Buckling of elastic structures under tensile loads

Transcription

1 Acta Mech 229, (2018) ORIGINAL PAPER F. G. Rammerstorfer Bucking of eastic structures under tensie oads This paper is dedicated to the memory of Franz Zieger Received: 30 March 2017 / Revised: 18 May 2017 / Pubished onine: 29 December 2017 The Author(s) This artice is an open access pubication Abstract Typicay, structures fai due to bucking if oaded by compression. However, it is important to notice that especiay in ightweight structures there are severa situations in which instabiities, such as bucking or wrinking, can be observed under tensie oads. In the present paper, a number of probems, deaing with bucking under tensie oads, are presented. Some soutions aready contained in former papers of the author are reconsidered, compared to recent resuts, and extended. Further new resuts are presented. Bifurcation bucking under tensie oading of beams, pates (with and without cut-outs), roed meta strips, thin ce was of meta foams, and of thin metaic fims on poymer substrates is treated in this paper. It is made cear that in a cases of bucking under tensie oads eventuay compressive stresses are responsibe for the oss of stabiity. Thus, one shoud carefuy differentiate between bucking under tension and bucking under tensie oads. Nonconservative oads as we as materia instabiities under tension, such as necking, are not considered in this paper. 1 Introduction In structura mechanics, oss of stabiity of equiibrium is generay associated with compressive oading. However, there are severa instabiities associated with goba tensie oading. In F. Zieger s famous book on Mechanics of Soids and Fuids [1] the bifurcation of a beam, simpy supported at both ends and oaded by a rigid bar in a way that the beam is under tension, is treated as an exampe. This exampe, which was earier studied by H. Zieger [2],has motivated the authorof this paper to choose the topic Bucking ofeastic structures under tensie oads for this paper in memory of Professor Franz Zieger, who was his esteemed teacher and, ater on, his encouraging and inspiring coeague for many years. Further motivation for choosing this topic for this paper ies in the fact that some of the author s papers on stabiity probems have been either co-authored or at east infuenced by F. Zieger; see, for exampe, [3]. As the first Ph.D. student of Franz Zieger, to whom this paper is devoted, the author understands the presentations in this paper (some sort of revisiting reated materia and extending it by new resuts) as devotion to Professor Franz Zieger, who died in 2016 after a rich and fruitfu ife as great scientist and teacher in Mechanica Sciences. Thinking on exampes for instabiity under tension brings necking of a tensie specimen into mind. Simiar forms of materia instabiities under tensie stresses may arise in meta forming of thin pates or shes if certain forming imits are surpassed. For instance, the formation of periodicay arranged necks (ocaized pastic deformations) during the conica expansion of a thin circuar cyindrica she (simiar to faring of a tube [4]) represents a bifurcation from the trivia, i.e. axisymmetric deformation process. F. G. Rammerstorfer (B) Institute of Lightweight Design and Structura Biomechanics, Vienna University of Technoogy (TU Wien), Vienna, Austria E-mai: ra@isb.tuwien.ac.at

2 882 F. G. Rammerstorfer In contrast to the mentioned materia instabiities, stabiity oss of structures under tensie oads is not that much known. Thin rectanguar sheets with a cut-out or a through-crack may ocay bucke in the surrounding of the cut-out and the crack, respectivey, when the sheet is stretched; see, for exampe, [5,6]. This is due to transverse compressive membrane forces being activated where the free edges, formed by the cut-out or crack, are predominanty oriented perpendicuary to the oading direction. Bucking under goba tension is aso observed, if a thin square pate is stretched diagonay between two corners [7] or a thin pate is stretched by nonuniform oads [5,8]. The rather surprising bucking or wrinking phenomenon appearing if rectanguar pates without any cut-out are stretched is treated in [8,9]. Simiar situations may appear on the micro-eve of cosed ce foams with thin ce was (as, for exampe, meta foams [10,11]) if the foam is under tension. In the cod-roing process of thin strip meta, undesired waviness can emerge frequenty athough the strip is under goba tension [12]. In materias sciences, some other, probaby ess known, but interesting exampes of instabiities under goba tension can be found. For instance, if a strip consisting of a thin metaic fim (around 100 nm thick) on a poymeric substrate (some μm thick) is stretched, oca fim bucking accompanied by deamination and progressive upift of the fim from the substrate can be observed [13]. In addition to the demonstration of the possibe appearance of instabiities under goba tension, it is the aim of this paper to show how stabiity imits can be systematicay determined. In most cases, dimensioness formuations, based on Buckingham s π-theorem [14], coud be achieved. Furthermore, the perhaps unexpected post-critica behaviours reated to these stabiity probems, cacuated by computationa methods, are discussed. In the foowing sections, bifurcation bucking under tensie oading of beams, pates (with and without cut-outs), roed meta strips, thin ce was of foams, and of some nanometres thick metaic fims on poymeric substrates is treated. It is shown that in a cases of bucking under tensie oads eventuay compressive stresses are responsibe for the oss of stabiity. Hence, it is important to differentiate between bucking under tension and bucking under tensie oads. 2 Bucking of beam systems under tensie oad Euer bucking of beams is the paragon for bifurcation bucking of eastic structures under compression oads. One hardy woud accept it, if one affirmed that straight beams may bucke under tensie oading. Nevertheess, in Zieger s famous book on Mechanics of Soids and Fuids [1] one finds in Section Stabiity of Equiibrium on pages a brief description of the exampe of a straight, simpy supported beam, oaded in tension by an axia force acting on the structure via a rigid rod, and the critica tensie force is cacuated there. 2.1 Zieger s beam Figure 1 shows a simpe beam rod system, which probaby the first time has been treated by H. Zieger [2] under Bucking by Tension and has aso been used by F. Zieger [1] as an exampe. Since this system is associated with the name Zieger (H. and F. Zieger, respectivey), in the present paper, this system sha be caed Zieger s beam. In [1], the critica tensie force is expressed as F crit = α 2 EI, (1) where α is derived from the transcendenta equation tanh (α) = αe/(1 + e/), (2) Fig. 1 Beam system under tensie oading and schematic sketch with notations as used in [1]

