Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 0974-5645 10.17485/IJST/v14i1.1874 Stresses in a monoclinic elastic plate placed upon an irregular monoclinic elastic half space Savita savitadhankhar67@gmail.com 1 Sahrawat Ravinder Kumar 1 Malik Meenal 2 Department of Mathematics, Deenbandhu Chhotu Ram University of Science and Technology Murthal, 131039, 8397883690 India Department of Mathematics, All India Jat Heroes Memorial College Rohtak, 124001 India 14 1 2021 Abstract

Objective: To study the stresses due to strip load in a monoclinic elastic plate placed over an irregular monoclinic elastic half space. Method: Anti-plane strain problem with different interfacing boundary conditions namely “perfectly welded interfacing, smooth-rigid interfacing, rough-rigid interfacing’’ has been considered and Fourier Transformation is used on the equilibrium equations to obtain the solutions. Findings: The displacements and stresses are obtained for each boundary condition and the variation of shearing stresses with horizontal distance are studied due to different size of rectangular shaped irregularity at any point of the medium consisting of monoclinic elastic plate placed upon an irregular monoclinic elastic half space. The comparison between the stresses for different types of interfacing has been made graphically. Novelty: The static deformation of two or more connected elastic mediums with irregularities due to surface loads has been studied earlier but just a few have showed graphic representation of stresses with irregularity in different sizes. Keeping in mind the shortcomings of earlier work done, the present paper visualizes shearing stresses in both theoretical and graphical manners completely.

Keywords Monoclinic strip loading anti plane strain rectangular irregularity HRDG Council of Scientific & Industrial Research for sanctioning SRF scholarship with File No. (09/1063/0012/2017)
Introduction

In this paper, we have obtained closed form analytical expressions for the displacements and stresses in a horizontal infinite monoclinic elastic layer placed over an infinite irregular monoclinic elastic half space. The variation of shearing stresses has been studied graphically with different sizes of irregularity.

Basic equation

The equilibrium equations in the cartesian co-ordinate system for zero body forces are

τxx,x+τxy,y+τxz,z=0 τyx,x+τyy,y+τyz,z=0 τzx,x+τzy,y+τzz,z =0

where τxx,τyy,τzz are normal stresses and τxy,τyz,τzx are shearing stresses.

The stress-strain relations for a monoclinic elastic medium in co-ordinate plane are given by Crampin 1

τxx=c11exx+c12eyy+c13ezz+c14eyzτyy=c12exx+c22eyy+c23ezz+c24eyzτzz=c13exx+c23eyy+c33ezz+c34eyzτyz=c14exx+c24eyy+c34ezz+c44eyzτzx=c55ezx+c56exyτxy=c56ezx+c66exy

exx=ux, eyy=vy, ezz=wz,  exy=12uy+vx,  eyz=12vz+wy,ezx=12wx+uz

exx,eyy,ezz are normal strain components and exy,eyz,ezx are shearing strain components. u,v,w are the displacement components in cartesian coordinate plane. The two suffix quantities ciji,j=1,2,3,4,5,6 are the elastic constants.

The displacements and stresses for a monoclinic elastic material in anti-plane strain equilibrium state in the yz-plane is

u=u(y,z),  v=0,  w=0exx=eyy=ezz=eyz=0exy=12uy,  exz=12uzτxy=c56uz+c66uyτzx=c55uz+c56uy

2uy2+2c56c662uyz+c55c662uz2=0

fx,y as

f-x,s=-fx,yeiysdy

So, the Inverse Fourier Transform is defined as

fx,y=12π-f-x,se-iysds
Formulation of the problem

We consider the cartesian co-ordinate system ( x,y,z). Here, z-axis has been taken horizontally and y-axis along vertically downward. An infinite monoclinic elastic plate of thickness ‘ Y’ lying horizontally over an irregular infinite monoclinic elastic half-space. The irregularity is assumed to be rectangular in shape. The origin of Cartesian co-ordinate system ( x,y,z) is taken at the upper boundary of the plate. The plate occupying the region ( 0y Y) is described as medium I whereas the region y>Y is described as medium II. (as shown in Figure 1).﻿

When the interfacing between the plate and half-space at y=ϵfz is welded contact, then condition (12), (14), (15), (16), (19), (20), and (21) yield

