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Length scale interactions in the folding of sandwich structures
Tectonophysics 335 (2001) 111±120
www.elsevier.com/locate/tecto
Length scale interactions in the folding of sandwich structures G.W. Hunt a,*, M.A. Wadee a, A. Ord b a
Department of Mechanical Engineering, University of Bath, Bath, UK b Division of Exploration and Mining, CSIRO, Perth, Australia
Abstract Compressed sandwich structures, comprising thin stiff faces separated by a thick layer of a soft-core material exhibit behaviour analogous to the geological phenomenon of parasitic folding in multi-layered media. Earlier cross-fertilization of ideas from structural mechanics into structural geology is extended, by considering the interactions of folding on two different length scales Ð overall buckling comprising a single half wave over the length of the structure, combined with short wavelength local wrinkling of a single face or layer. A new theory for such interactions is brie¯y presented and reviewed in the geological context. In the present problem, we see folding on different length scales at the same moment in time, implying a coupling of the instabilities associated with overall and local bucking. q 2001 Elsevier Science B.V. All rights reserved. Keywords: sandwich structure; localized buckling; length scale interaction
1. Introduction The recent cross-fertilization of ideas between structural geology and nonlinear structural mechanics has proved most fruitful, for example in describing the onset of localized buckling or folding of integral layers in an embedding medium (Hunt et al., 1996a). However, important distinctions between typical systems and methods of analysis from the two ®elds, particularly in the validity or otherwise of underlying assumptions, have started to emerge. One striking feature of geological systems not generally found in man-made structures is that of self-similarity: the same phenomenon may be observed at a number of different magni®cations. As can be observed in the photograph of Fig. 1, there is, for example, the prospect of a large-scale fold applying extra compression to a particular layer, such that the layer itself buckles or folds on a smaller scale; the innermost * Corresponding author. E-mail address: [email protected] (G.W. Hunt).
layers of the fold have suffered extra compression and buckled locally. For man-made structures, in which either global instability (for instance Euler-type overall buckling) or local instability (local bucking or material breakdown) usually represents failure, instability on different scales is probably the exception rather than the rule. Such behaviour is not altogether unknown however, and is often associated with highly ef®cient structural elements such as sandwich panels. One purpose of the present contribution is to demonstrate the complexity of modelling such a process. As well as the problem being inherently nonlinear, much of the engineers' standard kitbag of analytical techniques and simplifying assumptions turns out to be inadequate. One example of such inadequacy comes from a cornerstone of bending theory, the assumption that plane sections remain plane. This is known to be too restrictive for sandwich structures (Allen 1969), and by immediate implication for many geological systems. It imposes an overall structural integrity
0040-1951/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0040-195 1(01)00051-8
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G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
Fig. 2. A sandwich panel showing localized failure contributing to overall instability. For clarity, dotted lines have been superimposed on the edges of the faces.
Fig. 1. A multi-layer structure exhibiting folding on two scales. After Price and Cosgrove (1990).
which the structure may not be able to sustain, effectively denying it the possibility of differential distortion on the local scale. In the case of sandwich structures this removes the possibility of the local buckling of a single face, while for the geological structure of Fig. 1 it omits the shearing between layers that is clearly part-and-parcel of the localized parasitic folding (see for example Hobbs et al., 1976, pp 167±168). The present paper describes a failure in a compressed sandwich panel involving interaction between buckling on two different scales Ð an overall (Euler) mode and local buckling of a single face. Fig. 2 shows a typical panel, comprising thin metal faces separated by a lighter (foam) core material, having failed in compression. It is clear that local buckling, that has moreover localized to the centre of one face, has contributed signi®cantly to loss of integrity in the overall sense; local failure of the face initiates overall failure as the structure loses stiffness and shortens. Although the system has permanently deformed, plastic hinges having formed in the face, it is likely that the dimension of the observed plastic mechanism is governed in the ®rst instance by the wavelength of a local elastic buckle, as described here.
The response re¯ects a number of advanced geological features Ð nonlinearity, localization, plastic hinge lines (as perhaps in chevron folding, see also Handin et al., 1972), and most signi®cantly, buckling on different length scales (parasitic folding and quasiperiodicity). Contrasting sharply with the more familiar periodic folding as determined in much of Biot's work (e.g. Biot, 1965) involving dominant wavelengths, it follows a trend expressed by Hobbs (1993), in seeking speci®cally structural mechanisms that can lead naturally to such phenomena (see also Hobbs et al., 1991; MuÈhlhaus et al., 1994; Ord, 1994); localization without the need of large initial perturbations (Hunt et al., 1996a,b), and quasiperiodicity from two-dimensional plate effects (MuÈhlhaus et al., 1998) are recent contributions in the same vein. This paper demonstrates how both localization, and folding on different length scales can be reproduced in the same structure, but only by invoking a complex modelling process. The interaction reduces to a system of two coupled nonlinear ordinary differential equations, equivalent to a single sixth-order equation with a set of nonconstant coef®cients to represent the contribution from overall Euler bucking. We have recently developed the mathematical background for this system via the calculus of variations (Hunt and Wadee, 1998); here we give a reduced version, together with a simpli®ed approach based on body force arguments, and an idealized sense explore appropriate geological analogies.
G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
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Fig. 4. A single layer on an elastic foundation.
2.2. Body force description of localized buckling
Fig. 3. Sway and tilt components of overall mode.
2. Modal description 2.1. Overall buckling mode Previous work on sandwich structures has shown ®rst that overall buckling usually precedes localized failure, and secondly that, contrasting with the normal assumption of engineers' bending theory, shearing displacement in the core cannot be ignored. We allow for this in the classical manner of Allen (1969) by introducing separate sway and tilt components of overall buckling in the two degree-of-freedom formulation shown in Fig. 3, where, from simple linear theory, lateral displacement W(x)and rotation u (x) are represented by, W x qs L sin
px ; L
u x qt p cos
px : L
1
A linear eigenvalue analysis can determine the ratio of amplitudes qs/qt. Shear angles from the two components combine to give the shear strain,
gxy
2W px 2 u qs 2 qt p cos : 2x L
2
Standard engineers' bending theory in which shear strain is taken to be absent is therefore represented by the constraint qs qt. While both components contribute to the shear in the core, only the tilt, represented by qt, contributes to the strain energy in the faces; thus only qt appears in the simpli®ed description of face buckling that follows.
