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Contact buckling and postbuckling of thin rectangular plates
Journal of the Mechanics and Physics of Solids 49 (2001) 209–230 www.elsevier.com/locate/jmps
Contact buckling and postbuckling of thin rectangular 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 buckling response of unilaterally constrained plates under monotonically increasing edge thrust. Real time observations, together with a wide range of plate aspect ratio and a large load level facilitate deep physical insight into the general behavior of this class of problems. The interaction of the plate with the rigid restraining plane following buckling leads to interesting deformation sequences, characterized by the development of asymmetric bulges and contact zones following by a possible plate snapping. The latter is motivated by a secondary buckling evolving gradually from a contact zone(s) or a bulge(s). These two instability mechanisms are competitive, being dictated by the plate aspect ratio and other system parameters. The critical load for plate snapping agrees well with a finite element prediction based on an asymmetric deformation choice that minimizes the strain energy in the plate. A semi analytic relation for predicting the onset of secondary instability in the contact area and subsequent plate snapping is developed based on the numerical results. Finally, the present work seems to add a new dimension into the fracture of coatings and laminated composites containing near-surface defects. 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 interlaminar defects 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: S 0 0 2 2 - 5 0 9 6 ( 0 0 ) 0 0 0 3 8 - 7
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loading conditions may buckle into a wavy pattern. However, the natural development of this deformation may be altered by interaction with the parent laminate, which may affect the stiffness and fracture resistance of the global structure. This phenomenon 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 et al., 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 concern 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 of this class of problems, a more comprehensive treatment, particularly in the post buckling regime, seems to be warranted. The present work involves the evolution of secondary buckling in the film/substrate contact zones or the uplifting portions of the plate, and the snapping of the plate to new equilibrium states. The following literature survey focuses on these aspects. It is instructive to consider first relevant one-dimensional configurations as they afford analytically tractable solutions from which a valuable insight into more general contact buckling problems can be gained. Chateau and Nguyen (1991), and Adan et al. (1994) show that when a column positioned a distance from a flat wall is compressed, contact zones may develop leading to buckling mode transition. More recent works dealing with the bilaterally constrained column (Chai, 1998; Domokos et al., 1997; Holmes et al., 1999) have exposed more relevant characteristics of the contact buckling problem, including a sequential mode transition process, an inherent asymmetry of the deformation pattern, and a unique hysteresis in the load vs axial displacement 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 rigid lateral constraint may increase the buckling resistance by as much as 33% relative to the unrestrained plate. Shahwan and Waas (1994) have extended this work to material orthotropy and various boundary conditions. The buckling strength due to unilateral constrain of finite size plates was considered by Wright (1995) and Smith et al. (1999) using finite element and Rayleigh–Ritz approaches, respectively. The effect of lateral constraint on the plate deformation was vividly demonstrated by Comiez et al. (1995) with the aid of the shadow Moire technique. However, the response of the plate could not be separated from the overall structure in these tests because the two were firmly attached at the plate boundary. Plate contact studies in the post buckling regime are scarce. Hhatake et al. (1980), using a finite element scheme coupled with the penalty method, have provided information on the evolution of contact with compression load. No plate snapping was noted in this study, which was 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 attributed to secondary buckling in the may plate. Secondary buckling also occurs in axisymmetrically compressed circular plates (e.g. Keller and Reiss, 1958). In this case, the deformation is characterized by wrinkles around a narrow circumferential strip near the plate edge. Hutchinson et al. (1992) provide a vivid experimental dem-
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onstration of the evolution of circumferential buckling lubes in circular films on rigid substrates under the combined action of in-plane compression and transverse loading. The onset of secondary buckling in circular or rectangular plates have been studied analytically by a number of authors, mostly using variational energy principles based on perturbation from a given post buckling state (e.g. Ceho and Reiss, 1974; Nakamura and Uetani, 1979). Such analyses are complex and generally do not provide information on the new equilibrium state following secondary buckling. In this work, the response of unilaterally constrained plates due to edge thrust is studied experimentally and analytically. To simulate near surface debonding problems, all four edges of the plates are clamped. A systematic variation of the plate aspect ratio and load level, together with real time observation of the outward plate deformation via the shadow Moire technique, allow for general conclusions to be made. 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 secondary instability evolving from the contacting zones. A large strain, large deformation finite element analysis incorporating a frictionless contact algorithm is employed to model the plate response. The analysis provides quantitative insight into the post-buckling behavior, including secondary buckling and plate snapping. The buckling stage is discussed in Section 2. The experimental program is reported in Section 3, where the phenomenological aspects of the deformation process are exposed. This information is then used to construct appropriate numerical models (Section 4). Some discussions and conclusions are given in Sections 5 and 6, respectively. 2. Problem definition 2.1. Nomenclature Thin, flat rectangular plates of width b, length a and thickness t are subjected to a 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 to use the following thin-plate normalization K⫽
12(1−n2)b2 e0 p2t2
(1)
where n is Poisson’s ratio of the plate and e0 is defined as e0⬅Va/a.
(2)
In the following, K will be referred to as “load”. Note that in the case of thin-film delamination 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 determined by Levy (1942). The variation of the buckling load with the plate aspect ratio, R, is shown in Fig. 1, where
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Fig. 1. Finite element prediction (triangular symbols) for normalized buckling strain vs aspect ratio for a clamped, unilaterally constrained plate under axial edge displacement. Also shown are results for unrestrained plate (Levy, 1942).
R⬅a/b.
(3)
The plate buckles into a half wave if R⬍R0 (=1.07); for larger aspect ratios, the number of bulges increases with R. In the case of a unilaterally constrained plate, R0 has a special meaning because for larger aspect ratios, contact between the plate and the support occurs. In this range, the constrained plate buckles into a multitude of half waves, each having the same profile as the one for R0. It is clear that the number of bulges in the plate, n, relates to the aspect ratio according to the following rule n⫽smallest closest integer of the ratio R/R0
(4)
(e.g. if R/R0 =1.7 or 2.3, there will be one bulge and two bulges, respectively). The buckling load for R⬎R0 remains fixed at the same value as for R0, i.e. K=10. Thus, the constraint increases the buckling strength relative to the unrestrained plate by an amount that increases with the plate aspect ratio. For RÀ1, this amounts to 30% [which is similar to the simply supported case (Siede, 1958)]. Note that for R⬎R0, the bulge(s) may freely move within the span of the plate without affecting the energy state in the system (assuming a frictionless contact). This feature, also common to
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bilaterally constrained columns (Chai, 1998), may greatly affect the post buckling response.
