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Theory of stability of multilayered continua in finite anisotropic elasticity
T H E O R Y OF STABILITY OF M U L T I L A Y E R E D C O N T I N U A IN F I N I T E A N I S O T R O P I C ELASTICITY BY
M. A. BlOT I ABSTRACT
The writer's general equations for the mechanics of continua under initial stress are applied to the formulation of a rigorous theory of stability of multilayered elastic media in a state of finite initial strain. The medium is assumed incompressible. It is either isotropic or anisotropic. The problem is analyzed in the context of the writer's earlier discussionsshowing the existence of internal and interfacial instability. The results provide a rigorous solution of the problem of buckling of sandwich plates. Recurrence equations are derived for the interracial displacements. It is shown that they are equivalent to a variational principle expressed in terms of these displacements. A matrix multiplication procedure is also developed for automatic computing of critical values when a large number of layers is involved, 1. INTRODUCTION
T h e t h e o r y of s t a b i l i t y of a c o n t i n u u m u n d e r initial stress was dev e l o p e d b y t h e w r i t e r m o r e t h a n t w e n t y y e a r s ago (1, 2, 3). 2 M o r e rec e n t l y it was applied t o p r o b l e m s of s t a b i l i t y of elastic a n d viscoelastic m e d i a . S t a b i l i t y p r o b l e m s h a v e been solved a n d discussed for the following cases: t h e isotropic h o m o g e n e o u s a n d n o n h o m o g e n e o u s half space of elastic a n d viscoelastic p r o p e r t i e s (4, 5) t h e e m b e d d e d layer w i t h elastic a n d viscoelastic i s o t r o p y (4, 6), a n d t h e surface i n s t a b i l i t y a n d t h e b u c k l i n g of a t h i c k sIab of r u b b e r in finite initial s t r a i n (7, 8). T h e a n a l y s i s was r e s t r i c t e d to m a t e r i a l s w h i c h r e t a i n i s o t r o p y under initial stress for i n c r e m e n t a l plane strain. A n o t h e r series of papers i n t r o d u c e d elastic a n i s o t r o p y , w h e t h e r i n d u c e d b y t h e initial finite s t r a i n or i n h e r e n t originally in t h e stress-free m a t e r i a l . T h e phenome n o n of i n t e r n a l i n s t a b i l i t y implicit in t h e w r i t e r ' s earlier paper on w a v e p r o p a g a t i o n (9) was given a d e t a i l e d discussion a n d analysis (10). F u r t h e r d e v e l o p m e n t s i n c l u d e d t h e solutions for t h e p r o b l e m s of surface a n d interfacial s t a b i l i t y u n d e r c o n d i t i o n s of a n i s o t r o p y in finite elast i c i t y (11, 12). T h e s e results p r o v i d e d t h e essential f o u n d a t i o n for t h e t r e a t m e n t of t h e m o r e c o m p l e x p r o b l e m s a n a l y z e d in t h e p r e s e n t paper. In p a r ticular, it was n e c e s s a r y to clarify t h e n a t u r e of i n t e r n a l instability a~ well as surface a n d interfacial i n s t a b i l i t y . T h i s is q u i t e a n a l o g o u s to the a n a l y s i s of w a v e p r o p a g a t i o n in l a y e r e d m e d i a , w h e r e a t h o r o u g h under" 1Shell Development Company, New York, N. Y. The boldface numbers in parentheses refer to the references appended to this paper. I28
Aug., 1963.]
IV~ULTILAYERED C O N T I N U A IN t~LASTICITY
12 9
standing of the behavior of body waves, Rayleigh and Stoneley waves is essential before any sound t r e a t m e n t of more complex structures'can be undertaken. This paper analyzes the stability of an elastic m e d i u m made of a superposition of layers of finite or infinite thickness. The m e d i u m is assumed to be incompressible. T h e assumption of incompressibility does not affect essentially the generality of the results and introduces considerable simplification in the algebra. The theory applies to an elastic c o n t i n u u m in a state of homogeneous finite strain. The state of initial stress, which m a y be different in each layer, has principal directions which are the same in all layers, one of these being perpendicular to the layers. The material in each layer may be orthotropic with the same planes of s y m m e t r y as the initial stress or it m a y be isotropic in the original stress-free state. In the latter case the initial stress will induce an elastic anisotropy with the same planes of symmetry. The incremental elastic coefficients for the case of a m e d i u m isotropic in finite strain were derived recently (10, la). Some basic results derived in previous work are briefly outlined in Section 2. T h e y refer to general solutions for the anisotropic m e d i u m under initial stress and a discussion of the elastic coefficients for laminated materials and for an elastic c o n t i n u u m which is isotropic in finite strain. These results are applied in Section 3 to a single anisotropic plate under initial stress. Equations are derived for the surface displacements under normal and tangential forces. Limiting cases are discussed in Section 4. These include the case of infinite thickness and the classical degenerate case of an isotropic m e d i u m with vanishing initial stress. The multilayered system is treated in Section 5. The results for the single plate are used to derive a general formulation of the stability equations for multilayered media. Recurrence and matrix equations are obtained which are well suited for numerical solution with a u t o m a t i c computers. The matrix multiplication procedure is suggested by a method proposed by Haskell for the analogous case of wave propagation. The recurrence equations also lead to a variational principle expressed in terms of interracial displacements as arbitrary variables. 2. FUNDAMENTALEQUATIONS AND GENERAL SOLUTIONS
Consider the two-dimensional deformation of an anisotropic continuum under initial stress. The deformation is in the x, y plane and the initial stress is a uniform compression P parallel with the x direction. Earlier work has shown t h a t the incremental stresses S~l, s22, s12 satisfy the equilibrium conditions
13o
M.A.
BlOT
[J. F.I.
Os~lq._Osl_ ~_pO_W___ = 0 Ox Oy oy
os~2+Os~2_pO~ Ox
= O.
Oy
Ox
1 ( 0v ox
0u~ 7y /
(1)
] ' h e local rotation ~ is =
(2)
where u and v are t h e components of displacement. T h e incremental stress components are referred to r o t a t e d axes. T h e general equations (A1) for plane strain are w r i t t e n out in the Appendix. E q u a t i o n s 1 are obtained b y putting X=Y=O S ~ = $1~ = 0
(3)
$11 = - P in Eqs. (A1). A principal stress S~3 m a y or m a y not be present in a direction perpendicular to the x, y plane. This c o m p o n e n t of the initial stress m a y be disregarded since it does not appear explicitly in t h e plane strain theory. T h e m e d i u m is assumed to be incompressible and orthotropic with t h e directions of s y m m e t r y parallel with the coordinate axes. For an elastic m e d i u m the incremental stress-strain relations are sit -- s = 2 N e ~ s~ - s = 2Ne~ $12
=
(4)
2Qe~.
T h e left-hand side of these equations represents the incremental stress deviator in two dimensions. It is n o t t h e same as t h e three-dimensional deviator. T h e strain components are
T h e condition of incompressibility m u s t be a d d e d : + % = 0.
(6)
T h e stress-strain relations (4) have been written with elastic coefficients N and Q. T h e case of an orthotropic viscoelastic medium is
Atlg., i963.]
MULTILAYERED CONTINUA IN J22LASTICITY
131
formally identical with the elastic case. By the correspondence principle introduced by this writer the elastic coefficients are replaced by operators N* and Q*. T h e algebra is the same for both cases and m a y be carried out with the elastic coefficients. Results are readily extended to the viscoelastic m e d i u m by substituting the operators in the final results. An i m p o r t a n t expression in the present problem is t h a t of the incremental force acting on a deformed surface which is initially a plane perpendicular to the y axis. The force is expressed per u n i t i n i t i a l area before deformation. It is considered to act on the b o t t o m half space as shown in Fig. 1. The x and y components of this force were derived
I///////////~ ×
BEFORE DEFORMATION
fy
Y
fx IP- x
AFTER DEFORMATION
FIG. 1. Illustration of the incremental force
components~f,~, &fu. in earlier work and m a y be found by applying Eqs. A2 of the Appendix. They are Af~ = (2Q + P)ex,~ dxJ~ = s + 2Ney~. (7) Equations 1 and 4 m a y be transformed to a simpler set by introducing a scalar ¢, since the condition of incompressibility (6) implies o¢ oy o4, ox
(8)
i32
IV[. A. BIoT
[J. F. L
W i t h this expression, the equations reduce to a s y s t e m with two unknowns, Os Ox
ay
#
-Ox ~
2
Oy~ A
=0 (9)
oy + ~
2x
- (2 -
oy--~
=0
~;
and expression (7) become
Oy 2 ]
Ox 2
(10) zXfy = s + 2 N ~02q) OxOy"
Eliminations of s between the two equations (9) yields
(Q-P)
o~,b + ( Q + P ) o ' ~ = o .
°`4,+2(2N-Q) Ox-N
Ox2Oy-----i2
~
Oy~
Solutions will be sought which are sinusoidal along x.
(11) Hence, let
(12)
chl~ = f ( l y ) sin lx s -- F(ly) cos lx
(13)
u = U(ly) sin lx v = V ( l y ) cos lx
Af~ = r(/y)
(14)
sin lx q(ly) coslx.
Afy --
E q u a t i o n 11 becomes an ordinary differential equation. f(O) satisfies the equation f""
T h e function
(15)
- 2 m f " q- k~f = O.
T h e prhnes denote derivatives with respect to t h e a r g u m e n t 0 = Iy. Let
Q m
--
2N-
p
Q
(2+- 2
k~ _
P 2 p"
Q+~
(16)
Aug., 1 9 6 3 , i
~/['ULTILAYERED CONTINUA IN ELASTICiT3/
i33
in previous work and also in the discussion which follows tim parameter P = -2Q was considered.
(17)
With this parameter, let 2N/Q 1+~k2 _
1 (18)
l+r
Substituting q~ and s in the first of Eqs. 9 determines F in terms of f
F= (2N-Q+P)f'-(Q+P)f
'''
(18a)
All quantities are then expressed in terms of f, t h a t is, Ul
~-
~ f]
V! = f
L ~ff=
--
f" --f
(19)
(2m + 1)f'
f'".
Let ~=Q+~.
P
(20)
The physical significance of this coefficient is brought out by considering a deformation which is a uniform shear displacement u = ~y V ---~ O.
(21)
The force on any plane perpendicular,to the y direction is given by Eq. 7. As a result, z~f~ -- L~ (22) ~L, -- 0. This represents a tangential shearing force aft. The elastic coefficient L represents a sliding rigidity on this particular plane under the existing state of initial stress. This has been referred to as the slide modulus.
I34
M.A.
A n o t h e r coefficient is (8,
BIOT
[J. F. I.
10) P
(23)
Using these coefficients, 2M
-
L
L
(24) ks
L - P
-
L For a m e d i u m which is isotropic in finite strain, Q is given by (I0, 13) 0 = !p
2
}k2A -~- •12
M2 _ Xl2
(25)
where M and M are the finite extension ratios in x and y directions in the state of initial stress. T h e slide modulus in this case is ~2 2
L = PX2 2 -
X/
(26)
For a compression P in the x direction Xl < 1. If, in addition to being isotropic in finite strain, the material is incompressible and obeys the finite stress-strain relations of an idealized r u b b e r - t y p e material, P = ~o(X22 - M2). This material remains isotropic for plane incremental strain. coefficients N and Q become equal, N = Q = ½u0(xl 2 + x ? ) .