Fig. 2 shows that the local mode is aperiodic and apparently con®ned to the middle of one face. The buckling of a single face supported by the core might be modelled in the ®rst instance as an elastic layer of bending stiffness EI resting on a nonlinear elastic foundation with linear stiffness k with respective quadratic and cubic stiffness components, k1 and k2, as seen in Fig. 4. If the resisting springs provide a force of F kw 2 k1 w2 1 k2 w3 per unit length, the response of the layer of Fig. 4 could be described by the equation, EI w 1 Pw 1 kw 2 k1 w2 1 k2 w3 0:
3
(Potier-Ferry 1983; Hunt and Wadee, 1991), dots denoting differentiation with respect to x. For the sandwich face, however, this classical constant load formulation no longer suf®ces. The tilt component of the overall mode adds a long wave variation to the applied horizontal force P. This effect is introduced in this section as a parametrically varying body force, P0, enabling a description of the short wavelength buckling of a single face via a single differential equation. The tilt component alone generates a variable axial strain 1 x in the compression face of the system of Fig. 5 of,
1x u_ t
b _ bp2 px ; sin u 2qt 2 L 2L
4
ut being the associated horizontal displacement. Applying Hooke's Law and considering equilibrium of the differential face element of Fig. 5 gives a total replacement load at a position x on the
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G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
Fig. 5. The sandwich structure and its cross-section.
compression face Ptotal
P Zx P px 1 Pb x dx 1 P0 sin 2 2 L 0
3. Interactive buckling model 5
where P0 (Etcp 2/2L)qt represents the magnitude of the long wave component of the axial load arising from the overall long wave buckle. We see that the variable load P0 is additional to the constant load P in the system; when added to the classical formulation of the elastic layer it leads to the differential equation P px p px cos EI w 1 w 1 P0 sin w 1 w_ 1 kw 2 L L L 2 k1 w2 1 k2 w3 0;
(6)
EI being now the bending stiffness of one face and k being the transverse stiffness of the core. Nonlinearities (k1,k2 ± 0) model real situations of progressive destabilization and restabilization that occur when strain in the core become large (Gibson et al., 1982); these are introduced to maintain consistency with the interactive model of the next section. This is not a full interactive buckling model in the sense that the relative magnitudes of overall and local buckling cannot be determined directly; both P and P0 appear as parametric variations. It does however provide a useful check on the more complete model that follows.
The full sandwich panel is modelled as two thin plates or faces separated by a softer isotropic core material, with the dimensions of Fig. 5. Young's Moduli and Poisson's ratio of the face and core materials are E and v, and Ec and vc, respectively. Loading is provided by an axial compression P acting at the mid-section as shown, with rigid end plates to ensure that the load transfers directly and equally to the faces. Shear in the core is permitted to enable local buckling (as shown in Fig. 2) and has an associated modulus Gc. Overall buckling is represented by the two degrees of freedom of sway and tilt of Fig. 3. No preconceived shape is assumed for local buckling, which remains free to choose its minimum energy con®guration via the calculus of variations. The relative displacements of the local buckle are allowed by assuming functions for lateral and in-plane face displacements, w(x).and u(x) respectively, along with a pure compression degree of freedom, D , as shown in Fig. 6, where DL measures the shortening of the mid-section in the
Fig. 6. The principal functions used in the modelling of localized buckling.
G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
Fig. 7. Typical equilibrium diagram for sandwich struts: (i) fundamental path; (ii) critical path of overall buckling; (iii) secondary path of localized buckling.
absence of overall buckling. The face now has the freedom ®rst to buckle in a short wavelength periodic mode and secondly for the buckle to localize. The expected response of the panel under increasing applied (total) end shortening E/L is as shown in Fig. 7. (i) It initially follows a fundamental equilibrium path involving pure compression in the direction of the load. (ii) And buckles into the overall mode at C causing differential compression in the faces. (iii) This induces the face under greater compression to buckle into a second localized mode at secondary bifurcation point S. The combined outcome is then a highly unstable post-buckling response. The following is a brief summary of a more extensive formulation of Hunt and Wadee (1998). The result is a set of equations, solved numerically, which provides an explanation of this buckling model.
115
de¯ections from the overall and local buckling modes. By accumulating the strain energy contributions of bending, membrane action from compression and de¯ection of the face plates, core distortion and the work done by the load, we obtain a total potential energy functional V. Applying the calculus of variations on V, to ®nd the stationary points of the potential energy and thus the equilibrium equations, we obtain the governing differential equations (Hunt and Wadee, 1998): " ! bp2 px p px w 1 cos w_ sin EI w 1 D 2Dw 1 qt L L L L #
"
2
1G 2 2u w_ 1 2u_ w 1 3w_ w p2 px sin 1 qs 2 qt L L
2 u_ 2 w 3 b
# 1 kw 2 k1 w2 1 k2 w3
0; Du 2
(8)
G G u1 w_ 1 Dw_ w 2b b2
px q 2 qt bp2 G s 2 Dqt 2 1 p cos L b 2L
!
0;
9
where Etc G cb ;G c ; 2 2
I
ct3 : 12 1 2 v2
3.1. Equilibrium equations
D
The mechanical system under consideration is elastic and in equilibrium. Governing equations may thus be derived from consideration of forces or energy. In the formulation presented, stationary total potential energy is used as a mechanism for determining the equilibrium equations of the system and their stability. Total potential energy, V, is a summation of the contributions of strain energy, U, minus the work done by the load, PE
This is a pair of coupled nonlinear differential equations, equivalent to a single equation of sixth order in x, subject to the following boundary conditions:
V U 2 PE;
7
where E is the total end-shortening of the panel under the load P, comprising the contribution from pure compression D and the release from the out of plane
w 0 w L w 0 w L 0;
10
11
and 2
P 1 2 1 _ _ 1 w_ 2 L 2 D; u 01 w_ 0 2 D u L 4D 2 2 12
which minimize the potential energy, by assuming simple supports (Eq. (11)) and matching the applied stress at the boundaries (Eq. (12)). For equilibrium, V must also be stationary with respect to the degrees
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G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
of freedom, qs, qt and D . Thus minimizing V and rearranging these equations to make P, qs and D the subject 2p2 EI qs 2 qt P 1 2G qs L2 2G ZL px 2 13 1 cos w_ 2 u dx; qs pL 0 L b qs qt
" ! ZL 1 Dp2 b2 px cos 11 1 2 L 2GL 0 pL
Db px 1 u_ 1 w_ 2 2 sin L 2 GL2
D
2 u 2 w_ b
!
