Indentation of ultrathin elastic films and the emergence of asymptotic isometry

Transcription

1 Indentation of ultrathin elastic films and the emergence of asymptotic isometry Dominic Vella 1, Jiangshui Huang,3, Narayanan Menon, Thomas P. Russell 3 and Benny Davidovitch 1 Mathematical Institute, University of Oxford, Oxford OX 6GG, United Kingdom Physics Department, University of Massachusetts, Amherst, Massachusetts 13, USA 3 Polymer Science and Engineering Department, University of Massachusetts, Amherst, Massachusetts 13, USA We study the indentation of a thin elastic film floating at the surface of a liquid. We focus on the onset of radial wrinkles at a threshold indentation depth and the evolution of the wrinkle pattern as indentation progresses far beyond this threshold. Comparison between experiments on thin polymer films and theoretical calculations shows that the system very quickly reaches the Far from Threshold (FT) regime, in which wrinkles lead to the relaxation of azimuthal compression. Furthermore, when the indentation depth is sufficiently large that the wrinkles cover most of the film, we recognize a novel mechanical response in which the work of indentation is transmitted almost solely to the liquid, rather than to the floating film. We attribute this unique response to a nontrivial isometry attained by the deformed film, and discuss the scaling laws and the relevance of similar isometries to other systems in which a confined sheet is subjected to weak tensile loads. elastic films floating on a liquid as formed by vulcanization of a liquid polymer drop, in which case an unknown pre-stress is hypothesized [14], or by deposition, in which case the liquid surface tension acts at the film s edge [15]. Experiments on the latter system are better controlled than the former and show that indentation gives rise to a shape full of radial wrinkles that transform into sharp folds beyond a threshold indentation [15]. The striking difference between the observed wrinkled/folded shape and the nearly isometric d-cone, was interpreted in [15] as an indication of considerable strain in the film induced by the combination of indentation and boundary tension. Here we focus on the wrinkle pattern, and show that wrinkling reveals a new isometry of the film with the strain at the pre-indentation level. As a result, the indentation force exhibits a nontrivial dependence on the surface tension and density of the liquid, but is independent of the film s elastic moduli. This type of isometry is novel in the elasticity of thin bodies [], being achieved only in the doubly asymptotic limit of weak applied tension and small bending stiffness; we therefore refer to it as an asymptotic isometry. Our experimental setup consists of polystyrene films (Young s modulus E = 3.4 GPa, Poisson ratio ν =.33 and radius R film = 1.14 cm) floating at the surface of deionized water [16]. The interfacial tension,, was varied in the range 36 mn/m 7 mn/m using surfactant. The thickness of the film, t, satisfied 85 nm t 46 nm [35]. Stainless steel needles (tip radii r tip 5 µm, 135 µm) were used to impose a vertical displacement, δ, at the center of the film. Indentations up to δ.75 mm were applied and measured to within 1 µm. The deformed film was viewed from above using a microscope. Our theoretical study is based on the Föppl-von Karman (FvK) equations for an elastic film, with stretching modulus Y = Et and bending modulus B = Et 3 /1(1 ν ), floating on a liquid of density ρ l, subject to tension at its edge and a localized indentation force F causarxiv: v [cond-mat.soft] 6 Jan 15 PACS numbers: x,46.7.De,6..mq When an elastic sheet is subjected to external forces, it is often implicitly assumed that the work done is stored in the deformed sheet. Under purely tensile loads, the work is stored primarily by stretching energy. When the forces are purely compressive, as in uniaxial buckling, the strain is typically negligible, and the work is instead stored as bending energy [1]. Under more complicated compressive forces, such as those required to confine a sheet in a box [], the work is stored in localized (stressfocusing) zones that involve bending and stretching. In this Letter, we report a new response exhibited by the indentation of an elastic film floating at a liquid gas interface. We show that for sufficiently large indentations, only a negligible fraction of the work done by the indenter is stored as elastic energy the majority is stored in the gravitational and surface energies of the liquid. Interest in the indentation of elastic objects includes a range of metrological