- Email: [email protected]
Effect of interfacial contact on the propagation of flexural waves along a composite plate
J. Sound Vz3. (1965) z (z), 167-174
EFFECT
OF INTERFACIAL
OF FLEXURAL
CONTACT
WAVES R.
ALONG
A COMPOSITE
PLATE
E. N. THROWER
JONES AND
Road Research Laboratory,
ON THE PROPAGATION
Harmondsworth,
Middlesex, England
(Received IO November 1964) The wave propagation along a composite plate is shown to be considerably influenced by the interfacial coupling. Results are computed from the theory of the dispersion curves in composite plates of similar and dissimilar materials assuming either no slip or perfect slip at the interface. Comparison with experimental results obtained from the propagation of flexural waves on a road construction in which the surface and base layers could be treated as a free composite plate indicated that the “ no slip ” condition applied.
I. INTRODUCTION
A previous paper (I) has shown that the basic theory for the propagation of surface waves along road structures is closely linked with the problem of waves in plates and composite plates. An analysis for the propagation of flexural-type waves along a composite plate with no slip at the interface was given, and a few solutions were evaluated by desk computation for the lowest flexural wave branch which is required for the interpretation of experimental data obtained on roads (I). Desk computation of the solutions proved to be a long laborious procedure and a programme has since been prepared by one of the authors (E.N.T.) for use with a Pegasus Computer. A description of the programme and analysis is being given in a separate paper. Analytical long-wave approximations have also been worked out to give an easier computational solution. The present paper also extends the analysis to the case of composite plates in which there is perfect slip at the interface. Experimental results for the relation between the velocity and wavelength of flexural waves on road constructions having surface layers of known properties are then compared with theoretical relations for each type of interfacial contact. As in the previous analysis (I) the case of plane waves propagating in perfectly elastic materials is considered; the effect of deviations from these assumptions in practice were discussed in the previous publication.
2. BASIC
THEORY
It is assumed that the plane wave is propagating in the x-direction and that the depth dimension is in the x-direction. The interface is taken to be at x = o and the surfaces of the composite plate are at z = -Hi and z = +H2 respectively. The nomenclature is as used in the previous publication (I) and is as follows : c
phase velocity of the vibrations,
L
wavelength of the vibrations,
k
2_rr/L,
Hr and H2
G
are the thicknesses of the layers in a two-layer composite plate, is the shear modulus of elasticity of the material, 1’57
168
R. JONES AND E. N. THROWER p
is the density of the material,
p
is the velocity of shear waves in the material,
cc is the velocity of compressional y s2 ?? 9.2
waves in the material,
is the velocity of Rayleigh surface waves on a semi-infinite medium, =
C2
I---,,
fi =
C2
I--,
??
b =
012
s2
I+-.
k2
Subscript I is given to those symbols which refer to the material in the layer bounded by the planes z = o and z = -Hr. Subscript z refers to the material lying between z = o and z = H2. u, w are the displacements of the vibrations in the x and z directions respectively and are both taken to be proportional to exp [ik(x - ct)]. It can be shown (2) that the equations of motion in any one medium reduce to: 3
a22
=
a”lCr =
r24
9 az2
s2*
,
(1)
where
The normal stress is (3) The shear stress is P._=G$+$
( 1
(4)
In each medium convenient forms for the solutions to equation (I) are as follows : $ = :[Asinh(rz)+Bcosh(rz)]
exp[ik(x-ct)],
# = T[Ccosh(sz)+Dsinh(sz)]exp[ik(x-ct)],
(6)
in which the values of A, B, C and D appropriate to each of the two layers have to be determined from the conditions imposed upon the stresses and displacements at the surfaces and the interface. (a)
CASE OF NO INTERFACIAL
SLIP
Here the boundary conditions are as follows : atz
= -HI,
(P*,)l
= 0,
(7)
(Pm)1
=
(8)
0;
FLEXURAL WAVES ALONG A COMPOSITE
at z = 0,
(L)l
at z = H,,
=
PLATE
169
(~za)*~
(9)
(P*z)l = (%)2,
(IO)
a1 =
u2,
WI
=
w2;
(P*r)Z
=
0,
(‘3)
(Pm)2
=
0.
