PSI - Issue 41

Umberto De Maio et al. / Procedia Structural Integrity 41 (2022) 598–609 Author name / Structural Integrity Procedia 00 (2019) 000–000

601

4

The nonlinear constitutive behavior of the cohesive interfaces is expressed by a traction-separation law, reported in Fig. 1b, written in the following form:

(2)

  0 (1 )  K

coh

t

D  

where 0 K is the initial stiffness parameter of the cohesive elements, and    is the displacement jump occurring along the boundaries of the mesh.

Fig. 1. (a) Schematic representation of the diffuse interface model; (b) adopted exponential-type traction-separation law.

The symbol D , reported in Equation (2), represents a scalar damage variable with an exponential evolution law that involves the following effective mixed-mode displacement jump:

2

2

(3)





n    s

m

being n  , s  the normal and tangential components of the displacement jump    . Moreover, the interfacial elastic stiffness 0 K reported in Equation (2) plays the role of penalty parameter to enforce the inter-element kinematic constraint, without having a physical meaning. Correct values of its components, i.e. 0 n K and 0 s K , have been computed, adopting the micromechanics-based calibration technique proposed by some of the authors in (De Maio et al., 2020b, 2020c), through the following expressions:

0 S n K E K K L     0 0 mesh , n

(4)

where E is the Young’s modulus of the material while  and  are dimensionless stiffness parameters obtained by the adopted calibration technique, as a function of the desired Young’s modulus reduction E R and the Poisson’s ratio of the bulk material. The mixed-mode initiation and propagation are governed by the following stress- and energy based criteria, respectively:

2

2

t

t

I G G G G Ic

n                s c  c

II

1,

1

(5)

 

IIc