3. Experimental 3.1. Apparatus Tests are carried out to elucidate the response of clamped rectangular plates under axial compression. Fig. 2 details the testing apparatus. To increase the range of elastic deformation 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.
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to geometric imperfections, a high degree of precision in the machining and alignment of all relevant components of the test apparatus is employed. The plates are machined to specific dimensions from 1 mm thick Polycarbonate sheets having Young’s modules (E), Poisson’s ratio and proportional limit of 2.3 GPa, 0.35 and approx. 1%, in that order. A range of plate dimensions are examined, i.e. b=25.4, 50.8 and 76.2 mm, and t=0.5, 0.625, 0.7 and 1 mm. The aspect ratio varies from 0.75 to 6 (or from 0.7R0 to 5.6R0). The plates rest upon a 38 mm thick aluminum block which is firmly attached to the test fixture. The longitudinal edges of the plates are constrained by flat steel bars. The latter are adjusted prior to testing so as to produce intimate yet unconstraining contact. The lower edge of the plate is bolted to the aluminum block while the upper one is attached to a steel block that travels along 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 prior to testing to reduce friction. The out of plane deformation in the plate are obtained via the shadow Moire technique; the optical test set up is as described in Chai et al. (1983). A 6 lines/mm Moire grid is placed several millimeters from the sample. Illumination is achieved by a collimated light beam directed at 29° to the plate normal. The sample is viewed in the normal direction, giving a fringe constant, f, of 0.3 mm. The upper edge of the plates is compressed at a slow rate to well over the buckling load using a screw driven testing machine (Instron). The axial load, P, and the end shortening, Va, obtained using an LVDT, are recorded during the tests. The evolving fringe pattern is monitored using a video camera. 3.2. Test results Figs. 3–5 show three Moire sequences representative of the deformation pattern obtained 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 Section 2.2, buckling occurs at K=10. The post buckling behavior is greatly affected by the aspect 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 of secondary buckling near the vertical edges of the bulge (II), the final stage of bulge collapse (III) and the splitting of the bulge into two separate bulges (IV). Print V corresponds to unloading, just before the transition from two to one bulge; note the associated load is well less than that in (IV), indicating a pronounced elastic hysteresis. 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 development of a second bulge in the contact area below the first bulge leads to a transition into a two-bulge configuration (II). The next significant event is the onset of secondary buckling at the vertical edges of both bulges (III). Note that the plate response thereafter was stable (i.e. no bulge splitting). This behavior was common over the range 1.3⬍R⬍2R0
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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 normalized edge displacement. Prints I–IV correspond to loading while print V to unloading.
Fig. 5, R=2.9 — Buckling starts with two randomly positioned bulges (I), in consistency with the buckling behavior discussed in Section 2.2. A third bulge develops near 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, but not 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 behavior typifies the range 2R0⬍R⬍3R0, with the production of the fourth bulge being suppressed in favor of secondary buckling in the three uplifting bulges if R is less than about 2.6. For longer plates, more bulges emerge at buckling, in consistency with Eq. (4). For example, for R=6 (b, t =25.4 mm, 0.5 mm, results not shown), five bulges initially emerged. Similarly to the process described earlier, a transition to six and then seven bulges 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
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Fig. 4. Moire fringe sequence for a unilaterally constrained plate under axial compression; b, t=76.2 mm, 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 instability is 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), in particular at the edges of the bulge, analogously to secondary buckling or wrinkling in unrestrained plates. (3) These two instability mechanisms, to be referred to as “secondary contact buckling” and “secondary circumferential buckling”, respectively, are competitive; the onset of one delays or prevents the other. (4) Friction is expected 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 transition. Three plates each of a given aspect ratio were tested, each one repeatedly loaded a total of three times. Thus, nine data point are produced for each aspect ratio. Fig. 6 (symbols) displays the critical loads obtained; the data for R=0.75 and R=1.2 correspond to the onset of secondary circumferential buckling (i.e. no snapping with increasing load) while the rest of the data to buckling mode transition or snapping. In this case, the data for R⬍2R0 and R⬎2R0 correspond to a jump from one to two bulges and two to three bulges, respectively. The results exhibit some scatter, with no pronounced difference between different specimens or repeated tests on the same specimen. These data will be discussed further following the development of the analysis.
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Fig. 5. Moire fringe sequence for a unilaterally constrained plate under axial compression; b, t=50.8 mm, 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, Version 5.3) with a built in contact algorithm is employed. A four node rectangular plate element (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 of the fundamental characteristics of the problem.) In consistency with the tests, the outward deflection, w, and the plate rotation, dw/dl, where l denotes the normal to the plate edge, are assumed zero along all four edges of the plate. Also, the in-plane displacements on the lower edge and the horizontal displacement on the upper edge are assumed zero. The upper edge is given a uniform vertical displacement that is applied in load steps. The center of the expected bulge(s) is (are) given a small initial outward displacement to help obtain a smooth transition into the buckling regime. This auxiliary displacement is removed once the load exceeds about 50% of the buckling load. The plate response is studied as a function of aspect ratio, up to R=3R0; the behavior for larger R can be inferred, to some extent, from these results. As discussed
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Fig. 6. Normalized critical loads as a function of plate aspect ratio. Symbols and curves denote experimental results and finite element predictions, respectively. Experiments: for each aspect ratio, three different samples were tested, each one repeatedly loaded three times. Solid and open symbols correspond to secondary circumferential buckling and plate snapping, respectively. Finite element: shown are results for the three deformation models defined in Fig. 7; solid and dashed curves correspond to plate snapping and onset of secondary circumferential buckling, respectively.
in Section 2.2, for R⬍2R0 only one bulge forms at buckling while for 2R0⬍R⬍3R0, two bulges initially form; in both cases, the bulge(s) is of width b and aspect ratio R0. Because in the range R⬎R0 the position(s) of the initial bulge(s) within the plate is (are) inherently random, a few (three) extreme configurations are analyzed, namely symmetric, off-symmetric and asymmetric, see Fig. 7: Symmetric: If R⬍2R0, a single outward displacement is initially forced at the plate center; if 2R0⬍R⬍3R0, two initial displacements are forced, each a distance R/4 from a plate edge. Off-symmetric: This configuration is defined only for R⬎2R0. In this case, two initial displacements are placed a distance R0/2 from each edge of the plate. Asymmetric: If R⬍R0, one initial displacement is placed at the plate center; if
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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 once the load exceeds about 50% of the buckling load.