!# dx;
(14)
P 1 ZL 1 1 u_ 1 w_ 2 dx: 4D 2L 0 2
15
Finally, a simple linear eigenvalue analysis yields the critical load for overall buckling that agrees with the classical analysis of Allen (1969): 2 3 PC Gc cb 1
6 7 2pEI G2c cb 6 7 2 6 7: 4 Etbp2 5 L2 1 G c 2L2
16
At the secondary bifurcation point, the relationships between qs and qt, P, and D simplify to: Dp2 b2 qt ; 2GL2
P PC ;
D
G 2 u 2 w_ 0; 2Db b
19
describing the more compressed face. As these equations are nonautonomous the position of the secondary bifurcation is best determined numerically. We thus vary qt until a nontrivial solution with a slight hump at the centre is found. Physically, this occurs when the effective load at midspan is greater than the local buckling load. 3.3. Rescaling of equations It is useful at this point to rescale the equations in terms of nondimensional variables. This process is vital if the equations are to be successfully analysed by the numerical continuation package AUTO (Doedel et al., 1995), which is only programmed in nondimensional terms. The scalings used for the principal functions are as follows: x~ !
2 x; L
w~ !
2 w; L
u~ !
2 u: L
20
The equations are adjusted accordingly to accommodate this change of scale, and are implemented in auto. 4. Numerical experimentation
3.2. Secondary bifurcation for local buckling
qs 2 qt
u 2
PC : 4D
17
Differential Eqs. (8) and (9), can also be linearized to give ! PC 2G w 2 EI w1 3 2 1
Dp2 b px p px G qt sin cos w 1 w_ 1 u_ L L L L b
1
Dp4 b2 px 1 kw 0; qt sin 3 L 2L
(18)
Having formulated the necessary equations, we now present the fully interactive solutions found from the numerical continuation package auto. Simply supported boundary conditions (Eq. (11)) are chosen at the ends of the face in question, and the symmetric-section condition at the centre x~ 1 u0 w_ w
21
ensures that overall symmetry is not lost. The solutions are presented as follows: ®rst, the position of the secondary bifurcation on the critical path for overall buckling (see Fig. 7) is found for a range of different lengths; secondly, for the worst case in which this occurs closest to the initial bifurcation, the full postbuckling response is determined for the case of a core with a cubic transverse resistance as shown in Eq. (6); ®nally, results of the interactive model are compared with those found from the body force approach at the same qt value.
G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
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the position of the secondary bifurcation point. Fig. 8 shows the relation between total relative endshortening at the position of secondary bifurcation, E q2 p2 s 1 D; L 4
Fig. 8. Relative end-shortening at secondary bifurcation.
4.1. Results We choose a set of practical sandwich component properties used as earlier example (Hunt et al., 1988). Flange Young's modulus: E 68.95 GPa, core Young's modulus: Ec 198.57 MPa, core shear modulus: Gc 82.74 MPa, face thickness: t 0.508 mm, core thickness: b 5.08 mm, strut width: c 254 mm, face Poisson's ratio: v 0.3, core Poisson's ratio: vc 0.2, with the length of the structure (L) left as a possible parameter to vary. 4.1.1. Location of secondary bifurcation Although linear, the secondary bifurcation Eqs. (18) and (19) are nonautonomous and most easily handled numerically. Solutions are ®rst sought at constant qt by employing a numerical shooting method (Press et al., 1990), searching for solutions that maintain the symmetry of Eq. (21) using an implicit symplectic Runge±Kutta scheme. These are then introduced into AUTO so that numerical continuation can be used to reduce parametrically the value of qt until no solution can be found. This de®nes
22
with length. Table 1 compares this quantitatively with similar results from periodic (Rayleigh±Ritz) formulations, for two struts of the same dimensions as above with lengths of 50.8 and 508 mm (SimoÄes Da Silva 1989). In each case the secondary bifurcation that leads to localization occurs well inside that of the periodic model. Over the range 50 # L # 150, the end-shortening at secondary bifurcation is relatively small. The curve reaches a minimum at approximately L 100 mm, and we therefore choose this length for detailed post-buckling study below. This has an L/b ratio of around twenty, which for a sandwich structure is of realistic proportions. 4.1.2. Full post-buckling study The near-optimum length of L 100 mm is next reviewed in post-buckling reÂgime by solving Eqs. (8) and (9) subject to the integral constraints Eqs. (13±15). In the example chosen, we concentrate on a strut with a cubic constitutive relationship for the core where the transverse resistance, F, is given by F kw 2 k1 w2 1 k2 w3 . This chosen example gives a good indication of the relative in¯uence of the core nonlinearities. Results are given for through-core displacements of the order of b/2. Fig. 9 shows the results for a realistic set of coef®cients k 10341 N mm 22, k1 6728 N mm 23 and k2 1346 N mm 24. Fig. 9a shows the central amplitude beginning to dominate as the load reduces. The
Table 1 Comparison of the position of secondary bifurcation for the interactive model and earlier Rayleigh±Ritz results. For L 508 mm, the periodic results are interpolated from graphical data L (mm)
Measure at secondary bifurcation
Periodic Rayleigh±Ritz model (SimoÄes Da Silva 1989)
Interactive localized buckling model
50.8
qs E/L
0.140 0.053
0.116 0.038
508
qs E/L
0.46 0.53
0.304 0.228
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G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
Fig. 9. Post-buckling responses for strut of L 100 mm with cubic core: (a) localized buckle pro®les; (b) load-end-shortening curve; (c) relative magnitudes of overall and localized modal amplitudes; (d) transverse resistance of core.
load-end-shortening relationship of Fig. 9b shows how the quadratic and cubic core nonlinearities carry increasing in¯uence as the de¯ection develops. After an initial fall of load capacity governed largely by the linear term k, the central peak begins to dominate, a phenomenon caused by quadratic softening term k1. The system subsequently restabilizes at P < 0.40P C, when the in¯uence of the hardening cubic nonlinearity k2 begins to dominate over the quadratic term. The destabilization±restabilization of the core is shown in Fig. 9d, which re¯ects qualitatively the form of some real core behaviours (Gibson et al., 1982). 4.1.3. Comparison with the body force approach By relating analogous forms for both P and qt, comparisons between the body force approach and the results of the fully interactive model can be drawn up. Core shear appears in the latter but not the former, so P of Eq. (6) in the body force model is related to the equivalent term of Eq. (18) for the fully interactive model by ! PC 2G C P BF 2 23 3 Int 2
such that loads match at the secondary bifurcation point. Varying the values of P and qt in Eq. (6) accordingly, the following observations are made from Fig. 10. In view of the differences in the formulations, comparisons, for this geometry anyway, are satisfactory. For the maximum amplitude, wavelength and degree of localization, correlation is excellent in the case of the quadratic core (k2 0), as seen in Figs. 10a±c. Agreement is good for the cubic case (Figs.10d±f) although it tends to overestimate the maximum amplitude at lower loads; here the greater complexity of nonlinearity presumably makes it more dif®cult for the simpli®ed body force approach to match the more complete interactive model. 5. Concluding remarks The paper presents a theoretical development, involving buckling or folding on two different length scales that is in many ways far removed from direct geological application. It does however carry certain important messages. The system is purely elastic, and comprises essentially only three layers. However, as in the evolution of localized single-layer buckling
G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
119
Fig. 10. Comparison of body force approach with interactive buckling model for strut of L 100 mm: dashed line-body force model; solid lineinteractive model: (a)± (c) quadratic core; (d)± (f) cubic core: (a) P 0.95 P C, qt 0.2447; (b) P 0.50 P C, qt 0.0433; (c) P 0, qt 0.0022; (d) P 0.95 P C, qt 0.2461; (e) P 0.85 P C, qt 0.2311; (f) P 0.75 P C, qt 0.2088.
