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. λ {\displaystyle \lambda } and μ {\displaystyle \mu } ) 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 Francis Dominic 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 l {\displaystyle l} , m {\displaystyle m} , and n {\displaystyle n} . 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.
Note that the setting in this text as well as in is isothermal, and no reference is made to thermodynamics.
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 P {\displaystyle {\boldsymbol {P}}} (which is the transpose of the nominal stress tensor P T = N {\displaystyle {\boldsymbol {P}}^{T}={\boldsymbol {N}}} ), the constitutive equation for a compressible hyper elastic material can be expressed in terms of the Lagrangian Green strain ( E {\displaystyle {\boldsymbol {E}}} ) as:
P = F ⋅ ∂ W ∂ E or P i j = F i k ∂ W ∂ E k j , i , j = 1 , 2 , 3 , {\displaystyle {\boldsymbol {P}}={\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {E}}}}\qquad {\text{or}}\qquad P_{ij}=F_{ik}~{\frac {\partial W}{\partial E_{kj}}},\qquad i,j=1,2,3~,}
where F {\displaystyle {\boldsymbol {F}}} is the deformation gradient tensor, and where the second expression uses the Einstein summation convention for index notation of tensors. W {\displaystyle W} 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 ρ 0 {\displaystyle \rho _{0}} of the reference configuration. Assuming that the scalar strain energy density function W ( E ) {\displaystyle W({\boldsymbol {E}})} can be approximated by a Taylor series expansion in the current strain E {\displaystyle {\boldsymbol {E}}} , it can be expressed (in index notation) as:
W ≈ C 0 + C i j E i j + 1 2 ! C i j k l E i j E k l + 1 3 ! C i j k l m n E i j E k l E m n + ⋯ {\displaystyle W\approx C_{0}+C_{ij}E_{ij}+{\frac {1}{2!}}C_{ijkl}E_{ij}E_{kl}+{\frac {1}{3!}}C_{ijklmn}E_{ij}E_{kl}E_{mn}+\cdots }
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. W ( E i j = 0 ) = 0 {\displaystyle W(E_{ij}=0)=0} ) it is clear that there are no constant or linear term in the strain energy function, and thus:
W ≈ 1 2 ! C i j k l E i j E k l + 1 3 ! C i j k l m n E i j E k l E m n + ⋯ , {\displaystyle W\approx {\frac {1}{2!}}C_{ijkl}E_{ij}E_{kl}+{\frac {1}{3!}}C_{ijklmn}E_{ij}E_{kl}E_{mn}+\cdots ,}
where C i j k l {\displaystyle C_{ijkl}} is a fourth-order tensor of second-order elastic moduli, while C i j k l m n {\displaystyle C_{ijklmn}} is a sixth-order tensor of third-order elastic moduli. The symmetry of E i j = E j i {\displaystyle E_{ij}=E_{ji}} together with the scalar strain energy density function W {\displaystyle W} implies that the second order moduli C i j k l {\displaystyle C_{ijkl}} have the following symmetry:
C i j k l = C j i k l = C i j l k , {\displaystyle C_{ijkl}=C_{jikl}=C_{ijlk},}
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
C i j k l = C k l i j , {\displaystyle C_{ijkl}=C_{klij},}
which further reduce the number of independent elastic constants to 21. The same arguments can be used for the third order elastic moduli C i j k l m n {\displaystyle C_{ijklmn}} . These symmetries also allows the elastic moduli to be expressed by the Voigt notation (i.e. C i j k l = C I J {\displaystyle C_{ijkl}=C_{IJ}} and C i j k l m n = C I J K {\displaystyle C_{ijklmn}=C_{IJK}} ). The deformation gradient tensor can be expressed in component form as
F i j = ∂ u i ∂ X j + δ i j , {\displaystyle F_{ij}={\frac {\partial u_{i}}{\partial X_{j}}}+\delta _{ij},}
