Harmonic wave propagation in anisotropic laminated strips

Journal of Sound and Vibration (1990) 139(2), 313-324

HARMONIC

WAVE PROPAGATION IN ANISOTROPIC LAMINATED STRIPS G.

R.

LIU

AND

J.

TANI

Institute of Fluid Science, Tohoku University, Sendai 980, Japan

K. WATANABE Department of Mechanical Engineering,

Yamagata University, Yonezawa 992, Japan AND

T. OHYOSHI Department of Mechanical Engineering, Akita University, Akita 010, Japan (Received 3 January 1989, and in final form 4 September 1989)

A numerical method is presented for investigating harmonic wave propagation in arbitrary anisotropic laminated strips. The laminated strip is divided into N plate elements in the thickness. The displacement field within each element is approximated by a linear expansion in the thickness direction and by a series in the width direction. The principle of virtual work is applied to establish the eigenvalue equations, and since the number of the equations is usually much smaller than that obtained by the conventional finite element method, the computing time is much shorter for solution of comparable accuracy. As an example, the dispersion relations are determined for unidirectional carbon/epoxy laminated strips and hybrid composite laminated ones which consist of carbon/epoxy and glass/epoxy layers.

1.

INTRODUCTION

Characteristic solutions for elastic plane wave propagation in an infinite laminated medium are well known, and numerical solutions [l-5] and exact solutions [6] for an anisotropic unbounded laminated plate have been obtained. Approximate solutions for wave propagation in a strip with non-circular cross-section also have been obtained. The Ritz method [7,8] and the finite element method [9, lo] have been used to investigate the problems for isotropic or orthotropic strips. In this paper, a numerical method similar to the finite strip method [ 1l] is presented for the analysis of harmonic waves propagating along anisotropic laminated strips. In the method, the laminated strip is divided into N elements in the thickness direction, and the displacement field within each element is approximated by a linear expansion in the thickness direction and by a series in the width direction. The principle of virtual work is applied to establish the eigenvalue equations. As an example, the dispersion relations are determined for unidirectional carbon/epoxy laminated strips and hybrid composite laminated ones which consist of carbon/epoxy and glass/epoxy layers. 2. FORMULATION Consider a laminated strip which consists of layers of arbitrary anisotropic materials as shown in Figure 1. ‘The thickness and width of the strip are denoted by H and b, 313

0022460X/90/110313+12$03.00/O

@ 1990Academic

Press Limited

314

R. LIU ET AL.

G.

b

x(u)

4

Figure 1. A laminated strip and the co-ordinate system.

respectively. The strip is divided into N plate elements. The thickness, elastic coefficient matrix and density of the nth element are denoted by h,, [Cl, = (cG)” (i, j = 1,. . . ,6) and pn, respectively, as shown in Figure 2. The wave propagates in the x direction. The equations of motion for each element are

{W}=p{ij}-[LIT{a}={O},

(1)

hl, {*I nth interface Ll

nth

hn

element, (n+l)th

E4

pn,

(C/j),

interface

{4

JFigure 2. Isolated element.

where

[LIT= [ MT=

{e.x

ayy

MT={sxx

ffzz

00

a/ax

0

0

a/ay

@yz ax,

0

0

a/az

alay a/ax

0

a/az

0

a/az

alay

a/ax

QJ,

&yJ’ Ez* eyz e,z

id

=

0

[Cl{&), {U}‘={u

e,yI,

,1

(2)

{&I= [LI{v,

(3-5)

0

(697)

W}.

