Contact buckling and postbuckling of thin rectangular platesherzl/2001/Contact buckling and... · Contact buckling and postbuckling of thin rectangular plates Herzl Chai * Department

Journal of the Mechanics and Physics of Solids49 (2001) 209–230

www.elsevier.com/locate/jmps

Contact buckling and postbuckling of thinrectangular plates

Herzl Chai*

Department of Solid Mechanics, Materials and Structures, Faculty of Engineering, Tel Aviv University,Ramat Aviv, Tel Aviv 69978, Israel

Received 18 January 2000; received in revised form 19 May 2000

Abstract

A combined experimental/finite element effort is carried out to elucidate the post bucklingresponse of unilaterally constrained plates under monotonically increasing edge thrust. Realtime observations, together with a wide range of plate aspect ratio and a large load levelfacilitate deep physical insight into the general behavior of this class of problems. The interac-tion of the plate with the rigid restraining plane following buckling leads to interesting defor-mation sequences, characterized by the development of asymmetric bulges and contact zonesfollowing by a possible plate snapping. The latter is motivated by a secondary buckling evolv-ing gradually from a contact zone(s) or a bulge(s). These two instability mechanisms arecompetitive, being dictated by the plate aspect ratio and other system parameters. The criticalload for plate snapping agrees well with a finite element prediction based on an asymmetricdeformation choice that minimizes the strain energy in the plate. A semi analytic relation forpredicting the onset of secondary instability in the contact area and subsequent plate snappingis developed based on the numerical results. Finally, the present work seems to add a newdimension into the fracture of coatings and laminated composites containing near-surfacedefects. 2001 Elsevier Science Ltd. All rights reserved.

Keywords:A. Buckling; B. Plates; B. Contact mechanics; C. Stability and bifurcation; B. Elastic materials

1. Introduction

This work was originally motivated by buckling-induced growth of interlaminardefects in layered composites. A delaminated surface layer of certain geometry and

* Fax: 00972 3 6407617.E-mail address:[email protected] (H. Chai).

0022-5096/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S0022 -5096(00 )00038-7

210 H. Chai / Journal of the Mechanics and Physics of Solids 49 (2001) 209–230

loading conditions may buckle into a wavy pattern. However, the natural develop-ment of this deformation may be altered by interaction with the parent laminate,which may affect the stiffness and fracture resistance of the global structure. Thisphenomenon has been considered, albeit in a limited scope, by a number of authors(e.g. Chai and Babcock, 1985; Chai, 1990a,b; Whitcomb, 1988; Giannaakopoulos etal., 1995; Sekine et al., 2000), mainly as it relates to fracture or delamination growth.Contact between flexible beam/plate elements and hard substrates is an issue of con-cern in a variety of other technological applications, including civil engineering (e.g.reinforcements) and materials science (e.g. coatings). In view of the basic nature ofthis class of problems, a more comprehensive treatment, particularly in the postbuckling regime, seems to be warranted. The present work involves the evolutionof secondary buckling in the film/substrate contact zones or the uplifting portionsof the plate, and the snapping of the plate to new equilibrium states. The followingliterature survey focuses on these aspects.

It is instructive to consider first relevant one-dimensional configurations as theyafford analytically tractable solutions from which a valuable insight into more generalcontact buckling problems can be gained. Chateau and Nguyen (1991), and Adan etal. (1994) show that when a column positioned a distance from a flat wall is com-pressed, contact zones may develop leading to buckling mode transition. More recentworks dealing with thebilaterally constrained column (Chai, 1998; Domokos et al.,1997; Holmes et al., 1999) have exposed more relevant characteristics of the contactbuckling problem, including a sequential mode transition process, an inherent asym-metry of the deformation pattern, and a unique hysteresis in the load vs axial dis-placement curve.

Siede (1958) was apparently the first to study contact effects in buckled plates.In his study of infinitely long, simply supported plates. Seide found that a rigidlateral constraint may increase the buckling resistance by as much as 33% relativeto the unrestrained plate. Shahwan and Waas (1994) have extended this work tomaterial orthotropy and various boundary conditions. The buckling strength due tounilateral constrain of finite size plates was considered by Wright (1995) and Smithet al. (1999) using finite element and Rayleigh–Ritz approaches, respectively. Theeffect of lateral constraint on the plate deformation was vividly demonstrated byComiez et al. (1995) with the aid of the shadow Moire technique. However, theresponse of the plate could not be separated from the overall structure in these testsbecause the two were firmly attached at the plate boundary. Plate contact studies inthe post buckling regime are scarce. Hhatake et al. (1980), using a finite elementscheme coupled with the penalty method, have provided information on the evolutionof contact with compression load. No plate snapping was noted in this study, whichwas limited to the early posbuckling stage.

