advertisement

ComputationalMechanics16(1995)17o-1899 Springer-Verlag1995 Strength reduction and delamination growth in thin and thick composite plates under compressive loading B. P. Naganarayana, S. N. Atluri 17o Abstract In this paper, the coupled local-global buckling behavior in laminated composite plates with elliptic delaminations and the associated mechanisms of delamination growth under compressive loads are critically examined. The J-integral technique is used for delamination growth prediction in terms of pointwise energy release rate distribution along the delamination edge. A Multi-plate model, in conjunction with a 3-noded quasi-conforming shell element, is used to model the delaminated plates. The incremental equilibrium equations are set up based on total Lagrangian formulation. The solution strategy incorporates Gauss elimination in a cycle of Newton-Raphson iterations and is augmented with automated arc-length controled load incrementation and equilibrium iterations; and with automated post-buckling path tracing based on a linearised asymptotic solution. The effects of structural parameters such as delamination thickness, size and shape, on the post-buckling behavior and on the delamination growth are critically examined. 1 Introduction It is known that delaminations are the most frequent causes of failure of laminated structures, particularly under compressive loads. The presence of delaminations leads to a reduction in the overall buckling strength of the structure. In addition, the delaminations tend to grow rapidly under post-buckling loads, causing a further reduction in structural strength, and finally, leading to fatal structural failure. In this paper, in an effort towards gaining an understanding of the possible strength reduction due to the presence of the delaminations, and the mechanisms of their growth, a parametric study of the influence of the thickness, shape and size of elliptic delaminations in plates is undertaken. The Reissner-Mindlin theory of plate flexure is used to model the undelaminated plate and the delaminated sublaminates Communicated by S. N. Atluri, 10 January 1995 B. P. Naganarayana, S. N. Afluri Computational Modeling Center, Georgia Institute of Technology, Atlanta GA 30332-0356, USA This work was supported by the FAA to the Center of Excellence for Computational Modeling of Aircraft Structures at Georgia Institute of Technology, and in part by a grant from ONR. These supports are gratefully acknowledged. The first author also acknowledges the support by the INDO-US Science and Technology Fellowship under the auspices of United States Agency for International Development and Department of Science and Technology, India Correspondence to: B. P. Naganarayana independently; and the junction between the delaminated sublaminates and the undelaminated plate is modelled by using a two-dimensional generalisation of the so called Multi-Domain Model, presented earlier for one-dimensional problems by Chai, Babcock and Knauss (1981) [Naganarayana and Atluri (1994)]. A quasi-conforming 3-noded plate element [Huang, Shenoy and Atluri (1994)] is used to model the delaminated structure. An automated incremental post-buckling solution strategy, incorporating an arc-length controlled load incrementation, and branch switching based on a linearised asymptotic solution [Huang and Atluri (1994)], is utilised while using the displacement type finite element model. The stresses are post-processed to obtain pointwise energy release rate distribution along the delamination front, by using the adapted J-integral approach [Naganarayana and Atluri (1994)] for each load increment. In this paper, the influence of the structural parameters such as the location of the delamination with reference to the plate mid-surface, the shape and the size of the elliptic &lamination, and the structural thickness, material orthotropy, fiber orientation and lay-up sequence in general laminates - on the coupled local-global buckling phenomena, reduction in structural strength, and mechanisms of delamination growth is critically examined as the delaminated plate is subjected to compressive inplane loads. 2 Multi-Plate model In this paper, we shall consider a general laminated composite plate with a single delamination of an arbitrary shape and location subjected to arbitrary buckling loads (Fig. 1). The structure is modelled using the multi-plate model [Naganarayana and Atluri (1994)] where the delaminated plate is assumed to be assembled with three distinct plates-(1) Laminate: the undelaminated zone 12~ (2) Delaminate: the thinner side of the delaminated zone 12(2) and (3) Base: the thicker side of the delaminated zone 12(3/. The three plates have midsurface areas sr thicknesses t(~ boundaries 812{i);and midsurface boundaries 8 d (~ for i = 1, 2, 3 respectively (Fig. 1). The delamination edge is denoted by F. The Reissner-Mindlin theory of plate bending is used for modeling each of the plates and the interface between them. Thus, for each plate, the 3-dimensional displacement field (U {U1U2U3}) can be expressed in terms of the corresponding midsurface displacement (u - {u~u2u3}) and rotation (0 = {01020}) fields = as, u (') (x~, x3) = u {') (x~) - x~i) 0 (~)(x~) (1) # DeLaminote fl~2~ Delaminotion"~F I X2 L ; Ondetominote f~m Iw N Bose Assumed deformotion at pLote-junction (U I~l = UI~I= Ul~l)r Xl a, IX _ (2>~ L~ ~ • . T / 3 171 I / " - - "1 iI I I i I Iq)A t - X3 T/ XF, - - - x , I (3) h' -l/._ Ft:l.0N/mm t t t / , Fig. la-r . . . I r I e },1 . iI ! L I*[ I I I XF~ b - - t! //U:31 t" _ /I / Um I Multi-plate model, a Delamination in an arbitrary plate, b Clamped square plate with central elliptic delamination, c Section x-x where x~~ (~ = 1, 2) are the inplane coordinates and x~~ is the thickness coordinate for the ?h (i = 1, 2, 3) plate. The structural continuity at the delamination front F is maintained by assuming the deformation to be unique at the junction of the three plates i.e. U ~ = U (2) = U (3) on F. For each plate, the inplane components (en, e=, q2) and the transverse shear components (e~3, e23) of the 3-dimensional Green-Lagrange strain tensor (including large deformations) can be expressed in terms of the membrane, flexural and transverse shear (engineering) strain c o m p o n e n t s - e , ~, 7 respectively - as, where the material constitutive parameters E!!) are functions of the thickness coordinate of each plate x~i/~JGenerally, for a laminate with orthotropic layers, E!! 1 are assumed to be U piecewise constants over the laminate thickness. Now, for each plate, the strain energy density (per unit volume) can be calculated using Eqs. (2), (3) and (4) as, 1 ,•(i) ~a=~[u ~+ ~(u) = Ufl,. + u~y~,e, 1(i) W(~)= [o-. ( e + and the total strain energy stored in the i ~hplate is given by, ~(1 = l [ u .,fl -F uB,. -I- u3,=u3,fl] (i) - x(i) !2 tr~ o:,fl -I- O fl,~.](i) ~ _ ~ [u3,~ ~[ 3,~+u~,3]~'~-~ - 0J0 (3) respectively where, again, ~, fl = 1, 2 and i = 1, 2, 3. Assuming orthotropic material behaviour for each lamina, the inplane stresses ~r/~)= {G~ cr22cr12}l~/and the transverse shear stresses r (~ = {z13z23}(~ are related to the corresponding strain components as, f:t i) _~ "En EI2 El6 0 0'(i) El2 E22 E26 0 0 El6 E26 E66 0 0 0 0 0 E44 % 0 0 0 E45 Ess / (6) i= 1 \~r (2) 1 v(i) = ~.~ -(2i 89 N - e + M - r + Q. r)U/d~r = ~(i) __ _ _1 U Q) ~ ~ W~Od i=1 \ ~_ ~(i) 4 - "v(i)~"(i) (5) x3x)](~) + [r. r] (i) where N - {N u N= N12} are the inplane stress resultants; M -{M11M22M~2} are the bending moment resultants; and Q = {Q13Q23}are the transverse shear stress resultants given, for each plate, by, (i) .