where u i {\displaystyle u_{i}} is the displacement of a material point P {\displaystyle P} from coordinate X i {\displaystyle X_{i}} in the reference configuration to coordinate x i {\displaystyle x_{i}} 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 E k l {\displaystyle E_{kl}} with the expansion given on the finite strain tensor page yields (note that lower case u {\displaystyle u} have been used in this section compared to the upper case on the finite strain page) the constitutive equation
P i j = C i j k l ∂ u k ∂ X l + 1 2 M i j k l m n ∂ u k ∂ X l ∂ u m ∂ X n + 1 3 M i j k l m n p q ∂ u k ∂ X l ∂ u m ∂ X n ∂ u p ∂ X q + ⋯ , {\displaystyle P_{ij}=C_{ijkl}{\frac {\partial u_{k}}{\partial X_{l}}}+{\frac {1}{2}}M_{ijklmn}{\frac {\partial u_{k}}{\partial X_{l}}}{\frac {\partial u_{m}}{\partial X_{n}}}+{\frac {1}{3}}M_{ijklmnpq}{\frac {\partial u_{k}}{\partial X_{l}}}{\frac {\partial u_{m}}{\partial X_{n}}}{\frac {\partial u_{p}}{\partial X_{q}}}+\cdots ,}
where
M i j k l m n = C i j k l m n + C i j l n δ k m + C j n k l δ i m + C j l m n δ i k , {\displaystyle M_{ijklmn}=C_{ijklmn}+C_{ijln}\delta _{km}+C_{jnkl}\delta _{im}+C_{jlmn}\delta _{ik},}
and higher order terms have been neglected (see for detailed derivations). For referenceM by neglecting higher order terms in ∂ u k / ∂ X l {\displaystyle \partial u_{k}/\partial X_{l}} this expression reduce to
P i j = C i j k l ∂ u k ∂ X l , {\displaystyle P_{ij}=C_{ijkl}{\frac {\partial u_{k}}{\partial X_{l}}},}
which is a version of the generalised Hooke's law where P i j {\displaystyle P_{ij}} is a measure of stress while ∂ u k / ∂ X l {\displaystyle \partial u_{k}/\partial X_{l}} is a measure of strain, and C i j k l {\displaystyle C_{ijkl}} 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 X {\displaystyle {\boldsymbol {X}}} while the same point has the coordinate vector x {\displaystyle {\boldsymbol {x}}} 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 x ′ {\displaystyle {\boldsymbol {x'}}} . The total displacement of the material points (under influence of both a static pre-stress and a dynamic acoustic disturbance) can then be described by the displacement vectors
u = u ( 0 ) + u ( 1 ) = x ′ − X , {\displaystyle {\boldsymbol {u}}={\boldsymbol {u}}^{(0)}+{\boldsymbol {u}}^{(1)}={\boldsymbol {x'}}-{\boldsymbol {X}},}
where
u ( 0 ) = x − X , u ( 1 ) = x ′ − x {\displaystyle {\boldsymbol {u}}^{(0)}={\boldsymbol {x}}-{\boldsymbol {X}},\qquad {\boldsymbol {u}}^{(1)}={\boldsymbol {x'}}-{\boldsymbol {x}}}
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 u ( 1 ) {\displaystyle {\boldsymbol {u}}^{(1)}} can then be derived in terms of the intermediate Lagrangian deformation u ( 0 ) {\displaystyle {\boldsymbol {u}}^{(0)}} assuming that the small-on-large assumption
| u ( 1 ) | ≪ | u ( 0 ) | {\displaystyle |{\boldsymbol {u}}^{(1)}|\ll |{\boldsymbol {u}}^{(0)}|}
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
Div P = ρ 0 x ′ ¨ . {\displaystyle \operatorname {Div} {\boldsymbol {P}}=\rho _{0}{\ddot {{\boldsymbol {x}}'}}.}
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 ( t {\displaystyle t} ) derivative of the variable, and Div {\displaystyle \operatorname {Div} } is the divergence operator with respect to the Lagrangian coordinate system X {\displaystyle {\boldsymbol {X}}} . The right hand side (the time dependent part) of the law of motion can be expressed as
