equations with stress, velocity, heat flow, and temperature as dependent variables. This system of equations is analyzed by the method of characteristics, yielding the characteristics and the characteristic equations. Procedures of numerical integration along the characteristics are established and carried out for several generalized and classical thermoelastic wave problems in homogeneous materials, composite materials, nonhomogeneous materials, and nonlinear elastic solids.     

Thermal Stress Analysis of an Embedded Crack in a Graded Orthotropic Coating-Substrate Structure

An analysis of a coupled plane thermoelastic problem for a graded orthotropic coating-substrate structure is performed under thermomechanical loading conditions. The crack direction is parallel to the free surface. Applying the superposition principle and Fourier integral transform, the heat conduction and plane elasticity equations lends themselves to the derivation of two sets of Cauchy-type singular integral equations. The thermal stress intensity factors are defined and evaluated. In the numerical results, the effects of the orthotropy parameters, thermoelastic non-homogeneity parameters, and dimensionless thermal resistance on the temperature distribution and the thermal stress intensity factors (TSIFs) are studied. The obtained results can be used to design graded orthotropic coating-substrate structures under thermomechanical loading.     

THERMAL SINGULAR STRESSES NEAR A CIRCUMFERENTIAL EDGE CRACK IN A SPHERICAL CAVITY UNDER UNIFORM AXIAL HEAT FLOW

The linear thermoelastic problem of a spherical cavity with a circumferential edge crack is solved. The thermal stresses are caused by a uniform heat flow disturbed by the presence of the crack and the cavity. The surfaces of the crack and the cavity are assumed to be insulated. Integral transform techniques are used to reduce the problem concerning the temperature and thermoelastic fields to that of solving two singular integral equations of the first kind. The integral equations are solved numerically and the variation of the thermal stress intensity factor with the crack depth and the crack opening displacement are shown graphically.     

THERMOELASTIC PROBLEM OF STEADY-STATE HEAT FLOW DISTURBED BY A CRACK PERPENDICULAR TO THE GRADED INTERFACIAL ZONE IN BONDED MATERIALS

The plane thermoelasticity equations are used to investigate the steady-state nonisothermal crack problem for bonded materials with a graded interfacial zone. The interfacial zone is modeled as a nonhomogeneous interlayer having continuously varying thermoelastic moduli in the exponential form between the dissimilar, homogeneous half-planes. A crack is assumed to exist in one of the half-planes oriented perpendicular to the nominal interface, disturbing a uniform heat flow. Based on the method of Fourier integral transform, formulation of the crack problem is reduced to solving two sets of Cauchy-type singular integral equations for temperature and thermal stress fields. The heat-flux intensity factors and the thermally induced mode II stress intensity factors are defined in order to characterize the singular behavior of temperature gradients and thermal stresses, respectively, in the vicinity of the crack tips. In the numerical results, the values of heat-flux and thermal-stress intensity factors are presented for various combinations of material and geometric parameters of the dissimilar media bonded through a thermoelastically graded interfacial zone. The influence of crack-surface partial conductance on the near-tip temperature and thermal stress fields is also addressed.     

THERMOELASTIC PROBLEM OF BONDED ORTHOTROPIC SEMI-INFINITE AND INFINITE STRIPS

The stationary thermoelastic problem for an orthotropic semi-infinite strip bonded to an orthotropic infinite strip is investigated. By using Fourier transforms, the heat conduction problem is reduced to a singular integral equation for the temperature gradient. The associated elasticity problem leads to a system of singular integral equations for strain and stress. Numerical solution of these equations yields temperature, temperature gradient, and stress distributions along the interface, caused by prescribed boundary temperatures.     

THERMAL STRESSES IN A CIRCULAR CYLINDER WITH A CIRCUMFERENTIAL EDGE CRACK UNDER UNIFORM HEAT FLOW

The linear thermoelastic problem of an infinitely long circular cylinder with a circumferential edge crack is solved. The thermal stresses are caused by a uniform heat flow disturbed by the presence of the crack. The crack surfaces and the cylindrical surface are assumed to be insulated. Integral transform techniques are used to reduce the problem to that of solving two singular integral equations of the first kind. The equations are solved numerically, and the variation of the stress intensity factor with the crack depth is shown graphically.     

