Perturbed rigid body motions of an elastic rectangle
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Perturbed rigid body motions of an elastic rectangle Julius Kaplunov and Onur S ¸ ahin
Abstract. Plane and anti-plane dynamic problems for an elastic rectangle loaded along its sides are considered. Low-frequency perturbations to rigid body translations are calculated. The derivation involves the solutions of non-homogeneous boundary value problems for harmonic and bi-harmonic equations. The explicit solution for the harmonic problem for transverse antiplane translation is expressed through Fourier series. The bi-harmonic problem corresponding to the longitudinal in-plane translation is studied in greater detail for an elongated rectangle, which also may be treated using the elementary theory for plate extension. The derived perturbations incorporate the variations of displacements and stresses over the interior of the rectangle, including the case of self-equilibrated loading. The latter is obviously outside the range of validity of the classical rigid body framework. Mathematics Subject Classification. 74B05, 74H45, 74G10. Keywords. Low-frequency, Perturbation, Rigid body motion, Elastic rectangle.
1. Introduction Newton’s second law for a material point is also valid for calculating the accelerations at each point of a solid of a finite size provided the solid performs a rigid body motion, i.e. its deformation is negligible, e.g. [1]. This assumption usually appears to be pretty natural at sufficiently low vibration frequencies but inevitably fails at higher frequencies when internal structure and related resonance phenomena have to be taken into consideration. In the latter case, Newton’s law can be generalized by introducing a frequency-dependent anisotropic mass density [2]. The concept of a negative mass appears to be fruitful for theoretical and experimental investigations of metamaterials, e.g. see [3–7]. At the same time, even over a low-frequency range well below a resonance domain, corrections to rigid body dynamics seem to be of interest. The point is that the classical set-up excludes the possibility of incorporating the effects of self-equilibrated loading, internal dissipation, inhomogeneity and deformability. Evaluation of all of them is inspired by modern engineering applications, including railway transport. In particular, the role of self-equilibrated longitudinal forces may be essential for modelling of high-speed impacts of freight cars, especially those subject to substantial compression, e.g. see [8–10]. Low-frequency perturbations to rigid body rotation and translations are deduced in [11] for dimensional bending and extension of an inhomogeneous viscoelastic bar. For a general linear viscoelastic kernel, the sought for corrections to the convention equations in rigid body dynamics is expressed through integral-differential operators acting on prescribed end longitudinal and transverse forces and bending moments. The derived equations allow calculating the variations of displacements along the bar subject to self-equilibrated loading. We
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