Issue 61

H. Mazighi et alii, Frattura ed Integrità Strutturale, 61 (2022) 154-175; DOI: 10.3221/IGF-ESIS.61.11

Figure 1: Scheme of the domains of the problem.

The crack propagation states the minimization of the total potential energy stored ψ :

d e s ψ = ψ + ψ - ψ

(1)

where ψ d is the critical energy release during the fracture process, ψ e is the elastic energy, and ψ S are the external sources of energy due to body and surface loads. The critical energy release during the fracture propagation equals:    F F d c c l Ω Ω ψ = G dS= G γ d Ω (2) where G c is the Griffith critical energy release rate for mode-I of fracture, and γ l is the crack surface density per unit volume of the solid, given by Miehe et al. [48]:

2

    ,

l



2

0



γ

= + 2l

(3)

l

2

0

where l 0 is the length scale parameter, and ϕ is the phase-field which satisfies the following condition:

   0, if the material is intact 1, if the material is cracked

  

x,t =

(4)

Therefore, ψ d equals:

2

  2 l



d

0 + )d Ω

(5)

ψ = G (

c

2l

2

0

Ω

F

Borden et al. and Zhang et al. [65,66] proposed an analytical solution for computing the length scale parameter, l 0 , which depends on the resistance tensile strength, f t , the Young’s Modulus, E, and the Griffith critical energy release rate, G c , as follows:

c

0 t l = 27EG 256f 2

(6)

156