R0⬍R⬍2R0, one initial displacement is placed a distance R0/2 from the lower plate edge; if 2R0⬍R⬍3R0, two initial displacements are placed at distances R0/2 and 3R0/2 from the lower plate edge. The symmetric model represents an upper bound for the snapping load because it affords the smallest contact zones possible from which a local instability leading to plate snapping may ensue. As will be seen latter, the results for the two nonsymmetrical models are indistinguishable, and they lead to the lowest bound. To take advantage of symmetry, only half of the plate region is modeled; for the symmetric and off symmetric configurations, only a quarter of the plate is modeled. Analyses are performed for a range of plate dimensions. A systematic mesh refinement study shows that as many as 1000 rectangular elements are necessary to insure convergence of the post buckling solution over the range of parameters studied. Results from different geometries (i.e. b=50.8 mm and 76.2 mm, and t=0.35 mm and 0.7 mm) show that the post buckling solution (i.e. stresses, deflections) can be 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 choice b=50.8 mm and t=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 the buckling load, Kcr. The following technique for securing accurate solution in the early buckling stage was applied: load up to 50% above Kcr, unload to nearly Kcr, 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 data thus obtained are displayed in Fig. 1 as triangular symbols. The results agree with
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those of Levy (1942), and extends the latter down to R=0.22. The buckling load for R⬍0.5 seem to obey well a power law of the form Kcr⫽4.4/R2.05.
(5)
4.3. Post buckling Figs. 8–10 show the evolution of outward displacement contours with load for the three experimental case studies depicted in Figs. 3–5; the normalized fringe constant (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 deformation patterns generally well resemble the test results, notably in the development of local instability, either near the vertical edges of the uplifting bulge(s) or at the contact zone(s), and in the rapid mode transition that follows. A closer examination reveals that, for a given load, the displacements in a bulge are independent of the deformation model [e.g. compare print II of Fig. 9(a) with print I of Fig. 9(b)], being numerically consistent with the respective test results. A brief exposition of the main features follows. Fig. 8, R=1.2 — In consistency with the test results (Fig. 3), the asymmetric model is shown. Major events such as onset of secondary circumferential buckling (II), rapid bulge distortion (III) and bulge splitting (IV) seem to resemble well the test results, 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 bulge in Fig. 9(a) is seen to initiate from the contact area below the first bulge. Its evolution is gradual, but the final transition to a two-bulge configuration is rapid, tending toward a symmetric deformation (III). Upon increasing K to 95, secondary buckling occurs near the vertical edges of each of the two bulges. This, however, did not lead to 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 with R=1.2. K is the normalized edge displacement. Normalized fringe constant, f/t, is 0.43, as in Fig. 3.
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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 symmetrically to the first (II). However, once these bulges mature, they quickly push out the central bulge (III), ending in two symmetric bulges (IV); the rest of the loading 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 discussion in Section 2.2, two bulges initially form (I). The rest of the loading is much the same as discussed for Fig. 9(a), i.e. emergence of an additional bulge over the largest contact area (II), and onset of a secondary circumferential buckling in each bulge (III). In contrast to its experimental counterpart (Fig. 5), however, no fourth bulge 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
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Fig. 10. Outward displacement contours for asymmetric finite element model, R=2.9. Normalized fringe constant, f/t, is 0.48, as in Fig. 5.
increased from K=37.3 to K=69; the fourth bulge emerges from the largest contact area. Such unevenness is also observed in print III, but it is apparently insufficient to cause secondary contact buckling before secondary circumferential buckling intervenes. This demonstrates the complex interplay between these two secondary buckling mechanisms, and how imperfections and frictional effects may possibly play an important role on the post buckling response. The onset of equilibrium loss in the plate, determined from inspection of the contour plots, is displayed in Fig. 6 as a function of R for the three models analyzed. The data for the asymmetric and off symmetric models are virtually indistinguishable; we shall thus omit reference to the latter in the followings. The plate response seems sort of repeatable with increasing the aspect ratio by R0. Within each such range, the critical loads for plate snapping (solid lines) decrease with increasing aspect ratio, nearly approaching the buckling load when R becomes a multiplication of R0. The secondary circumferential buckling load (dashed lines) is favored for relatively small
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aspect ratios, being at least six times the buckling load (as compared to 7.5 for the analogous circular plate problem, see Ceho and Reiss, 1974). The asymmetric model leads to a much smaller snapping load than the symmetric one, making it the controlling configuration for design purposes. An interesting observation concerns loading beyond the snapping load. Assuming the post transition deformation pattern remains symmetric with increasing load, the subsequent instability event can be evaluated. For example, in the case of Fig. 10, secondary circumferential buckling following the emergence of the third bulge should be as for R=2.9/3=0.97. This gives (Fig. 6) K=76, as compared with K=68.3 in Fig. 10. Fig. 6 shows that the symmetric and asymmetric model bound well the experimental 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 of energy considerations. To this end, Fig. 11 shows the variation of the strain energy in the plate, U, with K for the symmetric and asymmetric models of Fig. 9; the results are normalized by the strain energy for an unbuckled plate, U0 U0⫽
abhEe20 . 2
(6)
Fig. 11. Finite element prediction for the variation of normalized strain energy in the plate with normalized load; symmetric and asymmetric models, R=1.7.
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The abrupt breaks in the slope of the curves help identify the onset of mode transition. As shown, following buckling the strain energy for the symmetric model well exceeds that for the asymmetric one, which is consistent with the observed tendency for asymmetry in the tests (Figs. 3–5). Following plate snapping, the energy level for both models coincides, indicating the deformation for the new equilibrium state is symmetric. 4.4. Secondary contact buckling The applied load needed to cause buckling in the contact area, Kcb, can be determined from inspection of the contour plots. Fig. 12 (solid symbols) shows the dependence of Kcb, normalized by Kt, the snapping or mode transition load, on the plate aspect ratio. The ratio Kcb/Kt seem fairly insensitive to R or to the deformation model, being approx. 0.84. To gain an analytical insight into this detrimental local buckling phenomenon, the contact area is assumed a flat rectangular plate of width b and length d, having clamp support on all four edges. Moreover, the horizontal boundary of 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 local buckling, Kcb, normalized by the snapping load, Kt, and of the ratio Kd/4.4/(d/b)2.05. Kd is the finite element prediction for the buckling load of the effective contact area.