(27) The (28)
T h e coefficient ~0 is the shear modulus at vanishing initial stress (Xi = X2 = 1). A t t e n t i o n is called to an i m p o r t a n t property of the p a r a m e t e r s (17) and (18) for such a material. T h e y become ~tk22 -- ~kl2
~,22 q-- X12 1 1+~"
M~ + M2 2X2 z ~kx2
~-~22"
(29)
Aug,, ~963.]
]~/~ULTILAYEREDCONTINUA IN ELASTICITY
135
Hence, they are independent of the rigidity coefficient go and are completely determined by the m a g n i t u d e of the finite initial strain. In the case of a laminated m e d i u m composed of alternate layers of different properties, one type of material is defined by the elastic coefficients L1 and M1. The layers of this material are of equal thickness and under the compressive stress PI. T h e y occupy a fraction a~ of the total thickness. Layers of another material of coefficients L~, M.2 are y U I V; T I
h12 h12
ql
p ~' ~. L~/////////x.
-
"//////////)
u z v~ r 2 %
(a)
~
i
(b) "~
1
.... '~'x
x
FIG. 2. Antisymmetric (a) and symmetric (b) deformation of an isolated layer with an initial compressive stress P.
sandwiched between the first and occupy a fraction a~ of the total thickness. T h e y are also of equal thickness, and the compressive stress in them is P=. Obviously, ~l+a.o = 1 (30) and the total average initial stress in the composite m e d i u m is
alP1 + a2P2
=
P.
(31)
The properties of this m e d i u m m a y be approximated by considering an anisotropic continuum of coefficients (10)
M = Mlal -k M=,~2 L-
Oll 012 L--~I + L--2
(32)
I36
M.A.
BIOT
[J. F. I.
However, there are limitations to this approximation. It is valid only for deformations whose wave length is large enough relative to the thickness of the layers. Combining Eqs. 23 and 30 and the value for M from Eq. 32, the coefficient N of the laminated medium can be derived as
N = Nlal + Nsa~.
(33)
The coefficient Q may also be derived from Eqs. 20 and 32. 3. ANALYSIS OF AN ISOLATED LAYER
The next step is to consider an isolated layer of thickness h. The x axis is located half way between the two faces (Fig. 2). Analysis of the most general deformation of this plate (which is sinusoidal along the x direction) is most conveniently accomplished by superposition of two types of deformation, one antisymmetric with respect to the x axis, as shown in Fig. 2a, and the other symmetric with respect to the same axis, as shown in Fig. 2b. In the antisymmetric deformation the function f(O) is an even function of the argument 0 = ly. The solution of the differential equation (15) which corresponds to this case is f = C1 cosh ~10 + Cs cosh 520.
(34)
The constants of integration C1, Cs are determined by the boundary conditions. Let 5~ and ~2 denote the roots of the characteristic equation />-2m¢~ s+k s =0.
(35)
Hence, (36) At this point something should be said about the choice of these roots, since there is a sign ambiguity involved in the definition of the square roots. The value of fl may be real, complex or imaginary. The following possible cases must be considered: 1. m > 0 m s -- k2 > 0.
Two subcases must be distinguished :
(a) k 2 > 0. In this case both roots ~1 and jSs are real, and their positive values m a y be chosen. (b) k 2 < 0. Here the root ~1 is real, and the root/~2 is pure imaginary. Again, positive values m a y be chosen for/~1 and B2/i. 2. m < 0 rns - k s > 0. In this case both roots are imaginary, and positive values are chosen for ~1/i and ~s/i.
Aug., 1963,]
MULTILAYERED CONTINUA IN ELASTICITY
i37
3. m 2 - k 2 < 0, with m assuming a n y real value, positive, negative or zero. In this case the roots are complex conjugates, and values for 51 and 52 are chosen with positive real parts. T h e two limiting cases m = 0 and m 2 - k ~ = 0 m u s t also be considered. T h e first does not give rise to a n y difficulty. T h e second case yields double roots and is discussed in detail in Section 4. Two of these cases, 1 (b) and 2, correspond to internal buckling (10). This can be recognized by p u t t i n g ~2 = _ (2 and writing the characteristic equation (35) in the form L~ 4 + 2 ( 2 M - L ) ~
2+L-P
= O.
(37)
Internal buckling corresponds to the existence of one or two real roots ~ for this equation (10). Therefore, internal buckling is represented by cases for which at least one root ~5 is imaginary. If the parameters remain outside the range of internal buckling, t h e roots Oh, 52 are either real or complex conjugate. In the latter case the solution (34) remains real if the constants of integration C, and C2 are also chosen to be complex conjugates. In all other cases the solution (34) remains real with real values of the constants. Designate b y the subscript a the values of U, V, r, and g relative to the a n t i s y m m e t r i c solution. T h e y are functions of the a r g u m e n t 0 and are designated as Us(O), Va(O), ra(O), and qa(O). Substituting solution (34) for f into relations (19) determines the displacements lUg(O) = - Ctflt sinh Oh0 - C2fl2 sinh ¢~20 lVa(O) = C; cosh fllO + C~ cosh j320
(38)
and the forces Ta(0) -L
q°(0) L
C1(~12 "~ 1) cosh /310 -- C2(~22 -~- 1) cosh ¢~20
(39)
- C,~(522 q- 1) sinh ¢hO q- C252(512 -~ ,1) sinh 520.
In deriving these expressions use has been m a d e of the relation ~12 + ~22 = 2m.
(40)
Next, write the values of these quantities at the upper face of the layer, that is, for o =
lh
=
(41)
~38
M.A.
BlOT
[J. F. I.
[:or simplicity, tim variables at this point are designated by Uo = u o ( u ) v,~ =
Va('y)
Ta =
Ta(~/)
~a =
~a('~]) •
(42)
Their expressions are given by the following equations, U~I
=
C151 sinh 517
-
-
C~¢~ sinh
¢t27
V~I = C1 cosh/317 + C2 cosh ¢/27 "]'a
L
C1B12cosh ~17 - C2/~22cosh 52y -
-
V.l
(43)
q~ = /~152(C1#2 sinh ¢h7 + C251 sinh ~2"/) -- Ual. L Solving the first two equations for C1 and C2 and substituting in the last two give 1 (U~ q- Vd52tanh/32~) ~ . cosh/3~y l C2 = ( U . + VoW1t a n h ~17) ~. cosh ~27
Ct =
(44)
and "]'a
-- = anU. lL
+ a12Va
(45) q__2 =
1L
a12U.
+
a22V..
Let /~12 Cbl~ - -
~
~2 2
A. (46)
a,22
=
/31/32
(~? LX~ -- fl~2) tanh/3F/tanh/3~'y
and A. = fl, t a n h ~ 7 - j3. tanh/32%
(47)
Aug., I963.]
MULTILAYERED
CONTINUA
IN
ELASTICITY
I39
It is also convenient to write these expressions in a more conlpact form by letting x~ = fl~ t a n h fl,~/ x= = /32 t a n h 52V.
(48)
This gives ~2 2
/•12
- -
a11 ----
a~2 =
X l - - X2
(~12 4 - I ) X 2 - - (Z2 2 "7[- 1 ) X l
(49)
X l - - X2
~22 =
Xl ~
=
X2
~llXlX2-
The displacements and stresses on the b o t t o m side of the layer, obtained by substituting 0 = - "r into relations (38) and (39), are uo(-
v) =
-
Va(-'r)
= Vo
,o(-'l)
=
~a(--
uo
(so)
~.
"/) "~ - - qa,
By a similar procedure the solution is obtained for the s y m m e t r i c deformation illustrated in Fig. 2b. In this case the function f(O) is odd, that is, f(O) = C1 sinh ill0 4- C= sinh fi~0.
(51)
The variables for this case are designated b y the subscript s. T h e y are Us(O), V,(O), r0(O), q~(O). Their values at the upper face of the layer are denoted by y.~(~) = us
v~(v) = v.
(52)
q . ( z ) = q..
For this case we derive r-2~ = bn U. 4- bl~ V~ lL
(s3) q-~ = b12U. 4- b22V~
IL
I4o
M.A.
BIOT
[J. F. I.
with f~l 2 - - f.~22
511 --
A~
tanh fl,y t a n h ~ y
Aa
(.54)
~ i2
- - f~22
A~ Values of the variables on the b o t t o m side of the layer are uo(-
-y) = uo
v.,(-
~/) = -
v., (s5)
q o ( - ~) = ¢~.
Next, superpose the two solutions obtained above. Denote by U1, V1, rl, ql, the values of the variables at the top of the layer and by U2, V2, r2, q2 the values of the same variables on the b o t t o m side (Fig. 2). For the top side, write U1-~
Ua'~- Us
V1 = Va-~- Vs
r l = ro -k r.~
(56)
ftl = q~ + q,~,
and for the b o t t o m side, U2=
-
V2 = V o -
Uaq-U.
T2 ~ g a -
Ts
fl,~ = -
V0
(57)
qo ÷ q.~.
Then, U a ...: 1 ( U 1 -
u,~ = ~ ( u 1 + u2)
U2)
Vo = ½(V~ + V2)
vo = ~ (v1 -
v~).
(58)
Substituting these values in expressions (45) and (53), the values of ra, qa, r,, q8 are obtained in terms of the top and b o t t o m displacements. Finally, introducing these values into Eqs. 56 and 57 gives A1
T1
ql T2
q2
= 1L
A2 A4 -As
A~ Aa
-A4
A5
U1
--As
A6
V1
A5
-A1
A2
U2 V2
--AG
A2
--Aa
(59)
A u g . , 1963.]
MULTILAYERED
CONTINUA
IN
141
]~LASTICITY
There are six distinct coefficients in this matrix expressed in terms of the six coefficients (49) and (54). T h e y are n i
=
+
=
A~
~(an + b1~)
A3
~ (a2~ + b~)
-
b.)
(60)
A5 = ½(a12 -- bl~)
A more convenient way to write Eq. 59 is to introduce an invariant which represents a q u a n t i t y proportional to the potential energy. Let I = ca~lU~2 + 2a12U~V~ + a1~Va 2 + b~lU~ ~ + 2bl~U.~V~ + b22VA
(61)
Substituting Eqs. 58 into Eq. 61 gives 1 2 + U~2) I--- 2AI(U~
j
A4U~U~+½A3(V~ 2 + V2=) +A6V~V~
+ A2(U1VI -
U~V~) + A ~ ( U ~ V 2 -
U2V1).
