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A higher order dynamic theory for viscoelastic plates and layered composites
Journal of Sound and Vibration (1984) 92(3), 31 l-320
A HIGHER ORDER DYNAMIC THEORY FOR VISCOELASTIC PLATES AND LAYERED COMPOSITES Y. MENGI Department of Civil Engineering,
Cukurova University, Adana,
Turkey
AND
D. TURHAN Department of Civil Engineering, King Saud University, Riyadh, Saudi Arabia (Received 27 October 1982)
By using a new technique proposed by the first author [l] approximate theories are developed for the dynamic response of viscoelastic plates and layered composites. The originality of the new technique lies in the fact that it permits the approximate theory to satisfy correctly the lateral boundary conditions of a plate, or the interface (continuity) conditions of a layered composite. This, in turn, enables the approximate theory to describe accurately the geometric dispersion of waves propagating in a plate or layered composite. The approximate equations of a single viscoelastic plate are first derived by making use of the new technique. To develop the approximate theory for viscoelastic layered composites made of two alternating layers it is noted that the approximate equations of a single plate already established also hold in each layer of the composite. The theory is completed by adding the continuity conditions to these equations and using a smoothing operation. The equations thus obtained constitute a continuum (homogeneous) model (CM) which simplifies the determination of the dynamic response of viscoelastic layered composites when the number of layers in the composite is large. The proposed approximate theories are open to improvement in the sense that their region of validity in the wave number-frequency plane can be enlarged as much as one wishes by increasing the orders of the theories and continuity conditions.
1. INTRODUCTION In recent publications [l, 21 a new technique has been proposed for the development of approximate higher order dynamic theories for thermoelastic plates and composites. In the present work, this new technique is used to establish dynamic theories for viscoelastic plates and layered composites. In the analysis the material is assumed to be linear. The layered composite is made of two alternating layers having different thicknesses and physical properties. The layers are assumed to be perfectly bonded along their interfaces. The new technique, which is based on a modified version of the Galerkin method, permits two types of variables to appear in the approximate theory. The first type represents the weighted averages of stresses and displacements over the thickness of the layer. They are called the generalized variables (GV). The second type represents the displacements and stress vector components at the lateral faces of the layer. They are called the face variables (FV). The novelty of the new technique lies in the fact that it enables the approximate theory to contain not only GV’s but also FV’s as unknown variables, which is in contrast with other theories [3-91 in which only GV’s appear as unknown variables. The inclusion of FV’s as field variables in an approximate theory is very important because the appearance of FV’s makes it possible to satisfy correctly the lateral boundary conditions 311 0022-460X/84/030311
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of a plate, or the interface conditions of a layered composite. This, in turn, enables the approximate theory to describe accurately the geometric dispersion of waves propagating in a plate or layered composite. It is to be noted that in the other theories cited [3-91 there are some discrepancies between the displacement distribution assumed over the thickness of the layer and lateral boundary conditions of the plate or interface conditions of the layered composite. To compensate for the error caused by these discrepancies some of the researchers have introduced so-called correction factors into the approximate theory. They have determined the values of the correction factors by adjusting certain properties and exact dispersion curves. Determination of the values of these correction factors depends on the availability of exact or experimental data and involves lengthy computations, so it would be preferable if their use could be avoided. In the first part of the present study, the approximate theory is established for a single viscoelastic plate. The order of the theory is kept arbitrary. Thus, the frequency range over which the theory is valid can be enlarged as much as one wishes by increasing the order of the theory. In the development of the approximate theory, a set of distribution functions {4i(Xz)} is chosen, where x2 is the distance measured perpendicular to the midplane of the plate. The analysis is carried out for general 4,‘s. The equations of the theory are composed of two types. The first type is obtained by taking the weighted averages of the exact equations of viscoelasticity over the thickness of the plate, the 4i’s being used as weighting functions. The second type, which can be interpreted as the constitutive equations for the FV’s, is derived by expanding the displacements in terms of the 4i’s and taking into account the exact constitutive equations of viscoelasticity. The equations of the approximate theory thus derived contain some constants whose values can be computed if the 4i’s are specified. Here they are taken to be Legendre polynomials and the values of these constants are calculated and presented for this case. In the second part of the study an approximate theory for viscoelastic layered composites consisting of two alternating layers is proposed. In the development of the theory it is first noted that the governing equations of a single plate, as established in the first part, also hold in each phase of the composite. The theory is completed by adding the continuity conditions to these equations. The continuity conditions are derived in a unified and systematic manner by taking advantage of the fact that the FV’s appear as field variables in the equations of a single plate, and by using the assumption that the layers are perfectly bonded. The equations thus obtained constitute a discrete (heterogeneous) model (DM) for viscoelastic layered composites. By using a smoothing operation the DM model equations are then homogenized and put into a continuum form. The resulting model, called the continuum (homogeneous) model (CM) simplifies the determination of the dynamic response of viscoelastic layered composite when the number of layers in the composite is large. The CM model developed for the composite possesses a generality that it is open to improvement by increasing the orders of the theory and continuity conditions (CC) (in the analysis the orders of the theory and CC are kept arbitrary). An increase in the orders of the theory and CC enlarges respectively the frequency and wave number intervals over which CM is valid.
2. APPROXIMATE
2.1.
INTEGRATION
OF
THE
THEORY
FIELD
FOR A VISCOELASTIC
PLATE
EQUATIONS
The plate is assumed to be made of an isotropic, linear viscoelastic material and to have a uniform thickness 2h. In a Cartesian co-ordinate system (x,, xZ, xJ the x,.x3 plane
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coincides with the midplane of the plate. The fundamental equations of linear viscoelasticity are the equations of motion and the constitutive equations, respectively: ajTji
~ij=
/.L
+fi
=
(1)
Piii,
d(aiuj + ajUi)+ 6ijA* dakUk.
