A comparative study of deformation field due to center of dilatation and center of rotation in a viscoelastic half-space model

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Highlights

  • The explicit expressions for the displacements and stresses in an elastic and a viscoelastic half-space due to two axially symmetric sources, namely, the center of dilatation and center of rotation have been obtained by using Galerkin vector approach.

  • The results are also compared graphically for displacement and stress fields for different values of Poisson’s ratio and three viscoelastic models, namely, Kelvin, Maxwell and SLS.

  • Comparative study of both these sources has been done graphically.

  • Our analysis revealed that center of dilatation is more efficient than center of rotation in generating radial displacement.

  • The numerical values of radial displacement and uplift assume the maximum value in the case of Maxwell and minimum in case of Kelvin model for both the sources.

Abstract

Closed-form analytic expressions for the displacement and stress fields due to two axially symmetric sources, namely, the center of dilatation, and center of rotation in an elastic half-space are obtained using Galerkin vector approach. The correspondence principle of linear viscoelasticity is used to obtain the viscoelastic displacement and stress fields. A comparative study of both these sources has been done graphically. Our analysis revealed that the center of dilatation is more efficient than the center of rotation in generating radial displacement. The results are valid for arbitrary values of the Poisson's ratio and relaxation time.

Introduction

Indirect measurements of rotational motions using a seismometer array have been studied by several investigators (e.g., Spudich et al., 1995; Singh et al., 1997; Li et al., 2001; Huang, 2003). Cowsik et al. (2009) have determined that, the basic design concept of using a torsion balance to detect rotational motions, as a filter is validated and can be implemented for the construction of rotational seismometers. Lee et al. (2009) have demonstrated that to study the earthquake, with the three components of translational motion, we also need to simultaneously measure the three components of rotational motion and components of strains.

Sun et al. (2012) have presented a model to study the three-dimensional viscoelastic interactions between the center of dilatation and a penny-shaped interfacial crack in an infinite bi-material. Han et al. (2015) have observed the coseismic compression/dilatation and viscoelastic uplift/ subsidence associated with the 2012 Indian Ocean earthquakes. Igel et al. (2015) have given that the concept of rotational motions due to seismic sources have applications in the fields of earthquake engineering, fundamental physics, Ocean generated noise, seismic instrumentation, and seismic tomography, etc.

Bonafede et al. (1986) have obtained analytical expressions for the displacement and stress fields due to a center of dilatation in an elastic and a viscoelastic half-space. The static and quasi-static deformation of a uniform half-space due to various seismic sources has been obtained by Singh and Singh (1989). Deformation field due to a center of dilatation has been compared with the corresponding field due to four axially-symmetric sources, namely, a vertical force, a vertical dipole, a tensile dislocation on a horizontal fault and a compensated linear vector dipole (CLVD) in an elastic half-space by Singh et al. (1998). Deformation field in two welded half-spaces due to an inclined shear and tensile point dislocations and a center of dilatation has been obtained by Singh et al. (2000).

Verma et al. (2017) have obtained analytical expressions for the displacements and stresses due to a center of rotation in an elastic and a viscoelastic half-space.

The present paper aims to study the comparison between displacement and stress fields due to two axially symmetric sources, namely, the center of dilatation and center of rotation. It is assumed that the half-space z ≥ 0 is stress-free at z = 0 and the point source is located at height c above the free surface (Fig. 1). We use a cylindrical coordinate system (r, θ, z) with the z-axis vertically upwards.

Section snippets

Displacement and stress fields due to a center of dilatation in elastic medium

A center of dilatation is equivalent to three equal mutually orthogonal dipoles (Fig. 1).

Taking the moment of each of the dipoles as F1, the elastic displacement field is given below has been obtained from Singh and Singh (1989) by taking Galerkin vector G through the relation2μu=21σ2G.Gwhere u is the displacement vector, μ is the modulus of rigidity and σ=3k2μ23k+μ is the Poisson's ratio.

The radial displacement and uplift at the surface of the half-space (z = 0) are given by:ur=12σF

Displacement and stress fields due to a center of rotation in elastic medium

If we combine two coplanar single couples with moments equal in magnitude and sense, we get a torque or center of rotation, in which the net force is zero, whereas the net moment is non-zero, as shown in Fig. 2.

If the strength of center of rotation is F1, the elastic displacement field given below has been obtained by Verma et al. (2017)ur=F12πμRcR2+12σR+cuz=F1r2πμR3τr=F1rπR31σ12σ3cR2+1σ2R+cR+c2

Viscoelastic solution

The correspondence principle of linear viscoelasticity is used to obtain the viscoelastic displacement and stress fields from the elastic solution. The bulk modulus κ and the shear modulus μ, which are appearing in an elastic solution, are replaced by κ˜s and μ˜s in viscoelastic solution, which depends on particular rheology of material considered. The source function f(t) is replaced by its Laplace transform f˜s and the resulting expression is the Laplace transformed viscoelastic solution. The

Numerical results and discussion

We define dimensionless epicentral distance D, dimensionless radial displacement U, dimensionless uplift W and dimensionless radial stress Γr by the relations D=rc,U=Mcur,W=Mcuz, and Γr=Mμτr where M=πμc3F1 is dimensionless constant.

Conclusions

The explicit expressions for the displacements and stresses in an elastic and a viscoelastic half-space due to two axially symmetric sources, namely, the center of dilatation and center of rotation have been obtained by using Galerkin vector approach. The results are also compared graphically for displacement and stress fields for different values of Poisson's ratio and for three viscoelastic models, namely, Kelvin, Maxwell, and Standard Linear Solid (SLS). A comparative study of these sources

CRediT authorship contribution statement

Nishu Verma:Conceptualization, Methodology, Software, Formal analysis, Data curation, Writing - original draft, Writing - review & editing, Visualization, Funding acquisition, Validation.Kuldip Singh:Validation, Writing - review & editing, Visualization, Supervision.

Declaration of competing interest

The authors declare that there is no conflict of interest.

Acknowledgements

One of the authors Ms. Nishu Verma is grateful to the University Grants Commission, New Delhi (India) for financial support.

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