Acoustoelastic effect

The acoustoelastic effect is how the sound velocities (both longitudinal and shear wave velocities) of an elastic material change if subjected to an initial static stress field. This is a non-linear effect of the constitutive relation between mechanical stress and finite strain in a material of continuous mass. In classical linear elasticity theory small deformations of most elastic materials can be described by a linear relation between the applied stress and the resulting strain. This relationship is commonly known as the generalised Hooke's law. The linear elastic theory involves second order elastic constants (e.g. formula_1 and formula_2) 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) 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.
The acoustoelastic effect was investigated as early as 1925 by Brillouin. 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 1937 Murnaghan presented a mathematical theory extending the linear elastic theory to also include finite deformation in elastic isotropic materials. This theory included three third-order elastic constants formula_3, formula_4, and formula_5. In 1953 Huges and Kelly 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.
Non-linear elastic theory for hyperelastic materials.
The acoustoelastic effect is an effect of finite deformation of non-linear elastic materials. A modern comprehensive account of this can be found in. This book treats the application of the non-linear elasticity theory and the analysis of the mechanical properties of solid materials capable of large elastic deformations. The special case of the acoustoelastic theory for a compressible isotropic hyperelastic material, like polycrystalline steel, is reproduced and shown in this text from the non-linear elasticity theory as presented by Ogden.
Constitutive relation – hyperelastic materials (Stress-strain relation).
A hyperelastic material is a special case of a Cauchy elastic material in which the stress at any point is objective and determined only by the current state of deformation with respect to an arbitrary reference configuration (for more details on deformation see also the pages Deformation (mechanics) and Finite strain). However, the work done by the stresses may depend on the path the deformation takes. Therefore, a Cauchy elastic material has a non-conservative structure, and the stress cannot be derived from a scalar elastic potential function. The special case of Cauchy elastic materials where the work done by the stresses is independent of the path of deformation is referred to as a Green elastic or hyperelastic material. Such materials are conservative and the stresses in the material can be derived by a scalar elastic potential, more commonly known as the Strain energy density function.
The constitutive relation between the stress and strain can be expressed in different forms based on the chosen stress and strain forms. Selecting the 1st Piola-Kirchhoff stress tensor formula_6 (which is the transpose of the nominal stress tensor formula_7), the constitutive equation for a compressible hyper elastic material can be expressed in terms of the Lagrangian Green strain (formula_8) as:
where formula_10 is the deformation gradient tensor, and where the second expression uses the Einstein summation convention for index notation of tensors. formula_11 is the strain energy density function for a hyperelastic material and have been defined per unit volume rather than per unit mass since this avoids the need of multiplying the right hand side with the mass density formula_12 of the reference configuration.
Assuming that the scalar strain energy density function formula_13 can be approximated by a Taylor series expansion in the current strain formula_8, it can be expressed (in index notation) as:
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. formula_16) it is clear that there are no constant or linear term in the strain energy function, and thus:
where formula_18 is a fourth-order tensor of second-order elastic moduli, while formula_19 is a sixth-order tensor of third-order elastic moduli.
The symmetry of formula_20 together with the scalar strain energy density function formula_11 implies that the second order moduli formula_18 have the following symmetry:
which reduce the number of independent elastic constants from 81 to 36. In addition the power expansion implies that the second order moduli also have the major symmetry
which further reduce the number of independent elastic constants to 21. The same arguments can be used for the third order elastic moduli formula_19. These symmetries also allows the elastic moduli to be expressed by the Voigt notation (i.e. formula_26 and formula_27).
The deformation gradient tensor can be expressed in component form as
where formula_29 is the displacement of a material point formula_30 from coordinate formula_31 in the reference configuration to coordinate formula_32 in the deformed configuration (see Figure 2 in the finite strain theory page). Including the power expansion of strain energy function in the constitutive relation and replacing the Lagrangian strain tensor formula_33 with the expansion given on the finite strain tensor page yields (note that lower case formula_34 have been used in this section compared to the upper case on the finite strain page) the constitutive equation
where
and higher order terms have been neglected
(see for detailed derivations).
For referenceM by neglecting higher order terms in formula_37 this expression reduce to
formula_38
which is a version of the generalised Hooke's law where formula_39 is a measure of stress while formula_37 is a measure of strain, and formula_18 is the linear relation between them.
Sound velocity.
Assuming that a small dynamic (acoustic) deformation disturb an already statically stressed material the acoustoelastic effect can be regarded as the effect on a small deformation superposed on a larger finite deformation (also called the small-on-large theory). Let us define three states of a given material point. In the reference (un-stressed) state the point is defined by the coordinate vector formula_42 while the same point has the coordinate vector formula_43 in the static initially stressed state (i.e. under the influence of an applied pre-stress). Finally, assume that the material point under a small dynamic disturbance (acoustic stress field) have the coordinate vector formula_44. The total displacement of the material points (under influence of both a static pre-stress and an dynamic acoustic disturbance) can then be described by the displacement vectors
where
describes the static (Lagrangian) initial displacement due to the applied pre-stress, and the (Eulerian) displacement due to the acoustic disturbance, respectively. 
