Mathematical Problems in Engineering

Volume 2018, Article ID 7371016, 11 pages

https://doi.org/10.1155/2018/7371016

## Transient Analysis of a Functionally Graded Ceramic/Metal Layer considering Lord-Shulman Theory

Department of Mechanics, Laboratory of Testing and Materials, School of Applied Mathematical and Physical Science, National Technical University of Athens, Zografou Campus, 15773 Athens, Greece

Correspondence should be addressed to Efstathios E. Theotokoglou; rg.autn.lartnec@sihtats

Received 27 October 2017; Revised 25 February 2018; Accepted 26 March 2018; Published 28 May 2018

Academic Editor: Michael Vynnycky

Copyright © 2018 Antonios M. Nikolarakis and Efstathios E. Theotokoglou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The transient displacement, temperature, and stress fields in a functionally graded ceramic/metal layer under uniform thermal shock conditions at the upper surface are numerically studied based on the Lord-Shulman model, employing a direct finite element method. The Newmark method is employed for the time integration of the problem. A Matlab finite element code is developed for the numerical analysis of the one-dimensional problem under consideration. The Voigt model (rule of mixture) is used for the estimation of the effective properties inside the functionally graded layer and the variation of the volume fraction of the materials follows the sigmoid function in terms of the introduced parameter . Furthermore, a parametric study with respect to the parameter follows, where three different combinations of ceramic/metal materials are considered. It is concluded that the value , which corresponds to a linear variation of the properties, minimizes the maximum (tensile) stress applied at the middle of the functionally graded layer.

#### 1. Introduction

Ceramic materials are used as thermal barrier coatings for the thermal protection of metals in high-temperature environments. A common failure mechanism of those composite configurations is the spallation of the ceramic coating close to the interface with the metal substrate, mainly due to their thermal expansion mismatch [1]. A possible solution for this problem is the use of functionally graded materials (FGMs), which are advanced materials with gradually varying properties [2]. In FG thermal barrier coatings, an intermediate FGM layer connects the ceramic coating and the metal substrate. The thermomechanical properties of the FGM layer vary from the properties of the ceramic material to the properties of the metal material in a continuous way, thus eliminating the material discontinuities.

On the other hand, the classical theory of thermoelasticity predicts that the thermal disturbances propagate through a solid medium with infinite speed. During the last 50 years, generalized theories of thermoelasticity have been formulated that predict finite-speed thermal waves and overcome this physical paradox. The wave type heat propagation is frequently described as* second sound*. Although the effects of second sound are short-lived, they become important in thermal shock applications [3].

Ceramic/metal FG thermal barrier coatings are used as parts in machines that operate in high-temperature environments. Combustion chambers, exhaust pipes, power generators, aircraft engines, and space shuttles are typical examples of such machines. In these applications, the study of the thermomechanical response of a ceramic/metal FGM layer under thermal shock conditions using generalized thermoelasticity theories is of great importance.

The first generalized thermoelasticity theory was formulated by Lord and Shulman [4]. The Lord-Shulman theory modifies the classical Fourier’s law of heat conduction by introducing a relaxation time, which can be interpreted as the time required to establish steady-state heat conduction in a volume element when a temperature gradient is suddenly imposed on that element [3, 4]. Another important generalized theory is the Green-Lindsay theory [5], which introduces two relaxation times to modify the stress-strain relations and the entropy equation, respectively. Furthermore, Green and Naghdi proposed three models of thermoelasticity [6], which are labelled as models I, II, and III. The Green-Naghdi model II in particular assumes no dissipation of thermal energy. Bagri and Eslami [7] suggested a unified generalized thermoelasticity theory, which incorporates the theory of classical coupled thermoelasticity and the generalized thermoelasticity theories of Lord-Shulman, Green-Lindsay, and Green-Naghdi as special cases.

There are several studies available that analyze the stresses developed in FGMs by employing generalized theories of thermoelasticity, which regard different structural configurations such as disks [8, 9], cylinders [10–13], and spheres [14, 15]. Researches regarding generalized thermoelasticity analyses in FGM layered structures are rarer in literature. Bagri et al. [16] studied the problem of a FGM layer under thermal shock conditions in the context of Lord-Shulman theory, where the variation of the thermomechanical properties in the FGM layer followed a power law function. They used a transfinite element method to solve the coupled system of equations, where time is eliminated using the Laplace transform. The inverse Laplace transform was based on a numerical scheme. Youssef and El-Bary [17] studied the problem of a symmetric 3-layered strip under thermal shock conditions, in the context of Lord-Shulman theory. Each layer was made of homogeneous, isotropic, and thermoelastic material, where they also assumed that the thermal conductivity was a function of temperature. They also used the Laplace transform to solve the coupled system of equations. Hosseini Zad et al. [18] used a direct finite element method to study the problem of a 2-layered one-dimensional media under uniform thermomechanical loading at the free surfaces. Each layer consisted of homogeneous, isotropic and thermoelastic material. Nowruzpour Mehrian et al. [19] studied the dynamic thermoelastic response of a functionally graded plate subjected to thermal shock based on the Lord-Shulman theory, using the state space approach and the Laplace transform. Heydarpour and Aghdam [20] employed the Lord-Shulman theory to study the transient thermoelastic behavior of rotating functionally graded conical shells.

In most of the aforementioned studies the Laplace transform is employed for the solution of the coupled system of partial differential equations. However, especially in the case of FGMs, the inverse Laplace transform is based on numerical methods, which have numerical stability issues in problems with fast transient loading such as thermal shock applications [21]. Furthermore, in almost all of the above studies the variation of the properties in the FGM layer follows a power law function [8–16].

In the present paper the standard Galerkin finite element method [22] is used to study the transient fields of displacement, temperature, and stress in a ceramic/metal FGM layer under uniform thermal shock loading, based on the generalized theory of Lord-Shulman. The time marching scheme employed is based on the Newmark method [22]. The effective properties of the FGM layer are estimated based on the Voigt model (rule of mixture) [23] and the through thickness volume fraction of the materials is based on a sigmoid function in terms of parameter [24]. Moreover, a parametric study with respect to the parameter follows, where three different combinations of ceramic/metal materials are considered and the minimization of the maximum stress applied at the middle of the FGM is used as the optimization criterion. The properties of the materials are assumed to be temperature-independent. A Matlab finite element code [25] is developed for the numerical analysis of the problem under consideration.

#### 2. Governing Equations

Consider a ceramic/metal FGM layer of total thickness , as shown in Figure 1, which is subjected to uniform thermomechanical loading at the upper and the lower surfaces. The other two dimensions of the layer are assumed to be infinite compared to its thickness, and the properties of the FGM layer depend only on the depth , where . At the FGM is fully ceramic, while at the FGM is fully metal. The material is considered to be isotropic and linear thermoelastic. Initially, the layer is stress-free, undeformed, and at uniform temperature .