- Email: [email protected]
End fixity effects on the buckling and post-buckling of delaminated composites
Composites Science and Technology 34 (1989) 113-128
End Fixity Effects on the Buckling and Post-buckling of Delaminated Composites G. A. Kardomateas General Motors Research Laboratories, Engineering Mechanics RMB 256, Warren, Michigan 48090-9055, USA (Received 26 October 1987; accepted 27 May 1988)
ABSTRACT A study of the effects ofendfixity (clamped-clamped vs simply-supported) on the buckling and post-buckling of delaminated composites is performed. The study includes the case of a delaminated composite beam-plate on an elastic foundation. First, the analytical solution for the post-buckling behavior of delaminated composites with simply-supported ends, derived via the perturbation technique, is presented. For high values of the foundation modulus the end fixity has little effect on the instability point. For the low values the simply-supported case requires less load for instability. Similarly, the endfixity has 6ttle effect on the slope of the energy release rate vs applied load curves for high values of the foundation modulus. For the low values the simply-supported case results in less steep curves, which suggests that the configuration can withstand more load beyond the eritical point in the initial post-buckling stage before delamination growth takes place.
1 INTRODUCTION Problems of delamination growth in composites have received considerable attention in recent years. Analytical models have been developed that deal with the basic premises of the problem. ~-a An ability to understand the mechanics o f this damage mechanism is useful, not only because delaminations can lead to an undesirable strength or stiffness degradation but also because they m a y have a desirable energy absorption capacity as a result of the relatively large deflections involved. 3 113 Composites Science and Technology 0266-3538/89/$03"50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain
114
G. A. Kardomateas
In a previous article, 3 an explicit quantitative description of the postbuckling behavior was obtained for a one-dimensional delamination in a compressively loaded laminate. In another related study, 4 the buckling and post-buckling behavior was investigated for the case of a laminate on an elastic foundation. This would simulate the compressive face of a composite boxed beam filled with a soft elastic medium such as foam or a sandwich beam consisting of two thin fiber-reinforced sheets separated by a low stiffness core, which are subjected to a bending load. In those studies the case of a clamped-clamped beam-plate was treated. End fixity can influence both the critical and the post-critical characteristics. In this article we shall first present the analytical post-buckling solutions for the case of a simplysupported beam-plate. The results will then be contrasted to those for a clamped-clamped beam. Both the usual configuration and the case of a beam-plate on an elastic foundation will be treated. Our method of solution is based on the asymptotic analysis of post-buckling behavior as a perturbation series with respect to the slope of the section at the delamination interface.
2 ANALYTICAL F O R M U L A T I O N
2.1 Instability modes and governing equations The geometry of the problem is defined in Fig. 1. A homogeneous, orthotropic beam-plate of thickness T, length L, and unit width, containing a strip delamination at a depth H ( H < T/2) from the top surface of the plate, is subjected to an axial compressive force, P, at the ends. The plate is simply supported. In the general case there may be a Winkler-type elastic foundation attached. The delamination extends over the interval 0 < x < l = 2a. Over this region the laminate consists of two parts, the part above the delamination ('upper' part, of thickness H) and the part below the delamination ('lower' part, of thickness T - H ) . The remaining laminate outside the delamination interval and of thickness T is referred to as the 'base' laminate. The coordinate systems for the separate parts are shown in Fig. 2. For each part we denote by Di its bending stiffness: Et.3,
Di = 12(1 - -
•13•31)
E being the modulus of elasticity along the x axis and ti being the thickness of the corresponding part; i = u (upper), 1(lower) or b (base) parts. Although we assume one value for the modulus throughout the composite, different values for the different parts could be considered.
Buckling and post-buckling of end-fixed delaminated composites
I~ __t'~
~-~-
L ~= ~
"l ~.-'~i=
"~
14 P~
L ~
~
~/~~.~L~}~. Fig. 1.
115
~] P
~.,~.L
Definitionof the delamination bucklinggeometry.The systemmay includean elastic foundation.
Three possible modes of instability can take place, 3 depending on the geometric configuration. Global buckling of the whole beam m a y occur before any other deflection pattern takes place, typically for small delaminations. The critical load for this limiting mode can be found from an energy formulation. ~ In terms o f an integer, m, that is specified so as to
• ~
L
u
~
_
~
I
y
x .
,
.
Me \P
e
/
~
~ -.....~~-~,
~
Y Fig. 2. Definition of the coordinate systems for the different parts.
G. ,4. Kardomateas
1 16
render the load a minimum, the global instability load for the simplysupported case that assumes a mode shape, y = am sin (mnx/L), is
pglo_rC2Db ( L2
flL4 ~ m ~ + mZr~4Db,]
(1)
where/3 is the modulus of the foundation. F o r example, for/3L4/(n*Db) < 4 then m = 1, for 4
(2)
considered next), transverse deflections for as well as the base plate may occur. the deflections of the lower (i = 1) and base plate can be written 5
__
d2yi
i dx4
dx 2
D d~yi + Pi
-
/3Yi
(3)
For the upper part (i= u) we should set/3 = 0. This equation is solved in conjunction with the following conditions: (1)
a condition of c o m m o n deflection, ~, and of equal slope at the interface section (where the delamination starts) (Fig. 2): Y.lx=o.~ = Y,lx=o,~ = Y~lx=to -- ~ t
(4a)
t
Y'.lx = o = Y,lx= o = Y~I~.= to (2)
(4b)
the end fixity condition for the base plate: Y~lx=o = Y£1x=o = 0
(3)
(5)
axial and shearing force and moment equilibrium at this section: Pu + P, = Pb = P
Vu + V~= V~
(6a)
M u + M, + P u ( T - n ) / 2 - P~H/2 = M~ (4)
(6b)
a condition of compatible shortening of the upper and lower parts: Pul (1 -- vx3v31) ~
, .~ P~I ~ (1 -- v~3v3~) E ( T - - H )
ill
y; dx-
ill
y', dx=
' 0 TY.x=
17)
Buckling and post-buckling of end-fixed delaminated composites
117
2.2 Solution procedure and buckling load The perturbation method is used to solve the problem defined above. The pre-buckling state of stress is a state of pure compression: Yi,o : 0
Pu.o = PoH/T
Mi, 0 : 0
Vi, 0 = 0
P,,o = Po( T - H)/T.
P~,o = Po
Let the angle at the interface of the delaminated and base plate be denoted by 49 (Fig. 1). The deflection and load quantities at each part, yi(x), Pi, Vi, Mi, are developed into ascending perturbation series with respect to 49:
Yi(x) = 49Yi,1(x) + 492yi,2(X) + " " M i = 49Mi, 1 + 492M,, 2 + " "
Pi = Pi,o + 49Pia + 492Pi,2 + " " (8a) Vi = 49Vi, ~ + 492 V~,2 + ' " (8b)
By the definition of the series, at this interface
Y'i,~ = 1
Y'i,2 = Y~,a . . . . .
0
(9)
Substituting eqns (8) into the differential equation (3) and equating like powers of 49 leads to a set of linear differential equations and,boundary conditions for each part. In the first approximation the terms in the first power of 49 are equated. The solution of the first-order equatior~ for the upper and lower part can be found in Refs 3 and 4. In the following we give the solution for the base plate. The corresponding equation is d*yb, a d 2Y~,l . h~--d-~x4 + Po--d-~x2 +flyb, 1 = 0
(lOa)
Define k 2b,o = eb,o/D
=
10b)
The solution can be expressed in terms of trigonometric functions only or a combination of trigonometric and hyperbolic functions depending on the magnitude of the modulus of the foundation. For k~4,o>42~ the solution to the first-order problem equation (10a) satisfying the end fixity conditions of eqn (5) is given by
yb,l= ~
C~jsinfjx
(11)
j=l,2
where 6j are defined by 6~,2 = x / l ( - k~2,o_+ ~/k~,o -42~)/21
¢12)
118
G. A. Kardomateas
In terms of Q = ~j
(-1)~6~ sin 63 _flo COS6flo
(13a)
j=l,2
the constants C11, C~z are found from eqns (4a) and (9) to be C1j = (-- 1}/(sin 63 -rio -- ~163 - j cos 63_fl0)/Q
(13b)
The first-order end shear can be expressed in the form ,,, + kb,oYb,1)lx=to 2 , = V5 Vb,1 = --Db(Yb,1 b,1
r ~1 V,b,1
(14)
V~, 1 = (Db/Q)6~6 ~ ) j ( - 1}i6~ sin 6flo cos 63_ri0
(15a)
"~
where
j=1,2
V'fb,1 (Db/Q)6~6~(6z~ - 6~)cos 6~l o cos 6~l o =
(15b)
The first-order m o m e n t at the delamination section can be similarly expressed as M~,,I
=
-
D
" c ~,Y~,,~ ~,=~o= M~,,~ + ~M~,~
(16)
where
M~, 1 = (Db/Q)(6 ~ - 6~) sin 6,lo sin 62/o
2
b,1 = (Db/Q)6162
( - 1) 3 -J6~ sin 6flo cos 63 -~lo
(17a) (17b)
j=l,2
For k~4,o< 42~ the solution for the first-order equation (10a) is found in terms of 61 and 62 defined as r= ~ 61 = x/~ cos (0/2)
0 = arccos [ - kZ~,o/(2r)] 62 = x/~ sin (0/2)
(18a) (lSb)
to be Yb, ~ = C 11 sinh 61 x cos 62 x "-~ C 12 cosh 61 x sin 62x
(19)
Q = (62 sinh 261/o - 61 sin 262/o)/2
(20a)
By setting the constants Clj are obtained from eqns (4a) and (9) as follows: C11 = [(1(62 cosh 611o cos 62l o + 61 sinh 6~lo sin 62/o) - cosh 611o sin 62lo]/Q
(20b)
C12 = [(1(62 sinh 611o sin 62/o - 61 cosh 61lo cos 62/o) + sinh 6~lo cos 621o]/Q
(20c)
Buckling and post-buckling of end-fixed delaminated composites
119
The terms defining the first-order shear in eqn (14) are given in this case by V5b , 1
~
- D ~ ( 3 ~ + 3z2)(3~ sin 232• 0 + 32 sinh 26~lo)/(2Q)
V~fA= (DdQ)(3~ + 322)23t32(sinh 2 3~lo + cos 2 32lo)
(21a) (21b)
and the terms for the first-order end moment in eqn (16) are given by M~,, ~ = - (Db/Q)26 t 62( sinh2 6 t lo + sin 2 6210) M~. x = D~(3~ + 3~)(6~ sinh 23x/o + 3~ sin 232lo)/(2Q)
(22a) (22b)
2.2.1 Characteristic equation
The condition of shear equilibrium (6a), written for the first-order terms, produces an expression for (1 (note that VuA = 0): ~1 = (Vie,1- V~,l)/(Vl~.l- Vlf.1)
(23)
The associated equilibrium and compatibility equations (6b) and (7) up to the order q~ are given in terms of the moments at the interface: M.,~ + Mr, 1 -- M ~ A = P , , I H / 2 -- P . , ~ ( T - H ) / 2 T
EH(T-
H)
4a(1 -- vt3v31 )
= Pu, x ( T - H ) / 2 -- e l , ~ H / 2
(24) (25)
The end moments, Mi, ~ = -Diy~'A, are given from eqns (16) and (22) in terms of the zero subscript quantities and the quantity ~ which was found in eqn (23). Substituting these expressions into the above two equations, and eliminating the quantity P t A H / 2 - P , ~ , I ( T - - H ) / 2 , gives an equation for the critical buckling load, Po, as follows: Duke, o cot k~,oa + M 1,1 ¢
-
-
¢ - Vd.a)(M,A ¢ r - M~,A)/(V~,., f r -- VIA r ) M bc, , + (V~,, = -- T E H ( T - - H)/[4a(1 - vx3v3x)] (26)
where the quantities V/,~I, V/fA,M ~ and M ~ , i = 1, b, in the above equation are given in eqns (15), (21), (17) and (22). This is the characteristic equation for the critical instability load. 2.2.2 Case o f no elastic f o u n d a t i o n
The solution to the first-order equation (10a) for fl = 0 is sin kb,oX Yb,1 = k~,o cos k~,olo k~, o = Po/D~
l o = ( L -- l)/2
(27a) (27b)
The characteristic equation is again derived by eliminating the quantity
120
G. A. Kardomateas
P~,~H/2- P u , ~ ( T - H ) / 2 in eqns (24) and (25), and is found to be (see also Ref. 3)
H3k~,,o cot (k,,,ol/2) + ( T- H)3k,,o cot (k~,ol/2)
-- T3kb,o tan kb,olo + 6 T H ( T - H)/I = 0
(28)
2.3 Post-buckling solution and delamination growth characteristics When the terms in ~b2 are equated, the second-order differential equations are obtained. The solution for the upper and lower parts can be found in Refs 3 and 4. The solution for the base plate, which is simply supported, will be developed in the following. The differential equation is expressed as tt
v¢
~'b.Yb,/-I ,,(4)2 "q- PoYb,2+
flYb,2
=
--Pu,~Yb,~
(29)
The solution to the above equation is a superposition of the general solution and a particular solution. For k~4,o> 42~,the solution to the second-order equation (29) satisfying the end fixity conditions (5), with 6~. defined in eqn (12), is found to be Yb,2
~
P1 C2J~b sin 6ix +
=
P1
B2j--xcos6JXDb
(30a)
j=l,2
(-- 1)Jc~joj B~j - 2(62~ _ 6~)
(30b)
The constants C2j are determined from the conditions (4a) and (9). In terms of Q defined by eqn (13a) and A~ = ) ,
B2jlocosfjl o
(31a)
B2j(lo6j sin 6flo -- cos 6jlo)
(31b)
j=l,2
A 2 = ~, j=l,2
these constants are given by Czj = Q - ~[~2(Db/P063 - j cos 33 -rio -- A163 -j cos 33 -rio + Az sin 33 -rio] (31c)
The second-order end shear can be written in the form ttt
2
t
Vb,2 -= --Db(yb,2 + k~,oY~,z)Jx=to - P~ Vbg,2 "-1-(2 Vbf,2
(32)
Buckling and post-buckling of end-fixeddelaminatedcomposites
121
where
V.b,2 ~ = --(A I/B)6a62(62~ + 6~)cos 61lo cos 62lo + ~,
(A2/Q)6 ~ sin 63 -rio cos 6flo
j=l,2
+ B2./6](3 cos 6flo - lo6./sin 6fl~)
V.b,2 f = (D~,/Q)6~62(62~ + 6~) cos 61lo cos 6210
(33a) (33b)
The second-order end moment can also be written in the form
M~,,2 = -- D bYb,2 " x=to = P1Mg,2 + (2M~f.2
(34)
where
M~,.2 = (A2/Q)(6~ + 6~) sin 61l o sin 62lo B2~6~(2sin 6flo + lo6./cos 6/0)
+ 2 ./=1,2
-- (A 1/Q)6f63 _./sin 6/0 cos 63 -rio
Mfb'2= 2
(Do/B)6]63 -j sin 6slo cos 63_rio
