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Full Paper Proc. of Int. Conf. on Recent Trends in Transportation, Environmental and Civil Engineering 2012

Solving Initial Value Problem in Micropolar Plate Theory Solmaz Dehghanmarvasty1, Asadollah Noorzad2 and Mandana Abbasi3 1

MSc student of structural engineering, School of civil engineering, University of Tehran, Tehran, Iran Email: solmaz_dehghanm@ut.ac.ir 2 Assistant Professor, School of civil engineering, University of Tehran, Tehran, Iran Email: noorzad@ut.ac.ir 3 PhD student of structural engineering, School of civil engineering, University of Tehran, Tehran, Iran Email: mandanabbasi@ut.ac.ir

Index Terms—initial value problem, impulse displacement, micropolar theory, thin plate, Hankel transform, sizedependent material

Firstly, Eringen presented the theory of micropolar plates [10], after that the researches has been conducted on some fundamental issues in this theory. Wang presented a systematic approach for deriving equation of thin plate [11]. Ariman [12] analyses the symmetric bending of laterally loaded micropolar plate. Schavone and Constanda studied on the problems of existence and uniqueness in the micropolar plate theory [13]. Also Erbay presented an asymptotic theory of thin micropolar plates [14]. Ciarlet [15] has derived the von karman plate equation corresponding to geometrically nonlinear, but materially linear. In this study, we considered flexural equations of micropolar and classical plate and solved them under impulse displacement using numerical Hankel transform and their inverse transformation to obtain displacement of plate in different radius. Finally, we compared the results of micropolar plate with classical plate theory.

I. INTRODUCTION

II. REVIEW OF CLASSICAL PLATE THEORY

Abstract— In this paper, the size-dependent plate is solved on basis of micropolar theory. The micropolar continuum theory is a non-classic theory capable of considering the small-scale size that effect on the mechanical behavior of structure. But the classical continuum theory is unable to predict the precise act of material especially in high frequency problem. In this study, nonlocal micropolar theory is used for investigating of the impulse displacement in plates using numerical Hankel transform and compared with classical theory result. The results show that the pattern of deflection of the micropolar plate is similar to the classical plate theory but the deflection is higher. It also can be observed that the higher wavelength is appeared in micropolar plate in comparison to the classical theory which can be related to the micropolar continuum’s nature.

The solids in civil and structural engineering have a composite and granular nature. Because the classical continuum mechanics cannot predict this characteristic of material; there should be a theory that predicts sizedependent nature of solids [1-4]. This motivated many researchers to develop material model using continuum elasticity that contains an additional material constant [4]. In recent years, micromechanical models of solids like soil, rock, concrete, and various granular materials have been developed [2]. Based on these models, various theories have been introduced to predict the behavior of materials in which the grain size becomes dominant. One of these theories is micropolar theory [3-5]. In this theory, four higher order material length scale parameters appear in constitutive equations rather than two lame constants [6]. In addition, in this theory the stress depends on strains at all points of the continuum [3,5]. Also micropolar continuum supposes that each particle can rotate individually without depending on its surrounding medium [3,4,7-9]. Consequently, this medium has three translational motions and three rotational motions which called micro-rotation. Because of micropolar continuum’s characteristic, this theory is used for high frequency and size-dependent problem to achieve accurate result. 25 © 2012 ACEE DOI: 02.TECE.2012.2. 5

Consider a thin elastic plate of thickness h, defined using a Cartesian coordinate system Oxyz (Fig. 1). The fundamental assumptions in theory of thin plates are A) the thickness of plate is small as compared to any characteristic length in median plate. B) Stress and displacement fields do not change violently across thickness. So the wave equation for transverse vibration of an isotropic, undamped plate which is subjected to a uniform load is as in (1) [16]: D 4 w( x, y, t )  h where

D

 2 w( x, y , t )  2t Eh3

 q ( x, y, t )

(1)

12(1   )

Where w(x,y,t) is the displacement along the z direction at the point (x,y), D is the stiffness of the plate in bending, ν is the Poisson ratio, E is elasticity modules (Young’s modulus), h is the plate thickness and q(x,y,t) is the lateral load. In terms of polar coordinates, this equation is given by (2): D 4 w(r , , t )  h

 2 w(r , , t )  2t

 q(r , , t )

(2)

Full Paper Proc. of Int. Conf. on Recent Trends in Transportation, Environmental and Civil Engineering 2012

1 u ,  u , 2  u,   



e  

 (6)

That εαβγ is the permutation symbol, i. e.