3 Bucking of eastic structures under tensie oads 883 and EI is the bending stiffness of the beam; for e and see Fig. 1. Let us now consider this simpe system in more detai. In order to compy with the intention of dimensioness descriptions, this Zieger s beam is now reformuated using dimensioness quantities: p = F crit 2 EI = β 2,η= e/, ϕ(0or) = w x With these notations, the characteristic equation reads: (x = 0or). (3) (1 + η) tanh β βη = 0. (4) When cacuating the root(s) β of Eq. (4), the dimensioness critica tensie oad p = β 2 is determined. p is shown in Fig. 2 in dependence of the dimensioness quantity η. Apparenty, the beam buckes under tensie stress. However, from a different point of view one might argue that it is not the beam, which buckes, but the rigid rod of ength e, which is easticay pinned at one end with a rotationa spring and oaded by an axia compression force F at the free end, oses the stabiity of its trivia equiibrium state. The rotationa spring stiffness is contributed by the eastic beam as γ = M (x = ) ϕ (x = ). (5) For the simpe probem of bifurcation bucking of an easticay hinged rigid rod of ength e, the critica force coud easiy be derived as F crit = γ/e, and in dimensioness form p = β 2 = F crit 2 EI = γ ηei. (6) If the rotationa spring stiffness is cacuated from the differentia equation w = M/EI with M (x) resuting from a torque M acting at x =, one gets γ = 3EI/ and p = 3/η. The soution obtained this way is aso sketched in Fig. 2, and one can see that, especiay for sma vaues of η, it hardy matches with the correct soution. One obvious reason for this discrepancy is the fact that the rotationa spring stiffness γ must be cacuated with M (x) depending on F according to 2nd order theory. This eads to w (x = ) = ϕ() = ( βη coth β η) ϕ() (7) From this equation and the requirement of a nontrivia soution, i.e. ϕ () = 0, we come up with the eigenvaue equation for β (η): βηcoth β (1 + η) = 0. (8) Recasting this equation renders (1 + η) tanh β βη = 0, and comparison with Eq. (4) showsthat β = β. Hence, the treated bucking probem under tensie oad can be interpreted as bucking of an easticay pinned rigid rod under compression oading. The.h.s. of Eqs. (4)and(8), respectivey, is antisymmetric w.r.t. β = 0 and β = 0, respectivey, and has just one positive and one negative root, both having the same absoute vaue and ead, because of p = β 2, to the same critica oad. This fact underpins the argument according to which it is the hinged rod, which as a system with just one degree of freedom becomes unstabe under compression rather than the beam under tension. (Remark: As wi be expained by considering the post-bucking behaviour ater on in this subsection, a coser ook shows that the virtua system of an easticay pinned rod has two degrees of freedom, a fact which has no consequence regarding the critica oad; see aso the Appendix.) In order to show the fundamenta difference in the bucking behaviour of this system under tensie and compression oading, respectivey, et us consider the configuration according to Zieger s beam with the exception that the externa force is now acting in the opposite direction, i.e. the beam is under compression and the rod under tension. Here, the foowing differentia equation is obtained for ϕ (0) 1 from the distribution of the bending moment M (x) according to 2nd order theory by using w = M/EI: w + ( ) 2 ˆβ w = ( ) ˆβ ϕ(0)(e ηx). (9)

4 884 F. G. Rammerstorfer Fig. 2 Dimensioness critica tensie oads p, p, ˆp as functions of η = e/ for Zieger s beam (see Fig. 1); inserts show bucking modes (a) (b) Fig. 3 Eigenvaue equations for Zieger s beam: a under tensie oad and b under compression oad; curve parameters are different vaues of η = e/ The genera soution is w = A cos ( ) ( ) ˆβ ˆβ x + B sin x + ϕ (0) (e ηx). (10) From the boundary conditions, w(x = 0) = 0andw(x = ) = 0 foows that A = e ϕ (0) and B = e ϕ (0) cot ˆβ. Using these resuts in (10) and deriving w (x), one gets from the condition w (x = 0) = ϕ(0) the characteristic equation (1 + η) sin ˆβ ˆβηcos ˆβ = 0 (11) for a nontrivia ϕ (0) = 0. In contrast to the uniqueness of the critica oad of Zieger s beam, the corresponding eigenvaue equation for the system with the externa force acting such that the beam is under compression and the rod under tension has an infinite number of eigenvaues (i.e., roots of Eq. (11)), see Fig. 3b, corresponding to the infinite number of bucking modes of the beam, which here is the part of the structure which buckes. The dimensioness critica bucking oad, ˆp = ˆβ 2, is aso shown in Fig. 2. By the way, here the critica oad approaches the cassica Euer bucking oad for η 0. The fundamenta difference between tensie and compression oading of Zieger s beam gets evident by considering the eigenvaue equations for both systems as shown in Fig. 3. In order to demonstrate that this simpe system exhibits a quite strange post-bucking behaviour, the resuts of a fuy geometricay noninear finite eement anaysis are shown in Fig. 4 for the foowing specific choice

