Issue 29

G. Gianbanco et alii, Frattura ed Integrità Strutturale, 29 (2014) 150-165; DOI: 10.3221/IGF-ESIS.29.14

 K E

h

/

where

i K is the stiffness matrix for interfaces, defined as

.

i

i

Making the following positions

4











T





T T T b b K S Φ C E CΦ S b b b 

 







d

d

(41)

Φ S Φ S K Φ S Φ S

i

i

b b

i

i

i

b b



k

1





b

k

4

T T i i   G S Φ Ψ S i 



d

(42)

i

1  

k

k

4

   F Φ S Φ S K u    m T p i i b b i  

p



d

(43)

1  

k

k

4

4









T T U S Ψ u 

T T S Ψ D ε

T Q ε

 

 

d

d

(44)

m

i

i

m

i

i

m M

M





k

k

1

1





k

k

the (40) becomes  T 



 0



T T     U KU GR R G U U 

(45)

m

m

m

m

m  U and  R , the governing equations of the UC are obtained, in the

Since Eq. (45) has to hold for any arbitrary

following matrix form:

m                   p m

T K G U F G 0 R U

(46)

 

In order to evaluate the integrals (41)-(44), the Gauss quadrature rule is adopted with nine sample points in the block domain and three sample points for each interface. Furthermore, it is assumed i i  Ψ Φ . In order to be introduced in a step-by-step algorithm, the governing Eq. (46) need to be rewritten in a discrete formulation. A Newton-Raphson procedure is employed at the mesoscale to find the solution of the BVP for a single load step. Each iteration of the load step is divided in an elastic trial predictor stage and a plastic corrector stage.

Consistent tangent stiffness matrix evaluation In the time step , 1 n n t t    

m  U is assigned. The inverse form of equation

 the increment of boundary displacements

system (46) reads





11     U B B F R B B U separable in the two expressions 12 21 22 m              

  

p

m

11 m      U B F B U 12 m p

(47)

21 m      R B F B U (48) It is clear how nodal displacements and reactions are dependent on two contributes that can be assumed equal to an elastic part (function of m  U ) and a plastic part (function of p  F ). Recalling (13), (39), (44) and (48) it is simply obtained that 22 p

  

     

  

1

1

T

  σ

 

 Q B F

(49)

Q B Q ε

M

M

p

22

21





UC

UC

The macroscopic tangent stiffness matrix is defined as:

158