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The variation, with the Poisson ratio, of Lamb modes in a free plate, III: Behaviour of individual modes
Journal of Sound and Vibration
(1990) 137(2), 249-266
THE VARIATION, WITH THE POISSON RATIO, OF LAMB
MODES
III: BEHAVIOUR
IN A FREE
PLATE,
OF INDIVIDUAL
MODES
A. FREEDMAN 65 Mount Pleasant Avenue South, Weymouth, Dorset DT3 SJF, England (Received
4 November
1987, and in revised form 24 May 1989)
The way individual, real-valued Lamb mode branches of a free, elastic plate vary with the Poisson ratio, a, is studied, the coverage extending over the whole range, both positive and negative, of the latter parameter. After examining the zero order antisymmetric and symmetric branches, an investigation is made of the non-zero order branches of arbitrary order. The overall behaviour is shown to be largely influenced by two cyclicities; the first of these applies over the full range of u and occurs when the normal displacements within the plate are entirely dilatational, while the second cyclicity applies only over the negative range of u and occurs when the normal displacements within the plate are entirely shear. Illustrations covering the full range of o are presented for the first few non-zero order modes. Additionally, guidance supplementing Mindlin’s rules of bounds is proffered for the easy sketching (without need for any computations) of sets of Lamb mode spectra, at any positive or negative value of a, onto grids of Mindlin’s bounds. 1. INTRODUCTION
A set of Lamb mode spectra at fixed values of the Poisson ratio, o, covering, effectively, the full range of a, has been presented in a companion paper [l], and in a second companion paper [2] problems linked with the above spectra at special values of the Poisson ratio have been examined. However, from the set of spectra of reference [l], it is not easy to discern how any one branch varies as a function of LT.Therefore, based on the information from references [l] and [2], the way real branches for Lamb modes of arbitrary order vary over the full range of the Poisson ratio is investigated here, and illustrations are presented for each of the first few, low order modes. Additionally, in two appendices, guidance, supplementing Mindlin’s rules of bounds [ 1,3,4], is proffered for the rapid and easy sketching of Lamb mode spectra onto grids of Mindlin’s bounds. The same nomenclature and conventions are used in this paper as in references [l] and [2]. In addition to a conventional Ah and S, terminology (where h = 0, 1,2, . . . ) for designating the real Lamb modes, this involves the use of the following alternative nomenclature. The A,, and S,,, designations, where q = 1,2,3, . . . , refer to those real, antisymmetric and symmetric Lamb branches which start at cut-off as thickness-shear modes, while the A, and S,, designations, where p = 1,2,3,. . . , refer to those real Lamb branches which start as thickness-stretch modes. 2. SKETCHING
OF THE LAMB MODE SPECTRA
The characteristic equations for a free, infinite, isotropic, elastic plate may be written as 4LStanh[?~S(d/A,)(k/k,)]-(l+S~)~tanh[1rL(d/h,)(k/k,)]=O,
(I)
249
0022-460X/90/050249+ 18
$03.00/O
@ 1990 Academic
Ress
Limited
250
A. FREEDMAN
4LScoth[?rS(d/A,)(k/k,)]-(l+S*)*coth[~L(d/A,)(k/k,)]=O,
(2)
where L=[l-(Q/C,)-2(k/k,s)-2]“2,
S=[l-(k/ks)-*I”*,
(394)
and d is the plate thickness, A, and k, are the wavelength and wavenumber of shear waves in the plate material, k = k, + iki is the modal wavenumber, and c, and cd are the velocities of shear and dilatational waves in the plate material. The solutions of equations (1) and (2) represent antisymmetric and symmetric Lamb modes, respectively. The presentation of Lamb spectra generally used in references [l] and [2] consists of real and imaginary solutions of the above equations superimposed upon a grid of Mindlin’s bounds [3,4] plotted on a d/A,, k/k, co-ordinate system at given u. For the non-zero order modes, Appendix A provides full guidance for easy sketching of such spectra, apart from the decision whether a real branch starts with negative or positive slope. Guidance on that point is provided in Appendix B, which derives tables showing the ranges of cd/c, (and hence of a) over which the various real, low order branches have negative initial slope. 3. THE ZERO ORDER MODES 3.1. THE A, MODE Curves of the A0 branch at values of cd/c,, covering virtually the full range of the Poisson ratio, are shown in Figure 1. The only notable effect of varying cd/c, is a gentle decrease in the level of k,/k, as cd/& increases. In the thick plate limit the levels tend to the normalized Rayleigh wavenumber, kR/ k,. The latter’s variation with cd/c, (and with the Poisson ratio) is shown in Figure 2. 3.2. THE S, MODE The curves for the S,, mode are shown in Figure 3. In the thin plate limit they start at k,,/ k,, given by equations (7) and (8) of reference [l] and illustrated in Figure 7(a) of that reference. One can note, from equations (7) and
01
0
I
I
1
2
d/A, Figure 1. The variation of the
A,,
branch with cd/q.
LAMB
Figure
2. Variation
of normalized
MODE
Rayleigh
wavenumber
01
3. The variation
(30) of reference
with cd/c,
with cd/q
and with the Poisson
ratio.
I
I
of the S, branch
251
III
1 d/X,
0
Figure
SPECTRA,
2
.
, Common
point for all positive
u values.
[l], that k,lk, = (d/A,),+.,=,:
(5)
i.e., that the value of k,/k, at the thin plate limit of the S, branch is equal to the d/As value at the intersection of the m = 0 and n = 1 bounds and, hence, to half the separation between successive intersections of symmetric Lamb modes with the line k,/k, = k,/k, (kd being the wavenumber of dilatational waves in the plate material). From equations (56) and (57) of reference [3], the S, branch crosses the line k,/ k, = 1 at d/As = [2(1-
r)l”‘rlT
(6)
where y is the root of (tanh y)/ y = (1 - a)/2.
(7)
The variation with u (and cd/c,) of d/h, at the crossing point is illustrated in Figure 4. In the thick plate limit the levels of the S, curves tend to the normalized Rayleigh wavenumber illustrated in Figure 2.
252
A. FREEDMAN
Figure 4. Variation, with u (and with cd/c,), of d/A\, at crossing by the S, branch of the line k,/ k, = 1.
For positive u (i.e., for cd/c, > 2”2), all the S, branches go through the point d/h, = 2- lf2, k,/k, = 2-1’2, at which the motion is purely shear. As each branch at positive u passes through that point its slope varies continuously. The branch for (T= 0 also passes through the above common point, but the slope there changes discontinuously. At negative u the branches do not pass through a common point. Apart from at the point common to all the curves for positive a, the level of k,/ k, at given d/h, decreases as c~/c,~increases. For cd/c, > 2’j2, the slope of each branch at the common point is the same as that for the n = 1 bound at that point, namely unity, and that applies also to the second part of the branch at cd/es = 21i2 (i.e., at c = 0). For all branches with positive a, the ratio, c,/c, of group velocity to phase velocity at the common point is 0.5 and the normalized group velocity, c,/c,, equals 2-1’2. At that point, c,/cs for the u = 0 branch changes discontinuously from 2”2 to 2-“2 as d/As increases. 4.