5 Bucking of eastic structures under tensie oads 885 -u F F u Fig. 4 Deep post-bucking behaviour of Zieger s beam; inserts show deformed configuration; F in mm and w in N w of parameters: = 1.0m, e = 0.15 m, EI = 0.5Nm 2. With these geometrica parameters, i.e. η = 0.15, the eigenvaue equation (4) eads to β = 7.667; hence, p = β 2 = 58.76, and with the given bending stiffness EI, the critica tensie oad is F crit = 29.4N. Two facts to be gained from Fig. 4 shoud be emphasized: (i) The support at that end of the beam, to which the rigid rod is attached, moves for a whie oppositey to the orientation of the oad, but moves back towards the initia position in the deep post-bucking range. Because of the motion of the support, the above-mentioned virtua system of an easticay pinned rod must be buit with two degrees of freedom; see the Appendix. (ii) The transverse dispacement of the midspan of the beam starts increasing after surpassing the critica tensie oad. However, after some further oad increase the defection diminishes more and more. One might argue that a simiar behaviour can be observed for the simpy supported eastica under compression oad. However, one shoud notice that in the case of the tensie oaded beam a quite different deformation mechanism eads to the reduction in the defection, namey some sort of smoothening down, instead of forming a oop as it happens in the case of the eastica. This smoothening effect in the post-bucking behaviour is in a sense characteristic for bucking under tensie oad (see pate bucking under goba tension in Sect. 3). 2.2 Some other beam rod systems Inspired by the coser investigation of Zieger s beam, two more beam rod systems under tensie oading wi be considered in brief: the simpy supported beam system oaded symmetricay at both ends and the cantiever beam system, both under tensie oading. (a) Assuming a symmetric defection of the simpy supported beam system oaded symmetricay at both ends as depicted in the insert (a) in Fig. 5, the foowing differentia equation is obtained: ( ) β 2 ( ) β 2 w w = e ϕ(0). (12) There, the genera soution is ( ) ( ) β β w = A cosh x + B sinh x + e ϕ (0), (13) from which, with the boundary conditions, w (x = 0) = 0andw (x = ) = 0, the characteristic equation is obtained as

6 886 F. G. Rammerstorfer (b) (a) Fig. 5 Dimensioness critica tensie oads p as functions of η = e/, for the systems according to the inserts in this figure, showing corresponding bucking modes (c) sinh β ηβ (cosh β 1) = 0. (14) The roots β ead, with the definitions (3), to the dimensioness critica tensie oad p (η), as it is shown in Fig. 5. However, one has to notice that there is another deformation mode of the beam; namey, an S-shaped one (see insert (b) in Fig. 5) might be activated, too, and the above soution might be not the reevant one. Thus, et us consider this aternative system, for which the foowing differentia equation is obtained: w ( β ) 2 w = ( β ) 2 (2ηx e) ϕ (0), (15) which with its genera soution and the above boundary conditions eads to the characteristic equation (1 + 2η) sinh β ηβ (cosh β + 3) = 0. (16) The corresponding dimensioness critica tensie oads, depending on η, are shown in Fig. 5. Finay, for the cantiever beam system as shown in Fig. 5 (insert c) the differentia equation in anaogy to Eq. (12) reads now w ( ) β 2 w = ( ) β 2 (e ϕ () w) (17) with w = w (x = ). Formuating the genera soution and fufiment of the boundary conditions, w (x = 0) = 0, w (x = 0) = 0 with w (x = ) = ϕ() the foowing characteristic equation is achieved: Figure 5 shows p for the cantiever beam system, too. ηβ sinh β cosh β = 0. (18) 3 Bucking of rectanguar pates under uniaxia in-pane tensie oading In contrast to beams, which never bucke when an externa axia tensie oad is directy appied to the end of the beam axis, bucking of pates under externa in-pane tensie edge oading is possibe. There is some iterature on oca bucking in the area around cut-outs of stretched pates or strips; see [5,15]. Some papers are deaing with oca bucking in the area of cracks in pates under tension [6,16]. A systematic treatment of these phenomena, especiay in terms of dimensioness quantities (based on Buckingham s π- Theorem [14]), is provided in Sect. 3.2 together with the consideration of the post-bucking behaviour.

