Anisotropic Blistering Instability of Highly ... - Semantic Scholar
PRL 112, 094302 (2014)
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PHYSICAL REVIEW LETTERS
Anisotropic Blistering Instability of Highly Ellipsoidal Shells 1
Hamid Ebrahimi,1 Amin Ajdari,2 Dominic Vella,3 Arezki Boudaoud,4 and Ashkan Vaziri1,*
Department of Mechanical and Industrial Engineering, Northeastern University, Boston, Massachusetts 02115, USA 2 Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA 3 Mathematical Institute, University of Oxford,Woodstock Road, Oxford OX2 6GG, United Kingdom 4 Laboratoire Reproduction et Développement des Plantes & Laboratoire Joliot-Curie, INRA, CNRS, ENS, Université de Lyon, 46 Allée d’Italie, F-69364 Lyon Cedex 07, France (Received 2 September 2013; revised manuscript received 9 January 2014; published 6 March 2014) The formation of localized periodic structures in the deformation of elastic shells is well documented and is a familiar first stage in the crushing of a spherical shell such as a ping-pong ball. While spherical shells manifest such periodic structures as polygons, we present a new instability that is observed in the indentation of a highly ellipsoidal shell by a horizontal plate. Above a critical indentation depth, the plate loses contact with the shell in a series of well-defined “blisters” along the long axis of the ellipsoid. We characterize the onset of this instability and explain it using scaling arguments, numerical simulations, and experiments. We also characterize the properties of the blistering pattern by showing how the number of blisters and their size depend on both the geometrical properties of the shell and the indentation but not on the shell’s elastic modulus. This blistering instability may be used to determine the thickness of highly ellipsoidal shells simply by squashing them between two plates. DOI: 10.1103/PhysRevLett.112.094302
PACS numbers: 46.70.De
As every ping-pong player knows, spherical elastic shells suffer an unusual mode of instability when hit too hard: a series of vertices appears, connected by linear ridges forming an inverted polygon [see Fig. 1(a)]. These polygonal deformations are an example of the localization of the stress state that occurs in more controlled experimental settings such as the indentation of a shell by a plate [2], at a point [3,4], or with indentors of size comparable to the radius of the shell [5]. This localization of stress is important not only because of its visual appearance but also because the focusing of stresses is more likely to lead to irreversible, plastic deformations—again a feature familiar to ping-pong players. The ubiquity of thin elastic shells also leads to a wide range of applications in which this asymmetric buckling can be important, from the folding of pollen grains [6] to the buckling of drying colloidal droplets [7] or osmotically shrinking polymeric capsules [8]. Previously, much interest has focused on the indentation of spherical shells [9]. In this setting, the determination of the force-displacement relationship is important because of its application to atomic force microscopy (AFM) measurements of viruses [10] as well as yeast and other cells [11–13]. Recently, this work has been extended to the indentation of ellipsoidal shells from both theoretical [14] and experimental points of view [15]. However, all of these studies implicitly assume that the deformation of the shell retains the symmetry of the undeformed shell. In fact, it has long been known that at a critical indentation depth (proportional to the thickness of the shell, t, [16,17]), an asymmetric deformation profile with a well-defined wave number 0031-9007=14=112(9)=094302(5)
appears [18,19]. As indentation continues past this critical depth, the number of vertices of the polygonal pattern that is observed changes [1,4,16]. In this Letter, we report a new type of localized deformation that occurs during the indentation of highly ellipsoidal shells by a flat plate. For sufficiently small compressions, the deformed region of the shell remains in contact with the plate and retains an elliptical shape, reflecting the symmetry of the undeformed shell. However, at a critical indentation depth, contact is lost within an approximately rectangular region— a blister appears, Fig. 1(c). As indentation continues beyond this point, new portions lose contact with the plate along the length of the original contact