# Transient Thermal Stress Problem of a Functionally Graded Magneto-Electro-Thermoelastic Hollow Sphere

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Analysis

_{i}is the outer radius of the ith layer. Throughout this article, the indices i (=1,2,…,N) are associated with the ith layer of a composite hollow sphere from the inner side.

#### 2.1. Heat Conduction Problem

_{a}and T

_{b}with relative heat transfer coefficients h

_{a}and h

_{b}, respectively. Then, the temperature distribution is one-dimensional, and the transient heat conduction equation for the ith layer is written in the following form:

_{i}is the temperature change; t is time; λ

_{ri}is the thermal conductivity in the radial direction; κ

_{ri}is the thermal diffusivity in the radial direction; and T

_{0}, λ

_{0}and κ

_{0}are typical values of temperature, thermal conductivity, and thermal diffusivity, respectively. To solve the fundamental equation (1), we introduced the Laplace transformation with respect to the variable τ as follows;

_{0}() and y

_{0}() are zeroth-order spherical Bessel functions of the first and second kind, respectively. Furthermore, A

_{i}and B

_{i}are unknown constants determined from the boundary conditions. Substituting Equation (11) into the boundary conditions in the transformed domain from Equations (3)–(6), these equations are represented in matrix form as follows:

_{i}and B

_{i}can be determined from Equation (12). Then the temperature solution in the transformed domain is

_{kl}], and the coefficients ${\overline{A}}_{i}$ and ${\overline{B}}_{i}$ are defined as determinants of a matrix similar to the coefficient matrix [a

_{kl}], in which the (2i − 1)th column or 2ith column is replaced with the constant vector {c

_{k}}, respectively. Using the residue theorem, we can accomplish the inverse Laplace transformation on Equation (13). Because the single-valued poles of Equation (13) correspond to p = 0 and the roots of Δ = 0, in which the residue for p = 0 gives a solution for the steady state. Accomplishing the inverse Laplace transformation of Equation (13), the solution of Equation (1) is written as follows:

_{kl}], and the coefficients ${{\overline{A}}^{\prime}}_{i}$ and ${{\overline{B}}^{\prime}}_{i}$ are defined as determinants of a matrix similar to the coefficient matrix [e

_{kl}], in which the (2i − 1)th column or 2ith column is replaced with the constant vector {c

_{k}}, respectively. The nonzero elements of the coefficient matrices [a

_{kl}] and [e

_{kl}] and the constant vector {c

_{k}} are given from the Equations (3)–(6). In Equation (14), ${\Delta}^{\prime}({\mu}_{j})$ is

_{j}is the jth positive root of the following transcendental equation

#### 2.2. Thermoelastic Problem

_{ri}is the displacement in the r direction; α

_{ri}are the coefficients of linear thermal expansion; C

_{kli}are the elastic stiffness constants; D

_{ri}is the electric displacement in the r direction; B

_{ri}is the magnetic flux density in the r direction; e

_{ki}are the piezoelectric coefficients; η

_{1i}is the dielectric constant; P

_{1i}is the pyroelectric constant; q

_{ki}are the piezomagnetic coefficients; μ

_{1i}is the magnetic permeability coefficient; d

_{1i}is the magnetoelectric coefficient; m

_{1i}is the pyromagnetic constant; and α

_{0}, Y

_{0}and d

_{0}are typical values of the coefficient of linear thermal expansion, Young’s modulus, and piezoelectric modulus, respectively.

_{ri}, electric potential φ

_{i}, and magnetic potential ψ

_{i}in the dimensionless form are written as

_{ki}($k=1,2,\cdots ,7$) are unknown constants. We have the following relation.

## 3. Numerical Results

_{3}, and the magnetostrictive material is made up of CoFe

_{2}O

_{4}. Numerical parameters of heat conduction and shape are presented as follows:

_{p}and the magnetostrictive phase V

_{m}for other layers are given by the relations

_{p}in ith layer is obtained by calculating the value of V

_{p}in Equation (34) at the centre point of each layer defined by $\overline{r}=({R}_{i-1}+{R}_{i})/2$. To estimate the material properties of FGM, we apply the simplest linear law of mixture. The material constants considered for BaTiO

_{3}and CoFe

_{2}O

_{4}are shown in the paper [20]. The typical values of material parameters such as κ

_{0}, λ

_{0}, α

_{0}, Y

_{0}, and d

_{0}, used to normalize the numerical data, based on those of BaTiO

_{3}are as follows:

**Figure 3.**Variation of the thermal stresses (M = 1, N = 10): (

**a**) normal stress ${\overline{\sigma}}_{rr}$; (

**b**) normal stress ${\overline{\sigma}}_{\theta \theta}$.

**Figure 6.**Variation of the thermal stresses (N = 10): (

**a**) normal stress ${\overline{\sigma}}_{rr}$; (

**b**) normal stress ${\overline{\sigma}}_{\theta \theta}$.

**Figure 9.**Numerical results for the two-layered hollow sphere (N = 2): (

**a**) normal stress ${\overline{\sigma}}_{rr}$; (

**b**) normal stress ${\overline{\sigma}}_{\theta \theta}$; (

**c**) electric potential; (

**d**) magnetic potential.

## 4. Conclusions

_{3}and magnetostrictive CoFe

_{2}O

_{4}, and examined the behaviors in the transient state for temperature change, displacement, stress, electric potential and magnetic potential distributions. We investigated the effects of the nonhomogeneity of material on the stresses, electric potential, and magnetic potential. Furthermore, the effect of relaxation of stress values in functionally graded magneto-electro-thermoelastic hollow sphere was investigated. We conclude that we can evaluate not only the thermoelastic response of the functionally graded magneto-electro-thermoelastic hollow sphere, but also the electric and magnetic fields of functionally graded magneto-electro-thermoelastic hollow sphere quantitatively in a transient state.

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**MDPI and ACS Style**

Ootao, Y.; Ishihara, M.
Transient Thermal Stress Problem of a Functionally Graded Magneto-Electro-Thermoelastic Hollow Sphere. *Materials* **2011**, *4*, 2136-2150.
https://doi.org/10.3390/ma4122136

**AMA Style**

Ootao Y, Ishihara M.
Transient Thermal Stress Problem of a Functionally Graded Magneto-Electro-Thermoelastic Hollow Sphere. *Materials*. 2011; 4(12):2136-2150.
https://doi.org/10.3390/ma4122136

**Chicago/Turabian Style**

Ootao, Yoshihiro, and Masayuki Ishihara.
2011. "Transient Thermal Stress Problem of a Functionally Graded Magneto-Electro-Thermoelastic Hollow Sphere" *Materials* 4, no. 12: 2136-2150.
https://doi.org/10.3390/ma4122136