On the Validity of Several Previously Published Perturbation Formulas for the Acoustoelastic Eﬀect on Rayleigh Waves T ⁎ P

Ultrasonics 91 (2019) 114–120

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Ultrasonics

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On the validity of several previously published perturbation formulas for the acoustoelastic eﬀect on Rayleigh waves T ⁎ P. Mora, M. Spies

Fraunhofer-Institute for Nondestructive Testing IZFP, Campus E3 1, 66123 Saarbrücken, Germany

ARTICLE INFO ABSTRACT

Keywords: This article revisits the evaluation by a perturbation theory of the modiﬁcation of the Rayleigh wave velocity Acoustoelastic eﬀect under a static loading varying with depth. Two derivations, that have been exposed in the past and presented as Rayleigh wave comparable, are questioned. A new derivation of the perturbation formula is given by adapting Auld’s approach. Dispersion Validation with exact calculations is provided. The examples cover depth-varying static stress as well as depth- Perturbation theory varying third order elastic properties. Residual stresses

1. Introduction This article is organized as follows. First, arguments are given to prove that neither the formula derived by Hirao et al. nor Husson’s can The slight modification of sound wave velocities when the propa- cover arbitrary profiles of loading, and steps in both demonstrations gation medium is statically stressed has been extensively studied in the referring to this fact are identified. Second, a new derivation of the past [1,2]. This effect is known as the acoustoelastic effect. The possi- perturbation formula is given by adapting Auld’s approach. A general bility of using it to monitor the state of residual stresses inside a ma- formula is given, and then applied to an initially isotropic half space. terial has been widely considered, and numerous applications have Finally, the several sets of formulas are compared numerically on di- been developed in fields where either unwanted tensile stresses or de- verse examples, together with a validation by an exact calculation. liberately generated compressive stresses play a major role on the lifetime of mechanical components. Previous works covered virtually 2. Preliminary arguments all types of waves (bulk waves as well as surface or other guided S waves). Because the strains involved are small, perturbation theory has In what follows, εij refers to the static strain, k to the wavenumber, been a dominant approach to predict the magnitude of the effect. x1 is the coordinate in the direction of propagation, x3 is the vertical We shall in this work focus on the Rayleigh surface wave. This field coordinate and the over-bar means a value at the surface (x3 = 0). has taken benefit from other communities which were already involved The formula derived by Hirao et al. expresses the variation of ve- in studying the influence of depth-dependent texture on the dispersive locity of the Rayleigh wave ΔvR as a linear combination of S S 2S 2 S character of the Rayleigh wave. A milestone was Auld’s [3] perturba- εεkεkii,/,/∂∂3 ii3 ii , and integrals of εxii ( 3) over the half-space weighted tion theory, which lays on reciprocity relationships under a first order by decreasing exponentials. The formula derived by Husson has some Born approximation (see Szabo [4] and Tittmann et al. [5] for early common and some different features. It expresses the variation of ve- S S examples of application). The first work to deal with depth-varying locity as a linear combination of ε11 and integrals of εxii ( 3) over the half- loadings was probably that of Hirao et al. [6]. These authors used a space weighted by decreasing exponentials. In both cases, the presence S perturbation approach to derive a formula which predicts a frequency- of terms that explicitly depend on the value at the surface of εij and its fi dependent behavior in the velocity of the Rayleigh wave, also providing rst two derivatives is problematic. Indeed, if we consider a loading fi experimental evidence in the case of a stress growing with depth. A few which is located near the surface, i.e. which has a nite extent in depth, years later, Husson [7] addressed the same problem by using another then the integral terms can be shown to tend to zero at low frequencies. way to derive the perturbation formula, based on an adaptation of The predicted low frequency behavior would then be of the form Δvββkβk(LF) =+// + 2, which has a non-null, potentially divergent Auld’s methodology. Ditri and Hongerholt [8] later corrected typo- R 01 2 k graphical errors. Both articles of Hirao et al. and of Husson are today value for → 0. This is in contradiction with the physical intuition that widely cited. Still, they do not agree. for a localized loading the low frequency limit of the velocity should be