THE CYCLICITIES UNDERLYING THE NON-ZERO ORDER BRANCHES
4.1. THE CYCLICITY AT CUT-OFF As mentioned in reference [2], over the whole range of a, the spectrum in the (cd/c,, d/A, at cut-off) plane of a real Ah or S,, branch can be regarded as made up of a number of cycles corresponding to passage along a path such as ABC in Figure 6(b) of reference [2], in addition to a partial cycle at each end of the locus. Coupled with the information on initial negative slope from Figure Bl of this paper (see Appendix B), this yields the typical cycle of Figure 5(a), which governs the type of behaviour of the branches in the vicinity of the k,/ k, = 0 axis as follows. For an Ah or S,, branch of order h = (p + q - 1)/2, as cd/c* increases from (q-2)/p to q/p, the branch starts at a cut-off value of d/As increasing from (q - 2) x 0.5 to q x 0.5, while, as cd/c, increases from q/p to q/( p - 2), the cut-off value remains constant at q x 0.5. The initial slope of the branch after departure from the cut-off point is positive, except in the region D to E of Figure 5(a). When u is positive, the Ah and S,, branches cross the line k,/k, = 2-“* where d/As equals 2h x 2-“2 and (2h + 1) x2-“*, respectively [ 11, and then follow closely above the bounds n = 2h and n = 2h + 1, respectively. Thus, for all positive values of a, the Ah (or S,,) branches for given h will pass through a common point, beyond which they lie so closely together that they are virtually represented by a single curve. This, combined with the cyclicity described above, and with the undulations associated with intersections pf branches of opposite symmetries, governs the overall type of behaviour of the real branches for positive u. That behaviour is illustrated in Figures 6(a) and (b) for an Ah and an Sh mode, respectively. Only the branches corresponding to boundaries between A_ and A+ or Ssq and S,, modes and to the lower and upper limits of positive u are shown. The branch for cd/c, = 2’/2 can be either of sq or of dp type, while that for cd/c, -*cc is of sq type,
LAMB
MODE
SPECTRA,
253
111 (b)
(a)
4
t
q/(p-2)
c
------
[Cd~Cs]m-2.n I
ii‘
1
-----7””
s(q_:lpf t l:“_2~::fjI-
h-2)x
qxo.5
(q-21x0.5
d/A, at k,/k,
Figure 5. Cycles relating to cd/c, the vicinity of k,/ k, = 2-I”.
d/A,
= 0
spectra
of real A,, and S,, branches.
24’2
n x 2-1’2
at k,/k,
=2-“2
(a) In the vicinity
of k,/k, = 0; (b) in
I
(a)
242
-_______-__-______-_---
0
.I
2-V’
I
h-j-0.5
I
I
/ ,
h-j-1.5
I I
I I
h-05
2h x 2-“2
I
h- 2.5
h-:.5
--------_
0
I
I h-j-t
hlj
ti2
/
I
I it
,:
(2Al
(2h+i)x2-‘”
d/X,
Figure 6. Schematic illustration of variation over the range of positive an A,, mode; (b) for an S,, mode. /yA, Region of A,,, or S, branches; 0, common point.
(r of an A,, and an S,, branch. (a) For N, region of A, or S,, branches;
254
A. FREEDMAN
both branches lying within the limits indicated by the adjacent dash-dot lines. In Figure 6(a) or 6(b), j denotes the highest positive integer not exceeding (h -0*5)/( 1 +2”2) or (h + 2”‘)/( 1 + 2”2), respectively. The undulations associated with intersections of branches are relatively small and are not illustrated in Figure 6. As can be seen both from Figure 6 of reference [2] and the present Figure 6, the number of complete cycles tends to increase as h increases. For the low order modes the number of complete cycles may be down to zero, and only the partial cycles at the two limits remain. At each coincidence at S/k, = 0 of modes of like symmetry the conditions for thicknessshear and for thickness-stretch modes are both satisfied, and so each branch forming a boundary between two modal regions represents both an A,, and an A, (or an S,, and an SdP) mode. This applies at each of the two branches starting with slopes of opposite sign at an axial coincidence value of cd/ c,~. The general manner in which the branches within a single cycle vary is examined next. The deduced variations take account of the need to avoid a discontinuity in modal behaviour, despite a jump of initial slope as cd/c, is varied through each value at which there is coincidence at cut-off of a pair of branches of like symmetry. (See equations (27) and (28) of reference [2].) A schematic, generalized representation of the sequence within a cycle is presented in Figure 7, in which the letters A to E correspond to those on the cycle of Figure 5(a). At A the branch starts at an initial angle of tan-’ (7r/2), and the initial slope of each branch remains positive between A and D. At D a boundary has been reached between branches with positive and branches with negative initial slope. Between D and E the initial slope is negative. Additionally, between D and B, a point of inflection gradually moves towards the cut-off point, while the initial radius of curvature also decreases from infinity to zero. At B the initial curved part of the branch and its radius of curvature both vanish, producing an initial angle of -tan-’ (7r/2). Between B and E the sequence is the reverse of that between D and B, and at E a boundary has been reached between branches with negative and branches with positive initial slope. Between E and C the sequence is the reverse of that between A and D, so that at C the branch starts at an initial angle of tan-’ (7r/2). Thus, at each coincidence value, the transitions have been achieved smoothly and without discontinuity in modal behaviour.
i\ 2
0
(4 -2) x 0.5
q x 0.5 d/X,
Figure 7. Generalized representation of behaviour of initial parts of an A, or S,, branch over a cycle of cd/c, B, cd/c, = q/p; C, cd/c, = q/(p -2); D and E, values corresponding to Figure 5(a). A, cd/c, = (q-2)/p; boundaries between those branches with positive and those with negative initial slope.
A corresponding cycle of curves for the ratio, c,/c, of group velocity to phase velocity is shown in Figure 8(a) and for the normalized group velocity, c,/c,, in Figure 8(b). Because it avoids confusing overlap of several curves, these cycles of group velocity are presented half a cycle out of phase with the cycle of Figure 7, thus covering B, to C or A, and A to Bz. The curves of Figure 8 were derived from those of Figure 7 by an
LAMB
MODE
SPECTRA,
III
255
0.4
0.2 9 e
0 I
I I (q-21 x0.5
qxo.5
(q-21X0.5
q x0.5
-1
I
-0.2 d/i,
Figure 8. Generalized representation of behaviour of group velocity over the initial parts of an Ah or S,, branch for a cycle of cd/c, values. B,, ~~/c,=(9-22)/(~+2); C and A, cd/cs=(9-2z)/P; I%, %/c,=~/P; E and D, boundaries between those branches which initially have group and phase velocities in opposite directions and those which do not. (a) For c,/c; (b) for c,/c,.
approximate graphical procedure based on the equations presented in Figure Al(b) of reference [l] and, as the latter equations involve the slope of the line from the origin to each point on the curves of Figure 7, q was arbitrarily set to 8. At the cut-off points, the group velocity is zero for all the branches, except for those at coincidences of modes of like symmetry, for which c,/cs = f4/7rq. However, even there the total group velocity remains zero. Despite the discontinuities in c,/cs, the curves of Figure 8(b) illustrate how there is a smooth transition at each coincidence value of cd/c,. 4.2. THE CYCLICITY AT kJ k, = 2-“2 Over the range of negative a, the spectrum in the (c,,/c,. d/As at k,/k, =2-l’*) plane can be regarded as made up of a number of cycles corresponding to passage along a path such as FGH in Figure 13(a) of reference [2], in addition to a partial cycle at each end of the locus. As with the cyclicity at k,/ k, = 0, the number of complete cycles tends to increase as h increases and, in the case of low order modes, may be down to zero. Letting the value of cd/c, at which there is intersection of the m and n bounds at k,/k, = 2-l’* a cycle of the above type is shown in Figure S(b). Between be denoted by [Q/C,],,,, the Ah or S,, branch of order (m + n - 1)/2 crosses the line CC~/C~I~,~-~ and Ccd~~l~,~
256
A. FREEDMAN
k,/ k, = 2-“2 at points for which the value of d/h, varies from (n - 2) x 2-‘12 to n x 22”*, while between [c~/c.~]_ and [c~/c,~],,_~,~ it crosses that line at a common point for which d/A., equals n x 2-‘12. For a given Ah mode and varying cd/c,, equations (34) and (35) of reference [2] yield the maximum and minimum values of n at which there are coincidences at k,/ k,%= 2-“2 with the A,,+, or A,,_, mode as nmax= 2h and nmin equals the lowest even integer not smaller than (2h + 1)/(2-l’* + 1). Similarly, for an S,, mode, nmax= 2h + 1 and nminequals the lowest odd integer not smaller than (2h+ 1)/(2-l”+ 1). At each value of n, the corresponding m value is given by m = 2h + 1 - n. For an Ah and also for an S, mode, schematic illustrations of the variation over the range of negative u of the part of the branch near k,/ k, = 22”* are presented in Figure 9, the parameter used being cd/c,. Apart from the branches at the upper and lower limits
I,
,,
IO)
2-’ l/2
_
/
Cd/cy]2j+l,2(h-j-I) /
I n
zL
(h-j-l)X2’fi
I
I
I\
I ,
[CdfC.J6,2(h-21
(h-j )x21/i!
