Issue 29

G. Gianbanco et alii, Frattura ed Integrità Strutturale, 29 (2014) 150-166; DOI: 10.3221/IGF-ESIS.29.14



  

      

  



F

1

1

σ

p

ct

T





(50)

M

E

Q B Q

Q B

22

21

  ε

 

ε

M

UC

UC

M

In the previous equation the second term has to be explicitly evaluated. After mathematical manipulation it is possible to write that

  m

  m       F F u ε u ε  p p M

 

U



(51)

m

α

ε

M

M

where

     p m m u u

    

    



4 3  1 1 k j  







T





 Φ S Φ S

w

(52)

α

Φ S Φ S K

Gpj

i

i

b b

i

i

i

b b



x

x

Ppj

Ppj

From Eq. (47)

 

 

F

U

p

T





(53)

m

B

12 B Q

11

ε

ε

M

M

which, substituted in (51), furnishes:   1 11 12 p T M      F α I B α B Q ε

(54)

The consistent macroscopic tangent stiffness matrix is therefore written as the sum of an elastic and a plastic part: ct ct ct e p   E E E (55) with

1

ct

T



(56)

22  E QB Q e

UC

1



 1 

ct

T





(57)

21  E QB α I B α B Q 11 12 p

UC

Localization of deformation at a single macroscopic quadrature point Let us consider, at a macroscopic quadrature point, a discontinuity band characterized by an incipient loss of strain continuity (Fig. 6).

Figure 6 : Body split by a potential discontinuity band.

159