(62)
Equations 59 are then written OI
rl = l L - O U1
To. =
Of -- I L - -
OI
O U2
(63)
OI
4. SOME LIMITING CASES
In this section some particular cases of special interest will be examined. An obvious simplification should result by considering the layer thickness to be infinite. This should yield the solution for the half space. (11) In order to retain its physical significance, the solution m u s t be such t h a t internal buckling is excluded. This corresponds to cases l(a) and 3 discussed in Section 3, and the roots ¢tl and fi2 are real and positive or complex conjugate with positive real parts. Since ~, = ½1h, (64) assuming an infinite thickness a m o u n t s to setting _"
(65)
The same limiting case is also obtained for a finite thickness when 1 = ~, that is, when the wave length becomes very small.
(66) Either case results,
42
M.A.
BIOT
[1. F. I.
of course, from t h e geometric interpretation of t h e p a r a m e t e r 7 as proportional to the ratio of the thickness to the wave length. "Faking into a c c o u n t the positive signs of the real parts in the values of/31 and ~ , lim t a n h ¢M' = i 3~--* o0
(67) lira t a n h ~27 = 1. Hence, t h e limiting values of the coefficients (Eqs. 49 and 54) are an=
bll=fh+52
ax~ = bl: = ¢h~2 - 1 a~2 = b~2 = ¢h~=(~, + ~=).
(68)
T h e coefficients (Eqs. 60) become A1 =
~ii
(69)
A 2 ~--- a12
A a = a2~ and A4 = A~
=
A6
=
0.
(70)
Reference to Eq. 59 shows t h a t t h e top and b o t t o m sides of the plate are now & c o u p l e d . T h e r o o t s / ~ and ¢~2do not have to be e v a l u a t e d separately. Because a sign has been chosen for the roots, it is possible to write without ambiguity ~1~2 = k f~t -t- f~2 - 4 2 ( m + k)-. (71) T h e relation between surface forces and displacement at the top of the infinitely thick layer becomes 7"
- - = a l l U + ar2V 1L
q--- -=
lL
(72)
a12U + a22V,
with t h e values of Eqs. 68 for the coefficients. This resu~ coincides with one obtained earlier by t h e a u t h o r (11) for the lower half space. Similar expressions are obtained for the upper half space simply by
Aug., I963.]
MULTILAYEREDCONTINUA IN ELASTICITY
243
reversing t h e sign of a,i and a~,. This is readily seen by using the last two equations of the m a t r i x equation (59). It is convenient to express Eqs. 72 for the lower half space by using the invariant I~ = ½a~,U2 + a ~ U V + ½a22V2. (73) In this expression the coefficients a~j are given by Eqs. 68 and 71. T h e y d e p e n d only on the elastic coefficients and the initial stress in the m e d i u m , and are i n d e p e n d e n t of the wave length. E q u a t i o n s 72 m a y then be written 0I~ OU OI,
(74)
= I L o--V"
Similarly for t h e upper half space, the invariant, fu
=
½~/'11 U 2 - -
(I~12UV JI- 1(l,1~V2
(75)
is introduced. Relations between surface forces and displacements for the upper half space are w r i t t e n as OI, OU =
-
lk
(76)
a-v
A n o t h e r limiting case ls obtained when the characteristic equation (35) has a double root. This is obtained when m 2 -- k 2 = 0.
(77)
51 = 52 = 4 ~ = H.
(78)
The double root is
Exclusion of internal buckling in this case requires t h a t m be positive. When fll = f12, the coefficients a~i and bo in Eqs. 46 and 54 become undetermined, and their true limiting value m u s t be found. To do this, let fll = fl + ~
(79)
52=5-~ and expand the hyperbolic functions of expressions (46) and (54) to the first order in e. After cancellations of c o m m o n factors in n u m e r a t o r and denominator, the limiting values are derived by p u t t i n g ~ = O.
144
M.A.
BIOT
[J. F. I.
F o r the a n t i s y m m e t r i c case, 4/3 cosh 2137 a n = sinh 2/37 + 2/3~/ /3~ sinh 2/37 - 2/37 -1 sinh 2/37 + 2t37
at~ =
(80)
4/3a sinh 2/3-/ a=~ = sinh 2137 + 2/37" In the s y m m e t r i c case the limiting v a l u e s are 4/3 sinh 2/~7 sinh 2/33' - 2/~7 512 ~---
/3~ sinh 2/37 -t- 2/37 -1 sinh 2/37 - 2/3-/
522 ~-
4/3~ cosh ~/37 sinh 2/37 - 2/37"
(81)
C o n d i t i o n 77 for t h e existence of d o u b l e roots m a y be w r i t t e n p2
4N(N
-
Q) + --~ = O.
I t is interesting to n o t e t h a t it c a n n o t o c c u r for N > O. case
P=O
N=O
(82) The particular (83)
r e p r e s e n t s the classical case of an isotropic m e d i u m free of initial stress. T h e c h a r a c t e r i s t i c r o o t s in this case d e g e n e r a t e into t h e v a l u e s /31 =
/32 =
/3 =
1,
(84)
S u b s t i t u t i n g /3 = 1 into Eqs. 80 a n d 81, t h e coefficients for t h e antis y m m e t r i c case b e c o m e
all=
4 cosh 2 7 sinh 27 + 27
45' al~ = -- sinh 27 + 27 4 sinh 2 7
a22 = sinh 23' + 2"),
(85)
Atlg., I963.]
~/[ULTILAYERED
CONTINUA
IN
ELASTICITY
~45
and for t h e symmetric case 4 sinh 2 7
sinh 23,
-- 2?
43, b12 = sinh 23, - 23, b22 =
(86)
4 cosh 2 3, sinh 23, - 23,"
These coefficients are identical with those obtained by solving directly the classical problem of the isotropic and incompressible elastic plate without initial stress. The basic equations for this classical case are, of course, derived immediately by p u t t i n g N = Q and _P = 0 into Eqs. 9, 10, and 11. It is seen t h a t the latter degenerates into the wellknown biharmonic equation. 5. GENERAL EQUATIONS FOR THE STABILITY OF MULTILAYERED SYSTEMS
The previous results make it possible to obtain very simply and in a systematic way the stability equations for a system of superposed layers under initial stress. In the preceding section it was assumed t h a t the state of initial stress in the x, y plane is reduced to a single principal stress c o m p o n e n t S n = - P acting in a direction parallel with the layer. In order to deal with more general problems, such as the case of layers embedded in an infinite medium, the case where the initial stress includes a principal c o m p o n e n t $22 acting perpendicularly to the layer (Fig. 3) m u s t be examined. S22
//..'/i/./////////////
~S22
FIG. 3. S t a t e of initial stress in a layer w h e n a n initial c o m p o n e n t $2~ is present.
As will now be shown, all expressions obtained in the previous section are valid for this case, provided P is defined as P = S~ - $11.
(87)
With this definition for P, the equilibrium equations (1) are obviously valid as can be seen by applying the more general equations (AI) of the Appendix.
146
M . A . BlOT
U.F.I.
Equations 63 also remain valid when P is given the value in Eq. 87. However, the physical significance of r and q must be given a new interpretation. This can be seen as follows. The incremental forces given by Eqs. A2 in the Appendix are Af~
=
Si2 -- $22oj -- Siie~j
(88)
This m a y be written Af~ = st~ + Pe,~ -- S
3v
(89) Ox
where P = The terms
$22 -
$11.
A'fx = s12 + P e ~ = (2Q + P ) G y A~fy = s22 = s + 2Ne~y
(90)
coincide with expressions (7), except t h a t P is now the difference (Eq. 87) of the principal stresses. Equations 89 show t h a t A~f, and A~f~ are the incremental stress components along directions which are tangent a n d normal to the deformed surface. Note t h a t these stresses are now referred to unit areas after deformation. It is concluded t h a t Eqs. 45, 53 and 63 are valid in the more general case where $22 ¢ 0, provided r and q are interpreted as representing A~.
=
rsinlx
A~fy = q cos lx.
(91)
Consider now a system of superposed layers with perfect adherence along parallel interfaces and embedded between two semi-infinlte media
{j}
F~& 4.
Superposed layers embedded between two seml-infinite media.
Aug., 1 9 6 3 . ]
MULTILAYERED CONTINUA IN ELASTICITY
I47
(Fig. 4). The y axis is oriented normally to the layers. There are n layers numbered from 1 to n starting at the top. The state of initial stress is represented by a normal c o m p o n e n t S~ the same for all layers. The x c o m p o n e n t S n of the initial stress m a y be different in each layer. Designate by S n ~;~ its value in the jth layer. In this layer the stress difference is -Pj = $22
-
(92)
X l l (j).
-
The displacements of the upper face of the jth layer are denoted by Uj, Vy and t h a t of the lower face by Uj+~, V~.+,. The properties of the f h layer are completely defined by the invariant (62). Its value for the jth layer is written Ij = ½A~(Ufl + U j + I 2) - A4]UjUj+I -Jr- ½A3J(Vj 2 -~- Vj+I 2) + A~iViVj+l + A 2 J ( U j V j -
Uj+IVj+I)
+ AsJ(UjVj+I -
Uj+~Vs).
(93)
The coefficients A / , A~' etc., are the values of A~, A2 etc., for the jth layer as defined by Eqs. 60. T h e y are functions of the following parameters : ( a ) T h e two elastic coefficients Nj and Qj of the jth layer. (b) T h e stress difference P~- = $22 - S n (~) of the jth layer. (c) T h e variable ~'s = ½1h~ where hj is the thickness of the jth layer. Denote by Lj = Qj + ½Pj the slide modulus of the same layer. layer are given by Eqs. 63,
(94)
T h e stresses at the b o t t o m of the
OIi "rj+l' = -- lLi 0 Uj+I
(95) qi+*' = -- ILj
OI~ 0 Vj+I"
The stresses at the top of the (j + 1) t~' layer are given by the same Eqs. 63, ri+l = IL~+I OIj+1 0 Ui+, Of j+ I
(96)
148
M . A . BlOT
[J. F. I.
Since the stresses A'y. and A~/ygiven by Eq. 91 must be continuous at an interface, expressions (95) and (96) must be equal. This yields 0 OUj+~ ( L j I j + Lj+Ilj+O = 0 (97) 0 OVj+~ (L~Ij + Li+~Ii+,) = O. These two recurrence equations relate six variables, namely the two displacement components U~ and Vj at three successive interfaces. At the top and bottom interfaces the equations t a k e s special form because of the infinite thickness of the upper and lower media. It is necessary to introduce the previously defined invariants Iz and I~ for the lower and upper half spaces as given by Eqs. 73 and 75. In the present case they are written I, = 1~anU,~+l ~ + a12 U~+IV,~+I+ ½a22Vn+~2 1 , U12 - a 12' U1V1 + ~-a, , 2 2,"r 2 -~an V1.
I~ =
(98)
As for the other interfaces, the continuity of the stress A'f~ and A'fy at the top and bottom interfaces can be expressed, by four additional equations : 0
-(L~I1 + 0 U1
0
0 U~+I
(LJ,
L'I~)
= 0
+ LI,) = 0
0
-(LII~ + 0 V1
0
~v~T'+O -
!
L I~)
= 0
(99) (LnI, q- LIz) = O.