(14)
(11) (14
Equations (7) to (14) yield eight simultaneous homogeneous equations in the eight unknowns AI, Br, Cl, Dr, AZ, B2, C2, and D2. This system of equations has solutions different from zero if
) ((99e*R,-(32(g2e2Ql)
(15) where for c < fi2, /I1 0 =
2 -
cash (YH) cash (sH) + (&+Z)
2
sinh (rH) sinh (sH)
i.2 = (&-T)sinh(rH)sinh(sH) (16) x = g sinh(sH) cash (rH) - $sinh
(rH) cash (sH)
A = $sinh
(sH) cash (rH)
(rH) cash (sH) - $sinh
Approximations to these functions at wavelengths which are long in relation to H, and H2 can be written as follows:
e=
“$tfx{l+($+g)
X =
q(~+(;+-$$)(kH)~)
A=
(kH)2)
(17) @p(I+~;+~)(kN)2)
-’
1 +;(kH)2
R. JONES AND E. N. THROWER
170 where
2
=
b2-g
=
c’ p4
x
=
4
s2
r2
Y=F+p v=
(18)
*
s2 72 2 p-K2 (1,
When PI > c > BZthen Or, Szr, x1 and A 1 have the same form as equations (16) but 02, Q2, x2 and A 2 are as follows : d2 =
Q2
=
cash (r2H2)cos(s2H2)+
2-2
-
(
GA2 ~-__
43.2~2
43.2~2
b;k2
sinh (r2 H2) sin (s2 H2) )
1
$$+~)sinh(r2H2)sin(s2H2) 2
*. (19)
b2k .
~2 = 2~2 sm (~2H2) cash (r2 H2) - Fk
sinh (r2 H2)
cos (s2 ff2)
A2 = ~sinh(r2H2)cos(s2H2)+~ksin(s2H2)cosh(r2H2) 2
2
When c > /3r, j?, there exists a branch which is analogous to the principal mode of the symmetrical waves of Lamb (2) which has a long wavelength solution 2(I+~)[f&+f3+2(+)+2fh(~-1)]
=
(kH,)2(~-I)~(I-~(I-5$)+4(~-‘),
where h = H,/H,, t = G2/G1. For infinitely long waves equation
c2
(20)
(20)
reduces to
(I-$++$) (21)
-2 = 4
A
Equation (21) reduces to the velocity of long longitudinal waves in the remaining plate if HI or H2 is zero.
w
CASE OF INTERFACIAL
SLIP
Here the appropriate boundary conditions are : atz=
-HI,
(Paz)1 = 0,
(24
(Px*)l =
(23)
0;
FLEXURAL
WAVES
at z = 0,
ALONG
(%)*
Equations equation :
(22)
=
PLATE
17’ (24)
(PzA2,
(P.w)* = 0,
(25)
(P&Y,>2
=
0,
(26)
=
w,;
(27)
Pz*>2
=
0,
(28)
(Psz)z
=
0.
(29)
\
WI
at z = H,,
A COMPOSITE
to (29) yield a relation between L and c which reduces to the following =9,X*+ (2-b)
$&e2x1
(30)
= 0.
For c < 8, and c > &, 0 and x are given by equations (16) and their long wavelength approximations by equations (I 7). For c > j12 and c > fll, &’and x are given by equation (19) with appropriate suffixes. The long wavelength solution of equation (30) for the principal mode of waves corresponding to Lamb’s (2) symmetrical branch for a plate has the following form:
E-+$)][&4(‘-$)][d&+f’2&] = 0.
