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Transient wave propagation in a viscoelastic layered composite—An approximate theory
Journal of Sound and Vibration (1987) 113(1), 141-153
TRANSIENT WAVE PROPAGATION IN A VISCOELASTIC LAYERED COMPOSITE-AN APPROXIMATE THEORY G. A.
BIRLIK
Department of Engineering Sciences, Middle East Technical University, Ankara, Turkey AND
Y.
MENGI
Department of Civil Engineering, Cukurova University, Adana, Turkey (Received 21 August 1985, and in revised form 3 April 1986)
The approximate theory proposed in reference [1] for viscoelastic layered composites is appraised by applying it to a transient wave propagation problem. The problem involves a viscoelastic slab subjected at one end to a dynamic pressure which has either step or trapezoidal variation in time while its other end is kept fixed. The faces of the slab are parallel to the layering. For the case in which the composite material is elastic, which can be obtained from viscoelastic case when viscous terms vanish, the wave profiles for normal stress and particle velocity are determined by using exact and approximate theories, and they are compared. It is found that even the lowest order approximate theory is capable of predicting the essential dynamic characteristics of the layered composite correctly. Further, the influence of the viscosity on wave profiles is studied by using the approximate theory. For the time integration of the approximate equations a numerical algorithm based on FFT is employed. 1. INTRODUCTION
By using a new technique a higher order approximate theory was developed in reference [1] for the dynamic behavior of viscoelastic layered composites. The present study is concerned with the assessment of this theory in cases of transient wave propagation. In recent years, a new method has been proposed to study the geometric dispersion of waves in a layered composite (see, e.g., references [2-4]). This method, called the "viscoelastic analogy", was developed only for cases in which the direction of waves is perpendicular to layering and it involves the replacement of the layered material by a fictitious viscoelastic one. The material functions of the viscoelastic material are determined in such a way that the viscoelasticity theory describes the geometric dispersion properties of the composite. The evaluation of material functions in the viscoelastic analogy requires the solution of some integral equations and accordingly involves lengthy computations. In the aforementioned studies the predictions of viscoelastic analogy were compared with those of the exact theory when the constituent materials of the composite are elastic and a close agreement between these two is found. The object of the present study is to appraise the approximate theory proposed in reference [1] with regard to its ability for predicting transient waves propagating in a viscoelastic layered composite. It may be noted that this approximate theory is three dimensional and is valid for any type of wave propagating in an arbitrary direction in a layered composite. All of the material constants and functions appearing in the theory 141 0022-460X/87/040141+ 13 $03.00/0
©
1987 Academic Press Inc. (London) Limited
142
G. A. BIRLIK AND Y. MENGI
are known in terms of constituent properties of the layered composite. A basic feature of the theory is that it satisfies the interface conditions between the layers exactly. Accordingly, one expects that the approximate theory should be capable of describing correctly the geometric dispersion of waves in the layered composite. For the assessment of the theory the transient waves propagating perpendicular to the layering are considered. For the generation of these waves the following problem involving a composite slab is chosen. The composite slab is made of two alternating viscoelastic layers. One end of the slab is subjected to pressure having a time variation which is either step or trapezoidal. The other end of the slab is kept fixed. The faces of the slab are parallel to the layering so that the propagation direction of the waves generated by the applied pressure will be normal to the layers. The equations of the approximate theory are integrated with respect to time by employing a numerical technique in which use is made of the FFT (Fast Fourier Transform). The proposed technique is discussed in detail within the text of the paper. The approximate solutions are obtained by using zeroth order theory and first order continuity conditions. The numerical results are obtained in terms of wave profiles for normal stress and particle velocity, and for the cases in which the slab is made of four and eight pairs of alternating viscoelastic layers. For the elastic case, which can be obtained from viscoelastic case as its special case, the predictions of approximate and exact theories are compared. It is found that the match between the two is good in spite of our using the lowest order approximate theory. The match improves, though slightly, as the number of pairs in the slab increases. This is expected, because the smoothing operations used in the development of the approximate theory implies that the macrodimension of the composite is large compared to its microdimension. For the viscoelastic case, wave profiles are obtained for various values of viscosities by using the approximate theory, and are compared in several figures. It is found that as viscosity increases the wave profiles get smoother and the amplitudes decrease. As the number oflayers in the composite slab increases the influence of viscosity on wave profiles becomes less effective. 2. FORMULATION OF THE PROBLEM USING APPROXIMATE THEORY
The viscoelastic layered composite under study is a slab of finite thickness H' made of two alternating layers perfectly bonded at their interfaces. The faces of the slab are parallel to the layering. The two different layers are indicated by the circled numbers 1 and 2 as shown in Figure 1. The layers 1 and 2 are assumed to be made of linear viscoelastic material. The thicknesses, relaxation moduli and mass densities of the layers 1 and 2 are respectively (2h l , J.tl' AI' PI) and (2h 2 , j.L2, A2, P2)' The relaxation moduli A" and IJ-" (0: = 1, 2) are some functions of time t which reduce to Lame's constants in the elastic case. It is assumed that the first pair of the slab starts with layer 1. One end of the slab is subjected to a uniform dynamic pressure while its other end is kept fixed. The slab is referred to an (XI, X2, x 3 ) Cartesian co-ordinate system in which the X IX3 plane coincides with the midplane of the first layer of the first pair. To describe the viscoelastic behavior of the composite, in the mathematical formulation of the problem, the Voigt model is adopted, as it is the model usually preferred in structural dynamics. The problem described above can be formulated by using the approximate theory developed in reference [1]. The proposed theory is general. Its order is arbitrary, and it is valid for any kind of waves propagating in an arbitrary direction of a layered composite. The symmetry conditions of the problem suggest that the variables of the approximate theory depend on the normal distance X2 and time t only. For simplicity, in the discussions
TRAl','SIE:-.ITS IN A VISCOELASTIC COMPOSITE
143
I
H' IE----~-----~..
hz
1E-----~1t-H-----:>t..1 @I
®
I
~
I
®
~~
®
IFixed
I
I I
I I I I
I
First
Second
pai'
poir
-
I J
I
Mth pair
or
(i i ) Trapezoidal looding
(i ) Step loading
Figure I. Description of the problem involving the layered composite slab.