Buckling mode transition has been reported for unrestrained plates. Stein (1959)was apparently the first to observe this phenomenon in long plates, which he attri-buted to secondary buckling in the may plate. Secondary buckling also occurs inaxisymmetrically compressed circular plates (e.g. Keller and Reiss, 1958). In thiscase, the deformation is characterized by wrinkles around a narrow circumferentialstrip near the plate edge. Hutchinson et al. (1992) provide a vivid experimental dem-

211H. Chai / Journal of the Mechanics and Physics of Solids 49 (2001) 209–230

onstration of the evolution of circumferential buckling lubes in circular films on rigidsubstrates under the combined action of in-plane compression and transverse loading.The onset of secondary buckling in circular or rectangular plates have been studiedanalytically by a number of authors, mostly using variational energy principles basedon perturbation from a given post buckling state (e.g. Ceho and Reiss, 1974; Naka-mura and Uetani, 1979). Such analyses are complex and generally do not provideinformation on the new equilibrium state following secondary buckling.

In this work, the response of unilaterally constrained plates due to edge thrust isstudied experimentally and analytically. To simulate near surface debonding prob-lems, all four edges of the plates are clamped. A systematic variation of the plateaspect ratio and load level, together with real time observation of the outward platedeformation via the shadow Moire technique, allow for general conclusions to bemade. The tests show that a lateral constraint may lead to a buckling mode transition.Unlike for common plates, however, this transition may be driven by secondaryinstability evolving from the contacting zones. A large strain, large deformation finiteelement analysis incorporating a frictionless contact algorithm is employed to modelthe plate response. The analysis provides quantitative insight into the post-bucklingbehavior, including secondary buckling and plate snapping. The buckling stage isdiscussed in Section 2. The experimental program is reported in Section 3, wherethe phenomenological aspects of the deformation process are exposed. This infor-mation is then used to construct appropriate numerical models (Section 4). Somediscussions and conclusions are given in Sections 5 and 6, respectively.

2. Problem definition

2.1. Nomenclature

Thin, flat rectangular plates of widthb, lengtha and thicknesst are subjected toa uniform downward displacement,Va. The plates are clamped on all four edges,with the unloaded edges stress free and the bottom edge fixed. It is customary touse the following thin-plate normalization

K512(1−n2)b2

p2t2e0 (1)

wheren is Poisson’s ratio of the plate ande0 is defined as

e0;Va/a. (2)

In the following, K will be referred to as “load”. Note that in the case of thin-filmdelamination problems,e0 can be interpreted as the far-field strain in the substrate.

2.2. The buckling stage

The buckling strength of a clamped, unrestrained plate was apparently first determ-ined by Levy (1942). The variation of the buckling load with the plate aspect ratio,R, is shown in Fig. 1, where


Fig. 1. Finite element prediction (triangular symbols) for normalized buckling strain vs aspect ratio fora clamped, unilaterally constrained plate under axial edge displacement. Also shown are results for unre-strained plate (Levy, 1942).

R;a/b. (3)

The plate buckles into a half wave ifR,R0 (=1.07); for larger aspect ratios, thenumber of bulges increases withR. In the case of a unilaterally constrained plate,R0 has a special meaning because for larger aspect ratios, contact between the plateand the support occurs. In this range, the constrained plate buckles into a multitudeof half waves, each having the same profile as the one forR0. It is clear that thenumber of bulges in the plate,n, relates to the aspect ratio according to the follow-ing rule

n5smallest closest integer of the ratioR/R0 (4)

(e.g. if R/R0 =1.7 or 2.3, there will be one bulge and two bulges, respectively). Thebuckling load forR.R0 remains fixed at the same value as forR0, i.e. K=10. Thus,the constraint increases the buckling strength relative to the unrestrained plate by anamount that increases with the plate aspect ratio. ForRÀ1, this amounts to 30%[which is similar to the simply supported case (Siede, 1958)]. Note that forR.R0,the bulge(s) may freely move within the span of the plate without affecting the energystate in the system (assuming a frictionless contact). This feature, also common to


bilaterally constrained columns (Chai, 1998), may greatly affect the post bucklingresponse.