By using equations (4) and (5), equation (1) can be expressed as

~~~=P~~~-~~lT~cI~~l~u~,=~o~.

(8)

Next, the operator matrix can be written as

IL1= [Lx1a/ax

+ [L,] a/ay+

[L,] a/az.

(9)

The matrices [LX], [IL,,] and [L,] are obtained easily by inspection of equation (2). This definition leads to the expansion of the product [ LIT[ C][ L] as [~]T[C][~]=[D,]a2/ax2+2[0,,]az/axa~+2[0,,]a2/axaz + [D,,]

a2/ay2 + 2[ D,,] a2/ay az + [IL]

a2/az2,

(10)

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315

STRIPS

in which the matrices [Dij] (i, j = x, y, z) are defined by

ED~l~~1/2~~~LilT~~1~~jl+~LjlT~~I~LiII~

i,j=x,y,z.

(II)

These matrices are given in the Appendix. The interface stresses {S,,,} (at z = 0, h, ) can be written as

{SIT= {{SJT -{SAT),

(12)

LITb?l~ = [LIT[a~Iwhn.

(13)

where {S*) =

Subscripts m = 1 and 2 indicate the stresses on the upper and lower surfaces of the element, respectively. If one writes the tractions at’both surfaces of the element as {T]T=U-JT the boundary

{QTL

(14)

condition at the surfaces of the element can be expressed as {S) = {7.1.

(IS)

3. ANALYSIS The

displacement

field within the element is approximated

{U}=

as

F [N],{d},exp(ikx-iot)=[N]{d}exp(ikx-iot),

(16)

m=l

where i = a.

t, k and w are time, wavenumber and frequency, respectively, and

INI =[[Nl,

[n

***

[ml=r.~z1[Ylfl,

[WMI,

[Ml = lIdME WT=WI:

(4;

..*

Wid,

(m=l,...,M),

(17)

Cl- zlfJ)[m

id>‘, = {Ml’,

1

(m=l,...,W

b&IT,)

(m = 1,. . . ) M),

I%=[ F irny!].

(18)

In equation (17) the subscript n is omitted for convenience and [E] denotes the 3 x 3 unit matrix. From equation (16) the displacement at the upper and lower surfaces of the element can be written as {V}T={{V,}T

{V,}‘}=[Y]{d}exp(ikx-iwt),

(19)

where [J’l=KY~l, [&I2 *. . [&hl. With the help of equations (12) and (16), the stresses {S} can be expressed as {S}= F [G],(d),

exp (ikx-iot)=[G]{d}exp

m=l

(ikx-iwr),

(20)

where

[Gl=[[Gl,

[Glz

...

[GIMI,

~~lm=~~~~~Q~1+~9~ll~~~~~l,+~Q~1~~~1,1, (21)

G.

316

R. LIU ET AL.

in which Y& is the derivative of Yd with respect to y and the matrices [ Qi] (i = 1,2,3) are given in the Appendix. The approximate displacement field as expressed in equation (16) cannot satisfy the wave equation (8) and boundary condition (15) exactly. By making use of the principle of virtual work, one has 6{

I0

V}‘{ T}

dy =

JJ b

b

b

a{ VT@] dy + I 0

0

h

S{ U}‘{ W} dz dy:

(22)

0

that is, the virtual work performed by the external tractions {T} is equal to that performed by the interface stresses {S} and the unbalanced internal body forces { W}. Assuming the external tractions {T} to be of the form {T}={T}exp(ikx-iwt),

(23)

substituting equations (8), (16), (19), (20) and (23) into equation (22), and then integrating over the element, one has {U=

[a*{4

- ~2mflwL

(24)

(F]=JbpqTgldy.

(25)

where

[a* = [PI - [Kl,

0

In equation (25), [P] is given by

(26)

where

[eImn= Jb[ylL[~jl[y]n

dy

0

(j = 1,2,3).

(28)

If the elements of matrices [Z], [A] and [B] are denoted by 4, av and b, respectively, the calculations between the elements of the matrices, zii = aVx b, is defined by [Z] = [A] * [B]. By using this definition, [ 4 I,,,, can be expressed as [~,lm?l

= [CA1* [~,lm 9

rmnn = [Q21*

[mm

9

[mm

= IQ31 * [al”,

(29-31) where

[~olmn WI [lllmn= [ [O] [lolmn1,

b y”, y”n y”,y”” ~tm~tm ymyvn O [ YWnY”. YWlY”.

[~o~mn= J

dy. 1

yw,,yw, ymywn

(32)

YW,YW”

The matrix [IZlmn can be obtained by replacing Yi, Yin(i = u, v, w) by Yi, Yy”, and the matrix [ IJIm” can be obtained by replacing YimY;., by Yi, YI,.

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STRIPS

In equation (24), the matrix [M] can be written as

[N]T[ N] dz dy =

[MI=P

[WI,

[Ml,,

...

[Wmf

[yl=

[ylz2

. ’*

rM:]zM

[Mj,,

kl,:

...

[km,