(35a) (35b)
j=l,2
For k~,o < 42 b the second-order solution satisfying eqns (29) and (5) is found to be P1
Yb.2 = C21 ~
PI
sinh 61xcos dizX + C22 - ~ cosh 61X sin 62x
PI P1 + B2~ - - x cosh 61x cos 62x + Bz2 - ~ x sinh 61x sin 62x D~
B2./ =
- - [ C 1163
_j + ( - 1)Jc126j]/(861(~2)
(36a) (36b)
where 6./are defined in eqns (18). The constants C2~ and C22 are found from eqns (4a) and (9). In terms of Q defined by eqn (20a) and
A 1 = B211o cosh 61lo cos 62l 0 + B2fl o sinh 6110 sin 62l0 A2 = B21 cosh 61lo cos 62/o + (B2161 + B2262)l o sinh 61lo cos 6zl o + (B2261 -- B2162)locosh6~lo sin6210 + B22 sinh611osin6210
(37a) (37b)
these constants are given as follows: C21 = Q- 1{A2 cosh 61lo sin 62lo + [(2(D~/P1) -- A 1] x (61 sinh 61lo sin 62/o + 62 cosh 6~1o cos 62/o)}
(37c)
122
G. A. Kardomateas
Cz2 = Q - I {[~2(Db/PI) -- A 1](82 sinh 8110 sin 8zl o - 81 cosh 81lo cos 8~lo) - A 2 sinh 81l o cos 82lo} (37d)
The second-order end shear is expressed by eqn (32) with the definitions V,b,2 ~ = Q - 1[A2(812 + 82~)(81 sin 28zl o + 82 sinh 281lo)/2 --.A 1(812 + 8~)28~S2(sinh 2 8~lo + cos 2 82/o)] + [O21(8~ - - 812) - - 6B228182] cosh 81lo cos 82/0 + [B22(8 ~ - 8~) + 6B2~8~82] sinh 8~lo sin 821o - (82~ + 82~)lo[(B2282 - B 2180 sinh 8~1o cos 82l o (Bzx82 + B2281)cosh 811o sin 82lo]
(38a)
V~.2 = D~(812 + 822)Z8182(sinh 2 811o + cos z 8zlo)/Q
(38b)
-
Likewise, the second-order end m o m e n t is given by eqn (34) with the definitions M~,2 = Q-l[A228182(sinh2 81lo + sin 2 82/o) - A~(812 + 8~)(8~ sin 282l o + 82 sinh28flo)/2 ] + [B2~(82~ -- 812) -- 2B2~6~82]l o cosh 6~1o cos 82/o + [B22(8 ~ - 812)+ 2B~6~82]lo sinh 81l o sin 82l o - 2(B 2181 + Bz282) sinh 6~lo cos 82/o + 2(Bz~82 - B2281) cosh 81/o sin 6flo
(39a)
M b,2 r = Db(812 + 8~)(81 sin 282/o + 82 sinh 28flo)/(2Q)
(39b)
2.3.1 Solution The shear condition (6a) allows determining ( 2 in terms of the first-order (yet unknown) forces (note that Vu,2 = Pu.1) as follows:
K c -- 1 V5 -- V5 ~2 - V,rb'z Pu 1 + b,2 ~,2 p~ 1 1,2
-
V. r
b,2
,
V,r
1,2
-
V, f
b,2
(40)
,
The m o m e n t equilibrium equation (6b) for the second-order terms is M..~ + M,.~ - M~,~ = e , . ~ / 2 - ~ . . ~ ( r - g ) / 2
(4~)
while the geometric compatibility equation (7) for the second-order terms is given by l f~ y .,z, ~ d x - ~ l fo Y'~Yld x = 2(1-vl3v31)l [ P , , 2 H / 2 - P . ~ "( T - H ) / 2 ] EH(T-H)
~
(42)
Notice that the second-order moments, M~,z = -D~y~',z, at the interface are given from eqns (34), (35) and (39) in terms of the (yet undetermined) firstorder end forces. Thus eliminating the quantity P ~ , ~ H / 2 - P . , z ( T - H ) / 2 from eqns (41) and (42), and taking into account eqn (40), gives the following
Buckling and post-buckling of end-fixed delaminated composites
123
equation for the first-order forces PuA and PLI: [-Mu~,2 -- M~,2 + ( V~g,2 - 1)(M~ 2, -- Mb,z)/(f V'f,,2 -- V~,2)]Pu,1 + [m,g,2 - M ' b.2 + (V,b.2 g - V~g,2)(Mif,2 -M~,.z)/(VL2f f V~2)]P,
(2/¢o,0a-_sin_Zko,oa =\
4ku.osinZk..oa
.
E/-m'- m
-a.,)4a(l_Vl~V~O
(43)
The second equation needed for finding P.,1 and P~.I is the first-order equilibrium equation (24) at the interface, namely
P~, ~H/2 - P.. I ( T - H)/2 = Fo( P., o, P~.o)
(44)
where Fo is the left-hand side of eqn (26) which depends only on the zeroorder quantities. In the above equations V~,~2,V[.2,M~2 and M[.z, i = 1, b, take the values given in eqns (33), (38), (35) and (39) depending on the relative magnitude of the foundation modulus; S~ is the shortening of the lower part due to first-order deflections and is given in Refs 3 and 4. The above system of linear equations allows finding Pu,1 and P~,I, and hence the first-order applied end force PI = Pu, 1 + PL 1.
2.3.2 Case of no elastic foundation The case of fl = 0 admits as a solution to the second-order equation: Pb,1 Yb.2 = cb,2 sin kb.oX + 2DbkZb.O cos kb,ol o x cos kb.oX
(45)
and, from eqns (9), Pb, l
Cb.2 = 2Dbk3~.o cos2 (kb.olo) (Iokb. o sin kb.ol o - cos kb.olo)
(46)
Again, eliminating the quantity P m H / 2 - Pu.2(T-H)/2 from eqns (41) and (42) gives the following equation for the first-order forces Pu.l and Pt.l:
p, F cos(kuol/2) lcos2(ku.ol/2) "lL2k~.o-S~n~/2)-4sin2(ku,ol/2 ) -
-
~ [- cos (k, o//2) + r~'l L z k ~ 2 )
l 4
sin kb,o/o 2kb. 0 cos kb,ol o /cos 2 (k~ o//2) 4 sin 2 (k,',ol/2)
-
-
sin kb.ol o 2kb, o cos kb,ol o
-sin (k~,o l) - k,.ol = _kt, ° sin 2 (kLol/2)
1o sin 2 kb.olo 2 cos 2 kb.ol o
lo7
l°sin2kb'°l° 2 cos 2 kb.ol o
l~1
l 4
sin (k u ol) --.ku ol] E H ( T - H) ~ J 8/(1 - - V 1 3 Y 3 1 )
(47)
124
G. A. Kardomateas
The second equation needed for finding Pu,1 and P~.I is the first-order equilibrium equation (24) at the interface, namely Pu,o
P,,o
Po t a n kb,ol o
P~.~H/2 - P u , I ( T - H)/2 - ku.o tan(ku,ol/2) ~- k~, o tan (k~.ol/2)
kb, o
(48) The above system of linear equations allows the finding of Pu, ~ and P~,~,and hence of the first-order applied force P~ = Pu.~ + P~.~. 2.3.3 Growth characteristics
The post-buckling solution that has been obtained can be used to study the post-critical state of deformation. In addition, the initiation of delamination growth can now be analyzed on the basis ofa Griffith-type fracture criterion. Predicting whether the delamination will grow requires an evaluation of the energy release rate. This quantity, which is the differential of the total potential energy with respect to the delamination length, can be easily calculated by using the path-independent J-integral expression in terms of the axial forces and bending moments acting across the various crosssections adjacent to the tip of the delamination. 6 In terms of the quantities P* = P ( H / T ) -
Po
M* = M u
M * * = P* T/2 - M *
(49)
the energy release rate per unit width is expressed as G = 12(1- v13v3, ) {p.2 + 1 2 ( M . / H ) 2 p . Z + 1 2 [ M * * / ( T - H ) ] 2} 2E H + T- H
(50)
The J-integral method that is used here does not distinguish between the Mode I and II components. It should be noted that future work should consider this question, since there may be a dependence of the fracture toughness on the ratio of those two components.