applications. Just as one tests an object s stiffness by poking it, controlled indentation is used in the measurement of internal pressure within polymeric [3] and biological [4 8] capsules and to determine the modulus of thin membranes [9]. These applications motivated theoretical studies of indented spherical shells, which suggested that mirror-buckling [1] (fig. 1a) occurs in the presence of an internal pressure [8]. Mirror buckling is the simplest possible isometric (i.e. strainless) deformation of an infinitely thin shell so the work done in indenting the shell is nearly independent of the elastic moduli; instead it goes into compressing the gas within the shell [8]. In contrast to shells, the indentation of elastic sheets is highly sensitive to tension. If a sheet is not under tension, indentation typically leads to the formation of a developable cone ( d-cone ) [11 13], which is isometric everywhere except within a small region around the indenter (fig. 1b). The formation of this nearly isometric shape involves large vertical deflections of the initially planar sheet and is therefore unattainable when vertical displacements are penalized. This is the case for thin

2 (a) F lem. For depth very small δ, the response is similar to that of a fluid membrane: the stress tensor σ ij is homogenous p and isotropic throughout the film, σ rr σ θθ,and the vertical deformation ζ(r) δ = δ decays (Y/γ over lv a) horizontal, p distance l c (fig. 1c). As δ is increased, l c the indentationinduced (1) strain, which scales as (δ/l c ), leads to a notice- able inhomogeneity which determines the stress the(fig. stress a): profiles the radial fully. stressas increases σ rr (r) above decreases a threshold monotonically (b) c, towards analysis offor the r FvK l c.in equations shows contrast, that the the hoop hoop stress stress σ θθ (r) decreases becomesmore compressive sharply, (σ θθ < ) overshooting within a before narrow approaching annulus γ (blue lv from solid below. curve, Intuitively, the rapid decrease of the hoop stress occurs fig. a). As because these indentation films are causes verymaterial thin, acircles compressive to be pulled hoop stress causes inwards wrinkling and become (fig. relatively b). For compressed; a film with thisinfinite effect isradius, numericalby analysis the confinement of the FvK ratio equations (δ/l c ) /( /Y yields )be- c 11.75, in governed tween good the indentation-induced agreement withstrain, our experiments and the purelyfor ten-sile strain caused by surface tension. We therefore intro- range of film (c) thicknesses, tensions and indenter sizes (fig. 3a). l c duce the dimensionless indentation depth Two crucial phenomena occur as the indentation amplitude is increased δ = δ (Y/ above ) 1/, c. First, the (1) wrinkled zone l δ l δ /l curv expands: a detailed c calculation [17] shows that the outer curv whichradius determines of the wrinkled rescaled stress profiles fully. As δ increases above a threshold δ zone L O /l c δ 3/ so that wrinklesshows reachthat the the film s hoop edge stress when becomes (R compres- film /l c ) /3. Second, c, analysis of the FvK F equations sive (σthe θθ < thinness ) within of a narrow the film annulus means (blue that solid the curve, compressive hoop FIG. 1: (a) Schematic illustration of mirror buckling, the fig. a). stress As these is completely films are veryrelaxed thin, a compressive by wrinkling: hoop σ θθ (r), a FIG. isometric 1: (a) Schematic deformation illustration of an infinitely of the thinindentation spherical shell of[9] a verystress qualitative leads to wrinkling change (fig. b). fromfor the anprebuckled infinite film, numerical analysis of the FvK equations yields δ that characterizes the response to indentation of a very thin and compressive thin axisymmetric shell with or without an internal pressure, (but unwrinkled) profiles [the solid c red 11.75, and blue curves, whichaxisymmetric tends to an shell isometric with or without mirror anbuckling internal pressure deformation [8]. in very good agreement with our experimental measure- of the critical indentation amplitude, which col- (b) Indenting a sheet with free boundaries leads to the formation of a d-cone, which is isometric everywhere except close respectively, in Fig. a). We therefore use the Far-from- [8, 1]. (b) Indenting a sheet with free boundaries leads to aments lapse onto Threshold a curve corresponding (FT) approach, to δ to the indentation point. (c) Schematic illustration showing c valid 1 (fig. in the 3a). singular limit of d-cone, which is isometric everywhere except close to the indenter the evolution [1, 13]. of (c) a Schematic Two crucial features of this problem occur as the indentation amplitude is increased above its critical value, δ c. floating filmillustration subject to increasing showingdimen- sionless indentation, δ: pre-indentation state ( δ =,upper, the evolution of a floating film subject to increasing zero bending stiffness [18, 19]. These two phenomena are flat); small indentation ( δ δ indentation, δ: preindentation state (upper, flat); at small indentation ( δ δ c, ability, ɛ 1, [] of the film, where: characterized by the dimensionless radius, R, and bend- c,secondfromtop)thetension is approximately homogeneous and isotropic; at interrow annulus. A detailed calculation [15] shows that the First, the wrinkled zone expands beyond the original nar- second from top) the tension mediate indentation ( δ c < δ is approximately uniform; at R /3,secondfrombottom), outer radius of the wrinkled zone L O δ 3/ so that wrinkles reach the edge of the film film /l intermediate wrinkles develop indentation an annular ( δ c < region (light R /3 blue), second in which from the bottom), R = R when c, δ ɛ 1 = γ (R film /l lv l c ) c/b. () /3. hoop the stress hoopisstress compressive; is compressive at large indentation an annular ( δ Rwrinkled /3, Second, the thinness of the film means that the com- region bottom), (light blue); wrinkles at(light large blue) indentation cover the entire ( δ film R /3 except, bottom), in pressive For hoop ourstress experiments, is completely ɛ relaxed 1 5. by wrinkling: an inner core of radius L I (black). wrinkles cover the entire film except for r < L I (black). σ θθ (r) In, the a qualitative FT approach changethe fromenergy the pre-buckled is written U = U dom + profile U(red sub solid withcurve U sub inthe fig. subdominant a). We therefore energy use the governed by the Far-from-Threshold bending cost(ft) of wrinkling, approach, valid which in the vanishes singular as ɛ, and ment, δ, at the center of the film, r = (using needles limit of zero bending stiffness [16, 17]. These two phenomena dom the dominant energy, which remains finite as ɛ. U with r tip < 5 µm led to plastic flow on the time scale of are characterized by the dimensionless radius R ing athe vertical experiment). displacement Indentation δ at depths r = up. towe δ assume.75 mmthat and bendability Minimization ϵ 1 of [18] Uof sub the determines film, where: the number of wrinkles. the film s were applied radiusand R film measured is much to larger within 1 than µm. the Thecapillary deformed film is viewed from below using a microscope. R = R film /l c, ϵ 1 = l c /B. () In the current study we employ tension field theory [18] length l c = ( /K f ) 1/, where K f = ρ l g. (minimizing U dom ) to determine the mean deflection profilexperiments, ζ(r) and the ϵ 1 extent 5. of the wrinkles. Our theoretical study is based on the Föppl-von Karman useful (FvK) toequations identifyfor the andimensionless elastic film withgroups stretching the In thewe FT approach write the we write axisymmetric the energy of FvK the system equations using an For our It is problem modulus by describing Y = Et and the bending characteristic modulus B = behavior Et 3 /1(1 of theas U = U dom + U sub with U sub the subdominant energy film as ν ), δ that increases floats on (fig. a liquid 1c). For of density very small ρ l, is subject δ, the to response an Airy potential ψ (so that σ governed by the bending cost of wrinkling, rr = ψ/r and σ which vanishes as vertical ϵ, force and Ubalance dom the dominant reads energy, which θθ = ψ ). The interfacial tension at its edge and to a localized indentation to that forceof F athat fluid causes membrane: a vertical displacement the stress remains δ at remains finite as ϵ. Minimization of U sub deter- is similar closeits tocenter. its pre-indentation We further assume state, thatσthe rr film s σ θθ radius R f is much larger than the capillary length l c =, andmines the number B of wrinkles. In the current study we the vertical deformation ζ(r) decays over a horizontal ζ 1 ( d ψ dζ ) = K f ζ F δ(r), (3) /K f. employ tension field theory r dr [16] (corresponding dr to minimizing U dom ) to determine the mean deflection profile πr distance Before l c (fig. discussing 1c). As the FvK δ is equations increased, it is the useful indentationinducescribe to degeneitdentation strain, the characteristic (δ/l c ), behavior leads toof athe noticeable film as theinhomo- in- ζ(r) and where the extent F is the of the point-like wrinkled indentation zone. force, found as part in thedepth stress δ is(fig. increased a): (Fig. the radial 1c), andstress thereby σ We write