Acoustic Methods of Evaluating Elastic Properties or, Will the Real Young’s Modulus Please Stand Up?
Measurement of the elastic moduli using ultrasound has become fairly routine. One measures a speed of sound, c, the density, ρ, and calculates a modulus, E, from a formula of the type c 2=E/ρ. Now, E describes a static response whereas c describes a dynamic response of the material. The connection between the two, E= ρc2, is strictly valid for an ideally elastic and homogeneous material; it remains valid for heterogeneous materials, e.g. composites, so long as a key assumption is satisfied, namely, the wavelength is large compared to any characteristic length of the material. In metals, a characteristic length is the grain size and generally the wavelength is large compared to the grain size. By their very definition, composites are heterogeneous materials and have one or more characteristic lengths. The objective of this paper is to demonstrate that when the wavelength becomes of the order of the characteristic length of a composite, large errors may occur and the dynamic measurement of E may differ from its true static value by as much as 200 percent; hence the subtitle: ‘Will the real Young’s modulus please stand up?’
KeywordsPhase Velocity Group Velocity Characteristic Length Particulate Composite Heterogeneous Material
List of Symbols
Young’s modulus, GPa
velocity of sound in water, mm/μs
inclusion radius, mm
longitudinal wavespeed, mm/μs
shear wavespeed, mm/μs
a characteristic length, mm
travel time, s
specimen thickness, mm
normalized frequency, k 1 a
aggregate property of composite
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