(h-2)X2”*
1
(h-1)X2’”
/ II
(2h. -2j-3)~2-'~ .(2h-2j-1)x2m1’2
I I
I I
I
/
I
@h-5)X2-‘“(2h-J)XZ
I I
hX2”’
I lR (2h-l)x2-“p
I I
(2h+l)x2-v2
Figure 9. Schematic illustration of variation over the range of negative w of the parts of a non-zero Lamb branch near k,/k, =2-“2. (a) For an Ah mode; (b) for an S, mode; 0, common point.
order
258
A. FREEDMAN 5. SPECTRA
OF THE
NON-ZERO,
LOW ORDER
MODES
With cd/c, as the parameter,
the families of spectral curves for each of the first four in Figures 10 and 11, respectively. These sets of curves were assembled from spectra at fixed values of cd/c, sketched by using the techniques described in Appendices A and B. Ah and S,, modes are presented
2
1.0
I
[ (cl
Figure11. The variationof the S,, 0, common
point.
branches with cd/c,.
I
-
I
(a) S, mode; (b) S, mode; (c) Sa mode; (d) S, mode;
LAMB
MODE
SPECTRA, III
259
6. CONCLUSION To supplement the examination in two companion papers [l, 21 of the Lamb mode spectrum of a free, isotropic, elastic plate at fixed values of the Poisson ratio, a, in this paper the variation of individual real branches with m over the complete range of the latter parameter has been investigated. The zero order antisymmetric branch, Ao, shows only gentle changes of level over the whole range of g (see Figure l), the normalized wavenumber, k,/ k,, at given normalized plate thickness, d/h,, decreasing as u increases. For all positive u values the zero order symmetric branch, S,,, goes through a common point (see Figure 3) at which the normal displacements in the plate are entirely equivoluminal. For negative u values there is no common point and the normal displacements are never completely equivoluminal. Apart from at the common point, k,/k, at given d/h, decreases as u increases. For the non-zero order antisymmetric and symmetric branches, Ah and S,,, where h is a positive integer, the general shapes of the branches are largely influenced by the following two cyclicities. (i) At k,/k, = 0 (i.e., where the normal displacements within the plate are entirely dilatational), the branches for which the ratio of dilatational to shear velocities, cd/c,, lies between n/m and n/(m -2) (where m, n = 1,2,3,. . . , n/m 3 (4/3)“‘, h= (m + n - 1)/2) pass through a common point at which d/A, = n/2 (see Figure 6). (ii) At kJ k, = 2-l’* (i.e., where the normal displacements within the plate are entirely shear), the Ah and S,, branches for all cd/c, values down to 21’2{2h/[1+(2h)‘]“2} and 2 1’2,respectively, pass through a common point at d/As equal to 2h x 2-“2 and (2h + 1) x 2- 1/2, respectively, while at lower values of cd/c, there are cyclic groupings passing through other common points (as indicated in Figures 9(a) and (b)). The number of cycles of each of the two types tends to increase with h, and at low values of h the number of complete cycles may be down to zero. Families of branches covering the complete range of positive and negative u have been provided (in Figures 10 and 11) for the Ah and S,, modes at values of h from 1 to 4. Additionally, supplementing Mindlin’s rules of bounds, guidance has been proffered (in Appendices A and B) for easy sketching (without need for any computations) of the real and imaginary Lamb mode spectra on a grid of bounds at any positive or negative value of u.
ACKNOWLEDGMENT I am grateful to Mr P. G. Redgment for critical reading of the manuscript helpful discussions.
and for
REFERENCES 1. A. FREEDMAN 19% Journal of Sound and Vibration 137,209-230. The variation, with the Poisson ratio, of Lamb modes in a free plate, I: General spectra. 2. A. FREEDMAN 1990Journal of Sound and Vibration 137,231-247. The variation, with the Poisson ratio, of Lamb modes in a free plate, II: At transitions and coincidence values. 3. R. D. MINDLIN 1957 in Roceedings of the 11th Annual Symposium on Frequency Control, U.S. Army Signal Engineering Laboratories, Fort Monmouth, New Jersey, l-40. Mathematical theory of vibrations of elastic plates. 4. R. D. MINDLIN 1960 in First Symposium on Naval Structumi Mechanics, 1958 (J. N. Goodier and N. J. Hoff, editors), 199-232. Oxford: Pergamon Press. Waves and vibrations in isotropic, elastic plates.
260
A. FREEDMAN
APPENDIX
A: GUIDELINES
FOR SKETCHING
THE LAMB SPECTRA
By making use of equations (Bl)-(B4) of [l], a grid of bounds for given o can readily be produced on the d/h,, k/k, co-ordinate system by using an automatic plotter. Rapid and easy sketching on that grid of the Lamb mode spectrum (excluding the zero order modes) is facilitated by the following considerations. Except where stated otherwise, these considerations apply to c > 0.263 in the d/A,, k,/ k, plane, but to arbitrary values of u in the d/h,, ki/ k, plane. The regions of the spectrum requiring careful decisions on how the branches should be drawn are those bordering the line k/k, = 0. Once those parts of the curves have been sketched, the other parts follow straightforwardly. In particular, when sketching the spectrum, one is faced with the need to decide, for the start of each real branch, whether the curve should proceed to the right or to the left of the real bound starting at the same point. While that could be determined, as described by Mindlin [3], by finding the local ratio of curvatures of branch and bound, the following simple guidance, which calls for no calculations, suffices. In the region bordering on k/k, = 0, the latter axis and the m and n bounds divide the d/h,, k,/k, plane or the d/A,, ki/ks plane into three-, four- or five-sided areas of the types shown by the thin lines in Figures AI-A3. Figure Al applies to the d/A,, k,/k, plane for general values of m, while Figure A2 applies to the particular case where m = 0. Figure A3 is for the d/h,, ki/k,y plane. In Figures Al-A3, each intersection of bounds for which m + n is even is denoted by a large dot, and this gives rise to two diagrams for each area of similar shape. The heavy lines show the layout of branches within each area and, as can be seen, in half the cases each type of area contains no branches. Symmetric and antisymmetric modes are denoted either by thick continuous and thick broken lines, respectively, or vice versa. Figure Al(a) is for c greater than O-263. As shown in Figure Al(b), for u smaller than O-263, before intersecting at the intersection of bounds, the branches may be constrained to go through the lower of the two “additional” points of intersection [l]. Figure A2(a) is also for u greater than O-263, but Figures A2(b) and (c) illustrate that, for values below O-263, the branches are constrained to go through both additional points of intersection. Figure A2(c) also illustrates that when u is negative the branches cross the kJ k, = 2-‘12 line between the two additional intersection points, and each branch is tangential at the crossing point to an n or m bound. The right-hand one
d/X,
Figure Al. axis, showing 0.263>crz-1.
Types of area formed, for m # 0, by the grid of bounds in the region adjacent to the k,/k, =O the location and general form of the Lamb branches within each ama. (a) 0.5~ a>0*263; (b) bound; ?? , intersection of bounds with m + n even; or - --, branch. -,
LAMB
MODE
SPECTRA,
261
III
d/X,
Figure A2. Types of area formed, for m = 0, by the grid of bounds in the region adjacent to the k,/k, = 0 axis, showing the location and general form of the Lamb branches within each area. (a) 0.5 > o > 0.263; (b) 0.263 > D > 0; (c) 0 > u 2 -1. Bound: -. Branch: -, antisymmetric; - - -, symmetric. ?? , intersection of bounds with m + n even.
m+lJ
m+l/“+’
d/X,
Figure A3. Types of area formed by the grid of bounds in the region adjacent to the ki/ks = 0 axis, showing the location and general form of the Lamb branches within each area. -, bound; 0, intersection of bounds with m + n even; or -- -, branch.
of each pair of real branches, such as shown in Figures Al and A2, may, after leaving the k,/ k, =0 axis, initially have either positive or negative slope. Where there is doubt, the decision on whether a real branch has negative initial slope (and, hence, its imaginary continuation has positive initial slope) can be ascertained as described in Appendix B. For the special cases where there are coincidences of bounds on the k/k, = 0 axis, (as discussed in reference [2],) the resulting situations are similarly illustrated in Figure A4. When cd/cS = n/m, where n and m are integers one of which is even and the other is odd, then two symmetric or two antisymmetric real branches start at the same cut-off point. These branches do not lie within the four-sided areas shown in Figures A4(b) and (d), but, for clarity, the convention adhered to in Figures Al-A3 and the rest of Figure A4 of only showing curves which lie within such areas is here disregarded. The typical behaviour of bounds and branches in the region where k,/k, is greater than kd/ k, is as previously portrayed in Figure 13 of reference [ 11.