where

EMlmn= P

I ,

(33)

[~lT,Wln dz dy = b [ YiJT,WI[Ydlndy, I0

(34)

[RI = P

(35)

The matrix [R] is given in the Appendix. The matrix [K] in equation (25) is defined by

(36)

where (37) The matrices [Ki],, (i = 1, . . . ,5) are given by

[Unn = [@I * [~dmn, [Umn

= [DJ

* [Z2lmn,

[K2lmn = [021*

[Ktlmn = [&I

* [Gnn,

[ZJmn,

[&lmn = [RI

* [Mmn,

where the [Di]mn (i = 1,. . . ,5) are given in the Appendix. By overlapping the matrices in equation (24) of adjacent elements just as one does in the finite strip element method [ 111, one can express the dynamic equilibrium equation as {F1, = [KlT{4,

- u2Wl,W,,

(38)

where the subscript t means total. If the strip is free from traction at the interfaces, the characteristic equation can be expressed by

[Kl:{d, -w2[W,{4,

= UN.

(39)

When the wavenumber k is given, the natural frequency can be obtained from equation (39). Consequently equation (39) is the dispersion relationship. 4. NUMERICAL

EXAMPLES

Two kinds of boundary conditions at both the edges (y = 0 and b) of the strip are considered: (a) simply supported (but no x-direction displacement given), u =o,

w = 0,

MY.” = 0;

(40)

w =o.

(41)

(b) rigidly fixed, u =o,

v = 0,

G.

318 For an orthotropic assumes

R.

ET AL.

LIU

laminate strip, the boundary

Y,,, = sin (mmy/ b),

Y,, = cos (mry/

condition (a) can be satisfied, if one Y,,,, = sin (mry/ b).

b),

(42)

For an arbitrary anisotropic laminated strip, the boundary condition (b) can be satisfied, if one assumes Y,, = sin (mry/ b),

Y,, = sin (mry/ b),

Y,,=sin(mry/b).

(43)

The validity of the numerical results obtained by the present method is demonstrated in Figure 3, which shows the calculated frequency spectra for a unidirectional carbon/epoxy strip with very large width (b = 100 H), and in Figure 4, which shows the analogous results obtained by using the method for an unbounded plate. When the wavenumber is large enough, it can be seen from Figures 3-4 that the curves for a strip of large width approach those for the unbounded plate. This comparison indicates the accuracy of the present method. Frequency spectra for orthotropic strips (unidirectional carbon/epoxy denoted by (CO)), orthotropic hybrid laminated ones (CO/G90/G90/CO) and general anisotropic hybrid laminated ones (CO/G+45/G_45/CO) (where C and G denote carbon/epoxy and glass/epoxy respectively, and the numbers 0, +45, -45 and 90 denote the angle of fiber-orientation to the x axis) have been calculated and are plotted in Figures 3-12 for real wavenumbers. The strips are two or ten times as wide as they are thick. Material properties of the composites are given in Table 1. TABLE

1

Material properties of composites (from reference [ 121)

Properties

Carbon/epoxy Glass/epoxy

h,(GPa)

6 (GPa)

G12(GPa)

h2

v23

142.17 38.49

9.255 9.367

4.795 3.414

0.3340 0.2912

O-4862 0.5071

Dimensionless

wavenumber,

P (g/cm3)

1.90 2.66

kH

Figure 3. Frequency spectra for the unidirectional strip with boundary condition (b); b = (CO) strip.

100 H,

laminate

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319

Dimensionless wovenumber, kH

Figure 4. As Figure 3, but b-co.

Dimensionless wovenumber, kH

Figure 5. Frequency spectra for the unidirectional strip with boundary condition (a); b = 2 H, laminate (CO) strip.

0

1 2 3 Dimensionless wovenumber, kH

4

Figure 6. Frequency spectra for the unidirectional strip with boundary condition (a); b = 10 H, laminate (CO) strip.

320 G. R. LIU

0

7. Fwuew

AI

ok--0

figure

ET

1 hwnsionless

spectra for the orthotropic

2 3 wovenumber, ku

laminated

4

strip with boundary

(CQ/G90/G90/CO) strip.

condition

(a); b=zH,

Dimensionless wavenumber, k~ Figure

Figure

stnp.

9.

Frequency

8.

As Figure 7, but b = 10 H.