(Hunt et al., 1996a; Budd et al., 1998), the elastic response can strongly in¯uence initial conditions for more realistic rheological models. If for example a Maxwell-type visco-elastic material is used to replace the elastic core, the initial response, from which the system would start to evolve, would coincide with one of the post-buckled states of Fig. 9. Thus an initial large-de¯ection localized perturbation, of direct signi®cance to the subsequent evolution (Cobbold, 1975), is automatically provided and de®ned by the initial elastic response. By extension, the analysis indicates how folding on a large scale in a multi-layered system can be accompanied by localized parasitic folding in the regions where extra compression applies. Such behaviour is found both experimentally and in the ®eld, but is seen to be dif®cult to model. To capture it satisfactorily in this relatively primitive (proto-type) system has required sophistication of analysis over and above that of conventional engineering. The model is necessarily nonlinear, and standard engineering assumptions of simple bending theory have had to be relaxed. It is interesting to compare and contrast this situation with developments in single-layer localized buckling. An elastic layer embedded in a viscous medium, under
the nonlinearity of prescribed end-shortening, has recently been demonstrated to have near self-similar localized solutions (Budd et al., 1998); after rescaling both space and time, it is shown that the same (or nearly the same) localized form can appear over a range of different length scales during the evolution. In the present problem we see folding on different length scales at the same moment in time, implying a coupling of the instabilities associated with overall and local buckling. Added complexity associated with nonlinearity may mean that practically relevant states will need sifting from the multiplicity of solutions that can arise (see for example Champneys et al., 1997), and highly selective path-following techniques may be required to ®nd them.
References Allen, H.G., 1969. Analysis and design of structural sandwich panels. Pergamon Press, Oxford. Biot, M.A., 1965. Mechanics of Incremental Deformations. Wiley, New York. Budd, C.J., Hunt, G.W., Peletier, M.A., 1998. Self-similar fold evolution under prescribed end-shortening. Math. Geol. 31 (8), 989±1005.
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Champneys, A.R., Hunt, G.W., Thompson, J.M.T., 1997. Localization and solitary waves in solid mechanics. Phil. Trans. R. Soc. Lond. A 355 (1732), 2073±2213 special issue. Cobbold, P.R., 1975. Fold propagation in single embedded layers. Tectonophysics 27, 333±351. Doedel, E.J., Wang, X.J., Fairgrieve, T.F. auto94: Software for continuation and bifurcation problems in ordinary differential equations. Technical Report CRPC-95-2, California Institute of Technology, 1995. Gibson, L.J., Ashby, M.F., Schjaer, G.S., Robertson, C.I., 1982. The mechanics of two-dimensional cellular materials. Proc. R. Soc. Lond. A 382, 25±42. Handin, J., Friedman, M., Logan, J.M., Pattison, L.J., Swolfs, H.S., 1972. Experimental folding of rocks under con®ning pressure: buckling of single-layer rock beams. Flow and Fracture of Rock, Heard, H.C., Borg, I.Y., Carter, N.L., Raleigh, C.B. (Eds.). Geophys. Monogr. Am. Geophys. Union 16, 1±28. Hobbs, B.E., 1993. The signi®cance of structural geology in rock mechanics. In: Hudson, J.A. (Ed.). Comprehensive Rock Engineering. Pergamon Press, Oxford, pp. 25±62. Hobbs, B.E., Means, W.D., Williams, P.F., 1976. An outline of structural geology. Wiley, New York. Hobbs, B.E., MuÈhlhaus, H-B., Ord, A., 1991. The fractal geometry of deformed rocks. The Geometry of Naturally Deformed Rocks, vol. 239b. Mitt. aus den Geolological Institute, ETH ZuÈrich, Neue Folge, p. 28. Hunt, G.W., Wadee, M.K., 1991. Comparative lagrangian formulations for localized buckling. Proc. R. Soc. Lond. A 434, 485± 502. Hunt, G.W., Wadee, M.A., 1998. Localization and mode interaction in sandwich structures. Proc. R. Soc. Lond. A 454 (1972), 1197±1216. Hunt, G.W., Da Silva, L.S., Manzocchi, G.M.E., 1988. Interactive
buckling in sandwich structures. Proc. R. Soc. Lond. A 417, 155±177. Hunt, G.W., MuÈhlhaus, H-B., Hobbs, B., Ord, A., 1996a. Localized folding of viscous layers. Geologische Rundschau (Int. J. Earth Sci.) 85 (1), 58±64. Hunt, G.W., MuÈhlhaus, H-B., Whiting, A.I.M., 1996b. Evolution of localized folding for a thin elastic layer in a softening viscoelastic medium. PAGEOPH (Pure Appl. Geophys.) 146 (2), 229±252. MuÈhlhaus, H-B., Hobbs, B.E., Ord, A., 1994. The role of axial constraints on the evolution of folds in single layers. In: Siriwardane, H.J., Zaman, M.M. (Eds.). Computer methods and advances in geomechanics, 1. A.A. Balkema, Rotterdam, pp. 223±231. MuÈhlhaus, H-B., Sakaguchi, H., Hobbs, B.E., 1998. Evolution of three-dimensional folds for a non-Newtonian plate in a viscous medium. Proc. R. Soc. London, Ser. A. 454, 3131±3143. Ord, A., 1994. The fractal geometry of patterned structures in numerical models of rock formation. In: Kruhl, J.H. (Ed.). Fractals and Dynamic Systems in Geoscience. Springer, Berlin, pp. 131±155. Potier-Ferry, M., 1983. Amplitude modulation, phase modulation and localization of buckling patterns. In: Thompson, J.M.T., Hunt, G.W. (Eds.). Collapse: the Buckling of Structures in Theory and Practice. Cambridge University Press, Cambridge, pp. 149±159. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T., 1990. Numerical recipes in C: the art of scienti®c computing. Cambridge University Press, Cambridge. Price, N.J., Cosgrove, J.W., 1990. Analysis of Geological Structures. Cambridge University Press, Cambridge. SimoÄes Da Silva, L.A.P., 1989. Modal interactions in bending and buckling of sandwich structures. PhD thesis, Imperial College of Science, Technology and Medicine, London.