A Plane Magnetothermoelastic Waves Reflection and Refraction Between Two Solid Media with External Heat Sources and Initial Stress

In this article, an estimation to illustrate the reflection and transmission of thermoelastic waves at the interface of two different thermoelastic solid half-spaces under initial stress, magnetic field, and external heat sources is made. The governing equations are considered for an isotropic and homogeneous generalized thermoelastic model under initial stress, heat sources, magnetic field, and perfect boundary condition. The reflected and transmitted waves amplitudes are obtained for the incidence of SV-waves. Approximate expressions are obtained as a special case from the present problem. The results obtained are computed numerically for the amplitude ratios under initial stresses and are displayed graphically. It appears that the effects of initial stress, magnetic field, and external heat sources are very pronounced on the propagation phenomena. If the magnetic field and heat sources are neglected, the results obtained then lead to the same results obtained by relevant authors.     

Twin Vector Fields and Independence of Spectra for Quadratic Vector Fields

The object of this paper is to address the following question: when is a polynomial vector field on \(\mathbb {C}^{2}\) completely determined (up to affine equivalence) by the spectra of its singularities? We will see that for quadratic vector fields, this is not the case: given a generic quadratic vector field there is, up to affine equivalence, exactly one other vector field which has the same spectra of singularities. Let us say that two distinct vector fields are twin vector fields if they have the same singular locus and the same spectrum at each singularity. Our main result is as follows: any two generic quadratic vector fields with the same spectra of singularities (yet possibly different singular locus) can be transformed by suitable affine maps to be either the same vector field or a pair of twin vector fields. Moreover, a generic quadratic vector field has exactly one twin vector field. We later analyze the case of quadratic Hamiltonian vector fields in more detail and find necessary and sufficient conditions for a collection of non-zero complex numbers to arise as the spectra of singularities of a quadratic Hamiltonian vector field. Lastly, we show that a generic quadratic vector field is completely determined (up to affine equivalence) by the spectra of its singularities together with the characteristic numbers of its singular points at infinity.     

Visualisation of heat transfer in 3D unsteady flows

Heat transfer in fluid flows traditionally is examined in terms of temperature field and heat-transfer coefficients at non-adiabatic walls. However, heat transfer may alternatively be considered as the transport of thermal energy by the total convective–conductive heat flux in a way analogous to the transport of fluid by the flow field. The paths followed by the total heat flux are the thermal counterpart to fluid trajectories and facilitate heat-transfer visualisation in a similar manner as flow visualisation. This has great potential for applications in which insight into the heat fluxes throughout the entire configuration is essential (e.g. cooling systems, heat exchangers). To date this concept has been restricted to 2D steady flows. The present study proposes its generalisation to 3D unsteady flows by representing heat transfer as the 3D unsteady motion of a virtual fluid subject to continuity. This unified ansatz enables heat-transfer visualisation with well-known geometrical methods from laminar-mixing studies. These methods lean on the property that continuity “organises” fluid trajectories into sets of coherent structures (“flow topology”) that geometrically determine the fluid transport. Decomposition of the flow topology into its constituent coherent structures visualises the transport routes and affords insight into the transport properties. Thermal trajectories form a thermal topology of essentially equivalent composition that can be visualised by the same methodology. This thermal topology is defined in both flow and solid regions and thus describes the heat transfer throughout the entire domain of interest. The heat-transfer visualisation is provided with a physical framework and demonstrated by way of representative examples.     