3 DISCUSSION OF N U M E R I C A L RESULTS Numerical examples are presented for the case of T / H = 6, L / H = 200. The characteristic equation (26) is solved for the critical load Poc. If Poc < (Pg~o,P~o~),with Pg~oand P,o~ given by eqns (1) and (2), then combined 'mixed' buckling involving out-of-plane deflections for both the delaminated layer and the base plate takes place. Otherwise the critical load is the minimum of the above, corresponding to global buckling (typical of very short delaminations) or local buckling (typical of very large delaminations). As the above analysis shows, the conditions of end attachment, by affecting the
Buckling and post-buckling of end-fixed delaminated composites
,-~ ~
(
~ = 10
125
c-c s-~
,~
~ 2
,3=0
.E ~ 0 0
~ 20
~0
60
D~,l~mitmtion Le~gth. ~
80
~00
~./11
Fig. 3. Buckling load vs delamination length for T/H=6, L ~ H = 2 0 0 and a set of foundation moduli, for both the simply-supported (solid line)and clamped-clamped (dashed line) cases. The usual case without an elastic foundation corresponds to ~ = 0.
equations that describe the deformation of the base plate, influence the performance of the whole system. To illustrate the effect of end fixity, the variation of the critical load, normalized with respect to the Euler load for the delaminated layer,/~o = Pol2/(4n2Du), vs delamination length, ~'= I/H, for both the simply-supported and clamped-clamped cases, is given in Fig. 3 for a set of values for/~. The foundation modulus is normalized as /~= 3131'~/(16n4Db). It can be concluded that for high values of the foundation modulus/~ the end fixity has a small effect on the instability point. This is because in these cases the instability is mostly controlled by the local buckling mode as opposed to the global one that is more pronounced for low values of/~. Notice that in both cases, as/~increases, the curves are shifted to the left, indicating the attainment of loads similar in magnitude to the local buckling ones for smaller delamination lengths. The combined effect of the location of the delamination through the thickness for different foundation moduli and the end fixity is indicated in Table 1, which gives the values of the delamination length, I/L, for which the characteristic equation has no roots less than the local buckling load given by eqn (2) (the 'local buckling threshold delamination lengths'). As the delaminated layer becomes thinner, local buckling is reached at smaller lengths, and an increase in the foundation modulus reduces this 'threshold length' more effectively. For example, for H/T= 1/15 and /~> 10 4, local buckling occurs for practically all delamination lengths. In addition, local buckling occurs in general at smaller delamination lengths in the clampedclamped case than in the simply-supported one. It should also be noted that, for even smaller lengths than the ones given in Table 1, the solution of the
126
G. A. Kardomateas
TABLE 1
Values of Delamination Length, l/L, beyond which the Characteristic Equation has no Roots less than the Local Buckling Load ('Local Buckling Threshold') H/T a
~ 0
1
10
10 ~
10 ~
10 ~
1/6 (C C) (S-s)
0.966 1.000
0.963 1.000
0.961 1.000
0.951 1.000
0.816 0.844
0.511 0.512
1/9 (C C) (S-s)
0.943 1.000
0.936 1.000
0.921 1.000
0.759 0.784
0.516 0.516
0.291 0.291
1/12 (C-C) (S-s)
0.603 1.000
0.471 0.475
0.279 0"288
0.161 0"161
0.089 0.089
0-05 I 0-051
1/15 (C-C) (S-s)
0.451 1.000
0.277 0"279
0.166 0" 169
0'096 0"096
0.054 0-054
0-031 0'031
"C C: Clamped-Clamped; S-s: Simply-supported.
0.307
,~-~ ~ .~
q3 II ~
0.310
0.313 0.316 0.319 0.322
~
0.305 o.o~
0.308 ~
~
J
I
I
0.311
0.314
0.317
l
~
l
~
I
~
1
0.325 ~ = 10, S-S, C-C ,
0.320 ~ .
I
]
0.323 ~
~ = 01 0.1, (~-C
~
= lo
~--
o,~
o.o~
¢~.. S-S
~
~
0.02
0.00 0.24~ i
0.302
I
0.2~ I
0.305
I
I
0.251 I
0.308
I
0.2~ *
I
0.311
0.2~ ,
I
0.314
I
'
i
'
0~ '
I
~
0.263 ~ :
0.317
I
0.320
v~
S-S
~
= O. 1, S-S
Applied Load, /5, = pL2/(4rc2D~,) Fig. 4. Strain energy release rate vs applied compressive force during the initial postbuckling stage for both the simply-supported (solid line) and clamped-clamped (dashed line) cases ( T / H = 6. L / H = 200, and delamination length I/H = 60). The different load scales are of the same length and correspond to the different initial buckling loads.
Buckling and post-buckling of end-fixed delaminated composites
127
characteristic equation is very close to (by less than 1% smaller than) the local buckling load, as also is evident from Fig. 3. To illustrate the effect of end fixity on the post-critical characteristics, the variation during the initial post-buckling stage of the normalized strain energy release rate, t7 = G/(ETS/L'*), vs the applied load normalized with respect to the Euler load for the entire beam with no elastic foundation, P = pL2/(4r~2D~,), is plotted in Fig. 4 for the example case of T / H = 6, L / H = 200, and delamination length l / H = 60, for a set of values of the foundation modulus. Both the simply-supported and clamped-clamped cases are considered. Since the critical load changes with/~, different scales (of the same length) are used on the load axis. As for the critical load, the end fixity does not much affect the slope of the G - P curves for the high values of the foundation modulus,/~. For the low values of/~ the simply-supported case results in less steep curves. This decreased slope means that (1) the laminate can withstand more load beyond the critical point in the initial postbuckling stage before delamination growth takes place, and (2) there could be less energy absorbed since growth would not be promoted as much. Notice also that for both cases the curves are steeper for a larger/~.
4 CONCLUSIONS The effects of end fixity (clamped-clamped vs simply, supported) on the buckling and post-buckling of delaminated composites under compressive loads are investigated. First, the perturbation technique is used to derive the analytical solution for the post-buckling behavior of delaminated composites with simply-supported ends. The results are compared with the characteristics of the clamped-clamped case. Both the usual case and that of a delaminated composite beam-plate on an elastic foundation are treated. End fixity effects are found to be more pronounced for lower values of the foundation modulus.
REFERENCES 1. Chai, H., Babcock, C. D. & Knauss, W. G., One-dimensional modelling of failure in laminated plates by delamination buckling. Int. J. Solids Structures, 17 (1981) 1069. 2. Yin, W.-L., Sallam, S. N. & Simitses, G. J., Ultimate axial load capacity of a delaminated beam-plate. AIAA Journal, 24 (1986) 123. 3. Kardomateas, G. A. & Schmueser, D. W., Effect of transverse shearing forces on buckling and postbuckling of delaminated composites under compressive loads.
128
G. A. Kardomateas
Proceedings 28th SMD A 1A A/A SME/A SCE/A HS Conf , Monterey, CA, 1987, p. 757; also in press, AIAA Journal, 26 (1988) 337. 4. Kardomateas, G. A., Effect of an elastic foundation on the buckling and postbuckling of delaminated composites under compressive loads. J. Appl. Mech., ~ (1988) 238. 5. Timoshenko, S. P. & Gere, J. M., Theory of Elastic Stability, McGraw-Hill, New York, 1961. 6. Yin, W. L. & Wang, J. T. S., The energy-release rate in the growth of a onedimensional delamination. J. Appl. Mech., ~1 (1984) 939.
