Original Research Article
IJCRR Section: General Science Sci. Journal Impact Factor 4.016 ICV: 71.54
Stresses in an Orthotropic Elastic Layer Lying Over an Irregular Isotropic Elastic Half-Space Dinesh Kumar Madan1, Poonam Arya2, N.R. Garg2, Kuldip Singh3 Department of Mathematics, Chaudhary Bansi Lal University, Bhiwani-127021, India; 2Department of Mathematics, Maharshi Dayanand University, Rohtak-124001, India; 3Department of Mathematics, Guru Jambeshwar University of Science and Technology, Hisar 125001, India. 1
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. Key Words: Orthotropic, Shear load, Anti-plane strain, Rectangular irregularity
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.
Corresponding Author: Dinesh Kumar Madan, Department of Mathematics, Chaudhary Bansi Lal University, Bhiwani-127021, India; E-mail:
[email protected] Received: 27.01.2017
Revised: 03.02.2017
Int J Cur Res Rev | Vol 9 • Issue 4 • February 2017
Accepted: 10.02.2017
15
Madan et.al.: Stresses in an orthotropic elastic layer lying over an irregular isotropic elastic half-space
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
with co-ordinate planes as planes of elastic symmetry are
σ 1 = C11e1 + C12 e2 + C13e3 σ 2 = C21e1 + C22 e2 + C23e3 σ 3 = C13e1 + C23e2 + C33e3 τ 23 = 2C44 e23 τ13 = 2C55e13 (6) τ12 = 2C66 e12
where e1 , e2 , e3 are normal strain components and e12 , e23 , e13
are normal strain components. The suffices Cij (i, j = 1, 2,3, 4,5, 6) 5 Consider a horizontal orthotropic elastic plate of thickness H are stiffnesses of an orthotropic elastic material. lying over an infinite isotropic elastic medium with x1–axis The strain - displacement relations are given as 5 vertically downwards. The origin of the Cartesian The strain coordinate - displacement relations are given as system (x1, x2, x3) is taken at the upper boundary of the plate. 1 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 + 2as � and 𝑒𝑒𝑒𝑒1 = 1 , etc. (7) 𝑒𝑒𝑒𝑒12 = �are 1given strain - displacement relations 2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 The orthotropic elastic plate occupies theThe region 5 0 ≤ x1 ≤ H 1 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 1
𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 2
𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 1
In terms of displacement components, can be written from equations = � of the + equilibrium � and equations 𝑒𝑒𝑒𝑒1 = , etc. (7) 𝑒𝑒𝑒𝑒12terms and is described as Medium I whereas the region x1 > H is In components, the equilibrium equa2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 displacement 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 (3) – (7) as: The strain displacement relations are given as the isotropic elastic half space over which plate is lying tions can bethe written from equations equationscan(3)be–written (7) as from : equations5 In the terms of displacement components, equilibrium 2 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 1 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 1 1 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 1 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 3 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 12 21 and is described as Medium II. (Fig. 1) (3) – (7) as: 𝐶𝐶𝐶𝐶11 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 + 𝐶𝐶𝐶𝐶66𝑒𝑒𝑒𝑒12𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥=2 +�𝐶𝐶𝐶𝐶55 +𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 +� (𝐶𝐶𝐶𝐶12 + (𝐶𝐶𝐶𝐶13 + 𝐶𝐶𝐶𝐶55 ) = 0 (7)5 and+ 𝐶𝐶𝐶𝐶66 𝑒𝑒𝑒𝑒1)= , etc. 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥
𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 2 1 1 3 1 2 2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 3 1 The strain - displacement relations are given as 2 𝑢𝑢𝑢𝑢 2 𝑢𝑢𝑢𝑢 2 𝑢𝑢𝑢𝑢 2 𝑢𝑢𝑢𝑢 (8) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 3 1 1 1 2 can be written from Suppose a shear load P per unit area is acting over the strip In terms of displacement components, the equilibrium equations (𝐶𝐶𝐶𝐶 (𝐶𝐶𝐶𝐶 ) ) + 𝐶𝐶𝐶𝐶 + 𝐶𝐶𝐶𝐶 + + 𝐶𝐶𝐶𝐶 + + 𝐶𝐶𝐶𝐶 =0 𝐶𝐶𝐶𝐶 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 1given 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 The strain - displacement relations 11 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 66 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 1 are 12 66 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 13 55 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 equations 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 32 2as 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 1 32 𝑒𝑒𝑒𝑒12strain =2 𝜕𝜕𝜕𝜕�2-𝑢𝑢𝑢𝑢55 + � and 𝑒𝑒𝑒𝑒 = , etc. (7) 1 2 2 2 1 The displacement relations are given as 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 3 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥12 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 2 (3) – (7) as: +𝐶𝐶𝐶𝐶12 ) 21 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 + 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 𝐶𝐶𝐶𝐶66 + 𝐶𝐶𝐶𝐶22 22 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 +1𝐶𝐶𝐶𝐶44 22 (𝐶𝐶𝐶𝐶23 + 𝐶𝐶𝐶𝐶44 ) =0 | x2 |≤ h of the surface x1 = 0 in the positive x1–direction. The (𝐶𝐶𝐶𝐶66The 1 2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥�1-𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥=2are given 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 (8) 2 1 3 𝑒𝑒𝑒𝑒 = + � and 𝑒𝑒𝑒𝑒 , etc. (7) strain displacement relations as 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 2 12𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕 2𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 1 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 2 2 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 1 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 be written 1 2 2 the 𝑢𝑢𝑢𝑢equilibrium 3 equations 𝑒𝑒𝑒𝑒12𝐶𝐶𝐶𝐶equations =𝜕𝜕𝜕𝜕 )2 𝑢𝑢𝑢𝑢�𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝑢𝑢𝑢𝑢112+can 𝑒𝑒𝑒𝑒𝜕𝜕𝜕𝜕1 = , etc. boundary condition at the surface x1 = 0 isIn terms of displacement 2� 2= 𝐶𝐶𝐶𝐶11 21 + 𝐶𝐶𝐶𝐶components, + 𝑢𝑢𝑢𝑢𝐶𝐶𝐶𝐶155 211 +𝜕𝜕𝜕𝜕 2(𝐶𝐶𝐶𝐶 +𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝜕𝜕𝜕𝜕(𝐶𝐶𝐶𝐶 +and 𝐶𝐶𝐶𝐶55 ) from 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 212 + 𝑢𝑢𝑢𝑢13 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝑢𝑢𝑢𝑢13 0 (9) 2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 66 66 2 2 2 2 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1(𝐶𝐶𝐶𝐶 +𝐶𝐶𝐶𝐶 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥)2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1)𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 