Transient disturbance of multilayered transversely isotropic media under buried thermal/mechanical sources

doi:10.1016/j.ijmecsci.2020.105928

International Journal of Mechanical Sciences

Volume 187, 1 December 2020, 105928

https://doi.org/10.1016/j.ijmecsci.2020.105928 Get rights and content

Highlights

•
Transient disturbance of transversely isotropic multilayered media under buried thermal/mechanical sources is studied.
•
The coupled thermoelastic governing equations involve the second sound effect based on the Lord-Shulman model.
•
Laplace-Hankel transform and the extended precise integration method are employed to solve governing equations.
•
The impacts of transient load type, material transverse isotropy and the relaxation time are investigated.

Abstract

This paper presents the coupled thermo-elastic analysis of the multilayered transversely isotropic media due to buried thermal/mechanical sources. The coupled thermo-elastic governing equations involve the second sound effect based on the Lord-Shulman model. We employ the Laplace-Hankel transform and introduce the extended precise integration method to solve these equations. Once the thermal/mechanical boundary conditions are given, the solutions for temperature and displacement in time domain can be achieved by implementing Laplace-Hankel numerical inversion procedure. Furthermore, comparisons with the existing solutions are made to verify the presented method. Numerical analysis is performed through a MATLAB platform to investigate the impacts of the transient load type, the mechanical transverse isotropy, the thermal transverse isotropy and the relaxation time on the transient thermo-elastic response.

Graphical abstract

Introduction

Heat transfer theory is widely applied in practical engineering, such as the disposal of the radioactive waste [1], [2], [3], nuclear blast environment [4], [5], [6], [7] and petroleum engineering [8], [9], [10]. The conventional theory of the coupled thermoelatsticity has been addressed by many researchers [11], [12], [13], [14], [15], [16]. However, in the conventional theory, there is an apparent paradox that thermoelastic waves propagate with a theoretically infinite velocity, which is physically unreasonable [17]. To eliminate this paradox, the generalized thermoelastic theory is proposed considering the second sound effect, i.e. thermelastic wave propagation with a finite velocity. In this respect, Lord and Shulman [18] modified the Fourier heat conduction law by introducing the concept of “relaxation time” in the thermal conductivity analysis. Since then, more attentions were paid to the second sound effect of materials such as sand and rocks, for example, the analytical solutions [19], [20], [21], [22] and experimental investigations [23], [24] for the generalized thermoelastic theory have been developed by many researchers.

Meanwhile, in some engineering cases [25], [26], materials such as soils and rocks display non-negilible anisotropy behavior, which may have a significant influence on thermoelastic responses. A large amount of research in generalized thermoelasticity was reviewed by Ignaczak [27] who emphasized the thermoelatsic coupling of anisotropic and inhomogeneous media. Banerjee and Pao [28] investigated the propagation of plane harmonic thermoelastic waves in an infinitely extended anisotropic solid considering the modified constitutive law of heat conduction. Dhaliwal and Sherief [29] derived the equations of generalized thermoelasticity behavior in anisotropic media and proved a unique theorem for theses equations. Sharma and Singh [30] obtained the explicit expressions describing the propagation of surface waves in the transversely isotropic thermoelastic half-space. Chandrasekharaiah and Keshavan [31] studied the plane waves in a linear homogeneous and transversely isotropic body on the basis of the generalized thermoelasticity. Sharma et al. [32] gave a detailed account of the plane harmonic generalized thermoelastic waves in orthorhombic materials. Chen et al. [33] introduced two displacement functions to simplify the basic equations and obtained an exact fundamental solution for the thermoelastic transversely isotropic medium. Hayati et al. [34] introduced the potential functions to obtain a general solution for the Lord-Shulman non-classical thermoelastic equations of three dimensional transversely isotropic solids.

The above research [28], [29], [30], [31], [32], [33], [34] aims at solutions for homogeneous media. However, the long-term tectonic movement, weathering and sedimentary environment bring about the layered properties of the rock masses and soils [35]. As regards the multi-layered media, the explicit solutions are increasingly complicated when the number of layer increases [36], [37], and these fundamental solutions are important for the boundary element method [38]. Numerous solutions for layered media based on the classical thermo-elastic theory have been obtained by the finite layer method [39], [40], the transfer matrix method [41], [42], the flexibility/stiffness matrix method [43], the analytical layer-element method [44], [45], the extended precise integration method [46], [47], and the dual variable and position method [48], [49]. However, in the context of the generalized thermoelasticity, the existing solutions considering both transverse isotropy and stratification are relatively rare. Hawwa and Nayfeh [50] examined the propagation of harmonic waves in a laminated composite of layered anisotropic plates by the transfer matrix method. Verma [51] provided an analysis of the wave propagation in a layered anisotropic medium in the context of generalized thermoelasticity. Verma [52] analyzed the generalized thermoelastic small-amplitude wave propagation in laminated structures subjected to an imposed pre-strain. However, these analyses [50], [51], [52] concern on the propagation of harmonic thermal waves and there exist requirements for dealing with numerical instabilities in the proposed method [51], [52]. Less work is available on the transient thermal/mechanical response of multilayered transversely isotropic media, and a numerically efficient and stable method should be introduced to analyze transient generalized thermo-elastic problem considering both transverse isotropy and stratification.

In this article, the coupled thermo-elastic analysis of multilayered transversely isotropic media under buried thermal/mechanical sources is investigated based on the Lord-Shulman model. The generalized thermoelastic governing equations are transferred into ordinary differential equations by utilizing the Laplace-Hankel transform technique. The extended precise integration method [46], [47], [48], [49], [50], [51], [52], [53], [54], [55] can provide an effective and numerically stable approach to the complete solutions in the transformed domain once the thermal and mechanical boundary conditions and layer interface conditions are introduced. Then the solutions in the time domain are acquired by a numerical transverse inversion. Several numerical examples are conducted to verify the proposed method. Subsequently, the illustrative examples and discussions on the effects of the transient load type, the mechanical transverse isotropy, the thermal transverse isotropy and the relaxation time are provided.

