PSI - Issue 6

Giulio Zuccaro et al. / Procedia Structural Integrity 6 (2017) 236–243 Author name / Structural Integrity Procedia 00 (2017) 000–000

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The classical way to compute the Eshelby tensor is to evaluate the integrals appearing in its component expression either numerically, e.g. Huang et al. (2009); Zhou et al. (2009) and Huang et al. (2011), or analytically, e.g. Nozaki and Taya (1997, 2001); Rodin (1996); Kawashita and Nozaki (2001) and Zou et al. (2010), i.e.

1 2 Ω

d ρ C klmn

d ρ + Ω

∂ 2 G jk ∂ρ l ∂ρ i

∂ 2 G ik ∂ρ l ∂ρ j

S i jmn ( P ) =

(1)

where ρ = p − d is the 2D vector connecting P , whose position is defined by the vector p = ( p 1 , p 2 ), and the generic point d within the inclusion. The quantities G ik and G jk are the components of the rank-two Green tensor whose expression, for the present case of plane strain, is

log ( ρ · ρ ) I ⇐⇒ G ik ( ρ ) = g

+ h log ( ρ · ρ ) δ ik

1 8 πµ (1 − ν )

3 − 4 ν 2

ρ ⊗ ρ ρ · ρ −

ρ i ρ k ρ j j ρ j j

G ( ρ ) =

(2)

where µ is the shear modulus, ν is the Poisson’s ratio and I is the two-dimensional identity tensor. For isotropic linear elasticity the components C klmn in (1) of the elasticity tensor C are given by C klmn = λδ kl δ mn + µ ( δ km δ ln + δ kn δ lm ) where λ is the first Lame´ constant. Formula (1) is classically arrived at by di ff erentiating the displacement field with respect to p j , u i ( p ) = − Ω ∂ G ik ( p − d ) ∂ p l d d C klmn ε ∗ mn = Ω ∂ G ik ( ρ ) ∂ρ l d ρ C klmn ε ∗ mn (3) 2 ∂ u i /∂ p j + ∂ u j /∂ p i . Invoking the generalized version of Gauss theorem, see, e.g., D’Urso (2012), we can transform the domain integral into a boundary integral as follows Ω ∂ G ik ( ρ ) ∂ρ l d ρ = ∂ Ω G ik ρ ( s ) v l ρ ( s ) ds (4) where s is the curvilinear abscissa along the boundary ∂ Ω of the inclusion and ν l the l − th component of outward unit vector ν orthogonal to ∂ Ω . Recalling (2) the boundary integral above becomes ∂ Ω G ik ρ ( s ) v l ρ ( s ) ds = g ∂ Ω ρ ( s ) ⊗ ρ ( s ) ⊗ ν ( s ) ds ρ ( s ) · ρ ( s ) + h ∂ Ω log ρ ( s ) · ρ ( s ) I ⊗ ν ( s ) ds ikl (5) For an inclusion having a polygonal boundary ∂ Ω the two integrals on the right-hand side can be evaluated analyt ically as function of the position vectors d c that define the vertices of ∂ Ω , where c ranges from 1 to the number N of vertices defining ∂ Ω . Thus, according to (4) and (5), the displacement field in (3) can be expressed as explicit function of p being p − d c = ρ c and ρ c the relative position of the field point P with respect to the c − th vertex of Ω . The explicit expression of the displacement field and of the Eshelby tensor induced by a polygonal inclusion with a uniform eigenstrain has been derived in Trotta et al. (2016) and Trotta et al. (2017a); for the reader’s convenience they are both reported in the sequel. In particular, we parametrize each edge of the boundary ∂ Ω by setting ∂ Ω ρ ( s ) ⊗ ρ ( s ) ρ ( s ) · ρ ( s ) ds = l c 0 ρ ( s c ) ⊗ ρ ( s c ) ρ ( s c ) · ρ ( s c ) ds c (6) in order to derive the infinitesimal strain tensor according to the well-knowm formula ε i j = 1

N c = 1