1 = 0 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 2 +𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝐶𝐶𝐶𝐶3equilibrium (3) – (7) as: 66components, 12 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 the 66 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 +𝑒𝑒𝑒𝑒 𝐶𝐶𝐶𝐶22 44 23 + 𝐶𝐶𝐶𝐶from =𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥�22 21+1 𝐶𝐶𝐶𝐶+ �32 2(𝐶𝐶𝐶𝐶 and 𝑒𝑒𝑒𝑒44 In terms of displacement can𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥be written 12 equations 1 =𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥equations 1 2 3 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥, 2etc. 2 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕 2terms 𝑢𝑢𝑢𝑢 1 𝜕𝜕𝜕𝜕 12 𝑢𝑢𝑢𝑢 2components, 𝑢𝑢𝑢𝑢 32 the𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥equilibrium 1 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 3 1 (8)be written from e In of displacement equations 2 2 2 2 (𝐶𝐶𝐶𝐶 ) ) (𝐶𝐶𝐶𝐶 +1 44 +𝜕𝜕𝜕𝜕 𝐶𝐶𝐶𝐶𝑢𝑢𝑢𝑢23 + 𝐶𝐶𝐶𝐶55 𝜕𝜕𝜕𝜕 2𝑢𝑢𝑢𝑢 2+ 𝐶𝐶𝐶𝐶44 2 + 𝐶𝐶𝐶𝐶33 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢233 = 0 can(9) 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢13 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 55 +𝐶𝐶𝐶𝐶 (3) – (7) as: − P | x2 |≤ h 1 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2+ 𝐶𝐶𝐶𝐶 ) 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2+ 𝐶𝐶𝐶𝐶 ) 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3 = 0 1 1 (𝐶𝐶𝐶𝐶 (𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶 + 𝐶𝐶𝐶𝐶 + 𝐶𝐶𝐶𝐶 + + 2 2 2 2 2 (3)66terms –𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 (7)2 as: 11 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 55 12 66𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 13 55 equations 2 In of displacement components, the equilibrium can be written from e τ 31 = 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 2 2 2 3 (1) 1 2 1 3 2 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢2) 21) 3 (𝐶𝐶𝐶𝐶 +2 𝑢𝑢𝑢𝑢𝐶𝐶𝐶𝐶 =0 (10) 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 1 (𝐶𝐶𝐶𝐶66𝜕𝜕𝜕𝜕+𝐶𝐶𝐶𝐶 𝑢𝑢𝑢𝑢 𝐶𝐶𝐶𝐶 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕 22𝑢𝑢𝑢𝑢 2+ 𝐶𝐶𝐶𝐶22 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝜕𝜕𝜕𝜕2𝜕𝜕𝜕𝜕22𝑢𝑢𝑢𝑢+ 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕 166 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3 23 + 𝐶𝐶𝐶𝐶44 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 32 + 2𝐶𝐶𝐶𝐶 23= 0 | x2 |> h (3) (7) as: ) 2 232+𝜕𝜕𝜕𝜕44 ) 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢23𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 (𝐶𝐶𝐶𝐶 1𝐶𝐶𝐶𝐶55 2+ 𝐶𝐶𝐶𝐶2𝜕𝜕𝜕𝜕 2+ 𝐶𝐶𝐶𝐶11 + 𝐶𝐶𝐶𝐶66–112 + + 𝐶𝐶𝐶𝐶55 02 (8) (𝐶𝐶𝐶𝐶 ) ) (𝐶𝐶𝐶𝐶 + + 𝐶𝐶𝐶𝐶 0 𝑢𝑢𝑢𝑢 1(𝐶𝐶𝐶𝐶112 ++𝜕𝜕𝜕𝜕𝐶𝐶𝐶𝐶2𝐶𝐶𝐶𝐶 𝑢𝑢𝑢𝑢55 𝑢𝑢𝑢𝑢44 𝜕𝜕𝜕𝜕= 𝑢𝑢𝑢𝑢𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 3 66 13 2 2 55 +𝐶𝐶𝐶𝐶 13 44 23 33 1 1 2 2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 3 2 𝐶𝐶𝐶𝐶𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝐶𝐶𝐶𝐶266 + (𝐶𝐶𝐶𝐶13 + 𝐶𝐶𝐶𝐶55 ) 113 material 12 + 𝐶𝐶𝐶𝐶661) 3𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 + 2 + 𝐶𝐶𝐶𝐶155 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥strain 2 + (𝐶𝐶𝐶𝐶 Consider the field equation of an orthotropic in𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 anti equilibrium state 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 