“Driving forces” and radiated fields for expanding/shrinking half-space and strip inclusions with general eigenstrain
Authors:
Xanthippi Markenscoff and Luqun Ni
Journal:
Quart. Appl. Math. 69 (2011), 529-548
MSC (2000):
Primary 74N20, 74H05, 74B99
DOI:
https://doi.org/10.1090/S0033-569X-2011-01224-4
Published electronically:
May 9, 2011
MathSciNet review:
2850744
Full-text PDF Free Access
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Abstract: A half-space constrained Eshelby inclusion (in an infinite elastic matrix) with general uniform eigenstrain (or transformation strain) is analyzed when the plane boundary is moving in general subsonic motion starting from rest. The radiated fields are calculated based on the Willis expression for constrained time-dependent inclusions, which involves the three-dimensional dynamic Green’s function in an infinite traction-free body, and they constitute the unique elastodynamic solution, with initial condition the Eshelby static fields obtained as the unique minimum energy solutions by a limiting process from the spherical inclusion. The mechanical energy-release rate and associated “driving force” to create dynamically an incremental region of eigenstrain (due to any physical process) is calculated for general uniform eigenstrain. For dilatational eigenstrain the solution coincides with the one obtained by a limiting process from a spherically expanding inclusion, while for shear eigenstrain the fields are due to the propagation of the rotation. The “driving force” has the same expression both for expanding and shrinking motions, resulting in expenditure of the energy rate for motion of the boundary in both cases. By superposition from the half-space inclusions, the fields and “driving force” for a strip inclusion with both boundaries moving are obtained. The “driving force” consists also of a contribution from the other boundary when it has time to arrive. The presence of applied loading contributes the counterpart of the Peach-Koehler force of dislocations, in addition to the self-force.
References
- Atkinson, C. and Eshelby, J. D., The flow of energy into the tip of a moving crack, Int. J. Fract. Mech., 4, 1968, pp. 3–8.
- Burridge, R. and Willis, J.R., The self-similar problem of the expanding crack in an anisotropic solid, Proc. Camb. Phil. Soc., 66, 1969, pp. 443–468.
- Chlieh, M., Avouac, J. P., Sieh, K., Natawidjaja, D.H., and Galetzka, J., Heterogeneous coupling of the Sumatran megathrust constrained by geodetic and paleogeodetic measurements, J. Geoph. Res., 113, 2008, BO5305
- Dundurs, J. and Markenscoff, X., Stress fields and Eshelby forces on half-plane inhomogeneities and strip inclusions meeting a free surface, Int. J. Sol. Str. 46, 2009, pp. 2481–2485.
- J. D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. London Ser. A 241 (1957), 376–396. MR 87326, DOI https://doi.org/10.1098/rspa.1957.0133
- J. D. Eshelby, Elastic inclusions and inhomogeneities, Progress in Solid Mechanics, Vol. II, North-Holland, Amsterdam, 1961, pp. 87–140. MR 0134510
- Eshelby, J.D., “Energy relation of the energy-momentum tensor in continuum mechanics” in Inelastic Behavior of Solids, eds. Kanninen, M.F. et al., 1970, pp. 77–115.
- Eshelby, J.D., The Mechanics of Defects and Inhomogeneities, Lecture on the elastic energy-momentum tensor, in Collected Works of J.D. Eshelby, pp. 907-931, Markenscoff, X. and Gupta, A., editors, Springer, Netherlands, 2006.
- Freund, L. B., Energy-flux into the tip of an extending crack in an elastic solid, J. Elast., 2, 1972, pp. 293–299.
- L. B. Freund, Dynamic fracture mechanics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Cambridge, 1990. MR 1054375
- Gavazza, S.D., Forces on pure inclusion and Somigliana dislocations, Scripta Metallurgica, 11, 1977, pp. 979–981.
- Green, H. W. II and Burnley, P. C., A new self-organizing mechanism for deep-focus earthquakes, Nature, 341, 1989, pp. 733–737
- Green, H. W. II, Shearing instabilities accompanying high-pressure phase transformations and the mechanics of deep earthquakes, Proceedings National Academy of Science, 104, 2007, pp. 9133–9138.
- Gupta, A. and Markenscoff, X. (submitted), A new interpretation for configurational forces
- R. Hill, Discontinuity relations in mechanics of solids, Progress in Solid Mechanics, Vol. II, North-Holland, Amsterdam, 1961, pp. 245–276. MR 0128115
- Houston, H. and Williams, Q.R., Fast-Rise Times and the physical Mechanism of deep Earthquakes, Nature, 352, 1991, pp. 520–522.