In Eqs. 99, L' denotes the slide modulus of the upper half space and L the slide modulus of the lower half space. Equations 97 and 99 constitute a system of 2 (n + 1) homogeneous (Q)
F I G . 5.
(b)
(a) Superposed layers resting on a half space and free at the top. (b) Superposed layers free at top and bottom.
equations for the 2(n + 1) displacement variables U~- and V~.. The characteristic stability equation is obtained by equating the determinant of this system to zero.
Aug., 1963.]
MULTILAYERED CONTINUA IN ELASTICITY
149
For layers resting on a semi-infinite half space and free at the top surface (Fig. 5a), L' = 0. Equations 99 are then replaced by 0Ii --=0
011 -0 VI
O U1 0
- (L.I. 0 U.+I
0
+ LI3
(100)
0
= 0
- (L.I,~ + LIg) = O. 0 V.+I
If the top and bottom surfaces of the layers are both free (Fig. 5b), then L ' = L = 0 and Eqs. 99 become 011 -0 0 U1
011 --=0 0 VI
OI,,
0I, --=0.
-0
(lOl)
A further sirnplification of the general formalism is obtained by introducing a single invariant which corresponds to the potential energy of the complete system. Let d
~q = L f z + Y: L~I~ + L'Iu.
(102)
The stability equations become 009 _ OUj 0
0~
OVy
=
O.
(103)
These 2 (n + 1) equations include all three cases of free and embedded layers formulated by Eqs. 99, 100, and 101. It is of interest to examine the case where the various materials obey the finite stress-strain relations (Eq. 27) of rubber-type elasticity. Here, a system originally.stress-free is brought to an initial state of finite homogeneous strain represented by the extension ratio M and M in the x, y plane. As shown by expressions (29), the parameters m, k2and f are then identical for all materials. They m a y be expressed in terms of a single variable, say Xd/M ~or 3. The only other parameter in the values of I j is % which for the j~l~ layer of thickness hj is written VJ = }lhj.
(104)
If the thickness of one of the layers is chosen as reference, say hi, then -ys = ( h # h l ) v
(10S)
15o
M . A . BlOT
[.L F. I.
where 3" = ½hll. Hence, the values of I~- contain the variable 3" and the ( n - 1) thickness ratios as parameters. Finally, taking into account Eqs. 26 and 27, Lj m a y be written as L~ = ~ojM ~
(106)
where ~0j is the stress-free modulus of the jtl~ layer. Hence, in all Eqs. 103, the c o m m o n factor M 2 cancels out. This brings out the ratios of the moduli /z0j as another set of parameters. W h e n these rigidity ratios are given along with the various thickness ratios, the characteristic equation becomes a relation between the two variables f and y. The first characterizes the initial stress and the second the wave length. T h e m i n i m u m value of f as a function of 3' yields the buckling stress and wave length. The preceding analysis has assumed perfect adherence. If the layers are allowed to slip w i t h o u t friction at the interface, the following condition m u s t be satisfied, T = 0,
(107)
since the tangential stresses m u s t remain zero. The displacements U~. in this case are not continuous. However, they m a y be eliminated from the equations by using condition (107) for perfect slip. This leaves a system of equations containing only the normal displacement Vj at the slipping interface. This normal c o m p o n e n t is continuous. A remark m u s t be made here regarding the continuity of q when such slippage occurs. Since this value is evaluated at the displaced points and since these points do not coincide due to slippage, the values of q are not exactly continuous. However, the difference is a second order quantity, and in a first order theory interracial values of q m a y be assumed equal to each layer. T h e case of perfect slippage is illustrated by an example in a forthcoming paper in this Journal. The recurrence equations derived above contain relatively simple coefficients and constitute a system of linear equations containing not more t h a n six variables at a time. It is possible to formulate the stability problem by a different procedure which m a y have advantages for numerical work with digital computers. The procedure was developed by Haskell for the formally similar problem of wave propagation in multilayered media (14). Equations 59 m a y be written in the form T1
ql lU1 IG
=
gg
q2
lU2 lV2
(108)
Aug., 1963.]
MULTILAYERED CONTINUA IN ELASTICITY
151
The matrix is B1
B.2
LB5
LB6
Ba
B4
-- L B 6
LB7
1
=
1
1 -- ]_.B~
(109)
1 f~B~o
-- B~
B~
The elements B~ are functions only of aCj and b~j. With =
(a~
-
b.)
2 -
(a.
-
bl,)(a..
-
b2~)
(11o)
they are B1
=X
S~
= ~ (a.b12 -
Ba
2 = ~ (alibi2 -- a22bm2)
2
al~biO
1
B4 = ~ [(&2 + B5
=
2 -A
b~2)(all
[a12=b" _ alibi22
-
b11) -
_
_
(a~
-
b12=)]
axlb11(a22 -- b22)]
2 J~6 --~- X E-- G'12512(a12 -- b12) + alla22bi2 -- a12511522~
B~
2 ~a22b22(an = -A
-
-
(111)
511) -~- a22612 ~ -- a122b2,J
2 2 B,
=
-
-a ( ~ "
2 B,o = ~ (a,, -
-
bl~)
b.).
The matrix equation (108) relates the values of the variables at two successive interfaces. Obviously the values at the top and b o t t o m interfaces satisfy the equations
15 2
M.A.
BIOT Tn+l
T1 n
q~ lU1 lV~
[J. F. I.
= II ~; s=~
g~+l 1Un+l
(112)
lV~+l
The product f l ~rcs of the ~Ygj matrices for each layer is a 4 X 4 j=l
matrix. If the top and bottom faces are free, let rl = ql = r,~+1 = q,+1 = 0, and Eq. 112 reduces to two homogeneous equations for the two unknowns U,+~ and V~+I. If the layers lie over a half space, r,,+~ and q,+~ are replaced by their values in terms of U~+, and V,+~ for the half space and are again led to a 2 X 2 system of equations. If the layers are embedded between two half spaces, the values of r~, q~, r~+~, q,+l are expressed in terms of U~, VI and U,~+I, V.~+I. This case results in a system of four equations. In using this procedure attention nmst be called to the case where the wave length is sufficiently small so that the top and bottom faces of a layer become decoupled. This happens when aij ~ b~s.
(113)
In this case the layer may be replaced by a half space. The procedure and formulas proposed here for the mechanics of layered media are quite general and are not restricted to incompressible media or stability problems. The matrix coefficients (Eqs. 111) are given a new form which is immediately applicable to a large category of problems, including wave propagation in anisotropic multilayered media. The recurrence equations (103) or the matrix multiplication process of Eqs. 112 provide a systematic way of programming the numerical work for the solution of problems in the mechanics of layered media when a large number of layers are involved and the use of a large capacity digital computer is available. A variational principle is readily obtained from Eqs. 103, that is, = o.
(114)
In this principle the interfacial displacement U~ and V~ are given arbit r a r y variations. Equation 114 is also a consequence of the physical interpretation of o~ as the total incremental energy of the multilayered system expressed in terms of the interfacial displacements. APPENDIX A brief outline is given here of previous results. The general equations were derived earlier (I, 2, 3, 9). For the case of plane strain and constant body force they were briefly rederived in a more recent paper (4). The results for this case are repeated here. The initial stresses are denoted by $11, $22, Sin. The incremental stresses referred to axes rotated by an angle ¢0 are denoted by sn, sin,s12. They satisfy the equilibrium conditions
At~g., i963.]
153
MULTILAYERED C O N T I N U A IN ELASTICITY
Ost~ Oa~ Ow osl~, + ~yy + p Y,o - 2&2 ~ x + (s1, - &2) ay Ox
+a&l 0s22
Ow
_ ( a s s , + as,~'~
+ as,~
( O&,2 + OS1;~
O&2 +~euu
&o
Ox
O&~
+--~y e ~ , - - \ Ox
oy/e~u
= O.
(At)
The strain components e,~, euu, e~j and the rotation o~ are defined as in Section 2. The body force components X and Y per unit mass are assumed to be constant2 The coordinate system is chosen clockwise. Incremental forces acting at a boundary per unit initial area (that is, per unit area as measured in the state of initial stress) are represented by the following x and y components ,~f~ = (sll -- Sl~w + Silevu - S12e~u) cos (n, x) + (s12 -- S~2w -- Sllexu + Sv2e~z) cos (n, y) af~ = (s~ + &~o, + &~e~ - &2e~) cos (n, x) + (s~2 + &,~o~ - Sl~e.y + &2e~) cos (n, y).
