Issue 59

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

    

















  x z x y t z x y t  y , , , ,

 , , , , , , u x y z t v x y z t w x y z t , , , 

 , , u x y t v x y t w x y t , , , ,   0 0









(1)







0

where   0 0 0 , , u v w is the displacement vector of a point on the plane  0 z ,  x is the rotations of a transverse normal around the x -axes and  y is the equivalent for the y -axes.

z

z k+1

z k

Layer k

Layer 1

Figure 1: Laminated plate - organization of layers in the thickness direction.

The strain energy is:

1 2

  U u B D B B D B B zD B B z D B dzudA    2 T k T T k T k T k m m m f f m f f

 

   A z

(2)

1 2

T T k c u B D B dzudA c c

    

 

A z

The components of the stiffness matrix include the membrane-bending coupling part, are written as follows:

           e e e e e e mm mf fm ff cc K K K K K K      

(3)

e mm K is the stiffness matrix of the membrane part, and e mf K , e

where

fm K are the membrane-bending coupling components,

e ff K represent the bending component and e

cc K represent the shear component. These matrices are detailed as follows:

n

            e mm e mf K K  





c

T

  B D B z z dA m k m k k 1

 k A n k A n k A n k A n              1 1 1 1 c c c c  k A 1

1 2 1 2 1 3

 

 

T

2

2

  B D B z z dA m k f k k 1

  e fm

T

2

2

  B D B z z dA k m k k 1 f

K

(4)

  





  e       ff e K 

T

3 k

3 k

 B D B z z dA



f

k f

1





T

 B D B z z dA

K





cc

c

c c

k

k

1

Here c n represents the number of layers across the z-axes.

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