21 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3 2 𝑢𝑢𝑢𝑢1 2 2 𝑢𝑢𝑢𝑢2 2 – plane 2 𝑢𝑢𝑢𝑢3 2 2 12𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 as:(9) 2 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 1 1 1 2 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 1 𝑢𝑢𝑢𝑢 2 𝑢𝑢𝑢𝑢 2 𝑢𝑢𝑢𝑢 2 (𝐶𝐶𝐶𝐶 (8) + 𝐶𝐶𝐶𝐶 ) 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 3 (10) The irregularity is assumed to be rectangular with length 2a (𝐶𝐶𝐶𝐶66 +𝐶𝐶𝐶𝐶 )) 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 3 + 𝐶𝐶𝐶𝐶+11𝐶𝐶𝐶𝐶66 2𝜕𝜕𝜕𝜕 + 𝐶𝐶𝐶𝐶66𝐶𝐶𝐶𝐶22 2𝜕𝜕𝜕𝜕 + 𝐶𝐶𝐶𝐶55𝐶𝐶𝐶𝐶44 2𝜕𝜕𝜕𝜕 + +𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶66 =(𝐶𝐶𝐶𝐶 0 13 12 + 55 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 𝑢𝑢𝑢𝑢 12 ) 44 2 𝑢𝑢𝑢𝑢+ 2 2(𝐶𝐶𝐶𝐶 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕2 2 + 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕= 𝜕𝜕𝜕𝜕(𝑥𝑥𝑥𝑥 𝑢𝑢𝑢𝑢 3 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 3 23 13𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 22 (11) 1 3 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 120, 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 𝑢𝑢𝑢𝑢 = 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1𝑢𝑢𝑢𝑢 = 2 3 ); 1 2 3 𝑢𝑢𝑢𝑢 , 𝑥𝑥𝑥𝑥 𝑢𝑢𝑢𝑢 2 2 𝑢𝑢𝑢𝑢0 3 (𝐶𝐶𝐶𝐶may )𝜕𝜕𝜕𝜕 23𝑢𝑢𝑢𝑢 2 1 in+2anti +𝐶𝐶𝐶𝐶13 ) 1of an 2orthotropic + 𝜕𝜕𝜕𝜕(𝐶𝐶𝐶𝐶𝑢𝑢𝑢𝑢44 𝐶𝐶𝐶𝐶𝜕𝜕𝜕𝜕552–𝑢𝑢𝑢𝑢 2plane 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕= and depth d. The equation of the rectangular irregularity 55 44𝑢𝑢𝑢𝑢𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 + 𝐶𝐶𝐶𝐶 2 + 𝐶𝐶𝐶𝐶33 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 state 1 + 𝐶𝐶𝐶𝐶23 2 equilibrium 3 as: Consider the field equation material strain 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 2 2 2 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3 2 (𝐶𝐶𝐶𝐶66𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥+𝐶𝐶𝐶𝐶 ) (𝐶𝐶𝐶𝐶 ) + 𝐶𝐶𝐶𝐶 + 𝐶𝐶𝐶𝐶 + 𝐶𝐶𝐶𝐶 + 𝐶𝐶𝐶𝐶 = 0 1 2 3 (9) 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 2 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 2 𝜕𝜕𝜕𝜕2 𝑢𝑢𝑢𝑢 1 44 𝜕𝜕𝜕𝜕 12 66 22 44 2 2𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 223 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 Consider fieldequilibrium equation anti 1the 2strain 3 in (𝐶𝐶𝐶𝐶166 +𝐶𝐶𝐶𝐶12 )of (𝐶𝐶𝐶𝐶23 + 𝐶𝐶𝐶𝐶44 ) 2an orthotropic + 𝐶𝐶𝐶𝐶663 2 +material 𝐶𝐶𝐶𝐶22 2 + 𝐶𝐶𝐶𝐶244 The non-zero stresses for an anti – plane state be represented as (10) 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 212𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥are 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 (𝑥𝑥𝑥𝑥 2 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢2 = 0, 2 𝑢𝑢𝑢𝑢1 , 𝑥𝑥𝑥𝑥2 ); = 𝑢𝑢𝑢𝑢 = 𝑢𝑢𝑢𝑢 (11) 𝑢𝑢𝑢𝑢–1𝜕𝜕𝜕𝜕plane 𝜕𝜕𝜕𝜕22𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢312 𝜕𝜕𝜕𝜕22𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢322 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢32 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢132 𝜕𝜕𝜕𝜕 3 3 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 (9) 1 2 as: + 𝐶𝐶𝐶𝐶66 22 ++𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶3322 22 = + 0𝐶𝐶𝐶𝐶44 2 (𝐶𝐶𝐶𝐶23 + 𝐶𝐶𝐶𝐶44 ) (𝐶𝐶𝐶𝐶55 +𝐶𝐶𝐶𝐶13 ) (𝐶𝐶𝐶𝐶44 +equilibrium +strain 𝐶𝐶𝐶𝐶23 ) (𝐶𝐶𝐶𝐶𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 +12state 𝐶𝐶𝐶𝐶)55𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 