Nonlinear effects of micro-cracks on acoustic surface and wedge waves
Micro-cracks give rise to non-analytic behavior of the stress-strain relation. For the case of a homogeneous spatial distribution of aligned flat micro-cracks, the influence of this property of the stress-strain relation on harmonic generation is analyzed for Rayleigh waves and for acoustic wedge wa...
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| Cite this: | Nonlinear effects of micro-cracks on acoustic surface and wedge waves / M. Rjelka, P.D. Pupyrev, B. Koehler, A.P. Mayer // Физика низких температур. — 2018. — Т. 44, № 7. — С. 946-953. — Бібліогр.: 31 назв. — англ. |
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| author | Rjelka, M. Pupyrev, P.D. Koehler, B. Mayer, A.P. |
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| citation_txt | Nonlinear effects of micro-cracks on acoustic surface and wedge waves / M. Rjelka, P.D. Pupyrev, B. Koehler, A.P. Mayer // Физика низких температур. — 2018. — Т. 44, № 7. — С. 946-953. — Бібліогр.: 31 назв. — англ. |
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| description | Micro-cracks give rise to non-analytic behavior of the stress-strain relation. For the case of a homogeneous spatial distribution of aligned flat micro-cracks, the influence of this property of the stress-strain relation on harmonic generation is analyzed for Rayleigh waves and for acoustic wedge waves with the help of a simple micro-mechanical model adopted from the literature. For the efficiencies of harmonic generation of these guided waves, explicit expressions are derived in terms of the corresponding linear wave fields. The initial growth rates of the second harmonic, i.e., the acoustic nonlinearity parameter, has been evaluated numerically for steel as ma-trix material. The growth rate of the second harmonic of Rayleigh waves has also been determined for micro-crack distributions with random orientation, using a model expression for the strain energy in terms of strain in-variants known in a geophysical context.
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7, pp. 946–953
Nonlinear effects of micro-cracks on acoustic surface
and wedge waves
M. Rjelka1, P.D. Pupyrev2,3, B. Koehler1, and A.P. Mayer3
1Fraunhofer IKTS, D-01109 Dresden, Germany
2Prokhorov General Physics Institute of the Russian Academy of Sciences, Moscow 119991, Russia
3HS Offenburg, University of Applied Sciences, D-77723 Gengenbach, Germany
E-mail: andreas.mayer@hs-offenburg.de
Received March 12, 2018, published online May 28, 2018
Micro-cracks give rise to non-analytic behavior of the stress-strain relation. For the case of a homogeneous
spatial distribution of aligned flat micro-cracks, the influence of this property of the stress-strain relation on har-
monic generation is analyzed for Rayleigh waves and for acoustic wedge waves with the help of a simple micro-
mechanical model adopted from the literature. For the efficiencies of harmonic generation of these guided
waves, explicit expressions are derived in terms of the corresponding linear wave fields. The initial growth rates
of the second harmonic, i.e., the acoustic nonlinearity parameter, has been evaluated numerically for steel as ma-
trix material. The growth rate of the second harmonic of Rayleigh waves has also been determined for micro-
crack distributions with random orientation, using a model expression for the strain energy in terms of strain in-
variants known in a geophysical context.
PACS: 43.35.+d Ultrasonics, quantum acoustics, and physical effects of sound;
46.40.Cd Mechanical wave propagation (including diffraction, scattering, and dispersion);
62.30.+d Mechanical and elastic waves; vibrations.
Keywords: micro-cracks, surface acoustic waves, wedge waves, nonlinearity, harmonic generation.
1. Introduction
Rayleigh waves and wedge waves are two different
types of guided acoustic waves. The strain field associated
with the prior (more generally: surface acoustic waves,
SAWs) is localized at the surface of a solid, i.e., a two-
dimensional manifold, whereas the latter have strains lo-
calized at a one-dimensional manifold, namely the apex
line of a solid elastic wedge, i.e., the intersection line of
two surfaces. Straight-crested time-harmonic SAWs are
characterized by a two-dimensional wave-vector, whereas
the wave-vector of a time-harmonic wedge wave is always
along the apex line and hence one-dimensional. Rayleigh
waves are studied since the pioneering work by Lord Ray-
leigh [1] and find technical applications in various fields of
science and engineering. Acoustic wedge waves were dis-
covered much later [2,3] and are yet awaiting practical use
in technical devices, while prototypes for such devices
have already been developed (see, for example, [4,5] for
recent reviews).
Both types of guided acoustic waves have in common
the property of being non-dispersive in the ideal case. This
means that their phase velocity is independent of their fre-
quency. Ideal means essentially a homogeneous elastic
medium, planar surfaces and a perfectly sharp wedge tip.
The absence of dispersion favors nonlinear effects, as it
guarantees phase matching for the growth of higher har-
monics.
