Issue 59

A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03

There exist three B matrices (membrane, the bending and the shear components), corresponding to the strain-displacements. They are described in detail as follows:

         m B

         

N

0 0 0 0

x

,

0 N N N

0 0 0 0 0 0

y

,



y

x

,

,

   

   

N

0 0 0

0

x

,

(5)

 

B

N N N

0 0 0 0 0 0 0



f

y

,

 

y

x

,

,

    

N N

0 0 0 0

0

  

  

 

x

,

B

c

0 N N

y

,





where , y N N y In this paper, we consider the dynamic version of the principle of virtual displacements. The equations of motion for the free vibration of symmetric cross-ply laminated plates can be expressed as:  , x N N x  and  

    

   

   

  

  

  

2

2

2

     x

 x



 





w

w

w





K A

A

I

 

s

55

44

0

2

2

2





x

x





x

y

t

   

   2

2





2

 y



2

2



 x

 

w

y

w x

 

 

x

      

 D D







    x

D

K A

I

(6)

      

 

s

11

12

66

55

2

2

2

2

 

 



x y

x y





x

y

t

   

2

2

2





 y

 y

 y



2

2

  

  





 x

 x



w











 y



D

D

D

K A

I

s

12

22

66

44

2 2

2

2

2

 



x y

y









y

y

x

t

Here, ଴ and ଶ are the tensor components for the mass inertia, and they are defined as:     3 0 2 ; 12 h I h I (7) where  and h are, respectively, the density and the total thickness of the composite plate. ௜௝ and ௜௝ are the extensional and bending stiffnesses, and they are expressed as in the following equations:                          1 1 3 3 1 1 N k ij ij k k k N k ij ij k k k A Q z z D Q z z (8)

where   k ij Q is the stress-reduced stiffness matrix for the transformed material plane in each layer k . In Eqn. (8), the matrix can be obtained as follows:

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