which follow, the normal distance is denoted by x, instead of x z- When zeroth order theory and first order continuity conditions are used the governing equations of the approximate theory, for this problem, reduce to the following: equations of motion: ex
=<
(1)
1, 2;
constitutive relations for interface variables: R-;;::: (3/ ha)(2/-La + Aa R~
) '"
deS: -2ua),
= (11 ha)(2/-La + A",) '" dS~,
ex
= 1, 2;
(2)
first order continuity conditions: L1B x(S;,. S1) =< 2(S;- + 8;:), L1B x(R; + R7) = 2(R l
L1a x(S2" - 8 1) = 2(81 - Si),
+ R;:),
L1a x(R;: - R l )::: 2(R; - Ri). (3) Here L1 = hi + h», axe ) =< a( )/ ax, and (") = aZ ( )at z. In these equations the Greek indices a and f3 take only the values 1 and 2 and they indicate respectively the variables and constants belonging to layers 1 and 2. For example, the variables PI and /-LI pertain to layer 1, and PZ and /-Lz to layer 2. Summation convention is not used in the analysis. In equations (3), * indicates the convolution integral defined by
G*dh:::G(t)h(O)+
'f 0
dh(z)
G(t-z)~dz=
It G'(t-z)h(z)dz, 0
(4)
where G and h are some functions of the time t and O'(z) :::dG/dz. If the actual displacement and normal stress distributions in the x direction are denoted by U'" and and the generalized variable U a appearing a , respectively, the interface variables
:r
S:, R:
144
G. A. BIRLIK AND Y. MENOI
in equations (1)-(3) can be defined in terms of
a",
and it" as
(5)
where X2 is the local co-ordinate axis measuring the perpendicular distance from the midplane of the a phase (the positive direction of X2 is chosen to be coincident with that of the x axis). The first two and last two of equations (3) describe respectively the interface continuity conditions for U'" and a"" and they were obtained in reference [1] using smoothing operations. In equations (3) the displacement and stress continuity conditions relate respectively the displacement and stress interface variables (S~ and R:) of the two constituent layers and have exactly the same forms. 2.1. BOUNDARY CONDITIONS The viscoelastic composite slab under study is subjected to a uniform dynamic pressure P( t) at the left face of the first layer while the right face of the last layer is held fixed (see Figure 1). The relevant boundary conditions can be written in terms of the actual variables 0-", and U'" as
at the left boundary at the right boundary
(x
= -hI)'
(x = H + h2 ) ,
UI =- P( t);
(6)
U2 = O.
(7)
u;
Here H = H' -..d. By using the relations u~ = (Rt - R~)/2 and = (S; + 8:;)/2, which can be obtained from the second and third lines of equations (5), the boundary conditions (6) and (7) can be written in terms of the field variables of the approximate theory as follows: at the left boundary, at the right boundary,
(Ri - R~)lx_o = -2P; (S~+s2)lx=H =0.
(8)
The boundary conditions (8) are written at x = 0 and x = H, instead of at x = -hI and x = H + ha- This is because the smoothing operations used in the derivation of the approximate theory imply that the interface variables and have physical meanings only at the midplanes of the a phase even though they are defined for all x. Study of the equations of the approximate theory, equations (1)-(3), shows that their integration with respect to the space variable x involves four integration constants. Accordingly, unique determination of the solution requires four boundary conditions. Two boundary conditions come from equations (8). The general procedure for establishing the remaining boundary conditions is explained first in what follows, and then the general procedure is used to obtain the remaining two boundary conditions for the problem. Let n be the order of the continuity conditions (c.c.) of the approximate theory considered in the analysis. One first writes down c.c, of the orders 0, 1, 2, ... , n -1. The remaining boundary conditions are established by assuming that those c.c, of the lower orders 0, 1,2, ... , n - 1 which are independent of c.c, of order n are to be satisfied at the boundaries. In the problem here the order of the c.c., equations (3), considered in the analysis is 1: i.e., n = 1. When one writes down the zeroth order c.c. and assumes that those of the zeroth order c.c. which are independent of equations (3) are to be satisfied at the boundaries
R:
S:
TRANSIENTS IN A VISCOELASTIC COMPOSITE
145
of the composite slab, the remaining two boundary conditions can be obtained as follows: at the left boundary, at the right boundary,
(Ri - R7)/x=o = 0;
(st - S7)lx=H = O.
(9)
2.2. INITIAL CONDITIONS Since the slab is at rest initially all of the field variables of the approximate theory should be zero at the initial time t = O. The relevant conditions therefore are (10)
u"lt=o= 0,
where (·)=ajat. The formulation of the problem is now completed. Equations (1)-(3) constitute 10 partial differential equations for the 10 unknowns u", S~, R:. The unknowns can be determined uniquely by solving these equations subject to the boundary conditions (8) and (9) and initial conditions (10). With these unknowns determined, the prediction of the approximate theory for the displacement and stress can be found at any point of a layer from the equations
a" = (2,.."" + A,,) * d(axau,,),
aX a
= ajax",
(11)
where
Here the cf>;'s are Legendre polynomials defined by (12)
cf>o= 1,
and X" = x"j h«, x" being the normal local co-ordinate axis of the layer under consideration. To determine the distributions of displacement U" and stress aCt in a certain layer of the a phase of the composite, first one evaluates u", at the midplane of that layer from the solution of the equations of the approximate theory; then, using equations (11), one finds the desired distributions.
S:
3. SOLUTION OF THE PROBLEM
The object in this study is to integrate the equations of the approximate theory for the problem under consideration, to assess the approximate theory by comparing its prediction for elastic case with that of exact theory and to study the influence of viscosity on the wave profiles by using the approximate theory. The exact solution of the elastic problem is obtained by the method of characteristics. Because the equations of the approximate theory are not hyperbolic the method of characteristics is not suitable for solving them. The approximate solution is obtained by employing the FFT procedure. The method of characteristics used in obtaining the exact elastic solution is well known and can be found in standard text books, such as reference [2]; therefore, it will not be discussed here. On the other hand, the FFT procedure used in the integration of approximate equations is relatively new and is based on the use of the discrete Fourier transform (DFT) and the FFT algorithm (for DFT and FFT see, e.g., references [6-8]). The FFT procedure requires first the determination of the solution in Fourier space.
146 3.1.
G. A. BIRLIK AND Y. MENGI APPROXIMATE SOLUTION IN FOURIER SPACE
To find this solution the Fourier transform is applied, with respect to time variable, to equations (1)-(3). When the initial conditions (10) are taken into account, after some manipulations, the following expressions are obtained in the Fourier space: F
P
F
R;:,=daSi"
sf =
Zj(ea j
+
F
Ri,=baS;:"
+ ea2ql )M cos
z2(eaj
F
F
Ua
ZjX - Zj(ea j
+ ea2q2)H cos Z2X -
R;:,
=- , ga
+ ea2ql)N sin ZjX
z2(ea j
+ ea2q2)J sin Z2X,
si = M sin ZjX + N cos ZjX + H sin Z2X + J cos Z2X,
S; = Mq, sin zjx+ Nq, cos Zjx+ Hq2 sin Z2X+Jq2 cos Z2X,
Q'
= 1,2,
(13)
where d.; = ga/(2+ ga/3ba),
ga = -2h erPa W2,
(2j.ta +A a)* =iw(2j.ta + Aa)F = Ea +iw1Ja,
b.; = (2J.1..a + A",)*/ h""
ell = A (d 2+ b j),
e2j=A(b j - d j),
e12= A (b2 - d2 ) ,
e22 = A(b2+ dl ) ,
A = L12(dj + d 2),
zi= (-S+,) S2-4QT)/2Q,
z~= (-s-Jr-S-=-2-- 4- Q- T) / 2 Q ,
S= (2b 2B+2G+2b 1 D - 2 K ) / A.d, T=(4b 2+4b j)/(A.d)2,
D=2b 2-d2 + dj,
B=2b 1 - d1+d2,
K = dibl-d l) - d j(d2+b j),
q",
Q = BG-DK,
= - (Bz~ + 2/ Ad)/ (Dz~
G= d 2(b2 + d j)- d j(b 2 - d 2),
+ 2/ AL1),
a = 1, 2.