⁎ Corresponding author. E-mail address: [email protected] (M. Spies). https://doi.org/10.1016/j.ultras.2018.07.020 Received 12 January 2018; Received in revised form 26 June 2018; Accepted 30 July 2018 Available online 31 July 2018 0041-624X/ © 2018 Elsevier B.V. All rights reserved. P. Mora, M. Spies Ultrasonics 91 (2019) 114–120 only determined by the unmodified substrate. Therefore, both formulas ΔvvRR//=− δϕvωL R(). By transforming some terms of δϕ into the ma- are restricted to some cases which exclude the low frequency limits of terial coordinate system, one can obtain the difference between both localized profiles of loading. definitions of phase shifts: The demonstration of Hirao et al. follows the strategy of first ob- ω KE− S ω S taining a perturbed solution to the wave equation. A wave field po- δϕ−=− δΦ ∫∫∂−N uVN ddEij ∂ juVi , P V 2 P V (1) tential F is decomposed into a zeroth order and a first order term, la- belled FF=+0 F1. The differential equation satisfied by F1 has a in which the power flow P and the densities of kinetic energy = 1 ρ v 2 and elastic energy ==∂,{.()}1 Re u∗ σ of the homogeneous part identical to the original wave equation, and an in- K 2 0 0 E EENN ij 2 i0 j 0 homogeneous part involving the static stress field and F 0. By using the dynamic field in the unperturbed medium have been defined. In the plane waves of the unperturbed medium, a particular solution for F1 is case of a Rayleigh wave in an isotropic medium, one can show that only constructed. Then, F is inserted into the boundary condition and a E11 and E33 are not null, with furthermore VLPv V V δ system is obtained whose determinant must vanish. This last step pro- ∫V EEK11d/,d0,()d==−=R ∫∫VV 33 E0, i.e. S vides an explicit expression for the variation of velocity. The method, ϕδ−=−Φ(/) ε11 ωLvR if the static strain is homogeneous. which is standard and, in principle, correct, is however truly cumber- some as it requires to, first, expand the inhomogeneous part of the 3. Basic equations (fourth order) differential equation satisfied by F1, second, construct explicitly a particular solution by integrating this inhomogeneous part Let us consider a half space which mechanical properties are in- multiplied by products of the (four) linearly independent solutions, and variant in the planar (xx12, ) directions but may vary in the vertical x3 finally insert the whole expansion into the boundary condition which direction. In its natural state, i.e. in absence of any mechanical de- involves several derivative operators. The expressions to deal with are formation, the medium is described by a mass density ρ()x3 , a stiffness thus growing considerably at each step, and it would be a true challenge tensor Cxijkl ( 3) and third order elastic moduli Cxijklmn ()3 , and its surface is to re-derive them to obtain an error-proof formula. Nevertheless, the isolated from any other medium. At first no assumption is made on the following mistake can be identified. Hirao et al. wrote the inertial term symmetry of the medium, although isotropy will be assumed in the next 2 2 2 S S fi in the wave equation μ VV/ T instead of ρ V , to anticipate a further sections. A static stress σij ()xCux3 =∂ijkl k l (3) is applied and de nes the division by μ. This is of no consequence for the unperturbed equation, initial state. Except when specified, the coordinates and derivatives but apparently misled them to write the corresponding first order var- refer to this state. Then, a mechanical wave of small additional am- 2 2 2 2 2 fi fi iation 2ΔVV0 / VT,0 ≡ zinstead of 2ΔVV0 / VT,0 +ΔρV0 // μ=− z εNN V0 VT,0. plitude is considered and de nes the nal state (referred to with su- f Indeed, in the Ri expression (see Eq. (35) in [6]), the factors of z should perscript ). We shall be interested in a wave guided by the surface and 2 2 (2) also be present multiplied by −rV0 0 / VT,0 in the Li constants (see Eq. propagating in the x1 direction. (36) in [6]), which is not the case. Ri is used to generate the particular Following Pao et al. [10], in the space coordinate