Arbitrary Thermal Loading Research Articles

Application of fractional order theory to a functionally graded perfect conducting thermoelastic half space with variable Lamé’s Modulii

In this work, the model of fractional magneto-thermoelasticity is applied to a one-dimensional thermal shock problem for a functionally graded half-space whose surface is assumed to be traction free and subjected to an arbitrary thermal loading. The Lame’s modulii is taken as functions of the vertical distance from the surface of thermoelastic perfect conducting medium in the presence of a uniform magnetic field. Laplace transform and the perturbation techniques are used to derive the solution in the Laplace transform domain. Numerical inversion of the Laplace transform is carried out to obtain the temperature, displacement, stress and induced magnetic and electric field distributions. Numerical results are represented graphically and discussed.

Thermoelectric spherical shell with fractional order heat transfer

The model of fractional magneto-thermoelasticity is applied to one-dimensional problems of a thermoelectric spherical shell subjected to an arbitrary thermal loading of its external boundary in the presence of a uniform magnetic field. By means of the Laplace transform and numerical Laplace inversion, the problems are solved. The distributions of the considered temperature, stress and displacement, are represented graphically. The theories of coupled and generalized magneto-thermoelasticity follow as limited cases. Some comparisons are shown in the figures to estimate the effects of the fractional order and ramping parameters.

A functionally graded magneto-thermoelastic half space with memory-dependent derivatives heat transfer

In this work, the model of magneto-thermoelasticity based on memory-dependent derivative (MDD) is applied to a one-dimensional thermal shock problem for a functionally graded half-space whose surface is assumed to be traction free and subjected to an arbitrary thermal loading. The Lame's modulii are taken as functions of the vertical distance from the surface of thermoelastic perfect conducting medium in the presence of a uniform magnetic field. Laplace transform and the perturbation techniques are used to derive the solution in the Laplace transform domain. A numerical method is employed for the inversion of the Laplace transforms. The effects of the time-delay on the temperature, stress and displacement distribution for different linear forms of Kernel functions are discussed. Numerical results are represented graphically and discussed.

Modeling of steel frame structures in fire using OpenSees

The OpenSees framework has been extended to deal with frame structures under fire conditions. OpenSees is an object-oriented, open source software framework developed at UC Berkeley and has so far been focused on providing an advanced computational tool for analyzing the non-linear response of structural frames subjected to seismic excitations. New classes defining time-dependent temperature distributions in the cross-section of members have been created and OpenSees material classes have been modified to include temperature dependent properties based on the Eurocode. New functions and interfaces have been added into existing element and section classes to calculate the member actions due to arbitrary thermal loading taking into account material degradation with increasing temperature for non-linear analyses. This paper reports on a number of benchmark tests to ascertain the performance of the new codes implemented in OpenSees for beams, frames, and plate structures. The analysis procedures being developed for structures exposed to fire in OpenSees will make it easier for users to define temperature-dependent material properties and allow for arbitrary non-uniform temperature distributions across and along an element by interfacing a fire and heat transfer analysis module also being developed for OpenSees. This work will also enable the modeling of earthquake damaged structural frames subjected to a subsequent fire.

Hamiltonian Approach to Analytical Thermal Stress Intensity Factors—Part 2 Thermal Stress Intensity Factor

The equations of thermoelasticity are first rewritten in Hamiltonian form where the variables are separable in spatial coordinates. The symplectic eigensolutions satisfying the boundary conditions along the crack faces are solved analytically from the homogeneous equations. The loading and the particular integral are expanded in the eigensolutions and the coefficients are determined by the geometric boundary conditions excluding the crack. The thermal stress intensity factors are obtained and the behavior of the thermal stress intensity factors under arbitrary thermal loading is investigated.

Response of spatially tailored structures to thermal loading

The paper presents the formulation and analysis of composite plates serving as STATs, i.e., spatially tailored advanced thermal structures where the distribution of the constituent phases varies throughout the surface as well as through the thickness. This is an extension of the well-known concept of functionally graded materials (FGM) and structures with the constituent phases varying only in the latter direction. As a result of two- or three-dimensional grading it is possible to optimize the response and properties of the structure providing multitask and multi-scale optimization. The response of plates with two- or three-dimensional grading to an arbitrary thermal loading is elucidated, including the conditions that result in thermal bending versus thermal instability.

