PSI - Issue 13

Zhengkun Liu et al. / Procedia Structural Integrity 13 (2018) 781–786 Z. Liu et al. / Structural Integrity Procedia 00 (2018) 000–000

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2

2. Brittle fracture at finite strains

2.1. Finite viscoelastic material models

A viscoelastic material model is studied, namely, the model proposed by Zienkiewicz et al. (2014). The decom position of the elastic energy density ψ e based on volumetric ψ vol e and deviatoric ψ dev e contributions is formulated as

vol e + ψ

dev e ,

(1)

ψ e = ψ

and the second Piola-Kirchho ff stress S is then expressed as S = S vol + S dev ,

(2)

where

∂ψ vol e

S vol = 2

∂ C ,

(3)

∂ψ dev e

S dev = 2

∂ C ,

here C is the right Cauchy-Green deformation tensor. Eq. 2 represents a split of the second Piola-Kirchho ff stress S into volumetric part S vol and deviatoric part S dev in the material configuration. The finite viscoelasticity is defined by S dev = μ 0 S dev + N i = 1 μ i Q ( i ) , (4) where N i = 0 μ i = 1 with μ i > 0 , (5) and the partial stress Q is obtained from the di ff erential equation ˙ Q ( n ) + 1 λ n Q ( n ) = ˙ S dev . (6) Here λ n is treated as a relaxation time. The phase-field model is based on the variational formulation of brittle fracture by Miehe et al. (2015). The global energy storage functional consists of the stored elastic strain energy and the energy release due to fracture. The energy density function ψ is given as ψ = ( s 2 + η )( ψ e − ψ c ) + ψ c + ψ f rac , (7) where ψ f rac = ψ c (1 − s ) 2 + 4 κ 2 |∇ ( X ) s | 2 with G c = 4 κ ψ c , (8) in which ψ e is the elastic energy density function. ψ c is a specific critical fracture energy per unit volume. G c is the fracture toughness. The fracture energy density ψ f rac is defined through the time-dependent crack phase-field s , which varies smoothly from 1 (intact material) to 0 (fully cracked material). The sti ff ness resistance η is used to ensure a numerically well-conditioned system for the totally broken phase ( s = 0). The crack width is characterized by the di ff use length scale κ . In order to ensure crack propagation under tensile or shear loading, The elastic energy density ψ e splits into the positive part ψ + e and the negative part ψ − e , namely ψ e = ψ + e + ψ − e , (9) 2.2. Viscoelasticity coupled with a phase-field approach for brittle fracture