7 Bucking of eastic structures under tensie oads 887 Fig. 6 Experimentay observed bucking patterns of stretched pates with and without cut-outs It is ess known that stretched pates without any cut-out or crack may bucke. This kind of bucking under tensie oading was treated in [9], and severa papers foowed afterwards; see, for exampe, the nice anaytica treatment in [17]. In [18]and[19], where the decrease of the heights of the buckes in the post-bucking regime are treated, too, a number of references to recent papers on stretch bucking of rectanguar pates can be found. This tensie bucking probem is shorty recaed here for the sake of competeness. Figure 6 shows the mentioned tensie bucking phenomena in experimenta considerations. There the postbucking deformations of the pates with a hoe and with a crack, respectivey, correspond to the second mode shape. Obviousy, this is due to predominanty antisymmetric imperfections, caused by the preparation of the specimens. 3.1 Stretched pates without cut-outs or cracks In this Subsection, bucking of stretched pates without any cut-outs or cracks is presented, based on resuts achieved in [9]. Figure 7 shows the notations used as we as the distribution of the in-pane stresses σ yy. One can see the appearance of areas of tensie stresses cose to each of the short edges, which resuts from the Poisson effect, caused by hindering the movement in y-direction, and an area of compressive stresses in between, appearing due to the requirement of equiibrium (consider, for exampe, a cut aong the symmetry axis in x-direction). For arger aspect ratios (ong strips), the area of compressive stresses breaks down forming two areas of significant compressive stresses near the tensie stress areas. The distribution of the bucking waves in the post-bucking domain refects this character of the compressive stress distribution; see the inserts in Fig. 8. It becomes obvious again that bucking under tensie oading is due to compressive stresses. In [9], from some anaytica considerations and from parametric computationa studies the dependence of the dimensioness bucking stress p = σ crit E (t/b) 2 ; p = p (ξ) with ξ = L B, (19) has been found as a function of ξ for a fixed vaue of the Poisson s ratio, ν = 0.33, which is typica for most of the metas used in ightweight design. In Eq. (19), σ crit is the goba tensie stress acting at the short edges as oading at the instant of bucking, t is the pate thickness and E is the Young s moduus. y tension compression tension tension compression B x L L Fig. 7 Geometric notations of the stretched rectanguar pate, camped at both short edges, and the distribution of the in-pane stresses σ yy ; see aso [9]

8 888 F. G. Rammerstorfer Fig. 8 Dimensioness critica tensie oad as function of the aspect ratio of the pate; inserts show typica bucking patterns for short and ong strips; compare [9] By the way, p represents the bucking factor k in the formuation σ crit = ke (t/b) 2, (20) as typicay used in the engineering pate bucking iterature; see, for exampe [20]. Bucking phenomena as described here can be aso observed on the micro-eve of ightweight cosed ce foams under tensie oad, eading to a macroscopic, i.e. homogenized stress strain behaviour appearing as if pastic deformations woud take pace; see Fig Stretched pates with circuar cut-outs or cracks For configurationsas shown in Fig. 10 one hardy can obtain anaytica estimates for critica tensie oads at the short edges of the pate. There computationa methods have been appied [21] for cacuating both the stress fied and the critica oad intensities. However, based on the π-theorem dimensioness quantitiescan be found goba hydrostatic compression goba uniaxia tensie stress goba tensie strain goba tension Fig. 9 Macroscopic tensie stress-strain diagram for a cosed ce meta foam; microbucking as possibe reason for pseudopastic behaviour. Ce bucking under hydrostatic pressure is shown, too; compare [10,11]

9 Bucking of eastic structures under tensie oads 889 Fig. 10 Geometric notations of stretched pates with a circuar cut-out or a crack Fig. 11 Distribution of the in-pane stresses σ yy and numbers represent dimensioness stress eves, normaized by the tensie stress oad σ at the short edges of the pate which aow resuts of parametric finite eement studies to be used in a genera way, as ong as inear easticity can be assumed. The foowing dimensioness quantities are used in addition to those aready defined in Eq. (19): ρ = r B ; ϑ = a B. (21) Figure 11 shows, for a given set of geometric parameters, areas of compressive stresses σ yy, which eventuay are responsibe for oca bucking under goba tensie oading. For obtaining the critica tensie oad, the foowing eigenvaue probem (K 0 + λ i K )φ i = 0, (22) has to be soved. The eigenvaue λ i corresponds to the ith bucking mode, represented by the eigenvector φ i. Here, K 0 is the tangent stiffness matrix at a oad intensity σ 0, chosen to be cose enough to the critica intensity σi in order to avoid the cacuation of negative eigenvaues as absoute smaest ones, which woud correspond to critica compression oadings. K is the change in the tangent stiffness matrix due to a chosen