In crystal lattices, the elastic nonlinearity stems from
the interatomic forces in the neighborhood of the rest posi-
tions of the atoms, which allows the potential energy of the
solid to be expanded in powers of the Green–Lagrange
strain tensor [6] and consequently, stress is an analytic
function of strain. The influence of the third-order terms in
this expansion on the propagation of surface and, to a less-
er extent of wedge acoustic waves, has been investigated in
theory and experiment (see the recent review [4], and [7]).
Defects like dislocations or micro-cracks are known to
strongly modify the nonlinear properties of acoustic waves
in solids, while they often change their linear properties
© M. Rjelka, P.D. Pupyrev, B. Koehler, and A.P. Mayer, 2018
mailto:andreas.mayer@hs-offenburg.de
Nonlinear effects of micro-cracks on acoustic surface and wedge waves
like their speed to a much lesser degree [8,9]. In the case of
bulk and surface acoustic waves, this fact is used for pur-
poses of non-destructive evaluation [10,11] as it provides a
viable tool for pre-fatigue detection, even if the size of the
micro-cracks is much smaller than the wavelength of the
acoustic waves. A central quantity in this context is the
acoustic nonlinearity parameter (ANP), which is a measure
of the efficiency of second harmonic generation of a time-
harmonic fundamental wave. In the case of acoustic wedge
waves, no experimental work is known to us that would
focus on the influence of dislocations or micro-cracks of
sub-wavelength size on the nonlinear propagation proper-
ties. However, recent experiments on wedge waves in a po-
lycrystalline aluminum sample containing residual stresses
point to unusual, “non-classical” nonlinear behavior [12].
Micro-cracks can render the stress-strain relation of the
solid non-analytic. Kinks in this relation are known to be
generated by flat micro-cracks of the “kissing bond”
type [9] due to the difference of their elastic response to
tensile and compressive stress. Non-analytic behavior of
the stress-strain relation has also been found in finite ele-
ment simulations of solids with micro-cracks that have an
internal structure [13].
Our goal in this contribution is to present a method of
calculating the growth of higher harmonics and of the
products for a time-harmonic Rayleigh or wedge wave in
an elastic medium with a homogeneous spatial distribution
of flat micro-cracks with sub-wavelength size. Concerning
their orientation, we consider the simple case of all micro-
cracks being aligned such that their surface normals are all
along the same direction. In the case of Rayleigh waves,
we choose this direction to be the propagation direction. In
the case of wedge waves propagating at a rectangular edge,
the alignment is chosen such that the surface normals of
the micro-cracks are vertical to the apex line and parallel to
one of the surfaces of the wedge. Based on an effective
stress-strain relation for this system that follows from a
simple micro-mechanical model [14–17] and is supported
by finite element simulations, an asymptotic expansion of
the displacement field is derived that yields a set of cou-
pled evolution equations for the slowly varying amplitudes
of the fundamental and higher harmonics. This approach
differs from an earlier study by Oberhardt et al. who used
the finite element method to simulate the nonlinear propa-
gation of surface acoustic waves in a medium containing
micro-cracks [18].
If the matrix material of the elastic medium with micro-
cracks is isotropic, a totally random distribution of micro-
crack orientations maintains the isotropy. An expression
for the density of potential energy set up by Lyakhovsky
and Myasnikov [19] applies to this situation and contains a
term that is not analytic in the strain invariants. We shall
apply this expression as a model for the elastic properties
of media with random micro-crack orientations. Our con-
tribution concludes with a short summary and discussion.
2. Stress-strain relation for an elastic medium
with micro-cracks
The following derivation will be confined to flat micro-
cracks that are homogeneously distributed in an isotropic
matrix material. In the first part of this section, we consider
the case of all micro-cracks having the same orientation
such that their surface normals are oriented along the
x1 direction. Micro-mechanical models of penny-shaped
micro-cracks in the context of nonlinear acoustic wave
effects [20,14–17] make use of an additive decomposition
of the total macroscopic infinitesimal strain parameters
αβε into the infinitesimal strain (0)
αβε generated by a macro-
scopic Cauchy stress αβσ in the absence of micro-cracks
and a separate contribution ( )MC
αβε of the micro-cracks.
Here and in the following, Cartesian indices are denoted by
small Greek letters, and we invoke the convention that
summation over repeated Cartesian indices is implied. Ap-
plying the approach in [15] to a distribution of fully
aligned micro-cracks in the static limit, we obtain
( ) ( )
( )
11 1 1 11
11 11
11 112 2 2 2
12 13 12 13
2 1 1 2 21 3 1 1 3 31
( )
1 ( ) 1 ( )
.