(14)
In equations (14), oi is the radian frequency related to the cyclic frequency f by ta = 21Tf, and i is the imaginary unit. The last term after the equality sign in the last equation of the second line in equations (14) is written in view of the assumption that the viscoelastic behavior of the composite slab is represented by the Voigt model. In this, term B", describes the elastic modulus and '17", the viscosity coefficient. The superscript F in equations (13) and (14) designates the Fourier transform of the variable over which it appears, M, N, Hand J in equations (13) are integration constants and are to be determined by satisfying the boundary conditions (8) and (9), which results in -d,y,
I I
b,
I
q, cos
I I
-d,yz
I I
b,
---------T---------T---------T--------o I b1-bzq, I 0 I b,-b 2
q\ sin z\ H +w cos z,H j
:
z.H
-WI sin ZtH
I
1
q2 sin zzH +Wzcos Z2H
I
:
qz cos zzH - w2 sin zzH
(;:=;I):O;z~:=~~--w~)~i;;-~H: (y~=;2)C~S-Z~HT=(~z-=-:J~;Z~H
M
-2pF
N
o
H
o
-J-"
.
(15)
o
In equation (15), the definitions Q'
=
1, 2,
(16)
are used. Equation (15) determines the integration constants as functions of frequency. Substitution of these into equations (13) establishes the variables of the approximate theory in Fourier space. pF in equation (15) designates the Fourier transform of the applied pressure.
TRANSIENTS IN A VISCOELASTIC COMPOSITE
147
4. NUMERICAL PROCEDURE
As indicated in the previous section, for the approximate solution, we have employed a numerical procedure which makes use of FIT. The procedure has four main steps: (i) determination of the solution in Fourier space; (ii) truncation of the Fourier spectrum of the quantity of interest at the cut-off frequency, and evaluation of the spectrum at discrete frequency points by using a frequency increment; (iii) extension of the Fourier spectrum beyond the cut-off frequency in such a way that the Fourier spectra on the left and on the right of the cut-off become complex conjugates of each other; (iv) inversion of the Fourier spectrum established in (iii) by using FFT, which determines the time variation of the quantity. The solution in Fourier space in step (i) can be obtained either analytically or numerically. For this problem it has been obtained analytically in the previous section. However, if the equations are complicated it should be obtained numerically by using methods available in the literature. Deciding about the cut-off frequency and frequency increment in step (ii) is a crucial step of-this procedure. The cut-off frequency can be chosen, after inspection of the Fourier spectrum of the quantity of interest, as the point beyond which the spectrum values are negligibly small. The frequency increment, however, should be selected in such a way that the frequency variation of the Fourier spectrum ofthe quantity is described adequately. The cut-off frequency fc and frequency increment J1f satisfy the relations
I; = 1/2dt,
d/= 1/T,
(17)
where J1t and T designate respectively, associated with the solution in time space, the time increment and the upper bound of the solution interval. The first of equations (17) relates J1t to fc and can be regarded as a formula giving an optimum value for J1t. Now a few remarks about the selection of Ie are in order. Let j denote the frequency beyond which the Fourier spectrum values of a quantity becomes negligibly small. To obtain the solution with sufficient accuracy the chosen value of fc should be at least equal to However, to be flexible, it is preferable to assign a value to fc which is greater than When this is the case, one can shorten the computations by not evaluating the spectrum values in the interval Ll.fc], and instead set them equal to zero there. Rough estimates for fe and !J.I can be found by studying frequency variations of the Fourier spectra of input functions, for this problem by examining the frequency variation of r". It may be noted that the Fourier transforms of the input functions need not be evaluated analytically; instead, they may be found with the aid of FIT. For the present problem and its solution the following numerical algorithm, based on the procedure described above, was used. It is first to be noted that the use of FFT requires the number of subdivisions N to be expressed by the formula
1
J
(18) The steps of the algorithm are as follows. (i) In view of the discussions given above, a proper value for fc and a value for m in equation (18) leading to a suitable value for df are chosen. It is to be noted that the chosen value of m determines the number of subdivisions to be used both in frequency and time spaces, which, in turn, determines the frequency increment df from the formula jjf= 2fc/ N
(19)
The chosen values of fc and !J.I fix the values of the time increment J1t and the upper bound T of the solution interval by the formulas (17).
148
G. A. BIRLIK AND Y. MENG!
(ii) With use of the values of m and T specified in (i), the Fourier spectrum pF of the input function is computed by FFf at the discrete frequency points having the increment tif. (iii) The Fourier spectra u~, ;;.~ of the normal displacement and stress at a certain point of the composite slab are computed at the frequency points of the interval [0,!c1 with the increment tif. In this calculation Fourier transforms of equations (11) are written at frequency points by taking into account equations (13), where the integration constants M, N, Hand J are computed from equation (15). (iv) The Fourier spectra of ;;.~, u~ are extended to the interval [0,2fc] in such a way that their values in the interval [!c,2fc] are complex conjugates of those in [OJc]. (v) With use of the values of m and tV specified in step (i), ;;.~ and u~ are inverted back into time space by FFT, which determines the values of normal stress and displacement at the time points with the increment tit over the time interval [0, T]. 5. NUMERICAL RESULTS
Numerical results have been obtained for a composite slab having the following properties: (20) For this choice of properties the elastic impedances and viscous properties of the constituent layers are very different. Accordingly, it is to be expected that, while attenuating, the waves will be influenced greatly by the geometric dispersion. The computations were carried out in terms of the non-dimensional time I defined by (21) where C 1 =.JE;Tii;.. The time variation of the applied pressure pet) for which the numerical results were obtained is shown in Figure 1. For step loading, it is zero at I = 0 and rises linearly to a constant value Po during a rise time of a, after which it remains constant. In case of trapezoidal loading, at I = f*, it decreases linearly to a zero value during a time interval of a. Figure 2 shows a comparison of the exact and approximate dispersion curves for the p waves propagating perpendicular to the layering in an infinite elastic composite with the properties given in the first three of equations (20) and 7]", = O. Here, cO and k are o- '\ .---,---.,---..,.--,---,---,.....---.----.-----,---.---. 0·3
i Figure 2. Approximate and exact fundamental dispersion curves for P waves propagating perpendicular to the layering for the elastic case. - , Exact theory; - x - , approximate theory.
149
TRANSIENTS IN A VISCOELASTIC COMPOSITE
non-dimensional frequency and wave number, which are related to, the frequency wand wave number k by (22) In view of this comparison one can expect the approximate theory to give good results for waves whose wave number and frequency are contained in the region of the spectral plane 0"", c.i:i "'" O: 30, 0""" k "'" O'70. It may be noted that, if desired, the comparison can be improved and the periodic pattern of the spectra can be obtained by increasing the orders of the theory and the continuity conditions. With the object of assessing the approximate theory, for the elastic case, the wave profiles for the normal stress at the midpoint Q of a four pair slab were found by using exact and approximate theories and are compared in Figures 3(a) and (b). The former figure displays the comparison for step loading, and the latter for trapezoidal loading. The non-dimensional viscous constants ga, stress s, slab thickness H and co-ordinate of the midpoint i appearing in the figures are defined by ga=1}acJEJhlo
u=a-/Po,
H=Hjh J,
i=xjh J.
(23)
-2,2 (0)
-2·0
-1·6
-1·2 Q,
Ib
-0·8
-0'4
0
-0·4 0
100
50
150
200
240
-1·4
-1'0
:6-
-0,6
s
I
-0,2
Ib
1920
r
0 -0,2
-0·6 -1'0 0
50
100
150
180
r Figure 3(a) and (b). Comparison of time variations of normal stress at the midpoint Q (i .. 11) of the four pair slab (H = 21) for the elastic case. (a) Step loading; (b) trapezoidal loading. - - . Exact theory; approximate theory.
