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In this article, the nonclassical transient heat propagation process in a cracked strip is investigated by Guyer–Krumhansl (G–K) model, which incorporates both the time lagging behavior and the spatially nonlocal effect. The impulsive thermal loading as well as cyclic loading exerted on the top bounding surface are examined to explore the non-Fourier thermal characteristics. By means of the Laplace transform and Fourier transform, the governing partial differential equations subjected to mixed boundary conditions are converted to a group of singular integral equations. With the aid of numerical Laplace inversion, the transient temperatures are calculated to make comparisons of thermal responses determined by Fourier's law, Cattaneo–Vernotte (C–V) equation, and G–K model. The numerical results display the specific thermal behaviors of G–K model in the cracked medium and demonstrate the G–K model's capabilities in eliminating the unrealistic phenomena accompanied by C–V equation. Our research would contribute to achieving a better understanding of the transient heat conduction in small-sized systems or composites at the macroscopic scale. [ABSTRACT FROM AUTHOR]

We view the (real) Laplace transform through the lens of linear algebra as a continuous analogue of the power series by a negative exponential transformation that switches the basis of power functions to the basis of exponential functions. This approach immediately points to how the complex Laplace transform is a generalisation of the Fourier transform where the pole of the transform realises the linear algebraic intuition. The exponential transformation also motivates the Taylor inversion of the real Laplace transform. [ABSTRACT FROM AUTHOR]

The state-space formulation for two-dimensional generalized thermoviscoelasticity has been formulated. In this formulation, the governing equations are transformed into a matrix equation whose solution enables us to write the solution of any two-dimensional problem in terms of the boundary conditions. The resulting formulation is applied to a 2 D problem of a half-space subjected to harmonically varying heating with constant angular frequency. [ABSTRACT FROM AUTHOR]

• Semi-analytic methods are used to solve 3D DPL problems. • Compared with FEM, the semi-analytic method has higher computational efficiency. • The phase lag of heat flow and the phase lag of temperature gradient have influence on the temperature response speed and the formation of temperature field. With the development of picosecond laser technology, the dual-phase-lagging (DPL) model is more suitable to describe the heat transfer problem with micro-scale effect and micro-time effect. In this paper, a three-dimensional dual-phase-lagging heat transfer model for picosecond laser processing is established. Then Fourier transform (FT) and Laplace transform (LT) are used to derive the frequency response functions (FRFs) of the problem in the frequency domain, and the inverse Laplace transform (ILT) and discrete-convolution fast Fourier transform (DC-FFT) algorithm are used to solve the temperature field. The calculation accuracy of this method is verified and the effect of phase lag constants on temperature field is studied. The results show that the semi-analytical method has a high accuracy for solving the dual-phase-lagging heat conduction model, and the phase lag of heat flux will affect the heating rate in the heating stage, and the phase lag of temperature gradient will delay the formation of temperature field. The effect of phase lag constants on temperature-time history is investigated and the related phenomena are explained by using the two-step heat transfer theory. This semi-analytical method provides a new approach to solve the problem of three-dimensional dual-phase-lagging heat transfer. [ABSTRACT FROM AUTHOR]