(31)
Equation (3 I) is interesting because it indicates that two limiting solutions are possible at infinitely long wavelengths, corresponding to the propagation of long plate waves in either of the two plates behaving independently of each other. 3. COMPOSITE PLATES OF THE SAME MATERIAL AND THICKNESSES An interesting case arises when the two layers of the plate have the same thickness and are composed of the same material. When there is no interfacial slip equation (I 5) reduces to Ax = o. (32) As was to be expected, these dispersion equations correspond to Lamb’s (2) antisymmetric (A = o) and symmetric (x = o) modes of propagation of the vibrations in a plate of thickness 2H, and the relations between phase velocity and wavelength for the principal modes are given by branches A(H) = o and x(H) = o in Figure I. The relative vertical displacements at the boundaries and interface for the principal mode are shown in Figure 2. When there is slip at the interface of two equal plates of the same material, equation (28) reduces to 8, = 0. (33) x = o again corresponds to the symmetric mode of waves in a plate of thickness 2H which has vertical particle displacements as shown in Figure 3. The equation 6 = o can be further split into A(H/2) = $sinhTcoshz-$sinhTcoshz
= o,
(34)
x(H/2) = ~~sinh$cosh’$-EsinhFcosh’:
= o.
(35)
01
R. JONES AND E. N. THROWER
172 2:.0,-
I*8-
x(H) = 0
I -6 -
I-4 -
No interfacial slip,branchesx(H) = 0, and A(H) =
No interfacial friction, branches x( H/2) = 0 AH/21 = 0 and x(H) = 0
I*2 Olh
,
01*8-
0
C
C
I
1
I
I
I
I
I
1
2
3
4
5
6
7
L s
Wavelength of vibrations o Thickness of composite pIarC
Figure I. Effect of interfacial contact on the phase velocity of vibrations in a composite platerof equal thicknesses (H) of the same material (Poisson’s ratio : 0~~5).
ia)
(b) Figure 2. Vertical displacements in a composite plate of two identical plates of equal thickness with no slip at their interface. (a) Symmetric type x = o. (b) Anti-symmetric type d = o.
Equation (34) gives the dispersion equation for flexural (or anti-symmetric) waves in a plate of thickness H. The wavelength with no interfacial friction is thus half of the wavelength of flexural waves of the same velocity in the same plates but with no interfacial slip (see Figure I). Equation (35) is the dispersion equation for symmetrical waves in each plate of thickness H so that the wavelength is half the wavelength of the other type of symmetrical wave
FLEXURAL
WAVES
ALONG
A COMPOSITE
PLATE
173
b(H) = o] in the same system, or of that in similar plates with no interfacial slip (see Figure I). The two types of symmetrical wave also give rise to different relative displacements of the boundaries and interface as can be seen from Figure 2, in which #f/z) = o produces anti-symmetric motion of the overall composite plate system.
(a)
(b)
Figure 3. Vertical displacements in same composite system as in Figure 2, but with interfacial slip. (a) Symmetric type x(H) = o; (b) Symmetric type x(H/z) = o; (c) Anti-symmetric type fl(H/z) = 0.
The three branches obtained when there is slip at the interface give a triple root at zero wavelength when c = y, implying Rayleigh waves on each free surface and also at the interface. There is also a double root at infinite wavelength for the x(H) = o and x(H/2) = o branches, as may also be deduced from equation (3 I) when suffix I is the same as suflix 2. 4.
COMPOSITE
PLATE
OF DISSIMILAR
MATERIALS
The case to be considered is the one for which solutions were obtained in an earlier paper (I). The two plates corresponding to the surface and base layers of a road had the following properties :
HI = 3’8in.,
H2 = 5.5 in.,
czl = a/3, = 2cr2 = 4fia.
(36)
The velocities (cc3 and pa) in the soil beneath the road were much lower than the velocities in the upper layers (cc1 = 8a3 = 20&) so that, to a first approximation, the surfacing and base were treated as a free composite plate (I). Further theoretical work is in hand to check the conditions under which the assumption is acceptable. It was noted previously that below p2, the wavelength of flexural vibrations in the composite plate (with no interfacial slip) agreed quite well with the sum of the wavelengths of flexural waves in the individual plates at the same velocity (I). Additional results by computer and from the long wavelength approximations in Figure 4 contirmed the agreement with the relation obtained by the additive procedure. Results obtained for the theoretical case of perfect interfacial slip are also given in Figure 4, and will be seen to be in agreement with the mean of the wavelengths from individual plates at corresponding velocities. The computed results for perfect slip are thus considerably different from the computed results assuming no slip.