(A2)
These forces are acting on a closed contour in the x, y plane and the normal direction chosen positive outward is designated by n. REFERENCES
(1) M. A. BLOT, "Theory of Elasticity with Large Displacements and Rotations," Proc. 5th Internat. Cong. Applied Mechanics, pp. 117-122 (1938). (2) M. A. BIOT, "Non-Linear Theory of Elasticity and the Linearized Case for a Body Under lnitial Stress," Phil. Mag., Vol. 27, pp. 468-489 (1939). (3) M. A. BLOT, "Elastizit~ts Theorie Zweiter Ordnung mit Anwendungen," Zeit. Ang. Math. und Mech., bd. 20, pp. 89-99 (1940). (4) M. A. BIOT, "Folding of a Layered Viscoelastic Medium Derived from an Exact Stability Theory of a Continuum Under Initial Stress," Quart. Appl. Math., Vol. X V I I , pp. 185-204 (1959). (5) M. A. BIOT, "Instability of a Continuously Inhomogeneous Viscoelastic Half Space Under Initial Stress," Jot3R. FRANKLIN INST., VOl. 270, pp. 190-201 (1960). (6) M. A. BIOT and H. OD~, "On the Folding of a Viscoelastic Medium with Adhering Layer Under Compressive Initial Stress," Quart. A p p l . Math., Vol. XIX, pp. 351-355 (1962). (7) M. A. BIoT, "Surface Instability of Rubber in Compression," Appl. Sci. Res. A, 12 (1963). (On press.) (8) M. A. BIoT, "Exact Theory of Buckling of a Thick Slab," Appl. Scl. Res. A, 12 (1963). (On press.) (9) M. A. BIOT, "The Influence of Initial Stress on Elastic Waves," Y. Appl. Phys., Vol. 11, pp. 522-530 (1940). (10) M. A. BraT, "Internal Buckling Under Initial Stress in Finite Elasticity," Proc. Roy. Soe., Ser. A, Vol. 273, No. 1354, pp. 306-328 (1963). (11) M. A. BraT, "Surface Instability in Finite Anisotropic Elasticity Under Initial Stress," Proc. Roy. Soc., Ser. A, Vol. 273, No. 1354, pp. 329-339 (1963). (12) M. A. BLOT, "Interracial Instability in Finite Elasticity Under Initial Stress," Proc. Roy. Soc., Set. A, Vol. 273, No. 1354, pp. 340-344 (1963). (la) M. A. BraT, "Incremental Elastic Coefficients of an Isotropic Medium in Finite Strain," App. Sci. Res., A, 12 (1963). (On press.) (14) A. N. HASKELL, "Dispersion of Surface Waves in Multilayered Media," Bull. Seism. Soc. Am., Vol. 43, p. 17 (1953). 3 The more general equations are found in (1, 2, 3, 7),
M. A. BlOT I ABSTRACT
The writer's general equations for the mechanics of continua under initial stress are applied to the formulation of a rigorous theory of stability of multilayered elastic media in a state of finite initial strain. The medium is assumed incompressible. It is either isotropic or anisotropic. The problem is analyzed in the context of the writer's earlier discussionsshowing the existence of internal and interfacial instability. The results provide a rigorous solution of the problem of buckling of sandwich plates. Recurrence equations are derived for the interracial displacements. It is shown that they are equivalent to a variational principle expressed in terms of these displacements. A matrix multiplication procedure is also developed for automatic computing of critical values when a large number of layers is involved, 1. INTRODUCTION
T h e t h e o r y of s t a b i l i t y of a c o n t i n u u m u n d e r initial stress was dev e l o p e d b y t h e w r i t e r m o r e t h a n t w e n t y y e a r s ago (1, 2, 3). 2 M o r e rec e n t l y it was applied t o p r o b l e m s of s t a b i l i t y of elastic a n d viscoelastic m e d i a . S t a b i l i t y p r o b l e m s h a v e been solved a n d discussed for the following cases: t h e isotropic h o m o g e n e o u s a n d n o n h o m o g e n e o u s half space of elastic a n d viscoelastic p r o p e r t i e s (4, 5) t h e e m b e d d e d layer w i t h elastic a n d viscoelastic i s o t r o p y (4, 6), a n d t h e surface i n s t a b i l i t y a n d t h e b u c k l i n g of a t h i c k sIab of r u b b e r in finite initial s t r a i n (7, 8). T h e a n a l y s i s was r e s t r i c t e d to m a t e r i a l s w h i c h r e t a i n i s o t r o p y under initial stress for i n c r e m e n t a l plane strain. A n o t h e r series of papers i n t r o d u c e d elastic a n i s o t r o p y , w h e t h e r i n d u c e d b y t h e initial finite s t r a i n or i n h e r e n t originally in t h e stress-free m a t e r i a l . T h e phenome n o n of i n t e r n a l i n s t a b i l i t y implicit in t h e w r i t e r ' s earlier paper on w a v e p r o p a g a t i o n (9) was given a d e t a i l e d discussion a n d analysis (10). F u r t h e r d e v e l o p m e n t s i n c l u d e d t h e solutions for t h e p r o b l e m s of surface a n d interfacial s t a b i l i t y u n d e r c o n d i t i o n s of a n i s o t r o p y in finite elast i c i t y (11, 12). T h e s e results p r o v i d e d t h e essential f o u n d a t i o n for t h e t r e a t m e n t of t h e m o r e c o m p l e x p r o b l e m s a n a l y z e d in t h e p r e s e n t paper. In p a r ticular, it was n e c e s s a r y to clarify t h e n a t u r e of i n t e r n a l instability a~ well as surface a n d interfacial i n s t a b i l i t y . T h i s is q u i t e a n a l o g o u s to the a n a l y s i s of w a v e p r o p a g a t i o n in l a y e r e d m e d i a , w h e r e a t h o r o u g h under" 1Shell Development Company, New York, N. Y. The boldface numbers in parentheses refer to the references appended to this paper. I28
Aug., 1963.]
IV~ULTILAYERED C O N T I N U A IN t~LASTICITY
12 9
standing of the behavior of body waves, Rayleigh and Stoneley waves is essential before any sound t r e a t m e n t of more complex structures'can be undertaken. This paper analyzes the stability of an elastic m e d i u m made of a superposition of layers of finite or infinite thickness. The m e d i u m is assumed to be incompressible. T h e assumption of incompressibility does not affect essentially the generality of the results and introduces considerable simplification in the algebra. The theory applies to an elastic c o n t i n u u m in a state of homogeneous finite strain. The state of initial stress, which m a y be different in each layer, has principal directions which are the same in all layers, one of these being perpendicular to the layers. The material in each layer may be orthotropic with the same planes of s y m m e t r y as the initial stress or it m a y be isotropic in the original stress-free state. In the latter case the initial stress will induce an elastic anisotropy with the same planes of symmetry. The incremental elastic coefficients for the case of a m e d i u m isotropic in finite strain were derived recently (10, la). Some basic results derived in previous work are briefly outlined in Section 2. T h e y refer to general solutions for the anisotropic m e d i u m under initial stress and a discussion of the elastic coefficients for laminated materials and for an elastic c o n t i n u u m which is isotropic in finite strain. These results are applied in Section 3 to a single anisotropic plate under initial stress. Equations are derived for the surface displacements under normal and tangential forces. Limiting cases are discussed in Section 4. These include the case of infinite thickness and the classical degenerate case of an isotropic m e d i u m with vanishing initial stress. The multilayered system is treated in Section 5. The results for the single plate are used to derive a general formulation of the stability equations for multilayered media. Recurrence and matrix equations are obtained which are well suited for numerical solution with a u t o m a t i c computers. The matrix multiplication procedure is suggested by a method proposed by Haskell for the analogous case of wave propagation. The recurrence equations also lead to a variational principle expressed in terms of interracial displacements as arbitrary variables. 2. FUNDAMENTALEQUATIONS AND GENERAL SOLUTIONS
Consider the two-dimensional deformation of an anisotropic continuum under initial stress. The deformation is in the x, y plane and the initial stress is a uniform compression P parallel with the x direction. Earlier work has shown t h a t the incremental stresses S~l, s22, s12 satisfy the equilibrium conditions
13o
M.A.
BlOT
[J. F.I.
Os~lq._Osl_ ~_pO_W___ = 0 Ox Oy oy
os~2+Os~2_pO~ Ox
= O.
Oy
Ox
1 ( 0v ox
0u~ 7y /
(1)
] ' h e local rotation ~ is =
(2)
where u and v are t h e components of displacement. T h e incremental stress components are referred to r o t a t e d axes. T h e general equations (A1) for plane strain are w r i t t e n out in the Appendix. E q u a t i o n s 1 are obtained b y putting X=Y=O S ~ = $1~ = 0
(3)
$11 = - P in Eqs. (A1). A principal stress S~3 m a y or m a y not be present in a direction perpendicular to the x, y plane. This c o m p o n e n t of the initial stress m a y be disregarded since it does not appear explicitly in t h e plane strain theory. T h e m e d i u m is assumed to be incompressible and orthotropic with t h e directions of s y m m e t r y parallel with the coordinate axes. For an elastic m e d i u m the incremental stress-strain relations are sit -- s = 2 N e ~ s~ - s = 2Ne~ $12
=
(4)
2Qe~.
T h e left-hand side of these equations represents the incremental stress deviator in two dimensions. It is n o t t h e same as t h e three-dimensional deviator. T h e strain components are
T h e condition of incompressibility m u s t be a d d e d : + % = 0.
(6)
T h e stress-strain relations (4) have been written with elastic coefficients N and Q. T h e case of an orthotropic viscoelastic medium is
Atlg., i963.]
MULTILAYERED CONTINUA IN J22LASTICITY
131
formally identical with the elastic case. By the correspondence principle introduced by this writer the elastic coefficients are replaced by operators N* and Q*. T h e algebra is the same for both cases and m a y be carried out with the elastic coefficients. Results are readily extended to the viscoelastic m e d i u m by substituting the operators in the final results. An i m p o r t a n t expression in the present problem is t h a t of the incremental force acting on a deformed surface which is initially a plane perpendicular to the y axis. The force is expressed per u n i t i n i t i a l area before deformation. It is considered to act on the b o t t o m half space as shown in Fig. 1. The x and y components of this force were derived
I///////////~ ×
BEFORE DEFORMATION
fy
Y
fx IP- x
AFTER DEFORMATION
FIG. 1. Illustration of the incremental force
components~f,~, &fu. in earlier work and m a y be found by applying Eqs. A2 of the Appendix. They are Af~ = (2Q + P)ex,~ dxJ~ = s + 2Ney~. (7) Equations 1 and 4 m a y be transformed to a simpler set by introducing a scalar ¢, since the condition of incompressibility (6) implies o¢ oy o4, ox
(8)
i32
IV[. A. BIoT
[J. F. L
W i t h this expression, the equations reduce to a s y s t e m with two unknowns, Os Ox
ay
#
-Ox ~
2
Oy~ A
=0 (9)
oy + ~
2x
- (2 -
oy--~
=0
~;
and expression (7) become
Oy 2 ]
Ox 2
(10) zXfy = s + 2 N ~02q) OxOy"
Eliminations of s between the two equations (9) yields
(Q-P)
o~,b + ( Q + P ) o ' ~ = o .
°`4,+2(2N-Q) Ox-N
Ox2Oy-----i2
~
Oy~
Solutions will be sought which are sinusoidal along x.
(11) Hence, let
(12)
chl~ = f ( l y ) sin lx s -- F(ly) cos lx
(13)
u = U(ly) sin lx v = V ( l y ) cos lx
Af~ = r(/y)
(14)
sin lx q(ly) coslx.
Afy --
E q u a t i o n 11 becomes an ordinary differential equation. f(O) satisfies the equation f""
T h e function
(15)
- 2 m f " q- k~f = O.
T h e prhnes denote derivatives with respect to t h e a r g u m e n t 0 = Iy. Let
Q m
--
2N-
p
Q
(2+- 2
k~ _
P 2 p"
Q+~
(16)
Aug., 1 9 6 3 , i
~/['ULTILAYERED CONTINUA IN ELASTICiT3/
i33
in previous work and also in the discussion which follows tim parameter P = -2Q was considered.
(17)
With this parameter, let 2N/Q 1+~k2 _
1 (18)
l+r
Substituting q~ and s in the first of Eqs. 9 determines F in terms of f
F= (2N-Q+P)f'-(Q+P)f
'''
(18a)
All quantities are then expressed in terms of f, t h a t is, Ul
~-
~ f]
V! = f
L ~ff=
--
f" --f
(19)
(2m + 1)f'
f'".
Let ~=Q+~.
P
(20)
The physical significance of this coefficient is brought out by considering a deformation which is a uniform shear displacement u = ~y V ---~ O.
(21)
The force on any plane perpendicular,to the y direction is given by Eq. 7. As a result, z~f~ -- L~ (22) ~L, -- 0. This represents a tangential shearing force aft. The elastic coefficient L represents a sliding rigidity on this particular plane under the existing state of initial stress. This has been referred to as the slide modulus.
I34
M.A.
A n o t h e r coefficient is (8,
BIOT
[J. F. I.
10) P
(23)
Using these coefficients, 2M
-
L
L
(24) ks
L - P
-
L For a m e d i u m which is isotropic in finite strain, Q is given by (I0, 13) 0 = !p
2
}k2A -~- •12
M2 _ Xl2
(25)
where M and M are the finite extension ratios in x and y directions in the state of initial stress. T h e slide modulus in this case is ~2 2
L = PX2 2 -
X/
(26)
For a compression P in the x direction Xl < 1. If, in addition to being isotropic in finite strain, the material is incompressible and obeys the finite stress-strain relations of an idealized r u b b e r - t y p e material, P = ~o(X22 - M2). This material remains isotropic for plane incremental strain. coefficients N and Q become equal, N = Q = ½u0(xl 2 + x ? ) .