region. The regular array of rectangular blisters thus formed are reminiscent of those formed by the uniaxial compression of a stiff, thin beam adhered to a soft substrate [20]. However, we emphasize that in this system there is no adhesive interaction between the ellipsoid and plate. Here, we study this anisotropic blistering instability and show that blistering is energetically favorable in comparison to both the planar state and the single elliptical delamination region that might be expected based on earlier experiments with a spherical shell [2]. We use a combination of computer simulations, experiments, and scaling analyses to understand the onset of directional delamination in ellipsoidal shells of revolution with axes a, b, and c ¼ b and thickness t [see Fig. 1(b)]. Our computer simulations were performed using the commercial finite element package ABAQUS (Simulia, Providence, RI). The results presented here were obtained with a Poisson ratio ν ¼ 0.3; further simulations with other values of ν show that
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FIG. 1 (color online). (a) Illustrations of the “ping pong ball instability” in a spherical shell of radius b indented by a rigid plate (gray circle in left panel). Here, a polygonal deformation is observed in simulations (left) and experiments on a plastic spherical cap (right) with dimensionless indentation depth Z0 ¼ Z=t ¼ 13.2, t the thickness, and t=b ¼ 0.01. (Adapted from Ref. [1].) (b) Axes of the ellipsoids considered here showing the rotational symmetry about the x axis. (c) Indentation of a highly ellipsoidal shell by a rigid plate causes “anisotropic blistering” of the shell—the contact becomes nonconformal. As the dimensionless indentation Z0 ¼ Z=t increases, the pattern develops with additional “blisters” forming. The left panel shows the energy density (high energy density corresponds to regions of compression) as computed in simulations. The right panel shows images from experiments performed on a shell made of polyvinylsiloxane using our mold and wetting technique.
quantitatively similar results are obtained with other values of ν [21]. As in the experiments described later, the half shell z ≥ 0 was simulated with clamped boundary conditions applied at z ¼ b. Our simulations used conventional thin shell elements with three nodes and quadratic interpolation; a mesh sensitivity study was performed to ensure that the results were minimally sensitive to the element size (see the Supplemental Material [21]). Experimentally, ellipsoidal shells were fabricated using a coating technique described in detail elsewhere [5]. In short, solid polyurethane molds were first fabricated using a 3D printer. A liquid silicon based elastomer, vinylpolysiloxane (VPS), was then used to wet the entire surface of the mold and the mold placed upside down to drain out
excessive polymer during curing. Half shells were formed in this way for aspect ratios a=b ¼ 0.3, 1, 3, 5; the Poisson ratio of VPS is ν ≈ 0.497. The shells have an approximately uniform thickness of t ¼ 0.32 0.05 mm (i.e., a 15% variation) which is determined by the polymer viscosity and curing time as well as the surface tension between VPS and polyurethane [5]. The ellipsoidal shells were indented by a flat plate moving at a constant speed of 10 mm= min controlled by an Instron machine. Images of the deformed shape of the shells were captured during the course of the indentation using a digital camera placed beneath the samples or from a side angle. The size of the blisters was measured from these digital images. Qualitative comparison between the deformation patterns observed in computer simulations and experiments seems promising at first [see Fig. 1(c)]. In particular, both show the onset of directional delamination at an indentation of Z0 ≈ 8 with similar numbers of blisters formed as indentation progresses. To understand the onset of the anisotropic blistering instability quantitatively, we use scaling arguments. Consider a prolate ellipsoid of revolution, with long and short axes of lengths 2a and 2b, respectively. The radii of curvature in the region of indentation are Rx ¼ a2 =b and Ry ¼ b in the long and short directions, respectively. We shall implicitly assume that the ellipsoid is very long, i.e., that a=b ≫ 1 and, hence, that Rx ⋙Ry . (The corresponding analysis for the case a ≪ b follows through upon replacing Rx ↔Ry .) When the ellipsoid is flattened by a vertical