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Length scale interactions in the folding of sandwich structures G.W. Hunt a,*, M.A. Wadee a, A. Ord b a
Department of Mechanical Engineering, University of Bath, Bath, UK b Division of Exploration and Mining, CSIRO, Perth, Australia
Abstract Compressed sandwich structures, comprising thin stiff faces separated by a thick layer of a soft-core material exhibit behaviour analogous to the geological phenomenon of parasitic folding in multi-layered media. Earlier cross-fertilization of ideas from structural mechanics into structural geology is extended, by considering the interactions of folding on two different length scales Ð overall buckling comprising a single half wave over the length of the structure, combined with short wavelength local wrinkling of a single face or layer. A new theory for such interactions is brie¯y presented and reviewed in the geological context. In the present problem, we see folding on different length scales at the same moment in time, implying a coupling of the instabilities associated with overall and local bucking. q 2001 Elsevier Science B.V. All rights reserved. Keywords: sandwich structure; localized buckling; length scale interaction
1. Introduction The recent cross-fertilization of ideas between structural geology and nonlinear structural mechanics has proved most fruitful, for example in describing the onset of localized buckling or folding of integral layers in an embedding medium (Hunt et al., 1996a). However, important distinctions between typical systems and methods of analysis from the two ®elds, particularly in the validity or otherwise of underlying assumptions, have started to emerge. One striking feature of geological systems not generally found in man-made structures is that of self-similarity: the same phenomenon may be observed at a number of different magni®cations. As can be observed in the photograph of Fig. 1, there is, for example, the prospect of a large-scale fold applying extra compression to a particular layer, such that the layer itself buckles or folds on a smaller scale; the innermost * Corresponding author. E-mail address: [email protected] (G.W. Hunt).
layers of the fold have suffered extra compression and buckled locally. For man-made structures, in which either global instability (for instance Euler-type overall buckling) or local instability (local bucking or material breakdown) usually represents failure, instability on different scales is probably the exception rather than the rule. Such behaviour is not altogether unknown however, and is often associated with highly ef®cient structural elements such as sandwich panels. One purpose of the present contribution is to demonstrate the complexity of modelling such a process. As well as the problem being inherently nonlinear, much of the engineers' standard kitbag of analytical techniques and simplifying assumptions turns out to be inadequate. One example of such inadequacy comes from a cornerstone of bending theory, the assumption that plane sections remain plane. This is known to be too restrictive for sandwich structures (Allen 1969), and by immediate implication for many geological systems. It imposes an overall structural integrity
0040-1951/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0040-195 1(01)00051-8
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G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
Fig. 2. A sandwich panel showing localized failure contributing to overall instability. For clarity, dotted lines have been superimposed on the edges of the faces.
Fig. 1. A multi-layer structure exhibiting folding on two scales. After Price and Cosgrove (1990).
which the structure may not be able to sustain, effectively denying it the possibility of differential distortion on the local scale. In the case of sandwich structures this removes the possibility of the local buckling of a single face, while for the geological structure of Fig. 1 it omits the shearing between layers that is clearly part-and-parcel of the localized parasitic folding (see for example Hobbs et al., 1976, pp 167±168). The present paper describes a failure in a compressed sandwich panel involving interaction between buckling on two different scales Ð an overall (Euler) mode and local buckling of a single face. Fig. 2 shows a typical panel, comprising thin metal faces separated by a lighter (foam) core material, having failed in compression. It is clear that local buckling, that has moreover localized to the centre of one face, has contributed signi®cantly to loss of integrity in the overall sense; local failure of the face initiates overall failure as the structure loses stiffness and shortens. Although the system has permanently deformed, plastic hinges having formed in the face, it is likely that the dimension of the observed plastic mechanism is governed in the ®rst instance by the wavelength of a local elastic buckle, as described here.
The response re¯ects a number of advanced geological features Ð nonlinearity, localization, plastic hinge lines (as perhaps in chevron folding, see also Handin et al., 1972), and most signi®cantly, buckling on different length scales (parasitic folding and quasiperiodicity). Contrasting sharply with the more familiar periodic folding as determined in much of Biot's work (e.g. Biot, 1965) involving dominant wavelengths, it follows a trend expressed by Hobbs (1993), in seeking speci®cally structural mechanisms that can lead naturally to such phenomena (see also Hobbs et al., 1991; MuÈhlhaus et al., 1994; Ord, 1994); localization without the need of large initial perturbations (Hunt et al., 1996a,b), and quasiperiodicity from two-dimensional plate effects (MuÈhlhaus et al., 1998) are recent contributions in the same vein. This paper demonstrates how both localization, and folding on different length scales can be reproduced in the same structure, but only by invoking a complex modelling process. The interaction reduces to a system of two coupled nonlinear ordinary differential equations, equivalent to a single sixth-order equation with a set of nonconstant coef®cients to represent the contribution from overall Euler bucking. We have recently developed the mathematical background for this system via the calculus of variations (Hunt and Wadee, 1998); here we give a reduced version, together with a simpli®ed approach based on body force arguments, and an idealized sense explore appropriate geological analogies.
G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
113
Fig. 4. A single layer on an elastic foundation.
2.2. Body force description of localized buckling
Fig. 3. Sway and tilt components of overall mode.
2. Modal description 2.1. Overall buckling mode Previous work on sandwich structures has shown ®rst that overall buckling usually precedes localized failure, and secondly that, contrasting with the normal assumption of engineers' bending theory, shearing displacement in the core cannot be ignored. We allow for this in the classical manner of Allen (1969) by introducing separate sway and tilt components of overall buckling in the two degree-of-freedom formulation shown in Fig. 3, where, from simple linear theory, lateral displacement W(x)and rotation u (x) are represented by, W x qs L sin
px ; L
u x qt p cos
px : L
1
A linear eigenvalue analysis can determine the ratio of amplitudes qs/qt. Shear angles from the two components combine to give the shear strain,
gxy
2W px 2 u qs 2 qt p cos : 2x L
2
Standard engineers' bending theory in which shear strain is taken to be absent is therefore represented by the constraint qs qt. While both components contribute to the shear in the core, only the tilt, represented by qt, contributes to the strain energy in the faces; thus only qt appears in the simpli®ed description of face buckling that follows.