150
G. A. BIRLIK AND Y. MENGI
The influence of reflections from the faces of the slab on the wave profiles is evident in these figures. Figures 3(a) and (b) indicate the capability of the approximate theory for duplicating the exact wave profiles closely. The comparison of the wave profiles for the stress a( Q) for different values of viscous constants g"" when a four pair slab is subjected to step loading, is given in Figure 4(a). The responses to trapezoidal input are displayed in Figure 4(b). As expected, as viscosity increases, the irregularities in the wave profiles smooth out and the amplitudes decrease. Figures Sea) and (b) give exactly the same comparison as in Figures 3(b) and 4(b), but this time for a slab which is composed of eight pairs of alternating layers. Figure 6 presents, for the elastic case, a comparison of particle velocities obtained from approximate and exact theories. In Figure 6, V designates the non-dimensional particle velocity defined by
v= BV/ cPo,
E
n",=h",/IJ.,
= (E I / nl)(E2/ n2)/[(Ed nl)+ (E 2/ n2)]' c=./E/p,
P=Plnl+p2n2'
-2'4
(0 ) -2·0
:.~
-1,6
§
I
-1·2
Ib
-0·8
-0,4
0 -0,2 0
50
150
100
240
200
r -1,4
(b) -\·0
:·t
-0,6
3 Ib
~ ,.
1920
r
-0'2
a 0·2
0·6 ',0
a
150
50
180
r Figure 4(a) and (b). Influence of the viscosity on the time variations of normal stress at the midpoint Q (x = 11) of the four pair slab (if =21). (a) Step loading; (b) trapezoidal loading. - , Approximate theory
(elastic); - - -, tl = 1, t z = 40; .. " (, = 5,
q2 = 200.
151
TRANSIENTS IN A VISCOELASTIC COMPOSITE -Hr------...,r------...,r---------r-------, -1'0
;.~
-0'6
2
'G Ib
r
-0·2
3640
0 0·2
0·6
1·0
H
100
0
200
300
360
300
360
j' -1'4
-1'0
:.~
-0'6
2.
s Ib
3640
1
-0'2
a 0'2
0'6
1·2
100
0
200
j'
Figure 5(a) and (b). (a) Comparison, for the elastic case of, and (b) influence of the viscosity on time variations of normal stress at the midpoint Q (i = 23) of the eight pair slab (if =45) (trapezoidal loading). (a); - - , exact theory (elastic); .. " approximate theory; (b); - - , approximate theory (elastic); - - -, ~ = 2, ~2 = 80; ... , ~l = 5, ~2 = 200. -1-2
-0,8
:.~
-0-4
1920
I
1 ~ I::..
0
0-4
0'6
-1-4
0
50
100
150
180
1 Figure 6. Comparison of time variations of particle velocity at the midpoint Q (.i = 11) of the four pair slab (H' = 21) for the elastic case (trapezoidal loading). - - , Exact theory (elastic); ... , approximate theory.
152
G. A. BIRLIK AND Y. MENGI
6. DISCUSSION AND CONCLUSIONS
The results presented in the previous section show that the approximate theory, even at its lowest order, predicts well the essential characteristics of laminated composites when they are subjected to dynamic loading. The following observations can be made, after study of the figures. When a laminated composite slab is subjected to compressive loading, first a compressive stress wave is induced in the body, which, as it propagates, undergoes dispersion due to reflections and refractions at interfaces. If the boundary at the opposite face of the slab is of clamped type (which is the case in this problem), the reflected pulse at this face will again be compressive. After a certain interval of time, the multiple reflections and transmissions at interfaces and at slab faces may give rise to tensile stresses, in particular, of considerable magnitude in case of trapezoidal loading. As is well known, in practice, these stresses, reducing the load bearing capacity of the composite, finally cause delamination of the composite. As seen in Figures 3-5 this important phenomenon is well predicted by the approximate theory. As the number of layers is increased, damping becomes less effective. This fact can be observed from the comparison of the results for gl = 5 and g2 = 200 in Figures 4(b) and 5(b). As seen from the study of these figures, for the aforementioned viscous constants, the amplitudes for the four pair slab are considerably smaller than those for the eight pair slab. Due to the viscoelastic character of the composite, attenuation is expected. With the increase of the viscous constants of the constituent layers attenuation in the amplitudes become more pronounced (see Figures 4(a) and (b)). Comparison of these two figures, which give the response to step and trapezoidal loading respectively, displays the fact that attenuation is comparatively higher for trapezoidal loading. This has to be expected since damping cannot be observed effectively in step loading due to continuing action of the applied pressure. In the static case the approximate theory gives the stress and displacement distributions exactly. This can be verified as follows for an elastic slab whose left face is subjected to uniform static pressure Po while its right face is kept fixed, as shown in Figure 1. For the static case U'" = 0 in equation (1), which leads to R-;' = O. When this is used in equations (2) and (3), boundary conditions (8) and (9) are taken into account and equations (11) are used, one finds, after some manipulations, *
[( 1
u",=Po
b
1) H - x
1+iJ2
1
-L.\-+iJ
X'" ]
2-iJ",
'
(24)
where b", = Bet / h.: Equation (24) represents the static distributions of stress and displacement predicted by the approximate theory. In equation (24), x is the global co-ordinate axis shown in Figure 1 and x'" = x"'/h"" x, being the local co-ordinate axis of a layer with the origin located at its midplane. To compute the displacement distribution in a certain layer of the slab one substitutes the x value of the midplane of that layer into equation (24)\. Then, for that fixed value of x one varies x"" which determines the desired distribution. It is clear that the stress distribution given in equation (24h is exact. On the other hand, the study of equation (24)\ shows that the displacement varies linearly over each layer; its close examination indicates further that it describes simply the exact displacement distribution. A final remark is now in order. The approximate theory in reference [1] was developed for three dimensional dynamic behavior of layered composites. In this study, the theory has been assessed for a one dimensional problem. This assessment gives an indication about the validity of the theory for the three dimensional case.
TRANSIENTS IN A VISCOELASTIC COMPOSITE
153
REFERENCES 1. Y. MENGI and D. TURHAN 1984 Journal of Sound and Vibration 92.311-320. A higher order
dynamic theory for viscoelastic plates and layered composites. 2. L. M. BARKER 1971 Journal of Composite Materials 5, 140-162. A model for stress wave propagation in composite materials . 3. R. M. CHRISTENSEN 1975 Journal of Applied Mechanics 42, 153-158. Wave propagation in layered elastic media. 4. T. C. T. TING and 1. MUKUNOKI 1979 Journal of Applied Mechanics 46. 329-336. A theory of viscoelastic analogy for wave propagation normal to the layering of a layered medium. 5. R. COURANT and D. HILBERT 1966 Methods of Mathematical Physics. New York : Interscience. 6. E. O. BRIGHAM 1974 The Fast Fourier Transform. Englewood Cliffs, NJ: Prentice-Hall. 7. J. W. COOLEY. P. A. W. LEWIS and P. D. WELCH 1969 Institute of Electrical and Electronic Engineers Transact ions on Education 12,27-34. The fast Fourier transform and its applications. 8. G. D. BERGLAND 1969 Institute of Electrical an Electronic Engineers Spectrum 6,41-52. A guided tour of the fast Fourier transform.
