An Investigation on Two Temperature Dual-Phase-Lag Model of Thermoelasticity Under Fuzzy Environment

  • Saroj MandalEmail author
  • Smita Pal Sarkar
  • Tapan Kumar Roy
Original Paper


In this work, the solution of the two temperature dual-lag-phase model in half space under the uncertain thermal loading at boundary has been discussed. A system of fuzzy partial differential equations arise due to study. System of fuzzy partial differential equations are solved by assuming all the partial derivatives as one differentiable. Laplace transformation is used to solve the partial differential equations related to the problem and using the eigen value approach, the system of ordinary differential equations in Laplace transformed domain is solved. The inverse of such Laplace transformations are carried out by the help of numerical computation. The observations based upon the numerical results are illustrated graphically for stress, strain, temperatures and displacement distributions.


Thermoelasticity Dual-phase-lag model Vector matrix differential equation Laplace transformation Fuzzy number Fuzzy partial differential equation 

List of symbols

\(\lambda , \mu \)

Lame’s constant

\(\rho \)




\(\phi \)

Conductive temperature

\(\theta \)

Thermodynamic temperature

\(\alpha _t\)

Coefficient of linear thermal expansion

\(\sigma _{ij}\)

Component of stress tensor


Component of strain tensor


Component of displacement

\(\alpha \)

Dimensionless mechanic coupling constant

\(\omega \)

Dimensionless two temperature parameter


Specific heat


Thermal conductivity

\(\tau _q\)

Phase lag of heat flux

\(\tau _T\)

Phase lag of temperature gradient

\(\tau _0\)

Relaxation time


Wave speed

\(\eta \)

Thermal viscosity


Two temperature parameter

\(\epsilon \)

Thermoelastic coupling constant

\(\beta \)

\(\alpha _t(3\lambda +2\mu )\)



We want to thank the all the referees for constructive suggestions, information and helpful comments. This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.


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© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of MathematicsSitananda CollegeNandigram, Purba MedinipurIndia
  2. 2.Department of MathematicsIIESTShibpur, HowrahIndia

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