Approximate Direct and Inverse Relationships for Thermal and Stress States in Thick-Walled Vessels Under Thermal Shock

Approximate solutions were derived for the transient through steady-state thermal-stress fields developed in thick-walled vessels subjected to a potentially arbitrary thermal shock. In order to accomplish this, Duhamel’s integral was first used to relate the arbitrary thermal loading to a previously derived unit kernel for tubular geometries. Approximate rules for direct and inverse Laplace transformations were then used to modify the resulting Volterra equation to an algebraically solvable and relatively simple form. The desired thermoelastic stress distributions were then determined using the calculated thermal states and elasticity theory. Good agreement was seen between the derived temperature and stress relationships and earlier analytical and finite-element studies of a cylinder subjected to an asymptotic exponential heating on the internal surface with convection to the outer environment. It was also demonstrated that the derived relationships can be used to approximate the more difficult inverse (deconvolution) thermal problem for both exponential (monotonic) and triangular (non-monotonic) load histories. The use of polynomial of powers tn∕2 demonstrated the feasibility of employing the method with empirical data that may not be easily represented by standard functions. For any of the direct and inverse cases explored, the resulting relationships can be used to verify, calibrate, and/or determine a starting point for finite-element or other numerical methods.

Thermoelastic Stresses in an Axisymmetric Thick-Walled Tube Under an Arbitrary Internal Transient

A closed-form axisymmetric solution was derived for the transient thermal-stress fields developed in thick-walled tubes subjected to an arbitrary thermal loading on the internal surface with convection to the surrounding external environment. Generalization of the temperature excitation was achieved by using a versatile polynomial composed of integral-and half-order terms. In order to avoid the difficult and potentially error prone evaluation of functions with complex arguments, Laplace transformation and a ten-term Gaver-Stehfest inversion formula were used to solve the resulting Volterra integral equation. The ensuing series representation of the temperature distribution as a function of time and radial position was then used to derive new relationships for the transient thermoelastic stress-states. Excellent agreement was seen between the derived temperature and stress relationships, existing series solutions, and a finite-element analysis when the thermophysical and thermoelastic properties were assumed to be independent of temperature. The use of a smoothed polynomial in the derived relationships allows the incorporation of empirical data not easily represented by standard functions. This in turn permits an easy analysis of the significance of the exponential boundary condition and convective coefficient in determining the magnitudes and distribution of the resulting stress states over time. Moreover, the resulting relationships are easily programmed and can be used to verify and calibrate numerical calculations.

THERMAL BUCKLING ANALYSIS OF INHOMOGENEOUS RECTANGULAR PLATE DUE TO UNIFORM HEAT SUPPLY

This paper is concerned with the thermal buckling analysis of an isotropic inhomogeneous rectangular plate subjected to the arbitrary thermal loads. The fundamental equations system is derived by introducing the technique of the newly defined position of the reference plane, which allows us to analyze the problem using an elementary plate theory. It is assumed that the material properties such as the coefficient of linear thermal expansion α, the thermal conductivity λ, and Young's modulus of elasticity, E, are changed in the thickness direction with the power law of the coordinate variable, whereas Poisson's ratio ν is assumed to be constant. As an illustrative example, we consider the thermal buckling problem of a simply supported inhomogeneous rectangular plate due to uniform heat supply. Numerical calculations are carried out for several cases taking into account the variations of the inhomogeneous material properties, aspect ratio, and width-to-thickness ratio.

