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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.

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