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Fig. 13) lying on the actual contact boundary a distance b/4 from the plate edge. The buckling of the effective contact area can now be evaluated analytically if the vertical displacement along the boundary line through S is known. The finite element results show that this displacement monotonically increases from the midpoint to the plate edge, with the relative difference increasing with load. As a first approximation, the vertical displacement at point S, Vd, or, in a non dimensional form, Kd, is taken to represent the entire boundary line. For this buckling model to be valid, the finite element values of the pair Kd and d at the onset of local buckling of the contact area must conform to Eq. (5); note we are concerned here only with relatively small aspect ratios in which Eq. (5) is valid. This conformation requires that Kd=4.4/(d/b)2.05. Fig. 12 (open symbols) shows that this condition holds true within approx. 10% error over the entire range of aspect ratios and deformation models used. The results above suggest a possible decoupling of the contact and uplifting regions of the plate. It is thus useful to construct empirical relations for the evolution of the properties of these regions with load. Figs. 13 and 14 show, respectively, the variations of the length of the uplifting bulge, c, and the local load parameter Kd with K for a number of plate aspect ratios within the range R0⬍R⬍2R0 ; the data
Fig. 13. Finite element prediction for the variation of the normalized length of the uplifting bulge with the normalized applied load for plates of various aspect ratios. Results for both symmetric and asymmetric models are shown. Solid line is a possible fit to the data.
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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 the contact zone seem to evolve fairly independently of the plate aspect ratio or deformation model, with c and Kd being well characterized by the following power law relations c/b⫽A⫺BK
(7)
Kd⫽CK D
(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 length d exist. Compatibility of axial dimensions requires that d⫽(a⫺c)/2,
(9)
or, using (7) d/b⫽(R⫺A⫹BK)/2,
(10)
where d/b denotes the aspect ratio of the effective contact zone. The applied load needed to buckle the contact area can now be obtained in terms of the plate aspect ratio alone by replacing R and Kcr in (5) with d/b and Kd, respectively. The result is
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K 0.366 (R⫺1.02⫹0.0058Kcb)⫺3.1⫽0, R0⬍R⬍2R0. cb
227
(11)
The mode transition load, Kt, is found by noting that Kcb=0.84Kt (see Fig. 12): K 0.366 (R⫺1.02⫹0.00487Kt)⫺3.3⫽0, R0⬍R⬍2R0. t
(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 the mobility of the bulge with load. The length of the contact zone in this case is dictated by a complex interplay between the motion of the bulge and its changing size with increasing load. Since this motion is affected by unavoidable friction, it is not felt worthwhile 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 analogous case of a bilaterally constrained column (Chai, 1998). It was shown analytically in that work that the position of the buckle(s) within the column has no affect on the energy state of the system throughout the post buckling regime. This characteristic led to very large scatter in the mode transition loads by virtue of the possible large variations in the relative sizes of the contact zones. The plate contact problem differs from the column case in that the bulge position in the post buckling range is not arbitrary, but is dictated by the evolving stress variations across the width of the plate. Fig. 11 shows that the preferred mode of deformation in this case is asymmetric. Several Finite Elements runs, performed on intermittent levels of asymmetry, show that the least strain energy obtainable corresponds to the most extreme form of 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 applications to debonding or delamination in layered structures. The present work encompasses such applications if K is interpreted as the normalized strain in the substrate. The phenomena of film/substrate contact and buckling mode transition observed in this work seem to add a new dimension into the mechanics of fracture in such systems. Contact near the plate boundaries eliminates mode I cracking but enhance shear fracture. From a material viewpoint, this may be of concern in ductile type 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 half the original one, being less than unity. Consequently, crack opening stresses form at the boundaries of each bulge following plate snapping. This, together with the dynamic energy released in this process, may lead to catastrophic delamination growth, especially for highly brittle interlayers. The mechanics of fracture in such
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events can be worked out relatively easily from a straightforward relations connecting the 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 monotonically increasing edge displacement is studied experimentally and analytically. The aspect ratios studied essentially cover the entire range of practical applications. A clamp type boundary support is considered due to its relevance to thin film debonding and delamination problems, but it is believed that the main conclusions will be only little affected by other choices of support conditions. The tests show that the interaction of the plate with the adjoining rigid substrate following buckling leads to some unique deformation sequences. This includes the formation of discrete (early buckling) or continuous (post buckling) contact zones, and the rapid transition of the buckling waveform to new equilibrium configurations following a gradual evolution of secondary buckling either at a contact area or an uplifting bulge. The specific details strongly depend on the plate aspect ratio and other system parameters. The plate deformation at buckling is quite random, but it tends toward an extreme form of asymmetry with increasing load. A geometrically nonlinear finite element scheme incorporating frictionless contact is used to elucidate the plate response. The analysis duplicates the main sequence of 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 post buckling behavior is characterized by buckling mode transitions that are dominated by two competitive secondary buckling mechanisms, one at a contact area and the other at the uplifting bulge. Because of the inherent randomness of the bulge position in the early stages of buckling, both symmetric and an extreme form of asymmetry of the buckling deformation are analyzed. These models bound well the experimental mode transition loads, with the asymmetric model providing a fairly good fit. This is consistent with the fact that the latter model yields the least strain energy. Within certain ranges of aspect ratios, the evolutions of the contact zone and uplifting bulge are essentially independent of the aspect ratio or the deformation model, which suggests a possible decoupling of these two regions constituting the plate area. Such decoupling forms the basis for developing simple semi-analytic relations for predicting local buckling and mode transition loads in the symmetric case; extension to the asymmetric model is more involved. The results provide some insight into the fracture behavior of imperfectly bonded bilayer 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, which can lead to catastrophic debonding in brittle type interlayers. The specific details can be worked relatively easily from available formulas relating stress intensity factors to boundary membrane forces and bending moments in the plate.
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Acknowledgements The author thanks Mr David Armoni for his help in constructing the test fixtures and test samples, and Mr Eyal Moses for discussions concerning the finite element 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 Mathematics 26, 490. Chai, H., Knauss, W.G., Babcock, C.D., 1983. Observation of impact damage growth in compressively loaded laminates. Experimental Mechanics 23, 329–337. Chai, H., Babcock, C.D., 1985. Two-dimensional modeling of compressive failure in delaminated laminates. 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 of Applied Mechanics 57, 989–994. Chai, H., 1998. The post-buckling behavior of a bi-laterally constrained column. Journal of the Mechanics and Physics of Solids 46, 1155–1181. Chateau, X., Nguyen, Q.S., 1991. Buckling of elastic structures in unilateral contact with or without friction. European Journal of Mechanics, A/Solids 1, 71–89. Comiez, J.M., Waas, A.M., Shahwan, K.W., 1995. Delamination buckling; experiment and analysis. International 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 plate theory by a penalty method — Part 2, approximation and numerical analysis. Computer Methods in Applied Mechanics and Engineering 24, 317–337. Holmes, P., Domokos, G., Schmitt, J., Szeber’enyi, I., 1999. Constrained Euler buckling: an interplay of computation 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 US National Congress of Applied Mechanics and Engineering, pp. 375–385. Levy, S., 1942. Buckling of rectangular plates with built-in edges. Journal of Applied Mechanics A171–A17. Nakamura, T., Uetani, K., 1979. The secondary buckling and post-secondary-buckling behaviours of rectangular plates. International Journal of Mechanical Sciences 21, 265–286. Sekine, H., Hu, N., Kouchakzadeh, M.A., 2000. Buckling analysis of elliptically delaminated composite laminates 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 constrained rectangular 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. Journal of the Aeronautical Sciences June, 382–394.