TRANSIENT WAVE PROPAGATION IN A VISCOELASTIC LAYERED COMPOSITE-AN APPROXIMATE THEORY G. A.
BIRLIK
Department of Engineering Sciences, Middle East Technical University, Ankara, Turkey AND
Y.
MENGI
Department of Civil Engineering, Cukurova University, Adana, Turkey (Received 21 August 1985, and in revised form 3 April 1986)
The approximate theory proposed in reference [1] for viscoelastic layered composites is appraised by applying it to a transient wave propagation problem. The problem involves a viscoelastic slab subjected at one end to a dynamic pressure which has either step or trapezoidal variation in time while its other end is kept fixed. The faces of the slab are parallel to the layering. For the case in which the composite material is elastic, which can be obtained from viscoelastic case when viscous terms vanish, the wave profiles for normal stress and particle velocity are determined by using exact and approximate theories, and they are compared. It is found that even the lowest order approximate theory is capable of predicting the essential dynamic characteristics of the layered composite correctly. Further, the influence of the viscosity on wave profiles is studied by using the approximate theory. For the time integration of the approximate equations a numerical algorithm based on FFT is employed. 1. INTRODUCTION
By using a new technique a higher order approximate theory was developed in reference [1] for the dynamic behavior of viscoelastic layered composites. The present study is concerned with the assessment of this theory in cases of transient wave propagation. In recent years, a new method has been proposed to study the geometric dispersion of waves in a layered composite (see, e.g., references [2-4]). This method, called the "viscoelastic analogy", was developed only for cases in which the direction of waves is perpendicular to layering and it involves the replacement of the layered material by a fictitious viscoelastic one. The material functions of the viscoelastic material are determined in such a way that the viscoelasticity theory describes the geometric dispersion properties of the composite. The evaluation of material functions in the viscoelastic analogy requires the solution of some integral equations and accordingly involves lengthy computations. In the aforementioned studies the predictions of viscoelastic analogy were compared with those of the exact theory when the constituent materials of the composite are elastic and a close agreement between these two is found. The object of the present study is to appraise the approximate theory proposed in reference [1] with regard to its ability for predicting transient waves propagating in a viscoelastic layered composite. It may be noted that this approximate theory is three dimensional and is valid for any type of wave propagating in an arbitrary direction in a layered composite. All of the material constants and functions appearing in the theory 141 0022-460X/87/040141+ 13 $03.00/0
©
1987 Academic Press Inc. (London) Limited
142
G. A. BIRLIK AND Y. MENGI
are known in terms of constituent properties of the layered composite. A basic feature of the theory is that it satisfies the interface conditions between the layers exactly. Accordingly, one expects that the approximate theory should be capable of describing correctly the geometric dispersion of waves in the layered composite. For the assessment of the theory the transient waves propagating perpendicular to the layering are considered. For the generation of these waves the following problem involving a composite slab is chosen. The composite slab is made of two alternating viscoelastic layers. One end of the slab is subjected to pressure having a time variation which is either step or trapezoidal. The other end of the slab is kept fixed. The faces of the slab are parallel to the layering so that the propagation direction of the waves generated by the applied pressure will be normal to the layers. The equations of the approximate theory are integrated with respect to time by employing a numerical technique in which use is made of the FFT (Fast Fourier Transform). The proposed technique is discussed in detail within the text of the paper. The approximate solutions are obtained by using zeroth order theory and first order continuity conditions. The numerical results are obtained in terms of wave profiles for normal stress and particle velocity, and for the cases in which the slab is made of four and eight pairs of alternating viscoelastic layers. For the elastic case, which can be obtained from viscoelastic case as its special case, the predictions of approximate and exact theories are compared. It is found that the match between the two is good in spite of our using the lowest order approximate theory. The match improves, though slightly, as the number of pairs in the slab increases. This is expected, because the smoothing operations used in the development of the approximate theory implies that the macrodimension of the composite is large compared to its microdimension. For the viscoelastic case, wave profiles are obtained for various values of viscosities by using the approximate theory, and are compared in several figures. It is found that as viscosity increases the wave profiles get smoother and the amplitudes decrease. As the number oflayers in the composite slab increases the influence of viscosity on wave profiles becomes less effective. 2. FORMULATION OF THE PROBLEM USING APPROXIMATE THEORY
The viscoelastic layered composite under study is a slab of finite thickness H' made of two alternating layers perfectly bonded at their interfaces. The faces of the slab are parallel to the layering. The two different layers are indicated by the circled numbers 1 and 2 as shown in Figure 1. The layers 1 and 2 are assumed to be made of linear viscoelastic material. The thicknesses, relaxation moduli and mass densities of the layers 1 and 2 are respectively (2h l , J.tl' AI' PI) and (2h 2 , j.L2, A2, P2)' The relaxation moduli A" and IJ-" (0: = 1, 2) are some functions of time t which reduce to Lame's constants in the elastic case. It is assumed that the first pair of the slab starts with layer 1. One end of the slab is subjected to a uniform dynamic pressure while its other end is kept fixed. The slab is referred to an (XI, X2, x 3 ) Cartesian co-ordinate system in which the X IX3 plane coincides with the midplane of the first layer of the first pair. To describe the viscoelastic behavior of the composite, in the mathematical formulation of the problem, the Voigt model is adopted, as it is the model usually preferred in structural dynamics. The problem described above can be formulated by using the approximate theory developed in reference [1]. The proposed theory is general. Its order is arbitrary, and it is valid for any kind of waves propagating in an arbitrary direction of a layered composite. The symmetry conditions of the problem suggest that the variables of the approximate theory depend on the normal distance X2 and time t only. For simplicity, in the discussions
TRAl','SIE:-.ITS IN A VISCOELASTIC COMPOSITE
143
I
H' IE----~-----~..
hz
1E-----~1t-H-----:>t..1 @I
®
I
~
I
®
~~
®
IFixed
I
I I
I I I I
I
First
Second
pai'
poir
-
I J
I
Mth pair
or
(i i ) Trapezoidal looding
(i ) Step loading
Figure I. Description of the problem involving the layered composite slab.