(i) (i) (7) and (4) (Akl;Bkl;DkS I= f E~i)(x3)(1;x3;x2)dx3 and G~ tl as k,l= 1,2,3 correspond to correspond to i, j = 4, 5. = ~E(o mk(x3)dx3 ti i,j= 1,2,6 and re, n=1,2 172 If f(0= {f~ f2 f3} (~ are reference body forces acting on D(~ F = {F~ F2 F3}/~>are the traction forces acting on ~d(~ and 2 is the current load factor, the potential of the current external forces is given by, (hAW(U) - A2bB (U) - (62W(U) -- A2c3B(U)) = 0).t(u,~) (14) 2 B ( U ) = 2 ~ =( ~1 f . 3q- [ K t ] . A q - A23q.F = 0 i) UdD@ ~.Q(i) F.Ud~) where, assuming that f and F3 do not vary over the plate thickness, "f(i)= ~ f(Odx3; ~(i) = ~ F(i)dx3; ~(i) = ~ F(i)x3dx3. ti ti For a typical finite element application, incremental equilibrium Eq. (14) can be written as, (15) where Aq are the incremental nodal displacements; F are the discretised reference nodal forces (typically as specified in the input for the problem); and [Kt] is the tangent stiffness of the system. The incremental equilibrium equations thus become, [Kt]-Aq - A2F = 0 (16) On the other hand, the total equilibrium for the system (Eq. (12)) can be written, in finite element context, as, ti Applying the minimum total potential energy principle (cSYi = c~( W (U) - )3 (U)) = 0), we obtain internal equilibrium equations: [Ks] .q + 2F = 0 (17) where [Ks] is the secant stiffness (or simply stiffness) matrix for the system. {N~/~,~+ j ~ = 0 ; M~,~+T~=O; T~,~+f3=0}~(~/ (9) For each incremented external load vector, the incremental equilibrium equations (Eq. (16)) are solved in a cycle of Newton the boundary conditions on the external boundary: type iterations and using the total equilibrium condition (Eq. (17)) for computing the residual load vector which in turn { G = 0 or N~.n~=F~; 0 ~ = 0 or M~.n~=lfI~; is used for convergence check. The load increment is optimised for each iteration using the constant arc-length criterion to u3=O or T..n.=F3}o:g(~, (10) assure faster and definite convergence of the solution even while and the equilibrium condition on the delamination front: traversing limit points on the solution path. The presence and location of an instability point is determined by the singular { : ~ ( N ) = 0 ; : ~ ( M ' ) = O; : ~ ( T ) = 0}r (11) nature of the tangent stiffness matrix at the instability point; and the nature of the instability point - whether limit point or where T~ = O~3 + N~u3 ~; (M')(~) = M(i) + h(~)N(~ ~ n (*) = (,)(1) __ bifurcation p o i n t - is determined in a heuristic sense by (,)(2) _ (,)(3) with (,)(~)'corresponding to the value of (,) at the comparing the load increment and increment in the generalised specified point on the delamination boundary of the i~hplate; deflection across the instability point. If the instability point and n is the unit vector normal to delamination front. identified is a bifurcation point, a cost-effective branch However, it should be noted here that the equilibrium Eq. (11) switching technique is used to follow the desired postbuckling may be violated when the delamination edge intersects the branch - the nonlinear fundamental state between two points structural boundary and when the delaminated plies come into in the neighborhood of the bifurcation point is linearised to contact after global buckling. obtain an asymptotic solution on the desired postbuckling solution branch. Refer to Huang and Atluri (1994) and 3 Naganarayana (1994) for the automated post-buckling solution Incremental solution algorithms. Here, we shall derive the incremental equilibrium equations for an elastic conservative system D under linear loading. The 4 generalised deflection B (U) is assumed to be a linear function J-integral adaptation for pointwise energy release rate along in U; and the load factor 2 is assumed to be an arbitrary constant. a planar delamination front In an incremental solution strategy, the solution is known at In the case of a delamination, the growth is assumed to be along the initial point (U, 2) of the current increment. Then by the the interlaminar zone parallel to the midsurface of the plate (i.e. principle of minimum total potential energy, we have, the crack cannot shear into the neighboring laminae). In other words, the crack is assumed to grow in a self-similar sense (3W(U) -- 2c3B (U) = 0 (12) [Naganarayana and Atluri (1994) ]. The pointwise energy release rate for self-similar Now, the equilibrium at the unknown point (U + AU, 2 + A2) 3-dimensional crack growth (~(_F)) is defined as [Atluri requires, (1986)1, 3W(U + AU) - (2 + A2) cSB(U + AU) = 0 (13) Expanding the incremented functions W and B and neglecting the higher order incremental terms; and by using the equilibrium conditions at the initial point (U, 2) (Eq. (12)) we get, N ( r ) A F = L i m i t ~ L[ W e l - G ,_ n p OU~q ~JdA e~O Ae A1 -- ~ 6,2~ A2 ~0~ dA + i G2 ff~x~ 891 (18) x2 r aX3 7 73 ,n / ~ - ~ ~ 1 ~ xl ,n F A2 where A i (j = 1-9) are the segments forming the surface area A e of the rectangular tube (Fig. 2c). Since the Reissner-Mindlin assumptions (#13,#23 are constant over plate thickness; and #33 = 0) are used for the element formulations as well as to achieve displacement continuity at the delamination edge; and since fi= {00 +_1} on the segments Aj, j = 4 - 9 (see Fig. 2c), the integral in Eq. (19) vanishes over the segments Aj, j -- 4-9. Now, since fi= { + 1 0 0} on the segment A~ and fi = { - 1 0 0} on the segments A2 and Ar Eq. (19) becomes, (20) AF-~0 Now, carrying out the integration through the thickness for each plate, we get, b As § .... A~ 6~(F)AF'= j" ffg[lk7,r- (]k[s:fi. 1+MI~O~, 1 q-(7)13U3,1)1dr' (21) AF ,._.2k_.a: x~l 11 {1)L.~. . . . *'-" . ._3. X ~, A~ Therefore, as A F ~ 0, the pointwise energy release rate at any p o i n t on the delamination front Ng' computed from the Gauss-point variables, in a finite element model using constant strain elements, is given by, i .... . _ .9.t- . -t xL!'A.I Xg'A31 . - Aa 13 Ng(F) = fig [ ~/-- (/q, d2~,~+ M~O~,l + Av Fig. 2a-c. J-Integral for delamination growth in a plate model where, c~, fi=1,2,3; A e is the area of the tube of radius e enclosing the crack front; A1and A2 are the areas covering the ends of the tube (Fig. 2a); and ri, #, 0 and o7are defined in the crack tip coordinate system 2 (Fig. 2b). For the present problem of delamination progression, it is assumed that the material is homogeneous along the :~1, such that (8W/8,~)e~p~, = 0, in other words, 8W 0XI 8G~ [~W~ O-aft~1AV ~ 1 (22) where, Yg (,) = ( * )g(1) - - ( * )g(2) - - ( * )g(3)and (*)g(,)corresponds to the quantities (,) evaluated at specified points on the annular surface. For example, in a finite dement analysis, these specified points would be preferably the optimal stress recovery points - normally the Gauss points corresponding to reduced integration [Barlow (1976); Naganarayana (1991)] - i n the adjoining element of the i'h plate nearest to the delamination front F. The local stress resultants (N, M, Q) and displacement gradients (~.~) can be obtained from their global Cartesian counterparts (N, M, Q; Urn), by applying the regular tensorial transformations between the reference coordinate system x and the crack tip coordinate system ~. 