PSI - Issue 33

Daniele Gaetano et al. / Procedia Structural Integrity 33 (2021) 1042–1054 Author name / Structural Integrity Procedia 00 (2019) 000–000

1046

5

    coh coh , D  t t u ,

(4)

D being a scalar state variable, properly defined to enforce the damage irreversibility of the interfaces. The constitutive law expressed by Eq. (4) is that of a bed of damageable elastic springs, and its components can be obtained after introducing a local coordinate system   , n s centered at the interface, as shown in the right portion of Fig. 2, where n and s are the normal and tangential unit vectors, respectively.

Fig. 2. Schematic representation of the BVP for a cracked discretized body and representation of the crack with related notations.

The equilibrium problem for such a model can be expressed in an extended variational framework, which is based on the hypotheses of quasi-static loading, small deformations, and linearly elastic bulk material, without loss of generality. From the last hypothesis it can be deduced that the only nonlinearity source is the constitutive behavior of embedded cohesive interfaces. Given that, the related nonlinear BVP can be expressed in the following weak form: Find u belonging to the space of admissible displacement fields such that:   coh coh coh coh \ \ N h h h d d d d                         u t u f u t u , (5) for all virtual variations  u belonging to the same space of admissible displacement fields. The second term of the left-hand side of Eq. (5) is an additional nonstandard term, representing the virtual work of the cohesive tractions. As nonlinear constitutive behavior of the cohesive interfaces, a mixed-mode traction-separation law of intrinsic type with linear softening is used in the present work, as shown in Fig. 3. This law can be written in the following matrix form:

    n s u          

0

K

 coh         coh     1 n s t t

  

  



n

D

(6)

,

0

K u

s

where the initial stiffness parameters

n K and

s K can be easily computed according to the calibration procedure

proposed by some of the authors (De Maio et al., 2020b) to avoid any artificial compliance effect.

Fig. 3. Intrinsic mixed-mode cohesive law with linear softening adopted for the present Diffuse Interface Model.