Abstract
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.
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References
Mura, T., Micromechanics of Defects in Solids, Dordrecht: Martinus Nijhoff Publishers, 1982.
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.
Markenscoff, X., Self–Similarly Expanding Regions of Phase Change Give Cavitational Instabilities and Model Deep Earthquakes, 2018 (submitted).
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.
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.
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.
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.2016.0256
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.
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.
Barenblatt, G.I., The Mathematical Theory of Equilibrium Cracks in Brittle Fracture, Adv. Appl. Mech., 1962, vol. 7, pp. 55–129.
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Russian Text ©X. Markenscoff, 2018, published in Fizicheskaya Mezomekhanika’ 2018, Vol. 21, No. 6, pp. 45–48.
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Markenscoff, X. Cracks as Limits of Eshelby Inclusions. Phys Mesomech 22, 42–45 (2019). https://doi.org/10.1134/S1029959919010077
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DOI: https://doi.org/10.1134/S1029959919010077