- Kanamori, Hiroo., Earthquake-physics, and real-time seismology, Nature, 451, doi:10.1038/nature 06585 (January,2008)
- Mitio Nagumo, Über das Verhalten der Integrale von $\lambda y”+f(x,y,y’,\lambda )=0$ für $\lambda \to 0$, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 529–534 (German). MR 1085
- Xanthippi Markenscoff, On the shape of the Eshelby inclusions, J. Elasticity 49 (1997/98), no. 2, 163–166. MR 1635797, DOI https://doi.org/10.1023/A%3A1007474108433
- Xanthippi Markenscoff and Luqun Ni, The energy-release rate and “self-force” of dynamically expanding spherical and plane inclusion boundaries with dilatational eigenstrain, J. Mech. Phys. Solids 58 (2010), no. 1, 1–11. MR 2576253, DOI https://doi.org/10.1016/j.jmps.2009.10.001
- Gérard A. Maugin, Material inhomogeneities in elasticity, Applied Mathematics and Mathematical Computation, vol. 3, Chapman & Hall, London, 1993. MR 1250832
- Mendlguren, J.A. and Aki, K., Seismic Mechanism of a deep Columbian earthquake, J. Geoph. Res., 55, 1978, pp. 539–556.
- Mura, T., Micromechanics of Defects in Solids, second edition, Martinus Nijhoff Publishers, The Hague, 1982.
- Luqun Ni and Xanthippi Markenscoff, The self-force and effective mass of a generally accelerating dislocation. I. Screw dislocation, J. Mech. Phys. Solids 56 (2008), no. 4, 1348–1379. MR 2404017, DOI https://doi.org/10.1016/j.jmps.2007.09.002
- Randall, M.J. and Knopoff, L., The Mechanism at the Focus of Deep Earthquakes, J. Geoph. Res., 75, 1970, pp. 4965–4976.
- Rice, J.R., A path-independent integral and the approximate analysis of strain concentrations by notches and cracks, ASME J. Appl. Mech., 35, 1968, pp. 379-388.
- Scholtz, C. H., The Mechanics of Earthquakes and Faulting, Second Edition, Cambridge University Press, 2002, pp. 330–331.
- Eli Sternberg, On the integration of the equations of motion in the classical theory of elasticity, Arch. Rational Mech. Anal. 6 (1960), 34–50 (1960). MR 119547, DOI https://doi.org/10.1007/BF00276152
- Stolz, C., Energetical Approaches in Nonlinear Mechanics, Lecture Notes 11, Center of Excellence for Advanced Materials and Structures, Warsaw, Poland, 2003.
- Truskinovsky, L., Equilibrium phase interfaces, Sov. Phys. Dokl., 27, 1982, pp. 551–553.
- Wiens, D. A. and Snider, N. O., Reactivation at Great Depth Repeating Deep Earthquakes: Evidence for Fault Reactivation at Great Depth, Science, 293, 2001, pp. 1463–1466.
- Willis, J.R., Dislocations and Inclusions, J. Mech. Phys. Solids, 13, 1965, pp. 377–395.
- S.-Y. Yang, J. Escobar, and R. J. Clifton, Computational modeling of stress-wave-induced martensitic phase transformations in NiTi, Math. Mech. Solids 14 (2009), no. 1-2, 220–257. MR 2485360, DOI https://doi.org/10.1177/1081286508092613
References
- Atkinson, C. and Eshelby, J. D., The flow of energy into the tip of a moving crack, Int. J. Fract. Mech., 4, 1968, pp. 3–8.
- Burridge, R. and Willis, J.R., The self-similar problem of the expanding crack in an anisotropic solid, Proc. Camb. Phil. Soc., 66, 1969, pp. 443–468.
- Chlieh, M., Avouac, J. P., Sieh, K., Natawidjaja, D.H., and Galetzka, J., Heterogeneous coupling of the Sumatran megathrust constrained by geodetic and paleogeodetic measurements, J. Geoph. Res., 113, 2008, BO5305
- Dundurs, J. and Markenscoff, X., Stress fields and Eshelby forces on half-plane inhomogeneities and strip inclusions meeting a free surface, Int. J. Sol. Str. 46, 2009, pp. 2481–2485.
- Eshelby, J.D., The determination of the elastic field of an ellipsoidal inclusion and related problems, Proc. Roy. Soc. London, A241, 1957, pp. 376–396. MR 0087326 (19:338d)
- Eshelby, J.D., Elastic Inclusions and Inhomogeneities, Prog. Solid. Mech., 2, 1961, pp. 89–140. MR 0134510 (24:B563)
- Eshelby, J.D., “Energy relation of the energy-momentum tensor in continuum mechanics” in Inelastic Behavior of Solids, eds. Kanninen, M.F. et al., 1970, pp. 77–115.
- Eshelby, J.D., The Mechanics of Defects and Inhomogeneities, Lecture on the elastic energy-momentum tensor, in Collected Works of J.D. Eshelby, pp. 907-931, Markenscoff, X. and Gupta, A., editors, Springer, Netherlands, 2006.