(27) The (28)
T h e coefficient ~0 is the shear modulus at vanishing initial stress (Xi = X2 = 1). A t t e n t i o n is called to an i m p o r t a n t property of the p a r a m e t e r s (17) and (18) for such a material. T h e y become ~tk22 -- ~kl2
~,22 q-- X12 1 1+~"
M~ + M2 2X2 z ~kx2
~-~22"
(29)
Aug,, ~963.]
]~/~ULTILAYEREDCONTINUA IN ELASTICITY
135
Hence, they are independent of the rigidity coefficient go and are completely determined by the m a g n i t u d e of the finite initial strain. In the case of a laminated m e d i u m composed of alternate layers of different properties, one type of material is defined by the elastic coefficients L1 and M1. The layers of this material are of equal thickness and under the compressive stress PI. T h e y occupy a fraction a~ of the total thickness. Layers of another material of coefficients L~, M.2 are y U I V; T I
h12 h12
ql
p ~' ~. L~/////////x.
-
"//////////)
u z v~ r 2 %
(a)
~
i
(b) "~
1
.... '~'x
x
FIG. 2. Antisymmetric (a) and symmetric (b) deformation of an isolated layer with an initial compressive stress P.
sandwiched between the first and occupy a fraction a~ of the total thickness. T h e y are also of equal thickness, and the compressive stress in them is P=. Obviously, ~l+a.o = 1 (30) and the total average initial stress in the composite m e d i u m is
alP1 + a2P2
=
P.
(31)
The properties of this m e d i u m m a y be approximated by considering an anisotropic continuum of coefficients (10)
M = Mlal -k M=,~2 L-
Oll 012 L--~I + L--2
(32)
I36
M.A.
BIOT
[J. F. I.
However, there are limitations to this approximation. It is valid only for deformations whose wave length is large enough relative to the thickness of the layers. Combining Eqs. 23 and 30 and the value for M from Eq. 32, the coefficient N of the laminated medium can be derived as
N = Nlal + Nsa~.
(33)
The coefficient Q may also be derived from Eqs. 20 and 32. 3. ANALYSIS OF AN ISOLATED LAYER
The next step is to consider an isolated layer of thickness h. The x axis is located half way between the two faces (Fig. 2). Analysis of the most general deformation of this plate (which is sinusoidal along the x direction) is most conveniently accomplished by superposition of two types of deformation, one antisymmetric with respect to the x axis, as shown in Fig. 2a, and the other symmetric with respect to the same axis, as shown in Fig. 2b. In the antisymmetric deformation the function f(O) is an even function of the argument 0 = ly. The solution of the differential equation (15) which corresponds to this case is f = C1 cosh ~10 + Cs cosh 520.
(34)
The constants of integration C1, Cs are determined by the boundary conditions. Let 5~ and ~2 denote the roots of the characteristic equation />-2m¢~ s+k s =0.
(35)
Hence, (36) At this point something should be said about the choice of these roots, since there is a sign ambiguity involved in the definition of the square roots. The value of fl may be real, complex or imaginary. The following possible cases must be considered: 1. m > 0 m s -- k2 > 0.
Two subcases must be distinguished :
(a) k 2 > 0. In this case both roots ~1 and jSs are real, and their positive values m a y be chosen. (b) k 2 < 0. Here the root ~1 is real, and the root/~2 is pure imaginary. Again, positive values m a y be chosen for/~1 and B2/i. 2. m < 0 rns - k s > 0. In this case both roots are imaginary, and positive values are chosen for ~1/i and ~s/i.
Aug., 1963,]
MULTILAYERED CONTINUA IN ELASTICITY
i37
3. m 2 - k 2 < 0, with m assuming a n y real value, positive, negative or zero. In this case the roots are complex conjugates, and values for 51 and 52 are chosen with positive real parts. T h e two limiting cases m = 0 and m 2 - k ~ = 0 m u s t also be considered. T h e first does not give rise to a n y difficulty. T h e second case yields double roots and is discussed in detail in Section 4. Two of these cases, 1 (b) and 2, correspond to internal buckling (10). This can be recognized by p u t t i n g ~2 = _ (2 and writing the characteristic equation (35) in the form L~ 4 + 2 ( 2 M - L ) ~
2+L-P
= O.
(37)
Internal buckling corresponds to the existence of one or two real roots ~ for this equation (10). Therefore, internal buckling is represented by cases for which at least one root ~5 is imaginary. If the parameters remain outside the range of internal buckling, t h e roots Oh, 52 are either real or complex conjugate. In the latter case the solution (34) remains real if the constants of integration C, and C2 are also chosen to be complex conjugates. In all other cases the solution (34) remains real with real values of the constants. Designate b y the subscript a the values of U, V, r, and g relative to the a n t i s y m m e t r i c solution. T h e y are functions of the a r g u m e n t 0 and are designated as Us(O), Va(O), ra(O), and qa(O). Substituting solution (34) for f into relations (19) determines the displacements lUg(O) = - Ctflt sinh Oh0 - C2fl2 sinh ¢~20 lVa(O) = C; cosh fllO + C~ cosh j320
(38)
and the forces Ta(0) -L
q°(0) L
C1(~12 "~ 1) cosh /310 -- C2(~22 -~- 1) cosh ¢~20
(39)
- C,~(522 q- 1) sinh ¢hO q- C252(512 -~ ,1) sinh 520.
In deriving these expressions use has been m a d e of the relation ~12 + ~22 = 2m.
(40)
Next, write the values of these quantities at the upper face of the layer, that is, for o =
lh
=
(41)
~38
M.A.
BlOT
[J. F. I.
[:or simplicity, tim variables at this point are designated by Uo = u o ( u ) v,~ =
Va('y)
Ta =
Ta(~/)
~a =
~a('~]) •
(42)
Their expressions are given by the following equations, U~I
=
C151 sinh 517
-
-
C~¢~ sinh
¢t27
V~I = C1 cosh/317 + C2 cosh ¢/27 "]'a
L
C1B12cosh ~17 - C2/~22cosh 52y -
-
V.l
(43)
q~ = /~152(C1#2 sinh ¢h7 + C251 sinh ~2"/) -- Ual. L Solving the first two equations for C1 and C2 and substituting in the last two give 1 (U~ q- Vd52tanh/32~) ~ . cosh/3~y l C2 = ( U . + VoW1t a n h ~17) ~. cosh ~27
Ct =
(44)
and "]'a
-- = anU. lL
+ a12Va
(45) q__2 =
1L
a12U.
+
a22V..
Let /~12 Cbl~ - -
~
~2 2
A. (46)
a,22
=
/31/32
(~? LX~ -- fl~2) tanh/3F/tanh/3~'y
and A. = fl, t a n h ~ 7 - j3. tanh/32%
(47)
Aug., I963.]
MULTILAYERED
CONTINUA
IN
ELASTICITY
I39
It is also convenient to write these expressions in a more conlpact form by letting x~ = fl~ t a n h fl,~/ x= = /32 t a n h 52V.
(48)
This gives ~2 2
/•12
- -
a11 ----
a~2 =
X l - - X2
(~12 4 - I ) X 2 - - (Z2 2 "7[- 1 ) X l
(49)
X l - - X2
~22 =
Xl ~
=
X2
~llXlX2-
The displacements and stresses on the b o t t o m side of the layer, obtained by substituting 0 = - "r into relations (38) and (39), are uo(-
v) =
-
Va(-'r)
= Vo
,o(-'l)
=
~a(--
uo
(so)
~.
"/) "~ - - qa,
By a similar procedure the solution is obtained for the s y m m e t r i c deformation illustrated in Fig. 2b. In this case the function f(O) is odd, that is, f(O) = C1 sinh ill0 4- C= sinh fi~0.
(51)
The variables for this case are designated b y the subscript s. T h e y are Us(O), V,(O), r0(O), q~(O). Their values at the upper face of the layer are denoted by y.~(~) = us
v~(v) = v.
(52)
q . ( z ) = q..
For this case we derive r-2~ = bn U. 4- bl~ V~ lL
(s3) q-~ = b12U. 4- b22V~
IL
I4o
M.A.
BIOT
[J. F. I.
with f~l 2 - - f.~22
511 --
A~
tanh fl,y t a n h ~ y
Aa
(.54)
~ i2
- - f~22
A~ Values of the variables on the b o t t o m side of the layer are uo(-
-y) = uo
v.,(-
~/) = -
v., (s5)
q o ( - ~) = ¢~.
Next, superpose the two solutions obtained above. Denote by U1, V1, rl, ql, the values of the variables at the top of the layer and by U2, V2, r2, q2 the values of the same variables on the b o t t o m side (Fig. 2). For the top side, write U1-~
Ua'~- Us
V1 = Va-~- Vs
r l = ro -k r.~
(56)
ftl = q~ + q,~,
and for the b o t t o m side, U2=
-
V2 = V o -
Uaq-U.
T2 ~ g a -
Ts
fl,~ = -
V0
(57)
qo ÷ q.~.
Then, U a ...: 1 ( U 1 -
u,~ = ~ ( u 1 + u2)
U2)
Vo = ½(V~ + V2)
vo = ~ (v1 -
v~).
(58)
Substituting these values in expressions (45) and (53), the values of ra, qa, r,, q8 are obtained in terms of the top and b o t t o m displacements. Finally, introducing these values into Eqs. 56 and 57 gives A1
T1
ql T2
q2
= 1L
A2 A4 -As
A~ Aa
-A4
A5
U1
--As
A6
V1
A5
-A1
A2
U2 V2
--AG
A2
--Aa
(59)
A u g . , 1963.]
MULTILAYERED
CONTINUA
IN
141
]~LASTICITY
There are six distinct coefficients in this matrix expressed in terms of the six coefficients (49) and (54). T h e y are n i
=
+
=
A~
~(an + b1~)
A3
~ (a2~ + b~)
-
b.)
(60)
A5 = ½(a12 -- bl~)
A more convenient way to write Eq. 59 is to introduce an invariant which represents a q u a n t i t y proportional to the potential energy. Let I = ca~lU~2 + 2a12U~V~ + a1~Va 2 + b~lU~ ~ + 2bl~U.~V~ + b22VA
(61)
Substituting Eqs. 58 into Eq. 61 gives 1 2 + U~2) I--- 2AI(U~
j
A4U~U~+½A3(V~ 2 + V2=) +A6V~V~
+ A2(U1VI -
U~V~) + A ~ ( U ~ V 2 -
U2V1).