End Fixity Effects on the Buckling and Post-buckling of Delaminated Composites G. A. Kardomateas General Motors Research Laboratories, Engineering Mechanics RMB 256, Warren, Michigan 48090-9055, USA (Received 26 October 1987; accepted 27 May 1988)
ABSTRACT A study of the effects ofendfixity (clamped-clamped vs simply-supported) on the buckling and post-buckling of delaminated composites is performed. The study includes the case of a delaminated composite beam-plate on an elastic foundation. First, the analytical solution for the post-buckling behavior of delaminated composites with simply-supported ends, derived via the perturbation technique, is presented. For high values of the foundation modulus the end fixity has little effect on the instability point. For the low values the simply-supported case requires less load for instability. Similarly, the endfixity has 6ttle effect on the slope of the energy release rate vs applied load curves for high values of the foundation modulus. For the low values the simply-supported case results in less steep curves, which suggests that the configuration can withstand more load beyond the eritical point in the initial post-buckling stage before delamination growth takes place.
1 INTRODUCTION Problems of delamination growth in composites have received considerable attention in recent years. Analytical models have been developed that deal with the basic premises of the problem. ~-a An ability to understand the mechanics o f this damage mechanism is useful, not only because delaminations can lead to an undesirable strength or stiffness degradation but also because they m a y have a desirable energy absorption capacity as a result of the relatively large deflections involved. 3 113 Composites Science and Technology 0266-3538/89/$03"50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain
114
G. A. Kardomateas
In a previous article, 3 an explicit quantitative description of the postbuckling behavior was obtained for a one-dimensional delamination in a compressively loaded laminate. In another related study, 4 the buckling and post-buckling behavior was investigated for the case of a laminate on an elastic foundation. This would simulate the compressive face of a composite boxed beam filled with a soft elastic medium such as foam or a sandwich beam consisting of two thin fiber-reinforced sheets separated by a low stiffness core, which are subjected to a bending load. In those studies the case of a clamped-clamped beam-plate was treated. End fixity can influence both the critical and the post-critical characteristics. In this article we shall first present the analytical post-buckling solutions for the case of a simplysupported beam-plate. The results will then be contrasted to those for a clamped-clamped beam. Both the usual configuration and the case of a beam-plate on an elastic foundation will be treated. Our method of solution is based on the asymptotic analysis of post-buckling behavior as a perturbation series with respect to the slope of the section at the delamination interface.
2 ANALYTICAL F O R M U L A T I O N
2.1 Instability modes and governing equations The geometry of the problem is defined in Fig. 1. A homogeneous, orthotropic beam-plate of thickness T, length L, and unit width, containing a strip delamination at a depth H ( H < T/2) from the top surface of the plate, is subjected to an axial compressive force, P, at the ends. The plate is simply supported. In the general case there may be a Winkler-type elastic foundation attached. The delamination extends over the interval 0 < x < l = 2a. Over this region the laminate consists of two parts, the part above the delamination ('upper' part, of thickness H) and the part below the delamination ('lower' part, of thickness T - H ) . The remaining laminate outside the delamination interval and of thickness T is referred to as the 'base' laminate. The coordinate systems for the separate parts are shown in Fig. 2. For each part we denote by Di its bending stiffness: Et.3,
Di = 12(1 - -
•13•31)
E being the modulus of elasticity along the x axis and ti being the thickness of the corresponding part; i = u (upper), 1(lower) or b (base) parts. Although we assume one value for the modulus throughout the composite, different values for the different parts could be considered.
Buckling and post-buckling of end-fixed delaminated composites
I~ __t'~
~-~-
L ~= ~
"l ~.-'~i=
"~
14 P~
L ~
~
~/~~.~L~}~. Fig. 1.
115
~] P
~.,~.L
Definitionof the delamination bucklinggeometry.The systemmay includean elastic foundation.
Three possible modes of instability can take place, 3 depending on the geometric configuration. Global buckling of the whole beam m a y occur before any other deflection pattern takes place, typically for small delaminations. The critical load for this limiting mode can be found from an energy formulation. ~ In terms o f an integer, m, that is specified so as to
• ~
L
u
~
_
~
I
y
x .
,
.
Me \P
e
/
~
~ -.....~~-~,
~
Y Fig. 2. Definition of the coordinate systems for the different parts.
G. ,4. Kardomateas
1 16
render the load a minimum, the global instability load for the simplysupported case that assumes a mode shape, y = am sin (mnx/L), is
pglo_rC2Db ( L2
flL4 ~ m ~ + mZr~4Db,]
(1)
where/3 is the modulus of the foundation. F o r example, for/3L4/(n*Db) < 4 then m = 1, for 4
(2)
considered next), transverse deflections for as well as the base plate may occur. the deflections of the lower (i = 1) and base plate can be written 5
__
d2yi
i dx4
dx 2
D d~yi + Pi
-
/3Yi
(3)
For the upper part (i= u) we should set/3 = 0. This equation is solved in conjunction with the following conditions: (1)
a condition of c o m m o n deflection, ~, and of equal slope at the interface section (where the delamination starts) (Fig. 2): Y.lx=o.~ = Y,lx=o,~ = Y~lx=to -- ~ t
(4a)
t
Y'.lx = o = Y,lx= o = Y~I~.= to (2)
(4b)
the end fixity condition for the base plate: Y~lx=o = Y£1x=o = 0
(3)
(5)
axial and shearing force and moment equilibrium at this section: Pu + P, = Pb = P
Vu + V~= V~
(6a)
M u + M, + P u ( T - n ) / 2 - P~H/2 = M~ (4)
(6b)
a condition of compatible shortening of the upper and lower parts: Pul (1 -- vx3v31) ~
, .~ P~I ~ (1 -- v~3v3~) E ( T - - H )
ill
y; dx-
ill
y', dx=
' 0 TY.x=
17)
Buckling and post-buckling of end-fixed delaminated composites
117
2.2 Solution procedure and buckling load The perturbation method is used to solve the problem defined above. The pre-buckling state of stress is a state of pure compression: Yi,o : 0
Pu.o = PoH/T
Mi, 0 : 0
Vi, 0 = 0
P,,o = Po( T - H)/T.
P~,o = Po
Let the angle at the interface of the delaminated and base plate be denoted by 49 (Fig. 1). The deflection and load quantities at each part, yi(x), Pi, Vi, Mi, are developed into ascending perturbation series with respect to 49:
Yi(x) = 49Yi,1(x) + 492yi,2(X) + " " M i = 49Mi, 1 + 492M,, 2 + " "
Pi = Pi,o + 49Pia + 492Pi,2 + " " (8a) Vi = 49Vi, ~ + 492 V~,2 + ' " (8b)
By the definition of the series, at this interface
Y'i,~ = 1
Y'i,2 = Y~,a . . . . .