123  231  312   213  132  321  1 (7) Based on Eringen [10], a thin plate of thickness 2h, having x3=0 as its median plate is considered. Because of the assumption that is considered in plate theory (is mentioned in previous) we may use the average and the first moments of various quantities over the thickness in integrating the field equations with respect to x3. By integration (3) we get: Figure 1. A plate element

  ( f  u )  0 2h

(k  1, 2)

mk , k  ek  t k  e3  t 3 

   (l  j  )  0 2h

t k ,k 

III. REVIEW OF ERINGEN’S MICROPOLAR PLATE THEORY

The equation of micropolar plate theory is well posed and mathematical foundations have been derived by Eringen [3]. Based on Eringen’s study, Theory of micropolar plate introduced with shear and rotatory inertia correction. However, no artificial “fudge factor” need to be introduced for shear correction like Timoshenko beam theory [10]. Such factors arise from inherent of micropolar continuum. In this section, we present the basic relations of micropolar elasticity, definition of its displacement and other terms. According to theory proposed by Eringen [10], the equation of balance of momentum and moment of momentum of micropolar elasticity are respectively given by (3) (   1, 2, 3 ):

Where quantities carrying a superposed bar indicate average, e.g. tk 



h

m

dx3

mk  mkl el  mk 3 e3 ,

(3)

l  ( f l  ul )  0 2h .. p t k 3,k   ( f 3  w)  0 2h ..  mkl ,k   kl (t 3k  t k 3 )  l  (l l  j l )  0 2h .. m mk 3, k   kl t kl    ( l 3  j 3 )  0 2h

ek  el  kl e3 , e3  ek   kl el  0,

12

 21  1

(13)

Equation (12) must be completed with equation of couple stress. By vector multiplication of (3-1) by x3e3 and integration over x3 this type of equation can be achieved. So we get:

(5)

In above equation, the coefficients λ, μ, χ, α, β and γ are elastic constant appropriate to the theory. The strain and micropolar strain tensors are respectively given by (6): 1 u ,  u , 2   u,    © 2012 ACEE DOI: 02.TECE.2012.2. 5

(12)

Where

(4)

11   22

( ,   1, 2,3)

(11)

t kl , k 

Where m  m e

m3  m3l el  m33e3

(10)

By substituting (9-11) in (8) the plate equation is obtained:

t  e    2(  )    m       ,  ,

(9)

h

tk  t kl el  tk 3e3 , t3  t3l el  t33e3

Where tα, ρ, f, u, mα, eα, l, j and φ are respectively the stress vectors, the mass density, the body force, the displacement vector, the couple stress vector, the rectangular unit base vectors, the body couple, the microinertia and micro-rotation vector. Also comma indicates the partial differentiation with respect to xα and a superposed dot indicate the partial derivative with respect to time. In micropolar elasticity, the equations of motion are supplemented by the following constitutive equations:



1 2h

  m3 ( x1 , x2 , h)  m3 ( x1 , x2 ,  h)  k ek  m e3

m,  e  t  (l  j )  0

t  t e ,

tk dx3 , mk 

  t3 ( x1 , x2 , h)  t3 ( x1 , x2 ,  h)  k ek  p e3

e 

1 2h

And

t  ,   ( f  u )  0 ..

