PSI - Issue 41

Andrea Pranno et al. / Procedia Structural Integrity 41 (2022) 618–630 Author name / Structural Integrity Procedia 00 (2019) 000–000

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2.1. Nonlinear traction-separation law for loading and unloading processes Concrete exhibits a linear elastic behavior in the pre-peak phase and a more complex softening behavior in the post-peak phase. According to well-known constitutive tensile found in the literature, tensile monotonic stresses have different constitutive relations, for example, a linear, bilinear and exponential softening are proposed in (Hillerborg et al., 1976), (Petersson, 1981) and (Sima et al., 2008), respectively. Other study were conducted for cycling loads in (Reinhardt et al., 1986) and (Foster and Marti, 2003). The proposed traction-separation law allows to consider the effects of damage in mixed-mode conditions, contact phenomena between crack surfaces and residual plastic deformations associated with the concrete damage behavior. The fracture initiation involves a quadratic stress criterion, the propagation is governed by a mixed-mode equation, while the damage is represented by an exponential damage parameter, which is determined by a function of the equivalent displacement, which is appropriate given the mixed-mode conditions. Furthermore, due to the partial closure of cracks, which is induced by the presence of concrete aggregates, a formulation which adapts the tangent stiffness is developed to take into account the inversion of the cohesive stresses from tensile to compression. The mixed-mode traction separation las is defined by the following expression:

   

    

max

n 





p

  

n       s t t

  

 

0

0

K

 1  



n

n

max    p

d

n

,

(1)

n

n

0

0

K

 

s

s 

max n n    is the plastic normal displacement, p

where s and n define tangential and normal directions, respectively,

max n  is the maximum normal displacement jump detected along the entire loading process and the scalar d is the damage variable. In terms of the latter, a linear-exponential evolution law can be defined as a function of the equivalent mixed-mode displacement jump:

2

2





  

,

(2)

m

n

s

n  yield the equivalent displacement independent from the compressive normal

where the Macaulay bracket

displacement jump:

max    0

         

0

m

m

      

      

  

  

max          0 f m m m 

  

0

1 exp

 

m

0



0      max m m m f

1   

1

d



m

,

(3)

  

max

1 exp



 

m

max   

f

1

m

m

with 0 m  denoting the equivalent displacement jumps at the onset of the damage and f p n K characterizes the unloading phase as reported in Fig.1: plastic stiffness

m  at the total decohesion. A

0 max

max (1 )

n n      p d K

p

K



(4)

n

n

n