Aem1|s|ϵf(z)+Be-m1|s|ϵf(z)eim2ϵf(z)s-ce-m1'|s|ϵf(z)eim2'ϵf(z)s=0 Aem1|s|ϵfeim2ϵfsT1S1-iϵf'm2T1S1-ϵf'm1-Be-m1|s|ϵfeim2εfsT1S1-iϵf'm2T1S1+ϵf'm1+Ce-m1'|s|ϵfeim2'εfsT1'S1-iϵf'm2'T1'S1+ϵf'm1'=0

Solving equations (18), (26) and (27), we have the values

A=2P0T1sin(sh)s|s|S1V-ϵf'V3+iϵf'S1V2e-2m1|s|εfS1-iϵf'S1V1+ϵf'V3-S1V-ϵf'V3+iϵf'S1V2e-2m1|s|εf B=2P0T1sin(sh)s|s|1-S1V-ϵf'V3+iϵf'S1V2e-2m1|s|εfS1-iϵf'S1V1+ϵf'V3-S1V-ϵf'V3+iϵf'S1V2e-2m1|s|εf C=4P0T1sin(sh)s|s|S11+iϵf'm2e-(α|s|+iβs)εfS1-iϵf'S1V1+ϵf'V3-S1V-ϵf'V3+iϵf'S1V2e-2m1|s|εf

where V=T-1T+1,V1=T1m2+T1'm2'T1+T1',V2=T1m2-T1'm2'T1+T1',V3=m1'-m1T1+T1'

T=T1T1',α=m1-m1',β=m2-m2'

Substituting the values for A, B, C from (28)-(30) in equations (14)-(16) for med. I and in equations (19)-(21) for med. II and also substituting the value for f(z) from (22), we will obtain the following results for displacements and stresses.

For med. I (i.e., for y Y)

uI=P0πT1-sin(sh)s|s|e-m1|s|y+n=1Vne-m1|s|(2nϵf-y)-n=1Vne-m1|s|(2nϵf+y)e-iz-m2ysds τxyI=-P0πtan-12hm1yz-m2y2+m12y2-h2-n=1Vntan-12hm1(2nϵf+y)z-m2y2+m12(2nϵf+y)2-h2+tan-12hm1(2nϵf-y)z-m2y2+m12(2nϵf-y)2-h2 τxzI=-P0m2πtan-12hm1yz-m2y2+m12y2-h2-n=1Vntan-12hm1(2nϵf+y)z-m2y2+m12(2nϵf+y)2-h2       +tan-12hm1(2nϵf-y)z-m2y2+m12(2nϵf)-y2-h2+m1P02πlogz-m2y+h2+m1y2z-m2y-h2+m1y2          -n=1Vnlogz-m2y+h2+m1(2nϵf+y)2z-m2y-h2+m1(2nϵf+y)2+logz-m2y+h2+m1(2nϵf-y)2z-m2y-h2+m1(2nϵf-y)2

For med. II (i.e. y>Y )

uII=  2P0πT1TT+1-sin(sh)s|s|e-m1|s|y-(α|s|+iβs)ϵf+         n=1Vne-m1|s|(2nϵf+y)-(α|s|+ιβs)ϵfe-iz-m2'zysds τxyII=-2P0π(1+T)tan-12hm1'y+αϵfz-m2'y+ιβϵf2+m1'y+αϵf2-h2         +n=1Vntan-12h2nm1+αϵf+m1'yz-m2'y+ιβϵf2+2nm1+αϵf+m1'y2-h2 τxzII=-2P0m2'(1+T)πtan-12hm1'y+αϵfz-m2'y+ιβϵf2+m1'y+αϵf2-h2        +n=1Vntan-12h2nm1+αϵf+m1'yz-m2'y+ιβϵf2+2nm1+αϵf+m1'y2-h2+P0m1'π(1+T)logz-m2'y+ιβϵf+h2+m1'y+αϵf2z-m2'y+ιβϵf-h2+m1'y+αϵf2++n=1Vnlogz-m2'y+ιβϵf+h2+m1'y+2nm1+αϵf2z-m2'y+ιβϵf-h2+m1'y+2nm1+αϵf2
Smooth rigid-interface

When the interfacing between plate and half-space at y=Y is smooth-rigid, without loss of generality, on taking T i.e, V=1 in equations (31)-(33) for med. I and in equations (34)-(36) for med. II. we will obtain the following results for displacements and stresses.