displacement Z, simple geometry dictates that the region affected is roughly elliptical, with a long axis ∼ðZRx Þ1=2 and short axis ∼ðZRy Þ1=2 . The area of this deformed ellipse is then S ∼ ZðRx Ry Þ1=2 . In principle, strain can occur in both the x and y directions and will be proportional to the principal curvature in each direction. However, since the ellipsoid is much more curved in the y direction, any residual strain in this direction will be much larger than that in the x direction. The elastic energy is most readily minimized by releasing the strain in the y direction and so the ellipsoid will deform in such a way that only strain in the x direction will remain, i.e., ϵ ¼ Z=Rx . For a shell of thickness t and Young’s modulus E, the energy density of the flattened configuration Es due to compression is then given in scaling terms by Es ∼ Etϵ2 :
(1)
If buckling occurs, the strain in the x direction, ϵ, would be released from those portions of the shell still in contact with the indenting plate. We assume that the regions of nonconformal contact, i.e., blisters, have a typical amplitude A and wavelength l. The strain within each blister would then scale according to ϵ ∼ ðA=lÞ2 and so A ∼ lϵ1=2 . By analogy, with the buckling of a thin strip constrained to have zero displacement at its edges [22], we would also
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expect the wavelength of the blistering to scale with the width of the delaminated region, i.e., l ∼ ðZRy Þ1=2 and hence, A ∼ ðZRy Þ1=2 ðZ=Rx Þ1=2 ∼ ZðRy =Rx Þ1=2 . To determine when blistering occurs, we compare the bending energy of the blistered state to the energy due to in-plane compression given in Eq. (1). We find that the energy density associated with blistering is Eb ∼ Et3 ðA=l2 Þ2 ∼ Et3 =Rx Ry :
(2)
The critical displacement, Zc , at which the blistering instability should be expected is then found by balancing the energy densities in Eqs. (1) and (2) to give Zc ∼ tðRx =Ry Þ1=2 ∼ ta=b. Introducing the normalized displacement Z0 ¼ Z=t, we find that the critical displacement may be written Z0c ∼ a=b:
Fig. 3(a). Determining the prefactor numerically, we find that l ≈ 4.499ðatÞ1=2 . To understand the appearance of further blisters, we note that the strain caused by compression depends on the local indentation. Along the x axis, we find that ϵ ∼ ðZ − x2 = 2Rx Þ=Rx . To make further progress, we assume that the position of the nth blister will be xn ∼ ðn − 1Þl (since the first blister is assumed to be at x ¼ 0) and that its width
(a) 103
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This result is in good agreement with the results of our simulations, as shown in Fig. 2. Using Eq. (3) in combination with our earlier results, we find that the typical size of the blisters is
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FIG. 2 (color online). Regime diagram showing the dependence of the type of deformation observed on the aspect ratio a=b and the indentation depth Z0 ¼ Z=t. For 0.5 < a=b ≲ 2, mirror buckling is followed by the formation of localized defects [the “ping pong ball instability”, Fig. 1(a)] while for a=b ≳ 2, anisotropic blistering occurs [as shown in Fig. 1(c)]. Points show the boundary between these behaviors determined from simulations for various shell thicknesses (as described in the legend); the dashed line shows a fit for the scaling law Eq. (3), which is valid in the limit of a=b ≫ 1.
FIG. 3 (color online). The properties of the blistering pattern. (a) The scaling of blister width as a function of the major axis of the shell (both normalized by the shell thickness) predicted by Eq. (4) is borne out by both numerical simulations (colored points) and experiments (black points). (b) Main figure: the number of blisters and the indentation depth at which that number are first observed approximately reproduce the expected scaling Eq. (5) (shown by the dashed line) both for numerical simulations (colored points) and experiments (black points). Inset: raw data for the number of blisters as a function of indentation depth. In both (a) and (b) different shell ellipticities are encoded by the symbol: a=b ¼ 3 (circle), a=b ¼ 4 (square), a=b ¼ 5 (diamond), a=b ¼ 6 (down triangle), a=b ¼ 8 (up triangle), and a=b ¼ 10 (star). Different shell thicknesses are encoded by the color: t=b ¼ 0.002 (red), t=b ¼ 0.005 (green), t=b ¼ 0.01 (blue), t=b ¼ 0.02 (magenta), and t=b ≈ 0.015 (black—experimental points).