MC
N
S
s H
s H H H
α βαβ
α β α β α β α β
ε = σ δ δ σ +
σ µ σ µ + + −σ × + −σ ×
σ + σ σ + σ
× δ δ + δ δ σ + δ δ + δ δ σ
(1)
In (1), H denotes the Heaviside step function, ,N Ss s are
coefficients which are proportional to the concentration of
the penny-shaped micro-cracks and depend on their diame-
ter and the elastic properties of the matrix material. The
quantity µ is the coefficient of friction between the adja-
cent faces of a micro-crack.
Inserting (1) in the stress-strain relation for the matrix
material without micro-cracks,
( )(0) (0)(0) (MC)C Cαβ µν µν µναβµν αβµνσ = ε = ε − ε , (2)
where (0)Cαβµν are the elastic constants of the matrix material,
we obtain for the diagonal elements of the Cauchy stress
and infinitesimal strain tensor:
(0) (0)
11 11 22 33 1111 12 ( )c cσ = ε + ε + ε + ∆σ , (3a)
(0) (0)
22 22 11 33 2211 12 ( )c cσ = ε + ε + ε + ∆σ , (3b)
(0) (0)
33 33 11 22 3311 12 ( )c cσ = ε + ε + ε + ∆σ , (3c)
where the stress deviations
11 0 0 1 0| |∆σ = τ ε + τ ε , (4)
(0)
12
22 33 11(0)
11
c
c
∆σ = ∆σ = ∆σ (5)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 947
M. Rjelka, P.D. Pupyrev, B. Koehler, and A.P. Mayer
result from the micro-cracks, and where we have defined
the quantity
(0)
12
0 11 22 33(0)
11
( )
c
c
ε = ε + ε + ε . (6)
(0) (0)
11 12,c c denote two independent elastic constants of
the matrix material in Voigt notation.
The constants 0 1,τ τ in (4) are related to the coefficients
,N Ss s in (1) via
(0)
(0) 11
0 11 (0)
11
21
2 1
N
N
c s
c
c s
+
τ =
+
, (7)
(0)
(0) 11
1 11 (0)
11
1
2 1
N
N
c s
c
c s
τ = −
+
. (8)
The occurrence of the modulus in (4) introduces nonlinear-
ity into this stress-strain relation.
In the case of shear stress / strain, we consider here only
two limiting cases concerning the friction between the fac-
es of the micro-cracks. When the friction is negligible
( 0µ → ), we obtain from (1)
12 66 12 13 66 13 23 44 232 , 2 , 2c c cσ = ε σ = ε σ = ε , (9)
where (0)
44 44c c= and
(0)
44
66 (0)
441 2 S
c
c
c s
=
+
. (10)
In this case, the relation between shear stress and shear
strain remains linear, but causes the macroscopic elastic
properties of the elastic medium to become anisotropic.
In the case of very large friction (µ → ∞), (9) is re-
placed by
( )
( )
(0)
12 2 0 1244
(0)
13 2 0 1344
(0)
23 2344
2 ( ) ,
2 ( ) ,
2 ,
c H
c H
c
σ = − τ ε ε
σ = − τ ε ε
σ = ε
(11)
involving the quantity
(0)
(0) 44
2 44 (0)
44
2
1 2
S
S
c s
c
c s
τ =
+
. (12)
Because of the Heaviside function in (11), the relation be-
tween shear stress and shear strain becomes nonlinear, too.
When dealing with nonlinear acoustic waves, it is more
convenient to work with the first Piola–Kirchhoff stress
tensor ( )Tαβ instead of the Cauchy stress tensor ( )αβσ .
When transforming from the latter to the prior, additional
nonlinear terms appear, which are partly contained in the
nonlinear contributions to ( )Tαβ resulting from the matrix
material alone. We shall neglect these terms in comparison
to the dominant nonlinearity arising from the non-analytic
parts (4), (5) and (11). For the purpose of applying pertur-
bation theory, we decompose the first Piola–Kirchhoff
stress tensor into the contribution of the matrix material
and a perturbation due to the presence of the micro-cracks,
(0)T T Tαβ αβαβ= + ∆ with (0) (0)T C µναβ αβµν= ε . The diagonal
elements of Tαβ∆ are given by (4) and (5), when σ is re-
placed by T ,
11 0 0 1 0| |T∆ = τ ε + τ ε , (13)
(0)
12
22 33 11(0)
11
c
T T T
c
∆ = ∆ = ∆ . (14)
For the off-diagonal elements, we find 23 32 0T T∆ = ∆ =
and
12 21 2 122T T∆ = ∆ = − τ ε , (15a)
13 31 2 132T T∆ = ∆ = − τ ε , (15b)
if friction is neglected, and
12 21 2 0 122 ( )T T H∆ = ∆ = − τ ε ε , (16a)
13 31 2 0 132 ( )T T H∆ = ∆ = − τ ε ε (16b)
in the limit of infinite friction coefficient µ.
After having established the stress-strain relation for a
medium containing micro-cracks that are fully aligned, we
comment on the case of a distribution of micro-cracks with
totally random orientations. Now, elastic isotropy of the
elastic medium is maintained, and the density of potential
energy of the medium, considered as a function of macro-
scopic strain, depends on the three invariants 1 ,I αα= η
2 ,I αβ αβ= η η 3I αβ βγ γα= η η η of the Green–Lagrange
finite strain tensor , , , ,( ) ( ) / 2u u u uαβ α β β α γ α γ βη = + + , only.
Lyakhovsky and Myasnikov [19] introduced an expression
for the density of potential energy that contains a non-
analytic term via its square-root dependence on the strain
invariant I2,
2
1 2 1 2
1
2 LMI I I IΦ = λ + µ + κ . (17)
We use this potential as a model for an elastic medium
containing defects like micro-cracks with perfectly random
orientation. In (17), λ and µ are the second-order Lamé
constants of the medium with defects, and LMκ is an addi-
tional constant which becomes zero in the limit of vanish-
ing defect concentration. This potential gives rise to the
following dependence of the first Piola–Kirchhoff stress
tensor on displacement gradients: (0)T T Tαβ αβαβ= + ∆ with
(0)Tαβ being the linear part of Tαβ and
1
, , , 2
2
( ) LM
IT u u u I
Iαβ αλ α λ γ λ γ β λβ λβ
∆ = δ + µ + κ δ + η
.
(18)
948 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7
Nonlinear effects of micro-cracks on acoustic surface and wedge waves
We approximate Tαβ∆ by keeping on the right-hand side of
(18) only the dominant non-analytic contributions,
LMT ζζ
αβ γλ λγ αβ αβ
γλ λγ
ε
∆ ≈ κ ε ε δ + ε
ε ε
. (19)
3. Rayleigh waves
For the investigation of the influence of nonlinearity on
the propagation of surface acoustic waves, one may follow
well-established methods that were pioneered in this con-
text by the authors of [21–25]. The displacement field ( )uα ,
depending on the material coordinates ,xα 1, 2, 3,α = is
written as an asymptotic expansion
( ) 2 ( ) 3( )I IIu u u Oα α α= ν + ν + ν , (20)
where the expansion parameter ν is a typical strain, and
a stretched coordinate 1X x= ν is introduced.
The first-order field is a linear superposition of a fun-
damental surface acoustic wave with phase velocity Rv ,
frequency Rkvω = and its higher harmonics,
( )
1 3 3 1
1
( , , ) ( ) exp ( ( )) c.c.I
Ru x x t w kx i k x v t A
∞
α α
=
= − +∑
(21)
Here, c.c. stands for the complex conjugate of the preced-
ing expression. We assume that the elastic medium fills the
half-space x3 > 0. Consequently, wα(z) decays exponential-
ly for z → ∞. For homogeneous elastic media without de-
fects, a nonlinear evolution equation for the displacement
amplitudes A
(with appropriate normalization) was de-
rived [21–25], which may be written in the form
1
2
1
( ) ( ) m m
m
A k m m F m A A
X
−
−
=
∂
= − −
∂ ∑
2 * *
1
2 ( ) ( ) ( )n
m m
m
k m m m F m A A
∞
−
= +
− −∑
(22)
with exponent n = 1. The kernel function F in this evolu-
tion equation depends on the ratio of two wavenumbers. Its
value for the argument 1/2 is directly related to the acous-
tic nonlinearity parameter (ANP) for surface acoustic
waves propagating in the elastic medium in the absence of
defects. We call this quantity βR. For isotropic media, fol-
lowing the definition of the ANP for Rayleigh waves in-
troduced by Herrmann et al. [10,11] and normalizing the
functions wα(z) such that
2 2
3 2 2(0) 1
2
R R
L T R
v vw i
v v v
= − −
−
, (23)
this relation is
2
8 (1 2)L
R
R
v F
v
β =
. (24)
In (23), (24), the quantities vL, vT, vR are the speeds of lon-
gitudinal and transverse bulk waves and of Rayleigh
waves, respectively. The evolution equation (22) implies
that the growth rate for the second harmonic of a time-
harmonic surface acoustic input wave is proportional to the
square of the input wave’s amplitude.
For isotropic media, explicit analytic expressions for
the kernel (and hence for the ANP) in terms of the two
decay constants of the Rayleigh waves and the two second-
order and three third-order elastic constants are given by
Parker [24], Zabolotskaya [25] and Knight et al. [26]. For
an isotropic material with density of potential energy
2 3 4
1 2 1 1 2 1 2 3 3
1 1 4 ( )
2 6 3
I I I I I I OΦ = λ + µ + ν + ν + ν + η , (25)
where 1 2 3, , , ,λ µ ν ν ν are the two second-order and three
third-order Lamé constants, the ANP has the form
0 1 1 2 2 3 3( ) ( ) ( )R b b b bβ = + ν µ + ν µ + ν µ . (26)
For a given normalization, the coefficients ,nb 0, ..., 3,n =
depend on the Poisson ratio only.