*
(2)
Here p is the mass density, A and p are Lame’s relaxation functions, Uiare the displacement components, 7ij are the stress components, and fi are the components of the body force. Equation (2) is an integral form of the constitutive equations of linear isotropic viscoelastic materials, in which * indicates the convolution integral defined by G * dh = G(t)h(O)+
G’(t-z)h(z)
G(t-z)$)dr=G(O)h(t)+
dz,
(3)
where G and h are some functions of the time t and G’(z) =dG/dz. To develop the approximate dynamic theory for plates, one first chooses a set of distribution functions {+,(Z,), n = 0, 1,2, . . . }, where Z2= x2/h. It is assumed that the 4” form a complete set. For developing an mth order theory one retains the elements (40, #Ji,. * * 7An, 4%+1, &+*} of the set. As will be seen later, keeping the last two elements $~,,,+iand c#+,,+~ in the set allows one to satisfy the lateral boundary conditions correctly. Without loss of generality one can also assume that 4” (n = 0, . . . , (m + 2)) is an even function of L, for n even and odd function of Z2 for n odd. At this stage of the analysis no assumption is made about the orthogonality of the functions &,. Next one takes the weighted average of the equations of motion over the thickness by m) as weighting functions (i.e., one applies the operator L” = using & (n=O,..., (1/2h) I?,( )&dx* to equation (1)). This gives
72ni=
LnT*i,
R: =el?,,
)%dx,, 2
;, ” = RI=Tli-TTi I
(5)
I Rt=Tzi+T,
TO establish the constitutive relations for T;i, T$i and 7zi, one uses equation (2). The operator L” is applied to the constitutive equations for 71i and T3i, and the operator L” to the constitutive equations for T2i, to obtain Tyl =(2~++A)*da,u;+h*d&u;+A T&=h
*
da,u;+(2p++)*
d(a,u;+S;-C;),
772=/L*
* d(S;-ii;),
d&u;+h T;2
=
7;~ = T;l = /_L* d(&u; +a,u,“) ?& =A * da,C;+A ?;I =/.L * d(a,ti; +s;
-a;),
* d(S;.-a;),
/_L *
d(&u; + S; - ii;),
(6)
(n = 0,. . . ) m),
* da&‘+(2~+A) +;,=~*d(ⅈ+S,“-d;)
* d(% -a;), (n =o,. . . ) m),
(7)
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In the fourth of equations (8) the prime denotes the derivative with respect to the non-dimensional distance 2,: i.e., 4; = d&/d&. The integration of the field equations is now complete. Equations (4), (6) and (7) constitute 12( m + 1) equations. Stress or displacement boundary conditions on the lateral surfaces of the plate give another six equations which can be expressed in terms of the face variables SF, Rf. In fact, on the faces of the plate, as exact boundary conditions, one specifies quantities composed of one member of each of the pairs (r&, UT), (T,, u;) (i=l,... ,3). In the approximate theory these six boundary conditions can be expressed in terms of face variables by using equations (55) and (8,). Thus the number of available equations is [12(m + 1) +6]. On the other hand, the number of unknowns 7Ti, ul, it;, Sy, Rf, Sf ) is [18(m + l)+ 121. Therefore, to complete the mathetTri, 7Y1~ matical model [6( m + 1) + 61 more equations are needed. 2.2. ADDITIONAL EQUATIONS The additional equations can be obtained by establishing the constitutive relations for the face variables Rf and for the generalized variables tir and a;. To this end one can expand the displacements Ui in terms of 4, (n = 0,. . . , (m +2)), m+2 &=
C k=O
a:+k(X2),
where the coefficients a; are functions of x1, x3 and t. When the operator L”( n = 0, . . . , m) is applied to equation (9) and equation (9) is substituted into the expressions defining ST, equation (8,), two sets of algebraic equations are obtained, one governing ai with even k and the other governing a; with odd k. They are, respectively, P+2
P+2
C”kUL c k=O. Z,...
=
uy
(n=0,2,...,p),
k=;2
_,, 4k(lbik
=s:/2>
(10)
. .
p’+2 c k= 1 ,3,...
cnka
i =uy k
(n=1,3
,...,
p’),
“i’ k=l,
p=m-1
7
p’=
(11)
for even m,
p’=m-1
p=m,
C$k(l)ul,=S;2,
3,...
m
for odd m,
(12)
fl c,k
=
Ln4k
=
;
dd’k
(13)
dx2.
I -1
Through the solution of equations (10) and (11) one can determine the u;‘s in terms of the variables ul and SF (which appear as field variables in the approximate theory), or, more explicitly, a; (with even k) in terms of SC and ur
(n = 032,. . ., PL
ui (withoddk)intermsofS;andun
(n=1,3
)...)
p’).
(14)
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Substitution of equation (9) into the constitutive relations of rzi and into the expressions of al and d 1, equations (8,) and ( 82), and the use of the expression definining R f , equation (5,)) give Rt=?
“i’
h
R; =;
Ii2 k
R--l
l
4;(1)(p*daL)+p
(i = 1,3),
*dais:
k=1,3....
#;(1)[(2p+h)*da:]+h
*d(a,S:+a,S:),
1,3,...
‘+* -hk=~2,,,~;(l)(r*da’,)+p*da,Sr
(i=l,3),
..
‘;’ +;(1)[(2p+A)
R, =?
h
* da;]+h
* d(a,S; +a3S;),
k=0,2,...
’ &k = Lk(d&/dx2),
&k = Lk(d2+,/dx;).
(16)
When the expressions for a; in terms of ul and SF are substituted into equations (15) and the property stated in equations (14) is used the additional equations are finally obtained in the forms R;
~2 h 1
k=~3,,,~k(~*du:)+y~(~*dSi)
(i=l,3),
+P*d&S:
’. 1
+A*d(hS:+a3S:),
(i=1,3),
gy
=
!
&u; + c;+$
for even n
C;kUf
for odd n
k=0,2,...
5
k=1,3,...
(17) +
c;-s,
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In eqUationS (17) yk, &, cik, y*, CL* and ci* are some constants whose explicit values can be determined through the computations described above whenever the functions 4, are specified. The [6( m + 1) + 61 equations in equations (17), and the equations derived in the previous section, equations (4)-(7), constitute the governing equations of the approximate theory. The first four of equations (17) represent the constitutive relations for the face variables. The derivation of these equations is based on the expansion, equation (9), and the field equations of viscoelasticity. The use of these equations together with the lateral boundary conditions permits one to satisfy the lateral boundary conditions correctly and eliminates any inconsistency between the assumed displacement shape and the lateral boundary conditions.