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FIG. 4 (color online). The indentation force, P, as a function of indentation depth, Z, for a=b ¼ 5, t=b ≈ 0.016. The results of numerical simulations with t=b ¼ 0.016, ν ¼ 0.497 (solid curve) are shown together with the points at which new blisters form (○). Note that the force is approximately linear for small displacements (prior to anisotropic blistering) but dips sharply at the point of blistering before increasing again. The linear regime is well described by the known result for the indentation of an ellipsoidal shell Eq. (6). Experimental results are shown for shells with t=b ¼ 0.0134 0.0025 (△) and t=b ¼ 0.0165 0.0025 (□); two runs were performed for each scenario (open and closed symbols) and indicative error bars (resulting from the heterogeneity in shell thickness) are shown.
l ∼ ðatÞ1=2 , as for the first blister [23]. Balancing the stretching or compression and bending energy densities, we find that the number of blisters n and the indentation depth at which the nth blister first appears, Z0n , are related by 1=2 a n−1∼ ðZ0n − Z0c Þ1=2 : (5) b This scaling is consistent with the experimental and numerical results shown in Fig. 3(b). We also consider the load P required to impose a given displacement. For sufficiently small indentation depths, the zone of contact between the plate and the shell is sufficiently small that the problem is analogous to the case of the point indentation of an ellipsoidal shell, which has been studied previously [14,15]. Based on our previous theoretical analysis [14], we expect the force to increase linearly with displacement; in particular, we expect the force and displacement to be related by P¼
4 Et2 4 Et3 0 Z ¼ Z; ½3ð1 − ν2 Þ1=2 ðRx Ry Þ1=2 ½3ð1 − ν2 Þ1=2 a
(6)
with ν the Poisson ratio. This result is borne out (prior to the onset of anisotropic blistering) by the numerical and
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experimental results shown in Fig. 4. A key feature of this data is that at the point of blistering, the force decreases dramatically, as has also been observed in the formation of localized dimples [4]. The force then increases rapidly, although a logarithmic plot reveals that the force law is not necessarily quadratic as first appears. Finally, we note that the precise force-displacement relationship that is observed depends sensitively on the thickness of the shell, though the indentation depth at which the force drop associated with blistering is observed is a robust feature of the instability. In conclusion, we have explored a novel instability of ellipsoidal shells deformed by contact with a horizontal plate: a series of rectangular blisters form aligned with the long axis of the ellipsoid. We found that the number of blisters formed grows with increasing indentation although the width of each blister remains approximately constant during indentation [see Fig. 1(c)]. This ability to form a variable number of blisters by controlling the indentation depth has potential application in a range of fields. For example, the blisters could be used as fluid channels in a microelectromechanical system giving a fluid carrying capacity that depends on the indentation in much the same way as the deformation of a single blister has been used to control such flows recently [24]. We also showed that the onset and form of the anisotropic blistering instability is sensitive to the shell thickness. Conventional AFM techniques [10,14] to measure the stiffness of an ellipsoidal object in indentation yield the combination Et2 =a. While the lateral dimension a is easy to measure directly, the combination Et2 is difficult to disentangle into its constituents E and t and is sensitive to heterogeneity in the thickness, as shown in the large error bars shown in Fig. 4. However, even in the absence of any force measurements, measuring the characteristic size of any blisters that form, l, and using the scaling law Eq. (4) would yield an estimate for the thickness of the shell, t. If force and/or indentation depth measurements are available then a more accurate estimate of thickness t may be determined by measuring the indentation depth Zc at which delamination first occurs, since this corresponds to a sudden dip in the applied force (see Fig. 4); from Fig. 2 and Eq. (3), we see that t ≈ Zc b=ð1.78aÞ. Such a method could prove useful for measurements of long thin cells that might be modeled as ellipsoids with aspect ratio a=b ≳ 2 (and, hence, be subject to the anisotropic blistering instability described here); this includes elongated epidermal cells as in the shoot or the root of plants [25] as well as fission yeast [26,27]. We are grateful to Pedro Reis and his group for valuable input into the experimental protocols used here. A. A., H. E., andA. V. arethankfulfor the U.S. AirForceOffice ofScientific Research under AFOSR YIP Grant No. FA 9550-10-1-0145 under the technical supervision of Dr. Joycelyn Harrison. D. V. is partially supported by a Leverhulme Trust Research Fellowship. H. E. and A. A. contributed equally to this work.
