Dynamic equivalence, self-equilibrated excitation and Saint-Venant’s principle for an elastic strip

Abstract

a b s t r a c tThe distinction between the near and far-fields for a semi-infinite, elastic strip has been exploited to derive conditions under which different dynamic excitations can be considered as equivalent. These different excitations are equivalent in the sense that they produce the same displacement field far from the excited end. It is shown that dynamically equivalent excitations degenerate to statically equivalent loads in the limit of a vanishing frequency. The no-radiation condition is derived, and its relation to self-equilibrated, dynamic and static loads is presented. Dynamically equivalent excitations are utilized to formulate a dynamic version of Saint-Venant's principle for symmetric excitations with frequencies below the first cut-off frequency of a strip. It has been shown that the requirement of self-equilibrium of a load for the decay of end effects for static fields can be deduced from the requirement of zero average power for the dynamic fields.

Study of dynamic end effects in an elastic strip with sliding boundary conditions

“…These loads are termed dynamically equivalent, and as such, illustrate the interpretation and the validity of DSVP suggested by Karp (2009). According to that interpretation, two dynamically equivalent loads will generate identical strain filed far from the loaded area.…”

“…That interpretation is not a simple extension of the classical SVP since in the static case the equivalence is judged upon static equivalents, while here, the equivalence is judged upon average power (intensity) of the excitation. Nevertheless, it was shown that both interpretations coincide in the limit of vanishing frequency (Karp, 2009). Since the dynamic response of the whole strip can not be identical for differently distributed loads, let us seek the differences in response of the strip subjected to any dynamically equivalent loads.…”

“…(64) exposes dynamic equivalence, according to which various forms of excitation can produce identical far field response (see Karp, 2009 for analogous derivation for a strip with free surfaces). Therefore, relying on measurements within the far field alone will not expose different forms of excitations.…”

“…In particular, small change in tightness level of each screw was easily detected by strain gauges located at the close vicinity to the end (near field) as compared to no change in reading of strain gauges located far from the end. Analytical studies of the effect of nonuniformity of end excitation are commonly limited to examination of the far-field response (see review by Karp, 2005, with recent results by Karp, 2008Karp, , 2009. Sensitivity of the near-field to the non-uniformity was investigated experimentally by Karp et al (2008Karp et al ( , 2009).…”

“…Emphasis is given to quantitative estimation of the phenomenon and to the possible usefulness of end effects as a structural signature for health monitoring of joints. The secondary objective is to illustrate the practical and theoretical aspects of dynamic Saint-Venant's principle (DSVP) as formulated in Karp (2009). These two aims are accomplished with the aid of analytical solution for the dynamic fields of an elastic, semi-infinite strip considered to be held in a plane strain condition with the lateral surfaces subjected to sliding conditions.…”