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Non-stationary electro-thermo-mechanical creep response and smart deformation control of thick-walled sphere made of polyvinylidene fluoride

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Abstract

This paper describes an analytical–numerical model developed for non-stationary electro-thermo-mechanical creep response of a smart sphere made of polyvinylidene fluoride (PVDF) using Burgers’ creep model. The piezoelectric properties of the PVDF are used to control creep deformation of the sphere. Time-dependent stresses, displacements, electric potential and strains are calculated using Mendelson’s method of successive approximation. The vessel is subjected to internal and external pressures, distributed temperature fields and an applied electric potential. Two loading combinations have been considered in this study. All mechanical, thermal and piezoelectric properties are selected from the experimental data existed in the literature. All creep parameters for Burgers’ model are functions of time, temperature and stress. Their functional relationship is obtained using a nonlinear regression. Therefore, this study is a nonlinear time-dependent analysis. Total strains are assumed to be the sum of elastic, thermal, electrical and creep strains. Creep strains are time, temperature and stress dependent. Using the equations of equilibrium, compatibility, Maxwell’s equation for free electric charge and stress–strain relations, a differential equation, containing creep strains, for spherical vessel is obtained. A new semi-analytical solution has been developed to find displacement, stresses, strains and electric potential in terms of time-dependent creep strains. Creep strain rates are related to the Burgers’ creep model and current stresses with the Prandtl–Reuss relations. Using Mendelson’s method of successive approximation, the histories of electric potential, radial, circumferential and effective stresses and strains are calculated. It has been found that by applying a suitable electric potential deformation of the sphere can be controlled.

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Abbreviations

u (r):

Radial displacement (m)

r :

Radius (m)

a, b :

Inner and outer radii of hollow sphere (m)

E :

Elastic modulus (N/m2)

e 33, e 13, e 11, e 12 :

Piezoelectric constants (C/m2)

∊ ii (i = 1, 2, 3):

Dielectric constant (C2/Nm2)

\({\sigma_i}\,(i = r,\theta,\varphi)\) :

Component of stresses (N/m2)

\({\varepsilon_{ii}}\,(i = r,\theta,\varphi)\) :

Component of strains

P i :

Internal pressure

P o :

External pressure

τ :

Retardation time in Burgers’ model

\(\Psi\) :

Dimensionless temperature

\({\varLambda_r}\) :

Dimensionless electric displacement

\({\dot \varepsilon_i}\,(i = r,\theta,\varphi)\) :

Creep strain rate (1/s)

\({\dot \varepsilon_{e}}\) :

Effective creep strain rate (1/s)

σ e :

Effective stress (N/m2)

\({D_i}\,(i = r,\theta,\varphi)\) :

Electric displacement component (C/m2)

ϕ (r):

Electric potential (W/A)

T (r):

Temperature distribution function (K)

\({\alpha_i}\,(i = r,\theta,\varphi)\) :

Thermal expansion coefficients (1/K)

υ :

Poisson’s ratio

E M :

Elastic modulus of Maxwell model

η M :

Viscosity of the components of Maxwell model

E K :

Elasticity of the components of the Kelvin model

η K :

Viscosity of the components of the Kelvin model

\(\Gamma\) :

Dimensionless electric potential

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Acknowledgments

The authors are grateful to University of Kashan for supporting this research work.

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Authors and Affiliations

  1. Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Islamic Republic of Iran

    A. Loghman & H. Tourang

Authors
  1. A. Loghman
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  2. H. Tourang
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Correspondence to H. Tourang.

Additional information

Technical Editor: Marcelo A. Savi.

Appendix

Appendix

$$\begin{aligned}{\rm Ke} & = \frac{E}{(1 + \nu)(1 - 2\nu)} \\ p_{11} & = {\rm Ke} \cdot \nu + E_{1}E_{2} \\ p_{12} & = \frac{\rm Ke}{\xi} + \frac{2E_{2}^{2}}{\xi} \\ p_{13} & = - {\rm Ke} \cdot \nu - E_{1} E_{2} \\ p_{14} & = - {\rm Ke} - 2E_2^2 \\ p_{15} & = - {\rm Ke} \cdot \nu - E_{1} E_{2} \\ p_{16} & = - {\rm Ke} - 2E_2^2 \\ p_{17} & = - E_{2},\quad{p_{21}} = {\rm Ke}(1 - \nu) + {E_1}^2\\ {p_{22}} &= \frac{{2\nu {\rm Ke}}}{\xi} + \frac{{2{E_1}{E_2}}}{\xi}\\ {p_{23}} &= {\rm Ke}(1 - \nu) - {E_1}^2\\ {p_{24}} &= - 2\nu {\rm Ke} - 2{E_1}{E_2}\\ {p_{25}} &= - {\rm Ke}(1 - \nu) - {E_1}^2\\ {p_{26}} &= - 2\nu {\rm Ke} - 2{E_1}{E_2}\\ {p_{27}} &= - {E_1}\end{aligned}$$

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Loghman, A., Tourang, H. Non-stationary electro-thermo-mechanical creep response and smart deformation control of thick-walled sphere made of polyvinylidene fluoride. J Braz. Soc. Mech. Sci. Eng. 38, 2547–2561 (2016). https://doi.org/10.1007/s40430-015-0415-4

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  • DOI: https://doi.org/10.1007/s40430-015-0415-4

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