262
A. FREEDMAN
i
z (c)
Antisymmetric n+l +/ ,
(d)
n(odd)/
4
m+l
nkven) /
n+l,
/’
d/X,
Figure A4. Behaviour of bounds and branches where there are coincidences on the k/k, =0 axis. Bound: -. Branch: or - - -. (a) For general values of m, when m + n is even; (b) for general values of m, when m + n is odd; (c) as (a) when m =O; (d) as (b) when m = 0.
The imaginary branch linking to the real A, branch appears to show little change over the whole range of a, and is exemplified in Figure 3 of reference [ 11. For u > O-263, the procedures for sketching the bulk of the Lamb mode spectrum are thus as follows. (1) On the grid of bounds for the given cd/c,, those intersections are marked in for which m + n is even. (2) For each applicable area adjacent to the k/k, =0 axis, the branches are sketched in as indicated by the corresponding area in Figure Al, A2, A3 or A4 (supplemented by the information from Appendix B on whether the initial slope is negative). (3) The sketching of these branches into successive adjacent areas enclosed between bounds is continued, noting that the branches can only exit each such enclosed area at a marked intersection of bounds, (with the exception of the antisymmetric branches crossing the m = 0 bound). (4) When a branch crosses the line k,/ k, = 2-l”, it does so tangentially to an n bound. For u c 0.263, the following modifications to the above procedures are needed. From Figure 7 of reference [l], or by use of equations (22) and (21) of reference (5) [l], the values of the co-ordinates of the intersections of branches additional to those at intersections of bounds are found, and these points are then marked onto the grid of bounds. (6) Within any one area enclosed by bounds (or by bounds and the k,/ k, =0 axis), and which contains such an additional intersection point, the branches are constrained to cross over at that point.
LAMB
MODE
SPECTRA,
111
263
For 0~0,
the following further modification to the above procedures applies. it does so tangentially to either an n or an m bound.
(7) When a branch crosses the line kJ k, = 2-l”,
APPENDIX
B: OCCURRENCE OF REAL BRANCHES NEGATIVE INITIAL SLOPE
WITH
In the region where a real branch has initial negative slope and the slope is less than -(k,/
k,)/( d/A,), the group velocity and the phase velocity are in opposite directions, and the group velocity drops to zero at the point at which the minimum of d/AS occurs. After transformation
from the co-ordinate
system, Mindlin’s
equations
(50)-(53)
system used by Mindlin of reference
TABLE
to the d/h,,
[3] yield the following
k/k, co-ordinate criteria for real
Bl
Ranges of cd/c, for given A,, modes over which A, initial slope
and A,, real branches have negative
cd/c,
Lower limit h
P
2 344 55 66 6 77 7 88 8 8 99 9 IO10
2 2-
11 -
11 12 13 -
2-
4 24 2 4 6 24 6 24 6 46 8 6 8 8 10 10 12 -
for
A,
1.1838 24452 3.4809
(4/3)"'t 4.4911 1.7104 5.4952 2.2319
(4/3y2t 6.4971 2.7402 1.4725 7.498 1 3.2441 1.8185 3*?i62 2.1578 1.3551 2.4943 1.6131 1.8673 1.2851 1.4903 1.2384
t Lower limits for A, or for to a coincidence value.
A,,, set
Value at coincidence (forms upper limit for A, and lower limit for A,,) 1.5000 2.5000 3.5000 1a2500 4.5000 1.7500 5.5000 2.2500 1.1667 6.5000 2.7500 1.5000 7.5000 3.2500 1.8333
(4/3Y2t
3.7500 2.1667 1.3750 2.5000 1.6250 1.8750 1.3000 1.5000 1.2500
by lower limit of range of cd/c,.
Upper limit for A,, 1.9442 2.8110 3.7287 1.3899 4.6792 1.8603 5.6470 2.3194 1 a2364 6.6245 2.8230 1.5575 7.6080 3.3120 1.8815 1.1669 3.8038 2.2078 1.4106 2.5358 1.6556 1.9017 1.3242 1.5212 1a2675 The limit for
A,,does
9 3 5 7 5 9 7 11 9 7 13 11 9 15 13 11 9 15 13 11 15 13 15 13 15 15 not correspond
264
A. FREEDMAN
branches to have initial negative slope and, hence, a point of zero group velocity: (a) for an A,, mode, (qr/4)
+
[4/(cdlcs)1
cot
[qv/2(cd/cs)l
q = 1,3,5,.
..;
(Bl)
tan
[q~/2(cd/c.v)l
q = 2,4,6,
...;
032)
(b) for an S,, mode, (q~/4)-[4/(cd/cs)l
TABLE
B2
Ranges of cd / c, for given S, modes ouer which Sd, and S,, real branches have negative initial slope
I
h 1 2344 55
3
Lower limit for S&
P
: 1 3 1 3 1 3
5
5
66
1 3
6
5
77 7 7 88 8 8 99 9 lo10 10 1111 1212 13 14 -
1 3 5 13 5 7 3 5 7 5 7 9 7 9 9 11 11 13
\
-
-
1.2162 3.9475 5.9849 1.1570 7.9937 1.9525 9.9968 2.6474 (4/3)“? 11.9982 3.3236 1.5671 13.9989 3.9944 1.9835 -
-
15.9993 4.6632 2.3906 1.4053 5.3310 2.7941 1.7010 3.1961 1.9917 1.3162 2.2802 1.5450 1.7706 1.2596 14458 1.2204
Value at coincidence (forms upper limit for S& and lower limit for S,,) 2.0000 4~0000 6.0000 1.3333 8.0000 2*0000 10~0000 2.6667 1.2000 12~0000 3.3333 1.6000 14~0000 4.0000 2*0000 (4/3p2t 16.0000 4.6667 2.4000 1.4286 5.3333 2.8000 1.7143 3.2000 2.0000 1.3333 2.2857 1.5556 1.7778
1.2727 1.4545 1.2308
I’ Lower limit for sd,, or for s,,, set by lower limit of range of cd/c,. to a coincidence value.
Upper limit for S,, 3.3904 4.7922 6.5373 1.5599 8.4045 2.1717 10.3239 2.7995 1.2956 12.2700 34406 1.6774 14.2315 4.0897 2.0655 1.1960 16.2026 4.7437 2.4535 1.4731 5.4008 2.8460 1.7521 3.2403 2.0327 1.3624 2.3144 1.5808 1 *WOO 1.2932 1.4726 1.2460
9 2 4 6 4 8 6 10 8 6 12 10 8 14 12 10 8 16 14 12 10 16 14 12 16 14 12 16 14 16 14 16 16
The limit for .Squdoes not correspond
LAMB
(c) for an A,
MODE
SPECTRA,
265
III
mode,
(Pp/4) -[4/(cd/~)~l
tan [P~(cdlc.Y)121 CO,
p=2,4,6
, ****9
(B3)
(d) for an S,, mode, p = 1,3,5, . . . .
(P~/4)+t4/(c~/cs)31cot~P~(c~lc~)/21~0,
(B4)
(Bl) and (B2) for the A,, and S,, modes are satisfied, for each value of to coincidence. Accordingly, relationships (B3) and (B4) for the A, and S,, modes are satisfied, for each value of p, over a range of cd/c, values with an upper limit at the value at coincidence. For given q and p, the two abutting ranges correspond to part of an Ah or Sh branch of order (p + q - 1)/2. For all coincidence points in the region in which 0 < d/A, 6 8, the ranges over which the A, and A,, branches have negative initial slope are shown in Table Bl, while the corresponding information for the S,, and S,, branches is provided Relationships
q, over a range of cd/es values with a lower limit that corresponds
I 12
21/i (4/3) ‘fi 6
Figuy _’
Bl. Regions
in which the real branches
exhibit
negative
initial slope.
266
A. FREEDMAN
in Table B2. On the basis of the presentation of Figure 6 of reference [2], the regions of negative initial slope are shown, in Figure Bl, plotted (as heavy lines) in the plane of cd/c, versus d/A, at cut-off. Such regions are larger in the case of the A,, and S,, than of the A, and S,, modes and they decrease with increase of both q and of p.