10 890 F. G. Rammerstorfer 60 ξ 60 ξ Fig. 12 Dimensioness critica tensie oads p in dependence of the geometrica parameters oad increment σ. Hence, the smaest critica oad intensity σ 1 = σ crit = σ 0 + λ 1 σ is reevant for bucking. This formuation has been chosen in order to avoid cacuation of eigenvaues corresponding to bucking under compression oading as resuting from formuations of eigenvaue probems common in inear bucking anayses under compression oading, (K L + ˆλ i K g ) ˆφ i = 0, (23) with K L being the inear initia stiffness matrix and K g the inearized geometrica stiffness matrix at reference oad. In Fig. 12, the dimensioness critica oads p, together with the corresponding bucking modes, are shown in dependence of ρ and ϑ, respectivey, with the pate s aspect ratio ξ as parameter. As one can see, for ξ>3.0 the infuence of the aspect ratio L/B diminishes, and again we come up with typica bucking factor diagrams, where, in contrast to usua bucking of pates, the dimensioness geometric parameters of the cut-out and the crack, respectivey, are vaues at the abscissa. Studying the deep post-bucking behaviour by fuy geometrica noninear anayses [21] reveas the interesting effect that at a certain oad intensity the ampitudes of the buckes start decreasing, an observation aready made in Sect. 2 for Zieger s beam and mentioned in Sect Whie this behaviour is shown in [18] and[19] for rectanguar pates without any cut-out or crack, in the foowing this has now been investigated here for a pate with a circuar cut-out, see Fig. 13a, and for a pate with a crack, see Fig. 13b. The pates considered here have a width of B = mm. The other geometrica parameters are for the pate with cut-out: L/B = 3.0, t/b = , r/b = and for the pate with crack: L/B = 2.5, t/b = , a/b = Simiar to the observation described for Zieger s beam in Sect. 2 of this paper, immediatey after bifurcation from the trivia equiibrium path, the ampitudes of waves in the post-bucking deformations grow rapidy with sowy increasing tensie oad. This first process is foowed by a reduced growth rate of the buckes, and finay, the waves start to fatten out. One shoud, however, note that the resuts presented in Fig. 13 are based on the assumption of inear eastic materia behaviour. Thus, they are rather of demonstrative vaue, because pastic or viscous effects woud come into pay and, at east, for the pate with a crack fracture might happen before reaching sufficienty arge oad intensities for observing the fattening. 4 Bucking of thin strips with residua stresses and goba tension In Sect. 3, strips, free of initia stresses, under goba tensie oading are considered. In the present section, the objects of consideration are thin (infinitey ong) strips with stress states resuting from residua stresses and goba tension. The treatment of such situations is of great importance for a proper contro of strip roing and eveing processes. Figure 14 demonstrates schematicay that deviations from a parae roing gap (in reaity, just a few hundredth of a miimetres are sufficient to produce probems) may ead to residua stresses, arge enough in order to ead to waviness of the strip if the goba strip tension is reduced to a critica vaue. Since here it is not the increase in tensie oading, but its decrease the reason for bucking, this kind of instabiity

11 Bucking of eastic structures under tensie oads 891 (a) (b) Fig. 13 Deveopment of the ampitude of the buckes specified exampe of a pate with circuar cut-out and of a pate with a crack; p := σ/ [ E (t/b) 2],ω:= W/t under goba tension is not reay in the same ine as the other stabiity probems discussed in this paper. Nevertheess, not ony in papers by the author, see, for exampe, [12], but aso in papers which have foowed, as, for exampe, [22] and citations therein, such phenomena are mentioned in reation to bucking under tension. This is the reason for incuding this sma section in this paper. In [12,24], the critica vaues of the goba tension, at which for different intensities of the residua stresses and shapes of their distribution bucking appears, are cacuated anayticay. The notations are shown in Fig. 15. centre waves edge waves Fig. 14 Schematic demonstration of the evoution of residua stresses during roing, eading to wavy strips during reease of the goba strip tension

12 892 F. G. Rammerstorfer Fig. 15 Notations used in the anaytica modes The strip has a bending stiffness K = Et3 (with E being Young s moduus, ν Poisson s ratio, t the 12(1 ν 2 ) thickness), which is oaded by a sef-equiibrating residua membrane force distribution R n xx (y) = N ĝ(y) and a constant goba tensie force N 0 (a being forces per unit ength); see Fig. 15. Hence, the membrane force distribution is given by n xx (y) = N ĝ(y) + N 0. (24) With η = y/b, B/2 y B/2, 1/2 η 1/2, (25) one gets ĝ(y) g(η), n xx n xx (η) = Ng(η) + N 0. (26) The distribution shapes g(η) of the residua membrane forces are described in [12,23] as foows: For situations prone to centre wave bucking, cosine-type distributions are described as: with g c (η) = 1 C m cos m (πη) with m = 1, 2,... and 1/2 η 1/2, (27) C m = 1 2 1/2 0 cos m (πη)dη 1, (28) in order to ensure that the residua stresses R n xx (η) are sef-equiibrated. Poynomia-type distributions, fufiing the equiibrium requirements, are given by: g p (η) = 1 [ (m + 1)(2 η ) m 1 ] with m = 1, 2,... (29) m Both distributions, i.e. the cosine and the poynomia one, ead to g(η =±1/2) = 1. Hence, N can be defined as intensity of the residua membrane force distribution. If bucking in terms of edge wave bucking is the probem, the typica residua membrane stress distributions can be described in the same way as for centre wave bucking probems if mutipied by 1. In the appied Ritz Gaerkin approach, the Ritz-ansatz with just a singe degree of freedom, q, forthe nontrivia dispacement fied is, depending on whether centre or edge wave bucking is considered, again described by severa parameters. For centre wave bucking, ( π x ) w (x,η) = qw a (x,η), w a (x,η) = cos (aπη) cos (30) or w (x,η) = q w n (x,η), w n (x,η) = ( 1 12η η 3) ( n π x ) cos, (31) and for edge wave bucking ( π x ) w (x,η) = q w n (x,η), w n (x,η) = (2 η ) n cos (sign η) k