(62)
Equations 59 are then written OI
rl = l L - O U1
To. =
Of -- I L - -
OI
O U2
(63)
OI
4. SOME LIMITING CASES
In this section some particular cases of special interest will be examined. An obvious simplification should result by considering the layer thickness to be infinite. This should yield the solution for the half space. (11) In order to retain its physical significance, the solution m u s t be such t h a t internal buckling is excluded. This corresponds to cases l(a) and 3 discussed in Section 3, and the roots ¢tl and fi2 are real and positive or complex conjugate with positive real parts. Since ~, = ½1h, (64) assuming an infinite thickness a m o u n t s to setting _"
(65)
The same limiting case is also obtained for a finite thickness when 1 = ~, that is, when the wave length becomes very small.
(66) Either case results,
42
M.A.
BIOT
[1. F. I.
of course, from t h e geometric interpretation of t h e p a r a m e t e r 7 as proportional to the ratio of the thickness to the wave length. "Faking into a c c o u n t the positive signs of the real parts in the values of/31 and ~ , lim t a n h ¢M' = i 3~--* o0
(67) lira t a n h ~27 = 1. Hence, t h e limiting values of the coefficients (Eqs. 49 and 54) are an=
bll=fh+52
ax~ = bl: = ¢h~2 - 1 a~2 = b~2 = ¢h~=(~, + ~=).
(68)
T h e coefficients (Eqs. 60) become A1 =
~ii
(69)
A 2 ~--- a12
A a = a2~ and A4 = A~
=
A6
=
0.
(70)
Reference to Eq. 59 shows t h a t t h e top and b o t t o m sides of the plate are now & c o u p l e d . T h e r o o t s / ~ and ¢~2do not have to be e v a l u a t e d separately. Because a sign has been chosen for the roots, it is possible to write without ambiguity ~1~2 = k f~t -t- f~2 - 4 2 ( m + k)-. (71) T h e relation between surface forces and displacement at the top of the infinitely thick layer becomes 7"
- - = a l l U + ar2V 1L
q--- -=
lL
(72)
a12U + a22V,
with t h e values of Eqs. 68 for the coefficients. This resu~ coincides with one obtained earlier by t h e a u t h o r (11) for the lower half space. Similar expressions are obtained for the upper half space simply by
Aug., I963.]
MULTILAYEREDCONTINUA IN ELASTICITY
243
reversing t h e sign of a,i and a~,. This is readily seen by using the last two equations of the m a t r i x equation (59). It is convenient to express Eqs. 72 for the lower half space by using the invariant I~ = ½a~,U2 + a ~ U V + ½a22V2. (73) In this expression the coefficients a~j are given by Eqs. 68 and 71. T h e y d e p e n d only on the elastic coefficients and the initial stress in the m e d i u m , and are i n d e p e n d e n t of the wave length. E q u a t i o n s 72 m a y then be written 0I~ OU OI,
(74)
= I L o--V"
Similarly for t h e upper half space, the invariant, fu
=
½~/'11 U 2 - -
(I~12UV JI- 1(l,1~V2
(75)
is introduced. Relations between surface forces and displacements for the upper half space are w r i t t e n as OI, OU =
-
lk
(76)
a-v
A n o t h e r limiting case ls obtained when the characteristic equation (35) has a double root. This is obtained when m 2 -- k 2 = 0.
(77)
51 = 52 = 4 ~ = H.
(78)
The double root is
Exclusion of internal buckling in this case requires t h a t m be positive. When fll = f12, the coefficients a~i and bo in Eqs. 46 and 54 become undetermined, and their true limiting value m u s t be found. To do this, let fll = fl + ~
(79)
52=5-~ and expand the hyperbolic functions of expressions (46) and (54) to the first order in e. After cancellations of c o m m o n factors in n u m e r a t o r and denominator, the limiting values are derived by p u t t i n g ~ = O.
144
M.A.
BIOT
[J. F. I.
F o r the a n t i s y m m e t r i c case, 4/3 cosh 2137 a n = sinh 2/37 + 2/3~/ /3~ sinh 2/37 - 2/37 -1 sinh 2/37 + 2t37
at~ =
(80)
4/3a sinh 2/3-/ a=~ = sinh 2137 + 2/37" In the s y m m e t r i c case the limiting v a l u e s are 4/3 sinh 2/~7 sinh 2/33' - 2/~7 512 ~---
/3~ sinh 2/37 -t- 2/37 -1 sinh 2/37 - 2/3-/
522 ~-
4/3~ cosh ~/37 sinh 2/37 - 2/37"
(81)
C o n d i t i o n 77 for t h e existence of d o u b l e roots m a y be w r i t t e n p2
4N(N
-
Q) + --~ = O.
I t is interesting to n o t e t h a t it c a n n o t o c c u r for N > O. case
P=O
N=O
(82) The particular (83)
r e p r e s e n t s the classical case of an isotropic m e d i u m free of initial stress. T h e c h a r a c t e r i s t i c r o o t s in this case d e g e n e r a t e into t h e v a l u e s /31 =
/32 =
/3 =
1,
(84)
S u b s t i t u t i n g /3 = 1 into Eqs. 80 a n d 81, t h e coefficients for t h e antis y m m e t r i c case b e c o m e
all=
4 cosh 2 7 sinh 27 + 27
45' al~ = -- sinh 27 + 27 4 sinh 2 7
a22 = sinh 23' + 2"),
(85)
Atlg., I963.]
~/[ULTILAYERED
CONTINUA
IN
ELASTICITY
~45
and for t h e symmetric case 4 sinh 2 7
sinh 23,
-- 2?
43, b12 = sinh 23, - 23, b22 =
(86)
4 cosh 2 3, sinh 23, - 23,"
These coefficients are identical with those obtained by solving directly the classical problem of the isotropic and incompressible elastic plate without initial stress. The basic equations for this classical case are, of course, derived immediately by p u t t i n g N = Q and _P = 0 into Eqs. 9, 10, and 11. It is seen t h a t the latter degenerates into the wellknown biharmonic equation. 5. GENERAL EQUATIONS FOR THE STABILITY OF MULTILAYERED SYSTEMS
The previous results make it possible to obtain very simply and in a systematic way the stability equations for a system of superposed layers under initial stress. In the preceding section it was assumed t h a t the state of initial stress in the x, y plane is reduced to a single principal stress c o m p o n e n t S n = - P acting in a direction parallel with the layer. In order to deal with more general problems, such as the case of layers embedded in an infinite medium, the case where the initial stress includes a principal c o m p o n e n t $22 acting perpendicularly to the layer (Fig. 3) m u s t be examined. S22
//..'/i/./////////////
~S22
FIG. 3. S t a t e of initial stress in a layer w h e n a n initial c o m p o n e n t $2~ is present.
As will now be shown, all expressions obtained in the previous section are valid for this case, provided P is defined as P = S~ - $11.
(87)
With this definition for P, the equilibrium equations (1) are obviously valid as can be seen by applying the more general equations (AI) of the Appendix.
146
M . A . BlOT
U.F.I.
Equations 63 also remain valid when P is given the value in Eq. 87. However, the physical significance of r and q must be given a new interpretation. This can be seen as follows. The incremental forces given by Eqs. A2 in the Appendix are Af~
=
Si2 -- $22oj -- Siie~j
(88)
This m a y be written Af~ = st~ + Pe,~ -- S
3v
(89) Ox
where P = The terms
$22 -
$11.
A'fx = s12 + P e ~ = (2Q + P ) G y A~fy = s22 = s + 2Ne~y
(90)
coincide with expressions (7), except t h a t P is now the difference (Eq. 87) of the principal stresses. Equations 89 show t h a t A~f, and A~f~ are the incremental stress components along directions which are tangent a n d normal to the deformed surface. Note t h a t these stresses are now referred to unit areas after deformation. It is concluded t h a t Eqs. 45, 53 and 63 are valid in the more general case where $22 ¢ 0, provided r and q are interpreted as representing A~.
=
rsinlx
A~fy = q cos lx.
(91)
Consider now a system of superposed layers with perfect adherence along parallel interfaces and embedded between two semi-infinlte media
{j}
F~& 4.
Superposed layers embedded between two seml-infinite media.
Aug., 1 9 6 3 . ]
MULTILAYERED CONTINUA IN ELASTICITY
I47
(Fig. 4). The y axis is oriented normally to the layers. There are n layers numbered from 1 to n starting at the top. The state of initial stress is represented by a normal c o m p o n e n t S~ the same for all layers. The x c o m p o n e n t S n of the initial stress m a y be different in each layer. Designate by S n ~;~ its value in the jth layer. In this layer the stress difference is -Pj = $22
-
(92)
X l l (j).
-
The displacements of the upper face of the jth layer are denoted by Uj, Vy and t h a t of the lower face by Uj+~, V~.+,. The properties of the f h layer are completely defined by the invariant (62). Its value for the jth layer is written Ij = ½A~(Ufl + U j + I 2) - A4]UjUj+I -Jr- ½A3J(Vj 2 -~- Vj+I 2) + A~iViVj+l + A 2 J ( U j V j -
Uj+IVj+I)
+ AsJ(UjVj+I -
Uj+~Vs).
(93)
The coefficients A / , A~' etc., are the values of A~, A2 etc., for the jth layer as defined by Eqs. 60. T h e y are functions of the following parameters : ( a ) T h e two elastic coefficients Nj and Qj of the jth layer. (b) T h e stress difference P~- = $22 - S n (~) of the jth layer. (c) T h e variable ~'s = ½1h~ where hj is the thickness of the jth layer. Denote by Lj = Qj + ½Pj the slide modulus of the same layer. layer are given by Eqs. 63,
(94)
T h e stresses at the b o t t o m of the
OIi "rj+l' = -- lLi 0 Uj+I
(95) qi+*' = -- ILj
OI~ 0 Vj+I"
The stresses at the top of the (j + 1) t~' layer are given by the same Eqs. 63, ri+l = IL~+I OIj+1 0 Ui+, Of j+ I
(96)
148
M . A . BlOT
[J. F. I.
Since the stresses A'y. and A~/ygiven by Eq. 91 must be continuous at an interface, expressions (95) and (96) must be equal. This yields 0 OUj+~ ( L j I j + Lj+Ilj+O = 0 (97) 0 OVj+~ (L~Ij + Li+~Ii+,) = O. These two recurrence equations relate six variables, namely the two displacement components U~ and Vj at three successive interfaces. At the top and bottom interfaces the equations t a k e s special form because of the infinite thickness of the upper and lower media. It is necessary to introduce the previously defined invariants Iz and I~ for the lower and upper half spaces as given by Eqs. 73 and 75. In the present case they are written I, = 1~anU,~+l ~ + a12 U~+IV,~+I+ ½a22Vn+~2 1 , U12 - a 12' U1V1 + ~-a, , 2 2,"r 2 -~an V1.
I~ =
(98)
As for the other interfaces, the continuity of the stress A'f~ and A'fy at the top and bottom interfaces can be expressed, by four additional equations : 0
-(L~I1 + 0 U1
0
0 U~+I
(LJ,
L'I~)
= 0
+ LI,) = 0
0
-(LII~ + 0 V1
0
~v~T'+O -
!
L I~)
= 0
(99) (LnI, q- LIz) = O.
In Eqs. 99, L' denotes the slide modulus of the upper half space and L the slide modulus of the lower half space. Equations 97 and 99 constitute a system of 2 (n + 1) homogeneous (Q)
F I G . 5.