(1990) 137(2), 249-266
THE VARIATION, WITH THE POISSON RATIO, OF LAMB
MODES
III: BEHAVIOUR
IN A FREE
PLATE,
OF INDIVIDUAL
MODES
A. FREEDMAN 65 Mount Pleasant Avenue South, Weymouth, Dorset DT3 SJF, England (Received
4 November
1987, and in revised form 24 May 1989)
The way individual, real-valued Lamb mode branches of a free, elastic plate vary with the Poisson ratio, a, is studied, the coverage extending over the whole range, both positive and negative, of the latter parameter. After examining the zero order antisymmetric and symmetric branches, an investigation is made of the non-zero order branches of arbitrary order. The overall behaviour is shown to be largely influenced by two cyclicities; the first of these applies over the full range of u and occurs when the normal displacements within the plate are entirely dilatational, while the second cyclicity applies only over the negative range of u and occurs when the normal displacements within the plate are entirely shear. Illustrations covering the full range of o are presented for the first few non-zero order modes. Additionally, guidance supplementing Mindlin’s rules of bounds is proffered for the easy sketching (without need for any computations) of sets of Lamb mode spectra, at any positive or negative value of a, onto grids of Mindlin’s bounds. 1. INTRODUCTION
A set of Lamb mode spectra at fixed values of the Poisson ratio, o, covering, effectively, the full range of a, has been presented in a companion paper [l], and in a second companion paper [2] problems linked with the above spectra at special values of the Poisson ratio have been examined. However, from the set of spectra of reference [l], it is not easy to discern how any one branch varies as a function of LT.Therefore, based on the information from references [l] and [2], the way real branches for Lamb modes of arbitrary order vary over the full range of the Poisson ratio is investigated here, and illustrations are presented for each of the first few, low order modes. Additionally, in two appendices, guidance, supplementing Mindlin’s rules of bounds [ 1,3,4], is proffered for the rapid and easy sketching of Lamb mode spectra onto grids of Mindlin’s bounds. The same nomenclature and conventions are used in this paper as in references [l] and [2]. In addition to a conventional Ah and S, terminology (where h = 0, 1,2, . . . ) for designating the real Lamb modes, this involves the use of the following alternative nomenclature. The A,, and S,,, designations, where q = 1,2,3, . . . , refer to those real, antisymmetric and symmetric Lamb branches which start at cut-off as thickness-shear modes, while the A, and S,, designations, where p = 1,2,3,. . . , refer to those real Lamb branches which start as thickness-stretch modes. 2. SKETCHING
OF THE LAMB MODE SPECTRA
The characteristic equations for a free, infinite, isotropic, elastic plate may be written as 4LStanh[?~S(d/A,)(k/k,)]-(l+S~)~tanh[1rL(d/h,)(k/k,)]=O,
(I)
249
0022-460X/90/050249+ 18
$03.00/O
@ 1990 Academic
Ress
Limited
250
A. FREEDMAN
4LScoth[?rS(d/A,)(k/k,)]-(l+S*)*coth[~L(d/A,)(k/k,)]=O,
(2)
where L=[l-(Q/C,)-2(k/k,s)-2]“2,
S=[l-(k/ks)-*I”*,
(394)
and d is the plate thickness, A, and k, are the wavelength and wavenumber of shear waves in the plate material, k = k, + iki is the modal wavenumber, and c, and cd are the velocities of shear and dilatational waves in the plate material. The solutions of equations (1) and (2) represent antisymmetric and symmetric Lamb modes, respectively. The presentation of Lamb spectra generally used in references [l] and [2] consists of real and imaginary solutions of the above equations superimposed upon a grid of Mindlin’s bounds [3,4] plotted on a d/A,, k/k, co-ordinate system at given u. For the non-zero order modes, Appendix A provides full guidance for easy sketching of such spectra, apart from the decision whether a real branch starts with negative or positive slope. Guidance on that point is provided in Appendix B, which derives tables showing the ranges of cd/c, (and hence of a) over which the various real, low order branches have negative initial slope. 3. THE ZERO ORDER MODES 3.1. THE A, MODE Curves of the A0 branch at values of cd/c,, covering virtually the full range of the Poisson ratio, are shown in Figure 1. The only notable effect of varying cd/c, is a gentle decrease in the level of k,/k, as cd/& increases. In the thick plate limit the levels tend to the normalized Rayleigh wavenumber, kR/ k,. The latter’s variation with cd/c, (and with the Poisson ratio) is shown in Figure 2. 3.2. THE S, MODE The curves for the S,, mode are shown in Figure 3. In the thin plate limit they start at k,,/ k,, given by equations (7) and (8) of reference [l] and illustrated in Figure 7(a) of that reference. One can note, from equations (7) and
01
0
I
I
1
2
d/A, Figure 1. The variation of the
A,,
branch with cd/q.
LAMB
Figure
2. Variation
of normalized
MODE
Rayleigh
wavenumber
01
3. The variation
(30) of reference
with cd/c,
with cd/q
and with the Poisson
ratio.
I
I
of the S, branch
251
III
1 d/X,
0
Figure
SPECTRA,
2
.
, Common
point for all positive
u values.
[l], that k,lk, = (d/A,),+.,=,:
(5)
i.e., that the value of k,/k, at the thin plate limit of the S, branch is equal to the d/As value at the intersection of the m = 0 and n = 1 bounds and, hence, to half the separation between successive intersections of symmetric Lamb modes with the line k,/k, = k,/k, (kd being the wavenumber of dilatational waves in the plate material). From equations (56) and (57) of reference [3], the S, branch crosses the line k,/ k, = 1 at d/As = [2(1-
r)l”‘rlT
(6)
where y is the root of (tanh y)/ y = (1 - a)/2.
(7)
The variation with u (and cd/c,) of d/h, at the crossing point is illustrated in Figure 4. In the thick plate limit the levels of the S, curves tend to the normalized Rayleigh wavenumber illustrated in Figure 2.
252
A. FREEDMAN
Figure 4. Variation, with u (and with cd/c,), of d/A\, at crossing by the S, branch of the line k,/ k, = 1.
For positive u (i.e., for cd/c, > 2”2), all the S, branches go through the point d/h, = 2- lf2, k,/k, = 2-1’2, at which the motion is purely shear. As each branch at positive u passes through that point its slope varies continuously. The branch for (T= 0 also passes through the above common point, but the slope there changes discontinuously. At negative u the branches do not pass through a common point. Apart from at the point common to all the curves for positive a, the level of k,/ k, at given d/h, decreases as c~/c,~increases. For cd/c, > 2’j2, the slope of each branch at the common point is the same as that for the n = 1 bound at that point, namely unity, and that applies also to the second part of the branch at cd/es = 21i2 (i.e., at c = 0). For all branches with positive a, the ratio, c,/c, of group velocity to phase velocity at the common point is 0.5 and the normalized group velocity, c,/c,, equals 2-1’2. At that point, c,/cs for the u = 0 branch changes discontinuously from 2”2 to 2-“2 as d/As increases. 4.