which follow, the normal distance is denoted by x, instead of x z- When zeroth order theory and first order continuity conditions are used the governing equations of the approximate theory, for this problem, reduce to the following: equations of motion: ex
=<
(1)
1, 2;
constitutive relations for interface variables: R-;;::: (3/ ha)(2/-La + Aa R~
) '"
deS: -2ua),
= (11 ha)(2/-La + A",) '" dS~,
ex
= 1, 2;
(2)
first order continuity conditions: L1B x(S;,. S1) =< 2(S;- + 8;:), L1B x(R; + R7) = 2(R l
L1a x(S2" - 8 1) = 2(81 - Si),
+ R;:),
L1a x(R;: - R l )::: 2(R; - Ri). (3) Here L1 = hi + h», axe ) =< a( )/ ax, and (") = aZ ( )at z. In these equations the Greek indices a and f3 take only the values 1 and 2 and they indicate respectively the variables and constants belonging to layers 1 and 2. For example, the variables PI and /-LI pertain to layer 1, and PZ and /-Lz to layer 2. Summation convention is not used in the analysis. In equations (3), * indicates the convolution integral defined by
G*dh:::G(t)h(O)+
'f 0
dh(z)
G(t-z)~dz=
It G'(t-z)h(z)dz, 0
(4)
where G and h are some functions of the time t and O'(z) :::dG/dz. If the actual displacement and normal stress distributions in the x direction are denoted by U'" and and the generalized variable U a appearing a , respectively, the interface variables
:r
S:, R:
144
G. A. BIRLIK AND Y. MENOI
in equations (1)-(3) can be defined in terms of
a",
and it" as
(5)
where X2 is the local co-ordinate axis measuring the perpendicular distance from the midplane of the a phase (the positive direction of X2 is chosen to be coincident with that of the x axis). The first two and last two of equations (3) describe respectively the interface continuity conditions for U'" and a"" and they were obtained in reference [1] using smoothing operations. In equations (3) the displacement and stress continuity conditions relate respectively the displacement and stress interface variables (S~ and R:) of the two constituent layers and have exactly the same forms. 2.1. BOUNDARY CONDITIONS The viscoelastic composite slab under study is subjected to a uniform dynamic pressure P( t) at the left face of the first layer while the right face of the last layer is held fixed (see Figure 1). The relevant boundary conditions can be written in terms of the actual variables 0-", and U'" as
at the left boundary at the right boundary
(x
= -hI)'
(x = H + h2 ) ,
UI =- P( t);
(6)
U2 = O.
(7)
u;
Here H = H' -..d. By using the relations u~ = (Rt - R~)/2 and = (S; + 8:;)/2, which can be obtained from the second and third lines of equations (5), the boundary conditions (6) and (7) can be written in terms of the field variables of the approximate theory as follows: at the left boundary, at the right boundary,
(Ri - R~)lx_o = -2P; (S~+s2)lx=H =0.
(8)
The boundary conditions (8) are written at x = 0 and x = H, instead of at x = -hI and x = H + ha- This is because the smoothing operations used in the derivation of the approximate theory imply that the interface variables and have physical meanings only at the midplanes of the a phase even though they are defined for all x. Study of the equations of the approximate theory, equations (1)-(3), shows that their integration with respect to the space variable x involves four integration constants. Accordingly, unique determination of the solution requires four boundary conditions. Two boundary conditions come from equations (8). The general procedure for establishing the remaining boundary conditions is explained first in what follows, and then the general procedure is used to obtain the remaining two boundary conditions for the problem. Let n be the order of the continuity conditions (c.c.) of the approximate theory considered in the analysis. One first writes down c.c, of the orders 0, 1, 2, ... , n -1. The remaining boundary conditions are established by assuming that those c.c, of the lower orders 0, 1,2, ... , n - 1 which are independent of c.c, of order n are to be satisfied at the boundaries. In the problem here the order of the c.c., equations (3), considered in the analysis is 1: i.e., n = 1. When one writes down the zeroth order c.c. and assumes that those of the zeroth order c.c. which are independent of equations (3) are to be satisfied at the boundaries
R:
S:
TRANSIENTS IN A VISCOELASTIC COMPOSITE
145
of the composite slab, the remaining two boundary conditions can be obtained as follows: at the left boundary, at the right boundary,
(Ri - R7)/x=o = 0;
(st - S7)lx=H = O.
(9)
2.2. INITIAL CONDITIONS Since the slab is at rest initially all of the field variables of the approximate theory should be zero at the initial time t = O. The relevant conditions therefore are (10)
u"lt=o= 0,
where (·)=ajat. The formulation of the problem is now completed. Equations (1)-(3) constitute 10 partial differential equations for the 10 unknowns u", S~, R:. The unknowns can be determined uniquely by solving these equations subject to the boundary conditions (8) and (9) and initial conditions (10). With these unknowns determined, the prediction of the approximate theory for the displacement and stress can be found at any point of a layer from the equations
a" = (2,.."" + A,,) * d(axau,,),
aX a
= ajax",
(11)
where
Here the cf>;'s are Legendre polynomials defined by (12)
cf>o= 1,
and X" = x"j h«, x" being the normal local co-ordinate axis of the layer under consideration. To determine the distributions of displacement U" and stress aCt in a certain layer of the a phase of the composite, first one evaluates u", at the midplane of that layer from the solution of the equations of the approximate theory; then, using equations (11), one finds the desired distributions.
S:
3. SOLUTION OF THE PROBLEM
The object in this study is to integrate the equations of the approximate theory for the problem under consideration, to assess the approximate theory by comparing its prediction for elastic case with that of exact theory and to study the influence of viscosity on the wave profiles by using the approximate theory. The exact solution of the elastic problem is obtained by the method of characteristics. Because the equations of the approximate theory are not hyperbolic the method of characteristics is not suitable for solving them. The approximate solution is obtained by employing the FFT procedure. The method of characteristics used in obtaining the exact elastic solution is well known and can be found in standard text books, such as reference [2]; therefore, it will not be discussed here. On the other hand, the FFT procedure used in the integration of approximate equations is relatively new and is based on the use of the discrete Fourier transform (DFT) and the FFT algorithm (for DFT and FFT see, e.g., references [6-8]). The FFT procedure requires first the determination of the solution in Fourier space.
146 3.1.
G. A. BIRLIK AND Y. MENGI APPROXIMATE SOLUTION IN FOURIER SPACE
To find this solution the Fourier transform is applied, with respect to time variable, to equations (1)-(3). When the initial conditions (10) are taken into account, after some manipulations, the following expressions are obtained in the Fourier space: F
P
F
R;:,=daSi"
sf =
Zj(ea j
+
F
Ri,=baS;:"
+ ea2ql )M cos
z2(eaj
F
F
Ua
ZjX - Zj(ea j
+ ea2q2)H cos Z2X -
R;:,
=- , ga
+ ea2ql)N sin ZjX
z2(ea j
+ ea2q2)J sin Z2X,
si = M sin ZjX + N cos ZjX + H sin Z2X + J cos Z2X,
S; = Mq, sin zjx+ Nq, cos Zjx+ Hq2 sin Z2X+Jq2 cos Z2X,
Q'
= 1,2,
(13)
where d.; = ga/(2+ ga/3ba),
ga = -2h erPa W2,
(2j.ta +A a)* =iw(2j.ta + Aa)F = Ea +iw1Ja,
b.; = (2J.1..a + A",)*/ h""
ell = A (d 2+ b j),
e2j=A(b j - d j),
e12= A (b2 - d2 ) ,
e22 = A(b2+ dl ) ,
A = L12(dj + d 2),
zi= (-S+,) S2-4QT)/2Q,
z~= (-s-Jr-S-=-2-- 4- Q- T) / 2 Q ,
S= (2b 2B+2G+2b 1 D - 2 K ) / A.d, T=(4b 2+4b j)/(A.d)2,
D=2b 2-d2 + dj,
B=2b 1 - d1+d2,
K = dibl-d l) - d j(d2+b j),
q",
Q = BG-DK,
= - (Bz~ + 2/ Ad)/ (Dz~
G= d 2(b2 + d j)- d j(b 2 - d 2),
+ 2/ AL1),
a = 1, 2.
