Cracks as Limits of Eshelby Inclusions

As limiting behaviors of Eshelby ellipsoidal inclusions with transformation strain, crack solutions can be obtained both in statics and dynamics (for self-similarly expanding ones). Here is presented the detailed analysis of the static tension and shear cracks, as distributions of vertical centers of eigenstrains and centers of antisymmetric shear, respectively, inside the ellipse being flattened to a crack, so that the singular external field is obtained by the analysis, while the interior is zero. It is shown that a distribution of eigenstrains that produces a symmetric center of shear cannot produce a crack. A possible model for a Barenblatt type crack is proposed by the superposition of two elliptical inclusions by adjusting their small axis and strengths of eigenstrains so that the singularity cancels at the tip.


INTRODUCTION
Mura [1] has presented a method for solving crack problems as Eshleby [2] ellipsoidal inclusion problems. Based on the constant stress Eshelby property Mura cancels the applied tractions on the faces of an ellipsoidal inclusion as the vertical axis of the ellipsoid tends to zero, so that the inclusion is flattened to a crack. The limit of the product of the eigenstrain times the vertical axis length, as the eigenstrain tends to infinity and the axes length tends to zero, is a finite quantity, the crack opening displacement.
Here is provided a complete analysis for the tension and shear Griffith cracks based on distributing centers of eigenstrain. In addition to the eigenstrains considered by Mura, are included eigenstrains so that all stresses vanish in the interior of the crack. The external field is obtained analytically, while Mura [1] only obtains the singularity based on the energy-release rate known independently. We show that a crack cannot be produced as a symmetric center of shear with eigenstrains * * , xx yy ε = −ε because the internal stresses cannot be cancelled by an applied stress field in this case.
Analogously to the static, Markenscoff [3] has shown that the self-similarly expanding elliptical crack Burridge and Willis [4] can be obtained from the limit of the self-similarly expanding ellipsoidal inclusion with transformation strain [57], since the constant stress Eshelby property is valid also in self-similarly expanding Eshelby inclusions. It may be noted that the dynamic Eshelby fields in respective limits give both the static Eshelby inclusion (and hence the static cracks) and also Rayleigh waves as the axis expansion speed of the elliptically expanding crack tends to the Rayleigh wave speed.

THE .LATTENED ELLIPTICAL CYLINDER WITH TRANS.ORMATION STRAIN AS A CRACK
We consider that the crack will be a distribution of eigenstrains inside a flattened elliptical cylinder. The interior stresses of a flattened elliptical cylinder 3 (a → , ∞ 2 1 ) a a = η with transformation strains * ij ε are given by (also, [1]) In order for , xx yy ε = −ηε Such eigenstrains were not considered by Mura, but they need to be considered to obtain all stress and displacement fields correctly. We give below the fields for a vertical center of eigenstrain * ,  (2 )), U r q r θ = −µ π κ + + θ from which we obtain where κ denotes the Kolosov constant and the integral in the sense of principal value was evaluated according to Kaya and Erdogan [8]. Requiring that on 1 , x a < ( , 0) , yy x T σ =− in the limit as 2 0 a → and e → ∞, we set  ( 1 )) , a q a T µ κ+ = superpose yy T σ = and obtain the total stresses  ( 1)) U r q θ = µ θ π κ + we have the fields x y q y y u x y r r q x x u x y r r and the displacements where we set , a a q S µ κ + = and we superpose xy S σ = to obtain the total stresses for the Griffith shear crack. We may note that ( , 0) y u x x = for 1 x a < means that the shear crack rotates.
.or completeness we present the fields of a horizontal center of eigenstrain and of the center of dilatation. Horizontal center of expansion: