MODELLING THE PROPAGATING EDGE BENDING WAVES
IN PLATES AND LAMINATES
D.D. Zakharov
ESBE, London South Bank University
ABSTRACT. A propagating wave of the bending nature that is localised near the
stress free edge of an anisotropic laminate is studied with a view to application in
NDT. For a small thickness-to-wavelength ratio, a semi-analytical low frequency
asymptotic approach is introduced suitable for modelling any anisotropy and layup.
The solution of the corresponding dispersion equation is presented. Applicability of
the higher order asymptotics to larger thickness-to-wavelength ratios is discussed. As
shown, in some cases it approximates the dispersion curves by simple analytical
equations valid for rather short waves.
Introduction.
This paper is devoted to the properties of the elastic bending wave propagating
along the stress free edge. As known, since the first paper by Konenkov (1) in 1960
many authors investigated this type of waves on the basis of classical Kirchhoff’s
plate theory. Thurston, McKenna (2) and Sinha (3) rediscovered them in 1974 and
Thurston, Boyd, McKenna (4) and Krylov (5) considered flexural waves guided by the
wedge tip with a transition to the limit cases of the infinitely thin wedge (i.e. to the
bending wave of thin plate). Then further progress has been focused on the interface
bending waves, which represent an analogue of the Stonely waves at the edge-by-edge
junction of the semi-infinite plates. Such waves have been studied by Zilbergleit and
Suslova (6) and Kouzov et al.(7) for different configurations of nodal contact.
The edge bending wave in structures, immersed in a relatively light fluid, has
been described by Krylov (8) (fluid-loaded wedge), and by Abrahams and Norris (9).
The latter has shown that in the immersed plate the edge waves exist in the limited
frequency band and may propagate without energy loss.
As far as anisotropic materials in vacuum are concerned, Norris (10) and
Belubekyan and Engibaryan (11) treated a simplest case of the orthotropic plate with
principal axis directed along the edge. More general case of anisotropy has been
studied by Thompson, Abrahams and Norris (12) and by Zakharov and Becker
(13,14), with the relevant analysis of the possible reverse power flow direction. Fu
(15) investigated formal aspects of the edge wave existence and uniqueness. The case
of anisotropic laminate with unsymmetrical layup has been investigated recently by
Zakharov (16) and Fu (17).
For what follows we also have to mention results of 3D finite element analysis
of the ridge surface waves, obtained by Lagasse (18) and by Burridge and Sabina
(19,20). The wave speed, experimentally measured by Lagasse and Oliner (21), is in a
good agreement with FEM calculation (21). Comparison of these results with those of
the classical Kirchhoff’s theory demonstrates a satisfactory prediction at low
frequencies only. For higher frequency the FEM calculation is presented in the paper
of Liu et al. (22).
As shown by Norris, Krylov and Abrahams (23) at the higher frequencies the
wave speed can be also predicted using Timoshenko-Reissner-Mindlin (TRM) theory.
Based on the rotation inertia assumption with respective shear correction of the plate
equations and boundary conditions this theory is not asymptotically justified but is
2-50
wide spread in engineering practice. The consistent procedure we refer below is to
consider an asymptotic series with respect to the small relative thickness, and to
deduce the high order solving equation using recurrent formulas resulting from 3D
dynamic elasticity. Boundary conditions are refined on analysing the boundary layer
behaviour. Among the first researchers, who developed such methods of asymptotic
integration, were Friedrichs and Dresller (24), Reissner (25), Green (26) and
Goldenweiser (27). In what follows we use this asymptotic method to clarify the
nature of edge bending wave in isotropic plate and its sensitivity to the perturbations
of the boundary conditions and of the main differential operator.
To begin with we consider the Kirchhoff’s type equations for the
unsymmetrical anisotropic laminate, which has an asymptotic error of the second
order for monoclinic anisotropy of materials. The respective classical boundary
conditions have the error of the first order and require additional term when
improving. In general this is an open question but for the case of an orthotropic plate
the desired solution of the boundary value problem with uniform asymptotic error is
obtained and analysed below. As in the classical case this solution contains a partial
wave mostly responsible for the localisation. We show that it is low sensitive to the
perturbed boundary condition but the influence of the iteration of the main differential
operator can be essential. This influence is investigated for the case of isotropic plate.
The trick of the suggested approach is in the asymptotic correction of this main part of
wave only. Finally we arrive at the wave speed formula valid for high frequencies or
relatively thick plate. These results are compared with experimental data and
discussed.
STATEMENT OF THE PROBLEM
of
Consider a semi-infinite thin elastic laminate of total thickness H = 2h , made
N anisotropic layers with stiffness constant matrices G j = g mn j
(m, n = 1,2,3,4,5,6)
and mass densities ρ j , and occupying a region x2 ≥ 0 ,
∞ < x1 < ∞ in their plane. The indices correspond to the substitution
1,2,3,4,5,6 ↔ 11,22,33,23,13,12 . According to the low frequency 2D theory of
laminates (28,29) introduce the normal deflection w = w(x, t ) , slopes θα = −∂α w
(α = 1,2) and the vector of the longitudinal displacements u = u 0 (x, t ) − z grad w ,
where z is a transverse coordinate. Assuming that the plate faces z = m h are stress
free the longitudinal stresses in each layer acquire the form
σ αβj = χ αβ (Γ j )(u 0 − z grad w ) ,
⎡γ 11
⎡ χ 11 ⎤
⎢ χ ⎥ (Γ ) ≡ ⎢γ
⎢ 16
⎢ 12 ⎥ j
⎢⎣γ 12
⎢⎣χ 22 ⎥⎦
where Γ j ≡ γ pq
j
( p, q = 1,6,2)
γ 16 γ 12 ⎤ ⎡ ∂ 1
γ 66 γ 26 ⎥⎥ ⎢⎢∂ 2
γ 26 γ 22 ⎥⎦ j ⎢⎣ 0
0⎤
∂ 1 ⎥⎥ ,
∂ 2 ⎥⎦
are the effective stiffness matrices whose elements
345
γ pq ≡ Gqp G0 equal the ratio of minors from G j ; namely G0 ≡ G345
and the minor
Gqp is obtained by adding p -th row q -th column to G0 at the bottom and at the right
hand side, respectively. Thus, the stress resultants Qαβ and couples M αβ are obtained
by integration across the thickness
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Qαβ = χ αβ (D1 )u 0 − χ αβ (D 2 ) grad w , M αβ = χ αβ (D 2 )u 0 − χ αβ (D 3 ) grad w ,
(
)
D k ≡ k −1 ∑ z kj+1 − z kj Γ j
j
and the transverse shear stress resultants Qαz = ∂ β M αβ . The membrane integral
stiffnesses D1 come from the plane stress problem, the bending integral stiffnesses
D 3 correspond to the bending problem and the nonzero mixed stiffnesses D 2 cause
the coupling of two problems. The equations of motion for this case is derived in (29)
∂ β χ αβ (D1 )u 0 − ∂ β χ αβ (D 2 ) grad w = 0
,
2
2
− ∂ αβ χ αβ (D 2 )u 0 + ∂ αβ
χ αβ (D 3 ) grad + ρH∂ t2 w = 0
[
]
where ρ is an average mass density. For the case of the symmetrical layup D 2 = 0
and the quasistatic plane stress problem is decoupled from the dynamic bending
problem. The boundary conditions to the system are formulated in terms of the
displacements (and slopes θ = −grad w ) and in terms of the longitudinal forces,
bending moments and Kirchhoff’s force N αz = Qαz + ∂ β M αβ (28,29).
CASE OF THE UNSYMMETRICAL LAYUP
Let the laminate oscillates with frequency ω under the assumption that its
edge x 2 = 0 is stress free. Consider the wave propagating along the edge with the
exponential decay away from the edge, i.e.,
w = w∗ei [k1 ( x1 + λx 2 )−ωt ] , u 0 = isu∗ei [k1 ( x1 + λx 2 )−ωt ] (Im λ ≥ 0 )
with constant magnitudes w ∗ , uα∗ . The desired solution satisfies the equations of
motion and the integral form of the boundary conditions at x 2 = 0 :
Q12 = Q22 = 0 , M 22 = 0 , N 2 z = 0 .
Introduce characteristic polynomials of the second order pαβ (λ ) , third order
pα 3 (λ ) and forth order p33 (λ ) , p4 (λ ) by the following substitution
∂ β χ αβ (D1 )u 0 = −ik13 pαβ (λ )uβ ,
p12 = p21 ,
p4 (λ ) ≡ p11 p22 − p12 p21 ,
2
− ∂ β χ αβ (D 2 ) grad w = ik13 pα 3 (λ )w , − ∂αβ
χ αβ (D2 )u 0 = − k14 pβ 3 (λ )uβ ,
2
∂αβ
χ αβ (D3 ) grad w = k14 p33 (λ )w .
Hence, the magnitudes and λ satisfies the linear system of algebraic equations
∗
− p13
p12
⎡ p11
⎤ ⎡ u1 ⎤
⎢ p
⎥ ⎢ u∗ ⎥ = 0
− p23
p22
⎢ 21
⎥⎢ 2 ⎥
2 −4 ⎢ ∗ ⎥
⎢⎣− p13 − p23 p33 − ρHω k1 ⎥⎦ ⎣ w ⎦
with the characteristic polynomial of eighth order p8 (λ ) ≡ det P ω =0 , which yields the
relations for magnitudes
p p − p23 p12 ∗
p p − p21 p13 ∗
u1∗ = 13 22
w , u2∗ = 11 23
w .
p4
p4
and the polynomial equation for λ
p8 (λ ) p8 (0) − Ω 4 p 4 (λ ) p 4 (0) = 0,
Ω 4 ≡ VK2 VB2 = ρHω 2 k1−4 p 4 (0 ) p8 (0 ) ,
2-52
VB2 ≡ k12 p8 (0) {ρHp4 (0 )} ,
where V K = ω k1 is the phase speed and V B is the speed of bending wave,
propagating along the edge but not satisfying the stress free conditions at its edge. It is
easily to show that p33 (λ ) , p4 (λ ) and p8 (λ ) are positively determined for real λ and
the polynomial equation may have complex root only when
⎧
p (λ ) p8 (0 ) ⎫
Ω ∗ ≡ ⎨sup 4
0 < Ω < Ω∗ ,
⎬ .
⎩ λ∈R p 4 (0) p8 (λ ) ⎭
The physical meaning of this restriction is the inequality for the wave speed
VK < VB Ω ∗2 . The roots are conjugated and we choose only those λ k (Ω ) , k = 1,2,3,4
whose imaginary part is positive. The boundary conditions in terms of stress resultants
and couples
(Qαβ , qαβ (λ ) ↔ M αβ , mαβ (λ ))
Qαβ = − k12 qαβ (λ )w∗ei [k1 ( x1 + λx 2 )−ωt ] ,
−1 4
Nαz = −ik13nαz (λ )w∗ei [k1 ( x1 + λx 2 )−ωt ]
yield another system of equations with respect to Ω
1
⎡ q12
q122
q123
q124 ⎤ ⎡ w1∗ ⎤
⎢ 1
2
3
4 ⎥⎢ ∗ ⎥
⎢ q 22 q 22 q 22 q 22 ⎥ ⎢ w2 ⎥ = 0 , Δ(Ω ) = 0 ,
2
3
4 ⎥⎢ ∗ ⎥
⎢m122 m22
m22
m22
w3
⎢ 1
2
3
4 ⎥⎢ ∗ ⎥
⎣⎢ n 2 z n 2 z n 2 z n 2 z ⎦⎥ ⎣⎢ w4 ⎦⎥
↔ mαβ , nαz ), w1∗ ≡ w∗ (λ k ) , and Δ(Ω ) is the determinant
of the system. The latter equation can be solved numerically.
Let us demonstrate the behavior of the wave speed for laminate made of typical
orthotropic materials: carbon/epoxy composite with parameters ρ = 1500 [ kg / m 3 ],
E1 = 0.128 × 1015 , E 2 = 0.84 × 1013 , G12 = 0.46 × 1013 [ N / m 2 ], ν 12 = 0.37 and
glass/epoxy composite with ρ = 1400 , E1 = 0.26 × 1014 , E 2 = 0.18 × 1014 ,
G12 = 0.230 × 1013 , ν 12 = 0.14 . The total thickness is H = 0.2 × 10 −2 [m] and we
consider the cross-ply laminates with angle ψ between the principal axis 1 of the
upper layer and the axis x1 . The respective plots of Ω(ψ ) are shown in Fig.1 (the
curve 1 and 2 correspond to the carbon/epoxy and glass/epoxy, respectively).
k
≡ qαβ (λ k )
where qαβ
(q
αβ
Fig.1
Fig.2
2-53
CASE OF THE SYMMETRICAL LAYUP
Being a particular (decoupled) case with D 2 = 0 it can be obtained from
previous with taking into account that χ αβ (D 2 ) ≡ 0 , pα 3 (λ ) ≡ 0 and
p8 (λ ) = p33 (λ ) p 4 (λ ) .
The
polynomial
equation
for
λ
is
rewritten
as
p33 (λ ) − ρHω 2 k1−4 = 0 and the problem dimension is reduced twice. The numerical
results for a single-layer plate made of the above mentioned materials are shown in
Fig. 2.
THE REFINED BOUNDARY CONDITIONS
Now let us overview the above from the asymptotical viewpoint. For this
purpose consider a longitudinal scale L (minimal wavelength), natural small
parameter ε = h / L << 1 and timescale T = O(ε −1 ) . As it follows from 3D elasticity
consideration (28,29) under such assumptions the displacement field can be
subdivided into two components: an internal stress and strain state (at the distance of a
few thickness apart from the edge) and a boundary layer (near the edge). The internal
components can be represented in the form of asymptotic series
w = Lε −3 w 0 + εw1 + K ,
uα = Lε −2 uα0 + εuα1 + K ,
where displacements and stresses of different orders satisfy the chain of recurrent
relations. All the relations above for the displacements and stress couples and
equations of motions hold for the leading terms of internal stress and strain state
which is necessarily long-wave within the relative truncation error O(ε 2 ) to be
neglected. The respective boundary conditions coincide with the so-called natural or
variational boundary conditions in accordance with presented stress couples and
resultants, and displacements as given linear functions of z . But the error of these
boundary conditions is O(ε ) and exceeds that of the equation of motion, i.e., the
model is asymptotically non uniform. It is remarkable that the first order boundary
condition operates with the modified shear stress couple. The refinement requires
incorporating the second order terms associated with the out of plane boundary layer
localised in the vicinity of the plate edge (see (24)). In general, for laminate and for
the arbitrary anisotropy this is an open question. But something can be done in the
case of isotropic plate and orthotropic single layer plate whose principal axes
coincide with coordinate axes.
As it follows from the boundary layer analysis (27,30) when decreasing the
asymptotic error down to O(ε 2 ) the refined boundary conditions look as follows
(
(
)
∞
χ = ∑ (2n − 1)−5 ≈ 1.26049775 ;
M 22 + χh G12 G13 ∂ 1 M 12 = 0 ,
n =1
)
N 2z = 0 .
Hence, in the correct asymptotic formulation within the square relative error the edge
wave satisfies these boundary conditions and the equations of motion. The procedure
is similar to previous, but the system of equations with respect to Ω = Ω ∗ is modified
by its last row. The analysis performed shows that correction is small, can be
represented as Ω∗2 = Ω 2 + O k1hν 4 and numerically is less than 1%.
(
)
2-54
Fig.3
Fig.4
In Figs.3-4 the values at the origin correspond to the classical boundary
conditions (dimensionless speed does not depend on k1h ). The curves 1-6 in Fig.4
correspond to different Poisson’s ratio ν = 0, 0.8(3), 0.1(6), 0.25, 0.3(3), 0.41(6), 0.5 of
isotropic materials.
HIGH ORDER ASYMPTOTICS FOR THE ISOTROPIC CASE
Further improvement in the plate modelling meet considerable technical
obstacles caused by taking into account both the out of plane boundary layer and the
in plane one for the boundary conditions of high-order iterations. At the meantime the
refinement of the homogeneous Kichhoff’s equation may be easily done. The
equation of plate bending within the error O(ε 8 ) is deduced in (31) using 3D
elasticity and recurrent formulas for the high order terms of asymptotic expansions,
and reduced to a final form in terms of the middle plane deflection w as follows:
⎧
⎫
h2
h4
dΔ2 w + 2 ρh ⎨1 + a 0 (ν )h 2 Δ + a1 (ν ) 2 ∂ t2 + a 2 (ν ) 2 ∂ t2 Δ ⎬∂ t2 w = 0 ,
c2
c2
⎩
⎭
where c 2 is a shear wave speed and a m (ν ) depends on Poisson’s ratio. To be brief
we omit the closed form for them (for details see (32)). The evident analogue with
Kirchhoff’s equation consists in the replacement
⎧
h2 2
h4 2 ⎫ 2
2
2
2 ρh∂ t w → 2 ρh ⎨1 + a 0 (ν ) h Δ + a1 (ν ) 2 ∂ t + a 2 (ν ) 2 ∂ t Δ ⎬∂ t w .
