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The effect of long-range forces on the dynamics of a bar. (English) Zbl 1122.74431
Summary: The one-dimensional dynamic response of an infinite bar composed of a linear “microelastic material” is examined. The principal physical characteristic of this constitutive model is that it accounts for the effects of long-range forces. The general theory that describes our setting, including the accompanying equation of motion, was developed independently by I. A. Kunin [Elastic Media with Microstructure I: One-dimensional models. Springer Series in Solid-State Sciences, Vol. 26. Berlin etc.: Springer-Verlag (1982; Zbl 0527.73002), D. Rogula (ed.) [Nonlocal Theory of Material Media. International Centre for Mechanical Sciences. Courses and Lectures, No. 268. Wien: Springer-Verlag (1982; Zbl 0494.00013)], and S. A. Silling [J. Mech. Phys. Solids 48, No. 1, 175–209 (2000; Zbl 0970.74030)], and is called the peridynamic theory. The general initial-value problem is solved and the motion is found to be dispersive as a consequence of the long-range forces. The result converges, in the limit of short-range forces, to the classical result for a linearly elastic medium. Explicit solutions in elementary form are given in a broad class of special cases. The most striking observations arise in the Riemann-like problem corresponding to a constant initial displacement field and a piecewise constant initial velocity field. Even though, initially, the displacement field is continuous, it involves a jump discontinuity for all later times, the Lagrangian location of which remains stationary. For some materials the magnitude of the discontinuity-jump oscillates about an average value, while for others it grows monotonically, presumably fracturing the material when it exceeds some critical level.

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74M25 Micromechanics of solids
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