5 Numerical experiments 8G~ explicit which case, the path-independence of the ]-integral is maintained [Atluri (1986)] for opening mode of crack growth and hence the tube can have a cross section of any shape. Consider a rectangular tube enclosing the delamination front and passing through the nearest stress recovery points (S(~ of the adjoining elements. Note that, the integrals over the areas A1 and A2 nearly cancel each other in a constant strain/stress element model, since the quantities like 6"and 80~/0~ do not vary in the neighbourhood of a point in an element domain. Then, Eq. (18) for the opening mode of delamination growth becomes, in (19) AFj=I Aj (~)13/~3,1)] In this section, we shall analyse a number of delaminated structures for their post-buckling behavior and the associated delamination growth. 5.1 Problem description An isotropic square plate of edge length L with a central elliptic delamination (Fig. lb), is considered for the following numerical experiments. The plate is subjected to biaxial compressive loads and its boundary is assumed to be clamped against out-of-plane deformations. One quarter of the plate is modelled by imposing appropriate symmetry conditions. 264 shell elements are used for the undelaminate plate and 192 elements each are used for the delaminate and base plates respectively. The reference applied biaxial compressive loads are assumed to be of unit intensity (Fs = 1.0) and the equilibrium equations are solved at each load step for an applied load F = 2Fs, where 2 is the corresponding load factor (Fig. lb). 173 200" strength of the delaminate plate (G( [] 'AnaLytical' solution / - "~l~rFx/t,))is given by (23) IOO The local buckling strength of the delaminate plate obtained from the finite element analysis compares very accurately with the analytical estimate (Eq. (23)) as shown in Fig. (3a). Further, assuming that the post-buckling deformation is axisymmetric and nearly linear in the neighbourhood of local buckling point, the pointwise energy release rate is given by [Evans and Hutchinson (1984)], .i.r 0" 0 174 0.1 0.2 0.3 t2/tl 0.~ 0.5 2-'o1 (ffrb(V) A,*: 3./.55 - - ( 1 - v2) t 2 (1.8-28-5~v)E (r _ 0-2) (24) where rr0( = 2FJq) is the actual stress level at which the energy release is being computed. The ratio, ffFe/~qrb, is plotted along the delamination 1.60 periphery, in Fig. (3b) for a case of very thin delaminate 1./.0 ~ 1.084 configuration (tE/t~ = 0.01). It can be observed that, fife is close ,%*= 2.105 to Nrb when the post-buckling loads are in the close vicinity of 1.20' x*= x/;% local buckling point (i.e. 2* = M2tcr-- 1.0). However, ffrb is 1.00 under-estimated when compared to ffF~ even when 2 -~ )]cr"This 0o 3bo 6'0~ 9'00 8 is because, in the present problem, though the &laminate plate is very thin when compared to the total laminate thickness (t2/t 1 = 0.01), the base plate is flexible as opposed to the rigid base as considered in Evans and Hutchinson (1984). Laminate is also thin when compared to its edge length, L (tJL = 0.05). Hence, the finite element model represents [ ] Represents'analyticaL' a reasonably flexible laminate and base plates. The deviation Energy Release increases as the buckling load increases beyond its critical value [ Evans and Hutchinson (198l,) ] for local buckling of the delaminate plate. This is because, the analytical solution (Eq. (24)) is based on the assumption of [ ] Represents 'actual' E o Energy Release a quasi-linear post-buckling behavior for the delaminate plate. Z But, in practice, particularly when the laminate is thin, post-buckling behavior of the delaminate plate is highly 'Normalised' displacement (A~) nonlinear. Accordingly, much higher energy-release-rates are Fig. 3a-c. Clamped square plate with central circular delamination expected when compared to Nrb as shown in Fig. (3c). Note that, (Validation of the model) a Criticallocal buckling strength. Comparison in Fig. (3c), actual stress ((r) and displacement (A) are of F. E. solution with 'analytical' solution, b Pointwise energy release 'normalised' by the critical stress G and the associated critical rate distribution Comparison of F. E. solution with 'analytical' inward radial displacement A c respectively. solution (t21t1=0.01; 2/r=0.125). C Effect of 'quasi-linear' post-buckling behavior for delaminate on energy-release-rate 5.3 [] Represents 'analytical' energy release [Evans and Hutchinson Effects of delaminate-plate-thickness (1984)] [] Represents 'actual' energy release The numerical experiment is repeated for varying delaminate thicknesses, for a fixed laminate thickness (tJg = 0.05), delamination size (a/L = 0.3), and aspect ratio of delamination (a/b = 1.5). The mechanisms are examined for both supplementary and complementary modes of post-buckling 5.2 deformations of the delaminated plate. Validation of the model The structure is assumed to be isotropic with Young's modulus 5.3.1 E = 6500 and Poisson's ratio v = 0.3. The laminate thickness is chosen as tl = 0.05 L. The numerical experiment is conducted Supplementary modes of post-buckling deformation For very thin delamination (t2/tl = 0.02), the delaminate plate for a near-surface circular delamination with tJtl = 0.01, undergoes a second mode of buckling before reaching the global a/b = 1.0 and all = 0.3. Assuming that the base plate and the undelaminated plates are infinitely stiff when compared to the buckling load level. Typical post-local-buckling modes of deformation are presented in Fig. (4a) and (b) for first and delaminated plate, the delaminate plate can be considered as second mode respectively. The deflection at the centroids of a clamped circular plate under the same radial compressive the delaminate and base plates are traced in Fig. (5). For very stress [Evans and Hutchinson (1984)]. Then, the buckling = 1.80 X * : 2.793 0 -0.5 -1.0 5O lo 2o ~u 30 ~ a, 10 u 50 0 0 -0.2 -0.4 -0.6 -0.8 0 b ~ 50 z,O 1.5 1.0 0.5 0 50 0 '-~ 30 , ~ ~ 10 Fig. 4a-c. Post-local and post-global buckling deformation a/b = 1.50; a/L = 0.30; tJL = 0.05. a Typical post-local-buckling deformation (First Mode) t2/t1 = 0.02; )~= 58.5. b Typical-local-buckling deformation (Second Mode) tE/t~= 0.02; )o= -48.8. c Typical post-global-buckling deformation tE/t~ =0.20; )0 = 218.0 thin delaminate (t2/t ~ = 0.02), there exists a load l i m i t p o i n t at 2 = 92.3 as the delaminate plate undergoes second mode of buckling (see Fig. (4b) and Fig. (5a)). However, for larger delaminate thickness, the delaminate plate does not undergo a second mode of buckling. On the other hand, the locally buckled delaminate plate acts as a geometric imperfection for the base plate and leads to premature global buckling of the structure. As expected, the local buckling strength of the delaminate plate increases as its thickness increases (see Fig. (5) and Fig. (6)). However, the global buckling strength of the structure decreases as the delaminate thickness increases (see Fig. (5) and Fig. (6)). The reduction in the global buckling strength due to the presence of delamination is nearly 50% (see Table 1). In fact, for t 2 =< 0.2t~, the global buckling strength of the structure is less than that of the plate with the delaminate plate completely removed. This is understandable since the locally buckled delaminate plate acts as a geometric imperfection and can lead to a highly premature structural failure. The energy release rate distribution is nearly uniform along the delamination edge for very thin elliptic delaminate plate in the vicinity of the local buckling point (Fig. (7)). However, the energy release distribution pattern changes as the load factor increases, and as the thickness increases. The maximum energy release rate is observed to be in the vicinity of the direction of maximum local-buckling-stiffness. In