Physical Mesomechanics

, Volume 22, Issue 1, pp 42–45 | Cite as

Cracks as Limits of Eshelby Inclusions

  • X. MarkenscoffEmail author


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.


Eshelby inclusions transformation strain cracks Barenblatt crack elasticity 


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  1. 1.
    Mura, T., Micromechanics of Defects in Solids, Dordrecht: Martinus Nijhoff Publishers, 1982.CrossRefGoogle Scholar
  2. 2.
    Eshelby, J.D., The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems, Proc. Roy. Soc. Lond. A, 1957, vol. 241, pp. 376–396.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Markenscoff, X., Self–Similarly Expanding Regions of Phase Change Give Cavitational Instabilities and Model Deep Earthquakes, 2018 (submitted).Google Scholar
  4. 4.
    Burridge, R. and Willis, J.R., The Self–Similar Problem of the Expanding Crack in an Anisotropic Solid, Proc. Camb. Philos. Soc., 1969, vol. 66, pp. 443–468.ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Ni, L. and Markenscoff, X., The Self–Similarly Expanding Eshelby Ellipsoidal Inclusion. I. Field Solution, J. Mech. Phys. Sol, 2016, vol. 96, pp. 683–695.ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Ni, L. and Markenscoff, X., The Self–Similarly Expanding Eshelby Ellipsoidal Inclusion. II. The Dynamic Eshelby Tensor for the Expanding Sphere, J. Mech. Phys. Sol, 2016, vol. 96, pp. 696–714.ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ni, L. and Markenscoff, X., The Dynamic Generalization of the Eshelby Inclusion Problem and Its Static Limit, Proc. Roy. Soc. Lond. A, 2016, vol. 472, pp. 256–270. doi 10.1098/rspa.2016.0256MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kaya, A.C. and Erdogan, F., On the Solution of Integral Equations with Strongly Singular Kernels, Quart. Appl. Math, 1987, vol. 45, pp. 105–122.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dundurs, J. and Markenscoff, X., Stress Fields and Eshelby Forces on Half–Plane Inhomogeneities with Eigenstrain and Strip Inclusions Meeting a Free Surface, Int. J. Sol. Struct, 2009, vol. 46, pp. 2481–2485.CrossRefzbMATHGoogle Scholar
  10. 10.
    Barenblatt, G.I., The Mathematical Theory of Equilibrium Cracks in Brittle Fracture, Adv. Appl. Mech., 1962, vol. 7, pp. 55–129.MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaSan Diego, La JollaUSA

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