0
(9)
Substituting eqns (8) into the differential equation (3) and equating like powers of 49 leads to a set of linear differential equations and,boundary conditions for each part. In the first approximation the terms in the first power of 49 are equated. The solution of the first-order equatior~ for the upper and lower part can be found in Refs 3 and 4. In the following we give the solution for the base plate. The corresponding equation is d*yb, a d 2Y~,l . h~--d-~x4 + Po--d-~x2 +flyb, 1 = 0
(lOa)
Define k 2b,o = eb,o/D
=
10b)
The solution can be expressed in terms of trigonometric functions only or a combination of trigonometric and hyperbolic functions depending on the magnitude of the modulus of the foundation. For k~4,o>42~ the solution to the first-order problem equation (10a) satisfying the end fixity conditions of eqn (5) is given by
yb,l= ~
C~jsinfjx
(11)
j=l,2
where 6j are defined by 6~,2 = x / l ( - k~2,o_+ ~/k~,o -42~)/21
¢12)
118
G. A. Kardomateas
In terms of Q = ~j
(-1)~6~ sin 63 _flo COS6flo
(13a)
j=l,2
the constants C11, C~z are found from eqns (4a) and (9) to be C1j = (-- 1}/(sin 63 -rio -- ~163 - j cos 63_fl0)/Q
(13b)
The first-order end shear can be expressed in the form ,,, + kb,oYb,1)lx=to 2 , = V5 Vb,1 = --Db(Yb,1 b,1
r ~1 V,b,1
(14)
V~, 1 = (Db/Q)6~6 ~ ) j ( - 1}i6~ sin 6flo cos 63_ri0
(15a)
"~
where
j=1,2
V'fb,1 (Db/Q)6~6~(6z~ - 6~)cos 6~l o cos 6~l o =
(15b)
The first-order m o m e n t at the delamination section can be similarly expressed as M~,,I
=
-
D
" c ~,Y~,,~ ~,=~o= M~,,~ + ~M~,~
(16)
where
M~, 1 = (Db/Q)(6 ~ - 6~) sin 6,lo sin 62/o
2
b,1 = (Db/Q)6162
( - 1) 3 -J6~ sin 6flo cos 63 -~lo
(17a) (17b)
j=l,2
For k~4,o< 42~ the solution for the first-order equation (10a) is found in terms of 61 and 62 defined as r= ~ 61 = x/~ cos (0/2)
0 = arccos [ - kZ~,o/(2r)] 62 = x/~ sin (0/2)
(18a) (lSb)
to be Yb, ~ = C 11 sinh 61 x cos 62 x "-~ C 12 cosh 61 x sin 62x
(19)
Q = (62 sinh 261/o - 61 sin 262/o)/2
(20a)
By setting the constants Clj are obtained from eqns (4a) and (9) as follows: C11 = [(1(62 cosh 611o cos 62l o + 61 sinh 6~lo sin 62/o) - cosh 611o sin 62lo]/Q
(20b)
C12 = [(1(62 sinh 611o sin 62/o - 61 cosh 61lo cos 62/o) + sinh 6~lo cos 621o]/Q
(20c)
Buckling and post-buckling of end-fixed delaminated composites
119
The terms defining the first-order shear in eqn (14) are given in this case by V5b , 1
~
- D ~ ( 3 ~ + 3z2)(3~ sin 232• 0 + 32 sinh 26~lo)/(2Q)
V~fA= (DdQ)(3~ + 322)23t32(sinh 2 3~lo + cos 2 32lo)
(21a) (21b)
and the terms for the first-order end moment in eqn (16) are given by M~,, ~ = - (Db/Q)26 t 62( sinh2 6 t lo + sin 2 6210) M~. x = D~(3~ + 3~)(6~ sinh 23x/o + 3~ sin 232lo)/(2Q)
(22a) (22b)
2.2.1 Characteristic equation
The condition of shear equilibrium (6a), written for the first-order terms, produces an expression for (1 (note that VuA = 0): ~1 = (Vie,1- V~,l)/(Vl~.l- Vlf.1)
(23)
The associated equilibrium and compatibility equations (6b) and (7) up to the order q~ are given in terms of the moments at the interface: M.,~ + Mr, 1 -- M ~ A = P , , I H / 2 -- P . , ~ ( T - H ) / 2 T
EH(T-
H)
4a(1 -- vt3v31 )
= Pu, x ( T - H ) / 2 -- e l , ~ H / 2
(24) (25)
The end moments, Mi, ~ = -Diy~'A, are given from eqns (16) and (22) in terms of the zero subscript quantities and the quantity ~ which was found in eqn (23). Substituting these expressions into the above two equations, and eliminating the quantity P t A H / 2 - P , ~ , I ( T - - H ) / 2 , gives an equation for the critical buckling load, Po, as follows: Duke, o cot k~,oa + M 1,1 ¢
-
-
¢ - Vd.a)(M,A ¢ r - M~,A)/(V~,., f r -- VIA r ) M bc, , + (V~,, = -- T E H ( T - - H)/[4a(1 - vx3v3x)] (26)
where the quantities V/,~I, V/fA,M ~ and M ~ , i = 1, b, in the above equation are given in eqns (15), (21), (17) and (22). This is the characteristic equation for the critical instability load. 2.2.2 Case o f no elastic f o u n d a t i o n
The solution to the first-order equation (10a) for fl = 0 is sin kb,oX Yb,1 = k~,o cos k~,olo k~, o = Po/D~
l o = ( L -- l)/2
(27a) (27b)
The characteristic equation is again derived by eliminating the quantity
120
G. A. Kardomateas
P~,~H/2- P u , ~ ( T - H ) / 2 in eqns (24) and (25), and is found to be (see also Ref. 3)
H3k~,,o cot (k,,,ol/2) + ( T- H)3k,,o cot (k~,ol/2)
-- T3kb,o tan kb,olo + 6 T H ( T - H)/I = 0
(28)
2.3 Post-buckling solution and delamination growth characteristics When the terms in ~b2 are equated, the second-order differential equations are obtained. The solution for the upper and lower parts can be found in Refs 3 and 4. The solution for the base plate, which is simply supported, will be developed in the following. The differential equation is expressed as tt
v¢
~'b.Yb,/-I ,,(4)2 "q- PoYb,2+
flYb,2
=
--Pu,~Yb,~
(29)
The solution to the above equation is a superposition of the general solution and a particular solution. For k~4,o> 42~,the solution to the second-order equation (29) satisfying the end fixity conditions (5), with 6~. defined in eqn (12), is found to be Yb,2
~
P1 C2J~b sin 6ix +
=
P1
B2j--xcos6JXDb
(30a)
j=l,2
(-- 1)Jc~joj B~j - 2(62~ _ 6~)
(30b)
The constants C2j are determined from the conditions (4a) and (9). In terms of Q defined by eqn (13a) and A~ = ) ,
B2jlocosfjl o
(31a)
B2j(lo6j sin 6flo -- cos 6jlo)
(31b)
j=l,2
A 2 = ~, j=l,2
these constants are given by Czj = Q - ~[~2(Db/P063 - j cos 33 -rio -- A163 -j cos 33 -rio + Az sin 33 -rio] (31c)
The second-order end shear can be written in the form ttt
2
t
Vb,2 -= --Db(yb,2 + k~,oY~,z)Jx=to - P~ Vbg,2 "-1-(2 Vbf,2
(32)
Buckling and post-buckling of end-fixeddelaminatedcomposites
121
where
V.b,2 ~ = --(A I/B)6a62(62~ + 6~)cos 61lo cos 62lo + ~,
(A2/Q)6 ~ sin 63 -rio cos 6flo
j=l,2
+ B2./6](3 cos 6flo - lo6./sin 6fl~)
V.b,2 f = (D~,/Q)6~62(62~ + 6~) cos 61lo cos 6210
(33a) (33b)
The second-order end moment can also be written in the form
M~,,2 = -- D bYb,2 " x=to = P1Mg,2 + (2M~f.2
(34)
where
M~,.2 = (A2/Q)(6~ + 6~) sin 61l o sin 62lo B2~6~(2sin 6flo + lo6./cos 6/0)
+ 2 ./=1,2
-- (A 1/Q)6f63 _./sin 6/0 cos 63 -rio
Mfb'2= 2
(Do/B)6]63 -j sin 6slo cos 63_rio