We now consider the influence of micro-cracks on the
nonlinear properties of Rayleigh waves. Treating the stress
deviations Tαβ∆ , introduced in (13)–(16), as perturbations,
we regard the coefficients , 0, 1, 2,n nτ = as being of first
order in the expansion parameter ν, 0 0ˆτ = ντ , 1 1ˆτ = ντ ,
2 2ˆτ = ντ . We note that these coefficients can be made arbi-
trarily small by reducing the concentration of defects.
For simplicity, we consider here only the case of negli-
gible friction within the micro-cracks. In this case, the off-
diagonal elements of Tαβ∆ give rise to a correction of the
second-order elastic constants, which is of first order in ν
and leads to anisotropy, but does not directly affect har-
monic generation or nonlinear combination processes to
leading order in ν . Therefore, we shall not account for these
off-diagonal components in the following.
When inserting the expansion (20), with (21) for the first-
order field, into the equation of motion for the displace-
ment field and keeping only terms of second order in ν ,
we obtain
2
(0)( ) ( )
2
II IIu C u
x xt
α γαβ γδ
β δ
∂ ∂ ∂
ρ − =
∂ ∂∂
(0) (0)
1 1C C
x xα γβ αβ γ
β β
∂ ∂
= + × ∂ ∂
( )
3 1
1
( ) exp( ( )) .I
Rw kx i k x v t A T
X x
∞
γ αβ
β=
∂ ∂
× − + ∆
∂ ∂∑
(27)
The quantity ( )ITαβ∆ is obtained by inserting into (13)–(16)
the displacement gradients ( )
,
Iuα β for αβε and dividing by ν .
The right-hand side of (27) is a 2π-periodic function of
1( )Rk x v tξ = − . In particular, ( )ITαβ∆ may be represented as
a Fourier series
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 949
M. Rjelka, P.D. Pupyrev, B. Koehler, and A.P. Mayer
( ) ( )
1 3 3( , , ) ( ) exp ( )IT x x t T x i
∞
αβ αβ
=−∞
∆ = ∆ ξ∑
. (28)
The same argument was used by Pecorari [17] in connec-
tion with non-collinear mixing of acoustic bulk waves.
A physically meaningful solution for ( )IIu has to be a bound-
ed 2π-periodic function of ξ, too, which may be written in
the form of a Fourier series,
( ) ( )
1 3 3( , , ) ( ) exp ( )IIu x x t U x i
∞
α α
=−∞
= ξ∑
. (29)
When inserting (28) and (29) in (27), multiplying both
sides of this equation by *
3( ) exp ( )w nkx inα − ξ , where n is a
positive integer, summing over α, integrating over ξ from
0 to 2π, over 3x from 0 to infinity, performing two integra-
tions by parts, thereby obeying the traction-free boundary
conditions at the surface, the following infinite set of dif-
ferential equations is obtained:
* ( )
3 3 3
0
( ) ( ) ( )n
R niN A D nk w nkx T x dx
X
∞
β α αβ
∂ = ∆ ∂ ∫ . (30)
Here we have introduced the linear operator
1 3 3( )D q iq d dxα α α= δ + δ and the coefficient
2 *
0
2 ( ) ( )R RN v w z w z dz
∞
α α= ρ ∫ . (31)
We note that the quantity (0)
11/RN c depends on the Poisson
ratio of the matrix material only.
In order to determine the initial growth rate of the nth
harmonic amplitude for an initially sinusoidal Rayleigh
wave, we have to insert the first-order solution (21) with
0A =
for 1≠ into (13) and evaluate the right-hand side
of (30). Defining ψ as the phase angle of 1A , we obtain
[( )
1 0 311 ˆ sin ( ) ( )I
RT A f kx∆ = −τ ξ + ψ +
]1 3ˆ | sin ( ) || ( ) |Rf kx+ τ ξ + ψ , (32)
where we have defined the depth profile wα of the Ray-
leigh waves such that 1(0)w is real, and we have defined
the real function
(0)
12
1 3(0)
11
( ) 2 ( ) ( )R
c df z w z i w z
dzc
= −
. (33)
With (33) and (34), we obtain from (30)
1| |A k A
X
∂
= γ
∂
(34)
with the coefficients γ
being equal to zero for 1> odd and
1
0
ˆ
e ( )| ( ) |
( 1)( 1)
i
R R
R
f z f z dz
N
∞
ψτ
γ =
− + π ∫
(35)
for even . Note that the modulus of γ
is independent of
the normalization of the depth profile of the Rayleigh wave
displacement field. Also, the quantity 2
1ˆ/Rvγ ρ τ
depends
on the Poisson ratio of the matrix material, only. This de-
pendence is shown in Fig. 1 for 2= .
