First Axisymmetric Problem of Micropolar Elasticity with Voids

Size: px

Start display at page:

Download "First Axisymmetric Problem of Micropolar Elasticity with Voids"

Johnathan Terence Gray
5 years ago
Views:

1 International Journal of Applied Science-Research and Review (IJAS) Original Article First Axisymmetric Problem of Micropolar Elasticity with Voids Navneet Rana* Dept. of Mathematics Guru Nanak College for Girls Sri Muktsar Sahib (Punjab) India A R T I C L E I N F O A B S T R A C T Received 16 Jan Received in revised form 04 Feb Accepted 10 Feb Keywords: Laplace Transform Hankel Transform Eigen values and vectors. The theory of elasticity in its broad aspects deals with a study of the behavior of these substances which possess the property of recovering their size and shape when forces producing deformation are removed. In theory of micropolar elasticity a body was assumed consisting of interconnected particles in the form of small rigid bodies undergoing translational motion as well as rotational motion. The presence of small pores or voids in the constituent materials can also be formally introduced in a continuum model. This paper is an attempt to explain an axisymmetric problem of a homogeneous isotropic micropolar elastic medium with voids subject to a set of normal point sources by employing the eigen value approach. The analytical expression of displacement components microrotation force stresses couple stresses and volume fraction field have been derived for micropolar elastic solid with voids. Corresponding author: Dept. Of Mathematics Guru Nanak College for Girls Sri Muktsar Sahib (Punjab) India address: rana.navneet2@gmail.com 2016 International Journal of Applied Science-Research and Review. All rights reserved

2 INTRODUCTION The classical theory of elasticity successfully explains the behavior of construction materials (various sorts of steel aluminum concrete) provided the stresses do not exceed the elastic limit and no stress concentration occurs. The linear theory of elastic materials with voids is one of the generalizations of the classical theory of elasticity. This theory has practical utility to investigate various type of geological biological and synthetic porous materials for which the elastic theory is inadequate. It is concerned with elastic materials consisting of a distribution of small pores (voids) in which the void s volume is included among the kinematic variables and in the Laplace transform in Cartesian co-ordinates Laplace transform [Debnath (1995)] of a function f((x z t) with respect to time variable t ( ) = { ( ))} = ( ) limiting case when the volume tends to zero this theory reduces to the classical theory of elasticity. The process of voids is known to affect the estimations of the physicalmechanical properties of the composite and also to weaken the bond as these pores (voids) get spread over a wide area. In this paper the axisymmetric deformation problem in micropolar elastic solid with voids is considered. A general solution of the equation of motion of a micropolar elastic medium with voids in cylindrical polar coordinate system is obtained to the axisymmetric deformation. The Laplace and Hankel transforms are used to solve the equations through eigen value approach. with p as the Laplace transform variable is defined as (1.1) Along with the following basic properties [Debnath (1995)] = ( ) ( 0) (1.2) = ( ) ( 0) (1.3) Also the Laplace transform of the Dirac delta function δ(t) is given as L{δ(t)}=1 (1.4) Laplace transform in polar co-ordinates The Laplace transform with respect to time variable t with p as the Laplace transform variable is defined as ( ) = { ( ))} = ( ) (1.5) Along with the following basic properties = ( ) ( 0) (1.6) = ( ) ( 0) (1.7)

3 Rana ISSN The Hankel transform The Hankel transform [Sneddon (1979)] of order n of ( ) with respect to variable r is defined as ( ) = ( ) = ( ) (1.8) Where is the Hankel transform variable J n (---) is the Bessel function of first kind of order n alongwith the following basic operational properties [Sneddon (1979)] = = (1.9) = = (1.10) We know that [Debnath (1995)] the Dirac delta function δ(- -) having the property H δ = 1 (1.11) BASIC EQUATIONS The field equations and the constitutive relations in micropolar elastic solid with voids without body force and body couples are given by Eringen (1968) and Iesan (1985) as ( 2 ) (. ) ( ) = (2.1) ( ) (. )) 2 = (2.2). = (2.3) = (2.4) = (2.5) Where λ μ K α β γ are the material constants ρ is the density j is the micro-intertia is the displacement vector is the microrotation vector φ * is the scalar microstretch t ij is the force stress tensor m ij is the couple stress tensor q * is the volume fraction field δ ij is the Kronecker delta ε ijr is the unit anti-symmetric tensor and α * β * ζ * ω * K * are the material constants due to presence of voids. The dynamic equations (2.1)-(2.3) in cylindrical polar co-ordinate system (r θ z) in component form become ( ) ( ) = (2.6)

4 ( ) ( ) ( ) = (2.7) ( ) ( ) = (2.8) ( ) 2 = (2.9) ( ) 2 = (2.10) ( ) 2 = (2.11) = (2.12) Where (u r u θ u z ) and (φ r φ θ φ z ) represent the cylindrical polar components of the Formulation and Solution of the Problem Here we consider the axisymmetric two dimensional problem in micropolar elastic solid with voids. Since we are considering twodimensional axisymmetric problem so we displacement vector and microrotation vector respectively. assume the components of displacement vector and microrotation vector of the form = (u r 0 u z ) = (0 φ θ 0 ) (2.13) And the quantities remain independent of θ so that 0. With these considerations and using (2.13) the equations (2.7) (2.9) and (2.11) become identically zero and equations (2.6) (2.8) (2.10) and (2.12) and constitutive relations (2.4)-(2.5) can be written as ( ) ( ) = (2.14) ( ) ( ) = (2.15) 2 = (2.16) = (2.17)