Interfacial stresses and fracture analysis of a two-phase composite under time-dependent heat flux

An analysis of a two-phase composite component under time-dependent heat flux is presented. The fundamental thermoelastic solution is obtained in terms of complex potentials via the technique of the analytical continuation in order to satisfy the continuous conditions on the interface. The hereditary integral associated with the Kelvin–Maxwell model is applied to simulate the thermoviscoelastic properties while a thermorheologically simple material is considered. Based on the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real-time solution can then be solved numerically by taking inverse Laplace transform. Some typical examples of interface stresses induced by various time-dependent heat flux are discussed. Finally, the solution of a crack embedded in the bi-material subjected to a uniform heat flux is also discussed.     

TRANSIENT THERMOELASTIC ANALYSIS FOR TWO-DIMENSIONAL DISSIMILAR MATERIALS BY THE BOUNDARY ELEMENT METHOD

This article deals with the transient thermoelastic analysis for dissimilar materials under the plane strain condition. In the process of the boundary element formulation, the time-dependent fundamental solution for the transient heat conduction problem and the thermoelastic displacement potential for the transient thermal stress problem are introduced. Consequently, domain integrals are completely eliminated. The discretization based on the domain combination method for these boundary integral equations is implemented, and the transient temperature and stress fields are analyzed numerically. The transient temperature at the lateral surface and the transient thermal stress at the interface are investigated for the three categories that have been determined according to the characteristic equation expressed by Dundurs parameters.     

Thermal fracture of cracked cylinders associated with nonclassical heat conduction: The effect of material property

The thermoelastic problem of a transversely isotropic hollow cylinder containing a circumferential crack is investigated in the present article based on the non-Fourier heat conduction theory. The temperature and stress fields are obtained by solving the coupled partial differential equations in the Laplace domain, and corresponding thermal axial stress with minus sign is then applied to the crack surface to form a mode I crack problem. Three different kinds of crack are considered, and the singular integral equation method is adopted to solve the fracture problem. Finally, with the definition of stress intensity factor, the effect of material properties, coupling parameter, and crack geometry on the hyperbolic thermal fracture responses of a transversely isotropic hollow cylinder excited by a thermal loading is visualized.     

Investigation of the thermal-elastic problem in cracked semi-infinite FGM under thermal shock using hyperbolic heat conduction theory

In this paper, a thermoelastic analytical model is established for a functionally graded half-plane containing a crack under a thermal shock in the framework of hyperbolic heat conduction theory. The moduli of functionally graded materials (FGMs) are assumed to vary exponentially with the coordinates. By employing the Fourier transform and Laplace transform, coupled with singular integral equations, the governing partial differential equations under mixed, thermo-mechanical boundary conditions are solved numerically. For both the temperature distribution and transient stress intensity factors (SIFs) in FGMs, the results of hyperbolic heat conduction model are significantly different than those of Fourier’s Law, which should be considered carefully in designing FGMs.     

Unified finite element approach for generalized coupled thermoelastic analysis of 3D beam-type structures,part 1: Equations and formulation

An innovative 1D finite element (FE) approach is developed to analyze the 3D static, transient, and dynamic problems in the coupled and uncoupled thermoelasticity for the nonhomogeneous anisotropic materials. The Galerkin method is directly applied to the governing equations to obtain a weak formulation of the thermoelasticity problems with arbitrary loads and boundary conditions. To surmount the restrictions of the classical beam theories, a 1D FE procedure is proposed in the context of the Carrera Unified Formulation (CUF). Since coupled thermoelastic analyses are computationally demanding, the proposed 1D FE approach can be used as a powerful means to simulate the generalized coupled thermoelastic behavior of structures. This methodology, indeed, reduces the 3D problems to 1D models with 3D-like accuracies and very low computational costs. The Lord-Shulman and the Green-Lindsay models are considered as the generalized theories of thermoelasticity. Furthermore, as simplified cases, the classical coupled, dynamic uncoupled, quasi-static uncoupled and steady-state uncoupled theories of thermoelasticity may be derived from the formulation. Moreover, effects of the structural damping can be taken into account in the present formulation. The accuracy of the formulation has been evaluated through numerical simulations and comparisons, which have been presented in a companion article (Part 2).