Cauchy's first law of motion (or balance of linear momentum) for the additional Eulerian disturbance formula_47 can then be derived in terms of the intermediate Lagrangian deformation formula_48 assuming that the small-on-large assumption
holds.
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
under the assumption that both the unstressed state and the initial deformation state are static and thus formula_55.
For the left hand side (the space dependent part) the spatial Lagrangian partial derivatives with respect to formula_56 can be expanded in the Eulerian formula_57 by using the chain rule and changing the variables through the relation between the displacement vectors as 
where the short form formula_59 has been used. Thus
Assuming further that the static initial deformation formula_48 (the pre-stressed state) is in equilibrium means that formula_62, 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 formula_63) between the static initial deformation formula_48 and the additional dynamic disturbance formula_65 as (see for detailed derivations)
where 
This expression is recognised as the linear wave equation. Considering a plane wave of the form
where formula_69 is a Lagrangian unit vector in the direction of propagation (i.e. parallel to the wave number formula_70normal to the wave front, ), formula_71 is a unit vector referred to as the polarization vector (describing the direction of particle motion), formula_72 is the phase wave speed, and formula_73 is a twice continuously differentiable function (e.g. a sinusoidal function). Inserting this plane wave in to the linear wave equation derived above yields
where formula_75 is introduced as the acoustic tensor, and depends on formula_69 as
This expression is called the propagation condition and determines for a given propagation direction formula_78 the velocity and polarization of possible waves corresponding to plane waves. The wave velocities can be determined by the characteristic equation
where formula_80 is the determinant and formula_81 is the identity matrix.
For a hyperelastic material formula_75 is symmetric (but not in general), and the eigenvalues (formula_83) are thus real. For the wave velocities to also be real the eigenvalues need to be positive. If this is the case, three mutually orthogonal real plane waves exist for the given propagation direction formula_69. From the two expressions of the acoustic tensor it is clear that
and the inequality formula_86 (also called the strong ellipticity condition) for all non-zero vectors formula_69 and formula_71 guarantee that the velocity of homogeneous plane waves are real. The polarization formula_89 corresponds to a longitudinal wave where the particle motion is parallel to the propagation direction (also referred to as a compressional wave). The two polarizations where formula_90 corresponds to transverse waves where the particle motion is orthogonal to the propagation direction (also referred to as shear waves).
Isotropic materials.
Elastic moduli for isotropic materials.
For a second order isotropic tensor (i.e. a tensor having the same components in any coordinate system) like the Lagrangian strain tensor formula_8 have the invariants formula_92 where formula_93 is the trace operator, and formula_94. The strain energy function of an isotropic material can thus be expressed by formula_95, or a superposition there of, which can be rewritten as
where formula_97 are constants. The constants formula_1 and formula_2 are the second order elastic moduli better known as the Lamé parameters, while formula_100 and formula_101 are the third order elastic moduli introduced by, which are alternative but equivalent to formula_102 and formula_5 introduced by Murnaghan.
Combining this with the general expression for the strain energy function it is clear that
where formula_105. Historically different selection of these third order elastic constants have been used, and some of the variations is shown in Table 1.
Example values for steel.
Table 2 and 3 present the second and third order elastic constants for some steel types presented in literature
Acoustoelasticity for uniaxial tension of isotropic hyperelastic materials.
A cuboidal sample of a compressible solid in an unstressed reference configuration can be expressed by the Cartesian coordinates formula_106, where the geometry is aligned with the Lagrangian coordinate system, and formula_107 is the length of the sides of the cuboid in the reference configuration. Subjecting the cuboid to a uniaxial tension in the formula_108-direction so that it deforms with a pure homogeneous strain such that the coordinates of the material points in the deformed configuration can be expressed by formula_109, which gives the
elongations
in the formula_32-direction. Here formula_112 signifies the current (deformed) length of the cuboid side formula_113 and where the ratio between the length of the sides in the current and reference configuration are denoted by
called the principal stretches. For an isotropic material this corresponds to a deformation without any rotation (See polar decomposition of the deformation gradient tensor where formula_115 and the rotation formula_116). This can be described through spectral representation by the principal stretches formula_117 as eigenvalues, or equivalently by the elongations formula_118.
For a uniaxial tension in the formula_108-direction (formula_120 we assume that the formula_121 increase by some amount. If the lateral faces are free of traction (i.e. formula_122) the lateral elongations formula_123 and formula_124 are limited to the range formula_125. For isotropic symmetry the lateral elongations (or contractions) must also be equal (i.e. formula_126). The range corresponds to the range from total lateral contraction (formula_127, which is non-physical), and to no change in the lateral dimensions (formula_128). It is noted that theoretically the range could be expanded to values large than 0 corresponding to an increase in lateral dimensions as a result of increase in axial dimension. However, very few materials (called auxetic materials) exhibit this property.