2.3. THE VALUES OF THE CONSTANTS WHEN &‘s ARE LEGENDRE POLYNOMIALS Tables for the values of the constants appearing in the approximate theory can be prepared for various selected &‘s and for various orders of the approximate theory. As an illustration the values of the constants when +,, ( X2) = P, (x2), where P, is the nth order Legendre polynomial, and for orders of the approximate theory m = 0, 1 and 2 are presented here. The orthogonality of the Legendre polynomials simplifies the solution of equations (10) and (11) and, consequently, the computation of the constants. The values of the constants are presented in Table 1. The dash in the table indicates that the corresponding constant does not exist in the approximate theory.
TABLE
1
Values of constants for zeroth, first and second order theories
Constant Order
‘y,,
0
-3 -3
1
2
-10
Yl
Y2
Y+
CL
Y-
C&-
Chl
c;-
CL
_
-
-15
-
312 312
I/2 3
0
() 0
-
-
I/h
-15
-35
5
3
0
0
3/h
0
l/h
Constant ci2
c1,+
0
-
-
1
-
Order
2
0
GO 0
0
0
0
0
C& -
0
I,+ co
Go
0 0
-
0
3/h2
I,+
n-
C22
C2
4,
Cl
-
-
-
-
-
-
0
0
0
0
0
0
3. APPROXIMATE EQUATIONS OF THE LAYERED COMPOSITE The layered composite under study is composed of two alternating layers perfectly bonded at their interfaces. The two different layers are indicated by the circled numbers 1 and 2 in Figure 1. The layers 1 and 2 are assumed to be made of a linear, isotropic, viscoelastic material and have the material constants or functions (p,, pI, A 1, etc.) and (p2, pz, AZ, etc.), and have the thicknesses 2h, and 2h2, respectively. In the figure the pairs of layers, each of which consists of two different phases, are numbered in increasing order k = 0, 1,2, etc. In Figure 1 two kinds of co-ordinate systems are shown. The first
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(k) (k)
Figure
1. Geometric
description
of the layered
composite.
is the (x1, x2, x3) global co-ordinate system whose (x,, x3) plane is parallel to the midplane of the layers. This co-ordinate system is employed to describe the location of a layer by DI(k) specifying the vertical distance of its midplane from the (x1, xj) plane. For example, x2 (cz = 1,2) describes the position of the czth constitutent of the kth pair (see Figure 1). A convention adopted throughout the study is that the Greek letters (Y,/?, etc., are used only to distinguish the two different phases of the composite and take the values 1 and 2. The second co-ordinate system is the local co-ordinate system (x,, i2, xg) whose (x,, xj) plane is chosen to coincide with the midplane of a particular layer. Two mathematical models for the layered composite, described above, can now be proposed: namely, the discrete model (DM) and the continuum model (CM). The DM equations can be obtained by using the equations of the single plate determined in the previous sections and the continuity conditions at the interfaces. The CM equations then can be established through the application of a smoothing operation to the DM equations. 3.1.
DISCRETE
MODEL
The governing equations of the layered composite consist of two types of equations: namely, the field equations, valid in each layer of the composite, and the continuity conditions at the interfaces. The equations obtained in the previous sections for a single plate hold also in each layer of the composite. Accordingly one can obtain the field equations by putting the index (Yin each variable appearing in the equations of the single plate. To keep the length of the paper shorr these equations are not rewritten here. Before writing the continuity conditions, it needs to be noted that there are two kinds of interfaces: one follows layer 1 and the other layer 2. As it is assumed that there is perfect bonding between the layers, the displacements Ui and the stress components 72i should be continuous across these interfaces: i.e., for the interfaces following layer 1 and layer 2, respectively: 1
2
u;=u;,
2 u;=u;,
1 T2i =
1
2 TZi =
2 _ T2i,
1 _ 72i.
(18) (19)
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The continuity conditions, equations (18) and (19), can be expressed in terms of face variables fii and 5: as, respectively,
The dependent variables &= (‘;r, $, Tyi, t:,, etc.) appearing in the field equations of the layered composite and equations (20) and (21) are functions of iik’: i.e., their values depend on the positions of the layers. Accordingly, the aforementioned equations form a discrete system of equations. In order to obtain the solutions by using this discrete model one should write the field equations in all layers and take into account the continuity conditions, equations (20) and (21) , at all interfaces. This kind of procedure involves lengthy computations and appears to be of no practical use. To simplify the analysis, in what follows the DM model is replaced by a continuum model by using a smoothing operation. 3.2. CONTINUUM MODEL To obtain the smoothed form of the field equations (valid in each layer of the composite) . the arguments of the variables 4 appearing in these equations u(k) m one first replaces x2
by x2. After this smoothing G is now defined for all x2 but has physical meaning only at the midplanes x2 = i ik). The smoothing operation leaves the field equations unchanged because all the field variables in these equations are defined at the midplane of the same layer: i.e., at x2 = gkk’. In accordant e with the idealization implied by the smoothing operation it is assumed that both types of layers exist simultaneously at every point of the continuum and accordingly the field equations with (r = 1 and (Y= 2 hold at the same point x2. The variables appearing in the continuity conditions, equations (20) and (21), do not belong to the same layer. Therefore the smoothed form of the continuity conditions will change and will be found through analysis. The analysis starts by referring to Figure 1 and writing the continuity equations for the interface which follows the layer 1, equations (20)) explicitly as $+(x1,
*(k) x2
, x3,
t) -
f-+(x,,
‘,k) x2
, $7
t)
=
h,
’ (k) 7 7-2 9 x3,
where it-* stands for either of the face variables (sit, ii:),
f) +
;‘-ix,,
‘(k) x2
> X3?
t),
(22)
i = 1,. . . ,3. It must be noted
that b f is defined at the midplane of layer 1 (i.e., at x2 = z&~‘)while + is defined at that ‘(k) ). To apply the smoothing operations to the continuity conditions, of layer 2 (i.e., at x2 = x2 equations (22) can be written first in a form in which all of the variables in it are defined at a single point. To this end a point M, in the interval (:ikJ, :\“) with distances p, A and p,A from the midplanes of layers 1 and 2, respectively, is chosen, where A = hl + h, and pm has the property p1 + p2 = 1 (see Figure 1). The x2 co-ordinate of this point is designated
319 l(k)_- x2(k)_ plA andx,‘(k).--x2 (kl + by xik) in the figure. By taking into account the relations x2 p,A the continuity condition, equations (22), now becomes A
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~+(~~~)+p~A)-~+(x:ll’-p,A)=~-(x:~’-p~A)+~~(x~~’+p~A).