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Smith, S.T., Bradford, M.A., Oehlers, D.J., 1999. Numerical convergence of simple and orthogonal polynomials for the unilateral plate buckling problem using the Rayleigh–Ritz method. International Journal 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.
Contact buckling and postbuckling of thin rectangular 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 buckling response of unilaterally constrained plates under monotonically increasing edge thrust. Real time observations, together with a wide range of plate aspect ratio and a large load level facilitate deep physical insight into the general behavior of this class of problems. The interaction of the plate with the rigid restraining plane following buckling leads to interesting deformation sequences, characterized by the development of asymmetric bulges and contact zones following by a possible plate snapping. The latter is motivated by a secondary buckling evolving gradually from a contact zone(s) or a bulge(s). These two instability mechanisms are competitive, being dictated by the plate aspect ratio and other system parameters. The critical load for plate snapping agrees well with a finite element prediction based on an asymmetric deformation choice that minimizes the strain energy in the plate. A semi analytic relation for predicting the onset of secondary instability in the contact area and subsequent plate snapping is developed based on the numerical results. Finally, the present work seems to add a new dimension into the fracture of coatings and laminated composites containing near-surface defects. 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 interlaminar defects 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: S 0 0 2 2 - 5 0 9 6 ( 0 0 ) 0 0 0 3 8 - 7
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loading conditions may buckle into a wavy pattern. However, the natural development of this deformation may be altered by interaction with the parent laminate, which may affect the stiffness and fracture resistance of the global structure. This phenomenon 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 et al., 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 concern 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 of this class of problems, a more comprehensive treatment, particularly in the post buckling regime, seems to be warranted. The present work involves the evolution of secondary buckling in the film/substrate contact zones or the uplifting portions of the plate, and the snapping of the plate to new equilibrium states. The following literature survey focuses on these aspects. It is instructive to consider first relevant one-dimensional configurations as they afford analytically tractable solutions from which a valuable insight into more general contact buckling problems can be gained. Chateau and Nguyen (1991), and Adan et al. (1994) show that when a column positioned a distance from a flat wall is compressed, contact zones may develop leading to buckling mode transition. More recent works dealing with the bilaterally constrained column (Chai, 1998; Domokos et al., 1997; Holmes et al., 1999) have exposed more relevant characteristics of the contact buckling problem, including a sequential mode transition process, an inherent asymmetry of the deformation pattern, and a unique hysteresis in the load vs axial displacement 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 rigid lateral constraint may increase the buckling resistance by as much as 33% relative to the unrestrained plate. Shahwan and Waas (1994) have extended this work to material orthotropy and various boundary conditions. The buckling strength due to unilateral constrain of finite size plates was considered by Wright (1995) and Smith et al. (1999) using finite element and Rayleigh–Ritz approaches, respectively. The effect of lateral constraint on the plate deformation was vividly demonstrated by Comiez et al. (1995) with the aid of the shadow Moire technique. However, the response of the plate could not be separated from the overall structure in these tests because the two were firmly attached at the plate boundary. Plate contact studies in the post buckling regime are scarce. Hhatake et al. (1980), using a finite element scheme coupled with the penalty method, have provided information on the evolution of contact with compression load. No plate snapping was noted in this study, which was 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 attributed to secondary buckling in the may plate. Secondary buckling also occurs in axisymmetrically compressed circular plates (e.g. Keller and Reiss, 1958). In this case, the deformation is characterized by wrinkles around a narrow circumferential strip near the plate edge. Hutchinson et al. (1992) provide a vivid experimental dem-
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onstration of the evolution of circumferential buckling lubes in circular films on rigid substrates under the combined action of in-plane compression and transverse loading. The onset of secondary buckling in circular or rectangular plates have been studied analytically by a number of authors, mostly using variational energy principles based on perturbation from a given post buckling state (e.g. Ceho and Reiss, 1974; Nakamura and Uetani, 1979). Such analyses are complex and generally do not provide information on the new equilibrium state following secondary buckling. In this work, the response of unilaterally constrained plates due to edge thrust is studied experimentally and analytically. To simulate near surface debonding problems, all four edges of the plates are clamped. A systematic variation of the plate aspect ratio and load level, together with real time observation of the outward plate deformation via the shadow Moire technique, allow for general conclusions to be made. 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 secondary instability evolving from the contacting zones. A large strain, large deformation finite element analysis incorporating a frictionless contact algorithm is employed to model the plate response. The analysis provides quantitative insight into the post-buckling behavior, including secondary buckling and plate snapping. The buckling stage is discussed in Section 2. The experimental program is reported in Section 3, where the phenomenological aspects of the deformation process are exposed. This information is then used to construct appropriate numerical models (Section 4). Some discussions and conclusions are given in Sections 5 and 6, respectively. 2. Problem definition 2.1. Nomenclature Thin, flat rectangular plates of width b, length a and thickness t are subjected to a 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 to use the following thin-plate normalization K⫽
12(1−n2)b2 e0 p2t2
(1)
where n is Poisson’s ratio of the plate and e0 is defined as e0⬅Va/a.
(2)
In the following, K will be referred to as “load”. Note that in the case of thin-film delamination 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 determined by Levy (1942). The variation of the buckling load with the plate aspect ratio, R, is shown in Fig. 1, where
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Fig. 1. Finite element prediction (triangular symbols) for normalized buckling strain vs aspect ratio for a clamped, unilaterally constrained plate under axial edge displacement. Also shown are results for unrestrained plate (Levy, 1942).
R⬅a/b.