3. Experimental

3.1. Apparatus

Tests are carried out to elucidate the response of clamped rectangular plates underaxial compression. Fig. 2 details the testing apparatus. To increase the range of elasticdeformation possible, very thin plates are used. Because this entails a great sensitivity

Fig. 2. Test fixture and test panel details. All dimensions are in mm.


to geometric imperfections, a high degree of precision in the machining and align-ment of all relevant components of the test apparatus is employed. The plates aremachined to specific dimensions from 1 mm thick Polycarbonate sheets havingYoung’s modules (E), Poisson’s ratio and proportional limit of 2.3 GPa, 0.35 andapprox. 1%, in that order. A range of plate dimensions are examined, i.e.b=25.4,50.8 and 76.2 mm, andt=0.5, 0.625, 0.7 and 1 mm. The aspect ratio varies from0.75 to 6 (or from 0.7R0 to 5.6R0). The plates rest upon a 38 mm thick aluminumblock which is firmly attached to the test fixture. The longitudinal edges of the platesare constrained by flat steel bars. The latter are adjusted prior to testing so as toproduce intimate yet unconstraining contact. The lower edge of the plate is boltedto the aluminum block while the upper one is attached to a steel block that travelsalong a tight, yet frictionless confinement. To reduce unwanted bending and twisting,the load is applied via a centered steel ball. All moving components are oiled priorto testing to reduce friction.

The out of plane deformation in the plate are obtained via the shadow Moiretechnique; the optical test set up is as described in Chai et al. (1983). A 6 lines/mmMoire grid is placed several millimeters from the sample. Illumination is achievedby a collimated light beam directed at 29° to the plate normal. The sample is viewedin the normal direction, giving a fringe constant,f, of 0.3 mm. The upper edge ofthe plates is compressed at a slow rate to well over the buckling load using a screwdriven testing machine (Instron). The axial load,P, and the end shortening,Va,obtained using an LVDT, are recorded during the tests. The evolving fringe patternis monitored using a video camera.

3.2. Test results

Figs. 3–5 show three Moire sequences representative of the deformation patternobtained over the range of aspect ratio studied. Corresponding normalized load,K,and normalized fringe constant,f/t, are noted in the figures; as indicated in Section2.2, buckling occurs atK=10. The post buckling behavior is greatly affected by theaspect ratio and the load level. A summary of the main features follows.

Fig. 3, R=1.2 — Intermediate stage of post buckling (I) followed by onset ofsecondary buckling near the vertical edges of the bulge (II), the final stage of bulgecollapse (III) and the splitting of the bulge into two separate bulges (IV). Print Vcorresponds to unloading, just before the transition from two to one bulge; note theassociated load is well less than that in (IV), indicating a pronounced elastic hyster-esis. Bulge splitting following secondary buckling typifies the range 1.15,R,1.3;for smaller aspect ratios, no splitting occurred after onset of secondary buckling,apparently due to a tight confinement of the uplifting bulge.

Fig. 4, R=1.7 — A single bulge forms near the upper edge (I). The developmentof a second bulge in the contact area below the first bulge leads to a transition intoa two-bulge configuration (II). The next significant event is the onset of secondarybuckling at the vertical edges of both bulges (III). Note that the plate response there-after was stable (i.e. no bulge splitting). This behavior was common over therange 1.3,R,2R0


Fig. 3. Moire fringe sequence for unilaterally constrained plate under axial compression;b=76.2 mm,t=0.7 mm,R=1.2, fringe constant=0.3 mm, normalized fringe constant,f/t, is 0.43.K represents normalizededge displacement. Prints I–IV correspond to loading while printV to unloading.

Fig. 5, R=2.9 — Buckling starts with two randomly positioned bulges (I), in con-sistency with the buckling behavior discussed in Section 2.2. A third bulge developsnear the upper edge of the plate (III), much the same as the second bulge in Fig. 4.Note that the extent of the contact zones between the bulges increase with load, butnot symmetrically. Two small humps develop in the largest of the contact zones,i.e. just below the central bulge (III), leading to a fourth bulge (IV). This behaviortypifies the range 2R0,R,3R0, with the production of the fourth bulge being sup-pressed in favor of secondary buckling in the three uplifting bulges ifR is less thanabout 2.6.

For longer plates, more bulges emerge at buckling, in consistency with Eq. (4).For example, forR=6 (b, t =25.4 mm, 0.5 mm, results not shown), five bulges initiallyemerged. Similarly to the process described earlier, a transition to six and then sevenbulges occurred with increasing load. At that stage, plasticity has become significant.

Several trends are apparent from these and other test results. (1) The bulges are


Fig. 4. Moire fringe sequence for a unilaterally constrained plate under axial compression;b, t=76.2mm, 1 mm,R=1.7, fringe constant=0.3 mm, normalized fringe constant,f/t, is 0.3.

generally asymmetric, with their position changing with load. (2) Local instabilityis the precursor to buckling mode transition; it may initiate from a contact region,analogously to the bilaterally constrained column case, or within the bulge(s), inparticular at the edges of the bulge, analogously to secondary buckling or wrinklingin unrestrained plates. (3) These two instability mechanisms, to be referred to as“secondary contact buckling” and “secondary circumferential buckling”, respect-ively, are competitive; the onset of one delays or prevents the other. (4) Friction isexpected to play a role on the plate response, but snapping tends to erase this effect.