5 = ( ) (2.18) = ( )( 2 ) (2.19) = (2.20) Where = (2.21) We define the non-dimensional quantities as = = = = = = = = = = (2.22) Where = =. And h is the constant having dimension of length. Using the quantities given by (2.22) the field equations (2.14) (2.17) and stressdisplacement relations (2.18) (2.20) may be recast into two dimensionless form (after suppressing the primes for convenience) as = (2.23) = (2.24) = (2.25) = (2.26) = (2.27) = (2.28) = (2.29) Where

6 = = (μ ) (λ μ) = = = = = = = = = = = = = = (2.30) Applying Laplace transform defined by expression (1.5) with help of results (1.6) and (1.7) on equations (2.23)-(2.29) we get = (2.31) = (2.32) = (2.33) = (2.34) = (2.35) = (2.36) = Where the initial displacements microrotation and volume fraction and their (2.37) corresponding velocities are assumed as zero throughout the medium that is ( 0) = = 0 ( 0) = = 0 ( 0) = = 0 ( 0) = = 0 (2.38) Applying the Hankel transform defined by (1.8) with help of results (1.9) and (1.10) on equations (2.31) (2.37) we obtain

7 = (2.39) = (2.40) = ( ) (2.41) = ( ) (2.42) = (2.43) = (2.44) = (2.45) Where ( ) = { ( )} ( ) = { ( )} (2.46) ( ) = { ( )} ( ) = { ( )} (2.47) ( ) = { ( )} ( ) = { ( )} (2.48) ( ) = { ( )} (2.49) The system of equations (2.39)-(2.42) can be written in the vector matrix differential equation form as Where ( ) = ( ) ( ) (2.50) = = = = = =

8 = With = = = = = = = = = = = = = = =. (2.51) To solve the equation (2.50) we take ( ) = ( ) (2.52) For some parameter q so that ( ) ( ) = ( ) (2.53) Which leads to eigen value problem. The characteristic equation corresponding to the matrix A is given by det( ) = 0 (2.54) Which on expansion provides us = 0 (2.55) Where = (2.56) = (2.57)

9 = (2.58) = (2.59) The eigen values of the matrix A are characteristic roots of the equation (2.55). The vectors X(ξ p) corresponding to the eigen [ ] ( ) = 0 values q s can be determined by solving the homogeneous equations (2.60) The set of eigen vectors ( ); = may be defined as ( ) ( ) = ( ) (2.61) Where ( ) = ( ) = (2.62) q=q s s=1234 l=s4 q=-q s ;s=1234 (2.63) = {( )( ) ( )( )} (2.64) = [( ){( )( ) = } ( )] (2.65) { } (2.66) = [( ) ( ){ ( )}]; = 1234 (2.67) Thus a solution of equation (2.50) as given by (2.52) become ( ) = { ( ) ( ) } (2.68) Where E 1 E 2 E 3 E 4 E 5 E 6 E 7 and E 8 eight arbitrary constants. are The equation (2.68) represents the solution of the general problem in the axisymmetric case of micropolar elastic medium

10 with voids and gives displacement microrotation and volume fraction components in the transformed domains which are obtained as ( ) = (2.69) ( ) = (2.70) ( ) = { } (2.71) ( ) = (2.72) Substituting the values of from equations (2.69)-(2.72) in equations (2.43)-(2.45) we obtain the tangential force stress normal force stress and tangential couple stress as ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (2.73) ( ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (2.74) = { } (2.75) NUMERICAL RESULTS AND DISCUSSION Following Eringen (1984) we take the following values of the relevant parameters for the case of Magnesium crystal as = / = 4 10 / = 1 10 / = 1.74 = = And the void parameters are = =

