International Journal of Current Research and Review
ISSN: 2231-2196 (Print)ISSN: 0975-5241 (Online)
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IJCRR - 9(4), February, 2017

Pages: 15-20

Date of Publication: 20-Feb-2017


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Stresses in an Orthotropic Elastic Layer Lying Over an Irregular Isotropic Elastic Half-Space

Author: Dinesh Kumar Madan1, Poonam Arya2, N.R. Garg2, Kuldip Singh3

Category: General Sciences

Abstract:Objective: The objective is to obtain the stresses due to strip loading in orthotropic plate lying over an irregular isotropic elastic medium.
Methods: Anti-plane strain problem with perfect bonding boundary conditions following by Fourier Transformation on the equilibrium equation are used to obtain the solution.
The deformation field due to shear line load at any point of the medium consisting of an orthotropic elastic layer lying over an irregular isotropic elastic medium is obtained. The anti-plane strain problem with the presence of rectangular irregularity is considered. In order to study the effect of irregularity present in the medium and of anisotropy of the layer, we computed shearing stresses in both the media graphically.

Keywords: Orthotropic, Shear load, Anti-plane strain, Rectangular irregularity

Full Text:

Introduction

It is well known that the upper part of the Earth is recognized as having orthorhombic symmetry. Orthorhombic Symmetry is also expected to occur in sedimentary basins as a result of combination of vertical cracks with a horizontal axis of symmetry and periodic thin layer anisotropic with a vertical symmetry axis. When one of the planes of symmetry in an orthorhombic symmetry is horizontal, the symmetry is termed as orthotropic symmetry and most symmetry systems in the Earth crust also have orthotropic orientations (Crampin1).

The problem of deformation of a horizontally layered elastic material due to surface loads is of great interest in geosciences and engineering. In material science engineering, the applications related to laminate composite material are increasing. Many works related to Earth, such as fills or pavements consist of layered elastic medium. When the source surface is very long, then a two-dimensional approximation simplifies the algebra and one can easily obtain a closed form analytical solution. The deformation field around mining tremors and drilling into the crust of the Earth can be analyzed by the deformation at any point of the media due to strip-loading. It also contributes for theoretical consideration of volcanic and seismic sources as it account for the deformation fields in the entire medium surrounding the source region. It may also find application in various engineering problems regarding the deformation of layered isotropic and anisotropic elastic medium (Garg et al2, Singh et al3).

The study of static deformation with irregularity present in the elastic medium due to continental margin, mountain roots etc is very important to study. Chattopadhyay4, Kar et al5, De Noyer6, Mal7, Acharya and Roy8 discussed the problems with irregular thickness. Love9 provided the solution of static deformation due to line source in an isotropic elastic medium. Salim10 studied the effect of rectangular irregularity on the static deformation of initially stressed and unstressed isotropic elastic medium respectively. The distribution of the stresses due to strip-loading in a regular monoclinic elastic medium had been studied by Madan et al11. The effect of rigidity and irregularity present in fluid-saturated porous anisotropic single layered and multilayered elastic media on the propagation of Love waves had been analyzed by Madan et al12 and Kumar et al13 respectively. Recently, Madan and Gabba14 studied two-dimensional deformation of an irregular orthotropic elastic medium due to normal line load.  

In this paper, we have obtained the closed-form expressions for the displacement and shearing stresses in a horizontal orthotropic elastic plate of an infinite lateral extent lying over an irregular isotropic base due to strip-loading. Numerically, at different sizes of irregularity, we have studied the variations of shearing stresses with horizontal distance and it has been observed that the shearing stresses show significant variation with horizontal distance at the different depth levels.

PROBLEM FORMULATION

Consider a horizontal orthotropic elastic plate of thickness H lying over an infinite isotropic elastic medium with x1- axis vertically downwards. The origin of the Cartesian coordinate system x1, x2,x3  is taken at the upper boundary of the plate. The orthotropic elastic plate occupies the region 0 x1H  and is described as Medium I whereas the region x1>H  is the isotropic elastic half space over which the plate is lying and is described as Medium II. (Fig. 1)

Suppose a shear load P  per unit area is acting over the strip |x2|h   of the surface x1=0  in the positive x3-  direction. The boundary condition at the surface x1=0  is

 

Numerical Results and Discussion

In this section, we intend to examine the effect of irregularity on the stresses due to shear line load acting at any point of the orthotropic elastic layer lying over an irregular isotropic half space. For numerical computation, we use the values of elastic constants of Topaz (Orthotropic) for Medium I and the values of elastic constants of Glass (Isotropic) for Medium II given by Love9.