(3)
The plate buckles into a half wave if R⬍R0 (=1.07); for larger aspect ratios, the number of bulges increases with R. In the case of a unilaterally constrained plate, R0 has a special meaning because for larger aspect ratios, contact between the plate and the support occurs. In this range, the constrained plate buckles into a multitude of half waves, each having the same profile as the one for R0. It is clear that the number of bulges in the plate, n, relates to the aspect ratio according to the following rule n⫽smallest closest integer of the ratio R/R0
(4)
(e.g. if R/R0 =1.7 or 2.3, there will be one bulge and two bulges, respectively). The buckling load for R⬎R0 remains fixed at the same value as for R0, i.e. K=10. Thus, the constraint increases the buckling strength relative to the unrestrained plate by an amount that increases with the plate aspect ratio. For RÀ1, this amounts to 30% [which is similar to the simply supported case (Siede, 1958)]. Note that for R⬎R0, the bulge(s) may freely move within the span of the plate without affecting the energy state in the system (assuming a frictionless contact). This feature, also common to
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bilaterally constrained columns (Chai, 1998), may greatly affect the post buckling response.
3. Experimental 3.1. Apparatus Tests are carried out to elucidate the response of clamped rectangular plates under axial compression. Fig. 2 details the testing apparatus. To increase the range of elastic deformation 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.
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to geometric imperfections, a high degree of precision in the machining and alignment of all relevant components of the test apparatus is employed. The plates are machined to specific dimensions from 1 mm thick Polycarbonate sheets having Young’s modules (E), Poisson’s ratio and proportional limit of 2.3 GPa, 0.35 and approx. 1%, in that order. A range of plate dimensions are examined, i.e. b=25.4, 50.8 and 76.2 mm, and t=0.5, 0.625, 0.7 and 1 mm. The aspect ratio varies from 0.75 to 6 (or from 0.7R0 to 5.6R0). The plates rest upon a 38 mm thick aluminum block which is firmly attached to the test fixture. The longitudinal edges of the plates are constrained by flat steel bars. The latter are adjusted prior to testing so as to produce intimate yet unconstraining contact. The lower edge of the plate is bolted to the aluminum block while the upper one is attached to a steel block that travels along 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 prior to testing to reduce friction. The out of plane deformation in the plate are obtained via the shadow Moire technique; the optical test set up is as described in Chai et al. (1983). A 6 lines/mm Moire grid is placed several millimeters from the sample. Illumination is achieved by a collimated light beam directed at 29° to the plate normal. The sample is viewed in the normal direction, giving a fringe constant, f, of 0.3 mm. The upper edge of the plates is compressed at a slow rate to well over the buckling load using a screw driven testing machine (Instron). The axial load, P, and the end shortening, Va, obtained using an LVDT, are recorded during the tests. The evolving fringe pattern is monitored using a video camera. 3.2. Test results Figs. 3–5 show three Moire sequences representative of the deformation pattern obtained 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 Section 2.2, buckling occurs at K=10. The post buckling behavior is greatly affected by the aspect 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 of secondary buckling near the vertical edges of the bulge (II), the final stage of bulge collapse (III) and the splitting of the bulge into two separate bulges (IV). Print V corresponds to unloading, just before the transition from two to one bulge; note the associated load is well less than that in (IV), indicating a pronounced elastic hysteresis. 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 development of a second bulge in the contact area below the first bulge leads to a transition into a two-bulge configuration (II). The next significant event is the onset of secondary buckling at the vertical edges of both bulges (III). Note that the plate response thereafter was stable (i.e. no bulge splitting). This behavior was common over the range 1.3⬍R⬍2R0
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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 normalized edge displacement. Prints I–IV correspond to loading while print V to unloading.
Fig. 5, R=2.9 — Buckling starts with two randomly positioned bulges (I), in consistency with the buckling behavior discussed in Section 2.2. A third bulge develops near 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, but not 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 behavior typifies the range 2R0⬍R⬍3R0, with the production of the fourth bulge being suppressed in favor of secondary buckling in the three uplifting bulges if R is less than about 2.6. For longer plates, more bulges emerge at buckling, in consistency with Eq. (4). For example, for R=6 (b, t =25.4 mm, 0.5 mm, results not shown), five bulges initially emerged. Similarly to the process described earlier, a transition to six and then seven bulges 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
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Fig. 4. Moire fringe sequence for a unilaterally constrained plate under axial compression; b, t=76.2 mm, 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 instability is 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), in particular at the edges of the bulge, analogously to secondary buckling or wrinkling in unrestrained plates. (3) These two instability mechanisms, to be referred to as “secondary contact buckling” and “secondary circumferential buckling”, respectively, are competitive; the onset of one delays or prevents the other. (4) Friction is expected 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 transition. Three plates each of a given aspect ratio were tested, each one repeatedly loaded a total of three times. Thus, nine data point are produced for each aspect ratio. Fig. 6 (symbols) displays the critical loads obtained; the data for R=0.75 and R=1.2 correspond to the onset of secondary circumferential buckling (i.e. no snapping with increasing load) while the rest of the data to buckling mode transition or snapping. In this case, the data for R⬍2R0 and R⬎2R0 correspond to a jump from one to two bulges and two to three bulges, respectively. The results exhibit some scatter, with no pronounced difference between different specimens or repeated tests on the same specimen. These data will be discussed further following the development of the analysis.
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Fig. 5. Moire fringe sequence for a unilaterally constrained plate under axial compression; b, t=50.8 mm, 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, Version 5.3) with a built in contact algorithm is employed. A four node rectangular plate element (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 of the fundamental characteristics of the problem.) In consistency with the tests, the outward deflection, w, and the plate rotation, dw/dl, where l denotes the normal to the plate edge, are assumed zero along all four edges of the plate. Also, the in-plane displacements on the lower edge and the horizontal displacement on the upper edge are assumed zero. The upper edge is given a uniform vertical displacement that is applied in load steps. The center of the expected bulge(s) is (are) given a small initial outward displacement to help obtain a smooth transition into the buckling regime. This auxiliary displacement is removed once the load exceeds about 50% of the buckling load. The plate response is studied as a function of aspect ratio, up to R=3R0; the behavior for larger R can be inferred, to some extent, from these results. As discussed
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Fig. 6. Normalized critical loads as a function of plate aspect ratio. Symbols and curves denote experimental results and finite element predictions, respectively. Experiments: for each aspect ratio, three different samples were tested, each one repeatedly loaded three times. Solid and open symbols correspond to secondary circumferential buckling and plate snapping, respectively. Finite element: shown are results for the three deformation models defined in Fig. 7; solid and dashed curves correspond to plate snapping and onset of secondary circumferential buckling, respectively.
in Section 2.2, for R⬍2R0 only one bulge forms at buckling while for 2R0⬍R⬍3R0, two bulges initially form; in both cases, the bulge(s) is of width b and aspect ratio R0. Because in the range R⬎R0 the position(s) of the initial bulge(s) within the plate is (are) inherently random, a few (three) extreme configurations are analyzed, namely symmetric, off-symmetric and asymmetric, see Fig. 7: Symmetric: If R⬍2R0, a single outward displacement is initially forced at the plate center; if 2R0⬍R⬍3R0, two initial displacements are forced, each a distance R/4 from a plate edge. Off-symmetric: This configuration is defined only for R⬎2R0. In this case, two initial displacements are placed a distance R0/2 from each edge of the plate. Asymmetric: If R⬍R0, one initial displacement is placed at the plate center; if
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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 once the load exceeds about 50% of the buckling load.
