Issue 61

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

In Eqn. (5), the critical energy release can be calculated through [67]:

 

2

  2 2 1c K 1- ν , K , 1c

plane strain

(7)

G = E

  

c

plane stress

E

where K 1c is the fracture toughness and υ is the Poisson’s ratio. The strain tensor is decomposed into two parts [68]:

 δ

 a=1 ε = ε n n ± a± a

(8)

a

where ε + and ε - are the tensile and compressive strain tensors, respectively, ε a is the a th principal strain, and n a is the principal direction of strain tensor ε a . The elastic energies are expressed as:

  ± ε ± λ ψ ε = tr( ε ) + μ tr( ε ) 2 2 2 ±

(9)

where λ and μ are Lamé constants, given by:

ν E

E

 

λ =

, μ =

plane strain

(1+ ν )(1-2 ν )

2(1+ ν ) E(1+2 ν )

(10)

  

ν E

λ =

, μ =

plane stress

(1+ ν )(1- ν )

3(1+ ν )(1- ν )

The strain tensor is related to the displacement field u by:

  T 1 ε = ( u+ u ) 2

(11)

We adopt the anisotropic formulation proposed by Miehe et al. [68], which states that fractures only propagate under tension, i.e., due to the positive part of the elastic energy. The total elastic energy is then expressed as:          ± + - ε ε ε ψ ε =g ψ ε + ψ ε (12) where g( ϕ ) = (1-k)(1- ϕ 2 ) + k is the degradation function, and k (0 < k << 1) is a parameter that avoids numerical singularities. The external energy functional, ψ S accounts for the body forces and the applied loads on the boundaries as follows:





s



(13)

 Ω ψ = f.u d Ω + t.u d E Ω

The fluid pressure inside the fracture, p f , exerts a force on the surface of the fracture. We include this load in the term of traction vector force as follows:







(14)

 t.u d = t.u d - 



p n.u d

f







Ω

Ω

N F Ω

E

N

We develop the second term in Eqn. (14) using the Divergence Theorem as follows:

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