13 Bucking of eastic structures under tensie oads 893 with k = 1 for the antisymmetric bucking mode and k = 2 for the symmetric bucking mode. The parameters c, p, m and a, n, k as we as the sti unknown haf waveength aow a quite arge variabiity of the shape of the residua stress distribution and of the bucking mode shapes, respectivey. The foowing further dimensioness quantities have been introduced: Ñ = NB2 K π 2, Ñ 0 = N 0 B 2 K π 2. (32) The critica intensity of the residua stress fied with a given distribution shape g(η) is determined by c, p, m, and a given goba tensie oad is obtained by deriving the strain energy in the system with U B = K 2 U M = 1 2 U = U B + U M + U N0 (33) { ( 2 ) 2 w x w y 2 2 (1 ν) [ 2 w 2 ( w 2 ) 2 ]} x 2 y 2 w d, (34) x y ( ) ( ) R w 2 n xx + N 0 d, (35) x U N0 = N 0 2LB, (36) 2Et and ooking for its stationary vaue by U = 0. (37) q This eads the strip tension for first appearance of a nontrivia soution. However, this critica goba strip tension is yet a function of the sti unknown waveength as we as the other parameters describing the Ritz-ansatz for the bucking mode. Determining the minimum w.r.t. these parameters eads to the critica intensity of the residua stresses Ñ(Ñ 0 ) and, vice versa, to the critica strip tension Ñ 0 (Ñ), for a given intensity of the residua stresses. Again, the formuae and diagrams have been derived in dimensioness form in order to aow a quite genera use of the achieved resuts. From Fig. 16, as derived in [23], the critica vaue of the goba strip tension can be deduced for given residua stresses (determined by the shape of their distribution and their intensity). This means that from these diagrams one immediatey can find to which eve the goba tension has to be reduced in order to cause instabiity of the pane configuration of the strip and deveopment of waviness with further reease of the strip tension. Furthermore, the bucking mode shape as we as the waveength can be deduced. Let us denote this procedure, which is described in detai in [12,24], as forward probem. However, of much more practica importance is the backward probem, i.e. achieving knowedge of the intensity and shape of distribution of the residua stresses in the roed strip from observing (measuring) the critica goba strip tension as we as the bucking mode shape and waveength. In addition to, but based on soutions presented in the above-mentioned papers, a procedure is demonstrated now for soving this inverse probem appicabe to controing the roing and eveing process in a way, which eads to reduced residua stresses and, hence, to improved quaity of the roed strip. Figure 17 demonstrates schematicay the process of soving the backward probem: The goba strip tension is reeased to the critica vaue, i.e. to the instant, at which the first time buckes are observed (measured). This can be done even during the strip roing process in the pant by a roer system with appropriatey controed roing speeds. This critica goba strip tension vaue, transformed into the dimensioness form, together with measured waveength (again expressed in dimensioness form) and the observed (measured) shape of the buckes, i.e. either centre or edge waves, is sufficient for estimating the intensity and the distribution shape of the residua stresses in the strip. Now, for the sake of competeness, as done so far in the previous sections of this paper, the post-bucking behaviour, i.e. the process of strip tension reease and the increase in the wave ampitude, sha be demonstrated here as some further new interesting resut. For this purpose, in Fig. 16 a tensie oad path is added as a horizonta ine at a chosen eve of intensity of the residua stresses Ñ, starting from a goba tensie oad Ñ 0, which is arge

14 894 F. G. Rammerstorfer different shape parameters anay ca resuts resuts of FE anayses for comparison reasons bucking β=1 post-bucking β=0 no bucking β = β* Fig. 16 Diagram showing curves representing critica combinations of residua stress intensities N and goba strip tension N 0 for different shapes of the distribution of the residua stresses aong the width of the strip, characterized by specific vaues of the shape parameters of the residua stress distribution shown for centre waves as exampe (compare [23]). The ine from β = 0 to β = 1 is used in Fig. 18 Output: different shape parameters c,p,m shape parameters c,p,m input input ~ output ~ different shape parameters c,p,m output ~ Fig. 17 Graphica representation of the soution strategy for soving the inverse probem (photograph by courtesy of Voestapine, Linz) enough to prevent the strip from showing wavy deformations (the β vaues correspond to those in Fig. 18), and moving aong the path from right to eft towards fu reease (i.e. β = 1.0). At the instant at which in Fig. 16 the path crosses the N ( N 0 ) ine, the instabiity happens. This situation represents the bifurcation point in the diagram in Fig. 18. During continued reease of the strip tension the

15 Bucking of eastic structures under tensie oads 895 (a) (b) Fig. 18 Deveopment of post-bucking deformations during reease of the goba strip tension; eft: centre waves (see aso Fig. 16); right: edge waves ampitude of the waves increases, and under certain circumstances post-bucking bifurcations may appear, as shown in Fig. 18. Another kind of post-bucking process, in which the strip is aid down onto a pane surface during tension reease (as done in practice for the sake of quaity inspection) is simuated in [24]. The post-bucking bifurcations in Fig. 18b are the consequence of initia imperfections introduced as a random pattern in the simuation. Here, the correctness of the statement according to which bucking under tensie oad is caused by oca compressive stresses is obvious and needs no further discussion. 5Thinfims In materias sciences, some other, probaby ess known, but interesting exampes of instabiities under goba tensie oad can be found. For instance, if a strip consisting of a thin metaic fim (some nanometres thick) on a poymeric substrate (some micrometres thick) is stretched, the foowing observations can be made. At a certain goba tensie strain of the strip specimen, the metaic fim starts cracking with so-caed channe cracks running perpendicuary to the oading direction. With further stretching, the crack density grows up to eventua saturation [25]. Further stretching eads to oca fim bucking, accompanied by deamination and progressive upift of the fim from the substrate; see Fig. 19. Certainy, aso in this bucking probem under goba tensie oading oca compressive stresses in the fim are the reason for the instabiities. Where do these, et s ca them transverse compressive stresses, come from? (i) One reason coud be a mismatch in the Poisson s ratios between fim and substrate, ν f = ν s (subscript f stands for fim and s stands for substrate). Based on experimenta observation and on computationa resuts [25], it can be assumed that, due to the camping boundary conditions, the stretched, but sti uncracked specimen does not show any curvature in the area under consideration (i.e., in the midde of the specimen) during the experiment. Thus, as ong as the fim, the substrate, and the interface are intact, in-pane strain couping exists between fim and substrate in both ongitudina and transverse directions. This, in combination with the equiibrium condition (no resutant force in transverse direction), aows the estimation of the transverse fim stress σ f as a function of the goba stretch ε L of the specimen in ongitudina direction: σ f = (ν f ν s ) E f 1 νf 2 + ( 1 νs 2 ) Ef ε t L. (38) f E s t s In situations, in which ν f <ν s, transverse compression stresses deveop, when stretching the specimen. However, much more important and neary independent of any Poisson s mismatch is the foowing reason for transverse compressive stresses in the fim.