R0⬍R⬍2R0, one initial displacement is placed a distance R0/2 from the lower plate edge; if 2R0⬍R⬍3R0, two initial displacements are placed at distances R0/2 and 3R0/2 from the lower plate edge. The symmetric model represents an upper bound for the snapping load because it affords the smallest contact zones possible from which a local instability leading to plate snapping may ensue. As will be seen latter, the results for the two nonsymmetrical models are indistinguishable, and they lead to the lowest bound. To take advantage of symmetry, only half of the plate region is modeled; for the symmetric and off symmetric configurations, only a quarter of the plate is modeled. Analyses are performed for a range of plate dimensions. A systematic mesh refinement study shows that as many as 1000 rectangular elements are necessary to insure convergence of the post buckling solution over the range of parameters studied. Results from different geometries (i.e. b=50.8 mm and 76.2 mm, and t=0.35 mm and 0.7 mm) show that the post buckling solution (i.e. stresses, deflections) can be 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 choice b=50.8 mm and t=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 the buckling load, Kcr. The following technique for securing accurate solution in the early buckling stage was applied: load up to 50% above Kcr, unload to nearly Kcr, 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 data thus obtained are displayed in Fig. 1 as triangular symbols. The results agree with
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those of Levy (1942), and extends the latter down to R=0.22. The buckling load for R⬍0.5 seem to obey well a power law of the form Kcr⫽4.4/R2.05.
(5)
4.3. Post buckling Figs. 8–10 show the evolution of outward displacement contours with load for the three experimental case studies depicted in Figs. 3–5; the normalized fringe constant (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 deformation patterns generally well resemble the test results, notably in the development of local instability, either near the vertical edges of the uplifting bulge(s) or at the contact zone(s), and in the rapid mode transition that follows. A closer examination reveals that, for a given load, the displacements in a bulge are independent of the deformation model [e.g. compare print II of Fig. 9(a) with print I of Fig. 9(b)], being numerically consistent with the respective test results. A brief exposition of the main features follows. Fig. 8, R=1.2 — In consistency with the test results (Fig. 3), the asymmetric model is shown. Major events such as onset of secondary circumferential buckling (II), rapid bulge distortion (III) and bulge splitting (IV) seem to resemble well the test results, 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 bulge in Fig. 9(a) is seen to initiate from the contact area below the first bulge. Its evolution is gradual, but the final transition to a two-bulge configuration is rapid, tending toward a symmetric deformation (III). Upon increasing K to 95, secondary buckling occurs near the vertical edges of each of the two bulges. This, however, did not lead to 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 with R=1.2. K is the normalized edge displacement. Normalized fringe constant, f/t, is 0.43, as in Fig. 3.
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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 symmetrically to the first (II). However, once these bulges mature, they quickly push out the central bulge (III), ending in two symmetric bulges (IV); the rest of the loading 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 discussion in Section 2.2, two bulges initially form (I). The rest of the loading is much the same as discussed for Fig. 9(a), i.e. emergence of an additional bulge over the largest contact area (II), and onset of a secondary circumferential buckling in each bulge (III). In contrast to its experimental counterpart (Fig. 5), however, no fourth bulge 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
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Fig. 10. Outward displacement contours for asymmetric finite element model, R=2.9. Normalized fringe constant, f/t, is 0.48, as in Fig. 5.
increased from K=37.3 to K=69; the fourth bulge emerges from the largest contact area. Such unevenness is also observed in print III, but it is apparently insufficient to cause secondary contact buckling before secondary circumferential buckling intervenes. This demonstrates the complex interplay between these two secondary buckling mechanisms, and how imperfections and frictional effects may possibly play an important role on the post buckling response. The onset of equilibrium loss in the plate, determined from inspection of the contour plots, is displayed in Fig. 6 as a function of R for the three models analyzed. The data for the asymmetric and off symmetric models are virtually indistinguishable; we shall thus omit reference to the latter in the followings. The plate response seems sort of repeatable with increasing the aspect ratio by R0. Within each such range, the critical loads for plate snapping (solid lines) decrease with increasing aspect ratio, nearly approaching the buckling load when R becomes a multiplication of R0. The secondary circumferential buckling load (dashed lines) is favored for relatively small
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aspect ratios, being at least six times the buckling load (as compared to 7.5 for the analogous circular plate problem, see Ceho and Reiss, 1974). The asymmetric model leads to a much smaller snapping load than the symmetric one, making it the controlling configuration for design purposes. An interesting observation concerns loading beyond the snapping load. Assuming the post transition deformation pattern remains symmetric with increasing load, the subsequent instability event can be evaluated. For example, in the case of Fig. 10, secondary circumferential buckling following the emergence of the third bulge should be as for R=2.9/3=0.97. This gives (Fig. 6) K=76, as compared with K=68.3 in Fig. 10. Fig. 6 shows that the symmetric and asymmetric model bound well the experimental 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 of energy considerations. To this end, Fig. 11 shows the variation of the strain energy in the plate, U, with K for the symmetric and asymmetric models of Fig. 9; the results are normalized by the strain energy for an unbuckled plate, U0 U0⫽
abhEe20 . 2
(6)
Fig. 11. Finite element prediction for the variation of normalized strain energy in the plate with normalized load; symmetric and asymmetric models, R=1.7.
