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Exact Electromagnetothermoelastic Solution for a Transversely Isotropic Piezoelectric Hollow Sphere Subjected to Arbitrary Thermal Shock

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Abstract

This paper presents analytical study for electromagnetothermoelastic transient behavior of a transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Exact solutions for the transient responses of stresses, perturbation of magnetic field vector, electric displacement and electric potential in the transversely isotropic piezoelectric hollow sphere are obtained by means of the Hankel transform, the Laplace transform and their inverse transforms. An interpolation method is used to solve the Volterra integral equation of the second kind caused by interactions among electric, magnetic, thermal and elastic fields. From the sample numerical calculations, it is seen that the present method is suitable for the transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Finally, the result can be used as a reference to solve other transient coupling problems of electromagnetothermoelasticity.

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Abbreviations

\(\vec{U},u\) :

Displacement vector and radial displacement [m]

c ij (i=r,θ,j=r,θ,φ):

Elastic constants [kg/ms2]

e ri (i=r,θ):

Piezoelectric constants [As/m2]

β rr :

Dielectric constant [A2s4/kgm2]

p r :

Pyroelectric coefficient [A2s/Km2]

σ i (i=r,θ),D r :

Components of stresses [kg/ms2]

and radial electric displacement [As/m2]

T(r,t):

Temperature change [K]

α i (i=r,θ):

Thermal expansion coefficients [1/K]

φ(r,t):

Electric potential [kgm2/As3]

ρ :

Mass density [kg/m3]

t :

Time variable [s]

r,θ :

Radial variable and circumferential variable [m]

\(\vec{H}\) :

Magnetic intensity vector

\(\vec{h}\) :

Perturbation of magnetic field vector

\(\vec{J}\) :

Electric current density vector

\(\vec{e}\) :

Perturbation of electric field vector

μ :

Magnetic permeability [kgm/A2s2]

H u :

Magnetic intensity [A/m]

f u :

Lorentz’s force [kg/m2s2]

a,b :

Internal and external radii of piezoelectric hollow sphere [m]

C L :

Electromagnetothermoelastic wave speed [m/s]

ω :

The inherent frequency of the piezoelectric hollow sphere [1/s]

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Authors and Affiliations

  1. Department of Engineering Mechanics, College of Mechanical & Vehicle Engineering, Hunan University, Changsha, 410082, China

    Hong-Liang Dai, Hong-Yan Zheng & Lei Yang

  2. Research and Development Center for Deep-sea Detection Equipment, Hunan University, Changsha, 410082, China

    Hong-Liang Dai, Hong-Yan Zheng & Lei Yang

Authors
  1. Hong-Liang Dai
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  2. Hong-Yan Zheng
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  3. Lei Yang
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Correspondence to Hong-Liang Dai.

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Dai, HL., Zheng, HY. & Yang, L. Exact Electromagnetothermoelastic Solution for a Transversely Isotropic Piezoelectric Hollow Sphere Subjected to Arbitrary Thermal Shock. J Elast 102, 79–97 (2011). https://doi.org/10.1007/s10659-010-9263-8

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  • DOI: https://doi.org/10.1007/s10659-010-9263-8

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