A systematic study was carried out to determine the critical loads for mode tran-sition. Three plates each of a given aspect ratio were tested, each one repeatedlyloaded a total of three times. Thus, nine data point are produced for each aspectratio. Fig. 6 (symbols) displays the critical loads obtained; the data forR=0.75 andR=1.2 correspond to the onset of secondary circumferential buckling (i.e. no snappingwith increasing load) while the rest of the data to buckling mode transition or snap-ping. In this case, the data forR,2R0 and R.2R0 correspond to a jump from oneto two bulges and two to three bulges, respectively. The results exhibit some scatter,with no pronounced difference between different specimens or repeated tests on thesame specimen. These data will be discussed further following the development ofthe analysis.


Fig. 5. Moire fringe sequence for a unilaterally constrained plate under axial compression;b, t=50.8mm, 0.625 mm,R=2.9, fringe constant=0.3 mm, normalized fringe constant,f/t, is 0.48.

4. Analysis

4.1. Finite element model

A large strain, large deformation, commercial finite element code (Ansys, Version5.3) with a built in contact algorithm is employed. A four node rectangular plateelement (Shell 63) and a three dimensional contact element (Contact 49) are used;contact between plate and support is assumed frictionless. (The effect of friction,while deemed significant, may only complicate the exposition and interpretation ofthe fundamental characteristics of the problem.) In consistency with the tests, theoutward deflection,w, and the plate rotation, dw/dl, wherel denotes the normal tothe plate edge, are assumed zero along all four edges of the plate. Also, the in-planedisplacements on the lower edge and the horizontal displacement on the upper edgeare assumed zero. The upper edge is given a uniform vertical displacement that isapplied in load steps. The center of the expected bulge(s) is (are) given a small initialoutward displacement to help obtain a smooth transition into the buckling regime.This auxiliary displacement is removed once the load exceeds about 50% of thebuckling load.

The plate response is studied as a function of aspect ratio, up toR=3R0; thebehavior for largerR can be inferred, to some extent, from these results. As discussed


Fig. 6. Normalized critical loads as a function of plate aspect ratio. Symbols and curves denote experi-mental results and finite element predictions, respectively. Experiments: for each aspect ratio, three differ-ent samples were tested, each one repeatedly loaded three times. Solid and open symbols correspond tosecondary circumferential buckling and plate snapping, respectively. Finite element: shown are resultsfor the three deformation models defined in Fig. 7; solid and dashed curves correspond to plate snappingand onset of secondary circumferential buckling, respectively.

in Section 2.2, forR,2R0 only one bulge forms at buckling while for 2R0,R,3R0,two bulges initially form; in both cases, the bulge(s) is of widthb and aspect ratioR0. Because in the rangeR.R0 the position(s) of the initial bulge(s) within the plateis (are) inherently random, a few (three) extreme configurations are analyzed, namelysymmetric, off-symmetricandasymmetric, see Fig. 7:

Symmetric: If R,2R0, a single outward displacement is initially forced at the platecenter; if 2R0,R,3R0, two initial displacements are forced, each a distanceR/4 froma plate edge.

Off-symmetric: This configuration is defined only forR.2R0. In this case, twoinitial displacements are placed a distanceR0/2 from each edge of the plate.

Asymmetric: If R,R0, one initial displacement is placed at the plate center; if


Fig. 7. Initial bulge configurations for the three deformation models used in the finite element analysis.These configurations are generated by forcing initial outward displacements. The latter are removed oncethe load exceeds about 50% of the buckling load.

R0,R,2R0, one initial displacement is placed a distanceR0/2 from the lower plateedge; if 2R0,R,3R0, two initial displacements are placed at distancesR0/2 and 3R0/2from the lower plate edge.

The symmetric model represents an upper bound for the snapping load becauseit affords the smallest contact zones possible from which a local instability leadingto plate snapping may ensue. As will be seen latter, the results for the two nonsym-metrical models are indistinguishable, and they lead to the lowest bound. To takeadvantage of symmetry, only half of the plate region is modeled; for thesymmetricand off symmetricconfigurations, only a quarter of the plate is modeled.

Analyses are performed for a range of plate dimensions. A systematic meshrefinement study shows that as many as 1000 rectangular elements are necessary toinsure convergence of the post buckling solution over the range of parameters stud-ied. Results from different geometries (i.e.b=50.8 mm and 76.2 mm, andt=0.35mm and 0.7 mm) show that the post buckling solution (i.e. stresses, deflections) canbe scaled in accordance with common thin plate problems [e.g. Eq. (1),w/t, etc.].Accordingly, the data to be presented, corresponding to the specific choiceb=50.8mm andt=0.35 mm, are given in a non dimensional form.

