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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4
where s c is the curvilinear abscissa along the c − th edge of ∂ Ω , edge having lenght l c . Defining λ c = s c / l c , we can parametrize the edge by setting ρ ( λ c ) = ρ c + λ c ρ c + 1 − ρ c = ρ c + λ c ∆ ρ c where ρ c and ρ c + 1 are the position vectors of the end vertices of ∂ Ω , assumed to be numbered consecutively by adopting a counter-clockwise sense along ∂ Ω . Thus, as shown in Trotta et al. (2017a), the displacement field in (3) assumes the following explicit expression u i ( p ) = A ( p ) ikl C klmn ε ∗ mn (7)
where the third-order tensor A is given by A ( p ) ikl = g N c = 1 I c α ρ c ⊗ ρ c + I c
δ I
ik
∆ ρ ⊥ c l
β ( ρ c ⊗ ∆ ρ c + ∆ ρ c ⊗ ρ c ) + I c
γ ∆ ρ c ⊗ ∆ ρ c + hI c
(8)
where ∆ ρ ⊥ c = ( − ∆ ρ c 2 , ∆ ρ c 1 ) if ∆ ρ c = ( ∆ ρ c 1 , ∆ ρ c 2 ) and the scalars I c α , I c β , I c γ and I c quantities p c = ∆ ρ c · ∆ ρ c , q c = ρ c · ∆ ρ c and u c = ρ c · ρ c by means of the following expressions I c α = 1 p c u c − q 2 c arctan p c + q c p c u c − q 2 c − arctan q c p c u c − q 2 c ; I c β = 1 2 p c log p c + 2 q c + u c u c − 2 q c I c α (9) I c γ = 1 p c − u c − 2 q 2 c p c I c α − q c p 2 c log p c + 2 q c + u c u c (10) δ are defined as function of the
α + 1 +
q c p c
q 2 c p c
q c p c
) I c
I c δ = − 2 + 2( u c −
(11)
log( p c + 2 q c + u c ) −
log u c
that corrects some misprints in Trotta et al. (2017a). Furthermore, setting R c ρ = ρ c ⊗ ∆ ρ ⊥ c , R c
∆ ρ = ∆ ρ c ⊗ ∆ ρ ⊥ c , R c
ρρρ = ρ c ⊗ ρ c ⊗ ρ c ⊗ ∆ ρ ⊥ c ; R c
∆ ρ ∆ ρ ∆ ρ = ∆ ρ c ⊗ ∆ ρ c ⊗ ∆ ρ c ⊗ ∆ ρ ⊥ c ,
R c ρρ ∆ ρ = ρ c ⊗ ρ c ⊗ ∆ ρ c ⊗ ∆ ρ ⊥ c + ρ c ⊗ ∆ ρ c ⊗ ρ c ⊗ ∆ ρ ⊥ c ∆ ρ c ⊗ ρ c ⊗ ρ c ⊗ ∆ ρ ⊥ c , R c ρ ∆ ρ ∆ ρ = ρ c ⊗ ∆ ρ c ⊗ ∆ ρ c ⊗ ∆ ρ ⊥ c + ∆ ρ c ⊗ ρ c ⊗ ∆ ρ c ⊗ ∆ ρ ⊥ c + ∆ ρ c ⊗ ∆ ρ c ⊗ ρ c ⊗ ∆ ρ ⊥ c and considering the scalars I 01 , I 11 , I 02 , I 12 , I 22 and I 32 introduced in Trotta et al. (2016), one can define
N l c = 1
N c = 1
∆ ρ I 11 ; Θ =
∆ ρ ∆ ρ ∆ ρ I 32
R c
ρ I 01 + R c
R c
ρρρ I 02 + R c
ρρ ∆ ρ I 12 + R c
ρ ∆ ρ ∆ ρ I 22 + R c
(12)
ι =
Hence, the Eshelby tensor can be expressed as follows
g 2 2 λι ll δ i j δ mn + 2 µ ( ι nm + ι mn ) δ i j + ( λ + 2 h λ )( ι i j + ι ji ) δ mn + + ( µ + 2 h µ )( ι in δ jm + ι jn δ im + ι im δ jn + ι jm δ in ) + − 4 λ Θ i jll δ mn − 4 µ ( Θ i jmn + Θ i jnm )
S i jmn ( P ) = −
(13)
The previous results have been obtained by by exploiting recent results for the Newtonian potential, D’Urso (2012, 2013), and subsequently applied to several problems ranging from geodesy, D’Urso (2014a,b, 2016); D’Urso and Trotta (2015), to geomechanics, Sessa and D’Urso (2013); D’Urso and Marmo (2013, 2015); D’Urso (2016); Marmo and Rosati (2016); Marmo et al. (2016a,b, 2017a,b), to geophysics, D’Urso (2015); D’Urso and Trotta (2017), and to heat transfer, Rosati and Marmo (2014).
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