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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3. Elastic fields in an infinite plane containing an elliptical cavity

Let us now apply the general formulation derived in the previous section to solve a classical problem in elasticity, namely stress concentration around an elliptical hole in a linear, isotropic infinite plane subjected to a remote loading. This problem, originally solved by Inglis (1913), has been further addressed by Maugis (1992) who applied the classical solution by Muskhelishvili (1953), based on the use of conformal mapping and complex potentials, to evalu ate also the displacement field. The solution by Maugis (2000), also reported in his book, has been recently rederived by Jin et al. (2014) by an ingenious and elegant application of the equivalent inclusion method. The essence of this method is to decompose the solution to an inhomogeneity problem as the sum of two auxiliary solutions, namely the one associated with a homogeneous material subjected to the applied loads, as the inhomogene ity were absent, and the corresponding inclusion solution: the disturbance of the applied stress caused by the presence of the inhomogeneity is equivalently simulated by the eigenstress field due to a corresponding inclusion. Thus, the stress within the inclusion is the one σ ◦ induced by remote loadings in the fictitiously homogeneous material plus the one associated with the elastic strain ε − ε = S ε − ε induced by the, still unknown, eigenstrain ε : σ ◦ i j + C i jkl ε − ε = C i jkl ε ◦ kl + S klmn ε mn − ε kl (14) Since the total stress within the cavity is trivially zero, the eigenstrain ε can be evaluated by equating to zero the previous expression, that is ε = − ( S − I ) − 1 C − 1 σ ◦ = − ( S − I ) − 1 ε ◦ (15)

In the case of uniform loading σ ◦ applied along the x 1 and x 2 axes, the strain tensor is spherical and its components are ε ◦ 11 = ε ◦ 22 = (1 − 2 ν ) σ ◦ / 2 µ since the elastic compliance tensor C − 1 in plane strain is:

1 2 µ   

0 2   

1 − ν 1 − ν 0 1 − ν 1 − ν 0 0

[ C ] − 1

(16)

V =

if the Voigt representation for stress [ σ ] V = [ σ 11 , σ 22 , σ 12 ] and strains [ ε ] V = [ ε 11 , ε 22 , 2 ε 12 ] is adopted. For an elliptical cavity the eigenstrain ε has been analytically evaluated by Jin et al. (2011) as

1 − ν µ

1 − ν µ

b a

a b

σ ◦ ;

σ ◦

ε

ε

(17)

22 =

11 =

Furthermore, setting ω = x 2

2 + b 2 , the stress field has been given by

1 − a

x 1 ω

2 − b 2

x 1 ω

2 + b 2

σ 11 = σ ◦

σ 22 = σ ◦

;

(18)

ω 3 / 2

ω 3 / 2

while the displacement field was reported explicitly as a sum of several terms. All the previous results have been obtained by rather involved computations, reported in Jin et al. (2011, 2014, 2016, 2017), involving potential functions holding separately inside and outside the inclusions. On the contrary we show that the formulas illustrated in the previous section allow one to obtain the same results more concisely and, above all, by means of a procedure that can be applied to arbitrary inclusions, though derivable from the elliptical one, having a general orientation in the plane. The necessity of addressing elliptical-like inclusions is required by the fact that the eigenstrain is rigorously constant only for this kind of inclusions.