system defined by 1 fi solution F , so this error impacts the nal formula, even though, as will the initial state, the incremental displacement ui and the difference be shown below, it seems to affect only slightly the predicted dispersion between the final state second Piola-Kirchhoff and initial Cauchy stress fi f S in the particular case considered by Hirao et al. We did not try to nd tensors TTσij =−ij ij are related by the wave equation and generalized further errors, nor to correct them, as we chose to follow a more Hooke’s law: compact method to obtain the perturbation formula. ∂ []Tσu+∂S =∂ ρuS 2 , Husson’s demonstration is more attractive in the sense that it avoids jij jk ki t2 i (2) dealing explicitly with most of the perturbed terms, and results in TC=∂S u, handling more compact expressions. By multiplying the perturbed and ij ijkl kl (3) unperturbed fields and integrating them over a well-chosen volume, the with variation of phase δΦ is expressed as an integral over the static stresses S S weighted by the unperturbed field. However, Husson’s derivation is ρ =−∂ρu(1m m ), (4a) done using the wave equation expressed in the material coordinate CCSS=−∂+(1 u ) C ∂ u S system, i.e. coordinates which are deformed together with the material. ijkl ijkl m m ijklmn m n SS As a consequence, the phase shift defined this way is bound to the +∂+∂CuCumjkl m i imkl m j deformation of the distances and must be transformed back into the SS +∂+∂CuCuijml m k ijkm m l . (4b) space coordinate system before the variation of velocity can be deduced S S from it. This final step is presented as ΔvvRR/(Δ/)Φ/()=− LLδvR ωL, If ρ or Cijkl are discontinuous at some depth, then the displacement S with ΔLL/ = ε11. It however happens to be a special case of a more and forces are continuous through this interface. This condition must be general formula, and is valid only if the strain is uniform. So, in prac- written in the final set of coordinates, in which the wave slightly ad- tice, Husson’s formula is limited to uniform deformations, unless the ditionally modifies the space. Let us refer to this incremental de- ff latter transformation is replaced by the general one. Notice that formation gradient Fxδδuij =∂j i =ik( kj +∂ j k) and to its determinant Husson’s article adapted its methodology from an earlier work by J ff= detF . The Cauchy stress (or true stress) tensor σ f is related to Tf Husson and Kino [9] on bulk waves propagating in inhomogeneously through σ ff1fftf= ()J − FT F. An oriented surface element is transformed strained media. This latter article should therefore also be considered following nFnffd()dsJ= ftf1− s. Using these relations, and considering carefully. To obtain a correct formula one strategy could be to derive n = t(001), the elementary force through the interface expresses as: this corrective term. Another one could be to re-derive the perturbation ff f S S σijns j d{()(=+++∂ σi3 Ti3 σk3 Tkki3 )}d u s. (5) formula from the wave equation expressed in the space coordinate S fi system, in which the velocity is measured. We have done both, although Remembering that σij is a static stress, which therefore satis es we decided to present the latter one in this article because it leads to continuity without the presence of the incremental wave field, and dispersion equations that can also be solved exactly using standard neglecting the term Tukki3 ∂ in Eq. (5), the following incremental numerical procedures. We will devote in the near future another article quantity to the derivation of the general form of the corrections. Meanwhile, as a ∼ TTσu=+∂S hint, we give here the general form of the relation between velocity ii33k3 k i (6) variation and phase shift expressed in the material coordinates. By is continuous through the interface. At the surface, the stress-free adapting Husson’s demonstration to the wave equation expressed in the ∼ condition of natural state expresses as Tix30|03= = . space coordinate system, one can define a phase shift δϕ such as We now suppose that the wave field is harmonic in time and in the

