Issue 51

R. Massabò et alii, Frattura ed Integrità Strutturale, 51 (2020) 275-287; DOI: 10.3221/IGF-ESIS.51.22

where 22 22 2 , , b g N M Q are axial force, bending moment and generalized shear force, given by (8) in the Appendix, where the tilde defines prescribed values at the beam edges and 2 1 n   is the second component of the outward unit normal. The equations have the same form of those of first order shear deformation theory and the effects of the local enrichments (zigzag and interfacial jumps) enter the generalized shear force, Eqn. (8), which is variationally consistent and coincides with the resultant of the a posteriori calculated shear stresses, 3 1 3 ( ) 2 23 3 1.. k k x k post g k n x Q dx       , as in classical structural theories. The constitutive equations of the homogenized beam have been derived in [18]:

S

S

 

     

0

0

N A B C C v 

22         22 zS b

02 22           2 22 0 222 , , , v w 

02 2           2 2 0 22 , , ; , w 

22

22





S

S

S

S

S

S

 

 

1

1

0

1

2

2

22 M B D C C M C C C C      22 0 1 2 22     S S S

2 44 2 Q A w   g

0 2   ,

C C C C 

(5)

S

2

with

n

3 k

x

( ) k   22 C

P C C 

S

2



44 A k 

, A B

3 3 x x dx

(

,D )

(1, ,

3 ) ;

22 22 22

44 44



3 k

1

x



k

1





( ) k

k C

( )

with ( ) k

ij C the coefficients of the 6×6 stiffness matrix in the layer k ; the remaining 44 k is a shear correction factor which can be set equal to 5/6, as for a



22 23 32 C C C C  /

where

22

33

coefficients are given in (10) in the Appendix;

homogeneous beam, since the effects of the layered structure are already accounted for through the zigzag functions. The equilibrium equations in terms of global displacement variables (for a beam subjected to transverse surface tractions, 3 2 ( ) f x ) are [19]:

S

0 C w S

0

(  

) , 





A v

22 B C

,

,

0

22 02 22

2 22

0 222

S

S

S

1   0 222 C C w A w  ( ) , ( , S S 2

0

1

2

22 B C v 

  

) , 

  

2 D C C

(

) ,

(

) 0

(6)

02 22

22

2 22

44 0 2 2

0 C v S

1 C C S

S

2 C w S

2

(  

) , 





, )    f 

A w

,

,

( ,

0

02 222

2 222

0 2222

44 0 22 2 2

3

They return the equilibrium equations of first order shear deformation theory in a n -layered beam with equal fully bonded layers, where all coefficients with upper index S vanish. And they return the equilibrium equations of the zigzag theory in [6] for fully bonded unequal layers.

L OCAL STRESS AND DISPLACEMENT FIELDS AND DELAMINATION FRACTURE OF N - LAYERED BEAMS Stress and displacement fields in n-layered beams subjected to mechanical loadings The equilibrium Eqns. (6) with boundary conditions in (4) are easily solved for n -layered beams and wide plates with continuous imperfect interfaces, shown in Fig. 1(b); the model is applicable to beams with various boundary conditions (simply supported, clamped) and loadings (distributed, concentrated loads). Exemplary closed form solutions have been presented in [19] for simply supported plates under thermo-mechanical loadings; in [21] closed form solutions have been derived for the wave propagation problem. The model accurately captures the transverse displacements in fully bonded beams while requires a correction in shear deformable beams with fully debonded or weakly bonded interfaces where the I n this section various applications of the model are presented in order to highlight its main features.

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