this case, the delaminat e plate exhibits maximum stiffness against buckling in the minor axis direction and hence the point of maximum pointwise energy release rate is found to be at an angle of 75 ~ with reference to the major axis. As the delaminate plate thickness increases the energy release rate becomes negative at certain zones. This indicates, in a qualitative sense, that the delaminate and base plates are in contact. This happens because, for higher delaminate thickness, the local buckling point is very close to the global buckling point (see Fig. (5f)). For very thin delaminates, the energy release rate increases upto the load limit point ()~ = 92.3); then decreases as the delaminate enters the second buckling mode whence the delamination starts closing; and takes a non-zero positive value for )~ = 0.0. Then as the traction forces become tensile in nature, the buckled delaminate plate starts unfolding from its second mode of deformation, and tends to open the delamination at its edge and hence the energy release rate increases from that point onward. The variation of the maximum and the average pointwise energy release rates with reference to the load factor are presented in Fig. (8). Since, for very thin delaminates, the energy release rate distribution is nearly even, the average energy release rate is very close to the maximum energy release rate. The variation of the maximum and the average pointwise energy release rates with reference to the load factor are presented for varying thicknesses in Fig. (9) and Fig. (10) respectively. It can be observed from Fig. (9) that, for thin delaminates, the energy release rate is predominantly due to the local flexural deformation of the delaminate plate after the load level crosses its buckling strength. In fact the energy release rate is very much negligible for 5[ < Z~crfor thin delaminates (see Fig. (9a)). However, for thick delaminates, the maximum energy release rate is positive - and comparable with that with thin delaminate in post-local-buckling conditions- even when the load level is lower than corresponding local buckling strengths. In this case, the growth mechanisms are predominantly due to the membrane action. However, the maximum pointwise energy release rate increases steeply as the global buckling of the structure approaches (see Fig. (gb)), irrespective of the delaminate thickness; and delamination growth is relatively more rapid for thinner delaminate configurations. For thin delaminates, the variation of the average pointwise energy release rate with the load factor is similar to that of the maximum pointwise energy release rate (see Fig. (9)). However, for a very thick delaminate, the delaminate and base plates are in contact for lower load factor on the average (i.e. the average energy release rate is negative) as depicted in the corresponding energy release rate distributions (see Fig. (7)). This is because, the local buckling strength becomes very close to the global buckling strength as delaminate thickness increases (see Fig. (5)). In fact, for t J t 1 = 0.40, local and global buckling occur almost simultaneously (see Fig. (50). Accordingly, the pointwise energy release rate increases steeply 175 1.5 a~ t2/t ~: 0.02 '.C u 7.5 I ~.o ! / b t2/t 1=0.05 w3 / e t2/t 1=0.10 / / 6.0 5.0 :// -~ o.s 8.084 / // VO W2 2.5 g 2.0 W3 (..2 - 100 176 - 50 0 50 100 5.0 84 d t2/tl:0.20 ? I 2.0 200 30O 6.0- / 54.0 o= 3.0 100 e t2/tl=0.30 / , Z,.O- / 5.0 84 260 300 f t2/tl= 0.40 4.0 84 2.0 [... 100 20 ~.o 0 0 100 200 Load footer ,L 300 0 I ----~0 16o 260 Load factor L 360 Ok 10o 260 Load factor X 360 Fig. 5. Effect of delaminate thickness on load-deflection response a/b = 1.50; a/L = 0.30; tt/L = 0.05 as the load factor crosses beyond the corresponding local critical value. It is observed that, the thicker delaminates reduce the overall global buckling strength of the structure (Fig. (6)), while the thinner delaminations are more critical from the energy release rate point of view (Fig. (9) and Fig. (10)). 5.3.2 Complementary modes of post-buckling deformation Normally the locally buckled delaminate plate acts as a geometric imperfection and causes a premature global buckling in a supplementary (signifying that the deflections of the delaminate and the base plates are in the same transverse direction) sense (Fig. (4c)). However, when the delaminate plate is thick, the eccentricity of the geometric imperfection is small; and, as a result, the delaminate and the base plates may snap into complementary (signifying that the delaminate and the base plates deflect in opposite directions) modes of post-global-buckling deformations (Fig. ( 11 ) and Fig. (12) ). The load-deflection curves are plotted for the centroids of the delaminate and the base plates in Fig. (12). The centroid of the delaminate plate undergoes a displacement limit point as the plate enters global buckling mode. Then both the base and the delaminate plates, with comparable thicknesses, enter higher order buckling modes. As they enter second buckling mode, the centroids of both the sublaminates pass a load limitpoint as shown in Fig. 12. It is also interesting to note that the critical load factors, triggering local and global buckling, remain the same whether the two modes are complementary or supplementary to each other. It is also interesting to study the energy release patterns in the complementarymodes. For thick delaminate configurations, the delaminate and base plates close the crack at many parts of the delamination periphery, in supplementary modes (Fig. (7)). However, they tend to open the crack at all points in the complementary modes. All-positive energy release rate distribution shown in Fig. (13) signify this observation. As a result, though the maximum energy release rate is slightly lower in case of the complementary mode (Fig. (14)), the crack tends to open at all points on the average (Fig. (15)). Note that, the crack tends to close on the average in case of supplementary modes for higher delaminate thicknesses in post-globalbuckling range of loads (Fig. (10) and Fig. (15)). Thus, in case of thicker delaminate plate configurations, the complementary modes of buckling pose a serious threat of rapid delamination growth and fatal failure. In case of thin delaminate plate configurations, the residual strength of the base plate is still comparable with the original strength; while both the base and the delaminate plates may face failure almost simultaneously (in case of complementary local and global buckling modes) leading to catastrophic failure of the whole structure. 5.4 Effects of aspect ratio of delamination The experiment is repeated for varying aspect ratios (a/b) of the elliptic delamination with specified thickness (t2/t ~= 0.1) and major axis length (a/L = 0.3) and fixed total plate thickness (tl/L = 0.05). A typical solution path for the transverse deflections at center of delaminate (w2) and base (w3) is shown in Fig. (5c) for an aspect ratio a/b = 1.5, depicting local and global buckling. It is observed that the local buckling load increases as aspect ratio increases (Fig. (16)). The global buckling strength also increases as the aspect ratio increases. However, global buckling strength Table 1. Reduction in global buckling strength of plate with elliptic delamination: Effects of the delaminate thickness 200 (;~~ (t = t~))o~- (;~)~. .... * 100 (;.