115 P. Mora, M. Spies Ultrasonics 91 (2019) 114–120

2 ∞ x1 direction - with phase velocity v - and is invariant in the x2 direction: ωρvR ∗ i(ωt− x1 / v) P = ∫ uui i d x3 uii= uxe()3 . Following Osetrov et al. [11], the six dimensional 2 0 (13) ∼ state vector ΓT= [,iω u ]t satisﬁes: 3 the power ﬂux per unit length. As explained earlier, Eq. (12) can be S S S S ∂33ΓF= i(,)ωxv Γ, (7) readily applied for the σ11,,σσ22 12 and σ23 components using S S S 2 ΔCCCijkl =−ijkl and Δρρρσv=−−/ . where the propagator F is ijkl 11 R S S 1 vvρBC−−12S I−+ A BC 1 Bt 4.2. General formula for the σ33 and σ13 components (,)xv= ⎡ ⎤, F 3 2 v ⎣⎢ vv21CCB−− 1t ⎦⎥ (8) Auld’s demonstration [3] may be adapted to F. After reproducing I being the identity matrix, and the steps, one shows that:

AA()xσ=+S S I ∞ † 3 11 ΔvR vR ⎡iω u⎤ ⎡ T3 ⎤ ()ω =− ∫ ΔF dx3, S S S S vR 4P 0 T3 iω u (14) ⎡Cσ11 + 11 C16 C15 ⎤ ⎣ ⎦ ⎣ ⎦ =⎢ CCσCS S + S S ⎥, † ⎢ 16 66 11 56 ⎥ where denotes the transpose-conjugate operator and where the dis- ⎢ S S S S ⎥ placement u and normal stress T are approximated by the unperturbed ⎣ CCCσ15 56 55 + 11⎦ (9) 3 state. Let us focus on the part of ΔF that requires this extended formula. S3313S BB()xσ=+S S I To this end, let us separate ΔFF=+ΔΔ F + Δ,Δ FFbeing the 3 13 ff S S S S di erence between application of Formula (8) with ⎡Cσ15 + 13 C14 C13 ⎤ S S2 SSS ρ −σv11/,R AB , , Cand the same formula with the unperturbed con- =⎢ CCσCS S + S S ⎥, 33 13 fi ⎢ 56 46 13 36 ⎥ stants, and ΔFF,Δ arising from the modi cation of the diagonals of ⎢ S S S S ⎥ B and C: ⎣ CCCσ55 45 35 + 13⎦ (10) S −−12 12 t σ33 ⎡vR BC() BC () B⎤ ()xσ=+S S ΔF33 =− , CC3 33 I 2 ⎢ 2 −−12 12 t ⎥ vR vvR ()CCBR () (15a) S S S S ⎣ ⎦ ⎡Cσ55 + 33 C45 C35 ⎤ ⎢ S S S S ⎥ S −−−111t = CCσC+ . σ vR CBCCB+ ⎢ 45 44 33 34 ⎥ Δ.F13 = 13 ⎡ ⎤ S S S S 2 −1 ⎢ ⎥ vR ⎢ 0CvR ⎥ (15b) ⎣ CCCσ35 34 33 + 33⎦ (11) ⎣ ⎦ In Eq. (15a) it has been made use of (CIC+≈−σσS )()−−11S C −12, and Several numerical procedures were developed in the past to obtain 33 33 the C and B matrices referred to are built from the unperturbed con- the dispersion relationships vω( ) of a guided wave from Eq. (7). One of stants. them [12], which has become classical, is based on approximating the In the general case, the dispersion of the Rayleigh wave produced by medium as piece-wise constant and finding the roots of the determinant a static stress can therefore be calculated by applying Eq. (12) with of the so-called “transfer matrix”. We will rely on this exact method to ΔρρρσvC=−−S S2/,Δ = C S − C, and then adding the extra con- check for the validity of the perturbation formula derived in the next 11 R ijkl ijkl ijkl tribution of σS and σ S by evaluating Eq. (14) with ΔF33 and ΔF13. section. 33 13 Auld [3] derived a perturbation theory that describes how a varia- 4.3. Isotropic case tion of mass density or elastic constants affects the phase velocity of such a guided wave. The formula will be recalled in the next section. At We now apply the theory to a medium which is isotropic in its first order, the theory expresses the dispersion as an integral of the unperturbed state. The field components of a Rayleigh wave propa- variations weighted by the unperturbed wave field. The demonstration gating in the x direction are [3]: of the formula relies on a reciprocity approach applied to a propagation 1 – equation formally similar to Eq. (8). It is to be noted that Eqs. (8) (11) ω ω ω ⎛ −−−ns x3312 nl x ⎞ i x ff ff S S u1 = ines ⎜⎟vR − eevRRv , indicate that no formal di erence distinguishes the e ect of the σ22, σ12 1 + n 2 S ⎝ s ⎠ (16a) and σ23 components from a variation of mass density and elastic con- S stants. This remark can even be extended to the σ11 component by u2 = 0, (16b) noting that the extra variation in the diagonal of the A matrix can 2 ω 1 n ω ω formally be transferred into an appropriately defined equivalent mass ⎛ −−−ns x331+ s nl x ⎞ i x u3 = ⎜⎟e vR − eevRRv , S S 2 density ρ −σv11/ . Therefore, in these four particular but very relevant ⎝ 2 ⎠ (16c) ’ cases, Auld s formula may be readily applied. Concerning the con- 22 S S with nvvnvvsRslRl=−1( /), =− 1( /), and with vs and vl being the tribution of the remaining σ33 and σ13 components, the formal variation bulk shear and longitudinal velocities. Note that v is such that in the diagonals of the B and C matrices have deeper consequences and R 4nn=+(1 n2 )2.Inafirst time, we assume that σS and σ S are null, i.e. an extended formula is necessary. This will be dealt with in the next sl s 33 13 the perturbed medium may be entirely described by an effective mass section. density and elastic tensor. Let us rewrite Eq. (12) in the form

