PSI - Issue 2_A

A. Taştan et al. / Procedia Structural Integrity 2 (2016) 261 – 268

264

4

Tastan et al./ Structural Integrity Procedia 00 (2016) 000–000

As illustrated in Fig.1, transverse shear angles at material points, ( ) j  , ( ) k  and the slope ( )( ) k j  can be written as:

( ) j w w 

( ) j k j j      (7) By using Eq. (7), the transverse shear angle between material points j and k can be defined as the average of the transverse shear angles at these material points and can be written as ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) cos sin . 2 2 j k x j x k y j y k k j j k w w               (8) The equations of motion have been derived in Diyaroglu et al. (2015) using the principle of virtual work and solving the Lagrange equations. The obtained formulas are: ( )( ) ( ) , ( ) k k j k      ( )( ) ( ) , ( ) k ( )( ) k j ( )( ) j k .   

 

( ) ˆ , k

(9a)

( ) k hw c   

V b 



( )( ) ( ) k j j

s

1

j



3

1 2

h

1      j

 



  

cos

cos

,

c

( ) j V c 

( ) j V b 







 



(9b)

( )

( )( ) k j

( )( )

( )

x k

b

s

jk k j

x k

12

1

j



3

1 2

h

 



  

( ) (9c) In Eq. (9), s c and b c are bond parameters related to transverse shear and bending deformations respectively. Those parameters are given by Diyaroglu et al. (2015): ( )( ) k j ( ) j V c  ( )( )    ( ) j V b  ( ) 1 1 sin sin . 12 y k b s jk k j y k j j c     

2 3 9 , E k 

E h     

2 2 3 27 . 80 4 k    2

c



(10)

c



s

b

4



2.3. Orthotropic Plate For orthotropic plates, Classical Laminate Theory (CLT) is commonly used in practical applications. CLT is an extension of Kirchoff’s Plate Theory. In CLT it is assumed that sections always stay perpendicular to the bending plane. The Mindlin Plate formulation in Diyaroglu et al. (2015) is modified with the assumption no shear deformation. This means that the shear angles ( ) j  and ( ) k  defined in Eq. (7) are equal to zero and then also the transverse shear angle between material point k and j ( )( ) k j  is equal to zero. So the motion equations Eq. (9b) and Eq. (9c) can be re rewritten introducing also the angle dependence of micromodulus c in the following terms: 3 ( ) ( ) ( ) ( ) 2 2 ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) cos sin cos , 12 x j x k y j y k x k j x k j j k j k h c V b x x y y                                               (11a) 3 ( ) ( ) ( ) ( ) 2 2 ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) cos sin sin . 12 x j x k y j y k y k j y k j j k j k h c V b x x y y                                               (11b) As stated in Eq. (11), the bond constant c depends on the orientation of the bond in for orthotropic plates. The dependence of the bond constant on the fiber orientation can be stated as (Oterkus and Madenci, 2012b): ( ) if , ( ) if . f m f m f c c c c             (12) Peridynamic moment density M and the associated bond strain energy density, ( )( ) k j U can be defined as:   ( )( ) ( )( ) ( )( ) ( ) ( ), k j k j k j c      M   2 ( )( ) ( )( ) ( )( ) ( )( ) 1 1 ( ) ( ) . 2 2 k j k j k j k j U c         M (13) where  is the bond length. It must be observed that the material point energy density is half of the bond energy density (Hu et al., 2011). The energy density due to bending evaluated at the material point k is