4.2. The buckling stage

The numerical solution converged only after the load exceeded about 50% of thebuckling load,Kcr. The following technique for securing accurate solution in theearly buckling stage was applied: load up to 50% aboveKcr, unload to nearlyKcr,and reload. The first and second loading paths (i.e.K vs central deflection) differ,but the second and third paths coincided, indicating convergence. The buckling datathus obtained are displayed in Fig. 1 as triangular symbols. The results agree with


those of Levy (1942), and extends the latter down toR=0.22. The buckling load forR,0.5 seem to obey well a power law of the form

Kcr54.4/R2.05. (5)

4.3. Post buckling

Figs. 8–10 show the evolution of outward displacement contours with load forthe three experimental case studies depicted in Figs. 3–5; the normalized fringe con-stant (f/t) and load (K), indicated in each print, are identical and nearly the same,respectively, to those of the corresponding test sequences. It is evident that the defor-mation patterns generally well resemble the test results, notably in the developmentof local instability, either near the vertical edges of the uplifting bulge(s) or at thecontact zone(s), and in the rapid mode transition that follows. A closer examinationreveals that, for a given load, the displacements in a bulge are independent of thedeformation model [e.g. compare print II of Fig. 9(a) with print I of Fig. 9(b)], beingnumerically consistent with the respective test results. A brief exposition of the mainfeatures follows.

Fig. 8,R=1.2 — In consistency with the test results (Fig. 3), the asymmetric modelis shown. Major events such as onset of secondary circumferential buckling (II),rapid bulge distortion (III) and bulge splitting (IV) seem to resemble well the testresults, both qualitatively and quantitatively. The bulge mobility with load is noted(i.e. print II compared to print I).

Fig. 9, R=1.7 — Both asymmetric (a) and symmetric (b) models are shown.Clearly, the former better simulates the test counterpart (Fig. 4). The second bulgein Fig. 9(a) is seen to initiate from the contact area below the first bulge. Its evolutionis gradual, but the final transition to a two-bulge configuration is rapid, tendingtoward a symmetric deformation (III). Upon increasingK to 95, secondary bucklingoccurs near the vertical edges of each of the two bulges. This, however, did not leadto bulge splitting with increasing load as in the previous case, apparently due to a

Fig. 8. Outward displacement contours for an asymmetric finite element model withR=1.2. K is thenormalized edge displacement. Normalized fringe constant,f/t, is 0.43, as in Fig. 3.


Fig. 9. Outward displacement contours for asymmetric (a) and symmetric (b) finite element models,R=1.7. Normalized fringe constant,f/t, is 0.3, as in Fig. 4.

tight spatial confinement. The behavior for the symmetric model (b) is different.Now two secondary bulges simultaneously develop over the contact regions sym-metrically to the first (II). However, once these bulges mature, they quickly pushout the central bulge (III), ending in two symmetric bulges (IV); the rest of theloading is thus similar to that discussed for the asymmetric case (a).

Fig. 10, R=2.9 — Shown is the asymmetric case. In consistency with the dis-cussion in Section 2.2, two bulges initially form (I). The rest of the loading is muchthe same as discussed for Fig. 9(a), i.e. emergence of an additional bulge over thelargest contact area (II), and onset of a secondary circumferential buckling in eachbulge (III). In contrast to its experimental counterpart (Fig. 5), however, no fourthbulge has developed upon further loading. In trying to understand this departure,one observes from Fig. 5 that the contact areas become uneven as the load is


Fig. 10. Outward displacement contours for asymmetric finite element model,R=2.9. Normalized fringeconstant,f/t, is 0.48, as in Fig. 5.

increased fromK=37.3 toK=69; the fourth bulge emerges from the largest contactarea. Such unevenness is also observed in print III, but it is apparently insufficientto cause secondary contact buckling before secondary circumferential buckling inter-venes. This demonstrates the complex interplay between these two secondary buck-ling mechanisms, and how imperfections and frictional effects may possibly play animportant role on the post buckling response.