Figures (2)-(4) and Figures (5)-(7) show the variation of shearing stresses τI31 and τI32  respectively, with horizontal distance x2  for different values of a=1, 1.2, 1.4, 1.6   and for different depth levels x1=0.5, 1, 1.5 . Figures (5)-(7) clearly show that for different values of a , the difference between shearing stresses in magnitude significantly decreases as the depth increases.

Figures (8)-(10) and Figures (11)-(13) show the variation of shearing stresses τII31 and τII32 respectively with horizontal distance x2  for different values of a=1, 1.2, 1.4, 1.6 . It has been found from the Figures (8)-(10) that for different values of a, the difference between shearing stresses τII31 in magnitude significantly increases as the depth increases.

Conclusions

The explicit expressions for the shearing stresses in an elastic medium consisting of orthotropic elastic layer lying over an irregular isotropic half space due to shear loading has been obtained. The results obtained are useful to study the static deformation around mining tremors and drilling into the crust of the Earth. The results are also useful to study the effect of irregularity present between the layer and the half-space. Graphically, it has been observed that the difference between the shearing stresses in magnitude in orthotropic elastic layer decreases as depth increases due to irregularity present.

Further, it has also been observed that in isotropic semi-infinite half-space, the difference between the stresses in magnitude increases with the increase of depth. Thus, it has been concluded that the stress distribution in a layer with irregularity present at the interface is affected by not only the presence of irregularity but also by anisotropy of the elastic medium as a result of shear load acting over the strip of an orthotropic elastic medium.

Acknowledgement

Authors acknowledge the immense help received from the scholars whose articles are cited and included in references of the manuscript. The authors are also grateful to authors/ editors/ publishers of all those articles, journals and books from where the literature for this article has been reviewed and discussed. The authors are also extremely thankful to the reviewers and editors for helping in the improvement of the paper.

References:

  1. Crampin S. Geophysical Prospecting 1989; 37: 753-770.
  2. Garg NR, Madan DK, Sharma RK. Two-Dimensional Deformation of an Orthotropic Elastic Medium due to Seismic Sources. Phys. Earth Planet. Inter., 1996; 94: 43-62.
  3. Singh K, Madan DK, Goel A, Garg NR. Two-Dimensional Static Deformation of Anisotropic Medium. Sadhana(India) 2005; 30: 565-583.
  4. Chattopadhyay A, Chakraborty M, Pal AK. Effect of irregularity on the propagation of guided SH-waves. J. Mechan. Theor. Applied 1983; 2: 215-225.
  5. Kar BK, Pal AK, Kalyani VK. Propagation of love waves in an isotropic dry sandy layer. Acta Geophysics 1986; 157:157-170.
  6. Noyer JD. The effect of variations in layer thickness of Love waves. Bull. Seism. Soc. Am. 1961; 51: 227-235.
  7. Mal AK. On the frequency equation of love waves due to abrupt thickening of crustal layer. Geofis. Pure Applied 1962; 52: 59-68.
  8. Acharya DP, Roy I. Effect of surface stress and irregularity of the interface on the propagation of SH-waves in the magneto-elastic crustal layer based on a solid semi space. Sadhana 2009; 34: 309-330.
  9. Love AEH. A Treatise on the Mathematical Theory of Elasticity. Dover Publication, New York 1944.
  10. Selim MM. Effect of Irregularity on Static deformation of Elastic Half Space. International Journal of Modern Physics 2008; 22:2241-2253.
  11. Madan DK, Chugh S, Singh K. Stresses in an anisotropic Elastic Plate due to Strip-Loading. International Journal of Mechanics 2011; 5: 57-62.
  12. Madan DK, Kumar R, Sikka JS. Love wave propagation in an irregular fluid saturated porous anisotropic layer with rigid boundaries. Journal of Applied Science and Research 2014; 10: 281-287.
  13. Kumar R, Madan DK, Sikka JS. Shear wave propagation in multilayered medium including an irregular fluid saturated porous stratum with rigid boundary. Advances in Mathematical Physics 2014; 10: 1-9.
  14. Madan DK, Gabba A. 2-Dimensional Deformation of an Irregular Orthotropic Elastic Medium. IOSR Journal of Mathematics 2016; 12: 101-113.

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A Study by Ese Anibor et al. "Evaluation of Temporomandibular Joint Disorders Among Delta State University Students in Abraka, Nigeria" from Vol 13 issue 16 received Emerging Researcher Award


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