(23)
For simplicity the arguments x ,, x3 and t of > are omitted in equation (23). The reduced form of the continuity condition for the interface which follows layer 2 can be obtained from equation (23) by replacing the layer index 1 by 2 and 2 by 1. It is ~+(~~~‘.+p~A)-f+(x~~‘-p~A)=~~(x:~’-p,A)+;.-(x~~‘+p,A).
(24)
To obtain the smoothed forms of the continuity equations, it is assumed that the two types of interfaces exist simultaneously at the same point of the continuum and xik) in equations (23) and (24) is replaced by x2. Expanding the terms of equations (23) and (24) in Taylor’s series about the point x2, and adding and subtracting the resulting equations, one finally obtains
s,k+ +s,k+=c,i+c,;:-,c2~+-c,i-+=s2~--s,h-, where s, and c, are the operators defined by
Equations (25) with ? = (us:, ii?), respectively, represent the smoothed form of the continuity conditions for displacement and stresses. The derivation of the equations of an mth order continuum model for a layered composite is now completed. The governing equations are composed of field equations and the continuity conditions, equations (25). It may be noted that all of the constants except p1 and p2 appearing in the CM equations can be computed in terms of the geometric and physical properties of the constituent layers. In view of findings established in reference [l] for elastic layered composites (CM proposed here for viscoelastic layered composites contains CM established in reference [l] for elastic layered composites as a special case) the use of p1 =p2 = O-5, which corresponds to taking M in Figure 1 at the midpoint of [&kk),;$“‘I, may be suggested. In fact, the numerical results presented in reference [l] indicated that the best match between exact and approximate dispersion curves for waves propagating in elastic layered composites is obtained when p1 = p2 = 0.5. 4. CONCLUSIONS AND DISCUSSIONS Higher order dynamic theories proposed here for viscoelastic plates and layered composites have the capability of satisfying correctly the lateral boundary conditions of the plate and the interface conditions of the layered composite. Accordingly, it may be anticipated that the proposed approximate theories should be able to predict correctly the geometric dispersive characteristics of viscoelastic plates and layered composites. This anticipation has been verified for elastic plates and layered composites in references [l] and [2] by comparing exact and approximate dispersion curves. This verification is
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expected to hold also for the present approximate theories established for viscoelastic plates and layered composites because the latter theories contain the theories proposed in references [l] and [2] as special cases (the theories for elastic case can be obtained from the present approximate theories by treating the relaxation functions p and A as constants). The order of the theory m for viscoelastic plates is kept arbitrary. The completeness of the distribution functions +,, assures that as m increases the frequency range over which the approximate theory is valid enlarges and as M approaches infinity the dynamic response of the plate predicted by the approximate theory approaches that determined from the exact theory. CM established for viscoelastic layered composites has two types of orders. One is the order of the theory, m. The other is the order of the continuity conditions, equations (25) (the order of the continuity conditions corresponds to the number of terms retained in the expansions of c, and s,, equations (26); for example, if one keeps the terms up to the pth power of K in equations (26) one says that the order of the continuity conditions is p). In the analysis the orders of both the theory and the continuity conditions are kept arbitrary. An increase in the orders of the theory and the continuity conditions enlarges the frequency and wave number ranges over which CM is valid, respectively. Two different approximate models, namely DM and CM, are proposed for viscoelastic layered composites. The use of CM should be preferred over that of DM when the number of layers making up the composite is large because the use of CM in this case simplifies and shortens the computations in the determination of the dynamic response of the composite. The proposed approximate theories should be assessed further by comparing their predictions with the exact or experimental data available in the literature for viscoelastic plates and layered composites. In order not to lengthen the paper this will be done in a separate future paper. The proposed approximate theories are general in the sense that they govern all of the deformation modes of plates and layered composites, and they are valid for all kinds of waves propagating in an arbitrary direction of the layered composite. REFERENCES 1. Y. MENGI 1980 International Journal of Solids and Structures 16, 1155-l 168. A new approach
2.
3. 4. 5. 6. 7. 8. 9.
for developing dynamic theories for structural elements, Part 1: Application to thermoelastic plates. Y. MENGI, G. BIRLIK and H. D. MCNIVEN 1980 International Journal of Solids and Structures 16, 1169-l 186. A new approach for developing dynamic theories for structural elements, Part 2: Application to thermoelastic layered composites. R. D. MINDLIN 1951 Journal of Applied Mechanics 18, 31-38. Influence of rotary inertia and shear on flexural vibrations of isotropic, elastic plates. E. REISSNER 1945 Journal of Applied Mechanics 12, 69-77. The effectof transverse shear deformation on the bending of elastic plates. R. D. MINDLIN and M. A. MEDICK 1959 Journal of Applied Mechanics 26, 561-569. Extensional vibrations of elastic plates. P. C. Y. LEE and Z. NIKODEN 1972 International Journal of Solids and Structures 8, 581-612. An approximate theory for high-frequency vibrations of elastic plates. R. D. MINDLIN and H. D. MCNIVEN 1960 Journal of Applied Mechanics 27, 145-151. Axially symmetric waves in elastic rods. C. T. SUN, J. D. ACHENBACH and G. HERRMANN 1968 Journal of Applied Mechanics 35, 467-475. Continuum theory for a laminated medium. A. BEDFORD, D. S. DRUMHELLER and H. J. SUTHERLAND 1976 Mechanics Today (Editor S. Nemat-Nasser) 3,1-54. On modelling the dynamics of composite materials. London: Pergamon Press.
