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. and ) 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.
The acoustoelastic effect was investigated as early as 1925 by Brillouin.[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.[3]
In 1937 Francis Dominic Murnaghan[4] 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 , , and . 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.
Non-linear elastic theory for hyperelastic materials edit
The acoustoelastic effect is an effect of finite deformation of non-linear elastic materials. A modern comprehensive account of this can be found in.[1] 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,[6] is reproduced and shown in this text from the non-linear elasticity theory as presented by Ogden.[1]
- Note that the setting in this text as well as in [1] is isothermal, and no reference is made to thermodynamics.
Constitutive relation – hyperelastic materials (Stress-strain relation) edit
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 (which is the transpose of the nominal stress tensor ), the constitutive equation for a compressible hyper elastic material can be expressed in terms of the Lagrangian Green strain ( ) as:
Assuming that the scalar strain energy density function can be approximated by a Taylor series expansion in the current strain , it can be expressed (in index notation) as:
The deformation gradient tensor can be expressed in component form as
Sound velocity edit
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).[8] Let us define three states of a given material point. In the reference (un-stressed) state the point is defined by the coordinate vector while the same point has the coordinate vector 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 . 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
- Note that the subscript/superscript "0" is used in this text to denote the un-stressed reference state, and a dotted variable is as usual the time ( ) derivative of the variable, and is the divergence operator with respect to the Lagrangian coordinate system .
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 can be expanded in the Eulerian by using the chain rule and changing the variables through the relation between the displacement vectors as [8]
For a hyperelastic material is symmetric (but not in general), and the eigenvalues ( ) are thus real. For the wave velocities to also be real the eigenvalues need to be positive.[1] If this is the case, three mutually orthogonal real plane waves exist for the given propagation direction . From the two expressions of the acoustic tensor it is clear that[10]
Isotropic materials edit
Elastic moduli for isotropic materials edit
For a second order isotropic tensor (i.e. a tensor having the same components in any coordinate system) like the Lagrangian strain tensor have the invariants where is the trace operator, and . The strain energy function of an isotropic material can thus be expressed by , or a superposition there of, which can be rewritten as[8]
Landau & Lifshitz (1986)[11] | Toupin & Bernstein (1961)[12] | Murnaghan (1951)[4] | Bland (1969)[13] | Eringen & Suhubi (1974)[14] | Standard | |
---|---|---|---|---|---|---|
Example values for steel edit
Table 2 and 3 present the second and third order elastic constants for some steel types presented in literature
Lamé constants | Toupin & Bernstein constants | ||||
---|---|---|---|---|---|
Material | |||||
Hecla 37 (0.4%C)[15] | 111±1 | 82.1±0.5 | −385±70 | −282±30 | −177±8 |
Hecla 37 (0.6%C)[15] | 110.5±1 | 82.0±0.5 | −134±20 | −261±20 | −167±6 |
Hecla 138A[15] | 109±1 | 81.9±0.5 | −323±50 | −265±30 | −177±10 |
Rex 535 Ni steel[15] | 109±1 | 81.8±0.5 | −175±50 | −240±50 | −169±15 |
Hecla ATV austenitic[15] | 87±2 | 71.6±3 | 34±20 | −552±80 | −100±10 |
Lamé constants | Murnaghan constants | ||||
---|---|---|---|---|---|
Material | |||||
Nickel-steel S/NVT[16] | 109.0±1 | 81.7±0.2 | −56±20 | −671±6 | −785±7 |
Rail steel sample 1 [17] | 115.8±2.3% | 79.9±2.3% | −248±2.8% | −623±4.1% | −714±2.7% |
Rail steel sample 4[17] | 110.7±2.3% | 82.4±2.3% | −302±2.8% | −616±4.1% | −724±2.7% |
Acoustoelasticity for uniaxial tension of isotropic hyperelastic materials edit
A cuboidal sample of a compressible solid in an unstressed reference configuration can be expressed by the Cartesian coordinates , where the geometry is aligned with the Lagrangian coordinate system, and is the length of the sides of the cuboid in the reference configuration. Subjecting the cuboid to a uniaxial tension in the -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 , which gives the elongations
For a uniaxial tension in the -direction ( we assume that the increase by some amount. If the lateral faces are free of traction (i.e., ) the lateral elongations and are limited to the range . For isotropic symmetry the lateral elongations (or contractions) must also be equal (i.e. ). The range corresponds to the range from total lateral contraction ( , which is non-physical), and to no change in the lateral dimensions ( ). 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 edit
If the strong ellipticity condition ( ) holds, three orthogonally polarization directions ( will give a non-zero and real sound velocity for a given propagation direction . 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 -direction, and deriving the sound velocities for waves propagating orthogonally to the applied tension (e.g. in the -direction with propagation vector ), one selection of orthonormal polarizations may be
Expanding the relevant coefficients of the acoustic tensor, and substituting the second- and third-order elastic moduli and with their isotropic equivalents, and respectively, leads to the sound velocities expressed as
Measurement methods edit
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
Example of ultrasonic testing techniques edit
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 edit
As explained above, a set of three orthonormal polarizations ( ) of the particle motion exist for a given propagation direction 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 edit
Engineering material – stress estimation edit
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 edit
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.[8]
Soft tissue – medical ultrasonics edit
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.
See also edit
References edit
- ^ a b c d e f Ogden, R. W., Non-linear elastic deformations, Dover Publications Inc., Mineola, New York, (1984)
- ^ Brillouin, Léon (1925). "Les tensions de radiation ; leur interprétation en mécanique classique et en relativité". Journal de Physique et le Radium. 6 (11): 337–353. doi:10.1051/jphysrad:01925006011033700. ISSN 0368-3842.