Dknensionless kH spectra for the unidirectional strip wovenumber, with boundary condition (b); b = 2 H, laminate (CO)

IN

WAVES

ANISOTROPIC

LAMINATED

1 Dmensimless

0

Figure

2 wovenumber,

321

STRIPS

3

1 4

kH

10. As Figure 9, but b = 10 H.

14

00 0

1 Dimensionless

Figure 11. Frequency (CO/G+45/G-45/CO)

spectra strip.

for the anisotropic

2

3

wovenumber,

kti

laminated

strip

with boundary

1 2 3 Dimensionless wovenumber, kH

0

Figure

12. As Figure

4

11, but b = 10 H.

4

condition

(b); b = 2 H,

322

G.

R. LIU

ET

AL.

The elastic coefficient matrices [C] for any layers can be obtained from Table 1 (see, e.g., reference [13]). In Figures 3-12, wH/C, is a dimensionless frequency, kH is a dimensionless wavenumber, and the dotted lines o = C,,k and o = C,k are the frequency spectra for an infinite body of carbon/epoxy material when the wave propagates in the fiber direction. Further, C, and C, are given by

C,=J&JE,

CS=&JK,

where c,,(,,) and ~6~~~) are the elastic coefficients of carbon/epoxy when the angle between the x axis and the fiber orientation is zero, and pc is the density of carbon/epoxy. The number of elements, N, and the number of terms in the series, M, can be chosen to give the required accuracy. If only the lowest few eigenvalues are required, the calculation is convergent with a few terms of the series (M = 3-6). It is recommended, however, that the element number N is chosen large enough to give sufficiently accurate results. For these laminated strips with b = 2 H, M = 6 and N = 16 have been chosen to determine the lowest ten eigenvalues to three significant figures. The results are shown in Figures 5,7,9 and 11. Figures 3,4,6,8,10 and 12 show the spectra obtained when using A4 = 6 and N = 16 for the other aspect ratio of the cross-section. The frequency spectra of strips (CO) are shown in Figures 5-6 for simply supported boundary conditions and in Figures 9 and 10 for perfectly clamped boundary conditions. The spectral features are clearly different on each side of the straight line w = Cok At a specified value of kH the natural frequencies of the strips with b = 2 H are higher than those of the strips with b = 10 H. In Figures 7 and 8 are shown the frequency spectra of orthotropic hybrid laminated strips (CO/G90/G90/CO) with simple support boundary conditions for b = 2 H and b = 10 H, respectively. In Figures 11 and 12 are shown the frequency spectra for general anisotropic strips (CO/G + 45/G-45/CO) with clamped boundary conditions for b = 2 H and b = 10 H, respectively. Generally, if functions of the series yi, (i = u, u, w) can be found which satisfy the boundary conditions as well as the natural boundary conditions, the frequency spectra can be obtained effectively. Instead of the trigonometrical function which has been used previously, the following beam function also can be used for the clamped boundary condition, Y,,=sin(m7r_y/b)+cos(m~y/b)+cu,sh(mrry/b)-ch(m~y/b),

(44)

where (i = u, v, w).

a,=[ch(mr)-cos(m~)]/sh(m~)

(45)

In Table 2 are listed the lowest seven natural frequencies obtained for unidirectional carbon/epoxy by the two kinds of functions for comparison. A close agreement between TABLE

2

Comparison of dimensionless natural frequencies obtained by using two kinds of functions (kH=2-0, b=2 H) Mode --1

Trigonometrical function Beam function

2

3

4

5

6

7

0.3822 0*5096 0.5342 0.6942 0.7104 0.7790 0.8738 0.3857 0.5162 0.5407 0.7064 0.7149 0.7864 0.8775

WAVES IN ANISOTROPIC LAMINATED STRIPS

323