For med. I (i.e., for y Y )

uI=PoπT1-sin(sh)s|s|e-m1|s|y+n=1e-m1|s|(2nϵf-y)-n=1e-m1|s|(2nϵf+y)e-iz-m2ysds τxyI=-P0πtan-12hm1yz-m2y2+m12y2-h2-n=1tan-12hm1(2nϵf+y)z-m2y2+m12(2nϵf+y)2-h2          +tan-12hm1(2nϵf-y)z-m2y2+m12(2nϵf-y)2-h2 τxzI=-P0m2πtan-12hm1yz-m2y2+m12y2-h2-n=1tan-12hm1(2nϵf+y)z-m2y2+m12(2nϵf+y)2-h2          +tan-12hm1(2nϵf-y)z-m2y2+m12(2nϵf-y)2-h2+m1P02πlogz-m2y+h2-m1y2z-m2y-h2+m1y2              -n=1logz-m2y+h2+m1(2nϵf+y)2z-m2y-h2+m1(2nϵf+y)2+logz-m2y+h2+m1(2nϵf-y)2z-m2y-h2+m1(2nϵf-y)2

For med. II (i.e. y>Y )

uII=2P0πT1-sin(sh)s|s|e-m1|s|y+n=1e-m1|s|y-(α|s|+ιβs)ϵf-n=1e-m1|s|y-(α|s|+ιβs)ϵfe-iz-m2ysds τxyII=0τxzII=0
Rough rigid-interface

When the interfacing between the plate and half-space at y=Y is rough-rigid, without loss of generality, on taking T0 i.e, V=-1 in equations (31)-(33) for med. I and in equations (34)-(36) for med. II. we will obtain the following results for displacements and stresses.

For med. I (i.e., for y Y )

uI=Po2πT1-sin(sh)s|s|e-m1|s|y+n=1(-1)ne-m1|s|(2nϵf-y)-       n=1(-1)ne-m1|s|(2nϵf+y)e-iz-m2ysds τxyI=-P0πtan-12hm1yz-m2y2+m12y2-h2-n=1(-1)ntan-12hm1(2nϵf+y)z-m2y2+m12(2nϵf+y)2-h2         +tan-12hm1(2nϵf-y)z-m2y2+m12(2nϵf-y)2-h2 τxzI=-P0m2πtan-12hm1yz-m2y2+m12y2-h2-n=1(-1)ntan-12hm1(2nϵf+y)z-m2y2+m12(2nϵf+y)2-h2         +tan-12hm1(2nϵf-y)z-m2y2+m12(2nϵf-y)2-h2+m1P02πlogz-m2y+h2+m1y2z-m2y-h2+m1y2-n=1(-1)nlogz-m2y+h2+m1(2nϵf+y)2z-m2y-h2+m1(2nϵf+y)2+logz-m2y+h2+m1(2nϵf-y)2z-m2y-h2+m1(2nϵf-y)2

For med. II (i.e. y>Y)

uII=0 τxyII=-2P0πtan-12hm1'y+αϵfz-m2'y+ιβϵf2+m1'y+αϵf2-h2         +n=1(-1)ntan-12h2nm1+αϵf+m1'yz-m2'y+ιβϵf2+2nm1+αϵf+m1'y2-h2 τxzII=-2P0m2'πtan-12hm1'y+αϵfz-m2'y+ιβϵf2+m1'y+αϵf2-h2         +n=1(-1)ntan-12h2nm1+αϵf+m1'yz-m2'y+iβϵf2+2nm1+αϵf+m1'y2-h2            +P0m1'πlogz-m2'y+iβϵf+h2+m1'y+αϵf2z-m2'y+iβϵf-h2+m1'y+αϵf2++n=1(-1)nlogz-m2'y+iβϵf+h2+m1'y+2nm1+αϵf2z-m2'y+iβϵf-h2+m1'y+2nm1+αϵf2 6.1 Particular cases

Orthotropic elastic layered half-space:

By putting c56=0  i.e m1=m3=c55c66 and m2=0 in the equations (31)-(36), we will get the deformation of an orthotropic elastic layered half space due to strip loading.

Isotropic elastic layered half space:

By putting c56=0 and c66=c55=μ i.e.  m2=0 and m1=m3=1 in the equations (31)-(36), we will get the deformation of an isotropic elastic layered half space due to strip loading.