Fig. 2 shows that the local mode is aperiodic and apparently con®ned to the middle of one face. The buckling of a single face supported by the core might be modelled in the ®rst instance as an elastic layer of bending stiffness EI resting on a nonlinear elastic foundation with linear stiffness k with respective quadratic and cubic stiffness components, k1 and k2, as seen in Fig. 4. If the resisting springs provide a force of F kw 2 k1 w2 1 k2 w3 per unit length, the response of the layer of Fig. 4 could be described by the equation, EI w 1 Pw 1 kw 2 k1 w2 1 k2 w3 0:
3
(Potier-Ferry 1983; Hunt and Wadee, 1991), dots denoting differentiation with respect to x. For the sandwich face, however, this classical constant load formulation no longer suf®ces. The tilt component of the overall mode adds a long wave variation to the applied horizontal force P. This effect is introduced in this section as a parametrically varying body force, P0, enabling a description of the short wavelength buckling of a single face via a single differential equation. The tilt component alone generates a variable axial strain 1 x in the compression face of the system of Fig. 5 of,
1x u_ t
b _ bp2 px ; sin u 2qt 2 L 2L
4
ut being the associated horizontal displacement. Applying Hooke's Law and considering equilibrium of the differential face element of Fig. 5 gives a total replacement load at a position x on the
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G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
Fig. 5. The sandwich structure and its cross-section.
compression face Ptotal
P Zx P px 1 Pb x dx 1 P0 sin 2 2 L 0
3. Interactive buckling model 5
where P0 (Etcp 2/2L)qt represents the magnitude of the long wave component of the axial load arising from the overall long wave buckle. We see that the variable load P0 is additional to the constant load P in the system; when added to the classical formulation of the elastic layer it leads to the differential equation P px p px cos EI w 1 w 1 P0 sin w 1 w_ 1 kw 2 L L L 2 k1 w2 1 k2 w3 0;
(6)
EI being now the bending stiffness of one face and k being the transverse stiffness of the core. Nonlinearities (k1,k2 ± 0) model real situations of progressive destabilization and restabilization that occur when strain in the core become large (Gibson et al., 1982); these are introduced to maintain consistency with the interactive model of the next section. This is not a full interactive buckling model in the sense that the relative magnitudes of overall and local buckling cannot be determined directly; both P and P0 appear as parametric variations. It does however provide a useful check on the more complete model that follows.
The full sandwich panel is modelled as two thin plates or faces separated by a softer isotropic core material, with the dimensions of Fig. 5. Young's Moduli and Poisson's ratio of the face and core materials are E and v, and Ec and vc, respectively. Loading is provided by an axial compression P acting at the mid-section as shown, with rigid end plates to ensure that the load transfers directly and equally to the faces. Shear in the core is permitted to enable local buckling (as shown in Fig. 2) and has an associated modulus Gc. Overall buckling is represented by the two degrees of freedom of sway and tilt of Fig. 3. No preconceived shape is assumed for local buckling, which remains free to choose its minimum energy con®guration via the calculus of variations. The relative displacements of the local buckle are allowed by assuming functions for lateral and in-plane face displacements, w(x).and u(x) respectively, along with a pure compression degree of freedom, D , as shown in Fig. 6, where DL measures the shortening of the mid-section in the
Fig. 6. The principal functions used in the modelling of localized buckling.
G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
Fig. 7. Typical equilibrium diagram for sandwich struts: (i) fundamental path; (ii) critical path of overall buckling; (iii) secondary path of localized buckling.
absence of overall buckling. The face now has the freedom ®rst to buckle in a short wavelength periodic mode and secondly for the buckle to localize. The expected response of the panel under increasing applied (total) end shortening E/L is as shown in Fig. 7. (i) It initially follows a fundamental equilibrium path involving pure compression in the direction of the load. (ii) And buckles into the overall mode at C causing differential compression in the faces. (iii) This induces the face under greater compression to buckle into a second localized mode at secondary bifurcation point S. The combined outcome is then a highly unstable post-buckling response. The following is a brief summary of a more extensive formulation of Hunt and Wadee (1998). The result is a set of equations, solved numerically, which provides an explanation of this buckling model.
115
de¯ections from the overall and local buckling modes. By accumulating the strain energy contributions of bending, membrane action from compression and de¯ection of the face plates, core distortion and the work done by the load, we obtain a total potential energy functional V. Applying the calculus of variations on V, to ®nd the stationary points of the potential energy and thus the equilibrium equations, we obtain the governing differential equations (Hunt and Wadee, 1998): " ! bp2 px p px w 1 cos w_ sin EI w 1 D 2Dw 1 qt L L L L #
"
2
1G 2 2u w_ 1 2u_ w 1 3w_ w p2 px sin 1 qs 2 qt L L
2 u_ 2 w 3 b
# 1 kw 2 k1 w2 1 k2 w3
0; Du 2
(8)
G G u1 w_ 1 Dw_ w 2b b2
px q 2 qt bp2 G s 2 Dqt 2 1 p cos L b 2L
!
0;
9
where Etc G cb ;G c ; 2 2
I
ct3 : 12 1 2 v2
3.1. Equilibrium equations
D
The mechanical system under consideration is elastic and in equilibrium. Governing equations may thus be derived from consideration of forces or energy. In the formulation presented, stationary total potential energy is used as a mechanism for determining the equilibrium equations of the system and their stability. Total potential energy, V, is a summation of the contributions of strain energy, U, minus the work done by the load, PE
This is a pair of coupled nonlinear differential equations, equivalent to a single equation of sixth order in x, subject to the following boundary conditions:
V U 2 PE;
7
where E is the total end-shortening of the panel under the load P, comprising the contribution from pure compression D and the release from the out of plane
w 0 w L w 0 w L 0;
10
11
and 2
P 1 2 1 _ _ 1 w_ 2 L 2 D; u 01 w_ 0 2 D u L 4D 2 2 12
which minimize the potential energy, by assuming simple supports (Eq. (11)) and matching the applied stress at the boundaries (Eq. (12)). For equilibrium, V must also be stationary with respect to the degrees
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G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
of freedom, qs, qt and D . Thus minimizing V and rearranging these equations to make P, qs and D the subject 2p2 EI qs 2 qt P 1 2G qs L2 2G ZL px 2 13 1 cos w_ 2 u dx; qs pL 0 L b qs qt
" ! ZL 1 Dp2 b2 px cos 11 1 2 L 2GL 0 pL
Db px 1 u_ 1 w_ 2 2 sin L 2 GL2
D
2 u 2 w_ b
!
!# dx;
(14)
P 1 ZL 1 1 u_ 1 w_ 2 dx: 4D 2L 0 2
15
Finally, a simple linear eigenvalue analysis yields the critical load for overall buckling that agrees with the classical analysis of Allen (1969): 2 3 PC Gc cb 1
6 7 2pEI G2c cb 6 7 2 6 7: 4 Etbp2 5 L2 1 G c 2L2
16
At the secondary bifurcation point, the relationships between qs and qt, P, and D simplify to: Dp2 b2 qt ; 2GL2
P PC ;
D
G 2 u 2 w_ 0; 2Db b
19
describing the more compressed face. As these equations are nonautonomous the position of the secondary bifurcation is best determined numerically. We thus vary qt until a nontrivial solution with a slight hump at the centre is found. Physically, this occurs when the effective load at midspan is greater than the local buckling load. 3.3. Rescaling of equations It is useful at this point to rescale the equations in terms of nondimensional variables. This process is vital if the equations are to be successfully analysed by the numerical continuation package AUTO (Doedel et al., 1995), which is only programmed in nondimensional terms. The scalings used for the principal functions are as follows: x~ !