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[email protected] [1] A. Vaziri, Thin-Walled Struct. 47, 692 (2009). [2] B. Audoly and Y. Pomeau, Elasticity and Geometry (Oxford University Press, New York, 2010). [3] L. Pauchard and S. Rica, Philos. Mag. B 78, 225 (1998). [4] A. Vaziri and L. Mahadevan, Proc. Natl. Acad. Sci. U.S.A. 105, 7913 (2008). [5] A. Nasto, A. Ajdari, A. Lazarus, A. Vaziri, and P. M. Reis, Soft Matter 9, 6796 (2013). [6] E. Katifori, S. Alben, E. Cerda, D. R. Nelson, and J. Dumais, Proc. Natl. Acad. Sci. U.S.A. 107, 7635 (2010). [7] N. Tsapis, E. R. Dufresne, S. S. Sinha, C. S. Riera, J. W. Hutchinson, L. Mahadevan, and D. A. Weitz, Phys. Rev. Lett. 94, 018302 (2005). [8] S. S. Datta, S.-H. Kim, J. Paulose, A. Abbaspourrad, D. R. Nelson, and D. A. Weitz, Phys. Rev. Lett. 109, 134302 (2012). [9] E. Reissner, J. Math. Phys. (Cambridge, Mass.) 25, 80 (1946). [10] M. Hernando-Pérez, R. Miranda, M. Aznar, J. L. Carrascosa, I. A. T. Schaap, D. Reguera, and P. J. de Pablo, Small 8, 2366 (2012). [11] J. Arfsten, S. Leupold, C. Bradtmöller, I. Kampen, and A. Kwade, Colloids Surf. B 79, 284 (2010). [12] D. Vella, A. Ajdari, A. Vaziri, and A. Boudaoud, J. R. Soc. Interface 9, 448 (2012). [13] R. Mercadé-Prieto, C. R. Thomas, and Z. Zhang, Eur. Biophys. J. 42, 613 (2013).
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[14] D. Vella, A. Ajdari, A. Vaziri, and A. Boudaoud, Phys. Rev. Lett. 109, 144302 (2012). [15] A. Lazarus, H. C. B. Florijn, and P. M. Reis, Phys. Rev. Lett. 109, 144301 (2012). [16] B. Budiansky and J. W. Hutchinson, AIAA J. 4, 1505 (1966). [17] D. Vella, A. Ajdari, A. Vaziri, and A. Boudaoud, Phys. Rev. Lett. 107, 174301 (2011). [18] N.-C. Huang, Unsymmetrical Buckling of Thin Shallow Spherical Shells, Office of Naval Research Technical Report, 1963. [19] J. R. Fitch, Int. J. Solids Struct. 4, 421 (1968). [20] D. Vella, J. Bico, A. Boudaoud, B. Roman, and P. M. Reis, Proc. Natl. Acad. Sci. U.S.A. 106, 10901 (2009). [21] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.094302 for results with other values of ν and details of the simulation technique. [22] Y. Pomeau, Philos. Mag. B 78, 235 (1998). [23] This approximation is consistent with the images in Fig. 1 and is correct in the limit that nb=a ≪ 1. [24] D. P. Holmes, B. Tavakol, G. Froehlicher, and H. A. Stone, Soft Matter 9, 7049 (2013). [25] P. S. Nobel, Physicochemical and Environmental Plant Physiology (Academic Press, San Diego, 1999). [26] N. Minc, A. Boudaoud, and F. Chang, Curr. Biol. 19, 1096 (2009). [27] R. Pyati, L. L. Elvir, E. C. Charles, U. Seenath, and T. D. Wolkow, J. Yeast Fungal Res. 2, 106 (2011).
