IJCRR - 9(19), October, 2017
Pages: 01-11
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Static and Quasi-Static Deformation of a Uniform Half-Space Due to a Center of Rotation
Author: Nishu Verma, Kuldip Singh, Renu Muwal
Category: General Sciences
Abstract:Objective: The objective is to obtain static and quasi-static deformation of a uniform half-space due to a center of rotation.
Methodology and Results: The Galerkin vector approach has been used to calculate deformation field at an arbitrary point of an elastic half-space. Closed form analytical expressions for the displacements and stresses caused by a center of rotation buried in a homogenous, isotropic, perfectly elastic half-space are derived. The quasi-static deformation field for a viscoelastic medium has been obtained by applying the correspondence principle of linear viscoelasticity to the associated elastic solution. Explicit expressions giving the quasi-static deformation of a uniform half-space caused by a center of rotation are obtained when the medium is elastic in dilatation and Kelvin, Maxwell or SLS type viscoelastic in distortion.
Conclusion: The explicit expressions for the displacements and stresses in an elastic and viscoelastic medium due to a center of rotation source have been obtained. Numerical results are shown graphically for displacements and stresses.
Keywords: Center of rotation, Static and quasi-static deformation, Correspondence principle, Viscoelastic, Maxwell
Full Text:
INTRODUCTION
Nuclei of strain are the concentrated sources of a displacement field and are built up from the simple superposition of single forces which are acting at a point in the elastic medium (Love). Therefore, the displacement field owing to nuclei of strain is necessary for applications to crustal deformation. Analytical expressions for three-dimensional static displacement fields owing to a step-type single force in an elastic half-space, have been obtained by Mindlinby using Galerkin’s method. Mindlin and Cheng
derived the solutions in the form of the Galerkin vector for various nuclei of strain in a uniform half-space by the process of superposition, differentiation and integration. Many theoretical formulations have been developed (Okada), which were describing the deformation of an isotropic, homogeneous, and semi-infinite medium. Analytical expressions for the quasi-static surface displacements due to a vertical strike-slip fault in a Kelvin or Maxwell viscoelastic half-space were determined by Singh and Rosenmanby applying the correspondence principle of linear viscoelasticity. The correspondence principle has been broadly used to calculate the quasi-static deformation of a viscoelastic half-space by a point or extended sources (see. e.g. [6-10]). Singh and Singhhave identified the combinations in which moduli occur in the expressions for the displacements, strains and stresses in a uniform elastic half-space due to buried sources. Using analytical integration, the displacement field in two welded elastic half-spaces due to a finite rectangular fault has been obtained by Singh et al and they have also compared it with the corresponding field in an elastic half-space and in an infinite medium. To model the ground deformation in volcanic areas, Singh et al used four axially-symmetric source models in an elastic half-space and also compared it with the corresponding field due to a center of dilatation.
Cochard et alshowed that the observations of seismic rotational motions will give important new information referring to the Earth’s surface and are complementary to those found from the observations of the translational motions of the Earth’s surface using conventional seismometers. Cowsik et al resolved that to detect rotational motions, the basic design concept of using a torsion balance as a filter is validated and can be implemented for the construction of rotational seismometers.
The static displacements from the seismic recordings and identifying translation signals caused by rotation can be estimated by using rotational motions (Trifunac and Todorovska). Rotational seismology is of great interest to a wide range of geophysical disciplines, including strong-motion seismology, seismic tectonics, earthquake engineering, and geodesy as well as to physicists using Earth-based observatories for detecting gravitational waves generated by astronomical sources (Lee et al). The solutions for the displacement field produced by a center of rotation is also useful for many purposes and in bio-mechanical research.
RESEARCH METHODLOGY
In this paper, we study the 3-D deformation of a uniform half–space caused by a center of rotation by using the Galerkin vector approach. Explicit expressions for the static strains can be easily obtained with the help of strain -displacement relations and the stresses follow immediately by using stress-strain relations. The correspondence principle of linear viscoelasticity has been used to obtain the quasi-static displacements, strains and stresses.
The paper has been divided into two parts. Part-A deals with the static deformation field while the quasi-static deformation field is considered in Part-B.
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