Section snippets

Basic equations

The multilayered transversely isotropic media under a buried thermal/mechanical source is considered within a cylindrical coordinate system in Fig. 1. The equilibrium differential equations of the axisymmetric problem without considering body forces can be given as $\frac{\partial σ_{r}}{\partial r} + \frac{\partial σ_{r z}}{\partial z} + \frac{σ_{r} - σ_{φ}}{r} = ρ \frac{\partial^{2} u_{r}}{\partial t^{2}}$ $\frac{\partial σ_{r z}}{\partial r} + \frac{\partial σ_{z}}{\partial z} + \frac{σ_{r z}}{r} = ρ \frac{\partial^{2} u_{z}}{\partial t^{2}}$ where σ_r, σ_φ and σ_z are the normal stress components in the r, φ and z directions, respectively; σ_rz represents the shear stress components in the r-z plane; u_r and u_z are the

Ordinary differential matrix for coupled generalized thermoelasticity

The integral transformation is applied to deal with the above partial differential equations. The m^th-order Laplace-Hankel transform and its inverse transform with respect to r and t can be defined as follows according to Sneddon [57] ${\tilde{f}}^{m} (ξ, z, s) = \int_{0}^{+ \infty} \int_{0}^{+ \infty} f (r, z, t) J_{m} (ξ r) r e^{- s t} d r d t$ $f (x, y, z) = \int_{c - i \infty}^{c + i \infty} \int_{0}^{+ \infty} {\tilde{f}}^{m} (ξ, z, s) J_{m} (ξ r) ξ e^{s t} d ξ d s$ where the ${\tilde{f}}^{m} (ξ, z, s)$ with the superscript ~ denotes the m^th-order Laplace-Hankel transform of f(r, z, t); ξ and s are the Hankel and Laplace transform parameters, respectively;

Numerical results and discussions

The foregoing derivations are conducted in the Laplace-Hankel transformed domain, and it is necessary to apply a numerical technique to obtain the solutions in the time domain. On the basis of the previous studies [58], [59], [60], the Stehfest method [61] is introduced to obtain the actual results. $f (t) = \frac{\ln 2}{t} \sum_{n = 1}^{N / 2} K_{n} \tilde{F} (n \ln 2 / t)$ $K_{n} = {(- 1)}^{n + N / 2} \sum_{k = [(n + 1) / 2]}^{\min (n, N / 2)} \frac{k^{N / 2} (2 k)!}{(N / 2 - k)! k! (k - 1)! (n - k)! (2 k - n)!}$ where the free parameter N can be determined by the convergence study, which reveals that the

Conclusions

In this paper, the extended precise integration method is applied to obtain the generalized thermoelastic solutions for multilayered transversely isotropic media due to buried thermal/mechanical sources. Numerical examples are given for the verification of the proposed method and discussions on the influences of the transient load type, the mechanical transverse isotropy, the thermal transverse isotropy and the relaxation time. The computed results mentioned above reveal the following

CRediT authorship contribution statement

Zhi Yong Ai: Conceptualization, Methodology, Writing - review & editing, Supervision. Zi Kun Ye: Data curation, Validation, Investigation, Writing - original draft. Zhong Hao Chu: Investigation, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research is supported by the National Natural Science Foundation of China (Grant No. 51639008). The authors would like to thank the editor and the reviewers for their valuable comments that contribute to improving the quality of this paper.

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Ai et al. (2020) conducted an axisymmetric coupled thermo-mechanical analysis of multilayered media under a buried heat source or mechanical load and studied the effects of transient load types, transverse isotropic properties, and relaxation times on the transient thermoelastic response. In general, existing solutions to the thermo-mechanical problems of multilayered media mostly focus on axisymmetric loads (Small and Booker, 1986; Wong and Zhong, 2000; Zhong and Geng, 2009; Ai et al., 2015, 2020), plane strain conditions (Small and Booker, 1989, Wang and Ai, 2015) or three-dimensional conditions (Wang and Ai, 2015), solutions for more general non-axisymmetric loads are rare. To further study the thermo-mechanical effect of media, it is significant to obtain the solution of multilayered thermoelastic materials under a non-axisymmetric load.
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Transient disturbance of multilayered transversely isotropic media under buried thermal/mechanical sources

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Basic equations

Ordinary differential matrix for coupled generalized thermoelasticity

Numerical results and discussions

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

Int J Mech Sci

Int J Solids Struct

Int J Eng Sci

Int J Mech Sci

Int J Rock Mech Min Sci

Prog Aerosp Sci

J Sound Vib

J Mech Phys Solids

Int J Mech Sci

Int J Rock Mech Min Sci Geomech Abstr

Int J Solids Struct

Int J Mech Sci

Int J Mech Sci

Int J Solids Struct

Phys Earth Planet Inter

Comput Geotech

Appl Math Model

Adv Eng Software

Compos Eng

Int J Eng Sci

Comput Math Appl

Appl Math Model

Int J Mech Sci

Eng Anal Bound Elem

J Pet Sci Eng

Appl Math Model

Eng Geol

Eur J Mech-A/Solids

A coupled thermoelastic model for saturated–unsaturated porous media

Géotechnique

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Int J Numer Anal Methods Geomech

An introduction to the theory of seismology

The effect of thermoelastic stresses on injection well fracturing

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J l'Ecole Polytech