~T(t = tl))oo~, %lc~ '< 100 Rb - ,100 (,t~ = t3))ooo~ (2~r(t = t))~aIF~= 5.30n2Et3/12 (1 --v 2) (Timoshenko (1940)] 0.1 0.2 t2/tl 0'.3 177 0.4 Fig. 6. Effect of delaminate thickness on local and global buckling strengths becomes saturated asymptotically as aspect ratio increases beyond 2.0 as seen in Fig. (16). The global buckling strength of the delaminated plate is compared with that of the undelaminated plate of full thickness and of base thickness in Table 2. It can be noted here, again, that the delaminate buckles locally prior to the global buckling; acts as a geometric eccentricity to the global behavior of the plate; and hence leads to premature failure of the plate. For the structure considered, the global buckling strength reduces to nearly half its original strength. In fact, the delaminated structure is nearly 30% weaker than the structure with the delaminate plate being completely peeled off. Thus, the reduction in the buckling strength of the delaminated plate is very high due to the presence of the delamination irrespective of its shape. The variations of the maximum and the average point-wise energy release rates with reference to the applied load are presented in Fig. (17) and Fig. (18) for different aspect ratios. As the aspect ratio increases, local buckling strength increases and hence the energy release rate is lower at lower load factors. But, as the external load increases, the energy release rate increases sharply for higher aspect ratios. Thus, for a delamination with higher aspect ratio, the onset of delamination growth may be delayed, but the delamination growth rate is much higher once the growth is set-in. The location of the point on the delamination edge where the point-wise energy release rate is maximum is traced in Fig. (19). It is observed that the point of maximum energy release rate occurs at an angle of 0m = 45 ~ with reference to the major axis when a/b = 1.0. It is observed that as the aspect ratio increases, maximum energy release rate is experienced at 0~ = 75 ~ This is, again, due to the larger crack opening at minor-axis-tips than at major-axis-tips for a given locally buckled configuration. 5.5 Effects of size of delamination Now, the numerical experiment is repeated for varying sizes of the elliptic delamination with specified delaminate thickness (tJt 1 = 0.1), aspect ratio of the delamination (a/b = 1.5) and plate thickness (q/L = 0.05). The major axis length of the elliptic delamination, a, is varied. As the delamination size increases, both the local and global buckling strengths decrease as t2/t 1 0.02 0.05 0.10 0.20 0.30 0.40 Rb 34.39 25.72 1.39 -37.08 - 108.16 Rf 43.75 45.48 49.50 52.98 55.04 presented in Fig. (20). The percentage reduction in the global buckling strength of the structure is given in Table 3 for varying delamination size. Again the delaminated structure is weaker in buckling when compared to undelaminated laminates. For a smaller delamination, the local buckling begins at a higher load level; and hence, the energy release rate becomes significant at higher load levels. However, the energy release rate increases steeply once the local buckling sets in. In the pre-global-buckling stages, both the average energy release rate (Fig. (21)) and the maximum energy release rate (Fig. (22)) are lower for smaller delaminations since the local buckling strength is higher. However, in post-global buckling stages, the smaller delaminations tend to grow more rapidly. Thus the smaller the delamination, the later the delamination growth sets-in; but once it sets in, the delamination growth will be much more rapid than with the larger delaminations. 5.6 Effects of total plate thickness Here, the numerical experiment is repeated for varying thicknesses ratio of the plate (tl/L), for a fixed delamination thickness ratio (t2/t 1 =0.1), delamination aspect ratio (a/b = 1.5), and delamination size (a/L = 0.3). The critical buckling load increases as the cube of the plate thickness (Fig. 23). Both local and global buckling strengths increase as the plate thickness increases. However, the increase in the critical load factors is slightly lower than that expected from the expected variation proportional to cubic variation of the thickness (Fig. (23)). This is because, the actual delaminate thickness also increases as the plate thickness increases for a fixed delaminate thickness ratio. Hence, the eccentricity of the delaminate plate with reference to the plate midsurface increases as the plate thickness ratio increases with fixed delaminate thickness ratio. Accordingly, the rate of reduction in the global buckling strength of the structure increases as the plate thickness increases. In case of delaminated plate, 30-60% strength reduction is observed with reference to undelaminated 0.3( 2.0 t2/t 1 : 0,02 0.2! ,?:o;o 1.8 . . . . . 1.6 0.20 1.4 0.15 1.0 1.2 0.8 0.10 0.6 0.4 0.2 0.05 t78 0 1; 20 3'o ~'o 5o 6o io 8o 9o 0.25 0 , % 25 t2/t 1: 0.20 0.20 15 0.10 10 0.05 5 0 0 -0.05 -5 i 0 1;~ 2o ='297.2 2O 0.15 -0.10 % 3o i ~o 5'0 i ~o ~o 8o ~o -10 t 2/tl : 0.40 xQg.9 0 ~ 10 ' 20' 30 ' ~0 60 ' 5'0 70 ' 80 ' 90 Fig. 7. Pointwise energy release rate distribution along delamination edge a/b = 1.50; a/L = 0.30; t,/L = 0.05 o,o6. 0.050.3" 0.04 i 0.03" " 0.2 0.02 1 2 3 4 t2/t~=O.02 tJt~:O.05 tJt~:O.lO t2/tl=0.20 5 t2/t1:0.30 5 t2/tl:0.40 i 1;/7 // // 0.01 0 0.1 0 a 10 20 30 z,O 50 60 Load factor X 70 80 30 J - 100 - 50 i i 0 50 100 ;k Fig. 8. Average and maximum pointwise energy release rate for very thin delamination a/b = 1.50; a/L = 0.30; tilL = 0.05; tJtl = 0.02 20 t2/tl=O.05 tjfl=0.10 t2/t1:0.20 2"/// 3//// &//4//L t2/tl: 0.30 IIII l tJtl:O.L 10 plate of thickness tl, while 1 0 - 4 5 % strength reduction is observed with reference to u n d e l a m i n a t e d plate of thickness t l - t2 (see Table 4). The n o r m a l i s e d average and m a x i m u m pointwise energy release rates - ~q~J)/cr and ~max/)Jcr -- are plotted in Fig. (24) and Fig. (25) respectively with reference to n o r m a l i s e d load factor 2/)/cr. As the plate thickness increases, even though the d e l a m i n a t i o n growth m a y start at m u c h higher load factors, the 0 0 16o 260 360 Load factor ~, Fig. 9a, b. Influence of delamination thickness on maximum energy release rate a/b = 1.50; a/L = 0.30; tall = 0.05. a Pre-global-buckling regime, b Post-global-buckling regime 0.1 300 0 /t21q: O.~'~~.3- g -0.1 ' 200 100 t79 -0.2 10 20 30 40 50 Load factor k 60 70 80 0 -6 -i 0 2 w /* ,. Fig. 12. Load-deflection paths at the centroids of delaminate and base plates. (In complementary mode), t2/t~ = 0.20, q/L = 0.05, a/b = 1.50, a/L = 0.30 9 0.6 0.5 0.4 90.3 0.2 -2 2;o o b 3oo Load factor: ;L 9 0.1 Fig. lOa, b. Influence of delaminate thickness on average energy release rate a/b = 1.50; a/L = 0.30; tilL = 0.05. a Pre-global-buckling regime, b Post-global-buckling regime 10 20 30 40 50 60 70 80 90 8 Fig. 13. Pointwise energy release rate distribution. (In complementary mode) t2/q = 0.20, till = 0.05, a/b = 1.50, a/L = 0.30 6.0 1.0 (1) In complementary mode (2) ]n supptementary mode /..5 / 50 3.0 Lu 30 ~ / / ~ 10 1.5 Fig. 11. A typical complementary post-buckling deformation mode. fi = 205.8, tz/t ~ = 0.20, tJL = 0.05, a/b = 1.50, a/L = 0.30 -1.5 e n e r g y release rate i n c r e a s e s v e r y steeply. Thus, t h e t h i c k e r the l a m i n a t e plate, the later t h e d e l a m i n a t i o n g r o w t h begins; a n d the m o r e d a n g e r o u s the d e l a m i n a t i o n g r o w t h will be. 5.7 Effects of the material orthotropy A n o r t h o t r o p i c s q u a r e plate of edge l e n g t h L a n d t h i c k n e s s q = 0.05L, w i t h a c e n t r a l elliptic d e l a m i n a t i o n (Fig. l b ) , is c o n s i d e r e d for the following n u m e r i c a l e x p e r i m e n t s . The 0 . . . . . . 