the axisymmetric FvK equations using an rr (r) of the solution for a given indentation. The compatibility identify the relevant dimensionless groups in the prob- Airy potential ψ (so that σ rr = ψ/r and σ θθ = ψ ). The decreases monotonically towards for r l c, while of strains in the unwrinkled zone (where both σ rr and σ θθ the hoop stress σ θθ (r) decreases more sharply, overshooting are tensile) gives [1] before approaching from below. Intuitively, this occurs because indentation causes material circles to r d [ ] 1 d be pulled inwards and become relatively compressed. If dr r dr (rψ) = 1 ( ) dζ Y. dr the indenter s tip is sufficiently small, this purely geometric (4) effect is governed only by the confinement ratio (δ/l c ) /( /Y ) between the indentation-induced strain, and the purely tensile strain caused by surface tension. We therefore introduce the dimensionless indentation We note that equations (3)-(4) are invariant under ζ ζ, F F ; our results therefore apply equally to the cases of pushing down on (considered here) and pulling up on [15] a floating membrane. Invoking tension field

3 3 σrr/γlv, σθθ/γlv r/l c δc (mm).4. (a) l c(/y ) 1/ (µm) 1 (b) r = L O r = L I L O = R film r = L I LI/lcurv 1 1 δ/r /3 1 1 FIG. : (Color online) The profiles of the hoop (solid curves) and radial (dashed curves) stresses within an indented, unwrinkled film (with R 1) at indentation depths δ = 7.5 (red curves) and δ = 15 (blue curves). Notice that σ θθ is negative for intermediate values of r when δ = 15 so that sufficiently thin films will, in fact, wrinkle. (b) Just beyond the onset of instability (δ =.48 mm) wrinkles are confined to an annulus L I r L O. (c) Ultimately wrinkles reach the edge of the film (here δ =.56 mm) and wrinkles occupy L I r R film. Here t = 85 nm. theory, we neglect the bending term in Eq. (3), and replace Eq. (4) by ψ = constant in the wrinkled zone (since σ θθ = ) [35]. We turn now to large indentations δ R /3, where the wrinkles cover the whole film except in < r < L I (see fig. c). Noting that σ rr (R film ) =, and that σ θθ in the wrinkled zone, we find that σ rr (r) = R film /r for L I < r < R film ; Eq. (3) then reduces to Airy s equation []: ζ(r) = A out Ai(r/l curv ), l curv = R 1/3 l c. (5) Here l curv, which increases with film size R 1/3 film, replaces l c as the decay length of membrane deflections. The prefactor A out and the inner radius, L I, are found by patching the wrinkled zone to the unwrinkled core (r < L I ). In the limit δ R /3 we find, using standard techniques [3 5, 35], that A out δ/ai() and the radial displacement at the edge of the film approaches a limiting value: u r (R film ).43 δ /l curv ; (6) a result whose importance will become apparent shortly. Our asymptotic calculations also reveal that L I 6. ( δ/r ) /3 LI R5/3 film γ5/3 lv δ. (7) l curv Y K /3 f FIG. 3: (Color online) (a) Experimentally measured threshold indentation for wrinkling, δ c, as a function of l c( /Y ) 1/. Experiments with varying film thickness (59 nm t 46 nm), = 7 mn/m are shown for indenter radii r tip = 135 µm ( ) and r tip = 5 µm ( ). Experiments with varying surface tension coefficient (36 mn/m 7 mn/m) and t = 11 nm ( ). The theoretical prediction for R 1, δ c 11.75, is also shown (dashed line). Good agreement with experiment justifies our neglect of indenter size and any (hypothesized) manufacture-dependent pre-stress, which were both attributed crucial roles previously [14]. (b) The inner wrinkle radius, r = L I, decreases with increasing indentation, δ, when wrinkles reach the film s edge. Experiments with = 7 mn/m and: t = 85 nm ( ), t = 11 nm ( ), t = 158 nm ( ), t = 7 nm ( ), t = 46 nm ( ). Experiments with t = 11 nm and: = 58 mn/m ( ), = 5 mn/m ( ) and = 4 mn/m ( ). The prediction of the FT theory (solid curve) and the asymptotic result (7) (dashed line) are also shown. The wrinkle number scales similarly to that found in other studies [1] (data not shown). At the scaling level, Eq. (7) can be understood by noting that in the tensile core the indentation-induced radial stress Y (δ/l curv ), whereas in the wrinkled zone σ rr = R film /r. Continuity of the radial stress at r = L I yields the scaling in (7). Figure 3b shows that this result agrees well with numerical solutions of the full problem and with experiments. Strikingly, Eq. (7) shows that the size of the tensile core is affected by all physical parameters in the problem (except the bending modulus). Our calculation also yields the indentation force F R /3 δ, consistent with previous measurements [15, 35]. Two features of the scaling F γ /3 lv K1/3 f R /3 δ, are surprising. Firstly, F δ, even though the FvK equations are highly non-linear. Secondly, the force is independent of the elastic moduli of the film. Understanding this mechanical response requires reconsideration of