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The abrupt breaks in the slope of the curves help identify the onset of mode transition. As shown, following buckling the strain energy for the symmetric model well exceeds that for the asymmetric one, which is consistent with the observed tendency for asymmetry in the tests (Figs. 3–5). Following plate snapping, the energy level for both models coincides, indicating the deformation for the new equilibrium state is symmetric. 4.4. Secondary contact buckling The applied load needed to cause buckling in the contact area, Kcb, can be determined from inspection of the contour plots. Fig. 12 (solid symbols) shows the dependence of Kcb, normalized by Kt, the snapping or mode transition load, on the plate aspect ratio. The ratio Kcb/Kt seem fairly insensitive to R or to the deformation model, being approx. 0.84. To gain an analytical insight into this detrimental local buckling phenomenon, the contact area is assumed a flat rectangular plate of width b and length d, having clamp support on all four edges. Moreover, the horizontal boundary of 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 local buckling, Kcb, normalized by the snapping load, Kt, and of the ratio Kd/4.4/(d/b)2.05. Kd is the finite element prediction for the buckling load of the effective contact area.
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Fig. 13) lying on the actual contact boundary a distance b/4 from the plate edge. The buckling of the effective contact area can now be evaluated analytically if the vertical displacement along the boundary line through S is known. The finite element results show that this displacement monotonically increases from the midpoint to the plate edge, with the relative difference increasing with load. As a first approximation, the vertical displacement at point S, Vd, or, in a non dimensional form, Kd, is taken to represent the entire boundary line. For this buckling model to be valid, the finite element values of the pair Kd and d at the onset of local buckling of the contact area must conform to Eq. (5); note we are concerned here only with relatively small aspect ratios in which Eq. (5) is valid. This conformation requires that Kd=4.4/(d/b)2.05. Fig. 12 (open symbols) shows that this condition holds true within approx. 10% error over the entire range of aspect ratios and deformation models used. The results above suggest a possible decoupling of the contact and uplifting regions of the plate. It is thus useful to construct empirical relations for the evolution of the properties of these regions with load. Figs. 13 and 14 show, respectively, the variations of the length of the uplifting bulge, c, and the local load parameter Kd with K for a number of plate aspect ratios within the range R0⬍R⬍2R0 ; the data
Fig. 13. Finite element prediction for the variation of the normalized length of the uplifting bulge with the normalized applied load for plates of various aspect ratios. Results for both symmetric and asymmetric models are shown. Solid line is a possible fit to the data.
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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 the contact zone seem to evolve fairly independently of the plate aspect ratio or deformation model, with c and Kd being well characterized by the following power law relations c/b⫽A⫺BK
(7)
Kd⫽CK D
(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 length d exist. Compatibility of axial dimensions requires that d⫽(a⫺c)/2,
(9)
or, using (7) d/b⫽(R⫺A⫹BK)/2,
(10)
where d/b denotes the aspect ratio of the effective contact zone. The applied load needed to buckle the contact area can now be obtained in terms of the plate aspect ratio alone by replacing R and Kcr in (5) with d/b and Kd, respectively. The result is
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K 0.366 (R⫺1.02⫹0.0058Kcb)⫺3.1⫽0, R0⬍R⬍2R0. cb
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(11)
The mode transition load, Kt, is found by noting that Kcb=0.84Kt (see Fig. 12): K 0.366 (R⫺1.02⫹0.00487Kt)⫺3.3⫽0, R0⬍R⬍2R0. t
(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 the mobility of the bulge with load. The length of the contact zone in this case is dictated by a complex interplay between the motion of the bulge and its changing size with increasing load. Since this motion is affected by unavoidable friction, it is not felt worthwhile 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 analogous case of a bilaterally constrained column (Chai, 1998). It was shown analytically in that work that the position of the buckle(s) within the column has no affect on the energy state of the system throughout the post buckling regime. This characteristic led to very large scatter in the mode transition loads by virtue of the possible large variations in the relative sizes of the contact zones. The plate contact problem differs from the column case in that the bulge position in the post buckling range is not arbitrary, but is dictated by the evolving stress variations across the width of the plate. Fig. 11 shows that the preferred mode of deformation in this case is asymmetric. Several Finite Elements runs, performed on intermittent levels of asymmetry, show that the least strain energy obtainable corresponds to the most extreme form of 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 applications to debonding or delamination in layered structures. The present work encompasses such applications if K is interpreted as the normalized strain in the substrate. The phenomena of film/substrate contact and buckling mode transition observed in this work seem to add a new dimension into the mechanics of fracture in such systems. Contact near the plate boundaries eliminates mode I cracking but enhance shear fracture. From a material viewpoint, this may be of concern in ductile type 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 half the original one, being less than unity. Consequently, crack opening stresses form at the boundaries of each bulge following plate snapping. This, together with the dynamic energy released in this process, may lead to catastrophic delamination growth, especially for highly brittle interlayers. The mechanics of fracture in such
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events can be worked out relatively easily from a straightforward relations connecting the 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 monotonically increasing edge displacement is studied experimentally and analytically. The aspect ratios studied essentially cover the entire range of practical applications. A clamp type boundary support is considered due to its relevance to thin film debonding and delamination problems, but it is believed that the main conclusions will be only little affected by other choices of support conditions. The tests show that the interaction of the plate with the adjoining rigid substrate following buckling leads to some unique deformation sequences. This includes the formation of discrete (early buckling) or continuous (post buckling) contact zones, and the rapid transition of the buckling waveform to new equilibrium configurations following a gradual evolution of secondary buckling either at a contact area or an uplifting bulge. The specific details strongly depend on the plate aspect ratio and other system parameters. The plate deformation at buckling is quite random, but it tends toward an extreme form of asymmetry with increasing load. A geometrically nonlinear finite element scheme incorporating frictionless contact is used to elucidate the plate response. The analysis duplicates the main sequence of 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 post buckling behavior is characterized by buckling mode transitions that are dominated by two competitive secondary buckling mechanisms, one at a contact area and the other at the uplifting bulge. Because of the inherent randomness of the bulge position in the early stages of buckling, both symmetric and an extreme form of asymmetry of the buckling deformation are analyzed. These models bound well the experimental mode transition loads, with the asymmetric model providing a fairly good fit. This is consistent with the fact that the latter model yields the least strain energy. Within certain ranges of aspect ratios, the evolutions of the contact zone and uplifting bulge are essentially independent of the aspect ratio or the deformation model, which suggests a possible decoupling of these two regions constituting the plate area. Such decoupling forms the basis for developing simple semi-analytic relations for predicting local buckling and mode transition loads in the symmetric case; extension to the asymmetric model is more involved. The results provide some insight into the fracture behavior of imperfectly bonded bilayer 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, which can lead to catastrophic debonding in brittle type interlayers. The specific details can be worked relatively easily from available formulas relating stress intensity factors to boundary membrane forces and bending moments in the plate.
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Acknowledgements The author thanks Mr David Armoni for his help in constructing the test fixtures and test samples, and Mr Eyal Moses for discussions concerning the finite element code.
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