100 9 , 9 200 , X Fig. 14. Average pointwise energy release rate distribution. (In complementary mode) t2/t ~= 0.20, t~/L = 0.05, a/b = 1.50, all = 0.30 d e l a m i n a t i o n c o n f i g u r a t i o n is fixed as: a/L = 0.3; a/b = 1.5; a n d t2/t 1 = 0.1. The l o a d i n g a n d b o u n d a r y c o n d i t i o n s , a n d t h e finite e l e m e n t m o d e l i n g are t h e s a m e as in the p r e v i o u s p r o b l e m s . The m a t e r i a l p r o p e r t i e s are t a k e n as: E 2 = 26000; vi2 = v~3 = / 15 (1) 10' In comptementarymode supplementary mode (2) In 0.03 [ (2 t/ 1 2 3 t, 0.02 • u E 5 j ~ ( 1 ] a/b:1.00 a/b=1:50 a/b=2.00 a/b =3.00 0.01 180 0 o 1;o x 200 so Fig. 15-. Maximum pointwise energy release rate distribution. (In complementary mode) tg/t~ = 0.20, t~/L = 0.05, a/b = 1.50, a/L = 0.30 80 220 16o 1so 2so L Fig. 17. Influence of aspect ratio on maximum energy release rate a/L = 0.30; tJL = 0.05; t2/t 1 = 0.10 i:iill lab100 III III 2 o/b:150 3 a/b~2100 60 I / 2t0 x? 40 200 20 0 1.0 , 1.5 , 2.0 a/b 2.5 190 3.0 Fig. 16. Effect of aspect ratio of delamination on local and global buckling strengths a/L = 0.30; tJL = 0.05; t2/t ~= 0.10 Olo 50 100 2,. 150 200" 250 Fig. 18. Influence of aspect ratio on average energy release rate a/L = 0.30; tJL = 0.05; t2/t 1 ~ 0.10 900 Table 2. Reduction in global buckling strength of plate with elliptic delamination: Effects of the aspect ratio a/b R~ Rf 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 31.86 27.40 25.70 23.52 22.81 22.46 22.25 22.14 22.11 50.33 47.07 45.49 44.24 43.73 43.47 43.32 43.24 43.22 800 70 o 600 500 o/b =1.0 z,0o v23 = 0.16; G12 = G13 = G 2 3 = 7500. T h e n u m e r i c a l e x p e r i m e n t s are c o n d u c t e d for v a r y i n g r a t i o o f Y o u n g ' s m o d u l i i , E1/E 2 = 1.0-10.0; while fiber o r i e n t a t i o n is fixed a l o n g t h e m a j o r axis o f t h e elliptic d e l a m i n a t i o n . 30 o 200 10 ~ 0o 1 50 100 150 200 250 X Fig. 19. Location of point of maximum pointwise energy release rate a/L = 0.30; tJL = 0.05; tg/tt = 0.10 B o t h the local a n d global b u c k l i n g s t r e n g t h s i n c r e a s e as E1/E 2 increases (Fig. (26)). The local b u c k l i n g s t r e n g t h (2~c,) o f the d e l a m i n a t e plate, as expected, increases linearly as E1 increases. However, the global b u c k l i n g s t r e n g t h (2~r) of the d e l a m i n a t e d 50 230 0.035 III 2 o:30 0.030 220 40 0.025 g > 210 -xu30 s,~ 0.020 0.015 20O 0.010 20 4~ ~ 19o 0.O05 0 10 20 2; 3o 35 ~0 0 180 a Fig. 20. Effect of delaminate size on local and global buckling strengths a/b = 1.50; tJL = 0.05; t i t 1 = 0.10 50 100 450 x 200 250 Fig. 22. Effect of delamination size on maximum energy release rate a/b = 1.50; tJL = 0.05; t2/t 1 = 0.10 100000 Table 3. Reduction in global buckling strength of plate with elliptic delamination: Effects of the delamination size 10 000 1 000 100 a/L Rb Rf 0.40 0.50 0.60 0.70 0.80 22.26 23.32 25.69 31.86 31.86 43.45 44.22 45.51 48.07 52.70 10 g o (~,gcr)FE 1 (2~)FE 0.1 0101 . . . 0.01 0.010 0.008 . . . . . . . . . . . . . . 0.1 L/L Fig. 23. Influence of the plate thickness (t~) on buckling strengths of the delaminated plate a/b = 1.50; a/L = 0.30; t2/t I = 0.10 II I I/I 1 Q--2o 2~ Table 4. Reduction in global buckling strength of plate with elliptic delamination: Effects of the laminate thickness 0.006 > a 0.00~, 0.002 50 100 150 200 250 k Fig. 21. Effect of delamination size on average energy release rate a/b = 1.50; tl/L = 0.05; t2/t ~ = 0.10 structure increases initially a n d reaches a saturation point as E 1 increases. Finite element solutions are obtained for the critical buckling strength of equivalent u n d e l a m i n a t e d plates of original laminate thickness tl a n d of base plate thickness t3: ~ r (t = tl) and 2~r(t = t3) respectively. As it can be observed from Fig. (26), the critical buckling strength of the actual delaminated plate is m u c h lower t h a n that of the original u n d e l a m i n a t e d tJL Rb Rf 0.01 0.05 0.10 0.15 0.20 0.25 7.05 25.71 33.85 45.12 42.11 37.91 31.93 45.50 51.77 59.99 57.80 54.73 plate. It can be observed that, the delaminated plate is actually weaker than the plate with the delaminated layers completely peeled off. Typical energy release rate distributions along the d e l a m i n a t i o n front are shown in Fig. (27) for E~/E 2 = 8.0. Again, the m a x i m u m energy release rate is observed to be in the vicinity of the direction of m a x i m u m local-buckling-stiffness. For the 109 40 t2/t 1=0.10 o/b =1.50 a/L:0.30 8 7 7<./ / ~82 1 t~/L:O.05 0 0 10 20 30 40 tl/L =0.01 ; 1; 50 60 70 ~0 90 X 15 Fig. 27. Typical p o i n t w i s e e n e r g y release rate d i s t r i b u t i o n for d e l a m i n a t e d o r t h o t r o p i c plate E1/E 2 = 8 (a/b = 1.50; a/L = 0.30; tl/L = 0,05; tE/t ~ = 0.10) Fig. 24. Influence of laminate t h i c k n e s s o n average energy release rate a/b = 1.50; a/L = 0.30; t2/t ~ = 0.10 // 50 ,oi / t2/t1=0.10 o/,:o.3o 8 / ~/ / / ~-/ 5 E / ..; ~X- ~x // 40 / ?/~ ~o 30 El/E2: 10 60 s~ ~/ ~.-I ,,'/.~ / 4 20 o 2 0 . 0 5 10 15 ,~,~ 1250 1000 ~< 750, ~ . ~ 500 1 250 1 ] 1 •Lcr " i 2 9 i 3 , i 4 9 J 5 600 800 t i t I = 0.i0 aspect ratio of the delamination (a/b = 1.5) and the material orthotropy (EllE 2 = 8.0) considered, the delaminate plate exhibits maximum stiffness against buckling in the fiber direction and hence the point of maximum pointwise energy release rate is found to be at an angle of 0 ~ with reference to the major axis. As E1 increases, the local buckling strength increases and hence the energy release rate become significant at higher compressive loads. However, the average and maximum pointwise energy release rates are higher for higher material orthotropy at local-post-buckling range of loads (Fig. (28-29)). 1750 0 , 400 X Fig. 28. Effect of orthotropic material properties on maximum pointwise energy release rate a/b = 1.50; a/L = 0.30; tl/L = 0.05; Fig. 25. Influence of l a m i n a t e thickness o n m a x i m u m e n e r g y release rate a/b = 1.50; a/L = 0.30; talt ~ = 0.10 1500 200 9 i 9 i 6 7 El/E2 9 i 8 9 i 9 , 10 Fig. 26. Influence of material o r t h o t r o p y o n b u c k l i n g s t r e n g t h s a/b = 1.50; a/L = 0.30; tJL = 0.05; ta/t ~ = 0.10 5.8 Effects of fiber orientation Now the numerical experiments are repeated for the problem considered in the previous section for varying fiber orientation (~b = O, 15, 30, 45, 60, 75, 90); and the material orthotropyis fixed at EllE 2 = 8.0. The post-buckling deformation is plotted in Fig. (30) for typical fiber orientations. When the fiber is along the major axis of the delamination, the delaminate plate stiffness against buckling along major axis is comparable with that along the El/E2: 10 12.5 S 10.0 B 6 2 4 7.5 -1.0 . . . . 