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

265

5

1 1 ( )

 

2 dH     ( )( ) k j ( ) ,

(14)

U

c

( )

b k

2 2 H

in which ( )( ) k j  is the curvature along the bond direction.

2.3.1 Orthotropic Plate, strain density energy The strain energy density due to bending for an orthotropic plate (see Mansfield, 1989 for more details) is reported in the following formula:

  

xy D D D .       

  

1 2

(15)

2 U D D     2

2 2    

D

 



 

 

11

12

22

16

26

66

b

xx

xx yy

yy

xy

xx

yy

2

where the curvatures are defined as 2 2 , xx w x     2 2 yy w , y     2 2 xy     

w ,

x y (16) In order to find the bond constants, the Strain Energy Density (SED) under the following curvature field will be considered , , . xx yy xy xy xy              (17) The proposed curvature field represents the one obtained applying a uniform external moment on a plate, x M . Adopting the discretized form of Eq. (14), the SED in a Peridynamic model can be written as follows: (18) In the previous formula Q is the number of fiber bonds while J is the number of matrix bonds within the horizon. The SED using the CLT can be written as: (Reissner notation for twisting) 2   2   ( ) ( )( ) ( )( ) ( ) k q k q q ( )( ) ( )( ) ( ) k j j 1 2 2 q  1 2 2 j  1 1 Q  1 1 J  . b k f m k j U c V c V  

1 2





2 D D D 

2   2

  k W D D D      11 12 2 2 CLPT xy xy

2

.



26 xy xy     xy

(19)

16

22

66

xy

Using Eq. (18) and Eq. (19), bond constants f c and m c can be obtained as   11 22 , f D D c    12 3 24 , m c D h       1 . 1 2 Q q q k q V     

(20)

ij D are the coefficients of the bending stiffness matrix in CLT.

In Eq. (19) and Eq. (20)

2.4. Failure criteria Using the proposed Kirchhoff plate bending theory for orthotropic materials and adopting a PD approach it is possible to evaluate the curvature of the plate at each material point. The adopted failure criteria is based on a limit curvature. Following the roadmaps of Hu et al. (2011), the limit curvature could also be direction dependent in the way described in Fig. 2

limit_matrix 

_ limit fiber 

_ limit fiber 

Fig. 2 Variation of limit curvature with respect to the angle between the bond and the fiber direction.