From (34) it also follows that a time-harmonic Rayleigh
input wave with amplitude 1A generates immediately all
even harmonics with initial growth rates being proportional
to 1| |A . We also note that the initial efficiency of higher
harmonic generation decreases, 2−γ ∝
for → ∞ , be-
cause the integral on the right-hand side of (35) becomes
proportional to 1−
for large .
So far, we have considered Rayleigh waves propagating
in a medium with micro-cracks fully aligned according to
the inset of Fig. 1. We now move to elastic media contain-
ing a distribution of micro-cracks with perfectly random
orientation. When adopting the Lyakhovsky–Myasnikov po-
tential (17) as a model for this case, we may proceed in the
same way as in the aligned case, using (19) instead of
(13)–(16) for the stress deviation ( )ITαβ∆ and insert there the
displacement gradients ( )
,
Iuα β corresponding to a time-har-
monic Rayleigh input wave. It is then straightforward to
show that the initial growth of higher harmonics is again
described by (34) with coefficients γ
which can be non-zero
only for even harmonics. The dependence of 2
2 /R LMvγ ρ κ
on the Poisson ratio of the matrix material is shown in Fig. 1.
4. Acoustic wedge waves
In this section, elastic wedges with opening angle θ are
considered that contain a spatially homogeneous distri-
bution of micro-cracks which are aligned such that the
normal of their faces are along the x1 direction. The x3 di-
rection points along the apex line of the wedge and is
the propagation direction of acoustic wedge waves. The
Fig. 1. The quantity 2| |γ in units of 2
10.1 Rvτ ρ for fully aligned
micro-cracks (diamonds) and in units of 2
LM Rvκ ρ for random
orientation (circles) as function of the Poisson ratio. Inset: Orien-
tation of aligned micro-cracks with respect to surface and propa-
gation direction of Rayleigh waves.
950 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7
Nonlinear effects of micro-cracks on acoustic surface and wedge waves
geometry with the orientation of the coordinate system is
shown in Fig. 2. The matrix material is assumed to be iso-
tropic.
In isotropic elastic wedges, acoustic modes may be cha-
racterized as either even or odd, depending on their be-
havior with respect to the reflection at the wedge’s mid-
plane (x1 = x2). For opening angles smaller than 90° and
Poisson ratios in the range of most materials of practical use,
only odd wedge waves (anti-symmetric flexural modes, ASF
modes) exist [27]. Unlike Rayleigh waves, ASF modes do
not resonantly generate even harmonics [28]. This means
in particular, that the growth rate of the second harmonic
of an ASF mode and hence the acoustic nonlinearity pa-
rameter of such modes is zero.
For wedge waves of even symmetry, an evolution equa-
tion for their slowly varying amplitudes A
was derived by
Krylov and Parker [29] which can be brought into the
form (22) with exponent n = 2. In analogy to the case of
Rayleigh waves, the growth rate of the second harmonic
and consequently an acoustic nonlinearity parameter for
even acoustic wedge waves would be proportional to the
value of the kernel function F in the evolution Eq. (22) at
the argument 1/2.
In the following, we shall confine our discussion to
ASF modes. The displacement field of a linear time-har-
monic wedge wave with wavenumber k and amplitude A
has the form
1 2 3 1 2 3( , , , ) ( , ) exp ( ( )) c.c.Wu x x x t w kx kx ik x v t Aα α= − + ,
(36)
where Wv is the phase velocity of the wedge wave. The
profile functions ( , )w x yα have to be normalized appropri-
ately and decay to zero for large distances from the apex
line, 2 2x y+ → ∞ . They may be determined numerically
via an expansion in a double series of Laguerre functions
after a conformal mapping of the wedge with opening an-
gle θ into a rectangular wedge [30]. This method was also
applied to compute the kernel function F in the evolution
Eq. (22) for certain propagation geometries in anisotropic
wedges [31].