ΔvR ∞ ω ω 4. Perturbation theory ()γ −ni v x3 ()ωγxfe= ∑ ∫ Δ()3 i R dx3 , vR 0 vR γρC= , ij (17) 4.1. Auld’s formula where n123===+2, nnnnnnslsl 2, , and We here ﬁrst recall the result of Auld’s [3] perturbation theory. If we P 1 222⎛ −−8 ⎞ let the perturbed state be deﬁned by = ρvR ns ⎜⎟2 ++− nsl n 2 . ω 4 ⎝ 1 + ns ⎠ (18) ρ →+ρρxCCΔ(33 ), ijkl → ijkl +Δ( Cx ijkl ), then the velocity of the Ray- leigh wave vvR + Δ R deviates from its original value by: Because in Eq. (16) the only non-null gradients are εε11, 33 and ε13 and

∞ because of the π/2 phase shift between some of these (which leads to ΔvR vR 2 ∗∗ ()ω =−∫ [Δρω ui ui + εij Δ Cijkl ε kl ]d, x3 terms having zero real parts), it can be seen from Eqs. (12) and (17) that vR 4P 0 (12) only the ΔγρCCC= Δ,Δ11 ,Δ 33 ,Δ 13 and ΔC55 contribute to the disper- ()ρ (C11 ) (C 33 ) (C 13) ()C55 with sion. The corresponding fi ,,,fffi i i and fi constants are

116 P. Mora, M. Spies Ultrasonics 91 (2019) 114–120

Table 1 Mass density (in g cm−3), Lamé’s and Murnaghan’s constants (in GPa) de- scribing the mild steel [6,8] sample used in Fig. 1, and the Ti-6246 [13] sample used in Figs. 2–4.

ρ λ μ lmn

Steel 7.837 107.4 81.9 −206.5 −600 −800 Ti 4.54 80.0 45.5 −201 −272 −356 given in the Appendix. As can be seen from Eq. (4b) applied to an S S S S ff isotropic medium, CCC11,,33 13 and C55 are only a ected by the principal S S S fl static strains εε11, 22 and ε33, from which we deduce that the in uence of S S fi ff S σ12 and σ23 on ΔvR are null at rst order. The e ect of σ13 is also zero at first order, because the contribution of ΔF13 in Eq. (15b) calculated with Eq. (14) only involves products of fields having a π/2 phase shift, therefore producing a zero real part. We are thus only left with the evaluation of the contribution of ΔF33 to cover all static stress components. After some tedious but straightforward calculations, Eq. (14) can finally be rewritten in a similar form as Eq. (17):