Thermal Stresses in an Infinite Slab Under an Arbitrary Thermal Shock

Thermal Stresses in an Infinite Slab Under an Arbitrary Thermal Shock

A solution of a non-homogeneous orthotropic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems

A solution of a non-homogeneous orthotropic elastic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions as well as the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to the homogeneous ones. Then by virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally presented, which can degenerate in a rather straightforward way to the solution for a homogeneous orthotropic cylindrical shell and isotropic solid cylinder as well as that for a non-homogeneous isotropic cylindrical shell. Using the present method, integral transform can be avoided. It is fit for a cylindrical shell with arbitrary thickness subjected to arbitrary thermal loads. It is also very convenient to deal with dynamic thermoelastic problems for different boundary conditions. Besides, the numerical calculation involved is very easy to be performed. Several examples are presented.

ANALYTICAL SOLUTION OF A PYROELECTRIC HOLLOW CYLINDER FOR PIEZOTHERMOELASTIC AXISYMMETRIC DYNAMIC PROBLEMS

By virtue of the separation of variables technique, the piezothermoelastic axisymmetric dynamic problem of a pyroelectric hollow cylinder is transformed to a second kind of Volterra integral equation about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solution of displacements, stresses, electric displacements, and electric potential are obtained. The present method is suitable for a pyroelectric hollow cylinder with an arbitrary thickness subjected to arbitrary thermal loads. Numerical results are presented.

511 一様加熱媒体による不均質長方形板の熱座屈解析

This paper is concerned with the thermal buckling analysis of an isotropic inhomogeneous rectangular plate subjected to the arbitrary thermal loads. The fundamental equations system is derived by introducing the technique of the newly defined position of the reference plane, which allows us to analyze the problem using an elementary plate theory. It is assumed that the material properties such as the coefficient of linear thermal expansion α, the thermal conductivity λ, and Young's modulus of elasticity, E, are changed in the thickness direction with the power law of the coordinate variable, whereas Poisson's ratio ν is assumed to be constant. As an illustrative example, we consider the thermal buckling problem of a simply supported inhomogeneous rectangular plate due to uniform heat supply. Numerical calculations are carried out for several cases taking into account the variations of the inhomogeneous material properties, aspect ratio, and width-to-thickness ratio.

Transient Brake Temperatures Found by use of Analytical Solutions for Finite Hollow Cylinders

A linear analytical model for calculating transient axisymmetric temperature distributions in finite hollow cylinders is established and adapted for practical use. A homogeneous isotropic material with temperature-independent thermal properties is assumed. Among possible applications are brake discs, brake drums and block braked railway wheels. The thermal power from braking is applied as prescribed heat influxes over parts of the lateral and radial surfaces of the cylinder. Bessel series solutions to the heat conduction equation are found where Newton's law for heat transfer is used in the boundary conditions. In addition to convection prescribed surface temperatures and insulated surfaces are also studied. Arbitrary thermal loading histories are treated by use of Duhamel's principle where step loadings are superposed. Convergence and limitations of the method and required computer times are discussed. Calculated numerical results for some typical braking operations are compared with those of other studies.

Arbitrary thermal loading of transversely orthotropic cylindrical bodies: Fibre Science and Technology, Vol 2, No 1 p 1 (July 1969)

Arbitrary thermal loading of transversely orthotropic cylindrical bodies: Fibre Science and Technology, Vol 2, No 1 p 1 (July 1969)

Arbitrary thermal loading of transversely orthotropic cylindrical bodies

This paper is concerned with the prediction of the stress state in transversely orthotropic cylindrical bodies that are subjected to thermal loadings that are functions of both the radial and longitudinal coordinates. The basic equations of orthotropic elasticity are combined to form a fourth-order partial differential equation in the shear stress. The homogeneous solution is obtained by a separation of variables technique and the particular solution is found for a temperature distribution that is described by the product of a periodic function of the longitudinal coordinate and a power series in the radial coordinate. A solid circular cylinder that has its centerline fixed at both ends and is subjected to a linear radial thermal distribution is considered as an example. An abridged form of the expressions for the stresses is found by assuming certain recurring terms are small compared to unity. These assumptions are evaluated and it is found that they restrict the magnitude of the ratio of the largest radial dimension to the length of the cylinder. It is concluded that the restrictions are not severe ones and that the results of the analysis are widely applicable.