The onset of equilibrium loss in the plate, determined from inspection of the con-tour plots, is displayed in Fig. 6 as a function ofR for the three models analyzed. Thedata for the asymmetric and off symmetric models are virtually indistinguishable; weshall thus omit reference to the latter in the followings. The plate response seemssort of repeatable with increasing the aspect ratio byR0. Within each such range,the critical loads for plate snapping (solid lines) decrease with increasing aspect ratio,nearly approaching the buckling load whenR becomes a multiplication ofR0. Thesecondary circumferential buckling load (dashed lines) is favored for relatively small


aspect ratios, being at least six times the buckling load (as compared to 7.5 for theanalogous circular plate problem, see Ceho and Reiss, 1974). The asymmetric modelleads to a much smaller snapping load than the symmetric one, making it the con-trolling configuration for design purposes. An interesting observation concerns load-ing beyond the snapping load. Assuming the post transition deformation patternremains symmetric with increasing load, the subsequent instability event can beevaluated. For example, in the case of Fig. 10, secondary circumferential bucklingfollowing the emergence of the third bulge should be as forR=2.9/3=0.97. This gives(Fig. 6) K=76, as compared withK=68.3 in Fig. 10.

Fig. 6 shows that the symmetric and asymmetric model bound well the experi-mental snapping loads, with the latter model generally providing a fairly good fit.It is tempting to examine the preference of one model over the other in terms ofenergy considerations. To this end, Fig. 11 shows the variation of the strain energyin the plate,U, with K for the symmetric and asymmetric models of Fig. 9; theresults are normalized by the strain energy for an unbuckled plate,U0

U05abhEe20

2. (6)

Fig. 11. Finite element prediction for the variation of normalized strain energy in the plate with nor-malized load; symmetric and asymmetric models,R=1.7.


The abrupt breaks in the slope of the curves help identify the onset of mode tran-sition. As shown, following buckling the strain energy for the symmetric model wellexceeds that for the asymmetric one, which is consistent with the observed tendencyfor asymmetry in the tests (Figs. 3–5). Following plate snapping, the energy levelfor both models coincides, indicating the deformation for the new equilibrium stateis symmetric.

4.4. Secondary contact buckling

The applied load needed to cause buckling in the contact area,Kcb, can be determ-ined from inspection of the contour plots. Fig. 12 (solid symbols) shows the depen-dence ofKcb, normalized byKt, the snapping or mode transition load, on the plateaspect ratio. The ratioKcb/Kt seem fairly insensitive toR or to the deformation model,being approx. 0.84. To gain an analytical insight into this detrimental local bucklingphenomenon, the contact area is assumed a flatrectangular plate of width b andlengthd, having clamp support on all four edges. Moreover, the horizontal boundaryof the effective contact area is assumed to intersect a certain point,S (see insert in

Fig. 12. Finite element prediction for the variation with plate aspect ratio of the load at onset of localbuckling,Kcb, normalized by the snapping load,Kt, and of the ratioKd/4.4/(d/b)2.05. Kd is the finite elementprediction for the buckling load of the effective contact area.


Fig. 13) lying on the actual contact boundary a distanceb/4 from the plate edge.The buckling of the effective contact area can now be evaluated analytically if thevertical displacement along the boundary line throughS is known. The finite elementresults show that this displacement monotonically increases from the midpoint to theplate edge, with the relative difference increasing with load. As a first approximation,the vertical displacement at pointS, Vd, or, in a non dimensional form,Kd, is takento represent the entire boundary line. For this buckling model to be valid, the finiteelement values of the pairKd and d at the onset of local buckling of the contactarea must conform to Eq. (5); note we are concerned here only with relatively smallaspect ratios in which Eq. (5) is valid. This conformation requires thatKd=4.4/(d/b)2.05. Fig. 12 (open symbols) shows that this condition holds true withinapprox. 10% error over the entire range of aspect ratios and deformation models used.

The results above suggest a possible decoupling of the contact and upliftingregions of the plate. It is thus useful to construct empirical relations for the evolutionof the properties of these regions with load. Figs. 13 and 14 show, respectively, thevariations of the length of the uplifting bulge,c, and the local load parameterKd

with K for a number of plate aspect ratios within the rangeR0,R,2R0 ; the data

Fig. 13. Finite element prediction for the variation of the normalized length of the uplifting bulge withthe normalized applied load for plates of various aspect ratios. Results for both symmetric and asymmetricmodels are shown. Solid line is a possible fit to the data.


Fig. 14. Finite element prediction for the variation of the normalized vertical displacement at point S(see insert in Fig. 13),Kd, with normalized applied load. Solid line is a possible fit to the data.

are limited to the onset of buckling of the contact area. The uplifting bulge and thecontact zone seem to evolve fairly independently of the plate aspect ratio or defor-mation model, withc and Kd being well characterized by the following power lawrelations

c/b5A2BK (7)

Kd5CKD (8)

where (A, B, C, D)=(1.02, 0.0058, 1.78, 0.75).For the symmetric model, two symmetrically positioned contact zones of equal

lengthd exist. Compatibility of axial dimensions requires that

d5(a2c)/2, (9)

or, using (7)

d/b5(R2A1BK)/2, (10)

whered/b denotes the aspect ratio of the effective contact zone. The applied loadneeded to buckle the contact area can now be obtained in terms of the plate aspectratio alone by replacingR andKcr in (5) with d/b andKd, respectively. The result is