A HIGHER ORDER DYNAMIC THEORY FOR VISCOELASTIC PLATES AND LAYERED COMPOSITES Y. MENGI Department of Civil Engineering,
Cukurova University, Adana,
Turkey
AND
D. TURHAN Department of Civil Engineering, King Saud University, Riyadh, Saudi Arabia (Received 27 October 1982)
By using a new technique proposed by the first author [l] approximate theories are developed for the dynamic response of viscoelastic plates and layered composites. The originality of the new technique lies in the fact that it permits the approximate theory to satisfy correctly the lateral boundary conditions of a plate, or the interface (continuity) conditions of a layered composite. This, in turn, enables the approximate theory to describe accurately the geometric dispersion of waves propagating in a plate or layered composite. The approximate equations of a single viscoelastic plate are first derived by making use of the new technique. To develop the approximate theory for viscoelastic layered composites made of two alternating layers it is noted that the approximate equations of a single plate already established also hold in each layer of the composite. The theory is completed by adding the continuity conditions to these equations and using a smoothing operation. The equations thus obtained constitute a continuum (homogeneous) model (CM) which simplifies the determination of the dynamic response of viscoelastic layered composites when the number of layers in the composite is large. The proposed approximate theories are open to improvement in the sense that their region of validity in the wave number-frequency plane can be enlarged as much as one wishes by increasing the orders of the theories and continuity conditions.
1. INTRODUCTION In recent publications [l, 21 a new technique has been proposed for the development of approximate higher order dynamic theories for thermoelastic plates and composites. In the present work, this new technique is used to establish dynamic theories for viscoelastic plates and layered composites. In the analysis the material is assumed to be linear. The layered composite is made of two alternating layers having different thicknesses and physical properties. The layers are assumed to be perfectly bonded along their interfaces. The new technique, which is based on a modified version of the Galerkin method, permits two types of variables to appear in the approximate theory. The first type represents the weighted averages of stresses and displacements over the thickness of the layer. They are called the generalized variables (GV). The second type represents the displacements and stress vector components at the lateral faces of the layer. They are called the face variables (FV). The novelty of the new technique lies in the fact that it enables the approximate theory to contain not only GV’s but also FV’s as unknown variables, which is in contrast with other theories [3-91 in which only GV’s appear as unknown variables. The inclusion of FV’s as field variables in an approximate theory is very important because the appearance of FV’s makes it possible to satisfy correctly the lateral boundary conditions 311 0022-460X/84/030311
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of a plate, or the interface conditions of a layered composite. This, in turn, enables the approximate theory to describe accurately the geometric dispersion of waves propagating in a plate or layered composite. It is to be noted that in the other theories cited [3-91 there are some discrepancies between the displacement distribution assumed over the thickness of the layer and lateral boundary conditions of the plate or interface conditions of the layered composite. To compensate for the error caused by these discrepancies some of the researchers have introduced so-called correction factors into the approximate theory. They have determined the values of the correction factors by adjusting certain properties and exact dispersion curves. Determination of the values of these correction factors depends on the availability of exact or experimental data and involves lengthy computations, so it would be preferable if their use could be avoided. In the first part of the present study, the approximate theory is established for a single viscoelastic plate. The order of the theory is kept arbitrary. Thus, the frequency range over which the theory is valid can be enlarged as much as one wishes by increasing the order of the theory. In the development of the approximate theory, a set of distribution functions {4i(Xz)} is chosen, where x2 is the distance measured perpendicular to the midplane of the plate. The analysis is carried out for general 4,‘s. The equations of the theory are composed of two types. The first type is obtained by taking the weighted averages of the exact equations of viscoelasticity over the thickness of the plate, the 4i’s being used as weighting functions. The second type, which can be interpreted as the constitutive equations for the FV’s, is derived by expanding the displacements in terms of the 4i’s and taking into account the exact constitutive equations of viscoelasticity. The equations of the approximate theory thus derived contain some constants whose values can be computed if the 4i’s are specified. Here they are taken to be Legendre polynomials and the values of these constants are calculated and presented for this case. In the second part of the study an approximate theory for viscoelastic layered composites consisting of two alternating layers is proposed. In the development of the theory it is first noted that the governing equations of a single plate, as established in the first part, also hold in each phase of the composite. The theory is completed by adding the continuity conditions to these equations. The continuity conditions are derived in a unified and systematic manner by taking advantage of the fact that the FV’s appear as field variables in the equations of a single plate, and by using the assumption that the layers are perfectly bonded. The equations thus obtained constitute a discrete (heterogeneous) model (DM) for viscoelastic layered composites. By using a smoothing operation the DM model equations are then homogenized and put into a continuum form. The resulting model, called the continuum (homogeneous) model (CM) simplifies the determination of the dynamic response of viscoelastic layered composite when the number of layers in the composite is large. The CM model developed for the composite possesses a generality that it is open to improvement by increasing the orders of the theory and continuity conditions (CC) (in the analysis the orders of the theory and CC are kept arbitrary). An increase in the orders of the theory and CC enlarges respectively the frequency and wave number intervals over which CM is valid.
2. APPROXIMATE
2.1.
INTEGRATION
OF
THE
THEORY
FIELD
FOR A VISCOELASTIC
PLATE
EQUATIONS
The plate is assumed to be made of an isotropic, linear viscoelastic material and to have a uniform thickness 2h. In a Cartesian co-ordinate system (x,, xZ, xJ the x,.x3 plane
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coincides with the midplane of the plate. The fundamental equations of linear viscoelasticity are the equations of motion and the constitutive equations, respectively: ajTji
~ij=
/.L
+fi
=
(1)
Piii,
d(aiuj + ajUi)+ 6ijA* dakUk.