6.2 Special cases

(i) By putting c56=0  i.e m2=0,  m1=m3=c55c66=m,  and c56'=0, c66'=c55'=μ, i.e m2'=0,  m1'=m3'=1  in equations (31)-(36) we will obtain the system of equations

uI=P0πT1-sin(sh)s|s|e-m|s|y+n=1Vne-m|s|(2nϵf-y)-n=1Vne-m|s|(2nf+y)e-izsdsτxyI=-P0πtan-12hmyz2+m2y2-h2-n=1Vntan-12hm(2nϵf+y)z2+m2(2nϵf+y)2-h2+tan-12hm(2nϵf-y)z2+m2(2nϵf-y)2-h2 τxzI=m1P02πlog[z+h]2+(my)2[z-h]2+(my)2-n=1Vnlog[z+h]2+[m(2nϵf+y)]2[z-h]2+[m(2nϵf+y)]2+log[z+h]2+[m(2nϵf-y)]2[z-h]2+[m(2nϵf-y)]2 uII=2P0πT1TT+1-sin(sh)s|s|e-m|s|y-((m-1)|s|)ϵf+n=1Vne-m|s|(2nϵf+y)-((m-1)|s|)ϵf)e-i(z)sds τxyII=-2P0π(1+T)tan-12h(y+(m-1)ϵf)(z)2+(y+(m-1)ϵf)2-h2+n=1Vntan-12h((2nm+(m-1))ϵf+y)(z)2+((2nm+(m-1))ϵf+y)2-h2τxzII=P0π(1+T)log(z+h)2+(y+(m-1)ϵf)2(z-h)2+(y+(m-1)ϵf)2+n=1Vnlog(z+h)2+(y+(2nm+(m-1))ϵf)2(z-h)2+(y+(2nm+(m-1))ϵf)2

which is similar to Madan 10.

(ii) By taking the lower monoclinic elastic half space as a simple base the obtained results (31)-(36), (37)-(41) and (42)-(47) can similar to Madan7 for all three different boundary conditions “perfectly welded interfacing, smooth-rigid interfacing, and rough-rigid interfacing’’ respectively.

Numerical results

Here, we want to describe the effect of rectangular irregularity on the stresses due to strip load P0=107 dynes/cm-2 acting on zh the upper layer of monoclinic elastic plate connected with monoclinic elastic half space. For numerical calculation, we take the following data for elastic constants from Tiersten 13.

(a) For monoclinic elastic layer,

c55'=94×109N/m2,c66'=93×109N/m2,c56'=-11×109N/m2,

(b) For monoclinic elastic half-space,

c55''=57.94×109N/m2,c66''=39.88×109N/m2,c56''=-17.91×109N/m2

Figure 2, Figure 3, Figure 4, Figure 5 represent the variation of shearing stresses τxy and τxz for Med. I and Figure 6, Figure 7, Figure 8 represent the variation of shearing stresses τxy and τxz for Med. II in perfectly welded contact, with horizontal distance `z’ and for strip size 'h' at depth level y. All figures (2)-(8) in perfectly welded contact are calculated for T1=T1'. Figure 9, Figure 10, Figure 11 represent the variation of shearing stresses τxy and τxz in smooth-rigid contact for Med. I, for different values of h=1, 1.5, 2 at y=1 respectively. Figure 12, Figure 13 represent the variation of shearing stress τxy and τxz in rough-rigid contact for Med. I and Figure 14, Figure 15, Figure 16, Figure 17 in case of Med. II. In figures (9)-(17) both shearing stresses τxy and τxz have been plotted for all series terms by taking the sum up to first ten terms of the series. As per our consideration, it is mentioned that the irregularity is present in the lower half space. It has been observed graphically that the stresses for Med. I (i.e monoclinic elastic layer) are affected by the change of loaded strip size but not by irregularity and this observation prevails in all the three interfacing boundary conditions. So, it has been concluded that the change in the size of irregularity has affected the stresses in lower half space. Also It has been found that the stresses τxy and τxz for Med.I at (h = -0.25, -0.5, 0.25, 0.5) have no discontinuity and the discontinuity comes in graphs for h>0.5, shown in figures (2)-(5) and the same has been observed for stresses in smooth or rough-rigid contacts. Moreover, it has been observed that the stresses for Med. II (i.e lower half space) have discontinuities due to irregularity as well as strip size (  zh) for h ≥1. Same pattern has been observed in lower half space, if a=0 in absence of rectangular irregularity (i.e in case of regular half space) with h0.5 the stresses are continuous, clearly in figures (8), (16). Graphically, it has been concluded that in both mediums the variation in shearing stresses is affected by anisotropy of material, size of the strip and size of an irregularity. Also, the difference between the shearing stresses in magnitude decreases as horizontal distance increases .