(35a) (35b)
j=l,2
For k~,o < 42 b the second-order solution satisfying eqns (29) and (5) is found to be P1
Yb.2 = C21 ~
PI
sinh 61xcos dizX + C22 - ~ cosh 61X sin 62x
PI P1 + B2~ - - x cosh 61x cos 62x + Bz2 - ~ x sinh 61x sin 62x D~
B2./ =
- - [ C 1163
_j + ( - 1)Jc126j]/(861(~2)
(36a) (36b)
where 6./are defined in eqns (18). The constants C2~ and C22 are found from eqns (4a) and (9). In terms of Q defined by eqn (20a) and
A 1 = B211o cosh 61lo cos 62l 0 + B2fl o sinh 6110 sin 62l0 A2 = B21 cosh 61lo cos 62/o + (B2161 + B2262)l o sinh 61lo cos 6zl o + (B2261 -- B2162)locosh6~lo sin6210 + B22 sinh611osin6210
(37a) (37b)
these constants are given as follows: C21 = Q- 1{A2 cosh 61lo sin 62lo + [(2(D~/P1) -- A 1] x (61 sinh 61lo sin 62/o + 62 cosh 6~1o cos 62/o)}
(37c)
122
G. A. Kardomateas
Cz2 = Q - I {[~2(Db/PI) -- A 1](82 sinh 8110 sin 8zl o - 81 cosh 81lo cos 8~lo) - A 2 sinh 81l o cos 82lo} (37d)
The second-order end shear is expressed by eqn (32) with the definitions V,b,2 ~ = Q - 1[A2(812 + 82~)(81 sin 28zl o + 82 sinh 281lo)/2 --.A 1(812 + 8~)28~S2(sinh 2 8~lo + cos 2 82/o)] + [O21(8~ - - 812) - - 6B228182] cosh 81lo cos 82/0 + [B22(8 ~ - 8~) + 6B2~8~82] sinh 8~lo sin 821o - (82~ + 82~)lo[(B2282 - B 2180 sinh 8~1o cos 82l o (Bzx82 + B2281)cosh 811o sin 82lo]
(38a)
V~.2 = D~(812 + 822)Z8182(sinh 2 811o + cos z 8zlo)/Q
(38b)
-
Likewise, the second-order end m o m e n t is given by eqn (34) with the definitions M~,2 = Q-l[A228182(sinh2 81lo + sin 2 82/o) - A~(812 + 8~)(8~ sin 282l o + 82 sinh28flo)/2 ] + [B2~(82~ -- 812) -- 2B2~6~82]l o cosh 6~1o cos 82/o + [B22(8 ~ - 812)+ 2B~6~82]lo sinh 81l o sin 82l o - 2(B 2181 + Bz282) sinh 6~lo cos 82/o + 2(Bz~82 - B2281) cosh 81/o sin 6flo
(39a)
M b,2 r = Db(812 + 8~)(81 sin 282/o + 82 sinh 28flo)/(2Q)
(39b)
2.3.1 Solution The shear condition (6a) allows determining ( 2 in terms of the first-order (yet unknown) forces (note that Vu,2 = Pu.1) as follows:
K c -- 1 V5 -- V5 ~2 - V,rb'z Pu 1 + b,2 ~,2 p~ 1 1,2
-
V. r
b,2
,
V,r
1,2
-
V, f
b,2
(40)
,
The m o m e n t equilibrium equation (6b) for the second-order terms is M..~ + M,.~ - M~,~ = e , . ~ / 2 - ~ . . ~ ( r - g ) / 2
(4~)
while the geometric compatibility equation (7) for the second-order terms is given by l f~ y .,z, ~ d x - ~ l fo Y'~Yld x = 2(1-vl3v31)l [ P , , 2 H / 2 - P . ~ "( T - H ) / 2 ] EH(T-H)
~
(42)
Notice that the second-order moments, M~,z = -D~y~',z, at the interface are given from eqns (34), (35) and (39) in terms of the (yet undetermined) firstorder end forces. Thus eliminating the quantity P ~ , ~ H / 2 - P . , z ( T - H ) / 2 from eqns (41) and (42), and taking into account eqn (40), gives the following
Buckling and post-buckling of end-fixed delaminated composites
123
equation for the first-order forces PuA and PLI: [-Mu~,2 -- M~,2 + ( V~g,2 - 1)(M~ 2, -- Mb,z)/(f V'f,,2 -- V~,2)]Pu,1 + [m,g,2 - M ' b.2 + (V,b.2 g - V~g,2)(Mif,2 -M~,.z)/(VL2f f V~2)]P,
(2/¢o,0a-_sin_Zko,oa =\
4ku.osinZk..oa
.
E/-m'- m
-a.,)4a(l_Vl~V~O
(43)
The second equation needed for finding P.,1 and P~.I is the first-order equilibrium equation (24) at the interface, namely
P~, ~H/2 - P.. I ( T - H)/2 = Fo( P., o, P~.o)
(44)
where Fo is the left-hand side of eqn (26) which depends only on the zeroorder quantities. In the above equations V~,~2,V[.2,M~2 and M[.z, i = 1, b, take the values given in eqns (33), (38), (35) and (39) depending on the relative magnitude of the foundation modulus; S~ is the shortening of the lower part due to first-order deflections and is given in Refs 3 and 4. The above system of linear equations allows finding Pu,1 and P~,I, and hence the first-order applied end force PI = Pu, 1 + PL 1.
2.3.2 Case of no elastic foundation The case of fl = 0 admits as a solution to the second-order equation: Pb,1 Yb.2 = cb,2 sin kb.oX + 2DbkZb.O cos kb,ol o x cos kb.oX
(45)
and, from eqns (9), Pb, l
Cb.2 = 2Dbk3~.o cos2 (kb.olo) (Iokb. o sin kb.ol o - cos kb.olo)
(46)
Again, eliminating the quantity P m H / 2 - Pu.2(T-H)/2 from eqns (41) and (42) gives the following equation for the first-order forces Pu.l and Pt.l:
p, F cos(kuol/2) lcos2(ku.ol/2) "lL2k~.o-S~n~/2)-4sin2(ku,ol/2 ) -
-
~ [- cos (k, o//2) + r~'l L z k ~ 2 )
l 4
sin kb,o/o 2kb. 0 cos kb,ol o /cos 2 (k~ o//2) 4 sin 2 (k,',ol/2)
-
-
sin kb.ol o 2kb, o cos kb,ol o
-sin (k~,o l) - k,.ol = _kt, ° sin 2 (kLol/2)
1o sin 2 kb.olo 2 cos 2 kb.ol o
lo7
l°sin2kb'°l° 2 cos 2 kb.ol o
l~1
l 4
sin (k u ol) --.ku ol] E H ( T - H) ~ J 8/(1 - - V 1 3 Y 3 1 )
(47)
124
G. A. Kardomateas
The second equation needed for finding Pu,1 and P~.I is the first-order equilibrium equation (24) at the interface, namely Pu,o
P,,o
Po t a n kb,ol o
P~.~H/2 - P u , I ( T - H)/2 - ku.o tan(ku,ol/2) ~- k~, o tan (k~.ol/2)
kb, o
(48) The above system of linear equations allows the finding of Pu, ~ and P~,~,and hence of the first-order applied force P~ = Pu.~ + P~.~. 2.3.3 Growth characteristics
The post-buckling solution that has been obtained can be used to study the post-critical state of deformation. In addition, the initiation of delamination growth can now be analyzed on the basis ofa Griffith-type fracture criterion. Predicting whether the delamination will grow requires an evaluation of the energy release rate. This quantity, which is the differential of the total potential energy with respect to the delamination length, can be easily calculated by using the path-independent J-integral expression in terms of the axial forces and bending moments acting across the various crosssections adjacent to the tip of the delamination. 6 In terms of the quantities P* = P ( H / T ) -
Po
M* = M u
M * * = P* T/2 - M *
(49)
the energy release rate per unit width is expressed as G = 12(1- v13v3, ) {p.2 + 1 2 ( M . / H ) 2 p . Z + 1 2 [ M * * / ( T - H ) ] 2} 2E H + T- H
(50)
The J-integral method that is used here does not distinguish between the Mode I and II components. It should be noted that future work should consider this question, since there may be a dependence of the fracture toughness on the ratio of those two components.
