IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 12, Issue 4 Ver. V (Jul. - Aug.2016), PP 101-113 www.iosrjournals.org DOI: 10.9790/5728-120405101113 www.iosrjournals.org 101 | Page 2-Dimensional Deformation of an Irregular Orthotropic Elastic Medium Dinesh Kumar Madan and Aanchal Gaba Department of Mathematics, The Technological Institute of Textile & Sciences, Bhiwani-127021 (INDIA) Abstract: In the present paper the closed form analytical expressions for the displacement and stresses at any point of an infinite irregular orthotropic elastic medium as a result of normal line-load have been obtained by using eigen-value approach. The irregularity of rectangular and parabolic shape has been considered. To examine the effect of different type of irregularities the variation of displacement and stresses with horizontal distance have been shown graphically by taking different sizes of irregularities. Also to study the effect of irregularities present in the medium the comparison between the displacement due to with and without irregularities have been made. Contour maps showing the displacement field for each type of irregularities are presented. Keywords: Orthotropic medium, Normal line-load, Irregularity, Rectangular, Parabolic, Eigen value I. Introduction Although isotropy is a good approximation in the Earth, it is sometimes important to consider departure from isotropy, i.e., anisotropy. From the study on earthquakes and earth structures, it has been observed that the Earth is anisotropic in nature. Most anisotropic medium of interest in seismology have at least approximately a horizontal plane of symmetry. Medium having three orthogonal planes of symmetry is called orthotropic medium. Since the orientation of stern in the Earth’s crust is usually orthotropic, ‘most symmetry systems in the crust of Earth have orthotropic orientation’. The most important anisotropic materials are Olivine and Orthoyroxenes, which comprise much of the deep crust and upper mental, exhibit orthotropic symmetry. Moreover the interfaces separating the different media of the Earth are not perfectly plane. To better understand the seismic behavior at continental margins, mountain rocks etc., the static deformation problem of an anisotropic elastic medium with irregularities present is very important. The problems of propagation of seismic waves with irregularities present in the elastic medium have been studied by many researchers like De Noyer [1], A.K. Mal [2] ,Kar et al. [3], Chattopadhyay [4], Acharya and Roy[5], and others. Madan et al. [6] and Kumar et al. [7] analyzed the effect of rigidity and irregularity on the propagation of Love waves in fluid saturated porous anisotropic single layered and multilayered elastic media. Garg et al. [8] studied the plane strain problem of infinite orthotropic elastic medium due to two- dimensional sources by considering distinct Eigen values. By using Eigen value method Singh et al. [9] obtained the deformation field for the monoclinic medium in the transformed domain with plane interface. Salim [10] studied the effect of rectangular irregularity on the static deformation of initially stressed and unstressed isotropic elastic medium respectively. In isotropic medium the Eigen values cannot be distinct. In the present paper we have considered the homogenous, orthotropic elastic medium to study the effect of rectangular and parabolic irregularities on the static deformation due to normal line load. Anisotropy resulted from a material being non-uniform or homogenous. The crystal structure of the mineral Olivine is homogeneous as it is composed of the same repeating groups of atoms, but acts anisotropic because its elastic properties vary in different directions relative to the crystal lattice. Numerically by considering the material Olivine, the effect of irregularities has been examined by drawing the variations of displacements and normal stresses with horizontal distance for different size of irregularities. Also the comparison of displacements for regular medium with rectangular and parabolic irregularities present in the medium has been made graphically. The variations of displacements with different sizes of two different irregularities i.e. rectangular and parabolic are depicted by drawing the contour. The present problem is an improvement of the earlier papers studied by Garg et al. [8] for regular orthotropic elastic medium and Salim [10] for irregular (rectangular) isotropic medium. The corresponding results for normal loading obtained by Garg et al [8] can be recovered from our results as a particular case. II. Formulation of the Problem We consider an infinite orthotropic elastic medium with rectangular x- axis vertically downwards and let the origin of the co-ordinate(x,y,z) be situated at x=0. Suppose that a normal line load , per unit, length, is acting vertically downward on a line parallel to the z- axis and passing through the points (H, 0). Let the