S ΔvR ∞ ()σ ω ω S γ −ni v x3 ()ωσxfe= ∑ ∫ γ ()3 i R d.x3 v 0 v R γ=11,22,33 R (19) Fig. 2. Dispersion of Rayleigh wave in Ti-6246 produced by a layer of static

()σS ()σS (σS ) stress localized at the surface. The stress is isotropic in the (1, 2) plane The f 11 , f 22 and f 33 constants appearing in Eq. (19) are given S S S i i i (σσ11 = 22) and has no vertical component (σ33 = 0). in the Appendix. Note that because the third order constants do not aﬀect the un- S j+1 perturbed state, some hypothesis can easily be weakened. If 1 S ()σγ ⎛ vR ⎞ Φj = ∑ σfγj, i j! ⎜⎟. v n (lmn,,)→ (,,)( lmn x3) are allowed to vary with depth, then R γ = 11, 22, 33 ⎝ i ⎠ (20b) SS ()σγγ ()σ fi → fxi ( 3) are depth-dependent and Eq. (19) is still valid. As a last comment let us say a word on the high-frequency asymp- totic of Eq. (19). Szabo [4] pointed out that the integral operator −1 5. Numerical examples transforms a series in x3 into a series in ω , and proposed to use this property for an inverse procedure, although he did not deal with static stresses. Ditri and Hongerholt [8] did the same in the context of static In the following, the perturbation theory derived in this work (see stresses, but relied on Husson’s formula [7], which is not correct. If we Eq. (19)) is validated against an exact method described earlier [11,12]. S S j Husson’s formula [7,8] and, for the ﬁrst example, the formula of Hirao proceed to the polynomial expansion σγ ()xσx3 = ∑j=0 γj, 3 , then: et al. [6] are also compared. The elastic constants that describe the

ΔvR Φj materials are given in Table 1. All comparisons are made for small ()ω = ∑ , S v ω j values of the static strain (εij ≈ 0.1%). The range of validity according to R j=0 (20a) this amplitude is not addressed in this work.

Fig. 3. Dispersion of Rayleigh wave in Ti-6246 produced by a buried layer of S S Fig. 1. Dispersion of Rayleigh wave caused by a static stress growing with static stress. The stress is isotropic in the (1, 2) plane (σσ11 = 22) and has no S depth, in a mild steel sample. vertical component (σ33 = 0).