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PHYSICAL REVIEW LETTERS
Anisotropic Blistering Instability of Highly Ellipsoidal Shells 1
Hamid Ebrahimi,1 Amin Ajdari,2 Dominic Vella,3 Arezki Boudaoud,4 and Ashkan Vaziri1,*
Department of Mechanical and Industrial Engineering, Northeastern University, Boston, Massachusetts 02115, USA 2 Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA 3 Mathematical Institute, University of Oxford,Woodstock Road, Oxford OX2 6GG, United Kingdom 4 Laboratoire Reproduction et Développement des Plantes & Laboratoire Joliot-Curie, INRA, CNRS, ENS, Université de Lyon, 46 Allée d’Italie, F-69364 Lyon Cedex 07, France (Received 2 September 2013; revised manuscript received 9 January 2014; published 6 March 2014) The formation of localized periodic structures in the deformation of elastic shells is well documented and is a familiar first stage in the crushing of a spherical shell such as a ping-pong ball. While spherical shells manifest such periodic structures as polygons, we present a new instability that is observed in the indentation of a highly ellipsoidal shell by a horizontal plate. Above a critical indentation depth, the plate loses contact with the shell in a series of well-defined “blisters” along the long axis of the ellipsoid. We characterize the onset of this instability and explain it using scaling arguments, numerical simulations, and experiments. We also characterize the properties of the blistering pattern by showing how the number of blisters and their size depend on both the geometrical properties of the shell and the indentation but not on the shell’s elastic modulus. This blistering instability may be used to determine the thickness of highly ellipsoidal shells simply by squashing them between two plates. DOI: 10.1103/PhysRevLett.112.094302
PACS numbers: 46.70.De
As every ping-pong player knows, spherical elastic shells suffer an unusual mode of instability when hit too hard: a series of vertices appears, connected by linear ridges forming an inverted polygon [see Fig. 1(a)]. These polygonal deformations are an example of the localization of the stress state that occurs in more controlled experimental settings such as the indentation of a shell by a plate [2], at a point [3,4], or with indentors of size comparable to the radius of the shell [5]. This localization of stress is important not only because of its visual appearance but also because the focusing of stresses is more likely to lead to irreversible, plastic deformations—again a feature familiar to ping-pong players. The ubiquity of thin elastic shells also leads to a wide range of applications in which this asymmetric buckling can be important, from the folding of pollen grains [6] to the buckling of drying colloidal droplets [7] or osmotically shrinking polymeric capsules [8]. Previously, much interest has focused on the indentation of spherical shells [9]. In this setting, the determination of the force-displacement relationship is important because of its application to atomic force microscopy (AFM) measurements of viruses [10] as well as yeast and other cells [11–13]. Recently, this work has been extended to the indentation of ellipsoidal shells from both theoretical [14] and experimental points of view [15]. However, all of these studies implicitly assume that the deformation of the shell retains the symmetry of the undeformed shell. In fact, it has long been known that at a critical indentation depth (proportional to the thickness of the shell, t, [16,17]), an asymmetric deformation profile with a well-defined wave number 0031-9007=14=112(9)=094302(5)
appears [18,19]. As indentation continues past this critical depth, the number of vertices of the polygonal pattern that is observed changes [1,4,16]. In this Letter, we report a new type of localized deformation that occurs during the indentation of highly ellipsoidal shells by a flat plate. For sufficiently small compressions, the deformed region of the shell remains in contact with the plate and retains an elliptical shape, reflecting the symmetry of the undeformed shell. However, at a critical indentation depth, contact is lost within an approximately rectangular region— a blister appears, Fig. 1(c). As indentation continues beyond this point, new portions lose contact with the plate along the length of the original contact region. The regular array of rectangular blisters thus formed are reminiscent of those formed by the uniaxial compression of a stiff, thin beam adhered to a soft substrate [20]. However, we emphasize that in this system there is no adhesive interaction between the ellipsoid and plate. Here, we study this anisotropic blistering instability and show that blistering is energetically favorable in comparison to both the planar state and the single elliptical delamination region that might be expected based on earlier experiments with a spherical shell [2]. We use a combination of computer simulations, experiments, and scaling analyses to understand the onset of directional delamination in ellipsoidal shells of revolution with axes a, b, and c ¼ b and thickness t [see Fig. 1(b)]. Our computer simulations were performed using the commercial finite element package ABAQUS (Simulia, Providence, RI). The results presented here were obtained with a Poisson ratio ν ¼ 0.3; further simulations with other values of ν show that
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FIG. 1 (color online). (a) Illustrations of the “ping pong ball instability” in a spherical shell of radius b indented by a rigid plate (gray circle in left panel). Here, a polygonal deformation is observed in simulations (left) and experiments on a plastic spherical cap (right) with dimensionless indentation depth Z0 ¼ Z=t ¼ 13.2, t the thickness, and t=b ¼ 0.01. (Adapted from Ref. [1].) (b) Axes of the ellipsoids considered here showing the rotational symmetry about the x axis. (c) Indentation of a highly ellipsoidal shell by a rigid plate causes “anisotropic blistering” of the shell—the contact becomes nonconformal. As the dimensionless indentation Z0 ¼ Z=t increases, the pattern develops with additional “blisters” forming. The left panel shows the energy density (high energy density corresponds to regions of compression) as computed in simulations. The right panel shows images from experiments performed on a shell made of polyvinylsiloxane using our mold and wetting technique.