(14)
In equations (14), oi is the radian frequency related to the cyclic frequency f by ta = 21Tf, and i is the imaginary unit. The last term after the equality sign in the last equation of the second line in equations (14) is written in view of the assumption that the viscoelastic behavior of the composite slab is represented by the Voigt model. In this, term B", describes the elastic modulus and '17", the viscosity coefficient. The superscript F in equations (13) and (14) designates the Fourier transform of the variable over which it appears, M, N, Hand J in equations (13) are integration constants and are to be determined by satisfying the boundary conditions (8) and (9), which results in -d,y,
I I
b,
I
q, cos
I I
-d,yz
I I
b,
---------T---------T---------T--------o I b1-bzq, I 0 I b,-b 2
q\ sin z\ H +w cos z,H j
:
z.H
-WI sin ZtH
I
1
q2 sin zzH +Wzcos Z2H
I
:
qz cos zzH - w2 sin zzH
(;:=;I):O;z~:=~~--w~)~i;;-~H: (y~=;2)C~S-Z~HT=(~z-=-:J~;Z~H
M
-2pF
N
o
H
o
-J-"
.
(15)
o
In equation (15), the definitions Q'
=
1, 2,
(16)
are used. Equation (15) determines the integration constants as functions of frequency. Substitution of these into equations (13) establishes the variables of the approximate theory in Fourier space. pF in equation (15) designates the Fourier transform of the applied pressure.
TRANSIENTS IN A VISCOELASTIC COMPOSITE
147
4. NUMERICAL PROCEDURE
As indicated in the previous section, for the approximate solution, we have employed a numerical procedure which makes use of FIT. The procedure has four main steps: (i) determination of the solution in Fourier space; (ii) truncation of the Fourier spectrum of the quantity of interest at the cut-off frequency, and evaluation of the spectrum at discrete frequency points by using a frequency increment; (iii) extension of the Fourier spectrum beyond the cut-off frequency in such a way that the Fourier spectra on the left and on the right of the cut-off become complex conjugates of each other; (iv) inversion of the Fourier spectrum established in (iii) by using FFT, which determines the time variation of the quantity. The solution in Fourier space in step (i) can be obtained either analytically or numerically. For this problem it has been obtained analytically in the previous section. However, if the equations are complicated it should be obtained numerically by using methods available in the literature. Deciding about the cut-off frequency and frequency increment in step (ii) is a crucial step of-this procedure. The cut-off frequency can be chosen, after inspection of the Fourier spectrum of the quantity of interest, as the point beyond which the spectrum values are negligibly small. The frequency increment, however, should be selected in such a way that the frequency variation of the Fourier spectrum ofthe quantity is described adequately. The cut-off frequency fc and frequency increment J1f satisfy the relations
I; = 1/2dt,
d/= 1/T,
(17)
where J1t and T designate respectively, associated with the solution in time space, the time increment and the upper bound of the solution interval. The first of equations (17) relates J1t to fc and can be regarded as a formula giving an optimum value for J1t. Now a few remarks about the selection of Ie are in order. Let j denote the frequency beyond which the Fourier spectrum values of a quantity becomes negligibly small. To obtain the solution with sufficient accuracy the chosen value of fc should be at least equal to However, to be flexible, it is preferable to assign a value to fc which is greater than When this is the case, one can shorten the computations by not evaluating the spectrum values in the interval Ll.fc], and instead set them equal to zero there. Rough estimates for fe and !J.I can be found by studying frequency variations of the Fourier spectra of input functions, for this problem by examining the frequency variation of r". It may be noted that the Fourier transforms of the input functions need not be evaluated analytically; instead, they may be found with the aid of FIT. For the present problem and its solution the following numerical algorithm, based on the procedure described above, was used. It is first to be noted that the use of FFT requires the number of subdivisions N to be expressed by the formula
1
J
(18) The steps of the algorithm are as follows. (i) In view of the discussions given above, a proper value for fc and a value for m in equation (18) leading to a suitable value for df are chosen. It is to be noted that the chosen value of m determines the number of subdivisions to be used both in frequency and time spaces, which, in turn, determines the frequency increment df from the formula jjf= 2fc/ N
(19)
The chosen values of fc and !J.I fix the values of the time increment J1t and the upper bound T of the solution interval by the formulas (17).
148
G. A. BIRLIK AND Y. MENG!
(ii) With use of the values of m and T specified in (i), the Fourier spectrum pF of the input function is computed by FFf at the discrete frequency points having the increment tif. (iii) The Fourier spectra u~, ;;.~ of the normal displacement and stress at a certain point of the composite slab are computed at the frequency points of the interval [0,!c1 with the increment tif. In this calculation Fourier transforms of equations (11) are written at frequency points by taking into account equations (13), where the integration constants M, N, Hand J are computed from equation (15). (iv) The Fourier spectra of ;;.~, u~ are extended to the interval [0,2fc] in such a way that their values in the interval [!c,2fc] are complex conjugates of those in [OJc]. (v) With use of the values of m and tV specified in step (i), ;;.~ and u~ are inverted back into time space by FFT, which determines the values of normal stress and displacement at the time points with the increment tit over the time interval [0, T]. 5. NUMERICAL RESULTS
Numerical results have been obtained for a composite slab having the following properties: (20) For this choice of properties the elastic impedances and viscous properties of the constituent layers are very different. Accordingly, it is to be expected that, while attenuating, the waves will be influenced greatly by the geometric dispersion. The computations were carried out in terms of the non-dimensional time I defined by (21) where C 1 =.JE;Tii;.. The time variation of the applied pressure pet) for which the numerical results were obtained is shown in Figure 1. For step loading, it is zero at I = 0 and rises linearly to a constant value Po during a rise time of a, after which it remains constant. In case of trapezoidal loading, at I = f*, it decreases linearly to a zero value during a time interval of a. Figure 2 shows a comparison of the exact and approximate dispersion curves for the p waves propagating perpendicular to the layering in an infinite elastic composite with the properties given in the first three of equations (20) and 7]", = O. Here, cO and k are o- '\ .---,---.,---..,.--,---,---,.....---.----.-----,---.---. 0·3
i Figure 2. Approximate and exact fundamental dispersion curves for P waves propagating perpendicular to the layering for the elastic case. - , Exact theory; - x - , approximate theory.
149
TRANSIENTS IN A VISCOELASTIC COMPOSITE
non-dimensional frequency and wave number, which are related to, the frequency wand wave number k by (22) In view of this comparison one can expect the approximate theory to give good results for waves whose wave number and frequency are contained in the region of the spectral plane 0"", c.i:i "'" O: 30, 0""" k "'" O'70. It may be noted that, if desired, the comparison can be improved and the periodic pattern of the spectra can be obtained by increasing the orders of the theory and the continuity conditions. With the object of assessing the approximate theory, for the elastic case, the wave profiles for the normal stress at the midpoint Q of a four pair slab were found by using exact and approximate theories and are compared in Figures 3(a) and (b). The former figure displays the comparison for step loading, and the latter for trapezoidal loading. The non-dimensional viscous constants ga, stress s, slab thickness H and co-ordinate of the midpoint i appearing in the figures are defined by ga=1}acJEJhlo
u=a-/Po,
H=Hjh J,
i=xjh J.
(23)
-2,2 (0)
-2·0
-1·6
-1·2 Q,
Ib
-0·8
-0'4
0
-0·4 0
100
50
150
200
240
-1·4
-1'0
:6-
-0,6
s
I
-0,2
Ib
1920
r
0 -0,2
-0·6 -1'0 0
50
100
150
180
r Figure 3(a) and (b). Comparison of time variations of normal stress at the midpoint Q (i .. 11) of the four pair slab (H = 21) for the elastic case. (a) Step loading; (b) trapezoidal loading. - - . Exact theory; approximate theory.