the corresponding frequencies is shown in Table 2. The results obtained by using the beam function are only a little larger than those obtained by using the trigonometrical function. 5. DISCUSSION AND CONCLUSIONS The solution for wave propagation in unbounded laminated plate consists of infinite harmonic waves in the thickness direction only. These harmonic waves can be developed through search procedures to find out the characteristic roots [6], but for a strip with rectangular section, the exact solution must be formulated by using the infinite harmonic waves in the two directions of the section. Unfortunately, it is very difficult to determine the large number of harmonic waves that satisfy the boundary conditions at all of the surfaces of the strip. Hence approximate methods have been developed, such as the Ritz method [7] and the finite element method [9, lo]. In these methods, the cross-section of the strip is divided into elements in two directions. In the alternative method presented here, the strip is divided into plate elements in the thickness direction only, and the variation of the displacement in the width is expressed by using the series of Yi,. This method enables one to obtain the desired results by using only a few terms of the series, and hence only comparatively small matrices need be used to obtain good results. When the strip width is much larger than its thickness, many elements would be needed in the finite element method to obtain good results, whereas with the present method a small number of plate elements can be used and the calculation is convergent with a few terms of the series. Hence the present method considerably reduces the computing work for solution of comparable accuracy. Furthermore, the application of the present method to the analysis for a laminated strip is particularly appropriate, because the strip is made of thin layers. REFERENCES 1. E. KAUSEL 1986 International Journal for Numerical Methods in Engineering 23, 1567-1578. Wave propagation in anisotropic layered media. 2. S. B. DONG and K. H. HUANG 1985 Journal of Applied Mechanics 52,433-438. Edge vibrations in laminated composite plates. 3. S. B. DONG and R. B. NELSON 1972 Journal of Applied Mechanics 39, 739-745. On natural vibrations and waves in laminated orthotropic plates. 4. R. B. NELSON and S. B. DONG 1973 Journal ofSound and Vibration X),33-44. High frequency vibrations and waves in laminated orthotropic plates. 5. S. B. DONG and K. E. PAULEY 1978 Journal of the Engineering Mechanics Division 104(EM4), 801-817. Plane waves in anisotropic plates. 6. G. R. LIU, J. TANI, K. WATANABE and T. OHYOSHI 1988 Journal of Applied Mechanics (to be published). The Lamb wave propagation in anisotropic laminates. 7. B. AALAMI 1973 Journal of Applied Mechanics 40, 1067-1072. Waves in prismatic guides of arbitrary cross section. 8. N. J. NIGRO 1966 Journal of the Acoustical Society of America 40(6), 1501-1508. Steady-state wave propagation in infinite bar of noncircular cross section. 9. P. E. LAGASSE 1973 Journal of the Acoustical Society of America 53(4), 1116-l 122. Higher-order finite-element analysis of topographic guides supporting elastic surface wave. 10. G. 0. STONE 1973IEEE Transactions on Microwave theory and Techniques MIT-21, 538-542.

High-order finite elements for inhomogeneous acoustic guiding structures. 11. Y. K. CHEUNG 1976 Finite Strip Method in Structural Analysis. Oxford: Pergamon Press. 12. K. TAKAHASHI and T. W. CHOU 1987 Journal of Composite Materials 21,396-407. Non-linear deformation and failure behavior of carbon/glass hybrid laminates. 13. R. M. JONES 1975 Mechanics of Composite Materials. Washington, D.C.: Scripta Book Company.

324

G. R. LIU

ET AL,

APPENDIX

cc,,+ C56)

2Cl6 (cl,+ %A raJ

[

2c45 1

sym.

2c15 [&I

[D

rD51 =[[D;:]

2c35

sym.

+d 2C56 (cz5 2c24

CC36 + c45) 2c34

I

1 -[D 1 -[D;]

’

65

sym.

I

[Rl=W4~;;

c56

[031’=

&ll],

Ifl=[

1

c52

c54

&6

c42

‘&I

c34

c32

c34

[

c45 c44

[Dzzl =

9

(c23+%J

sym.

I

(c45 + C36)f

2%

[

[

cc,,+ C56) (Cl3+ c55)

= (l/2)

[Qzl = (l/2)

(%5+%) ,

2C26

= (l/2)

,

IQ31 = [[Zjl’

i

8 8].

I