PSI - Issue 2_A

A. Taştan et al. / Procedia Structural Integrity 2 (2016) 261 – 268 Ta ş tan et al./ Structural Integrity Procedia 00 (2016) 000–000

263

3

where ( )  x , u  and ( , ) t b x denote mass density, spatial acceleration and body force density vector of the material point , x respectively. The spatial derivatives of stress tensor σ required in Eq. (1) are not valid at discontinuities. In PD, equation of motion does not require spatial derivatives and can be written as: ( ) ( , ) ( , ) ( , ). ' ' x u x f u u x x b x H t dH t        (2) In Eq. (2), H is the horizon of material point x with a radius of  , x ' is a family material point,  is the mass density, u and b denotes the displacement and body force vector fields, respectively. Note that ( ( ) ( ) ) f u x' - u x , x' - x is the pairwise force vector that the material point x exerts on the material point x' . It has been shown that, PD converges to classical theory of elasticity, when the length of the horizon goes to zero (Silling and Lehoucq, 2008). 2.2. Mindlin Plate Kinematics (Diyaroglu et al., 2015) As shown in Fig.1, ( ) j  and ( ) k  represent the rotations of material points j and k with respect to line of action between of material points j and k. The curvature between material points k and j ( )( ) k j  can be written as (Diyaroglu et al., 2015): ( ) ( ) ( )( ) ( )( ) . j k k j j k       (3) By applying coordinate transformation, the rotations of a two dimensional plate can be written as (Diyaroglu et al., 2015): ( ) ( ) ( ) cos sin , j x j y j        ( ) ( ) ( ) cos sin . k x k y k        (4)

( )( ) k j  can be rewritten

where  is the bond angle with respect to x axis direction (see Fig.1a). The plate curvature

using Eq. (4) as follows:

a)

b)

Fig. 1: a) Initial and current configurations of a Mindlin plate b) representation of rotation at material point j ( Diyaroglu et al., 2015)

   

   

   

   

( )    ( ) j x x  x j

( )    ( ) j y y  y j

( )

( )

x k

y k

2

2 sin , 

(5)

cos









( )( ) k j

( ) k

( ) k

where ( ) j x x

( )( ) cos , j k  

( )( ) sin . j k  

( ) j y y

( ) k  

( ) k  

(6)