Acoustoelastic effect
This is a non-linear effect of the constitutive relation between mechanical stress and finite strain in a material of continuous mass.
) and yields constant longitudinal and shear sound velocities in an elastic material, not affected by an applied stress.
The acoustoelastic effect on the other hand include higher order expansion of the constitutive relation (non-linear elasticity theory[1]) between the applied stress and resulting strain, which yields longitudinal and shear sound velocities dependent of the stress state of the material.
In the limit of an unstressed material the sound velocities of the linear elastic theory are reproduced.
[2] He found that the propagation velocity of acoustic waves would decrease proportional to an applied hydrostatic pressure.
However, a consequence of his theory was that sound waves would stop propagating at a sufficiently large pressure.
This paradoxical effect was later shown to be caused by the incorrect assumptions that the elastic parameters were not affected by the pressure.
In 1953 Huges and Kelly [5] used the theory of Murnaghan in their experimental work to establish numerical values for higher order elastic constants for several elastic materials including Polystyrene, Armco iron, and Pyrex, subjected to hydrostatic pressure and uniaxial compression.
), the constitutive equation for a compressible hyper elastic material can be expressed in terms of the Lagrangian Green strain (
Imposing the restrictions that the strain energy function should be zero and have a minimum when the material is in the un-deformed state (i.e.
In addition the power expansion implies that the second order moduli also have the major symmetry
with the expansion given on the finite strain tensor page yields (note that lower case
have been used in this section compared to the upper case on the finite strain page) the constitutive equation
Finally, assume that the material point under a small dynamic disturbance (acoustic stress field) have the coordinate vector
Cauchy's first law of motion (or balance of linear momentum) for the additional Eulerian disturbance
Using the Lagrangian form of Cauchy's first law of motion, where the effect of a constant body force (i.e. gravity) has been neglected, yields
The right hand side (the time dependent part) of the law of motion can be expressed as
For the left hand side (the space dependent part) the spatial Lagrangian partial derivatives with respect to
, and the law of motion can in combination with the constitutive equation given above be reduced to a linear relation (i.e. where higher order terms in
[1] If this is the case, three mutually orthogonal real plane waves exist for the given propagation direction
Historically different selection of these third order elastic constants have been used, and some of the variations is shown in Table 1.
The following will derive the sound velocities for óne selection of applied uniaxial tension, propagation direction, and an orthonormal set of polarization vectors.
Expanding the relevant coefficients of the acoustic tensor, and substituting the second- and third-order elastic moduli
Both these methods are dependent on the distance over which it measure, either directly as in the Time-of-flight, or indirectly through the matching number of wavelengths over the physical extent of the specimen which resonate.
In general there are two ways to set up a transducer system to measure the sound velocity in a solid.
As the industry strives to reduce maintenance and repair costs, non-destructive testing of structures becomes increasingly valued both in production control and as a means to measure the utilization and condition of key infrastructure.
Since the sound velocity of such non-linear elastic materials (including common construction materials like aluminium and steel) have a stress dependency, one application of the acoustoelastic effect may be measurement of the stress state in the interior of a loaded material utilizing different acoustic probes (e.g. ultrasonic testing) to measure the change in sound velocities.
The interior of the Earth is subjected to different pressures, and thus the acoustic signals may pass through media in different stress states.
The acoustoelastic theory may thus be of practical interest where nonlinear wave behaviour may be used to estimate geophysical properties.
[8] Other applications may be in medical sonography and elastography measuring the stress or pressure level in relevant elastic tissue types (e.g., [19] [20] [21] ), enhancing non-invasive diagnostics.