40 5.0 0 zu a 2.5 n 200 400 >, 600 30 ~ 0 50 0 10 183 S00 Fig. 29. Effect of orthotropic material properties on average pointwise energy release rate a/b = 1.50; a/L = 0.30; tt/L= 0.05; t2/t ~= 0.10 -0.2 5-0.4 --0 0..8 6~ C minor axis for the aspect ratio of delamination (a/b = 1.50) and 5O LO material orthotropy (E~/E 2 = 8.0) considered. Hence, the 0 lO 2o 20 postbuckling deformation conforms with the fundamental Lu 30 540~ ~ 0 0 10 b buckling mode in both x 1- and x2-directions. However, the delaminate plate becomes much weaker in the major axis direction against compressive loads as the fiber orientation deviates from the major axis direction. Accordingly, the delaminate plate deforms at a higher mode in the x~-direction 0 while it is undergoing postbuckling deformation in the -0.2 f u n d a m e n t a l mode in the x2-direction. Initially, the local buckling strength (21cr)slightly decreases as the fiber orientation ):-0.4 -0.6 (qb) increases; however, 2~crincreases for q5 > 30 ~ (Fig. (31)). The finite element solutions for the critical buckling strengths of 50 4C the equivalent undelaminated plates of thicknesses t~ and 0 t 3 -A~r(t = ti) and 2~ = t3) respectively- are found to be ~u 30 5L0~ ~ 0 0 10 maximum for fiber orientation ~b = 45 ~ The global buckling o strength of the actual delaminated plate (2gr) is, again, highly underestimated when compared to both the undelaminated configurations as shown in Fig. (31). In the supplementary modes of buckling (shown in Fig. (30)), the base plate comes in contact with the delaminate plate and -0.1 accordingly the energy release rate becomes partially negative -0.2 -0.3 along the delamination front. However, when the local and global post-buckling deformations are complementary, the 50 pointwise energy release rate is positive all along the 30 delamination front (Fig. (32)). Accordingly, the location of ,v 30 ~ 10 maximum energy release rate changes from 0 = 0 ~ to 0 = 75 ~ as the fiber direction changes from ~b = 0 ~to 0 = 90 ~ as shown in d Fig. (33). The average and the maximum energy release rates E1/E2:8.0; t2/t 1:0.10; a/b:1.50; tl/L=0.05; o/L:0.60 are plotted in Fig. (33) and Fig. (34) for typical fiber orientations Fig. 30a-d. Influenceoifiber orientation on postbuckling deformation and for varying load levels. For a typical post-buckling load EI/E 2 = 8.0; a/b = 1.50; a/L = 0.30; tJL = 0.05; t21t~= 0.10. a ~b=0~ level (Z = 740.0), the average and the maximum energy release 2 = 465.3. b ~b= 30~ ). = 259.8. c ~b= 60~ 2 = 287.5. d ~b= 90~ rates vary as fiber orientation varies as shown in Fig. (35). The ~. = 299.6 energy release rate is again found to be maximum in the vicinity of the direction of maximum-stiffness against compressive loads (Fig. (36)). an antisymmetric sense. The plate thickness is assumed as 5.9 t 1 = 0.05 L. The delamination configuration is fixed as: a/L = 0.3; Antisymmetric laminate with central elliptic delamination a / b = 1.50; and t2/t1 =0.10. The loading and boundary In the following numerical experiments, the plate is assumed conditions, and the finite element modeling are the same as in to be a laminated composite with orthotropic layers stacked in the previous numerical experiments. The material properties - - 3o'~ 70 X2 60 50 40 30 20 1 800 10 1 600 0 1 400 184 -10 25 1 200 ,4 b ~ = 30 ~ 1 000 20 800 15 600 400 0 10 x~r 200 o /s 6'o r 9o E~/E2:8.0, t 2 / t ~ = 0 . 1 0 , { a / b : l . 5 0 : t J L - - 0 . 0 5 : a / L : 0 . 6 0 0 25 Fig. 31. Influence of fiber orientation on local and global buckling strengths E~/E 2 = 8.0; a/b = 1.50; alL = 0.30; t J L = 0.05; t2/t ~ = 0.10 20 15 10 for each layer are taken as: E~ = 208000; E z = 26000; v12 = v~3 = v23 = 0.16; Ga2 = G13 = Gz3 = 7500. All layers are assumed to be of equal thickness. The following three lay-up sequences are considered: (i) the laminate is composed of 20 laminae stacked 0 in the recursive sequence 0~176 (ii) the laminate is composed 14 of 100 laminae stacked in the recursive sequence 0~176 and (iii) the laminate is composed of 20 laminae stacked in the 12 d 0:90 ~ recursive sequence - 450/45 ~ The numerical experiments are 10 conducted for symmetric and antisymmetric configurations for 8 the delaminate plate. The local and global buckling strengths of the delaminated 6 plates are plotted for varying delaminate thicknesses in 4 Fig. (37). It can be noted that local buckling strength increases as 2 the delaminate thickness increases and becomes nearly equal to or greater than the global buckling. The finite element 0 solution for the equivalent undelaminated plate with -2 base-plate-thickness is also plotted for comparison. The global 10' 20' 30' 4o" 5'o 60' 7b 8'0 90 @ strength of the actual structure is less than the equivalent undelaminated-base-plate-strength. The behavior is nearly E l / E 2 = 8.0: t2/t 1 =0.10; o / b =1.50, t l / L = 0 . 0 5 r a / L = 0.60 identical for case-i and case-ii. The threshold delaminate thickness for case-i and case-ii is found to be t2/t ~ = .175 while Fig. 32. Influence of fiber orientation on pointwise energy release rate distribution E1/E 2 = 8.0; t2/t 1 = 0.10; a/b = 1.50; tt/L = 0.05; that for case-iii is t2/tt = .08 below which critical buckling a/L = 0.30 strength of the actual structure is less than that of the base-plate. The residual drop of buckling strength is found to be considerably less in case-iii when compared to the other two cases. Even though, case-i and case-ii show identical results with symmetric delaminate configuration (tE/t 1 -- 0.15). It is found reference to their buckling strengths, the energy release rate that the energy release rate drops considerably when the number distribution is found to be quite different. The energy release of layers (for the same total laminate thickness) increases rates are plotted along the delamination front for the three cases (compare case-i and case-ii). Since the angle-ply configuration with antisymmetric delaminate configuration (t2/t ~ = 0.10) and (case-iii) makes the plate stiffer (note the rise in buckling 90 ~=0 ~ 10 60o 30~ 75' ---.60 j 6 45- 90 o 4 3015- 2 84 0 ' 0 i 01 0o , 100 200 300 400 X 500 600 700 800 . ," . . . . 150 300 9 450 . . . . 600 75 ~ 90~ * Fig. 33. Influence of fiber orientation on average pointwise energy release rate E J E 2 = 8.0; t21q = 0.10; a/b = 1.50; tJL = 0.05; a/L = 0.30 Fig. 36. Angle of m a x i m u m energy release rate for different fiber orientations E J E 2 = 8.0; t2/t ~ = 0.10; a/b = 1.50; tJL = 0.05; a/L = 0.30 0=0 ~ 20 18 5 1600 J ' N o 60" 30~ 12001 90~ ~ 1 5 84 600 o S o11 a i 1 i 200 400 600 BOO Fig. 34. Influence of fiber orientation o n m a x i m u m pointwise energy release rate E1/E2 = 8.0; t2/t ~ = 0.10; a/b = 1.50; tJL = 0.05; a/L = 0.30 ~.0 L X~cr 0'.2 ola o.a tJtl 2 000 .1200! ~ 8001 /// "". 4oo ~%=740.0 30 0 0'.1 b 012 t2/tl 0.3 0.4 ~'~20 j $ Fig. 37a, b. A n t i s y m m e t r i c laminate with central elliptic delamination. Influence of delaminate thickness on local a n d global buckling strengths, tJL = 0.05; a/L = 0.30; a/b = 1.50; a n d E1/E2 = 8 . 0 for each lamina, a 10 x (0o/90 ~ b 10 x ( - 45~ ~ -----... 10 9 0o i 150 . ' i 300 , i 45 ~ , J 60 ~ 9 i 75~ 90 ~ Fig. 35. Variation of average a n d m a x i m u m pointwise energy release rates with fiber orientation E1/E 2 = 8.0; t2/t 1 = 0.10; a/b = 1.50; tJL = 0.05; a/L = 0.30 s t r e n g t h s , Fig. (37b) w h e n c o m p a r e d to Fig. (37a), t h e e n e r g y r e l e a s e r a t e is c o n s i d e r a b l y l o w e r w h e n c o m p a r e d to c a s e - i f o r t h e s a m e l o a d level. T h e a v e r a g e a n d t h e m a x i m u m e n e r g y release rates are plotted for different cases considered here in Fig. (39). 16 18/ , ~"o/,~ 1/-. 14[ 12 ,>'i, 10 8 6 6 qqS.'~ 4 2 0 ]86 I -20o I 10~ I 20 ~ I I 30 o 40 ~ a, 50 o I 600 I I 70 o 50 ~ -2 e 0.010 / . . . i 90 ~ 0~ i 100 i i 20 ~ i 300 I /*0~ d i 50 o i 600 i 70 ~ 500 90" e 2.5 . 0.009 k 0.008 t ~ =87/..2 0.007 ~ " 2.0 0.006 Ix. 787.1 1.5 ~ 0.005 k~69~. 