R. JONES AND E. N. THROWER
‘74
The experimental results given in Figure 4, for a road construction having surface and base layers with properties corresponding to condition (36), will be seen to give excellent agreement with the computed results assuming no interfacial slip. This and other similar comparisons, have indicated that the layers of a road are well bonded in respect of elastic
@7
SO.6 t
IWBranch
from
I \
HI layer
alone
curve
I
0
I
2
4
,
t
6
Wavelength
8
I
IO
(ft)
Figure 4. Effect of inter-facial contact on the relation between velocity and wavelength of flexural waves in a composite plate (HI = 3.8 in., H2 = 5.5 in., aI = 26, = 2~9 = 4/32). ??Results from A Results from o Results from A Results from ? ?Experimental
exact solution, no slip at interface. exact solution, slip at interface. long wavelength approximation, no slip at interface. long wavelength approximation, slip at interface. results.
wave propagation, and, in subsequent work, only the no slip boundary condition will be used. The range of applicability of the additive procedure in closely bound composite plates of varying thicknesses and elastic moduli at velocities below &, and also the shape of the dispersion curves at velocities greater than /I2 have now been investigated; the analysis and results will be given in a later paper. ACKNOWLEDGMENT
The work described in this paper was carried out as part of the programme of the Road Research Board of the Department of Scientific and Industrial Research. The paper is published by permission of the Director of Road Research. REFERENCES I. R. JONES 1962 Brit.J.
uppl. Phys. 13 (I), 21-9. Surface wave technique for measuring the elastic properties and thickness of roads : theoretical development. 2. H. LAMB 1916 Proc. Roy. Sot. A., 93, I 16. On waves in plates.
EFFECT
OF INTERFACIAL
OF FLEXURAL
CONTACT
WAVES R.
ALONG
A COMPOSITE
PLATE
E. N. THROWER
JONES AND
Road Research Laboratory,
ON THE PROPAGATION
Harmondsworth,
Middlesex, England
(Received IO November 1964) The wave propagation along a composite plate is shown to be considerably influenced by the interfacial coupling. Results are computed from the theory of the dispersion curves in composite plates of similar and dissimilar materials assuming either no slip or perfect slip at the interface. Comparison with experimental results obtained from the propagation of flexural waves on a road construction in which the surface and base layers could be treated as a free composite plate indicated that the “ no slip ” condition applied.
I. INTRODUCTION
A previous paper (I) has shown that the basic theory for the propagation of surface waves along road structures is closely linked with the problem of waves in plates and composite plates. An analysis for the propagation of flexural-type waves along a composite plate with no slip at the interface was given, and a few solutions were evaluated by desk computation for the lowest flexural wave branch which is required for the interpretation of experimental data obtained on roads (I). Desk computation of the solutions proved to be a long laborious procedure and a programme has since been prepared by one of the authors (E.N.T.) for use with a Pegasus Computer. A description of the programme and analysis is being given in a separate paper. Analytical long-wave approximations have also been worked out to give an easier computational solution. The present paper also extends the analysis to the case of composite plates in which there is perfect slip at the interface. Experimental results for the relation between the velocity and wavelength of flexural waves on road constructions having surface layers of known properties are then compared with theoretical relations for each type of interfacial contact. As in the previous analysis (I) the case of plane waves propagating in perfectly elastic materials is considered; the effect of deviations from these assumptions in practice were discussed in the previous publication.