K0.366cb (R21.0210.0058Kcb)23.150, R0,R,2R0. (11)

The mode transition load,Kt, is found by noting thatKcb=0.84Kt (see Fig. 12):

K0.366t (R21.0210.00487Kt)23.350, R0,R,2R0. (12)

This prediction well duplicates the mode transition curve for the symmetric model(Fig. 6). The behavior for the asymmetric model is more involved owing to themobility of the bulge with load. The length of the contact zone in this case is dictatedby a complex interplay between the motion of the bulge and its changing size withincreasing load. Since this motion is affected by unavoidable friction, it is not feltworthwhile to pursue this issue any further in this work.

5. Discussion

5.1. One dimenensional case

Asymmetry in the post buckling deformation has been observed also in the anal-ogous case of a bilaterally constrained column (Chai, 1998). It was shown analyti-cally in that work that the position of the buckle(s) within the column has no affecton the energy state of the systemthroughoutthe post buckling regime. This charac-teristic led to very large scatter in the mode transition loads by virtue of the possiblelarge variations in the relative sizes of the contact zones. The plate contact problemdiffers from the column case in that the bulge position in the post buckling rangeis not arbitrary, but is dictated by the evolving stress variations across the width ofthe plate. Fig. 11 shows that the preferred mode of deformation in this case is asym-metric. Several Finite Elements runs, performed on intermittent levels of asymmetry,show that the least strain energy obtainable corresponds to the most extreme formof asymmetry possible, i.e. the models depicted in Fig. 7.

5.2. Applications to delamination growth

The use of a clamp type support for the plates studied here is motivated by appli-cations to debonding or delamination in layered structures. The present workencompasses such applications ifK is interpreted as the normalized strain in thesubstrate. The phenomena of film/substrate contact and buckling mode transitionobserved in this work seem to add a new dimension into the mechanics of fracturein such systems. Contact near the plate boundaries eliminates mode I cracking butenhance shear fracture. From a material viewpoint, this may be of concern in ductiletype interlayers. Contact tends to occur for aspect ratios greater than about unity.When the plate snaps, say into two buckles, the aspect ratio for each bulge is halfthe original one, being less than unity. Consequently, crack opening stresses format the boundaries of each bulge following plate snapping. This, together with thedynamic energy released in this process, may lead to catastrophic delaminationgrowth, especially for highly brittle interlayers. The mechanics of fracture in such


events can be worked out relatively easily from a straightforward relations connectingthe boundary membrane and bending stresses to the stress intensity factors (e.g.Chai, 1990a).

6. Summary and conclusions

The post buckling response of unilaterally constrained plates under monotonicallyincreasing edge displacement is studied experimentally and analytically. The aspectratios studied essentially cover the entire range of practical applications. A clamptype boundary support is considered due to its relevance to thin film debonding anddelamination problems, but it is believed that the main conclusions will be only littleaffected by other choices of support conditions.

The tests show that the interaction of the plate with the adjoining rigid substratefollowing buckling leads to some unique deformation sequences. This includes theformation of discrete (early buckling) or continuous (post buckling) contact zones,and therapid transition of the buckling waveform to new equilibrium configurationsfollowing a gradual evolution of secondary buckling either at a contact area or anuplifting bulge. The specific details strongly depend on the plate aspect ratio andother system parameters. The plate deformation at buckling is quite random, but ittends toward an extreme form of asymmetry with increasing load.

A geometrically nonlinear finite element scheme incorporating frictionless contactis used to elucidate the plate response. The analysis duplicates the main sequenceof events observed in the tests, and provides new insight into the plate response.The latter is found to be normalizeable as for unrestrained thin plates. The postbuckling behavior is characterized by buckling mode transitions that are dominatedby two competitive secondary buckling mechanisms, one at a contact area and theother at the uplifting bulge. Because of the inherent randomness of the bulge positionin the early stages of buckling, both symmetric and an extreme form of asymmetryof the buckling deformation are analyzed. These models bound well the experimentalmode transition loads, with the asymmetric model providing a fairly good fit. Thisis consistent with the fact that the latter model yields the least strain energy. Withincertain ranges of aspect ratios, the evolutions of the contact zone and uplifting bulgeare essentially independent of the aspect ratio or the deformation model, which sug-gests a possible decoupling of these two regions constituting the plate area. Suchdecoupling forms the basis for developing simple semi-analytic relations for pre-dicting local buckling and mode transition loads in the symmetric case; extensionto the asymmetric model is more involved.