- Freund, L. B., Energy-flux into the tip of an extending crack in an elastic solid, J. Elast., 2, 1972, pp. 293–299.
- Freund, L. B., Dynamic Fracture Mechanics, Cambridge University Press, Cambridge, 1990. MR 1054375 (92f:73044)
- Gavazza, S.D., Forces on pure inclusion and Somigliana dislocations, Scripta Metallurgica, 11, 1977, pp. 979–981.
- Green, H. W. II and Burnley, P. C., A new self-organizing mechanism for deep-focus earthquakes, Nature, 341, 1989, pp. 733–737
- Green, H. W. II, Shearing instabilities accompanying high-pressure phase transformations and the mechanics of deep earthquakes, Proceedings National Academy of Science, 104, 2007, pp. 9133–9138.
- Gupta, A. and Markenscoff, X. (submitted), A new interpretation for configurational forces
- Hill, R., Discontinuity relations in mechanics of solids, in Progress in Solid Mechanics, I. N. Sneddon and R. Hill, eds., North-Holland, Amsterdam, 1961. MR 0128115 (23:B1159)
- Houston, H. and Williams, Q.R., Fast-Rise Times and the physical Mechanism of deep Earthquakes, Nature, 352, 1991, pp. 520–522.
- Kanamori, Hiroo., Earthquake-physics, and real-time seismology, Nature, 451, doi:10.1038/nature 06585 (January,2008)
- Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. MR 001085 (6:79e)
- Markenscoff, X., On the Shape of the Eshelby Inclusions, J. Elasticity, 49, 1998, pp. 163–166. MR 1635797 (99d:73021)
- Markenscoff, X. and Ni, L., The energy-release rate and “self-force” of dynamically expanding spherical and plane inclusion boundaries with dilatational eigenstrain, J. Mech. Phys. Sol., 58, 2010, pp. 1–11. MR 2576253 (2001b:74016)
- Maugin, G. A., Material Inhomogeneities in Elasticity, Applied Mathematics and Mathematical Computation, Vol. 3, Chapman and Hall, London. 1993. MR 1250832 (95c:73001)
- Mendlguren, J.A. and Aki, K., Seismic Mechanism of a deep Columbian earthquake, J. Geoph. Res., 55, 1978, pp. 539–556.
- Mura, T., Micromechanics of Defects in Solids, second edition, Martinus Nijhoff Publishers, The Hague, 1982.
- Ni, L. and Markenscoff, X., The self-force and effective mass of an accelerating dislocation. I. Screw Dislocation, J. Mech. Phys. Solids, 56, 2008, pp. 1348–1379. MR 2404017 (2009f:74009)
- Randall, M.J. and Knopoff, L., The Mechanism at the Focus of Deep Earthquakes, J. Geoph. Res., 75, 1970, pp. 4965–4976.
- Rice, J.R., A path-independent integral and the approximate analysis of strain concentrations by notches and cracks, ASME J. Appl. Mech., 35, 1968, pp. 379-388.
- Scholtz, C. H., The Mechanics of Earthquakes and Faulting, Second Edition, Cambridge University Press, 2002, pp. 330–331.
- Sternberg, E., On the integration of the equations of motion in the classical theory of elasticity, Arch. Rat. Mech. Anal., 6, 1960, pp. 34–50. MR 0119547 (22:10308)
- Stolz, C., Energetical Approaches in Nonlinear Mechanics, Lecture Notes 11, Center of Excellence for Advanced Materials and Structures, Warsaw, Poland, 2003.
- Truskinovsky, L., Equilibrium phase interfaces, Sov. Phys. Dokl., 27, 1982, pp. 551–553.
- Wiens, D. A. and Snider, N. O., Reactivation at Great Depth Repeating Deep Earthquakes: Evidence for Fault Reactivation at Great Depth, Science, 293, 2001, pp. 1463–1466.
- Willis, J.R., Dislocations and Inclusions, J. Mech. Phys. Solids, 13, 1965, pp. 377–395.
- Yang, S.-Y., Escobar, J. and Clifton, R.J., Computational modeling of stress-wave-induced martensitic phase transformations in NiTi, Math. Mech. Solids, 14, 2009, pp. 220–257. MR 2485360
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Additional Information
Xanthippi Markenscoff
Affiliation:
Department of Mechanical and Aerospace Engineering, Mail code 0411, University of California, San Diego, La Jolla, California 92093
Luqun Ni
Affiliation:
Department of Mechanical and Aerospace Engineering, Mail code 0411, University of California, San Diego, La Jolla, California 92093
Received by editor(s):
February 1, 2010
Published electronically:
May 9, 2011
Additional Notes:
Partial support by NSF (grant # CMS 0555280) is gratefully acknowledged. The first author wishes to thank Professors Rodney Clifton and Lev Truskinovsky for suggesting the problem of the moving plane boundary and initial comments.
Article copyright:
© Copyright 2011
Brown University