2. BASIC
THEORY
It is assumed that the plane wave is propagating in the x-direction and that the depth dimension is in the x-direction. The interface is taken to be at x = o and the surfaces of the composite plate are at z = -Hi and z = +H2 respectively. The nomenclature is as used in the previous publication (I) and is as follows : c
phase velocity of the vibrations,
L
wavelength of the vibrations,
k
2_rr/L,
Hr and H2
G
are the thicknesses of the layers in a two-layer composite plate, is the shear modulus of elasticity of the material, 1’57
168
R. JONES AND E. N. THROWER p
is the density of the material,
p
is the velocity of shear waves in the material,
cc is the velocity of compressional y s2 ?? 9.2
waves in the material,
is the velocity of Rayleigh surface waves on a semi-infinite medium, =
C2
I---,,
fi =
C2
I--,
??
b =
012
s2
I+-.
k2
Subscript I is given to those symbols which refer to the material in the layer bounded by the planes z = o and z = -Hr. Subscript z refers to the material lying between z = o and z = H2. u, w are the displacements of the vibrations in the x and z directions respectively and are both taken to be proportional to exp [ik(x - ct)]. It can be shown (2) that the equations of motion in any one medium reduce to: 3
a22
=
a”lCr =
r24
9 az2
s2*
,
(1)
where
The normal stress is (3) The shear stress is P._=G$+$
( 1
(4)
In each medium convenient forms for the solutions to equation (I) are as follows : $ = :[Asinh(rz)+Bcosh(rz)]
exp[ik(x-ct)],
# = T[Ccosh(sz)+Dsinh(sz)]exp[ik(x-ct)],
(6)
in which the values of A, B, C and D appropriate to each of the two layers have to be determined from the conditions imposed upon the stresses and displacements at the surfaces and the interface. (a)
CASE OF NO INTERFACIAL
SLIP
Here the boundary conditions are as follows : atz
= -HI,
(P*,)l
= 0,
(7)
(Pm)1
=
(8)
0;
FLEXURAL WAVES ALONG A COMPOSITE
at z = 0,
(L)l
at z = H,,
=
PLATE
169
(~za)*~
(9)
(P*z)l = (%)2,
(IO)
a1 =
u2,
WI
=
w2;
(P*r)Z
=
0,
(‘3)
(Pm)2
=
0.
(14)
(11) (14
Equations (7) to (14) yield eight simultaneous homogeneous equations in the eight unknowns AI, Br, Cl, Dr, AZ, B2, C2, and D2. This system of equations has solutions different from zero if
) ((99e*R,-(32(g2e2Ql)
(15) where for c < fi2, /I1 0 =
2 -
cash (YH) cash (sH) + (&+Z)
2
sinh (rH) sinh (sH)
i.2 = (&-T)sinh(rH)sinh(sH) (16) x = g sinh(sH) cash (rH) - $sinh
(rH) cash (sH)
A = $sinh
(sH) cash (rH)
(rH) cash (sH) - $sinh
Approximations to these functions at wavelengths which are long in relation to H, and H2 can be written as follows:
e=
“$tfx{l+($+g)
X =
q(~+(;+-$$)(kH)~)
A=
(kH)2)
(17) @p(I+~;+~)(kN)2)
-’
1 +;(kH)2
R. JONES AND E. N. THROWER
170 where
2
=
b2-g
=
c’ p4
x
=
4
s2
r2
Y=F+p v=
(18)
*
s2 72 2 p-K2 (1,
When PI > c > BZthen Or, Szr, x1 and A 1 have the same form as equations (16) but 02, Q2, x2 and A 2 are as follows : d2 =
Q2
=
cash (r2H2)cos(s2H2)+
2-2
-
(
GA2 ~-__
43.2~2
43.2~2
b;k2
sinh (r2 H2) sin (s2 H2) )
1
$$+~)sinh(r2H2)sin(s2H2) 2
*. (19)
b2k .