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Variation of the stress component <inline-formula id="inline-formula-48a358c4ef65477bb185fa1fed48a93b"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> in perfectly welded contact with horizontal distance <inline-formula id="inline-formula-926d036dc6074cfd9aa1ceaf8a15fd4c"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi><mml:mo>,</mml:mo></mml:math></inline-formula> for Med. II

Variation of the stress component <inline-formula id="inline-formula-f46cb2ab076d405da1c97225e43be6a5"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> in perfectly welded contact with horizontal distance <inline-formula id="inline-formula-8ec039ee9894409c9f4cf613515e8aee"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi><mml:mo>,</mml:mo></mml:math></inline-formula> for Med. II

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Variation of the stress component <inline-formula id="inline-formula-35f1564d726143d2b14f73cf799507b2"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> in smooth-rigid contact with horizontal distance <inline-formula id="inline-formula-879f4b8586f34518ba10372965f8c114"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi><mml:mo>,</mml:mo></mml:math></inline-formula> for Med. I

Variation of the stress component <inline-formula id="inline-formula-84704493e73545118dc0855328103ba5"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> in smooth-rigid contact with horizontal distance <inline-formula id="inline-formula-a112c9e459a041d9ad17eb57de1ed2aa"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi><mml:mo>,</mml:mo></mml:math></inline-formula> for Med. I

Variation of the stress component <inline-formula id="inline-formula-5aa15306e75b42b1af3fed5dc9769812"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> in rough-rigid contact with horizontal distance <inline-formula id="inline-formula-325debbc61da4cafbbaf554af08627de"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi><mml:mo>,</mml:mo></mml:math></inline-formula> for Med. I

Variation of the stress component <inline-formula id="inline-formula-d3cbc84599004496bc30e0f2474223a4"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> in rough-rigid contact with horizontal distance <inline-formula id="inline-formula-528d8ce9b25d4f128ff44119655503c8"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi><mml:mo>,</mml:mo></mml:math></inline-formula> for Med.I

Variation of the stress component <inline-formula id="inline-formula-1850f803d25a4efb8a3bd2f73685cb8d"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> in rough-rigid contact with horizontal distance <inline-formula id="inline-formula-8ec3856abe304bc286b0ff3b85d9594e"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi><mml:mo>,</mml:mo></mml:math></inline-formula> for Med. II

Variation of the stress component <inline-formula id="inline-formula-7af00c84d9184b298d74cc0016949df2"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> in rough-rigid contact with horizontal distance <inline-formula id="inline-formula-22fccaecdd374805aa50193e654d8884"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi><mml:mo>,</mml:mo></mml:math></inline-formula> for Med. II

Variation of the stress component <inline-formula id="inline-formula-db01988ea8b94ba3bb489bbb5ae56f34"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> in rough-rigid contact with horizontal distance <inline-formula id="inline-formula-c510cf730c204eac878be39c4bd88be7"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi><mml:mo>,</mml:mo></mml:math></inline-formula> for Med. II

Variation of the stress component <inline-formula id="inline-formula-ee80843a2c774d7e83cb5060815b3f6e"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi><mml:mrow><mml:mi>x</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula> in rough-rigid contact with horizontal distance <inline-formula id="inline-formula-c2df0f0a0f4a4945ad34b81ab85ebc38"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi><mml:mo>,</mml:mo></mml:math></inline-formula> for Med. II Conclusions

The closed form expressions for the stresses in an elastic model consisting of monoclinic elastic layer lying over an irregular monoclinic elastic half space due to shear strip load has been obtained. Graphically, it has been concluded that the stresses in an infinite monoclinic layer interface with an irregular monoclinic half space are significantly affected by the presence of an irregularity and also by anisotropy of the elastic medium as a result of shear load. The irregularity may be rectangular, triangular, parabolic etc. Also the difference between the shearing stresses in magnitude decreases as horizontal distance increases. The results obtained are useful to study the static deformation near the surface of the earth where lithostatic pressure have disjoint cracks perpendicular to the maximum compressional stress. The corresponding results obtained by Madan 7 and by Madan10 can be obtained from our results as particular cases.

Acknowledgement

One of the authors (Savita) is thankful to HRDG Council of Scientific & Industrial Research for sanctioning SRF scholarship with File No. (09/1063/0012/2017). The authors are also great thankful to the unknown reviewers.