The distribution of micro-crack orientations shown in
Fig. 2 breaks the reflection symmetry with respect to the
wedge’s midplane, and resonant generation of even har-
monics will be allowed. Following the same procedure as
described in the previous section in the context of Rayleigh
waves, starting with an asymptotic expansion (20), where
the first-order field is now a superposition of wedge waves,
( )
1 2 3( , , , )Iu x x x tα =
1 2 3
1
( , ) exp ( ( )) c.c.Ww kx kx i k x v t A
∞
α
=
= − +∑
, (37)
and the stretched coordinate is X = ν x3. At second order in
the expansion parameter ν, we obtain an equation of the
form (27) with the right-hand side modified. In particular,
3 1( ) exp ( ( ))Rw kx i k x v tα − is replaced by 1 2( , )w kx kxα ×
3exp ( ( ))Wi k x v t× − and ( )ITαβ∆ depends on all three Car-
tesian coordinates. In analogy to the case of Rayleigh
waves, its dependence on x3 is via 3( )Wk x v tζ = − , and it is
a 2π-periodic function of this quantity and may therefore
be expanded in a Fourier series of the form (28) with Fou-
rier coefficients ( )
1 2( , )T x xαβ∆
. Proceeding now in the same
way as in the previous section (see also [31] for the appli-
cation of this approach), we obtain the analog of (30),
WiN A
X
∂
=
∂
* ( )
1 2 1 2 1 2( ) ( , ) ( , )
S
k D k w kx kx T x x dx dxβ α αβ = ∆ ∫∫
(38)
with
2 2 *
1 2 1 2 1 22 ( ) ( , ) ( , )W W
S
N v k w kx kx w kx kx dx dxα α= ρ ∫∫ .
(39)
In contrast to its definition in the previous section, the line-
ar operator ( )D qα in (38) is defined as 1 1( )D q xα α= δ ∂ ∂ +
2 2 3x iqα α+ δ ∂ ∂ + δ . The integrals in (38) and (39) have to
be carried out over the cross section S of the wedge in the
x1x2 plane. Note that WN is independent of k and .
We now determine the Fourier coefficients ( )Tαβ∆
for
the stress deviation Tαβ∆ given by the expressions (13)–(15),
which correspond to a distribution of aligned micro-cracks
as shown in Fig. 2. When inserting in these expressions the
first-order displacement field (37) with 0A =
for 2≥
and 1 1| | exp ( )A A i= ψ , (38) takes on the form (34). The
coefficients γ
are zero for being an odd integer number.
For even they are related to the displacement field of the
ASF wedge modes via
2
1ˆ e ( , )| ( , ) |
( 1)( 1)
i
W W
W S
i f x y f x y dxdy
N
ψτ
γ =
− + π ∫∫
,
(40)
where we have defined
(0)
12
1 2 3(0)
11
( , ) 2 ( , ) ( , ) ( , )W
c
f x y w x y w x y iw x y
x yc
∂ ∂ = + + ∂ ∂
.
(41)
Making use of the isotropy of the matrix material, we de-
fine the profile functions ( , )w x yα such that 1w and 2w are Fig. 2. Wedge geometry.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 951
M. Rjelka, P.D. Pupyrev, B. Koehler, and A.P. Mayer
real, while 3w is imaginary, and consequently Wf is real.
In the same way as in the case of Rayleigh waves, we find
that γ
is independent of the normalization of the profile
functions and that 2−γ ∝
for large . In Fig. 3, numerical
results are presented for the dependence of 2γ on the open-
ing angle θ in the range 55 90° ≤ θ ≤ °. The quantity 2γ is
the initial growth rate of the second harmonic of a time-
periodic wedge wave apart from the factor 1| |k A . The data
refer to wedges made of steel with micro-cracks aligned
according to Fig. 2(b). The phase angle ψ was chosen to be
–π/4. The inset of Fig. 3 shows the phase velocity Wv of
the wedge waves as function of θ. For opening angles
60θ ≤ °, a second branch of ASF modes exists. The results
in Fig. 3 demonstrate that the ASF modes of the branch
with higher speed have 2γ values that differ from those of
the branch with lower speed by their sign and have consider-
ably smaller magnitude than the latter.
5. Conclusions
In summary, a method has been presented for the calcu-
lation of the initial growth rates of higher harmonics of
time-harmonic Rayleigh and wedge waves propagating in
elastic media with a spatially homogeneous distribution of
sub-wavelength micro-cracks. It is based on the effective
stress-strain relation of the system, which is non-analytic in
the case of flat micro-cracks. Numerical results have been
presented for the initial growth rate of the second harmonic.
A basic difference between Rayleigh and anti-symmet-
ric flexural wedge waves pertains to the nonlinearity of the
pure matrix material. In the case of Rayleigh waves, the
matrix material contributes to the growth rate of the second
harmonic a term proportional to the square of the funda-
mental amplitude, which becomes dominant with decreas-
ing concentration of micro-cracks. In the case of wedge
waves, the growth rate of the second harmonic vanishes in
the absence of micro-cracks. Consequently, acoustic wedge
waves are expected to be particularly sensitive to textured
micro-crack distributions that break the reflection sym-
metry with respect to the wedge’s midplane. Since such
distributions cause the linear elastic properties of the
wedge to become anisotropic, too, there will be a small
contribution to the initial growth rate of the second har-
monic proportional to the square of the fundamental ampli-
tude, which decreases to zero in the limit of vanishing mi-
cro-crack concentration.