3 DISCUSSION OF N U M E R I C A L RESULTS Numerical examples are presented for the case of T / H = 6, L / H = 200. The characteristic equation (26) is solved for the critical load Poc. If Poc < (Pg~o,P~o~),with Pg~oand P,o~ given by eqns (1) and (2), then combined 'mixed' buckling involving out-of-plane deflections for both the delaminated layer and the base plate takes place. Otherwise the critical load is the minimum of the above, corresponding to global buckling (typical of very short delaminations) or local buckling (typical of very large delaminations). As the above analysis shows, the conditions of end attachment, by affecting the
Buckling and post-buckling of end-fixed delaminated composites
,-~ ~
(
~ = 10
125
c-c s-~
,~
~ 2
,3=0
.E ~ 0 0
~ 20
~0
60
D~,l~mitmtion Le~gth. ~
80
~00
~./11
Fig. 3. Buckling load vs delamination length for T/H=6, L ~ H = 2 0 0 and a set of foundation moduli, for both the simply-supported (solid line)and clamped-clamped (dashed line) cases. The usual case without an elastic foundation corresponds to ~ = 0.
equations that describe the deformation of the base plate, influence the performance of the whole system. To illustrate the effect of end fixity, the variation of the critical load, normalized with respect to the Euler load for the delaminated layer,/~o = Pol2/(4n2Du), vs delamination length, ~'= I/H, for both the simply-supported and clamped-clamped cases, is given in Fig. 3 for a set of values for/~. The foundation modulus is normalized as /~= 3131'~/(16n4Db). It can be concluded that for high values of the foundation modulus/~ the end fixity has a small effect on the instability point. This is because in these cases the instability is mostly controlled by the local buckling mode as opposed to the global one that is more pronounced for low values of/~. Notice that in both cases, as/~increases, the curves are shifted to the left, indicating the attainment of loads similar in magnitude to the local buckling ones for smaller delamination lengths. The combined effect of the location of the delamination through the thickness for different foundation moduli and the end fixity is indicated in Table 1, which gives the values of the delamination length, I/L, for which the characteristic equation has no roots less than the local buckling load given by eqn (2) (the 'local buckling threshold delamination lengths'). As the delaminated layer becomes thinner, local buckling is reached at smaller lengths, and an increase in the foundation modulus reduces this 'threshold length' more effectively. For example, for H/T= 1/15 and /~> 10 4, local buckling occurs for practically all delamination lengths. In addition, local buckling occurs in general at smaller delamination lengths in the clampedclamped case than in the simply-supported one. It should also be noted that, for even smaller lengths than the ones given in Table 1, the solution of the
126
G. A. Kardomateas
TABLE 1
Values of Delamination Length, l/L, beyond which the Characteristic Equation has no Roots less than the Local Buckling Load ('Local Buckling Threshold') H/T a
~ 0
1
10
10 ~
10 ~
10 ~
1/6 (C C) (S-s)
0.966 1.000
0.963 1.000
0.961 1.000
0.951 1.000
0.816 0.844
0.511 0.512
1/9 (C C) (S-s)
0.943 1.000
0.936 1.000
0.921 1.000
0.759 0.784
0.516 0.516
0.291 0.291
1/12 (C-C) (S-s)
0.603 1.000
0.471 0.475
0.279 0"288
0.161 0"161
0.089 0.089
0-05 I 0-051
1/15 (C-C) (S-s)
0.451 1.000
0.277 0"279
0.166 0" 169
0'096 0"096
0.054 0-054
0-031 0'031
"C C: Clamped-Clamped; S-s: Simply-supported.
0.307
,~-~ ~ .~
q3 II ~
0.310
0.313 0.316 0.319 0.322
~
0.305 o.o~
0.308 ~
~
J
I
I
0.311
0.314
0.317
l
~
l
~
I
~
1
0.325 ~ = 10, S-S, C-C ,
0.320 ~ .
I
]
0.323 ~
~ = 01 0.1, (~-C
~
= lo
~--
o,~
o.o~
¢~.. S-S
~
~
0.02
0.00 0.24~ i
0.302
I
0.2~ I
0.305
I
I
0.251 I
0.308
I
0.2~ *
I
0.311
0.2~ ,
I
0.314
I
'
i
'
0~ '
I
~
0.263 ~ :
0.317
I
0.320
v~
S-S
~
= O. 1, S-S
Applied Load, /5, = pL2/(4rc2D~,) Fig. 4. Strain energy release rate vs applied compressive force during the initial postbuckling stage for both the simply-supported (solid line) and clamped-clamped (dashed line) cases ( T / H = 6. L / H = 200, and delamination length I/H = 60). The different load scales are of the same length and correspond to the different initial buckling loads.
Buckling and post-buckling of end-fixed delaminated composites
127
characteristic equation is very close to (by less than 1% smaller than) the local buckling load, as also is evident from Fig. 3. To illustrate the effect of end fixity on the post-critical characteristics, the variation during the initial post-buckling stage of the normalized strain energy release rate, t7 = G/(ETS/L'*), vs the applied load normalized with respect to the Euler load for the entire beam with no elastic foundation, P = pL2/(4r~2D~,), is plotted in Fig. 4 for the example case of T / H = 6, L / H = 200, and delamination length l / H = 60, for a set of values of the foundation modulus. Both the simply-supported and clamped-clamped cases are considered. Since the critical load changes with/~, different scales (of the same length) are used on the load axis. As for the critical load, the end fixity does not much affect the slope of the G - P curves for the high values of the foundation modulus,/~. For the low values of/~ the simply-supported case results in less steep curves. This decreased slope means that (1) the laminate can withstand more load beyond the critical point in the initial postbuckling stage before delamination growth takes place, and (2) there could be less energy absorbed since growth would not be promoted as much. Notice also that for both cases the curves are steeper for a larger/~.
4 CONCLUSIONS The effects of end fixity (clamped-clamped vs simply, supported) on the buckling and post-buckling of delaminated composites under compressive loads are investigated. First, the perturbation technique is used to derive the analytical solution for the post-buckling behavior of delaminated composites with simply-supported ends. The results are compared with the characteristics of the clamped-clamped case. Both the usual case and that of a delaminated composite beam-plate on an elastic foundation are treated. End fixity effects are found to be more pronounced for lower values of the foundation modulus.
REFERENCES 1. Chai, H., Babcock, C. D. & Knauss, W. G., One-dimensional modelling of failure in laminated plates by delamination buckling. Int. J. Solids Structures, 17 (1981) 1069. 2. Yin, W.-L., Sallam, S. N. & Simitses, G. J., Ultimate axial load capacity of a delaminated beam-plate. AIAA Journal, 24 (1986) 123. 3. Kardomateas, G. A. & Schmueser, D. W., Effect of transverse shearing forces on buckling and postbuckling of delaminated composites under compressive loads.
128
G. A. Kardomateas
Proceedings 28th SMD A 1A A/A SME/A SCE/A HS Conf , Monterey, CA, 1987, p. 757; also in press, AIAA Journal, 26 (1988) 337. 4. Kardomateas, G. A., Effect of an elastic foundation on the buckling and postbuckling of delaminated composites under compressive loads. J. Appl. Mech., ~ (1988) 238. 5. Timoshenko, S. P. & Gere, J. M., Theory of Elastic Stability, McGraw-Hill, New York, 1961. 6. Yin, W. L. & Wang, J. T. S., The energy-release rate in the growth of a onedimensional delamination. J. Appl. Mech., ~1 (1984) 939.