~2 = 2~2 sm (~2H2) cash (r2 H2) - Fk
sinh (r2 H2)
cos (s2 ff2)
A2 = ~sinh(r2H2)cos(s2H2)+~ksin(s2H2)cosh(r2H2) 2
2
When c > /3r, j?, there exists a branch which is analogous to the principal mode of the symmetrical waves of Lamb (2) which has a long wavelength solution 2(I+~)[f&+f3+2(+)+2fh(~-1)]
=
(kH,)2(~-I)~(I-~(I-5$)+4(~-‘),
where h = H,/H,, t = G2/G1. For infinitely long waves equation
c2
(20)
(20)
reduces to
(I-$++$) (21)
-2 = 4
A
Equation (21) reduces to the velocity of long longitudinal waves in the remaining plate if HI or H2 is zero.
w
CASE OF INTERFACIAL
SLIP
Here the appropriate boundary conditions are : atz=
-HI,
(Paz)1 = 0,
(24
(Px*)l =
(23)
0;
FLEXURAL
WAVES
at z = 0,
ALONG
(%)*
Equations equation :
(22)
=
PLATE
17’ (24)
(PzA2,
(P.w)* = 0,
(25)
(P&Y,>2
=
0,
(26)
=
w,;
(27)
Pz*>2
=
0,
(28)
(Psz)z
=
0.
(29)
\
WI
at z = H,,
A COMPOSITE
to (29) yield a relation between L and c which reduces to the following =9,X*+ (2-b)
$&e2x1
(30)
= 0.
For c < 8, and c > &, 0 and x are given by equations (16) and their long wavelength approximations by equations (I 7). For c > j12 and c > fll, &’and x are given by equation (19) with appropriate suffixes. The long wavelength solution of equation (30) for the principal mode of waves corresponding to Lamb’s (2) symmetrical branch for a plate has the following form:
E-+$)][&4(‘-$)][d&+f’2&] = 0.
(31)
Equation (3 I) is interesting because it indicates that two limiting solutions are possible at infinitely long wavelengths, corresponding to the propagation of long plate waves in either of the two plates behaving independently of each other. 3. COMPOSITE PLATES OF THE SAME MATERIAL AND THICKNESSES An interesting case arises when the two layers of the plate have the same thickness and are composed of the same material. When there is no interfacial slip equation (I 5) reduces to Ax = o. (32) As was to be expected, these dispersion equations correspond to Lamb’s (2) antisymmetric (A = o) and symmetric (x = o) modes of propagation of the vibrations in a plate of thickness 2H, and the relations between phase velocity and wavelength for the principal modes are given by branches A(H) = o and x(H) = o in Figure I. The relative vertical displacements at the boundaries and interface for the principal mode are shown in Figure 2. When there is slip at the interface of two equal plates of the same material, equation (28) reduces to 8, = 0. (33) x = o again corresponds to the symmetric mode of waves in a plate of thickness 2H which has vertical particle displacements as shown in Figure 3. The equation 6 = o can be further split into A(H/2) = $sinhTcoshz-$sinhTcoshz
= o,
(34)
x(H/2) = ~~sinh$cosh’$-EsinhFcosh’:
= o.
(35)
01
R. JONES AND E. N. THROWER
172 2:.0,-
I*8-
x(H) = 0
I -6 -
I-4 -
No interfacial slip,branchesx(H) = 0, and A(H) =
No interfacial friction, branches x( H/2) = 0 AH/21 = 0 and x(H) = 0
I*2 Olh
,
01*8-
0
C
C
I
1
I
I
I
I
I
1
2
3
4
5
6
7
L s
Wavelength of vibrations o Thickness of composite pIarC
Figure I. Effect of interfacial contact on the phase velocity of vibrations in a composite platerof equal thicknesses (H) of the same material (Poisson’s ratio : 0~~5).
ia)
(b) Figure 2. Vertical displacements in a composite plate of two identical plates of equal thickness with no slip at their interface. (a) Symmetric type x = o. (b) Anti-symmetric type d = o.