16 896 F. G. Rammerstorfer 4 µm channe cracks goba strain Fig. 19 Cracking and bucking of a thin metaic fim on a poymeric substrate under goba tensie oading (photograph see [26] by courtesy of ESI, Leoben) (ii) Since in the experiments no oca bucking is observed before saturation of the crack density is achieved, another mode can be deveoped for expaining the appearance of transverse compressive fim stresses and determining their intensity as function of the amount of stretch. The fim cracking process eads to a continued stress redistribution in the system, and after saturation further stretching of the specimen wi not ead to significant increase in the ongitudina stretch in the fim substrips, each of them between two adjacent channe cracks, but simpy enarges the width of the channe cracks between them. Just for the sake of simpicity, et us assume that then the channe cracks are narrow enough so that the in-pane stresses in ongitudina direction of the specimen, L, are negigiby sma. Under this (certainy rather roughy simpifying) assumption, the transverse compression stresses deveop with further stretching the specimen as foows: σ f = E f ν f E f E s + ( 1 ν 2 f ) tf t s ε L. (39) Both Eqs. (38) and(39) overestimate the transverse compression stress. In (38), no cross-bendingis taken into account, and in (39) very narrow substrips are assumed, i.e. the minimum crack distance, which is determined by shear eg considerations (see [25] is sma enough to hinder the deveopment of substantia membrane stresses in the goba ongitudina orientation. Anyway, in the foowing considerations these simpifications are no onger used. The experimenta treatment and computationa simuation of the above-mentioned cracking and stress redistribution processes are pubished in detai in [25]. Figure 20 shows some new resuts of extended simuations. In [26], it is shown how from experimenta considerations in combination with an anaytica mode or, more precisey, using computationa modes [13] interface strength parameters can be deduced, again in terms of soving an inverse probem. The knowedge of the interface strength parameters of such compounds, which due to the sma geometrica conditions hardy can be measured directy, is crucia for the design and manufacture of eectronic devices, particuary of fexibe eectronic devices. In order to get the basis for determining the interface parameters the whoe compex process of stretching cracking crack density saturation oca bucking with deamination at the buckes, has to be simuated. Due to the strong noninearities and, especiay, because of the three-dimensiona character of the process, anaytica soutions (as were obtained in simpified two-dimensiona modes in [26]) are no onger sufficient, but noninear finite eement modeing and anaysis in combination with carefuy performed experiments are the proper choice of methods. As resuts of a muti-scae anaysis strategy, deveoped in [13,25], the goba stretch, which eads to fim bucking of a given fim substrate specimen, can be cacuated and compared

17 Bucking of eastic structures under tensie oads % 3.2% 4.2% 4.3% 5.0% 5.3% 6.0% 7.0% 10.1% 10.5% channe crack substrate fim average nomina fim stress in GPa ongitudina fim stress transversa fim stress goba nomina strain Fig. 20 Evoution of the crack pattern and voume-averaged in-pane stresses in the fim, simuated as some sort of stochastic process [25] (a) fim substrip between two channe cracks ε = 9.5 % goba strain (b) (d) 10 (c) ε = 11 % goba strain goba strain in % bucke hight in μm ε = 11.8 % goba strain Fig. 21 Process: mode II debonding (a) bucking and mode I debonding post-bucking (b, c); bifurcation diagram (d); compare [13] with the stretch vaue, at which in the experiment the instant of bucking is observed. Variation in the so ong unknown interface strength parameters, which are input to the cohesive zone eements in the simuation mode, unti coincidence between cacuated and measured critica stretch is achieved, eads to an estimate of the rea interface strength. This means that for the cohesive zone mode used in [13], the maximum norma debonding stress during upift, ˆσ, at which interface damage starts, and the Griffith energy reease rates for mode I, G c I, and for mode II, G c II, respectivey, have to be varied appropriatey. For more advanced cohesive zone modes for simuating debonding of the fim from the substrate, see [27]. Figure 21 shows, from simuation and experimenta resuts, what happens if the goba stretch approaches the critica vaue and surpasses it. In Fig. 21a, the edges of the substrips curve upwards and interface faiure according to fracture mode II is initiated. In Fig. 21b, it is shown how the increase in the specimen s stretch eads