117 P. Mora, M. Spies Ultrasonics 91 (2019) 114–120

5.1. Static stress growing with depth a surface treatment, such as shot-peening, laser shock-peening or low plasticity burnishing. Two cases are shown in Figs. 2 and 3, where As a first example, and to compare the aforementioned theories with Husson’s formula is compared to Eq. (19) and to an exact calculation. the set of formulas derived in this work, we re-consider the situation of Fig. 2 shows the transition in the variation of velocity from zero to a Hirao et al. These authors applied couples to a mild steel plate in such a value proportional to the acoustoelastic coefficient when the frequency S S way to generate a uni-axial static stress σ22 ()xσ3 =−22 (12/) xH3 , with is increased. Indeed, the lower frequency limit is only determined by S σ22 = 63.75[MPa] and H = 10mm. They showed that the predicted the unchanged substrate, while the wave is confined in a uniform stress dispersion was of the form region when the penetration is smaller than the depth of the layer. It can be seen that Husson’s formula predicts a non-physical lower fre- β ΔvR ⎛ 1 ⎞ ν S ()kβ=−⎜⎟+ σ , quency limit. Fig. 3 shows how band-limited is the variation of velocity v 0 kH E 22 R ⎝ ⎠ (21) caused by a buried layer. Here again, Husson’s formula predicts wrong results unless Eq. (1) is used. and provided the numerical values β0 =−0.99 and β1 = 3.55. Using Eq. ff (20b) one can calculate β0 =−1.05 and β1 = 4.19. The slight di erence (0.6%)inβ0 is compatible with a use by Hirao et al. of constants slightly different from those reported, within the numerical precision reported. 5.3. Depth-varying third order constants The discrepancy is however bigger for β and we could not reproduce 1 fi the value with Eq. (20b) without modifying the elastic constants outside As a nal set of examples let us consider the dispersion caused by a the given numerical precision. We therefore tend to consider it as a variation of the third order elastic constants with depth when a static numerical hint that the formula that they used is not valid for this ex- stress is uniformly applied. The aim of this last set of examples is ample, even if it is difficult to be really categorical (the qualitative twofold. First, according to a comment we made before, it is the only ’ demonstration given at the beginning of this article can not be applied case where Husson s formula can correctly predict a non-trivial varia- to this example because the profile is not localized). In the case of tion of velocity without requiring Eq. (1). The second reason is that this ’ ff situation seems to have been ignored in the past, while the surface Husson s formula (β0 =−1.05 and β1 = 10.20), the result is clearly far o treatments earlier referred to are known to affect the third order elastic for β1. When using the corrective term indicated in Eq. (1) the results perfectly agree with Eq. (20b). constants in the near-surface region as a by-product of plastic de- We present the comparison in Fig. 1. The discrepancies appear at formation. low frequencies. The curve which is labeled “exact” was obtained by Fig. 4 shows three sub-figures (a), (b) and (c) corresponding to the finding the root of the determinant of the Transfer Matrix after ap- three possible directions of the uni-axial and uniformly applied stress. proximating the stress in thin constant layers. The instabilities that are On each sub-figure, three dashed lines are displayed, corresponding to well known to appear at high frequencies have been assessed by three scenarios of variation of lm,,n by 100% of their value. The curve adapting the discretization to the wavelength. labeled l refers to a variation l → 2l while m and n are kept unchanged, We tried without success to reproduce the results of Hirao et al. by and so forth. The variation is confined to a layer of depth H, located at using their formula. One reason could be that their comprehensive set the surface. Because of linearity, any other variation of lm,,n can be of constants would have been reported with typographical issues. This obtained by a linear combination of these three scenarios and the re- is why only Husson’s formula is compared with the present work in the ference levels. The dispersion is calculated using Eq. (19), now having SS ()σγγ ()σ following examples. fi = fxi ()3 with the same formal expression (see Appendix). A perfect agreement was found with both the exact method and Husson’s 5.2. Static stress localized in a layer formula (in the (a) and (b) cases for which it was designed). It can be seen on Fig. 4 (a), (b) and (c) that none of the situations is sensitive to l. As a second set of examples let us consider stress profiles that are Thus, only mx()3 and nx()3 can be expected to be deduced from ex- confined to the near-surface region. This choice is made to illustrate perimental data, and the inversion would certainly require the effect to situations in which stress is deliberately introduced into the material by be measured in two directions.

S Fig. 4. Dispersion of Rayleigh wave in Ti-6246 produced by depth-varying (,,)lmn=+ (,,) lmnref Δ(,,)() lmn x3 , under a uniform uni-axial static stress, (a) σ11, (b) S S σ22, (c) σ33. The depth-dependency of Δ(,lmn , ) is a layer of thickness H localized at the surface. The continuous curve is the reference, i.e. when Δ(,lmn , )= 0. The three dashed curves correspond to three scenarios where one constant is increased by 100% while the other two are kept unchanged.

118 P. Mora, M. Spies Ultrasonics 91 (2019) 114–120

6. Conclusion perturbation amplitudes. Numerical examples include depth-varying third order elastic constants, which, to the best of our knowledge, had In this work, the acoustoelastic eﬀect on Rayleigh waves under not been considered before. depth-varying loading was addressed via a perturbation approach. Two theories published in the past and widely cited were shown to be dif- Acknowledgments ferent, contrarily to what was claimed. By adapting a demonstration by Auld, a corrected perturbation theory was derived in a general case, The authors would like to thank Mr. Hans Rieder and Mr. Sebastian covering general anisotropy and all components of loading, and explicit Hubel, Fraunhofer IZFP, for many valuable discussions. Special thanks formulas were given for an isotropic medium. Validation against exact to Dr. Joachim Bamberg and Dr. Roland Hessert, MTU Aero Engines computations showed perfect agreement under the hypothesis of low Munich, Germany, for many helpful discussions.