- ^ Tang, Sam (1967). "Wave propagation in initially-stressed elastic solids". Acta Mechanica. 4 (1): 92–106. doi:10.1007/BF01291091. ISSN 0001-5970. S2CID 121910597.
- ^ a b c Murnaghan, F. D. (1937). "Finite Deformations of an Elastic Solid". American Journal of Mathematics. 59 (2): 235–260. doi:10.2307/2371405. ISSN 0002-9327. JSTOR 2371405.
- ^ Hughes, D. S.; Kelly, J. L. (1953). "Second-Order Elastic Deformation of Solids". Physical Review. 92 (5): 1145–1149. Bibcode:1953PhRv...92.1145H. doi:10.1103/PhysRev.92.1145. ISSN 0031-899X.
- ^ "Anisotropy and Isotropy". Archived from the original on 2012-05-31. Retrieved 2013-12-07.
- ^ a b c Norris, A. N. (1997). "Finite-Amplitude Waves in Solids". In Hamilton, Mark F.; Blackstock, David T. (eds.). Nonlinear Acoustics. Acoustical Society of America. ISBN 978-0123218605.
- ^ a b c d e f Norris, A. N. (2007). "Small-on-Large Theory with Applications to Granular Materials and Fluid/Solid Systems" (PDF). In M. Destrade; G. Saccomandi (eds.). Waves in Nonlinear Pre-Stressed Materials. CISM Courses and Lectures. Vol. 495. Springer, Vienna. doi:10.1007/978-3-211-73572-5. ISBN 978-3-211-73572-5.
- ^ a b Eldevik, S., "Measurement of non-linear acoustoelastic effect in steel using acoustic resonance", PhD Thesis, University of Bergen, (in preparation)
- ^ a b c d e Ogden, R. W. (2007). "Incremental Statics and Dynamics of Pre-Stressed Elastic Materials" (PDF). In M. Destrade; G. Saccomandi (eds.). Waves in Nonlinear Pre-Stressed Materials. CISM Courses and Lectures. Vol. 495. Springer, Vienna. doi:10.1007/978-3-211-73572-5. ISBN 978-3-211-73572-5.
- ^ a b Landau, L. D.; Lifshitz, E. M. (1970). Theory of Elasticity (second ed.). Pergamon Press. ISBN 9780080064659.
- ^ Toupin, R. A.; Bernstein, B. (1961). "Sound Waves in Deformed Perfectly Elastic Materials. Acoustoelastic Effect". The Journal of the Acoustical Society of America. 33 (2): 216–225. Bibcode:1961ASAJ...33..216T. doi:10.1121/1.1908623. ISSN 0001-4966.
- ^ Bland, D. R., Nonlinear dynamic elasticity, Blaisdell Waltham, (1969)
- ^ Suhubi, E. S., Eringen, A. C., Elastodynamics, Academic press New York, (1974)
- ^ a b c d e Smith, R. T.; Stern, R.; Stephens, R. W. B. (1966). "Third-Order Elastic Moduli of Polycrystalline Metals from Ultrasonic Velocity Measurements". The Journal of the Acoustical Society of America. 40 (5): 1002–1008. Bibcode:1966ASAJ...40.1002S. doi:10.1121/1.1910179. ISSN 0001-4966.
- ^ Crecraft, D.I. (1967). "The measurement of applied and residual stresses in metals using ultrasonic waves". Journal of Sound and Vibration. 5 (1): 173–192. Bibcode:1967JSV.....5..173C. doi:10.1016/0022-460X(67)90186-1. ISSN 0022-460X.
- ^ a b Egle, D. M.; Bray, D. E. (1976). "Measurement of acoustoelastic and third-order elastic constants of rail steel". The Journal of the Acoustical Society of America. 59 (S1): S32. Bibcode:1976ASAJ...59...32E. doi:10.1121/1.2002636. ISSN 0001-4966.
- ^ Abiza, Z.; Destrade, M.; Ogden, R.W. (2012). "Large acoustoelastic effect". Wave Motion. 49 (2): 364–374. arXiv:1302.4555. Bibcode:2012WaMot..49..364A. doi:10.1016/j.wavemoti.2011.12.002. ISSN 0165-2125. S2CID 119244072.
- ^ Gennisson, J.-L.; Rénier, M.; Catheline, S.; Barrière, C.; Bercoff, J.; Tanter, M.; Fink, M. (2007). "Acoustoelasticity in soft solids: Assessment of the nonlinear shear modulus with the acoustic radiation force". The Journal of the Acoustical Society of America. 122 (6): 3211–3219. Bibcode:2007ASAJ..122.3211G. doi:10.1121/1.2793605. ISSN 0001-4966. PMID 18247733.
- ^ Jun Wu; Wei He; Wei-min Chen; Lian Zhu (2013). "Research on simulation and experiment of noninvasive intracranial pressure monitoring based on acoustoelasticity effects". Medical Devices: Evidence and Research. 6: 123–131. doi:10.2147/MDER.S47725. PMC 3758219. PMID 24009433.
- ^ Duenwald, Sarah; Kobayashi, Hirohito; Frisch, Kayt; Lakes, Roderic; Vanderby, Ray (2011). "Ultrasound echo is related to stress and strain in tendon". Journal of Biomechanics. 44 (3): 424–429. doi:10.1016/j.jbiomech.2010.09.033. ISSN 0021-9290. PMC 3022962. PMID 21030024.