150
G. A. BIRLIK AND Y. MENGI
The influence of reflections from the faces of the slab on the wave profiles is evident in these figures. Figures 3(a) and (b) indicate the capability of the approximate theory for duplicating the exact wave profiles closely. The comparison of the wave profiles for the stress a( Q) for different values of viscous constants g"" when a four pair slab is subjected to step loading, is given in Figure 4(a). The responses to trapezoidal input are displayed in Figure 4(b). As expected, as viscosity increases, the irregularities in the wave profiles smooth out and the amplitudes decrease. Figures Sea) and (b) give exactly the same comparison as in Figures 3(b) and 4(b), but this time for a slab which is composed of eight pairs of alternating layers. Figure 6 presents, for the elastic case, a comparison of particle velocities obtained from approximate and exact theories. In Figure 6, V designates the non-dimensional particle velocity defined by
v= BV/ cPo,
E
n",=h",/IJ.,
= (E I / nl)(E2/ n2)/[(Ed nl)+ (E 2/ n2)]' c=./E/p,
P=Plnl+p2n2'
-2'4
(0 ) -2·0
:.~
-1,6
§
I
-1·2
Ib
-0·8
-0,4
0 -0,2 0
50
150
100
240
200
r -1,4
(b) -\·0
:·t
-0,6
3 Ib
~ ,.
1920
r
-0'2
a 0·2
0·6 ',0
a
150
50
180
r Figure 4(a) and (b). Influence of the viscosity on the time variations of normal stress at the midpoint Q (x = 11) of the four pair slab (if =21). (a) Step loading; (b) trapezoidal loading. - , Approximate theory
(elastic); - - -, tl = 1, t z = 40; .. " (, = 5,
q2 = 200.
151
TRANSIENTS IN A VISCOELASTIC COMPOSITE -Hr------...,r------...,r---------r-------, -1'0
;.~
-0'6
2
'G Ib
r
-0·2
3640
0 0·2
0·6
1·0
H
100
0
200
300
360
300
360
j' -1'4
-1'0
:.~
-0'6
2.
s Ib
3640
1
-0'2
a 0'2
0'6
1·2
100
0
200
j'
Figure 5(a) and (b). (a) Comparison, for the elastic case of, and (b) influence of the viscosity on time variations of normal stress at the midpoint Q (i = 23) of the eight pair slab (if =45) (trapezoidal loading). (a); - - , exact theory (elastic); .. " approximate theory; (b); - - , approximate theory (elastic); - - -, ~ = 2, ~2 = 80; ... , ~l = 5, ~2 = 200. -1-2
-0,8
:.~
-0-4
1920
I
1 ~ I::..
0
0-4
0'6
-1-4
0
50
100
150
180
1 Figure 6. Comparison of time variations of particle velocity at the midpoint Q (.i = 11) of the four pair slab (H' = 21) for the elastic case (trapezoidal loading). - - , Exact theory (elastic); ... , approximate theory.
152
G. A. BIRLIK AND Y. MENGI
6. DISCUSSION AND CONCLUSIONS
The results presented in the previous section show that the approximate theory, even at its lowest order, predicts well the essential characteristics of laminated composites when they are subjected to dynamic loading. The following observations can be made, after study of the figures. When a laminated composite slab is subjected to compressive loading, first a compressive stress wave is induced in the body, which, as it propagates, undergoes dispersion due to reflections and refractions at interfaces. If the boundary at the opposite face of the slab is of clamped type (which is the case in this problem), the reflected pulse at this face will again be compressive. After a certain interval of time, the multiple reflections and transmissions at interfaces and at slab faces may give rise to tensile stresses, in particular, of considerable magnitude in case of trapezoidal loading. As is well known, in practice, these stresses, reducing the load bearing capacity of the composite, finally cause delamination of the composite. As seen in Figures 3-5 this important phenomenon is well predicted by the approximate theory. As the number of layers is increased, damping becomes less effective. This fact can be observed from the comparison of the results for gl = 5 and g2 = 200 in Figures 4(b) and 5(b). As seen from the study of these figures, for the aforementioned viscous constants, the amplitudes for the four pair slab are considerably smaller than those for the eight pair slab. Due to the viscoelastic character of the composite, attenuation is expected. With the increase of the viscous constants of the constituent layers attenuation in the amplitudes become more pronounced (see Figures 4(a) and (b)). Comparison of these two figures, which give the response to step and trapezoidal loading respectively, displays the fact that attenuation is comparatively higher for trapezoidal loading. This has to be expected since damping cannot be observed effectively in step loading due to continuing action of the applied pressure. In the static case the approximate theory gives the stress and displacement distributions exactly. This can be verified as follows for an elastic slab whose left face is subjected to uniform static pressure Po while its right face is kept fixed, as shown in Figure 1. For the static case U'" = 0 in equation (1), which leads to R-;' = O. When this is used in equations (2) and (3), boundary conditions (8) and (9) are taken into account and equations (11) are used, one finds, after some manipulations, *
[( 1
u",=Po
b
1) H - x
1+iJ2
1
-L.\-+iJ
X'" ]
2-iJ",
'
(24)
where b", = Bet / h.: Equation (24) represents the static distributions of stress and displacement predicted by the approximate theory. In equation (24), x is the global co-ordinate axis shown in Figure 1 and x'" = x"'/h"" x, being the local co-ordinate axis of a layer with the origin located at its midplane. To compute the displacement distribution in a certain layer of the slab one substitutes the x value of the midplane of that layer into equation (24)\. Then, for that fixed value of x one varies x"" which determines the desired distribution. It is clear that the stress distribution given in equation (24h is exact. On the other hand, the study of equation (24)\ shows that the displacement varies linearly over each layer; its close examination indicates further that it describes simply the exact displacement distribution. A final remark is now in order. The approximate theory in reference [1] was developed for three dimensional dynamic behavior of layered composites. In this study, the theory has been assessed for a one dimensional problem. This assessment gives an indication about the validity of the theory for the three dimensional case.
TRANSIENTS IN A VISCOELASTIC COMPOSITE
153
REFERENCES 1. Y. MENGI and D. TURHAN 1984 Journal of Sound and Vibration 92.311-320. A higher order
dynamic theory for viscoelastic plates and layered composites. 2. L. M. BARKER 1971 Journal of Composite Materials 5, 140-162. A model for stress wave propagation in composite materials . 3. R. M. CHRISTENSEN 1975 Journal of Applied Mechanics 42, 153-158. Wave propagation in layered elastic media. 4. T. C. T. TING and 1. MUKUNOKI 1979 Journal of Applied Mechanics 46. 329-336. A theory of viscoelastic analogy for wave propagation normal to the layering of a layered medium. 5. R. COURANT and D. HILBERT 1966 Methods of Mathematical Physics. New York : Interscience. 6. E. O. BRIGHAM 1974 The Fast Fourier Transform. Englewood Cliffs, NJ: Prentice-Hall. 7. J. W. COOLEY. P. A. W. LEWIS and P. D. WELCH 1969 Institute of Electrical and Electronic Engineers Transact ions on Education 12,27-34. The fast Fourier transform and its applications. 8. G. D. BERGLAND 1969 Institute of Electrical an Electronic Engineers Spectrum 6,41-52. A guided tour of the fast Fourier transform.