(b)
(a) Superposed layers resting on a half space and free at the top. (b) Superposed layers free at top and bottom.
equations for the 2(n + 1) displacement variables U~- and V~.. The characteristic stability equation is obtained by equating the determinant of this system to zero.
Aug., 1963.]
MULTILAYERED CONTINUA IN ELASTICITY
149
For layers resting on a semi-infinite half space and free at the top surface (Fig. 5a), L' = 0. Equations 99 are then replaced by 0Ii --=0
011 -0 VI
O U1 0
- (L.I. 0 U.+I
0
+ LI3
(100)
0
= 0
- (L.I,~ + LIg) = O. 0 V.+I
If the top and bottom surfaces of the layers are both free (Fig. 5b), then L ' = L = 0 and Eqs. 99 become 011 -0 0 U1
011 --=0 0 VI
OI,,
0I, --=0.
-0
(lOl)
A further sirnplification of the general formalism is obtained by introducing a single invariant which corresponds to the potential energy of the complete system. Let d
~q = L f z + Y: L~I~ + L'Iu.
(102)
The stability equations become 009 _ OUj 0
0~
OVy
=
O.
(103)
These 2 (n + 1) equations include all three cases of free and embedded layers formulated by Eqs. 99, 100, and 101. It is of interest to examine the case where the various materials obey the finite stress-strain relations (Eq. 27) of rubber-type elasticity. Here, a system originally.stress-free is brought to an initial state of finite homogeneous strain represented by the extension ratio M and M in the x, y plane. As shown by expressions (29), the parameters m, k2and f are then identical for all materials. They m a y be expressed in terms of a single variable, say Xd/M ~or 3. The only other parameter in the values of I j is % which for the j~l~ layer of thickness hj is written VJ = }lhj.
(104)
If the thickness of one of the layers is chosen as reference, say hi, then -ys = ( h # h l ) v
(10S)
15o
M . A . BlOT
[.L F. I.
where 3" = ½hll. Hence, the values of I~- contain the variable 3" and the ( n - 1) thickness ratios as parameters. Finally, taking into account Eqs. 26 and 27, Lj m a y be written as L~ = ~ojM ~
(106)
where ~0j is the stress-free modulus of the jtl~ layer. Hence, in all Eqs. 103, the c o m m o n factor M 2 cancels out. This brings out the ratios of the moduli /z0j as another set of parameters. W h e n these rigidity ratios are given along with the various thickness ratios, the characteristic equation becomes a relation between the two variables f and y. The first characterizes the initial stress and the second the wave length. T h e m i n i m u m value of f as a function of 3' yields the buckling stress and wave length. The preceding analysis has assumed perfect adherence. If the layers are allowed to slip w i t h o u t friction at the interface, the following condition m u s t be satisfied, T = 0,
(107)
since the tangential stresses m u s t remain zero. The displacements U~. in this case are not continuous. However, they m a y be eliminated from the equations by using condition (107) for perfect slip. This leaves a system of equations containing only the normal displacement Vj at the slipping interface. This normal c o m p o n e n t is continuous. A remark m u s t be made here regarding the continuity of q when such slippage occurs. Since this value is evaluated at the displaced points and since these points do not coincide due to slippage, the values of q are not exactly continuous. However, the difference is a second order quantity, and in a first order theory interracial values of q m a y be assumed equal to each layer. T h e case of perfect slippage is illustrated by an example in a forthcoming paper in this Journal. The recurrence equations derived above contain relatively simple coefficients and constitute a system of linear equations containing not more t h a n six variables at a time. It is possible to formulate the stability problem by a different procedure which m a y have advantages for numerical work with digital computers. The procedure was developed by Haskell for the formally similar problem of wave propagation in multilayered media (14). Equations 59 m a y be written in the form T1
ql lU1 IG
=
gg
q2
lU2 lV2
(108)
Aug., 1963.]
MULTILAYERED CONTINUA IN ELASTICITY
151
The matrix is B1
B.2
LB5
LB6
Ba
B4
-- L B 6
LB7
1
=
1
1 -- ]_.B~
(109)
1 f~B~o
-- B~
B~
The elements B~ are functions only of aCj and b~j. With =
(a~
-
b.)
2 -
(a.
-
bl,)(a..
-
b2~)
(11o)
they are B1
=X
S~
= ~ (a.b12 -
Ba
2 = ~ (alibi2 -- a22bm2)
2
al~biO
1
B4 = ~ [(&2 + B5
=
2 -A
b~2)(all
[a12=b" _ alibi22
-
b11) -
_
_
(a~
-
b12=)]
axlb11(a22 -- b22)]
2 J~6 --~- X E-- G'12512(a12 -- b12) + alla22bi2 -- a12511522~
B~
2 ~a22b22(an = -A
-
-
(111)
511) -~- a22612 ~ -- a122b2,J
2 2 B,
=
-
-a ( ~ "
2 B,o = ~ (a,, -
-
bl~)
b.).
The matrix equation (108) relates the values of the variables at two successive interfaces. Obviously the values at the top and b o t t o m interfaces satisfy the equations
15 2
M.A.
BIOT Tn+l
T1 n
q~ lU1 lV~
[J. F. I.
= II ~; s=~
g~+l 1Un+l
(112)
lV~+l
The product f l ~rcs of the ~Ygj matrices for each layer is a 4 X 4 j=l
matrix. If the top and bottom faces are free, let rl = ql = r,~+1 = q,+1 = 0, and Eq. 112 reduces to two homogeneous equations for the two unknowns U,+~ and V~+I. If the layers lie over a half space, r,,+~ and q,+~ are replaced by their values in terms of U~+, and V,+~ for the half space and are again led to a 2 X 2 system of equations. If the layers are embedded between two half spaces, the values of r~, q~, r~+~, q,+l are expressed in terms of U~, VI and U,~+I, V.~+I. This case results in a system of four equations. In using this procedure attention nmst be called to the case where the wave length is sufficiently small so that the top and bottom faces of a layer become decoupled. This happens when aij ~ b~s.
(113)
In this case the layer may be replaced by a half space. The procedure and formulas proposed here for the mechanics of layered media are quite general and are not restricted to incompressible media or stability problems. The matrix coefficients (Eqs. 111) are given a new form which is immediately applicable to a large category of problems, including wave propagation in anisotropic multilayered media. The recurrence equations (103) or the matrix multiplication process of Eqs. 112 provide a systematic way of programming the numerical work for the solution of problems in the mechanics of layered media when a large number of layers are involved and the use of a large capacity digital computer is available. A variational principle is readily obtained from Eqs. 103, that is, = o.
(114)
In this principle the interfacial displacement U~ and V~ are given arbit r a r y variations. Equation 114 is also a consequence of the physical interpretation of o~ as the total incremental energy of the multilayered system expressed in terms of the interfacial displacements. APPENDIX A brief outline is given here of previous results. The general equations were derived earlier (I, 2, 3, 9). For the case of plane strain and constant body force they were briefly rederived in a more recent paper (4). The results for this case are repeated here. The initial stresses are denoted by $11, $22, Sin. The incremental stresses referred to axes rotated by an angle ¢0 are denoted by sn, sin,s12. They satisfy the equilibrium conditions
At~g., i963.]
153
MULTILAYERED C O N T I N U A IN ELASTICITY
Ost~ Oa~ Ow osl~, + ~yy + p Y,o - 2&2 ~ x + (s1, - &2) ay Ox
+a&l 0s22
Ow
_ ( a s s , + as,~'~
+ as,~
( O&,2 + OS1;~
O&2 +~euu
&o
Ox
O&~
+--~y e ~ , - - \ Ox
oy/e~u
= O.
(At)
The strain components e,~, euu, e~j and the rotation o~ are defined as in Section 2. The body force components X and Y per unit mass are assumed to be constant2 The coordinate system is chosen clockwise. Incremental forces acting at a boundary per unit initial area (that is, per unit area as measured in the state of initial stress) are represented by the following x and y components ,~f~ = (sll -- Sl~w + Silevu - S12e~u) cos (n, x) + (s12 -- S~2w -- Sllexu + Sv2e~z) cos (n, y) af~ = (s~ + &~o, + &~e~ - &2e~) cos (n, x) + (s~2 + &,~o~ - Sl~e.y + &2e~) cos (n, y).
(A2)
These forces are acting on a closed contour in the x, y plane and the normal direction chosen positive outward is designated by n. REFERENCES
(1) M. A. BLOT, "Theory of Elasticity with Large Displacements and Rotations," Proc. 5th Internat. Cong. Applied Mechanics, pp. 117-122 (1938). (2) M. A. BIOT, "Non-Linear Theory of Elasticity and the Linearized Case for a Body Under lnitial Stress," Phil. Mag., Vol. 27, pp. 468-489 (1939). (3) M. A. BLOT, "Elastizit~ts Theorie Zweiter Ordnung mit Anwendungen," Zeit. Ang. Math. und Mech., bd. 20, pp. 89-99 (1940). (4) M. A. BIOT, "Folding of a Layered Viscoelastic Medium Derived from an Exact Stability Theory of a Continuum Under Initial Stress," Quart. Appl. Math., Vol. X V I I , pp. 185-204 (1959). (5) M. A. BIOT, "Instability of a Continuously Inhomogeneous Viscoelastic Half Space Under Initial Stress," Jot3R. FRANKLIN INST., VOl. 270, pp. 190-201 (1960). (6) M. A. BIOT and H. OD~, "On the Folding of a Viscoelastic Medium with Adhering Layer Under Compressive Initial Stress," Quart. A p p l . Math., Vol. XIX, pp. 351-355 (1962). (7) M. A. BIoT, "Surface Instability of Rubber in Compression," Appl. Sci. Res. A, 12 (1963). (On press.) (8) M. A. BIoT, "Exact Theory of Buckling of a Thick Slab," Appl. Scl. Res. A, 12 (1963). (On press.) (9) M. A. BIOT, "The Influence of Initial Stress on Elastic Waves," Y. Appl. Phys., Vol. 11, pp. 522-530 (1940). (10) M. A. BraT, "Internal Buckling Under Initial Stress in Finite Elasticity," Proc. Roy. Soe., Ser. A, Vol. 273, No. 1354, pp. 306-328 (1963). (11) M. A. BraT, "Surface Instability in Finite Anisotropic Elasticity Under Initial Stress," Proc. Roy. Soc., Ser. A, Vol. 273, No. 1354, pp. 329-339 (1963). (12) M. A. BLOT, "Interracial Instability in Finite Elasticity Under Initial Stress," Proc. Roy. Soc., Set. A, Vol. 273, No. 1354, pp. 340-344 (1963). (la) M. A. BraT, "Incremental Elastic Coefficients of an Isotropic Medium in Finite Strain," App. Sci. Res., A, 12 (1963). (On press.) (14) A. N. HASKELL, "Dispersion of Surface Waves in Multilayered Media," Bull. Seism. Soc. Am., Vol. 43, p. 17 (1953). 3 The more general equations are found in (1, 2, 3, 7),