Appendix A. Explicit expressions for isotropic media

S S ΔCij as a function of the static strains εi for an isotropic medium (Eq. (4b)):

S S S S ΔC11=+(3)()(), C111 C 11 ε1 +− C113 C 11 ε2 + ε3 S S S S Δ(CC33=+111 3)( Cε 11 3 +− CCε113 11 )(),1 + ε2 S S S S Δ(CC13=+113 Cε 13 )()(1 ++− ε3 CCε123 13 ),2 S S S S Δ(CC55=+155 Cε 55 )()(1 ++− ε3 C255 Cε 55 ).2

v2 C ∼()Cij Constants f ()ρ = R f ()ρ and f ()ij = 1 f appearing in Eq. (17): i 4/Pωi i 4/Pωi

()ρ 2 ∼()C11 2 fnfn1 =−(1 + s ),1 = s , ()ρ ns 2 ∼()C11 ns fnf=−(1 + l ),= , 2 nl 2 nl

()ρ nns + l ∼()C11 ns 2 fn3 ==4,s 2 f3 −(1),+ ns 1 + ns nl ∼∼()C332 ()C 13 2 fn1 ==s ,2 f1 − ns , ∼∼()C33()C 13 fnnf==3,(−1 1+ n2)2, 2 s l 2 2 s ()C ()C n 2 ∼∼331 2 3 13 2 1 + l fnfn3 =−(1 + s ) ,3 = 4s 2 , 4 1 + ns ()C ∼ 55 2 2 fn1 =+(1s ) , ()C ∼ 55 2 2 fn2 =+(1s ) , ()C ∼ 55 2 2 fn3 =−2(1 + s ) .

()σSS()σ ()σ S Constants f jj =+1 ∼gαfjj ∑ jj ()γ appearing in Eq. (19): i 4/Pω i γρC= , ij ()γ i S S S ααα()σ11 ===−()σ22 ()σ33 ρν(1− 2 ) , ()ρ ()ρ ()ρ E S S αα()σ11 = ()σ33 ()C11 ()C33 =+−−1 [(CCνCC 3 ) 2 ( )], E 111 11 113 11 S S S S α()σ11 === ααα()σ33 ()σ22 ()σ22 ()C33 ()C11 ()C11 ()C33 =−1 [(1νC )( −−+ C ) νC ( 3 C )], E 113 11 111 11 S S αα()σ11 = ()σ33 ()C13 ()C13 =−1 [(1νC )( +−− C ) νC ( C )], E 113 13 123 13 S S αα()σ11 = ()σ33 ()C55 ()C55 =−1 [(1νCCνCC )( +−− ) ( )], E 155 55 255 55 S αCCνCC()σ22 =−−+1 [( ) 2 ( )], ()C13 E 123 13 113 13 S αCCνCC()σ22 =−−+1 [( ) 2 ( )], ()C55 E 255 55 155 55 and:

119 P. Mora, M. Spies Ultrasonics 91 (2019) 114–120

S ∼()σ11 2 ()ρ gvfi =−R i , S ∼()σ22 gi = 0, S ()C 2 ()CCC() () ∼()σ33 C13 ∼∼∼∼13 C13 11 33 55 gf1 =− fff ++ C11 1 ()C11 1 1 1 2 2 C13 + ns ⎡ +−131,⎤− ⎣()C11 ⎦ S ()C 2 ()CCC() () ∼()σ33 C13 ∼∼∼∼13 C13 11 33 55 gf2 =− fff ++ C11 2 ()C11 2 2 2 2 + nn⎡ 1 C13 +−23,C13 ⎤ sl 2 C C ⎣ nl ()11 11 ⎦ S ()C 2 ()CCC() () ∼()σ33 C13 ∼∼∼∼13 C13 11 33 55 gf3 =− fff ++ C11 3 ()C11 3 3 3 2 2 ns C13 C13 1 −+(1nnns )⎡ ++s ()l ⎣ nll()C11 C11 n 2 −−ns 2].

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