*
(2)
Here p is the mass density, A and p are Lame’s relaxation functions, Uiare the displacement components, 7ij are the stress components, and fi are the components of the body force. Equation (2) is an integral form of the constitutive equations of linear isotropic viscoelastic materials, in which * indicates the convolution integral defined by G * dh = G(t)h(O)+
G’(t-z)h(z)
G(t-z)$)dr=G(O)h(t)+
dz,
(3)
where G and h are some functions of the time t and G’(z) =dG/dz. To develop the approximate dynamic theory for plates, one first chooses a set of distribution functions {+,(Z,), n = 0, 1,2, . . . }, where Z2= x2/h. It is assumed that the 4” form a complete set. For developing an mth order theory one retains the elements (40, #Ji,. * * 7An, 4%+1, &+*} of the set. As will be seen later, keeping the last two elements $~,,,+iand c#+,,+~ in the set allows one to satisfy the lateral boundary conditions correctly. Without loss of generality one can also assume that 4” (n = 0, . . . , (m + 2)) is an even function of L, for n even and odd function of Z2 for n odd. At this stage of the analysis no assumption is made about the orthogonality of the functions &,. Next one takes the weighted average of the equations of motion over the thickness by m) as weighting functions (i.e., one applies the operator L” = using & (n=O,..., (1/2h) I?,( )&dx* to equation (1)). This gives
72ni=
LnT*i,
R: =el?,,
)%dx,, 2
;, ” = RI=Tli-TTi I
(5)
I Rt=Tzi+T,
TO establish the constitutive relations for T;i, T$i and 7zi, one uses equation (2). The operator L” is applied to the constitutive equations for 71i and T3i, and the operator L” to the constitutive equations for T2i, to obtain Tyl =(2~++A)*da,u;+h*d&u;+A T&=h
*
da,u;+(2p++)*
d(a,u;+S;-C;),
772=/L*
* d(S;-ii;),
d&u;+h T;2
=
7;~ = T;l = /_L* d(&u; +a,u,“) ?& =A * da,C;+A ?;I =/.L * d(a,ti; +s;
-a;),
* d(S;.-a;),
/_L *
d(&u; + S; - ii;),
(6)
(n = 0,. . . ) m),
* da&‘+(2~+A) +;,=~*d(ⅈ+S,“-d;)
* d(% -a;), (n =o,. . . ) m),
(7)
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In the fourth of equations (8) the prime denotes the derivative with respect to the non-dimensional distance 2,: i.e., 4; = d&/d&. The integration of the field equations is now complete. Equations (4), (6) and (7) constitute 12( m + 1) equations. Stress or displacement boundary conditions on the lateral surfaces of the plate give another six equations which can be expressed in terms of the face variables SF, Rf. In fact, on the faces of the plate, as exact boundary conditions, one specifies quantities composed of one member of each of the pairs (r&, UT), (T,, u;) (i=l,... ,3). In the approximate theory these six boundary conditions can be expressed in terms of face variables by using equations (55) and (8,). Thus the number of available equations is [12(m + 1) +6]. On the other hand, the number of unknowns 7Ti, ul, it;, Sy, Rf, Sf ) is [18(m + l)+ 121. Therefore, to complete the mathetTri, 7Y1~ matical model [6( m + 1) + 61 more equations are needed. 2.2. ADDITIONAL EQUATIONS The additional equations can be obtained by establishing the constitutive relations for the face variables Rf and for the generalized variables tir and a;. To this end one can expand the displacements Ui in terms of 4, (n = 0,. . . , (m +2)), m+2 &=
C k=O
a:+k(X2),
where the coefficients a; are functions of x1, x3 and t. When the operator L”( n = 0, . . . , m) is applied to equation (9) and equation (9) is substituted into the expressions defining ST, equation (8,), two sets of algebraic equations are obtained, one governing ai with even k and the other governing a; with odd k. They are, respectively, P+2
P+2
C”kUL c k=O. Z,...
=
uy
(n=0,2,...,p),
k=;2
_,, 4k(lbik
=s:/2>
(10)
. .
p’+2 c k= 1 ,3,...
cnka
i =uy k
(n=1,3
,...,
p’),
“i’ k=l,
p=m-1
7
p’=
(11)
for even m,
p’=m-1
p=m,
C$k(l)ul,=S;2,
3,...
m
for odd m,
(12)
fl c,k
=
Ln4k
=
;
dd’k
(13)
dx2.
I -1
Through the solution of equations (10) and (11) one can determine the u;‘s in terms of the variables ul and SF (which appear as field variables in the approximate theory), or, more explicitly, a; (with even k) in terms of SC and ur
(n = 032,. . ., PL
ui (withoddk)intermsofS;andun
(n=1,3
)...)
p’).
(14)
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315
Substitution of equation (9) into the constitutive relations of rzi and into the expressions of al and d 1, equations (8,) and ( 82), and the use of the expression definining R f , equation (5,)) give Rt=?
“i’
h
R; =;
Ii2 k
R--l
l
4;(1)(p*daL)+p
(i = 1,3),
*dais:
k=1,3....
#;(1)[(2p+h)*da:]+h
*d(a,S:+a,S:),
1,3,...
‘+* -hk=~2,,,~;(l)(r*da’,)+p*da,Sr
(i=l,3),
..
‘;’ +;(1)[(2p+A)
R, =?
h
* da;]+h
* d(a,S; +a3S;),
k=0,2,...
’ &k = Lk(d&/dx2),
&k = Lk(d2+,/dx;).
(16)
When the expressions for a; in terms of ul and SF are substituted into equations (15) and the property stated in equations (14) is used the additional equations are finally obtained in the forms R;
~2 h 1
k=~3,,,~k(~*du:)+y~(~*dSi)
(i=l,3),
+P*d&S:
’. 1
+A*d(hS:+a3S:),
(i=1,3),
gy
=
!
&u; + c;+$
for even n
C;kUf
for odd n
k=0,2,...
5
k=1,3,...
(17) +
c;-s,
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In eqUationS (17) yk, &, cik, y*, CL* and ci* are some constants whose explicit values can be determined through the computations described above whenever the functions 4, are specified. The [6( m + 1) + 61 equations in equations (17), and the equations derived in the previous section, equations (4)-(7), constitute the governing equations of the approximate theory. The first four of equations (17) represent the constitutive relations for the face variables. The derivation of these equations is based on the expansion, equation (9), and the field equations of viscoelasticity. The use of these equations together with the lateral boundary conditions permits one to satisfy the lateral boundary conditions correctly and eliminates any inconsistency between the assumed displacement shape and the lateral boundary conditions.