11 = = = The variation of non-dimensional normal displacement non-dimensional normal force stress non-dimensional tangential couple stress and volume fraction with distance r may be shown graphically. CONCLUSION The goal of this paper is to find a general solution of the equation of motion of a micropolar elastic medium with voids in cylindrical polar co-ordinate system to the axisymmetric deformation problem. The solution is obtained through eigen value approach and the Laplace and Hankel transform are used to solve the equations. The expression of displacement microrotation volume fraction components in the transformed domain tangential force stress normal force stress and tangential couple stress are obtained. REFERENCES 1. CiarlettaM. and Scalia A. Variational theorem and minimum principal for micropolar porous soild; Attin. Sem. Mat. Fis.Univ. Modena 41 (1993) Dhaliwal R.S. and Chowdhury K.L. The axisymmetric Reissner-Sagoci problem in the linear micropolar elasticity; Bull. Acad. Polon. Sci. Ser. Sci. Techn. 19 (1971) Dhaliwal R.S. and Khan S.M. The axisymmetric Boussinesq problem for a semi-space in micropolar theory; Int. J. Engng. Sci. 14 (1976) Dyszlewicz J. and Kolodziej C. Fundamental solutions of micropolar elastodynamics. The first axisymmetrical problem. Method of superposition; Bull. Acad. Polon. Sci. Ser. Sci. Tech. 33 (1985) Eringen A.C. Linear theory of micropolar elasticity; J. Math. Mech. 15 (1966a) Eringen A.C. Theory of micropolar fluids; J. Math. Mech. 16 (1966b) Eringen A.C. Theory of micropolar elasticity in Fracture; Chapter 7 Vol. II Acadmic Press New York Ed. H. Leibowitz (1968) Eringen A.C. Plane waves in non-local micropolar elasticity; Int. J. Engng. Sci. 22 (1984) Eringen A.C. and Suhubi E.S. Non-linear theory of simple microelastic solids I; Int. J. Engng. Sci. 2 (1964) Iesan D.A Shock waves in micropolar elastic materials with voids; Analele Stiintifice ale Universitatu Al.l. Cuza din lasi Seria Noua Sectiunea la matematica.31 (1985) Khan S. M. and Dhaliwal R. S. Axi-symmetric problem for half-space in micropolar theory of elasticity ; J. Elasiticity 7 (1977b) Kumar R. and Choudhary S. Axi-symmetric problem in time harmonic sources in micropolar elastic medium; Indian. J. Pure Appl. Math. 33 (2002b) Kumar R. and Choudhary S. Interactions due to mechanical sources in a micropolar elastic medium with voids; J. Sound and Vibration 266 (2005) Kumar R. and Tomar S.K. Propagation of micropolar waves at boundary surface; Indian J. Pure Appl.Math. 27 (1996) Kumar R. and Ailawalia P. Interaction due to mechanical sources in micropolar cubic crystal; Int. J. Appl. Mech. Engng. 11 (2006a) Kumar R. and Ailawalia P. Deformation due to mechanical sources in micropolar elastic soild with voids; Int. J. Appl. Mech. Engng. 11 (2006b) Kumar R. and Barak M. Wave propagation in liqued-saturated pouros solid with micropolar elastic skelton at boundary surface; Appl. Math. and Mechanics 28 (2007) Mahalanabis R.K. and Manna J. Eigen value approach to linear micropolar elasticity; Indian J. Pure Appl. Math. 20 (1989) Melinte T.C A generalized thermodynamics theory for micropolar materials with voids; Analele Stiintifice ale Universitatu Al.l. Cuza din lasi Seria Noua Sectiunea la matematica. 36 (1990)

12 20. Nowacki W. The axial-symmetric problem in micropolar elasticity; Bull. Acad. Polon. Sci. Ser. Sci. Techn. 19 (1971b) Palmov M.A. Fundamental equations of theory of asymmetric elasticity; Prikl. Mat. Mekh. 28 (1964) Sharma J.N and Chand D. On the axisymmetric and plane strain problems of generalized thermoelasticity; Int. J. Engng. Sci. 33 (1992) Singh B.and KumarR. Reflection and refraction of micropolar elastic waves at an interface between pourous elastic solid and a micropolar elastic solid; Proc. Nat. Acad. Sci.India 70 (2000) Suhubi E.S. and Eringen A.C. Non-linear theory of microelastic solids II; Int.J. Engng. Sci. 2 (1964) Wang J. and Dhaliwal R. S. On some theorems in the non-local theory of micropolar elasticity; Int.J. of Solids and Structures. 30 (1993) Wang X.L. and Stronge W.L. Micropolar theory for two dimensional stresses in elastic honeycomb; R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.455 (1999) Wang X.L. and Stronge W.L. Micropolar theory for a periodic force on the edge of elastic honeycomb; Int.J. Engng. Sci. 39 (2001) Xiao H. On symmetries and anisotropies of classical and micropolar linear elasticity: a new method based upon complex vector basic and some systematic results; J. Elasticity 49 (1997) Zavarzina N.A. Listrov A.T. and Surinov Ju.A. Group in variant properties of the equations of a micropolar viscoelastic Maxwell model; Voronez. Gos. Univ. Sb. Naucn. Trudov Fak. Prikl. Mat. Mec. VUG 1 (1971) Singh R.Singh K. eigen value approach in micropolar elastic medium with voids-int. J. of Applied Mechanics and Engineering (2013) vol.18 No.2 pp

EFFECT OF DISTINCT CONDUCTIVE AND THERMODYNAMIC TEMPERATURES ON THE REFLECTION OF PLANE WAVES IN MICROPOLAR ELASTIC HALF-SPACE

U.P.B. Sci. Bull., Series A, Vol. 75, Iss. 2, 23 ISSN 223-727 EFFECT OF DISTINCT CONDUCTIVE AND THERMODYNAMIC TEMPERATURES ON THE REFLECTION OF PLANE WAVES IN MICROPOLAR ELASTIC HALF-SPACE Kunal SHARMA,