THE CYCLICITIES UNDERLYING THE NON-ZERO ORDER BRANCHES
4.1. THE CYCLICITY AT CUT-OFF As mentioned in reference [2], over the whole range of a, the spectrum in the (cd/c,, d/A, at cut-off) plane of a real Ah or S,, branch can be regarded as made up of a number of cycles corresponding to passage along a path such as ABC in Figure 6(b) of reference [2], in addition to a partial cycle at each end of the locus. Coupled with the information on initial negative slope from Figure Bl of this paper (see Appendix B), this yields the typical cycle of Figure 5(a), which governs the type of behaviour of the branches in the vicinity of the k,/ k, = 0 axis as follows. For an Ah or S,, branch of order h = (p + q - 1)/2, as cd/c* increases from (q-2)/p to q/p, the branch starts at a cut-off value of d/As increasing from (q - 2) x 0.5 to q x 0.5, while, as cd/c, increases from q/p to q/( p - 2), the cut-off value remains constant at q x 0.5. The initial slope of the branch after departure from the cut-off point is positive, except in the region D to E of Figure 5(a). When u is positive, the Ah and S,, branches cross the line k,/k, = 2-“* where d/As equals 2h x 2-“2 and (2h + 1) x2-“*, respectively [ 11, and then follow closely above the bounds n = 2h and n = 2h + 1, respectively. Thus, for all positive values of a, the Ah (or S,,) branches for given h will pass through a common point, beyond which they lie so closely together that they are virtually represented by a single curve. This, combined with the cyclicity described above, and with the undulations associated with intersections pf branches of opposite symmetries, governs the overall type of behaviour of the real branches for positive u. That behaviour is illustrated in Figures 6(a) and (b) for an Ah and an Sh mode, respectively. Only the branches corresponding to boundaries between A_ and A+ or Ssq and S,, modes and to the lower and upper limits of positive u are shown. The branch for cd/c, = 2’/2 can be either of sq or of dp type, while that for cd/c, -*cc is of sq type,
LAMB
MODE
SPECTRA,
253
111 (b)
(a)
4
t
q/(p-2)
c
------
[Cd~Cs]m-2.n I
ii‘
1
-----7””
s(q_:lpf t l:“_2~::fjI-
h-2)x
qxo.5
(q-21x0.5
d/A, at k,/k,
Figure 5. Cycles relating to cd/c, the vicinity of k,/ k, = 2-I”.
d/A,
= 0
spectra
of real A,, and S,, branches.
24’2
n x 2-1’2
at k,/k,
=2-“2
(a) In the vicinity
of k,/k, = 0; (b) in
I
(a)
242
-_______-__-______-_---
0
.I
2-V’
I
h-j-0.5
I
I
/ ,
h-j-1.5
I I
I I
h-05
2h x 2-“2
I
h- 2.5
h-:.5
--------_
0
I
I h-j-t
hlj
ti2
/
I
I it
,:
(2Al
(2h+i)x2-‘”
d/X,
Figure 6. Schematic illustration of variation over the range of positive an A,, mode; (b) for an S,, mode. /yA, Region of A,,, or S, branches; 0, common point.
(r of an A,, and an S,, branch. (a) For N, region of A, or S,, branches;
254
A. FREEDMAN
both branches lying within the limits indicated by the adjacent dash-dot lines. In Figure 6(a) or 6(b), j denotes the highest positive integer not exceeding (h -0*5)/( 1 +2”2) or (h + 2”‘)/( 1 + 2”2), respectively. The undulations associated with intersections of branches are relatively small and are not illustrated in Figure 6. As can be seen both from Figure 6 of reference [2] and the present Figure 6, the number of complete cycles tends to increase as h increases. For the low order modes the number of complete cycles may be down to zero, and only the partial cycles at the two limits remain. At each coincidence at S/k, = 0 of modes of like symmetry the conditions for thicknessshear and for thickness-stretch modes are both satisfied, and so each branch forming a boundary between two modal regions represents both an A,, and an A, (or an S,, and an SdP) mode. This applies at each of the two branches starting with slopes of opposite sign at an axial coincidence value of cd/ c,~. The general manner in which the branches within a single cycle vary is examined next. The deduced variations take account of the need to avoid a discontinuity in modal behaviour, despite a jump of initial slope as cd/c, is varied through each value at which there is coincidence at cut-off of a pair of branches of like symmetry. (See equations (27) and (28) of reference [2].) A schematic, generalized representation of the sequence within a cycle is presented in Figure 7, in which the letters A to E correspond to those on the cycle of Figure 5(a). At A the branch starts at an initial angle of tan-’ (7r/2), and the initial slope of each branch remains positive between A and D. At D a boundary has been reached between branches with positive and branches with negative initial slope. Between D and E the initial slope is negative. Additionally, between D and B, a point of inflection gradually moves towards the cut-off point, while the initial radius of curvature also decreases from infinity to zero. At B the initial curved part of the branch and its radius of curvature both vanish, producing an initial angle of -tan-’ (7r/2). Between B and E the sequence is the reverse of that between D and B, and at E a boundary has been reached between branches with negative and branches with positive initial slope. Between E and C the sequence is the reverse of that between A and D, so that at C the branch starts at an initial angle of tan-’ (7r/2). Thus, at each coincidence value, the transitions have been achieved smoothly and without discontinuity in modal behaviour.
i\ 2
0
(4 -2) x 0.5
q x 0.5 d/X,
Figure 7. Generalized representation of behaviour of initial parts of an A, or S,, branch over a cycle of cd/c, B, cd/c, = q/p; C, cd/c, = q/(p -2); D and E, values corresponding to Figure 5(a). A, cd/c, = (q-2)/p; boundaries between those branches with positive and those with negative initial slope.
A corresponding cycle of curves for the ratio, c,/c, of group velocity to phase velocity is shown in Figure 8(a) and for the normalized group velocity, c,/c,, in Figure 8(b). Because it avoids confusing overlap of several curves, these cycles of group velocity are presented half a cycle out of phase with the cycle of Figure 7, thus covering B, to C or A, and A to Bz. The curves of Figure 8 were derived from those of Figure 7 by an
LAMB
MODE
SPECTRA,
III
255
0.4
0.2 9 e
0 I
I I (q-21 x0.5
qxo.5
(q-21X0.5
q x0.5
-1
I
-0.2 d/i,
Figure 8. Generalized representation of behaviour of group velocity over the initial parts of an Ah or S,, branch for a cycle of cd/c, values. B,, ~~/c,=(9-22)/(~+2); C and A, cd/cs=(9-2z)/P; I%, %/c,=~/P; E and D, boundaries between those branches which initially have group and phase velocities in opposite directions and those which do not. (a) For c,/c; (b) for c,/c,.
approximate graphical procedure based on the equations presented in Figure Al(b) of reference [l] and, as the latter equations involve the slope of the line from the origin to each point on the curves of Figure 7, q was arbitrarily set to 8. At the cut-off points, the group velocity is zero for all the branches, except for those at coincidences of modes of like symmetry, for which c,/cs = f4/7rq. However, even there the total group velocity remains zero. Despite the discontinuities in c,/cs, the curves of Figure 8(b) illustrate how there is a smooth transition at each coincidence value of cd/c,. 4.2. THE CYCLICITY AT kJ k, = 2-“2 Over the range of negative a, the spectrum in the (c,,/c,. d/As at k,/k, =2-l’*) plane can be regarded as made up of a number of cycles corresponding to passage along a path such as FGH in Figure 13(a) of reference [2], in addition to a partial cycle at each end of the locus. As with the cyclicity at k,/ k, = 0, the number of complete cycles tends to increase as h increases and, in the case of low order modes, may be down to zero. Letting the value of cd/c, at which there is intersection of the m and n bounds at k,/k, = 2-l’* a cycle of the above type is shown in Figure S(b). Between be denoted by [Q/C,],,,, the Ah or S,, branch of order (m + n - 1)/2 crosses the line CC~/C~I~,~-~ and Ccd~~l~,~
256
A. FREEDMAN
k,/ k, = 2-“2 at points for which the value of d/h, varies from (n - 2) x 2-‘12 to n x 22”*, while between [c~/c.~]_ and [c~/c,~],,_~,~ it crosses that line at a common point for which d/A., equals n x 2-‘12. For a given Ah mode and varying cd/c,, equations (34) and (35) of reference [2] yield the maximum and minimum values of n at which there are coincidences at k,/ k,%= 2-“2 with the A,,+, or A,,_, mode as nmax= 2h and nmin equals the lowest even integer not smaller than (2h + 1)/(2-l’* + 1). Similarly, for an S,, mode, nmax= 2h + 1 and nminequals the lowest odd integer not smaller than (2h+ 1)/(2-l”+ 1). At each value of n, the corresponding m value is given by m = 2h + 1 - n. For an Ah and also for an S, mode, schematic illustrations of the variation over the range of negative u of the part of the branch near k,/ k, = 22”* are presented in Figure 9, the parameter used being cd/c,. Apart from the branches at the upper and lower limits
I,
,,
IO)
2-’ l/2
_
/
Cd/cy]2j+l,2(h-j-I) /
I n
zL
(h-j-l)X2’fi
I
I
I\
I ,
[CdfC.J6,2(h-21
(h-j )x21/i!