(8)

M kl ,k  2ht 3l  hl  ( Ll  I v l )  0



(6) 26

(14)

Full Paper Proc. of Int. Conf. on Recent Trends in Transportation, Environmental and Civil Engineering 2012 Where Mkl, Ll and vl are respectively the stress couple, the bode couple and the angular rotation which defined by (15): h

M kl 

t

x3 dx3 , Ll 

h h

Ivl 

 I (     ) 2   2[I  h(     )] 4 

f



x3 dx3 ,

4Gh      4 ..  2   I ( j  )  G/2 G/2

h

u

x3 dx3 & I 

h

2 3 h 3

2( hj 

(15)

I  2  k ,k  2h k , k 

I is the area moment of a normal cross section of plate with unit length and 2h thickness respect to median line. By gathering the (12) and (14) the balance equation of micropolar plate is present. Displacements and rotation of micropolar plate are described as follow:



w( x1 , x2 , t )e3

(    ) 4  2 2   j 2   k ,l

P  p  2hf 3 , Tk  hk  Lk , k 

.. D D 2 P  P 2h (G   / 2) 2 h(G   / 2) G  / 2 P  Tk ,k  0 G/2

1 (k  2hlk ),  2 A  A,kk 2h

An initial value problem in plate vibration will be solved here. We consider an infinite cylindrical plate subjected to impulse displacement with axisymmetric condition. We assume the initial condition as (24): ( r ) , w ( r , 0)  0 (24) 2 That gives the case of impulse in displacement. Now we solve the impulse displacement for both classical and micropolar plate and then compare the result with each other. w( r , 0) 

(18)

A. Classical Plate The governing equation of motion in classical plate is given in terms of polar coordinates: D 4 w(r , t )  h

.. 2G 2G  4    2   j 2  2w G/2 G/2



..   w P  kl  k ,l  0 G  / 2 2h (G   / 2)

 2 w(r , t )  2t

 q (r , t )

(25)

Which herein

q(r , t )  0 (26) The hankel transform will be used to solve problem. By applying this transform we obtain:

(19)

b2 4 w(, t )  b2 

(23)

IV. INITIAL VALUE PROBLEM OF INFINITE PLATE

.. 4Gh D  2 w  (  I ) 2 w G/2 G/2



(22)

where

(17) All of the above terms are function of x1, x2 and t, so the final form of micropolar plate theory are given by the following equivalent system of (18 to 22):

.. 4Gh I  2  2  G/2 G/2

(21)

 2V 

vk  v,k   klV,l , k  , k  kl ,l



 kl Tk ,l )  0 G/2

 4   2 2   j 2   2 w   kl  k ,l

The above definition of displacement and micro-rotation is substituted in to (12) and (14) and by assumption t33=0 the final form of micropolar equation is obtained. But according to Eringen plate theory this formula is too complicated for analytic work. So instead we decompose each of vk and φk into a gradient and a curl by (17):

..  2 I .... D w 2h w  4 G/2 G/2

(20)

(16)

  k ( x1, x 2, t ) ek  k ( x1, x 2, t )e3



.. I k , k G/2

 2 v 

u  [u k ( x1 , x2 , t )  x3 vk ( x1 , x2 , t )]ek 

 D 4 w 

.. .... I  2 jI  2   2  G/2 G/2

 2 w(, t )  2t D h

0

(27)

Full Paper Proc. of Int. Conf. on Recent Trends in Transportation, Environmental and Civil Engineering 2012 By solving (27), displacement is obtained

The analytical hankel transform and the inverse hankel transform are described as:

(28) w(, t )  A cos(b2 t )  B sin(b2 t ) The Hankel transform of the initial condition (24) are w(, 0)  1, w (, 0)  0 (29) Applying the initial condition to the solution (28) gives A  1, B  0 (30) Now to find the answer of problem, we should take inverse Hankel transform but we stop the calculation until this step. First describe the method of solving micropolar plate then discuss on Hankel inverse transform.





f 2 ()  2 f1 (r ) Jp(2r ) rdr 0



In numerical method, we change integration to discrete form. α pm ia the m-th root of p-th Bessel function.