The results provide some insight into the fracture behavior of imperfectly bondedbilayer structures. Boundary contact suppresses mode I in favor of mode II fracture.However, once a mode transition occurs, the opposite of this may take place, whichcan lead to catastrophic debonding in brittle type interlayers. The specific details canbe worked relatively easily from available formulas relating stress intensity factorsto boundary membrane forces and bending moments in the plate.


Acknowledgements

The author thanks Mr David Armoni for his help in constructing the test fixturesand test samples, and Mr Eyal Moses for discussions concerning the finiteelement code.

References

Adan, N., Sheinman, I., Altus, E., 1994. Post-buckling behavior of beams under contact constraints.Journal of Applied Mechanics 61, 764–772.

Ceho, L.S., Reiss, E.L., 1974. Secondary buckling of circular plates. Siam Journal of Applied Mathematics26, 490.

Chai, H., Knauss, W.G., Babcock, C.D., 1983. Observation of impact damage growth in compressivelyloaded laminates. Experimental Mechanics 23, 329–337.

Chai, H., Babcock, C.D., 1985. Two-dimensional modeling of compressive failure in delaminated lami-nates. Journal of Composite Materials 19, 67–98.

Chai, H., 1990a. Three-dimensional analysis of thin-film debonding. International Journal of Fracture 46,237–256.

Chai, H., 1990b. Buckling and post-buckling behavior of elliptical plates: part II — results. Journal ofApplied Mechanics 57, 989–994.

Chai, H., 1998. The post-buckling behavior of a bi-laterally constrained column. Journal of the Mechanicsand Physics of Solids 46, 1155–1181.

Chateau, X., Nguyen, Q.S., 1991. Buckling of elastic structures in unilateral contact with or withoutfriction. European Journal of Mechanics, A/Solids 1, 71–89.

Comiez, J.M., Waas, A.M., Shahwan, K.W., 1995. Delamination buckling; experiment and analysis. Inter-national Journal of Solids and Structures 32, 767–782.

Domokos, G., Holmes, P., Royce, B., 1997. Constrained Euler buckling. Journal of Nonlinear Science 7,281–314.

Giannaakopoulos, A.E., Nilsson, K.F., Tsamasphyros, G., 1995. The contact problem at delamination.Journal of Applied Mechanics 62, 989–996.

Hhatake, K., Oden, J.T., Kikuchi, N., 1980. Analysis of certain unilateral problems in von Karman platetheory by a penalty method — Part 2, approximation and numerical analysis. Computer Methods inApplied Mechanics and Engineering 24, 317–337.

Holmes, P., Domokos, G., Schmitt, J., Szeber’enyi, I., 1999. Constrained Euler buckling: an interplay ofcomputation and analysis. Computer Methods in Applied Mechanics and Mechanical Engineering 170,175–207.

Hutchinson, J.W., Thouless, M.D., Liniger, E.G., 1992. Growth and configurational stability of circular,buckling-driven film delaminations. Acta Metallurgica Materialla 40, 295–302.

Keller, H.B., Reiss, E.L., 1958. Non linear bending and buckling of circular plates, Proceedings 3rd USNational Congress of Applied Mechanics and Engineering, pp. 375–385.

Levy, S., 1942. Buckling of rectangular plates with built-in edges. Journal of Applied MechanicsA171–A17.

Nakamura, T., Uetani, K., 1979. The secondary buckling and post-secondary-buckling behaviours of rec-tangular plates. International Journal of Mechanical Sciences 21, 265–286.

Sekine, H., Hu, N., Kouchakzadeh, M.A., 2000. Buckling analysis of elliptically delaminated compositelaminates with consideration of partial closure of delamination. Journal of Composite Materials 34,551–573.

Shahwan, K.W., Waas, A.M., 1994. A mechanical model for the buckling of unilaterally constrainedrectangular plates. International Journal of Solids and Structures 31, 75–87.

Siede, P., 1958. Compressive buckling of a long simply supported plate on an elastic foundation. Journalof the Aeronautical Sciences June, 382–394.


Smith, S.T., Bradford, M.A., Oehlers, D.J., 1999. Numerical convergence of simple and orthogonal poly-nomials for the unilateral plate buckling problem using the Rayleigh–Ritz method. International Jour-nal of Numerical Methods in Engineering 44, 1685–1707.

Stein M., 1959. NASA Technical Report R-40.Whitcomb J.D., 1988. Instability-related delamination growth of embedded and edge delamination. NASA

Technical Memorandum 100655.Wright, H.D., 1995. Local stability of filled and encased steel sections. Journal of Structural Engineering

12, 1382–1388.

Documents

Contact buckling and postbuckling of thin rectangular platesherzl/2001/Contact buckling and... · Contact buckling and postbuckling of thin rectangular plates Herzl Chai * Department