(h-2)X2”*
1
(h-1)X2’”
/ II
(2h. -2j-3)~2-'~ .(2h-2j-1)x2m1’2
I I
I I
I
/
I
@h-5)X2-‘“(2h-J)XZ
I I
hX2”’
I lR (2h-l)x2-“p
I I
(2h+l)x2-v2
Figure 9. Schematic illustration of variation over the range of negative w of the parts of a non-zero Lamb branch near k,/k, =2-“2. (a) For an Ah mode; (b) for an S, mode; 0, common point.
order
258
A. FREEDMAN 5. SPECTRA
OF THE
NON-ZERO,
LOW ORDER
MODES
With cd/c, as the parameter,
the families of spectral curves for each of the first four in Figures 10 and 11, respectively. These sets of curves were assembled from spectra at fixed values of cd/c, sketched by using the techniques described in Appendices A and B. Ah and S,, modes are presented
2
1.0
I
[ (cl
Figure11. The variationof the S,, 0, common
point.
branches with cd/c,.
I
-
I
(a) S, mode; (b) S, mode; (c) Sa mode; (d) S, mode;
LAMB
MODE
SPECTRA, III
259
6. CONCLUSION To supplement the examination in two companion papers [l, 21 of the Lamb mode spectrum of a free, isotropic, elastic plate at fixed values of the Poisson ratio, a, in this paper the variation of individual real branches with m over the complete range of the latter parameter has been investigated. The zero order antisymmetric branch, Ao, shows only gentle changes of level over the whole range of g (see Figure l), the normalized wavenumber, k,/ k,, at given normalized plate thickness, d/h,, decreasing as u increases. For all positive u values the zero order symmetric branch, S,,, goes through a common point (see Figure 3) at which the normal displacements in the plate are entirely equivoluminal. For negative u values there is no common point and the normal displacements are never completely equivoluminal. Apart from at the common point, k,/k, at given d/h, decreases as u increases. For the non-zero order antisymmetric and symmetric branches, Ah and S,,, where h is a positive integer, the general shapes of the branches are largely influenced by the following two cyclicities. (i) At k,/k, = 0 (i.e., where the normal displacements within the plate are entirely dilatational), the branches for which the ratio of dilatational to shear velocities, cd/c,, lies between n/m and n/(m -2) (where m, n = 1,2,3,. . . , n/m 3 (4/3)“‘, h= (m + n - 1)/2) pass through a common point at which d/A, = n/2 (see Figure 6). (ii) At kJ k, = 2-l’* (i.e., where the normal displacements within the plate are entirely shear), the Ah and S,, branches for all cd/c, values down to 21’2{2h/[1+(2h)‘]“2} and 2 1’2,respectively, pass through a common point at d/As equal to 2h x 2-“2 and (2h + 1) x 2- 1/2, respectively, while at lower values of cd/c, there are cyclic groupings passing through other common points (as indicated in Figures 9(a) and (b)). The number of cycles of each of the two types tends to increase with h, and at low values of h the number of complete cycles may be down to zero. Families of branches covering the complete range of positive and negative u have been provided (in Figures 10 and 11) for the Ah and S,, modes at values of h from 1 to 4. Additionally, supplementing Mindlin’s rules of bounds, guidance has been proffered (in Appendices A and B) for easy sketching (without need for any computations) of the real and imaginary Lamb mode spectra on a grid of bounds at any positive or negative value of u.
ACKNOWLEDGMENT I am grateful to Mr P. G. Redgment for critical reading of the manuscript helpful discussions.
and for
REFERENCES 1. A. FREEDMAN 19% Journal of Sound and Vibration 137,209-230. The variation, with the Poisson ratio, of Lamb modes in a free plate, I: General spectra. 2. A. FREEDMAN 1990Journal of Sound and Vibration 137,231-247. The variation, with the Poisson ratio, of Lamb modes in a free plate, II: At transitions and coincidence values. 3. R. D. MINDLIN 1957 in Roceedings of the 11th Annual Symposium on Frequency Control, U.S. Army Signal Engineering Laboratories, Fort Monmouth, New Jersey, l-40. Mathematical theory of vibrations of elastic plates. 4. R. D. MINDLIN 1960 in First Symposium on Naval Structumi Mechanics, 1958 (J. N. Goodier and N. J. Hoff, editors), 199-232. Oxford: Pergamon Press. Waves and vibrations in isotropic, elastic plates.
260
A. FREEDMAN
APPENDIX
A: GUIDELINES
FOR SKETCHING
THE LAMB SPECTRA
By making use of equations (Bl)-(B4) of [l], a grid of bounds for given o can readily be produced on the d/h,, k/k, co-ordinate system by using an automatic plotter. Rapid and easy sketching on that grid of the Lamb mode spectrum (excluding the zero order modes) is facilitated by the following considerations. Except where stated otherwise, these considerations apply to c > 0.263 in the d/A,, k,/ k, plane, but to arbitrary values of u in the d/h,, ki/ k, plane. The regions of the spectrum requiring careful decisions on how the branches should be drawn are those bordering the line k/k, = 0. Once those parts of the curves have been sketched, the other parts follow straightforwardly. In particular, when sketching the spectrum, one is faced with the need to decide, for the start of each real branch, whether the curve should proceed to the right or to the left of the real bound starting at the same point. While that could be determined, as described by Mindlin [3], by finding the local ratio of curvatures of branch and bound, the following simple guidance, which calls for no calculations, suffices. In the region bordering on k/k, = 0, the latter axis and the m and n bounds divide the d/h,, k,/k, plane or the d/A,, ki/ks plane into three-, four- or five-sided areas of the types shown by the thin lines in Figures AI-A3. Figure Al applies to the d/A,, k,/k, plane for general values of m, while Figure A2 applies to the particular case where m = 0. Figure A3 is for the d/h,, ki/k,y plane. In Figures Al-A3, each intersection of bounds for which m + n is even is denoted by a large dot, and this gives rise to two diagrams for each area of similar shape. The heavy lines show the layout of branches within each area and, as can be seen, in half the cases each type of area contains no branches. Symmetric and antisymmetric modes are denoted either by thick continuous and thick broken lines, respectively, or vice versa. Figure Al(a) is for c greater than O-263. As shown in Figure Al(b), for u smaller than O-263, before intersecting at the intersection of bounds, the branches may be constrained to go through the lower of the two “additional” points of intersection [l]. Figure A2(a) is also for u greater than O-263, but Figures A2(b) and (c) illustrate that, for values below O-263, the branches are constrained to go through both additional points of intersection. Figure A2(c) also illustrates that when u is negative the branches cross the kJ k, = 2-‘12 line between the two additional intersection points, and each branch is tangential at the crossing point to an n or m bound. The right-hand one
d/X,
Figure Al. axis, showing 0.263>crz-1.
Types of area formed, for m # 0, by the grid of bounds in the region adjacent to the k,/k, =O the location and general form of the Lamb branches within each ama. (a) 0.5~ a>0*263; (b) bound; ?? , intersection of bounds with m + n even; or - --, branch. -,
LAMB
MODE
SPECTRA,
261
III
d/X,
Figure A2. Types of area formed, for m = 0, by the grid of bounds in the region adjacent to the k,/k, = 0 axis, showing the location and general form of the Lamb branches within each area. (a) 0.5 > o > 0.263; (b) 0.263 > D > 0; (c) 0 > u 2 -1. Bound: -. Branch: -, antisymmetric; - - -, symmetric. ?? , intersection of bounds with m + n even.
m+lJ
m+l/“+’
d/X,
Figure A3. Types of area formed by the grid of bounds in the region adjacent to the ki/ks = 0 axis, showing the location and general form of the Lamb branches within each area. -, bound; 0, intersection of bounds with m + n even; or -- -, branch.
of each pair of real branches, such as shown in Figures Al and A2, may, after leaving the k,/ k, =0 axis, initially have either positive or negative slope. Where there is doubt, the decision on whether a real branch has negative initial slope (and, hence, its imaginary continuation has positive initial slope) can be ascertained as described in Appendix B. For the special cases where there are coincidences of bounds on the k/k, = 0 axis, (as discussed in reference [2],) the resulting situations are similarly illustrated in Figure A4. When cd/cS = n/m, where n and m are integers one of which is even and the other is odd, then two symmetric or two antisymmetric real branches start at the same cut-off point. These branches do not lie within the four-sided areas shown in Figures A4(b) and (d), but, for clarity, the convention adhered to in Figures Al-A3 and the rest of Figure A4 of only showing curves which lie within such areas is here disregarded. The typical behaviour of bounds and branches in the region where k,/k, is greater than kd/ k, is as previously portrayed in Figure 13 of reference [ 11.