B. Micropolar Plate To avoid complexity in solve micropolar plate under impulse displacement; we apply three approximation theories on micropolar plate equation based on Eringen’s study.



f (r ) 

c pm 

v at

 2 w(, t )

0

 2t 2 D 4  D6    4h 4  C 4h  2h2 

 f (r )(

J 2p 1 (  pm ) 0

α pm 2πR

(36)

r ) rdr a

α pn 2πV

and frequency

and replace infinite to number N, we can achieve to

  pm f 2   2R

 1   V 2 



  pn f1   2V

 1   2  R



n 1 N

f1(  pn / (2V )) J 2p 1 ( pn ) f 2 (  pm / (2 R )) J 2p 1 ( pm )

n 1

  pn  pm  J p    2 VR    pn  pm J p   2VR

  pm  1 F2 (m)  f 2   J p 1 ( pm ) V  2R    pn  1 F1 (n)  f1   J p 1 ( pn ) R  2V 

(33) w(, t )  A cos( Ct )  B sin( Ct ) By applying inversed initial value condition on (33) we have (34)

  

(37)

(38)

Then the equations are reduced to following equation and we use these equations for solving plate problem. .

w(ξ,t) is too complicated to analyze directly..

So we employ numerical inverse Hankel transform to obtain displacement of plate in numerous radius and fixed time. First we describe this method then depict the result of solution.

F2 ( m) 

T

mn F1 ( n )

n 1 N

C. Hankel Numerical Inversion Transform We use the numerical method to take Hankel inverse transform. The analytic method is too complicated. This method is simpler and gets result by high accuracy. Based on Manuel Guizar-Sicairo’s studies [16] on quasi-discrete Hankel transforms we convert hankel inverse transform to discrete form. Then use matrix of conversion. The formulation is mentioned below.

Above equation can be rewritten in a symmetric form by defining the vectors as following

(32)

By solving (32), displacement is obtained

w(, t )  1  cos( Ct )

r ) 0  r a a

same accuracy as integration form

(31)

By Taking Hankel on (31), we get w in converted form: Cw(, t ) 

pm J p ( pm

If we evaluate the radius r at values

 4h 4 w  0

c m 1

We consider rotatory inertia  ρI and microinertia  ρj equal to zero. Transverse shear effect is assumed that negligible also (letting G’!”). Consequently by considering these approximations and eliminating Õ and w between equations, we achieve the uncoupled equation of w:  2 2  4     ( D w  2h w q)   

(35)



f1 (r )  2 f 2 () Jp(2 r )d 

F2 ( n) 

T

nm F2 ( m )

(39)

m 1

where Tmn 

2 J p ( pn  pm / 2VR) J p 1 ( pn ) J p 1 ( pm ) 2 VR

(40)

Full Paper Proc. of Int. Conf. on Recent Trends in Transportation, Environmental and Civil Engineering 2012 It means that the distance between two successive points of constant phase in micropolar continuum is higher. So, the micropolar plate is affected in greater radius under impulse displacement. These differences are based on the nature of each medium. In micropolar plate because of considering size effect of material, micro-rotations in governing equation and four lame constant more than classic medium, we achieve more accurate result especially in severe initial condition like impulse displacement.

V. RESULT The aim of this study is to investigate the capability of the micropolar theory to analyze elastic behavior of plates under impulse displacement. For this purpose, we consider an infinite plate and an impulse displacement. The required parameters for solving the problem were chosen as follows: h  0.1m, E  20Gpa ,   0.3, E   4 109 , G= ,  =1.93 108 2(1   )

(39)

REFERENCES

These quantities are presented in [18] and corresponded to the human pelvis’s bones, because the pelvis is also a type of plate. The final result of solving plate problems for both classical and micropolar continuum, are illustrated in Fig.2. CONCLUSIONS The comparison of the results of the micropolar plate with classical plate theory is shown in Fig. 2 at a given time. “Fig. 2” shows that the pattern of deflection of micropolar plate is similar to the classic plate. The oscillation behavior is appeared in both types of plates. Moreover, deflection in micropolar plate is more than classic plate, especially, in center of plate. Another point that can be mentioned is the pattern of decadence in both plates. In both type of plate, we meet decaying wave but in micropolate plate the wave is damped later with higher wavelength in comparison to the classical theory.

(a)

(b) Figure 2. Deflection of the plate under impulse displacement at (a) t=0.1 s, (b) t=1 s

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