8 0.00/,1 \ ~\ 0.003 612./, 1.0 0.5 o o. ooi 0 ~ 0 ~O01 I 0~ -- 10 0 I 20" I I ~ 30 ~ 50 ~ 60 ~ 70 ~ 50 ~ 90 ~ -0.5 0 o 4"0~ b e 10 ~ I I A 20 ~ 30 ~ I 400 ,('~ 10 30 I 60 ~ 70 o I 500 900 @ e 35 I 500 . . . . . . . 8 25 6 20 ~ ~.~15 10 4 2 5 0 0 -5 i 0~ 100 i i i I 20 ~ 30 o 40" 500 I i -2 i 600 70 ~ 80 ~ 900 8 i O" f 10 ~ I 20" i 300 i i 40 ~ 500 e i ~ i 600 70 ~ 50 ~ 90" Fig. 38a-f. Antisymmetric laminate with central elliptic delamination. Influence of delaminate thickness on pointwise energy release rate distribution a/b = 1.50; tl/L = 0.05; a l l = 0.30; E J E z = 8.0 for each lamina, a 10 x (00/90~ laminate tJt~ = 0.10. b 50 x (0o/90~ laminate t2/t I =0.10. r 10 x ( - 450/45 ~ laminate tz/t ~ = 0.10. d 10 • (0o/90~ laminate t2/t ~ = 0.15. e 50 x (00/90~ laminate t2/t ~ = 0.15. f 10 x ( - 45o/45 ~ laminate t2/t ~ = 0.15. 5.10 Symmetric laminate with central elliptic delamination In this section, the l a m i n a t e d plate considered in the previous example is assumed to be symmetrically constituted with 32 orthotropic laminae in the following fashion: (i) ( 0 1 9 0 / 4 5 1 - 45), a n d (ii) ( 4 5 / 9 0 / - 4 5 / 0 ) , . Material properties, laminate a n d delaminate configurations, loading a n d b o u n d a r y conditions, a n d finite element m o d e l i n g r e m a i n the same as in the previous example. The d e l a m i n a t i o n is assumed to be between the n th a n d the (n + 1) th layer from the top surface where n takes values from 1 to 15. The characteristic buckling strengths for the delaminated structure are plotted in Fig. (40) for both case-i a n d case-ii. It is found that the local buckling strength becomes nearly equal for n = 8 or t J t l = 0.25. The threshold delaminate thickness for which equivalent u n d e l a m i n a t e d base plate is stronger than the actual structure in buckling is found to be 0.125 and 0.15 for case-i a n d case-ii respectively. It is interesting to note that the global buckling strength does not vary m u c h with the delaminate thickness for case-ii. The average a n d the m a x i m u m energy release rates are plotted for case-i and case-ii for typical delaminate thickness (n < 8) in Fig. (41) and Fig. (42). It should be noted that local buckling does not occur when n > 8 a n d hence no delamination growth is expected because of buckling action. Since the delaminate alternates between symmetric a n d antisymmetric 2 000 1 800 1 600 1 4O0 200 / x~ 1 000 800 2a 6O0 4O0 ~--...~,~ . 187 200 , i 9 , , , 9 500 , , , , , 1000 1500 a 0 , , , , k 2 000 1 800 15 t 600 1400 1 200 ~10 ~ '~ I 000 E J 800 6O0 5 400 2000 , i , i , i , 500 b i 1000 , i , i 1 10,(0~ ~ [erninote 2 50,(0~ ~ tominute 3 lO,,(-/-,5~176 E~./E::8.0 (o) (o) (o) for Fig. 39a, b. Antisymmetric laminate with central elliptic delamination a/b = 1.50; tJL = 0.05; a/L = 0.30. a Average pointwise energy release rate. b Maximum pointwise energy release rate configurations as n increases, the energy release rate variation also changes for different delaminate thicknesses as shown in the figures. Variation of the average and the maximum energy release rates with the delaminate thickness is plotted for typical load levels in Fig. (41c-d) and Fig. (42c-d) respectively. It is observed that the average energy release rate is lower for case-i while the maximum energy release rate is lower for case-ii. 6 Concluding 0.05 b t 2 / t l : 0 . 1 0 (b) t2/tl=0.15 t2/tl=0.10 (b) t2/t~=0.15 t2/t~=0.10 (b) t2/tl=O.15 each Lomino remarks In this paper, the reduction in buckling strength of a plate with an elliptic delamination and the mechanisms of delamination growth under post-buckling conditions, were critically examined using a finite element method. A quasi-conforming 3-noded plate element based on Reissner-Mindlin plate theory and the multi-plate modelling technique are used to model the delaminated plate. Gauss elimination solution algorithm is used to solve the incremental equations in a cycle of NewtonRaphson iterations. Arc-length controled load incrementation and a linearised asymptotic solution for branch switching are used for effectively traversing both limit and bifurcation points. ,. 0 , 1500 X Xt 0.10 0.15 0.20 0.25 t2/tl Fig. 40a, b. Symmetric laminate plate with central elliptic delamination Buckling strength reduction for different delaminate thickness, a Case-i. b Case-ii a/b = 1.50; tt/L = 0.05; all = 0.30 The post-buckling solution and the energy release rate models are validated using a simple classical p r o b l e m isotropic square plate with central near-surface delamination. Influence of the following structural parameters on the local and global buckling strengths of the structure, and on the pointwise energy release rates along the delamination front, are critically studied: delaminate plate thickness (tJtl), aspect ratio of the delamination (a/b), size of delamination (a/L), total laminate thickness (tJL), material orthotropy (EJE2), fiber orientation (~b), antisymmetric laminate with symmetric/antisymmetric delaminate plates (viii) symmetric laminate with symmetric/asymmetric delaminate plates (i) (ii) (iii) (iv) (v) (vi) (vii) The influence of these parameters are in particular examined regarding the reduction of the structural strength against buckling, coupling of local and global post-buckling solutions, and pointwise energy release rate at the delamination front. Several observations are made regarding the post-buckling failure mechanisms. 15 !5 I . "1 aC 4 bC 10 15" > TO " 5 54 ~88 O] 400 800 1200 ., . . . . . . . . . . . 0 400 ~00 X 1200 x 0.4 25 e-]l Case -11 20 0.3 15 02 ~10 0.1 Fig. 41a-d. Symmetric composite plate with central elliptic delamination Average pointwise energy release rate a/b = 1.50; tJL = 0.05; a/L = 0.30 5 0 0 0 c At X- z,00 0.1 0.2 0 t2/tl 0.1 d At ;L =1300 Z,0 0.2 t2/tl 40 a Case-I [/ 5 30 / b Case-]I 30' 20 ~-~ 20' 10 10. 0 0 0 400 800 5 1200 9 i 0 9 i , r 9 i /,00 ' i 9 800 i ' 1200 x 4O 0.6 30 ~ - I 0.4 ~ 20 0.2 10 0.1 c At,L : 400 0.2 t~/tl 00.1 d At ;~ =1300 0.2 t,~/tl Fig. 42a-d. Symmetric composite plate with central elliptic delamination Maximum pointwise energy release rate a/b = 1.50; tl/L = 0.05; a/L = 0.30 of Excellence of Computational Modelling of Aircraft Structures, References Atluri, S. N. 1986: Energetic approaches and path independent integrals Georgia Tech., Atlanta Naganarayana, B. P. 1991: Consistency and correctness principles in in fracture mechanics. In: Atluri, S. N. (ed): Computational methods quadratic displacement type finite elements. Ph.D. Dissertation, Indian in mechanics of fracture, Amsterdam: North-Holland Barlow, ]. 1976: Optimal stress locations in finite element models. Int. Institute of Science, India Naganarayana, B. P.; Atluri, S. N. 1994: Energy release rate evaluation J. Numer. Meths. Engg. 10:243-251 for delamination growth prediction in a multi-plate model of a laminate Chai, H.; Babcock, C. D.; Knauss, W. G. 1981: One-dimensional modelling of failure in laminated plates by delamination buckling. Int. composite. Comput. Mech. (to appear) Naganarayana, B. P. 1994: Incremental iterative strategies for J. Solids and Struct. 17(11): 1069-1083 automated post-buckling analysis. (To be published) Huang, B-Z.; Shenoy, V. B.; Atluri, S. N. 1994: A quasi-conforming triangular laminated composite shell element based on a refined first Timoshenko, S. P. 1940: Theory of plates and shells. 1st Ed., McGraw Hill, New York, 1940 order theory. Comput. Mech. 13:295-314 Huang, B.-Z.; Atluri, S. N. 1994: A simple method to follow postbuckling paths in finite element analysis. Internal Report, FAA Center 189