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The work considered only the first eight 110 acoustic

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The work considered only the first eight 110 Acoustic Emission Testing P ART 1. Wave Propagation in Plates

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branches of wave propagation, thus limiting the accuracy to a frequency range below 800 kHz. That analysis is extended here to arbitrarily buried and arbitrarily oriented strain nuclei sources, corresponding to brittle growth of small cracks. Orientable sources like cracks will give rise to nonaxisymmetric waves. In principle, the problem posed here is much harder than the original problem that considered the axisymmetric problem of a normally oriented point surface force. 1,2 However, because interest is restricted to responses written in terms of normal surface displacements, the nonaxisymmetric part remains irrelevant. This is formally shown below, where the stated problem of the surface normal displacement response to a buried strain nucleus source is transformed using reciprocity into the entirely equivalent problem of the calculation of the strain response at a buried location to a step force on the plate surface. Below, the mathematical problem is formally posed as a partial differential equation with initial conditions and boundary conditions for a vector field u that represents the elastodynamic displacement response to a buried strain nucleus source. The Green’s dyadic for the plate is defined as the displacement response (at an arbitrary position) to an arbitrary, buried, concentrated step force. The posed problem is then solved formally and exactly in terms of that Green’s dyadic. It transpires that it suffices to consider the Green’s dyadic for the normal surface step force. The Green’s dyadic is then expressed in terms of the normal modes of the plate, resulting in an expression for the desired response as a sum over branches b, each of which involves an integral with respect to wavenumber k. The integrand is given as a closed form transcendental function of the symmetry or antisymmetry of the branch b ; the wavenumber k ; the distance r from source to receiver ; the depth of the buried strain nucleus source; the time, the strength and the orientation of that source and the implicit (and tabulated) solution of the dispersion relation ω b ( k ). Problem Formulation It is desired to calculate the linear elastodynamic response to a step strain nucleus in an infinite plate with stress free surfaces. 10,11 The notion of a strain nucleus is equivalent to a set of double forces, that is, double couples without moment. Such nuclei are expected to describe any pointlike source smaller than a wavelength. Of interest is the vertical (outward) normal displacement response on one surface of the plate. The position of the receiver is taken to be at the coordinate origin ( r = 0 in polar coordinates) on the upper ( z = + h ) side of the plate of thickness 2 h . The coordinate system is shown in Fig. 1. The source S is located at position x S , with polar coordinates r S = | x S |, θ = 0, z = z S . The dynamic force distribution f ( x ,t ) is given as a strain nucleus 12 (or moment tensor 13 ) by the following function of x : (1)
• Fall '19
• Acoustic Emission

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