Heat Conduction in a Functionally Graded Plate Subjected to Finite Cooling/Heating Rates: An Asymptotic Solution.

Zhihe Jin

doi:10.3390/ma4122108

Abstract

This work investigates transient heat conduction in a functionally graded plate (FGM plate) subjected to gradual cooling/heating at its boundaries. The thermal properties of the FGM are assumed to be continuous and piecewise differentiable functions of the coordinate in the plate thickness direction. A linear ramp function describes the cooling/heating rates at the plate boundaries. A multi-layered material model and Laplace transform are employed to obtain the transformed temperatures at the interfaces between the layers. An asymptotic analysis and an integration technique are then used to obtain a closed form asymptotic solution of the temperature field in the FGM plate for short times. The thermal stress intensity factor (TSIF) for an edge crack in the FGM plate calculated based on the asymptotic temperature solution shows that the asymptotic solution can capture the peak TSIFs under the finite cooling rate conditions.

Highlights

Graded materials (FGMs) represent a new concept of tailoring materials with microstructural and properties gradients to achieve optimized performance
The temperature solution at short times is useful because thermal stresses and thermal stress intensity factors may reach their peak values in a very short period of time and these peak values govern the thermal stress failure of materials
This paper extends the method in [5] to investigate 1-D heat conduction in an Functionally graded materials (FGMs) plate with continuous and piecewise differentiable properties subjected to finite cooling/heating rates at the plate boundaries

Summary

Introduction

Graded materials (FGMs) represent a new concept of tailoring materials with microstructural and properties gradients to achieve optimized performance. Ishiguro et al [2] analyzed the 1-D temperature distribution in an FGM strip using a multi-layered material model. Tanigawa et al [3] modeled an FGM plate by a laminated composite with homogeneous layers and obtained the solution of the 1-D temperature field. A layered material model was used by Jin and Paulino [4] to obtain an approximate short time solution of temperature field in an FGM strip. Jin [5] obtained a simple closed form short time asymptotic solution of temperature field in an FGM strip with continuous and piecewise differentiable material properties under sudden cooling boundary conditions. The Laplace transform with its asymptotic properties and an integration technique are employed to obtain a closed form, short time solution of temperature distribution. The thermal stress intensity factor for an edge crack in the FGM plate is calculated using the asymptotic temperature solution

Basic Equations

A Multi-Layered Material Model and the Discrete Temperature Solution

A Closed Form Short Time Solution

Concluding Remarks