Expansion of sound velocities.
If the strong ellipticity condition (formula_129) holds, three orthogonally polarization directions (formula_71 will give a non-zero and real sound velocity for a given propagation direction formula_69. The following will derive the sound velocities for óne selection of applied uniaxial tension, propagation direction, and an orthonormal set of polarization vectors. For a uniaxial tension applied in the formula_108-direction, and deriving the sound velocities for waves propagating orthogonally to the applied tension (e.g. in the formula_133-direction with propagation vector formula_134), one selection of orthonormal polarizations may be
which gives the three sound velocities
where the first index formula_113 of the sound velocities formula_138 indicate the propagation direction (here the formula_133-direction, while the second index formula_140 indicate the selected polarization direction (formula_141 corresponds to particle motion in the propagation direction formula_113 – i.e. longitudinal wave, and formula_143 corresponds to particle motion perpendicular to the propagation direction – i.e. shear wave).
Expanding the relevant coefficients of the acoustic tensor, and substituting the second- and third-order elastic moduli formula_18 and formula_19 with their isotropic equivalents, formula_146 and formula_147 respectively, leads to the sound velocities expressed as
where
are the acoustoelastic coefficients related to effects from third order elastic constants.
Measurement methods.
To be able to measure the sound velocity, and more specifically the change in sound velocity, in a material subjected to some stress state, one can measure the velocity of an acoustic signal propagating through the material in question. There are several methods to do this but all of them use one of two physical relations of the sound velocity. The first relation is related to the time it takes a signal to propagate from one point to another (typically the distance between two acoustic transducers or two times the distance from one transducer to a reflective surface). This is often referred to as "Time-of-flight" (TOF) measurements, and use the relation
where formula_153 is the distance the signal travels and formula_51 is the time it takes to travel this distance. The second relation is related to the inverse of the time, the frequency, of the signal. The relation here is
where formula_73 is the frequency of the signal and formula_1 is the wave length. The measurements using the frequency as measurand use the phenomenon of acoustic resonance where formula_5 number of wave lengths match the length over which the signal resonate. 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.
Example of ultrasonic testing techniques.
In general there are two ways to set up a transducer system to measure the sound velocity in a solid. One is a setup with two or more transducers where one is acting as a transmitter, while the other(s) is acting as a receiver. The sound velocity measurement can then be done by measuring the time between a signal is generated at the transmitter and when it is recorded at the receiver while assuming to know (or measure) the distance the acoustic signal have traveled between the transducers, or conversely to measure the resonance frequency knowing the thickness over which the wave resonate. The other type of setup is often called a "pulse-echo" system. Here one transducer is placed in the vicinity of the specimen acting both as transmitter and receiver. This requires a reflective interface where the generated signal can be reflected back toward the transducer which then act as a receiver recording the reflected signal. See ultrasonic testing for some measurement systems.
Longitudinal and polarized shear waves.
As explained above, a set of three orthonormal polarizations (formula_71) of the particle motion exist for a given propagation direction formula_69 in a solid. For measurement setups where the transducers can be fixated directly to the sample under investigation it is possible to create these three polarizations (one longitudinal, and two orthogonal transverse waves) by applying different types of transducers exciting the desired polarization (e.g. piezoelectric transducers with the needed oscillation mode). Thus it is possible to measure the sound velocity of waves with all three polarizations through either time dependent or frequency dependent measurement setups depending on the selection of transducer types. However, if the transducer can not be fixated to the test specimen a coupling medium is needed to transmit the acoustic energy from the transducer to the specimen. Water or gels are often used as this coupling medium. For measurement of the longitudinal sound velocity this is sufficient, however fluids do not carry shear waves, and thus to be able to generate and measure the velocity of shear waves in the test specimen the incident longitudinal wave must interact at an oblique angle at the fluid/solid surface to generate shear waves through mode conversion. Such shear waves are then converted back to longitudinal waves at the solid/fluid surface propagating back through the fluid to the recording transducer enabling the measurement of shear wave velocities as well through a coupling medium.
Applications.
Engineering material – stress estimation.
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. There are several measurement techniques to measure stress in a material. However, techniques using optical measurements, magnetic measurements, X-ray diffraction, and neutron diffraction are all limited to measuring surface or near surface stress or strains. Acoustic waves propagate with ease through materials and provide thus a means to probe the interior of structures, where the stress and strain level is important for the overall structural integrity.
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.
Granular and porous materials – geophysics.
seismology study the propagation of elastic waves through the Earth and is used in e.g. earthquake studies and in mapping the Earth's interior. 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.
Soft tissue – medical ultrasonics.
Other applications may be in medical sonography and elastography measuring the stress or pressure level in relevant elastic tissue types 
(e.g. 
 ), enhancing non-invasive diagnostics.