2.3. THE VALUES OF THE CONSTANTS WHEN &‘s ARE LEGENDRE POLYNOMIALS Tables for the values of the constants appearing in the approximate theory can be prepared for various selected &‘s and for various orders of the approximate theory. As an illustration the values of the constants when +,, ( X2) = P, (x2), where P, is the nth order Legendre polynomial, and for orders of the approximate theory m = 0, 1 and 2 are presented here. The orthogonality of the Legendre polynomials simplifies the solution of equations (10) and (11) and, consequently, the computation of the constants. The values of the constants are presented in Table 1. The dash in the table indicates that the corresponding constant does not exist in the approximate theory.
TABLE
1
Values of constants for zeroth, first and second order theories
Constant Order
‘y,,
0
-3 -3
1
2
-10
Yl
Y2
Y+
CL
Y-
C&-
Chl
c;-
CL
_
-
-15
-
312 312
I/2 3
0
() 0
-
-
I/h
-15
-35
5
3
0
0
3/h
0
l/h
Constant ci2
c1,+
0
-
-
1
-
Order
2
0
GO 0
0
0
0
0
C& -
0
I,+ co
Go
0 0
-
0
3/h2
I,+
n-
C22
C2
4,
Cl
-
-
-
-
-
-
0
0
0
0
0
0
3. APPROXIMATE EQUATIONS OF THE LAYERED COMPOSITE The layered composite under study is composed of two alternating layers perfectly bonded at their interfaces. The two different layers are indicated by the circled numbers 1 and 2 in Figure 1. The layers 1 and 2 are assumed to be made of a linear, isotropic, viscoelastic material and have the material constants or functions (p,, pI, A 1, etc.) and (p2, pz, AZ, etc.), and have the thicknesses 2h, and 2h2, respectively. In the figure the pairs of layers, each of which consists of two different phases, are numbered in increasing order k = 0, 1,2, etc. In Figure 1 two kinds of co-ordinate systems are shown. The first
A DYNAMIC
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317
(k) (k)
Figure
1. Geometric
description
of the layered
composite.
is the (x1, x2, x3) global co-ordinate system whose (x,, x3) plane is parallel to the midplane of the layers. This co-ordinate system is employed to describe the location of a layer by DI(k) specifying the vertical distance of its midplane from the (x1, xj) plane. For example, x2 (cz = 1,2) describes the position of the czth constitutent of the kth pair (see Figure 1). A convention adopted throughout the study is that the Greek letters (Y,/?, etc., are used only to distinguish the two different phases of the composite and take the values 1 and 2. The second co-ordinate system is the local co-ordinate system (x,, i2, xg) whose (x,, xj) plane is chosen to coincide with the midplane of a particular layer. Two mathematical models for the layered composite, described above, can now be proposed: namely, the discrete model (DM) and the continuum model (CM). The DM equations can be obtained by using the equations of the single plate determined in the previous sections and the continuity conditions at the interfaces. The CM equations then can be established through the application of a smoothing operation to the DM equations. 3.1.
DISCRETE
MODEL
The governing equations of the layered composite consist of two types of equations: namely, the field equations, valid in each layer of the composite, and the continuity conditions at the interfaces. The equations obtained in the previous sections for a single plate hold also in each layer of the composite. Accordingly one can obtain the field equations by putting the index (Yin each variable appearing in the equations of the single plate. To keep the length of the paper shorr these equations are not rewritten here. Before writing the continuity conditions, it needs to be noted that there are two kinds of interfaces: one follows layer 1 and the other layer 2. As it is assumed that there is perfect bonding between the layers, the displacements Ui and the stress components 72i should be continuous across these interfaces: i.e., for the interfaces following layer 1 and layer 2, respectively: 1
2
u;=u;,
2 u;=u;,
1 T2i =
1
2 TZi =
2 _ T2i,
1 _ 72i.
(18) (19)
318
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The continuity conditions, equations (18) and (19), can be expressed in terms of face variables fii and 5: as, respectively,
The dependent variables &= (‘;r, $, Tyi, t:,, etc.) appearing in the field equations of the layered composite and equations (20) and (21) are functions of iik’: i.e., their values depend on the positions of the layers. Accordingly, the aforementioned equations form a discrete system of equations. In order to obtain the solutions by using this discrete model one should write the field equations in all layers and take into account the continuity conditions, equations (20) and (21) , at all interfaces. This kind of procedure involves lengthy computations and appears to be of no practical use. To simplify the analysis, in what follows the DM model is replaced by a continuum model by using a smoothing operation. 3.2. CONTINUUM MODEL To obtain the smoothed form of the field equations (valid in each layer of the composite) . the arguments of the variables 4 appearing in these equations u(k) m one first replaces x2
by x2. After this smoothing G is now defined for all x2 but has physical meaning only at the midplanes x2 = i ik). The smoothing operation leaves the field equations unchanged because all the field variables in these equations are defined at the midplane of the same layer: i.e., at x2 = gkk’. In accordant e with the idealization implied by the smoothing operation it is assumed that both types of layers exist simultaneously at every point of the continuum and accordingly the field equations with (r = 1 and (Y= 2 hold at the same point x2. The variables appearing in the continuity conditions, equations (20) and (21), do not belong to the same layer. Therefore the smoothed form of the continuity conditions will change and will be found through analysis. The analysis starts by referring to Figure 1 and writing the continuity equations for the interface which follows the layer 1, equations (20)) explicitly as $+(x1,
*(k) x2
, x3,
t) -
f-+(x,,
‘,k) x2
, $7
t)
=
h,
’ (k) 7 7-2 9 x3,
where it-* stands for either of the face variables (sit, ii:),
f) +
;‘-ix,,
‘(k) x2
> X3?
t),
(22)
i = 1,. . . ,3. It must be noted
that b f is defined at the midplane of layer 1 (i.e., at x2 = z&~‘)while + is defined at that ‘(k) ). To apply the smoothing operations to the continuity conditions, of layer 2 (i.e., at x2 = x2 equations (22) can be written first in a form in which all of the variables in it are defined at a single point. To this end a point M, in the interval (:ikJ, :\“) with distances p, A and p,A from the midplanes of layers 1 and 2, respectively, is chosen, where A = hl + h, and pm has the property p1 + p2 = 1 (see Figure 1). The x2 co-ordinate of this point is designated
319 l(k)_- x2(k)_ plA andx,‘(k).--x2 (kl + by xik) in the figure. By taking into account the relations x2 p,A the continuity condition, equations (22), now becomes A
DYNAMIC
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LAYERED
COMPOSITES
~+(~~~)+p~A)-~+(x:ll’-p,A)=~-(x:~’-p~A)+~~(x~~’+p~A).