c2
c2
⎩
⎭
For a free plate vibration mode, which satisfies the condition Δw < 0 ,
expressing Δw upon this analogue
2
⎫
⎛ ωh ⎞
ω 3(1 − ν ) ⎧⎪
ωh
6 ⎪
⎟
⎜
+
1
b
(
ν
)
b
(
ν
)
O
ε
Δw = −
+
+
⎨
⎬
1
2
⎜c ⎟
hc2
2 ⎪
c2
2 ⎠
⎝
⎪⎭
⎩
and after substitution we arrive at the high order equation of a plate in terms of
effective inertia
( )
{dΔ − 2 ρhω }w = 0 → ω n2 = ω 2 ∑ Bk (ν )(ωh c2 )k .
n
2
2
n
k =0
For indices n = 0,1,2,3 the relative truncation error of the respective high-order
(
2 n+ 2
)
. Of course, in the high order model the displacements are
operator is O ε
polynomial with respect to the transversal coordinate z . The assumption Δw < 0
2-55
corresponds to many practically important cases. In particular, it holds for the
propagation vibration modes. For the latter, correction of the equation of motion
appears to be more important then that of boundary conditions and the last feature
may also be characteristic of the most important components of the sought for edge
wave since they obey the same inequality. With taking into account the very small
difference caused by the corrected boundary conditions we may use for edge wave the
result based on the classical theory (1) and reduce the equation of the effective inertia
to the approximate dispersion relation as follows
ωH ⎧⎪
⎛ ωH ⎞
⎟
⎨∑ Bk ⎜⎜
c2 ⎪k =0 ⎝ 2c2 ⎟⎠
⎩
3
k
⎫⎪
⎬
⎪⎭
−1 2
12
⎧
⎫ ⎛ VK ⎞
6
≈⎨
⎬ ⎜⎜ ⎟⎟ .
2
⎩ − 1 + 3ν + 2 1 − 2ν + 2ν ⎭ ⎝ c2 ⎠
Fig.5
2
Fig.6
The evolution of the dispersion curves is shown in Fig.5. The letter K marks
the curve due to the classical Kirchhoff’s theory and that with refined boundary
conditions. Numbers 1,2,3 correspond to the iterations of the dispersion relation using
effective inertia (number of terms in the left hand side). The normalisation by the
speed of Rayleigh wave is chosen in accordance with the paper (21) and balls
correspond to the measurement performed by Lagasse et al. (21). In Fig.6 the results
of FEM calculation from (22) is plotted against our asymptotic results. As one can see
there is a good agreement practically up to the first cut-off frequency.
Discussion and conclusion
The performed analysis exhibits some essential properties of the edge bending
waves in laminates and plates. First, it concerns the possibility to describe this wave
in the unsymmetrical and symmetrical anisotropic laminate at relatively low
frequency using 2D theory of coupled bending and stretching. Second, in the case
where we may improve the boundary conditions we have seen the low sensitivity to it
– both for isotropic and orthotropic materials. But result may significantly differ when
principal axes are not parallel and perpendicular to the edge. Third, we may describe
the edge wave in isotropic plate at high frequency, where 2D plate theories are used
very seldom. Forth, it clarifies the nature of this wave (32). To this end the leading
partial wave mostly responsible for the edge wave localisation can be singled out. Its
behaviour is determined by the differential equation of plate bending and by its
improved versions of the highest orders. The sensitivity to the improvement of the
boundary condition can be neglected since even the first correction has the order
( )
O ν 4 and is numerically small. Other partial waves arisen in this representation
must be very sensitive to the perturbation of boundary conditions but the description
2-56
of leading part is a key, quite sufficient to simulate the edge wave propagation. Fifth,
such an important acoustic characteristic as wave speed can be predicted very easy
with high accuracy by a closed form expression presented.
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