Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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``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
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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.


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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

DOI: https://doi.org/10.1090/S0033-569X-2011-01224-4
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


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