2 x; L
w~ !
2 w; L
u~ !
2 u: L
20
The equations are adjusted accordingly to accommodate this change of scale, and are implemented in auto. 4. Numerical experimentation
3.2. Secondary bifurcation for local buckling
qs 2 qt
u 2
PC : 4D
17
Differential Eqs. (8) and (9), can also be linearized to give ! PC 2G w 2 EI w1 3 2 1
Dp2 b px p px G qt sin cos w 1 w_ 1 u_ L L L L b
1
Dp4 b2 px 1 kw 0; qt sin 3 L 2L
(18)
Having formulated the necessary equations, we now present the fully interactive solutions found from the numerical continuation package auto. Simply supported boundary conditions (Eq. (11)) are chosen at the ends of the face in question, and the symmetric-section condition at the centre x~ 1 u0 w_ w
21
ensures that overall symmetry is not lost. The solutions are presented as follows: ®rst, the position of the secondary bifurcation on the critical path for overall buckling (see Fig. 7) is found for a range of different lengths; secondly, for the worst case in which this occurs closest to the initial bifurcation, the full postbuckling response is determined for the case of a core with a cubic transverse resistance as shown in Eq. (6); ®nally, results of the interactive model are compared with those found from the body force approach at the same qt value.
G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
117
the position of the secondary bifurcation point. Fig. 8 shows the relation between total relative endshortening at the position of secondary bifurcation, E q2 p2 s 1 D; L 4
Fig. 8. Relative end-shortening at secondary bifurcation.
4.1. Results We choose a set of practical sandwich component properties used as earlier example (Hunt et al., 1988). Flange Young's modulus: E 68.95 GPa, core Young's modulus: Ec 198.57 MPa, core shear modulus: Gc 82.74 MPa, face thickness: t 0.508 mm, core thickness: b 5.08 mm, strut width: c 254 mm, face Poisson's ratio: v 0.3, core Poisson's ratio: vc 0.2, with the length of the structure (L) left as a possible parameter to vary. 4.1.1. Location of secondary bifurcation Although linear, the secondary bifurcation Eqs. (18) and (19) are nonautonomous and most easily handled numerically. Solutions are ®rst sought at constant qt by employing a numerical shooting method (Press et al., 1990), searching for solutions that maintain the symmetry of Eq. (21) using an implicit symplectic Runge±Kutta scheme. These are then introduced into AUTO so that numerical continuation can be used to reduce parametrically the value of qt until no solution can be found. This de®nes
22
with length. Table 1 compares this quantitatively with similar results from periodic (Rayleigh±Ritz) formulations, for two struts of the same dimensions as above with lengths of 50.8 and 508 mm (SimoÄes Da Silva 1989). In each case the secondary bifurcation that leads to localization occurs well inside that of the periodic model. Over the range 50 # L # 150, the end-shortening at secondary bifurcation is relatively small. The curve reaches a minimum at approximately L 100 mm, and we therefore choose this length for detailed post-buckling study below. This has an L/b ratio of around twenty, which for a sandwich structure is of realistic proportions. 4.1.2. Full post-buckling study The near-optimum length of L 100 mm is next reviewed in post-buckling reÂgime by solving Eqs. (8) and (9) subject to the integral constraints Eqs. (13±15). In the example chosen, we concentrate on a strut with a cubic constitutive relationship for the core where the transverse resistance, F, is given by F kw 2 k1 w2 1 k2 w3 . This chosen example gives a good indication of the relative in¯uence of the core nonlinearities. Results are given for through-core displacements of the order of b/2. Fig. 9 shows the results for a realistic set of coef®cients k 10341 N mm 22, k1 6728 N mm 23 and k2 1346 N mm 24. Fig. 9a shows the central amplitude beginning to dominate as the load reduces. The
Table 1 Comparison of the position of secondary bifurcation for the interactive model and earlier Rayleigh±Ritz results. For L 508 mm, the periodic results are interpolated from graphical data L (mm)
Measure at secondary bifurcation
Periodic Rayleigh±Ritz model (SimoÄes Da Silva 1989)
Interactive localized buckling model
50.8
qs E/L
0.140 0.053
0.116 0.038
508
qs E/L
0.46 0.53
0.304 0.228
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G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
Fig. 9. Post-buckling responses for strut of L 100 mm with cubic core: (a) localized buckle pro®les; (b) load-end-shortening curve; (c) relative magnitudes of overall and localized modal amplitudes; (d) transverse resistance of core.
load-end-shortening relationship of Fig. 9b shows how the quadratic and cubic core nonlinearities carry increasing in¯uence as the de¯ection develops. After an initial fall of load capacity governed largely by the linear term k, the central peak begins to dominate, a phenomenon caused by quadratic softening term k1. The system subsequently restabilizes at P < 0.40P C, when the in¯uence of the hardening cubic nonlinearity k2 begins to dominate over the quadratic term. The destabilization±restabilization of the core is shown in Fig. 9d, which re¯ects qualitatively the form of some real core behaviours (Gibson et al., 1982). 4.1.3. Comparison with the body force approach By relating analogous forms for both P and qt, comparisons between the body force approach and the results of the fully interactive model can be drawn up. Core shear appears in the latter but not the former, so P of Eq. (6) in the body force model is related to the equivalent term of Eq. (18) for the fully interactive model by ! PC 2G C P BF 2 23 3 Int 2
such that loads match at the secondary bifurcation point. Varying the values of P and qt in Eq. (6) accordingly, the following observations are made from Fig. 10. In view of the differences in the formulations, comparisons, for this geometry anyway, are satisfactory. For the maximum amplitude, wavelength and degree of localization, correlation is excellent in the case of the quadratic core (k2 0), as seen in Figs. 10a±c. Agreement is good for the cubic case (Figs.10d±f) although it tends to overestimate the maximum amplitude at lower loads; here the greater complexity of nonlinearity presumably makes it more dif®cult for the simpli®ed body force approach to match the more complete interactive model. 5. Concluding remarks The paper presents a theoretical development, involving buckling or folding on two different length scales that is in many ways far removed from direct geological application. It does however carry certain important messages. The system is purely elastic, and comprises essentially only three layers. However, as in the evolution of localized single-layer buckling
G.W. Hunt et al. / Tectonophysics 335 (2001) 111±120
119
Fig. 10. Comparison of body force approach with interactive buckling model for strut of L 100 mm: dashed line-body force model; solid lineinteractive model: (a)± (c) quadratic core; (d)± (f) cubic core: (a) P 0.95 P C, qt 0.2447; (b) P 0.50 P C, qt 0.0433; (c) P 0, qt 0.0022; (d) P 0.95 P C, qt 0.2461; (e) P 0.85 P C, qt 0.2311; (f) P 0.75 P C, qt 0.2088.
