Memory-dependent derivative theory of ultrafast laser-induced behavior in magneto-thermo-viscoelastic metal films

Abstract

In this work, the new concept of “memory-dependent derivative” in heat transfer process in a perfectly conducting viscoelastic metal films was used to investigate the action of a thin metal film, irradiated by a femtosecond laser pulse. The temporal profile of the ultrafast laser was regarded as being non-Gaussian. Laplace transform technique is utilized to solve the problem. Analytical results have been obtained for the fields of temperature, displacement and stress fields, as well as induced magnetic and electrical in a metal film. We have attempted to show the significance of a kernel function and a time delay parameter characteristic of memory-dependent derivative heat transfer in field variables actions with the aid of numerical tests. In the absence of memory-dependent derivative parameters, associations are made with the results obtained. The effects of the time delay on all distributions in a metal film for various forms of kernel functions are examined.

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Abbreviations

x :

One-dimensional space variable

t :

Time

\( e_{ij} \) :

Components of strain deviator tensor

\( S_{ij} \) :

Components of stress deviator tensor

\( \sigma_{ij} \) :

Components of stress tensor

\( u_{i} \) :

Components of displacement vector

Q :

Energy source of the laser pulse

\( T \) :

Absolute temperature

\( T_{\text{o}} {\kern 1pt} \) :

Reference temperature

\( k \) :

Thermal conductivity

\( C_{\text{E}} \) :

Specific heat

\( t_{\text{p}} \) :

Time duration of the laser pulse

\( x_{\text{o}} \) :

Absorptive depth of the heat energy

\( R \) :

Rejectivity of the irradiated surface

H :

Magnetic intensity vector

E :

Electric intensity vector

J :

Conduction current density vector

L :

Non-dimensional film thickness

\( \lambda ,\mu \) :

Lamé constants

\( \alpha_{o} \) :

\( = \, \sqrt {\frac{{\mu_{o} \,H_{o}^{2} }}{\rho }} \) Alfven velocity

\( C_{\text{o}}^{ 2} \) :

= \( \frac{{K_{\text{o}} }}{\rho } \), longitudinal wave speed

\( c \) :

\( = \frac{1}{{\sqrt {\mu_{\text{o}} \varepsilon_{o} } }} \) the speed of light

\( \mu_{\text{o}} \) :

Magnetic permeability

\( \varepsilon_{\text{o}} \) :

Electric permeability

\( \rho \) :

Mass density

\( \theta \) :

Temperature deviation from \( T_{\text{o}} \)

\( \varepsilon_{ij} \) :

Components of strain tensor

\( \sigma_{ij} \) :

Components of stress tensor

\( \alpha_{\text{T}} \) :

Thermal expansion coefficient

\( K_{\text{o}} \) :

= \( \lambda \; + (2/3)\,\mu \), bulk modulus

γ :

\( = 3K_{\text{o}} \alpha_{T} \)

δ ij :

Kronecker delta function

ε :

\( = \gamma^{2} T_{\text{o}} /\rho \,C_{\text{o}}^{2} \eta_{\text{o}} k \)

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Acknowledgments

The authors wish to acknowledge the approval and the support of this research study by the Grant No. SCI-2018-3-9-F-7576 from the Deanship of Scientific Research in Northern Border University, Arar, KSA.

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Hendy, M.H., Amin, M.M. & Ezzat, M.A. Memory-dependent derivative theory of ultrafast laser-induced behavior in magneto-thermo-viscoelastic metal films. Indian J Phys (2020). https://doi.org/10.1007/s12648-020-01783-7

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Keywords

  • Magneto-thermo-viscoelasticity
  • Memory-dependent derivative (MDD)
  • Femtosecond laser
  • Thin metal film
  • Laplace transform
  • Numerical results