ρ 0 x ′ ¨ = ρ 0 ∂ 2 ∂ t 2 ( u ( 0 ) + u ( 1 ) + X ) = ρ 0 ∂ 2 u ( 1 ) ∂ t 2 {\displaystyle {\begin{aligned}\rho _{0}{\ddot {{\boldsymbol {x}}'}}&=\rho _{0}{\frac {\partial ^{2}}{\partial t^{2}}}({\boldsymbol {u}}^{(0)}+{\boldsymbol {u}}^{(1)}+{\boldsymbol {X}})\\&=\rho _{0}{\frac {\partial ^{2}{\boldsymbol {u}}^{(1)}}{\partial t^{2}}}\end{aligned}}}
under the assumption that both the unstressed state and the initial deformation state are static and thus ∂ 2 u ( 0 ) / ∂ t 2 = ∂ 2 X / ∂ t 2 = 0 {\textstyle \partial ^{2}{\boldsymbol {u}}^{(0)}/\partial t^{2}=\partial ^{2}{\boldsymbol {X}}/\partial t^{2}=0} . For the left hand side (the space dependent part) the spatial Lagrangian partial derivatives with respect to X j {\displaystyle X_{j}} can be expanded in the Eulerian x j {\displaystyle x_{j}} by using the chain rule and changing the variables through the relation between the displacement vectors as
∂ ∂ X j = ∂ ∂ x j + u k , j ( 0 ) ∂ ∂ x k + ⋯ {\displaystyle {\frac {\partial }{\partial X_{j}}}={\frac {\partial }{\partial x_{j}}}+u_{k,j}^{(0)}{\frac {\partial }{\partial x_{k}}}+\cdots }
where the short form u k , j ( 0 ) ≡ ∂ u k ( 0 ) / ∂ x j {\displaystyle u_{k,j}^{(0)}\equiv \partial u_{k}^{(0)}/\partial x_{j}} has been used. Thus
∂ P i j ∂ X j ≈ ∂ P i j ∂ x j + u p . j ( 0 ) ∂ P i j ∂ x p {\displaystyle {\frac {\partial P_{ij}}{\partial X_{j}}}\approx {\frac {\partial P_{ij}}{\partial x_{j}}}+u_{p.j}^{(0)}{\frac {\partial P_{ij}}{\partial x_{p}}}}
Assuming further that the static initial deformation u ( 0 ) {\displaystyle {\boldsymbol {u}}^{(0)}} (the pre-stressed state) is in equilibrium means that Div P ( 0 ) = 0 {\displaystyle \operatorname {Div} {\boldsymbol {P}}^{(0)}={\boldsymbol {0}}} , 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 u m , n ( 0 ) {\displaystyle u_{m,n}^{(0)}} ) between the static initial deformation u ( 0 ) {\displaystyle {\boldsymbol {u}}^{(0)}} and the additional dynamic disturbance u ( 1 ) ( x , t ) {\displaystyle {\boldsymbol {u}}^{(1)}({\boldsymbol {x}},t)} as (see for detailed derivations)
B i j k l ∂ 2 u k ( 1 ) ∂ x j ∂ x l = ρ 0 ∂ 2 u i ( 1 ) ∂ t 2 , {\displaystyle B_{ijkl}{\frac {\partial ^{2}u_{k}^{(1)}}{\partial x_{j}\partial x_{l}}}=\rho _{0}{\frac {\partial ^{2}u_{i}^{(1)}}{\partial t^{2}}},}
where
B i j k l = C i j k l + δ i k C j l q r u q , r ( 0 ) + C r j k l u i , r ( 0 ) + C i r k l u j , r ( 0 ) + C i j r l u k , r ( 0 ) + C i j k r u l , r ( 0 ) + C i j k l m n u m , n ( 0 ) . {\displaystyle B_{ijkl}=C_{ijkl}+\delta _{ik}C_{jlqr}u_{q,r}^{(0)}+C_{rjkl}u_{i,r}^{(0)}+C_{irkl}u_{j,r}^{(0)}+C_{ijrl}u_{k,r}^{(0)}+C_{ijkr}u_{l,r}^{(0)}+C_{ijklmn}u_{m,n}^{(0)}.}
This expression is recognised as the linear wave equation. Considering a plane wave of the form
u ( 1 ) ( x , t ) = m f ( N ⋅ x − c t ) , {\displaystyle {\boldsymbol {u}}^{(1)}({\boldsymbol {x}},t)={\boldsymbol {m}}\,f({\boldsymbol {N}}\cdot {\boldsymbol {x}}-ct),}
where N {\displaystyle {\boldsymbol {N}}} is a Lagrangian unit vector in the direction of propagation (i.e., parallel to the wave number k = k N {\displaystyle {\boldsymbol {k}}=k{\boldsymbol {N}}} normal to the wave front), m {\displaystyle {\boldsymbol {m}}} is a unit vector referred to as the polarization vector (describing the direction of particle motion), c {\displaystyle c} is the phase wave speed, and f {\displaystyle f} is a twice continuously differentiable function (e.g. a sinusoidal function). Inserting this plane wave in to the linear wave equation derived above yields
Q ( N ) m = ρ 0 c 2 m {\displaystyle {\boldsymbol {Q}}({\boldsymbol {N}}){\boldsymbol {m}}=\rho _{0}c^{2}{\boldsymbol {m}}}
where Q