(Hunt et al., 1996a; Budd et al., 1998), the elastic response can strongly in¯uence initial conditions for more realistic rheological models. If for example a Maxwell-type visco-elastic material is used to replace the elastic core, the initial response, from which the system would start to evolve, would coincide with one of the post-buckled states of Fig. 9. Thus an initial large-de¯ection localized perturbation, of direct signi®cance to the subsequent evolution (Cobbold, 1975), is automatically provided and de®ned by the initial elastic response. By extension, the analysis indicates how folding on a large scale in a multi-layered system can be accompanied by localized parasitic folding in the regions where extra compression applies. Such behaviour is found both experimentally and in the ®eld, but is seen to be dif®cult to model. To capture it satisfactorily in this relatively primitive (proto-type) system has required sophistication of analysis over and above that of conventional engineering. The model is necessarily nonlinear, and standard engineering assumptions of simple bending theory have had to be relaxed. It is interesting to compare and contrast this situation with developments in single-layer localized buckling. An elastic layer embedded in a viscous medium, under
the nonlinearity of prescribed end-shortening, has recently been demonstrated to have near self-similar localized solutions (Budd et al., 1998); after rescaling both space and time, it is shown that the same (or nearly the same) localized form can appear over a range of different length scales during the evolution. In the present problem we see folding on different length scales at the same moment in time, implying a coupling of the instabilities associated with overall and local buckling. Added complexity associated with nonlinearity may mean that practically relevant states will need sifting from the multiplicity of solutions that can arise (see for example Champneys et al., 1997), and highly selective path-following techniques may be required to ®nd them.
References Allen, H.G., 1969. Analysis and design of structural sandwich panels. Pergamon Press, Oxford. Biot, M.A., 1965. Mechanics of Incremental Deformations. Wiley, New York. Budd, C.J., Hunt, G.W., Peletier, M.A., 1998. Self-similar fold evolution under prescribed end-shortening. Math. Geol. 31 (8), 989±1005.
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Champneys, A.R., Hunt, G.W., Thompson, J.M.T., 1997. Localization and solitary waves in solid mechanics. Phil. Trans. R. Soc. Lond. A 355 (1732), 2073±2213 special issue. Cobbold, P.R., 1975. Fold propagation in single embedded layers. Tectonophysics 27, 333±351. Doedel, E.J., Wang, X.J., Fairgrieve, T.F. auto94: Software for continuation and bifurcation problems in ordinary differential equations. Technical Report CRPC-95-2, California Institute of Technology, 1995. Gibson, L.J., Ashby, M.F., Schjaer, G.S., Robertson, C.I., 1982. The mechanics of two-dimensional cellular materials. Proc. R. Soc. Lond. A 382, 25±42. Handin, J., Friedman, M., Logan, J.M., Pattison, L.J., Swolfs, H.S., 1972. Experimental folding of rocks under con®ning pressure: buckling of single-layer rock beams. Flow and Fracture of Rock, Heard, H.C., Borg, I.Y., Carter, N.L., Raleigh, C.B. (Eds.). Geophys. Monogr. Am. Geophys. Union 16, 1±28. Hobbs, B.E., 1993. The signi®cance of structural geology in rock mechanics. In: Hudson, J.A. (Ed.). Comprehensive Rock Engineering. Pergamon Press, Oxford, pp. 25±62. Hobbs, B.E., Means, W.D., Williams, P.F., 1976. An outline of structural geology. Wiley, New York. Hobbs, B.E., MuÈhlhaus, H-B., Ord, A., 1991. The fractal geometry of deformed rocks. The Geometry of Naturally Deformed Rocks, vol. 239b. Mitt. aus den Geolological Institute, ETH ZuÈrich, Neue Folge, p. 28. Hunt, G.W., Wadee, M.K., 1991. Comparative lagrangian formulations for localized buckling. Proc. R. Soc. Lond. A 434, 485± 502. Hunt, G.W., Wadee, M.A., 1998. Localization and mode interaction in sandwich structures. Proc. R. Soc. Lond. A 454 (1972), 1197±1216. Hunt, G.W., Da Silva, L.S., Manzocchi, G.M.E., 1988. Interactive
buckling in sandwich structures. Proc. R. Soc. Lond. A 417, 155±177. Hunt, G.W., MuÈhlhaus, H-B., Hobbs, B., Ord, A., 1996a. Localized folding of viscous layers. Geologische Rundschau (Int. J. Earth Sci.) 85 (1), 58±64. Hunt, G.W., MuÈhlhaus, H-B., Whiting, A.I.M., 1996b. Evolution of localized folding for a thin elastic layer in a softening viscoelastic medium. PAGEOPH (Pure Appl. Geophys.) 146 (2), 229±252. MuÈhlhaus, H-B., Hobbs, B.E., Ord, A., 1994. The role of axial constraints on the evolution of folds in single layers. In: Siriwardane, H.J., Zaman, M.M. (Eds.). Computer methods and advances in geomechanics, 1. A.A. Balkema, Rotterdam, pp. 223±231. MuÈhlhaus, H-B., Sakaguchi, H., Hobbs, B.E., 1998. Evolution of three-dimensional folds for a non-Newtonian plate in a viscous medium. Proc. R. Soc. London, Ser. A. 454, 3131±3143. Ord, A., 1994. The fractal geometry of patterned structures in numerical models of rock formation. In: Kruhl, J.H. (Ed.). Fractals and Dynamic Systems in Geoscience. Springer, Berlin, pp. 131±155. Potier-Ferry, M., 1983. Amplitude modulation, phase modulation and localization of buckling patterns. In: Thompson, J.M.T., Hunt, G.W. (Eds.). Collapse: the Buckling of Structures in Theory and Practice. Cambridge University Press, Cambridge, pp. 149±159. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T., 1990. Numerical recipes in C: the art of scienti®c computing. Cambridge University Press, Cambridge. Price, N.J., Cosgrove, J.W., 1990. Analysis of Geological Structures. Cambridge University Press, Cambridge. SimoÄes Da Silva, L.A.P., 1989. Modal interactions in bending and buckling of sandwich structures. PhD thesis, Imperial College of Science, Technology and Medicine, London.