quantitatively similar results are obtained with other values of ν [21]. As in the experiments described later, the half shell z ≥ 0 was simulated with clamped boundary conditions applied at z ¼ b. Our simulations used conventional thin shell elements with three nodes and quadratic interpolation; a mesh sensitivity study was performed to ensure that the results were minimally sensitive to the element size (see the Supplemental Material [21]). Experimentally, ellipsoidal shells were fabricated using a coating technique described in detail elsewhere [5]. In short, solid polyurethane molds were first fabricated using a 3D printer. A liquid silicon based elastomer, vinylpolysiloxane (VPS), was then used to wet the entire surface of the mold and the mold placed upside down to drain out
excessive polymer during curing. Half shells were formed in this way for aspect ratios a=b ¼ 0.3, 1, 3, 5; the Poisson ratio of VPS is ν ≈ 0.497. The shells have an approximately uniform thickness of t ¼ 0.32 0.05 mm (i.e., a 15% variation) which is determined by the polymer viscosity and curing time as well as the surface tension between VPS and polyurethane [5]. The ellipsoidal shells were indented by a flat plate moving at a constant speed of 10 mm= min controlled by an Instron machine. Images of the deformed shape of the shells were captured during the course of the indentation using a digital camera placed beneath the samples or from a side angle. The size of the blisters was measured from these digital images. Qualitative comparison between the deformation patterns observed in computer simulations and experiments seems promising at first [see Fig. 1(c)]. In particular, both show the onset of directional delamination at an indentation of Z0 ≈ 8 with similar numbers of blisters formed as indentation progresses. To understand the onset of the anisotropic blistering instability quantitatively, we use scaling arguments. Consider a prolate ellipsoid of revolution, with long and short axes of lengths 2a and 2b, respectively. The radii of curvature in the region of indentation are Rx ¼ a2 =b and Ry ¼ b in the long and short directions, respectively. We shall implicitly assume that the ellipsoid is very long, i.e., that a=b ≫ 1 and, hence, that Rx ⋙Ry . (The corresponding analysis for the case a ≪ b follows through upon replacing Rx ↔Ry .) When the ellipsoid is flattened by a vertical displacement Z, simple geometry dictates that the region affected is roughly elliptical, with a long axis ∼ðZRx Þ1=2 and short axis ∼ðZRy Þ1=2 . The area of this deformed ellipse is then S ∼ ZðRx Ry Þ1=2 . In principle, strain can occur in both the x and y directions and will be proportional to the principal curvature in each direction. However, since the ellipsoid is much more curved in the y direction, any residual strain in this direction will be much larger than that in the x direction. The elastic energy is most readily minimized by releasing the strain in the y direction and so the ellipsoid will deform in such a way that only strain in the x direction will remain, i.e., ϵ ¼ Z=Rx . For a shell of thickness t and Young’s modulus E, the energy density of the flattened configuration Es due to compression is then given in scaling terms by Es ∼ Etϵ2 :
(1)
If buckling occurs, the strain in the x direction, ϵ, would be released from those portions of the shell still in contact with the indenting plate. We assume that the regions of nonconformal contact, i.e., blisters, have a typical amplitude A and wavelength l. The strain within each blister would then scale according to ϵ ∼ ðA=lÞ2 and so A ∼ lϵ1=2 . By analogy, with the buckling of a thin strip constrained to have zero displacement at its edges [22], we would also