4 4 the dominant energy of the wrinkle pattern: U dom = (W indent + W surf ) + (U gpe + U stretch ). (8) Here W indent, W surf are the work done by the indentation force and surface tension acting at the edge of the film, respectively. U gpe, U stretch are the gravitational energy of the displaced liquid and the elastic energy of the film, respectively. The work done by the indentation force W indent = F dδ R /3 δ. One might assume that W indent would be transmitted to the elastic energy U stretch due to the tensile components of the compression-free stress field. However, integrating the strain energy density σij /Y we obtain: U stretch /W indent ( δ/r /3 ) 1. Indeed, using Eqs. (5,6) to evaluate the work W surf R film u r (R film ) of the surface tension, and the energetic cost U gpe Rfilm K f ζ r dr of the vertically-displaced liquid, we find the asymptotic relation: for ɛ 1 1, δ R /3 : W indent W surf + U gpe (9) This energetic structure describes a novel mechanical response of an elastic film, whereby the work of the indenter is transmitted predominantly to the subphase (increasing gravitational energy and uncovering surface area of the liquid), while an asymptotically negligible fraction is stored as elastic energy in the film. This simple energetic structure reflects a nontrivial geometric feature: the wrinkled film becomes isometric to its pre-indentation state in the doubly asymptotic limit of small bending modulus (ɛ 1) and small exerted tensile strain (since δ R /3 ( /Y ) u r (R film )/R film by eqn (6)). In this doubly asymptotic limit, the hoop strain ɛ θθ is and asymptotic isometry follows from Eqs. (5,6), which yield the elimination of radial stretching in the limit δr /3 (the apparent stretching, δ + l curv l curv, is completely cancelled by the lateral displacement u r (R film ) of the edge). Thus, the formation of wrinkles at negligible energetic cost enables the metric of the film to remain almost identical to its pre-indentation state, even though the film suffers a large deflection, Eq. (5), that is determined by indentation, gravity, and surface tension. In other words, the film lies in a no-man s-land too stiff to be stretched significantly (since the applied tensile strain /Y is small), and yet perfectly deformable (since the bending modulus B is also small). In conclusion, we have shown that an indented floating film starts with a purely tensile response but evolves, with the aid of wrinkles, into a state that is asymptotically isometric to its initial state. This demonstrates a novel mechanical response, in which the indenter does work mainly on the liquid with only a negligible fraction transmitted to the elastic film. This response also underlies the stability of the poked film to two common failure modes of floating objects: the film would sink if the displacement at the edge exceeds l c [6], but ζ(r film ) δai(r /3 ) l c (since R 1). Similarly, pulling-induced delamination will occur if the adhesive energy, γrfilm, is smaller than the alternative deformation energy [7]. Here, the alternative deformation energy, U dom, is barely affected by the elastic modulii of the sheet, so delamination is expected only for δ > γ/ R /3 film l1/3 c, which is beyond the reach of existing experiments [15] and the validity of our small slope theory. The concept of asymptotic isometry should be relevant to other systems, where a thin elastic object is forced into a curved, nondevelopable shape, in the presence of weak tensile loads. Representative examples include the wetting of a film by a liquid meniscus [8, 9] or its adhesion to a sphere [3, 31], and the twisting of a stretched ribbon [3, 33]. Such systems may also have parameter regimes in which the object is highly deformed yet nearly isometric to its undeformed state; consequently, the work done by external forces is not stored in the object itself. Finally, it is important to realize that asymptotically isometric states may not necessarily be wrinkled: the wrinkle-fold transition [15, 34] and other secondary instabilities may also exhibit a similar phenomenology. We hope that our work will provide a suitable framework for studying these phenomena. Acknowledgments We thank E. Hohlfeld for insightful discussions on asymptotic isometry. This research was supported a Leverhulme Trust Research Fellowship (D.V.), the W.M. 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