Non-Fourier heat conduction in an exponentially graded slab

The present article investigates one-dimensional non-Fourier heat conduction in a functionally graded material by using the differential transformation method. The studied geometry is a finite functionally graded slab, which is initially at a uniform temperature and suddenly experiences a temperature rise at one side, while the other side is kept insulated. A general non-Fourier heat transfer equation related to the functionally graded slab is derived. The problem is solved in the Laplace domain analytically, and the final results in the time domain are obtained by using numerical inversion of the Laplace transform. The obtained results are compared with the exact solution to verify the accuracy of the proposed method, which shows excellent agreement.     

Hybrid graded element model for transient heat conduction in functionally graded materials

This paper presents a hybrid graded element model for the transient heat conduction problem in functionally graded materials (FGMs). First, a Laplace transform approach is used to handle the time variable. Then, a fundamental solution in Laplace space for FGMs is constructed. Next, a hybrid graded element is formulated based on the obtained fundamental solution and a frame field. As a result, the graded properties of FGMs are naturally reflected by using the fundamental solution to interpolate the intra-element field. Further, Stefest’s algorithm is employed to convert the results in Laplace space back into the time-space domain. Finally, the performance of the proposed method is assessed by several benchmark examples. The results demonstrate well the efficiency and accuracy of the proposed method.     

Asymptotic averaging technique for heat conduction problems with phase transitions in layered media

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 36, No. 5, pp. 155–163, September–October, 1995.     

Transient heat conduction in functionally graded thick hollow cylinders by analytical method

In this article, transient heat conduction in a cylindrical shell of functionally graded material is studied by using analytical method. The shell is assumed to be in axisymmetry conditions. The material properties are considered to be nonlinear with a power law distribution through the thickness. The temperature distribution is derived analytically by using the Bessel functions. To verify the proposed method the obtained numerical results are compared with the published results. The comparisons of temperature distribution between various time and material properties are presented.     

Transient heat conduction in two-dimensional functionally graded hollow cylinder with finite length

In this paper a thick hollow cylinder with finite length made of two dimensional functionally graded material (2D-FGM) subjected to transient thermal boundary conditions is considered. The volume fraction distribution of materials, geometry and thermal boundary conditions are assumed to be axisymmetric but not uniform along the axial direction. The finite element method with graded material properties within each element is used to model the structure and the Crank–Nicolson finite difference method is implemented to solve time dependent equations of the heat transfer problem. Two-dimensional heat conduction in the cylinder is considered and variation of temperature with time as well as temperature distribution through the cylinder are investigated. Effects of variation of material distribution in two radial and axial directions on the temperature distribution and time response are studied. The achieved results show that using two-dimensional FGM leads to a more flexible design so that transient temperature, maximum amplitude and uniformity of temperature distributions can be modified to achieve required specifications by selecting a suitable material distribution profile in two directions.     

基于物理信息神经网络的功能梯度材料稳态/瞬态热传导分析

采用物理信息神经网络PINN(Physics-informed Neural Networks)求解稳态和瞬态功能梯度材料(FGMs)热传导问题。该方法利用控制方程、边界及初始条件的残差构造损失函数,在无任何响应数据的情况下得到了更具泛化能力的神经网络模型,同时避免了传统数值方法在求解计算力学问题时所需的微分、积分公式推导以及繁重的建模和划分网格等前处理工作。本文探究了PINN及其域分解的扩展物理信息神经网络XPINN(eXtended Physics-informed Neural Networks)在求解稳态和瞬态FGMs热传导问题时的适用性,讨论了网络结构对预测结果的影响。研究结果表明,PINN/XPINN在解决几何复杂的稳态和瞬态FGMs热传导问题时仍具有较高的可靠性和简洁的求解流程,同时,为极端环境下求解复杂多场耦合和夹杂等问题提供了新思路。     

A high order control volume finite element procedure for transient heat conduction analysis of functionally graded materials

A robust unstructured control-volume finite element method is presented for the solution of two-dimensional transient heat conduction in functionally graded materials (FGMs) with isotropic properties. The material properties at a point in the domain vary exponentially to spatial coordinates. A triangular mesh is chosen for spatial discretization and a fully implicit scheme is adapted for time discretization. Several problems are investigated and the results are successfully validated by using analytical and other numerical solutions available in the literature.     

Heat conduction and heat wave propagation in functionally graded thick hollow cylinder base on coupled thermoelasticity without energy dissipation

In this paper, heat wave propagation and coupled thermoelasticity without energy dissipation in functionally graded thick hollow cylinder is presented based on Green–Naghdi theory. The material properties are supposed to vary as a power function of radius across the thickness of cylinder. The cylinder is considered in axisymmetry and plane strain conditions and it is divided to many sub-cylinders (layers) across the thickness. Each sub-cylinder is considered to be made of isotropic material and functionally graded property can be created by suitable arrangement of layers. The Galerkin finite element method and Newmark finite difference method are employed to solve the problem. The time history of second sounds and displacement wave propagation are obtained for various values of power function. Computed results agree well with the published data.     