(23)
For simplicity the arguments x ,, x3 and t of > are omitted in equation (23). The reduced form of the continuity condition for the interface which follows layer 2 can be obtained from equation (23) by replacing the layer index 1 by 2 and 2 by 1. It is ~+(~~~‘.+p~A)-f+(x~~‘-p~A)=~~(x:~’-p,A)+;.-(x~~‘+p,A).
(24)
To obtain the smoothed forms of the continuity equations, it is assumed that the two types of interfaces exist simultaneously at the same point of the continuum and xik) in equations (23) and (24) is replaced by x2. Expanding the terms of equations (23) and (24) in Taylor’s series about the point x2, and adding and subtracting the resulting equations, one finally obtains
s,k+ +s,k+=c,i+c,;:-,c2~+-c,i-+=s2~--s,h-, where s, and c, are the operators defined by
Equations (25) with ? = (us:, ii?), respectively, represent the smoothed form of the continuity conditions for displacement and stresses. The derivation of the equations of an mth order continuum model for a layered composite is now completed. The governing equations are composed of field equations and the continuity conditions, equations (25). It may be noted that all of the constants except p1 and p2 appearing in the CM equations can be computed in terms of the geometric and physical properties of the constituent layers. In view of findings established in reference [l] for elastic layered composites (CM proposed here for viscoelastic layered composites contains CM established in reference [l] for elastic layered composites as a special case) the use of p1 =p2 = O-5, which corresponds to taking M in Figure 1 at the midpoint of [&kk),;$“‘I, may be suggested. In fact, the numerical results presented in reference [l] indicated that the best match between exact and approximate dispersion curves for waves propagating in elastic layered composites is obtained when p1 = p2 = 0.5. 4. CONCLUSIONS AND DISCUSSIONS Higher order dynamic theories proposed here for viscoelastic plates and layered composites have the capability of satisfying correctly the lateral boundary conditions of the plate and the interface conditions of the layered composite. Accordingly, it may be anticipated that the proposed approximate theories should be able to predict correctly the geometric dispersive characteristics of viscoelastic plates and layered composites. This anticipation has been verified for elastic plates and layered composites in references [l] and [2] by comparing exact and approximate dispersion curves. This verification is
320
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D. TURHAN
expected to hold also for the present approximate theories established for viscoelastic plates and layered composites because the latter theories contain the theories proposed in references [l] and [2] as special cases (the theories for elastic case can be obtained from the present approximate theories by treating the relaxation functions p and A as constants). The order of the theory m for viscoelastic plates is kept arbitrary. The completeness of the distribution functions +,, assures that as m increases the frequency range over which the approximate theory is valid enlarges and as M approaches infinity the dynamic response of the plate predicted by the approximate theory approaches that determined from the exact theory. CM established for viscoelastic layered composites has two types of orders. One is the order of the theory, m. The other is the order of the continuity conditions, equations (25) (the order of the continuity conditions corresponds to the number of terms retained in the expansions of c, and s,, equations (26); for example, if one keeps the terms up to the pth power of K in equations (26) one says that the order of the continuity conditions is p). In the analysis the orders of both the theory and the continuity conditions are kept arbitrary. An increase in the orders of the theory and the continuity conditions enlarges the frequency and wave number ranges over which CM is valid, respectively. Two different approximate models, namely DM and CM, are proposed for viscoelastic layered composites. The use of CM should be preferred over that of DM when the number of layers making up the composite is large because the use of CM in this case simplifies and shortens the computations in the determination of the dynamic response of the composite. The proposed approximate theories should be assessed further by comparing their predictions with the exact or experimental data available in the literature for viscoelastic plates and layered composites. In order not to lengthen the paper this will be done in a separate future paper. The proposed approximate theories are general in the sense that they govern all of the deformation modes of plates and layered composites, and they are valid for all kinds of waves propagating in an arbitrary direction of the layered composite. REFERENCES 1. Y. MENGI 1980 International Journal of Solids and Structures 16, 1155-l 168. A new approach
2.
3. 4. 5. 6. 7. 8. 9.
for developing dynamic theories for structural elements, Part 1: Application to thermoelastic plates. Y. MENGI, G. BIRLIK and H. D. MCNIVEN 1980 International Journal of Solids and Structures 16, 1169-l 186. A new approach for developing dynamic theories for structural elements, Part 2: Application to thermoelastic layered composites. R. D. MINDLIN 1951 Journal of Applied Mechanics 18, 31-38. Influence of rotary inertia and shear on flexural vibrations of isotropic, elastic plates. E. REISSNER 1945 Journal of Applied Mechanics 12, 69-77. The effectof transverse shear deformation on the bending of elastic plates. R. D. MINDLIN and M. A. MEDICK 1959 Journal of Applied Mechanics 26, 561-569. Extensional vibrations of elastic plates. P. C. Y. LEE and Z. NIKODEN 1972 International Journal of Solids and Structures 8, 581-612. An approximate theory for high-frequency vibrations of elastic plates. R. D. MINDLIN and H. D. MCNIVEN 1960 Journal of Applied Mechanics 27, 145-151. Axially symmetric waves in elastic rods. C. T. SUN, J. D. ACHENBACH and G. HERRMANN 1968 Journal of Applied Mechanics 35, 467-475. Continuum theory for a laminated medium. A. BEDFORD, D. S. DRUMHELLER and H. J. SUTHERLAND 1976 Mechanics Today (Editor S. Nemat-Nasser) 3,1-54. On modelling the dynamics of composite materials. London: Pergamon Press.