Equation (34) gives the dispersion equation for flexural (or anti-symmetric) waves in a plate of thickness H. The wavelength with no interfacial friction is thus half of the wavelength of flexural waves of the same velocity in the same plates but with no interfacial slip (see Figure I). Equation (35) is the dispersion equation for symmetrical waves in each plate of thickness H so that the wavelength is half the wavelength of the other type of symmetrical wave
FLEXURAL
WAVES
ALONG
A COMPOSITE
PLATE
173
b(H) = o] in the same system, or of that in similar plates with no interfacial slip (see Figure I). The two types of symmetrical wave also give rise to different relative displacements of the boundaries and interface as can be seen from Figure 2, in which #f/z) = o produces anti-symmetric motion of the overall composite plate system.
(a)
(b)
Figure 3. Vertical displacements in same composite system as in Figure 2, but with interfacial slip. (a) Symmetric type x(H) = o; (b) Symmetric type x(H/z) = o; (c) Anti-symmetric type fl(H/z) = 0.
The three branches obtained when there is slip at the interface give a triple root at zero wavelength when c = y, implying Rayleigh waves on each free surface and also at the interface. There is also a double root at infinite wavelength for the x(H) = o and x(H/2) = o branches, as may also be deduced from equation (3 I) when suffix I is the same as suflix 2. 4.
COMPOSITE
PLATE
OF DISSIMILAR
MATERIALS
The case to be considered is the one for which solutions were obtained in an earlier paper (I). The two plates corresponding to the surface and base layers of a road had the following properties :
HI = 3’8in.,
H2 = 5.5 in.,
czl = a/3, = 2cr2 = 4fia.
(36)
The velocities (cc3 and pa) in the soil beneath the road were much lower than the velocities in the upper layers (cc1 = 8a3 = 20&) so that, to a first approximation, the surfacing and base were treated as a free composite plate (I). Further theoretical work is in hand to check the conditions under which the assumption is acceptable. It was noted previously that below p2, the wavelength of flexural vibrations in the composite plate (with no interfacial slip) agreed quite well with the sum of the wavelengths of flexural waves in the individual plates at the same velocity (I). Additional results by computer and from the long wavelength approximations in Figure 4 contirmed the agreement with the relation obtained by the additive procedure. Results obtained for the theoretical case of perfect interfacial slip are also given in Figure 4, and will be seen to be in agreement with the mean of the wavelengths from individual plates at corresponding velocities. The computed results for perfect slip are thus considerably different from the computed results assuming no slip.
R. JONES AND E. N. THROWER
‘74
The experimental results given in Figure 4, for a road construction having surface and base layers with properties corresponding to condition (36), will be seen to give excellent agreement with the computed results assuming no interfacial slip. This and other similar comparisons, have indicated that the layers of a road are well bonded in respect of elastic
@7
SO.6 t
IWBranch
from
I \
HI layer
alone
curve
I
0
I
2
4
,
t
6
Wavelength
8
I
IO
(ft)
Figure 4. Effect of inter-facial contact on the relation between velocity and wavelength of flexural waves in a composite plate (HI = 3.8 in., H2 = 5.5 in., aI = 26, = 2~9 = 4/32). ??Results from A Results from o Results from A Results from ? ?Experimental
exact solution, no slip at interface. exact solution, slip at interface. long wavelength approximation, no slip at interface. long wavelength approximation, slip at interface. results.
wave propagation, and, in subsequent work, only the no slip boundary condition will be used. The range of applicability of the additive procedure in closely bound composite plates of varying thicknesses and elastic moduli at velocities below &, and also the shape of the dispersion curves at velocities greater than /I2 have now been investigated; the analysis and results will be given in a later paper. ACKNOWLEDGMENT
The work described in this paper was carried out as part of the programme of the Road Research Board of the Department of Scientific and Industrial Research. The paper is published by permission of the Director of Road Research. REFERENCES I. R. JONES 1962 Brit.J.
uppl. Phys. 13 (I), 21-9. Surface wave technique for measuring the elastic properties and thickness of roads : theoretical development. 2. H. LAMB 1916 Proc. Roy. Sot. A., 93, I 16. On waves in plates.