66 +𝐶𝐶𝐶𝐶 2 + 𝐶𝐶𝐶𝐶44 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥equilibrium 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2state 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥of 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 21 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥32in anti 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 Consider the field equation an material – plane strain as: 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3 orthotropic = 𝐶𝐶𝐶𝐶 (12) 𝜏𝜏𝜏𝜏 12 21 3 2 2 2 2 31 55 for an𝜕𝜕𝜕𝜕anti state 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 3are 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 3 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 3 𝑢𝑢𝑢𝑢 1 – plane strain equilibrium 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 d | x2 |≤ a The non-zero stresses (𝐶𝐶𝐶𝐶55 +𝐶𝐶𝐶𝐶13 ) + (𝐶𝐶𝐶𝐶44 + 𝐶𝐶𝐶𝐶23 ) 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥2𝜕𝜕𝜕𝜕1 2+ + 𝐶𝐶𝐶𝐶 𝜕𝜕𝜕𝜕 2= 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢 𝐶𝐶𝐶𝐶 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢+ 𝐶𝐶𝐶𝐶33 𝑢𝑢𝑢𝑢 30 (10) 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢 3 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1= 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1) 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2, 𝑥𝑥𝑥𝑥1 );55 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 12 + 𝐶𝐶𝐶𝐶44 )𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 22 2 + 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 32 3 2 = 0, (𝐶𝐶𝐶𝐶 (𝐶𝐶𝐶𝐶 (𝑥𝑥𝑥𝑥 +𝐶𝐶𝐶𝐶 + 𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶44 2 + 𝐶𝐶𝐶𝐶33 23 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢 = 𝑢𝑢𝑢𝑢 x f x ε = = 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢 55 13 44 23 55 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 + (11) 1 3 3 1 2 3 1 2 (2) 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 21 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 21 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 𝑢𝑢𝑢𝑢1 2 𝑢𝑢𝑢𝑢2 = 𝐶𝐶𝐶𝐶 (12) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 2 𝑢𝑢𝑢𝑢33 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 𝜕𝜕𝜕𝜕 𝑢𝑢𝑢𝑢 (13) 𝜏𝜏𝜏𝜏 31 55 2 3 as: 32 material 44 )𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 1 in 1anti – (10) 3 Consider the field equation of an orthotropic strain equilibrium state 0 | x2 |> a The + (𝐶𝐶𝐶𝐶plane 2 55 +𝐶𝐶𝐶𝐶 13 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 44 + 𝐶𝐶𝐶𝐶23 ) 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 + 𝐶𝐶𝐶𝐶55 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 + 𝐶𝐶𝐶𝐶44 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 + 𝐶𝐶𝐶𝐶33 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 2 non-zero stresses for an anti – plane(𝐶𝐶𝐶𝐶 strain equilibrium 1 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥 3 state are 1 2 1 2 3 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢for d non-zero an – plane strain equilibrium Consider theEquations field equation oforthotropic an – plane strain equilibrium state as: 3,in (𝑥𝑥𝑥𝑥 );dueanti 𝑢𝑢𝑢𝑢an (11) 𝑢𝑢𝑢𝑢2orthotropic = 0, 𝜏𝜏𝜏𝜏 elastic 𝑢𝑢𝑢𝑢stresses =𝐶𝐶𝐶𝐶𝑢𝑢𝑢𝑢medium 𝑥𝑥𝑥𝑥2anti Equilibrium forThe to anti –material plane strain deformation 1 = =material (13) 𝜕𝜕𝜕𝜕𝑢𝑢𝑢𝑢1of 3 an 32 3equation 443 𝜕𝜕𝜕𝜕𝑥𝑥𝑥𝑥