18 898 F. G. Rammerstorfer to ocaization of this interface faiure and fim upift starts by forming trianguar buckes. The interface faiure changes now from mode II to mode I; trianguar buckes expand and deveop to become rectanguar buckes accompanied by rapid deamination growth with increased specimen stretch; see Fig. 21c. The cacuated stretch upift diagram as shown in Fig. 21d demonstrates the character of the process as bifurcation bucking. 6 Concusions Bucking is a possibe mode of faiure of thin waed or sender structures not ony under compression, but aso under tensie oading. Nevertheess, it shoud be mentioned that, even if the externa oad is tensie, the stress state, which eads to instabiity, is aways compressive. Thus, one shoud be cear in wording and tak about bucking under tensie oads instead of simpy saying bucking under tension. Bucking phenomena under tensie oading have been demonstrated for beams with rigidy connected rods at the ends (starting with Zieger s beam ), on stretched pates with and without hoes or cracks, on roed meta strip with residua stresses caused by the roing process, and on thin metaic fims bonded to poymeric substrates. Whie bucking in most cases is an unwanted phenomenon, in the atter exampe bucking and post-bucking investigations are used for determining interface strength parameters which hardy can be measured directy. Acknowedgements Open access funding provided by TU Wien (TUW). Parts of the paper stem from a project financiay supported by the Austrian Science Funds (FWF) under the project number P Furthermore, the contributions from coeagues from the TU Wien, the Montanunversität (MU) Leoben, from the Erich Schmid Institute (ESI) of the Austrian Academy of Sciences, from the Max-Panck-Institut für Eisenforschung GmbH (Düssedorf), as we as from the industria partners (BWG Duisburg) to the papers are acknowedged. Their names are in the authors ists of the respective papers. Open Access This artice is distributed under the terms of the Creative Commons Attribution 4.0 Internationa License ( creativecommons.org/icenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the origina author(s) and the source, provide a ink to the Creative Commons icense, and indicate if changes were made. Appendix In Sect. 2, a virtua system of an easticay pinned rod with two degrees of freedom is mentioned. It is described in Fig. 22. The post-bucking behaviour of the rea system (Zieger s beam) shows that the moveabe support for a whie moves towards the other support and at a certain oad intensity turns back to the initia position. At the same time, the defection increases initiay and at the same certain oad intensity it starts decreasing. It is shown now that the virtua system described here quaitativey represents the behaviour of the rea system, what quite obviousy a rod, just pinned at one end cannot do. From the equiibrium conditions of the nontrivia configuration with 0 <ϕ 1: (S is the tensie force in the spring), one gets the critica oad F S = 0, Feϕ Sfϕ γϕ = 0 (40) F crit = γ/(e f ). (41) With respect to the critica oad F crit, the second degree of freedom, i.e. the mobiity of the support, does not pay any roe. Hence, the existence of a second degree of freedom does not perturb the argument stated in Fig. 22 Virtua system for Zieger s beam

19 Bucking of eastic structures under tensie oads α μ ω Fig. 23 Post-critica behaviour of the virtua two-degree-of-freedom system Sect. 2, according to which the simpe virtua pinned rod system with one degree of freedom underpins the fact, that it is not the beam which buckes, but the rod. However, in order to provide a simpe virtua system for the post-bucking behaviour of Zieger s beam, the second degree of freedom is essentia. Let us introduce the foowing dimensioness quantities: κ = e/f 1, α= F (e f ) /γ, μ = u/f, ω= w/f, and φ = γ/(ce 2 ). (42) With these notations, the dimensioness critica oad gets α crit = 1, and equiibrium in the post-bucking regime, i.e. α>1and0<ϕ π, eads to the foowing equations for the description of the post-bucking configuration: For given vaues of κ and φ and the rotation ange ϕ as function of the oad intensity α, determined by α sin ϕ ϕ = 0 (43) the post-critica movement of the system with monotonicay increased oad (α >1.0) is given by μ (α) = αφ 1 + cos ϕ (α), ω (α) = (1 + κ) sin ϕ (α). (44) κ These resuts are shown in Fig. 23 for κ = 2.0 andφ = 0.1. Even so, for the sake of simpicity, the springs are modeed as inear eastic, the comparison between Figs. 4 and 23 shows that the principa characteristics of the post-bucking behaviour are captured by the virtua system. References 1. Zieger, F.: Mechanics of Soids and Fuids, 2nd edn. Springer, Berin (1995) 2. Zieger, H.: Principes of Structura Satbiity, 1st edn. Baisde Pubishing Company, New York (1968) 3. Zieger, F., Rammerstorfer, F.G.: Thermoeastic stabiity. In: Hetnarski, R.B. (ed.) Therma Stresses III, pp North- Hoand Pubishing Company, Amsterdam (1989) 4. Daxner, T., Rammerstorfer, F.G., Fischer, F.D.: Instabiity phenomena during the conica expansion of circuar cyindrica shes. Comput. Methods App. Mech. Eng. 194, (2005) 5. Datta, P.K., Biswas, S.: Research advances on tension bucking behaviour of aerospace structures: a review. Int. J. Aeronaut. Space Sci. 12, 1 15 (2011) 6. Brighenti, R.: Bucking sensitivity anaysis of cracked thin pates under membrane tension or compression oading. Nuc. Eng. Des. 239, (2009) 7. Tomita, Y., Shindo, A.: Onset and growth of wrinkes in thin square pates subjected to diagona tension. Int. J. Mech. Sci. 30, (1988) 8. Segedin, R.H., Coins, I.F., Segedin, C.M.: The eastic wrinking of rectanguar sheets. Int. J. Mech. Sci. 30, (1988)

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