A remarkable feature, generated by the non-analyticity
of the stress-strain relation and already known for acoustic
bulk waves (see, for example, [17]), is the immediate
growth of infinitely many (even) harmonics with growth
rates depending linearly on the input wave’s amplitude.
The observability of harmonics higher than the second is
clearly limited by attenuation in the metallic samples with
defects.
In the case of aligned micro-cracks, the initial growth
rates are proportional to the parameter τ1 in the non-
analytic stress-strain relation. This parameter can be quan-
tified by finite-element simulations, which have been car-
ried out for homogeneous distributions of aligned flat mi-
cro-cracks in the high-strength metallic alloy IN718 [13].
For the ratio of |τ1| and the elastic constant c11 of IN718, a
value of more than 0.005 was found for a micro-crack con-
centration of 0.014 per diameter cubed.
Finally we note that the approach presented for the
quantitative determination of growth rates of higher har-
monics can easily be extended to nonlinear frequency mix-
ing with two guided input waves.
The authors dedicate this work to the memory of Ar-
nold Markovich Kosevich with deep admiration of the va-
luable contributions he made to the fields of crystal defects
and nonlinear waves, which this paper pertains to. One of
us (A.P.M.) had the pleasure of cooperating with him on
the latter subject.
Financial support by Deutsche Forschungsgemeinschaft
(Grant No. MA 1074/11) is gratefully acknowledged.
________
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1. Introduction
2. Stress-strain relation for an elastic medium with micro-cracks
3. Rayleigh waves
4. Acoustic wedge waves
5. Conclusions
|
| id | nasplib_isofts_kiev_ua-123456789-176201 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T16:03:21Z |
| publishDate | 2018 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Rjelka, M. Pupyrev, P.D. Koehler, B. Mayer, A.P. 2021-02-04T07:09:43Z 2021-02-04T07:09:43Z 2018 Nonlinear effects of micro-cracks on acoustic surface and wedge waves / M. Rjelka, P.D. Pupyrev, B. Koehler, A.P. Mayer // Физика низких температур. — 2018. — Т. 44, № 7. — С. 946-953. — Бібліогр.: 31 назв. — англ. 0132-6414 PACS: 43.35.+d, 46.40.Cd, 62.30.+d https://nasplib.isofts.kiev.ua/handle/123456789/176201 Micro-cracks give rise to non-analytic behavior of the stress-strain relation. For the case of a homogeneous spatial distribution of aligned flat micro-cracks, the influence of this property of the stress-strain relation on harmonic generation is analyzed for Rayleigh waves and for acoustic wedge waves with the help of a simple micro-mechanical model adopted from the literature. For the efficiencies of harmonic generation of these guided waves, explicit expressions are derived in terms of the corresponding linear wave fields. The initial growth rates of the second harmonic, i.e., the acoustic nonlinearity parameter, has been evaluated numerically for steel as ma-trix material. The growth rate of the second harmonic of Rayleigh waves has also been determined for micro-crack distributions with random orientation, using a model expression for the strain energy in terms of strain in-variants known in a geophysical context. The authors dedicate this work to the memory of Arnold Markovich Kosevich with deep admiration of the valuable contributions he made to the fields of crystal defects and nonlinear waves, which this paper pertains to. One of us (A.P.M.) had the pleasure of cooperating with him on the latter subject. Financial support by Deutsche Forschungsgemeinschaft (Grant No. MA 1074/11) is gratefully acknowledged. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Распространение волн в неоднородных системах Nonlinear effects of micro-cracks on acoustic surface and wedge waves Article published earlier |
| spellingShingle | Nonlinear effects of micro-cracks on acoustic surface and wedge waves Rjelka, M. Pupyrev, P.D. Koehler, B. Mayer, A.P. Распространение волн в неоднородных системах |
| title | Nonlinear effects of micro-cracks on acoustic surface and wedge waves |
| title_full | Nonlinear effects of micro-cracks on acoustic surface and wedge waves |
| title_fullStr | Nonlinear effects of micro-cracks on acoustic surface and wedge waves |
| title_full_unstemmed | Nonlinear effects of micro-cracks on acoustic surface and wedge waves |
| title_short | Nonlinear effects of micro-cracks on acoustic surface and wedge waves |
| title_sort | nonlinear effects of micro-cracks on acoustic surface and wedge waves |
| topic | Распространение волн в неоднородных системах |
| topic_facet | Распространение волн в неоднородных системах |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/176201 |
| work_keys_str_mv | AT rjelkam nonlineareffectsofmicrocracksonacousticsurfaceandwedgewaves AT pupyrevpd nonlineareffectsofmicrocracksonacousticsurfaceandwedgewaves AT koehlerb nonlineareffectsofmicrocracksonacousticsurfaceandwedgewaves AT mayerap nonlineareffectsofmicrocracksonacousticsurfaceandwedgewaves |