262
A. FREEDMAN
i
z (c)
Antisymmetric n+l +/ ,
(d)
n(odd)/
4
m+l
nkven) /
n+l,
/’
d/X,
Figure A4. Behaviour of bounds and branches where there are coincidences on the k/k, =0 axis. Bound: -. Branch: or - - -. (a) For general values of m, when m + n is even; (b) for general values of m, when m + n is odd; (c) as (a) when m =O; (d) as (b) when m = 0.
The imaginary branch linking to the real A, branch appears to show little change over the whole range of a, and is exemplified in Figure 3 of reference [ 11. For u > O-263, the procedures for sketching the bulk of the Lamb mode spectrum are thus as follows. (1) On the grid of bounds for the given cd/c,, those intersections are marked in for which m + n is even. (2) For each applicable area adjacent to the k/k, =0 axis, the branches are sketched in as indicated by the corresponding area in Figure Al, A2, A3 or A4 (supplemented by the information from Appendix B on whether the initial slope is negative). (3) The sketching of these branches into successive adjacent areas enclosed between bounds is continued, noting that the branches can only exit each such enclosed area at a marked intersection of bounds, (with the exception of the antisymmetric branches crossing the m = 0 bound). (4) When a branch crosses the line k,/ k, = 2-l”, it does so tangentially to an n bound. For u c 0.263, the following modifications to the above procedures are needed. From Figure 7 of reference [l], or by use of equations (22) and (21) of reference (5) [l], the values of the co-ordinates of the intersections of branches additional to those at intersections of bounds are found, and these points are then marked onto the grid of bounds. (6) Within any one area enclosed by bounds (or by bounds and the k,/ k, =0 axis), and which contains such an additional intersection point, the branches are constrained to cross over at that point.
LAMB
MODE
SPECTRA,
111
263
For 0~0,
the following further modification to the above procedures applies. it does so tangentially to either an n or an m bound.
(7) When a branch crosses the line kJ k, = 2-l”,
APPENDIX
B: OCCURRENCE OF REAL BRANCHES NEGATIVE INITIAL SLOPE
WITH
In the region where a real branch has initial negative slope and the slope is less than -(k,/
k,)/( d/A,), the group velocity and the phase velocity are in opposite directions, and the group velocity drops to zero at the point at which the minimum of d/AS occurs. After transformation
from the co-ordinate
system, Mindlin’s
equations
(50)-(53)
system used by Mindlin of reference
TABLE
to the d/h,,
[3] yield the following
k/k, co-ordinate criteria for real
Bl
Ranges of cd/c, for given A,, modes over which A, initial slope
and A,, real branches have negative
cd/c,
Lower limit h
P
2 344 55 66 6 77 7 88 8 8 99 9 IO10
2 2-
11 -
11 12 13 -
2-
4 24 2 4 6 24 6 24 6 46 8 6 8 8 10 10 12 -
for
A,
1.1838 24452 3.4809
(4/3)"'t 4.4911 1.7104 5.4952 2.2319
(4/3y2t 6.4971 2.7402 1.4725 7.498 1 3.2441 1.8185 3*?i62 2.1578 1.3551 2.4943 1.6131 1.8673 1.2851 1.4903 1.2384
t Lower limits for A, or for to a coincidence value.
A,,, set
Value at coincidence (forms upper limit for A, and lower limit for A,,) 1.5000 2.5000 3.5000 1a2500 4.5000 1.7500 5.5000 2.2500 1.1667 6.5000 2.7500 1.5000 7.5000 3.2500 1.8333
(4/3Y2t
3.7500 2.1667 1.3750 2.5000 1.6250 1.8750 1.3000 1.5000 1.2500
by lower limit of range of cd/c,.
Upper limit for A,, 1.9442 2.8110 3.7287 1.3899 4.6792 1.8603 5.6470 2.3194 1 a2364 6.6245 2.8230 1.5575 7.6080 3.3120 1.8815 1.1669 3.8038 2.2078 1.4106 2.5358 1.6556 1.9017 1.3242 1.5212 1a2675 The limit for
A,,does
9 3 5 7 5 9 7 11 9 7 13 11 9 15 13 11 9 15 13 11 15 13 15 13 15 15 not correspond
264
A. FREEDMAN
branches to have initial negative slope and, hence, a point of zero group velocity: (a) for an A,, mode, (qr/4)
+
[4/(cdlcs)1
cot
[qv/2(cd/cs)l
q = 1,3,5,.
..;
(Bl)
tan
[q~/2(cd/c.v)l
q = 2,4,6,
...;
032)
(b) for an S,, mode, (q~/4)-[4/(cd/cs)l
TABLE
B2
Ranges of cd / c, for given S, modes ouer which Sd, and S,, real branches have negative initial slope
I
h 1 2344 55
3
Lower limit for S&
P
: 1 3 1 3 1 3
5
5
66
1 3
6
5
77 7 7 88 8 8 99 9 lo10 10 1111 1212 13 14 -
1 3 5 13 5 7 3 5 7 5 7 9 7 9 9 11 11 13
\
-
-
1.2162 3.9475 5.9849 1.1570 7.9937 1.9525 9.9968 2.6474 (4/3)“? 11.9982 3.3236 1.5671 13.9989 3.9944 1.9835 -
-
15.9993 4.6632 2.3906 1.4053 5.3310 2.7941 1.7010 3.1961 1.9917 1.3162 2.2802 1.5450 1.7706 1.2596 14458 1.2204
Value at coincidence (forms upper limit for S& and lower limit for S,,) 2.0000 4~0000 6.0000 1.3333 8.0000 2*0000 10~0000 2.6667 1.2000 12~0000 3.3333 1.6000 14~0000 4.0000 2*0000 (4/3p2t 16.0000 4.6667 2.4000 1.4286 5.3333 2.8000 1.7143 3.2000 2.0000 1.3333 2.2857 1.5556 1.7778
1.2727 1.4545 1.2308
I’ Lower limit for sd,, or for s,,, set by lower limit of range of cd/c,. to a coincidence value.
Upper limit for S,, 3.3904 4.7922 6.5373 1.5599 8.4045 2.1717 10.3239 2.7995 1.2956 12.2700 34406 1.6774 14.2315 4.0897 2.0655 1.1960 16.2026 4.7437 2.4535 1.4731 5.4008 2.8460 1.7521 3.2403 2.0327 1.3624 2.3144 1.5808 1 *WOO 1.2932 1.4726 1.2460
9 2 4 6 4 8 6 10 8 6 12 10 8 14 12 10 8 16 14 12 10 16 14 12 16 14 12 16 14 16 14 16 16
The limit for .Squdoes not correspond
LAMB
(c) for an A,
MODE
SPECTRA,
265
III
mode,
(Pp/4) -[4/(cd/~)~l
tan [P~(cdlc.Y)121 CO,
p=2,4,6
, ****9
(B3)
(d) for an S,, mode, p = 1,3,5, . . . .
(P~/4)+t4/(c~/cs)31cot~P~(c~lc~)/21~0,
(B4)
(Bl) and (B2) for the A,, and S,, modes are satisfied, for each value of to coincidence. Accordingly, relationships (B3) and (B4) for the A, and S,, modes are satisfied, for each value of p, over a range of cd/c, values with an upper limit at the value at coincidence. For given q and p, the two abutting ranges correspond to part of an Ah or Sh branch of order (p + q - 1)/2. For all coincidence points in the region in which 0 < d/A, 6 8, the ranges over which the A, and A,, branches have negative initial slope are shown in Table Bl, while the corresponding information for the S,, and S,, branches is provided Relationships
q, over a range of cd/es values with a lower limit that corresponds
I 12
21/i (4/3) ‘fi 6
Figuy _’
Bl. Regions
in which the real branches
exhibit
negative
initial slope.
266
A. FREEDMAN
in Table B2. On the basis of the presentation of Figure 6 of reference [2], the regions of negative initial slope are shown, in Figure Bl, plotted (as heavy lines) in the plane of cd/c, versus d/A, at cut-off. Such regions are larger in the case of the A,, and S,, than of the A, and S,, modes and they decrease with increase of both q and of p.