PSI - Issue 18

Guido Borino et al. / Procedia Structural Integrity 18 (2019) 866–874 G. Borino, F. Parrinello / Structural Integrity Procedia 00 (2019) 000–000

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4

and constant Ω ∞ is the value of the integral given in eq.(4) when the integration is performed on the unbounded 2D domain. The weighting function W , defined in eq.(3), is symmetric with respect to x , y at any point in V and satisfy the following normality condition: ∫ V W ( x , y ) dV ( y ) = 1 . (5) Beside the constitutive relation (1) a nonlocal damage activation function is defined as

ϕ d ( ¯ Y , χ ) = ¯ Y − χ − Y 0 ≤ 0

(6)

where

¯ Y ( x ) ∫

W ( x , y ) Y ( y ) dV ( y )

(7)

V

where Y is the energy release rate, χ is an internal variable able to characterize the post peak stress-strain softening material response and Y 0 is the initial damage activation threshold.

1 2

ε T E ε

(8)

Y =

Following a damage softening law proposed by Comi and Perego (2004), the internal variable χ is related to the damage ω by the following state law χ = κ ln n ( c 1 − ω ) − κ ln n c (9) where κ, n , c are parameters that describes the post elastic stress-strain response. With reference to the uni-axial re sponse it is possible to identify the constant by the following relations (Comi and Perego (2004)) • Damage threshold: Y 0 ≡ 1 / 2 E ε 2 e = κ ln n c • Fracture Energy G f = Y 0 + cn κ exp[ − ( Y 0 /κ ) 1 / n ]( n − 1)! ∑ (1 / i !)( Y 0 /κ ) i / n The constant c ≥ 1 is related to the stress-strain slope at the final elastic strain ε e , and for c = e n / 2 an initial horizontal slope is achieved followed by negative slope, i.e. softening. The damage flow rules and the loading / unloading conditions complete the nonlocal damage constitutive frame work:

∂ϕ d ∂ ¯ Y

∂ϕ d ∂χ

˙ λ

˙ λ = −

˙ ω =

(10)

˙ λϕ d = 0 .

˙ λ ≥ 0 ,

ϕ d ≤ 0 ,

(11)