Transmission and band gaps of elastic SH waves in functionally graded periodic laminates

Time-harmonic plane elastic SH-waves propagating in periodically laminated composites with functionally graded (FG) interlayers are investigated in this paper. A finite stack of periodic layers between two identical elastic half-planes is considered. Two different power laws are used to describe the property variation of the FG interlayers within the unit-cell. Two different models are developed to deal with the FG interlayers, namely, the explicit FG model and the multilayer model. In conjunction with the transfer matrix method, the wave reflection and transmission coefficients, and band gaps of the FG periodic laminates are computed. Numerical results are presented and discussed to reveal the influences of the FG and homogeneous interlayers, the incidence angle of time-harmonic plane SH wave on the location and width of band gaps. The explicit FG model developed in this study is accurate and capable to simulate the full wave pattern within the periodic laminates, and it can be easily extended to periodic laminates with defects. The corresponding results presented in this paper may have important applications in optimizing and developing novel acoustic devices such as wave filters and noise insulators.     

On Fourier's law of heat conduction

A linear theory of fluid is considered in which the gradients of density, internal energy and velocity are among the constitutive variables. Thus the heat flux may be a linear combination of the gradients of density and internal energy. It is proved that this linear combination may be written as the gradient of temperature so that Fourier's law of heat conduction holds.     

The conduction of fluctuating heat flow

Summary It has previously been known that, for calculation of temperatures and heat flows that vary sinusoidally with time, homogeneous slabs or hollow cylinders may be considered as passive four-poles. Extension of this idea is now shown to be possible for non-infinite slabs and cylinders and for situations wherein the temperature or heat flow over the whole surface is not independent of the position in the surface.     

Hamiltonian description of the heat conduction

It is shown that the linear boundary value problems of the heat conduction in a homogeneous slab can be mapped on the initial value problem for a Hamiltonian motion whose phase-space trajectories are subject to an additional restriction, the “arrival condition”. The physical consequences of this formal analogy for the macroscopic heat conduction are discussed in detail. Received on 27 April 1998     

Transient heat conduction in non-homogeneous spherical systems

A comprehensive study encompassing a general analytical development and an archival presentation of results is made for transient heat conduction in thermally coupled spherical regions. The system consists of a sphere and its surrounding environment, each region having different thermal properties and different initial temperatures. The general closed form solution, which accommodates an arbitrary inital temperature distribution in the sphere, is specialized to apply to a number of interesting problems. Among these, the situation in which the sphere and the surroundings are initially at different uniform temperatures constitutes a basic problem in the theory of heat conduction. Correspondingly, a comprehensive presentation of transient temperature histories is made for various locations in the sphere and in the surroundings, with relevant thermal property ratios serving as parameters. Characteristics such as the duration of the transient period and the penetration depth of the temperature field into the surroundings are illuminated. Another interesting situation is that in which the thermal conductivity of the sphere is much greater than that of the surroundings, so that the sphere temperature may be regarded as being spatially uniform. In addition to a presentation of temperature histories, the conditions are identified under which the assumed spatial uniformity of the sphere temperature is valid. For the case of a metallic sphere situated in a gaseous environment, it is demonstrated that the transient response can be represented by a single universal curve.     

Experimental investigation of heat conduction in wet sand

There has been a lasting discussion in the literature on the validity of Fourier's heat conduction law. The main concern is the inherent infinite propagation velocity of temperature changes. Some authors hold that due to this problem the law should fail for certain materials or on extreme time scales. Then, to reconcile experiment with theory Fourier's equation would need an extension to a hyperbolic form. To contribute to this matter we report on unsteady heat conduction experiments in wet sand for which hyperbolic behaviour (Kaminski 1990) was documented. Our results clearly salvage Fourier's law without giving any explanation on why a theory with a serious inconsistency can work so well. Received on 3 November 1998     

A mechanical model for heat conduction

Continuum Mechanics and Thermodynamics - A theory of heat conduction in rigid heat conductors based entirely on mechanical concepts is proposed and compared with the traditional thermodynamic...     

A state space approach for exact analysis of composite laminates and functionally graded materials

A class of problems of composite laminates and functionally graded materials (FGM) under extension, twisting, and bending is formulated in the state space setting. A solution approach for exact analysis of the deformation and stress fields in the media is developed. Exact solutions for torsion of cross-ply laminates and certain FGM are derived, which satisfy exactly the equations of anisotropic elasticity, the end conditions, the traction-free boundary conditions on the bounding planes of the rectangular section, and the interfacial continuity conditions in multilayered composite laminates, regardless of the number of layers. The solutions serve as useful benchmarks for numerical modeling and material characterization of composite laminates and FGM.