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expect the wavelength of the blistering to scale with the width of the delaminated region, i.e., l ∼ ðZRy Þ1=2 and hence, A ∼ ðZRy Þ1=2 ðZ=Rx Þ1=2 ∼ ZðRy =Rx Þ1=2 . To determine when blistering occurs, we compare the bending energy of the blistered state to the energy due to in-plane compression given in Eq. (1). We find that the energy density associated with blistering is Eb ∼ Et3 ðA=l2 Þ2 ∼ Et3 =Rx Ry :
(2)
The critical displacement, Zc , at which the blistering instability should be expected is then found by balancing the energy densities in Eqs. (1) and (2) to give Zc ∼ tðRx =Ry Þ1=2 ∼ ta=b. Introducing the normalized displacement Z0 ¼ Z=t, we find that the critical displacement may be written Z0c ∼ a=b:
Fig. 3(a). Determining the prefactor numerically, we find that l ≈ 4.499ðatÞ1=2 . To understand the appearance of further blisters, we note that the strain caused by compression depends on the local indentation. Along the x axis, we find that ϵ ∼ ðZ − x2 = 2Rx Þ=Rx . To make further progress, we assume that the position of the nth blister will be xn ∼ ðn − 1Þl (since the first blister is assumed to be at x ¼ 0) and that its width
(a) 103
2
10
(3)
1
This result is in good agreement with the results of our simulations, as shown in Fig. 2. Using Eq. (3) in combination with our earlier results, we find that the typical size of the blisters is
ABAQUS
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l ∼ ðatÞ1=2 ;
independently of b. This result is in good agreement with both numerical simulations and experiments, as shown in
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8 Anisotropic Blistering
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Conformal Contact
t / b = 0.002
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t / b = 0.005 t / b = 0.01
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Mirror Buckling Followed by Localized Dimpling
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FIG. 2 (color online). Regime diagram showing the dependence of the type of deformation observed on the aspect ratio a=b and the indentation depth Z0 ¼ Z=t. For 0.5 < a=b ≲ 2, mirror buckling is followed by the formation of localized defects [the “ping pong ball instability”, Fig. 1(a)] while for a=b ≳ 2, anisotropic blistering occurs [as shown in Fig. 1(c)]. Points show the boundary between these behaviors determined from simulations for various shell thicknesses (as described in the legend); the dashed line shows a fit for the scaling law Eq. (3), which is valid in the limit of a=b ≫ 1.
FIG. 3 (color online). The properties of the blistering pattern. (a) The scaling of blister width as a function of the major axis of the shell (both normalized by the shell thickness) predicted by Eq. (4) is borne out by both numerical simulations (colored points) and experiments (black points). (b) Main figure: the number of blisters and the indentation depth at which that number are first observed approximately reproduce the expected scaling Eq. (5) (shown by the dashed line) both for numerical simulations (colored points) and experiments (black points). Inset: raw data for the number of blisters as a function of indentation depth. In both (a) and (b) different shell ellipticities are encoded by the symbol: a=b ¼ 3 (circle), a=b ¼ 4 (square), a=b ¼ 5 (diamond), a=b ¼ 6 (down triangle), a=b ¼ 8 (up triangle), and a=b ¼ 10 (star). Different shell thicknesses are encoded by the color: t=b ¼ 0.002 (red), t=b ¼ 0.005 (green), t=b ¼ 0.01 (blue), t=b ¼ 0.02 (magenta), and t=b ≈ 0.015 (black—experimental points).
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PRL 112, 094302 (2014) 200 180 160 140 120 100 80 60 40 20 0
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FIG. 4 (color online). The indentation force, P, as a function of indentation depth, Z, for a=b ¼ 5, t=b ≈ 0.016. The results of numerical simulations with t=b ¼ 0.016, ν ¼ 0.497 (solid curve) are shown together with the points at which new blisters form (○). Note that the force is approximately linear for small displacements (prior to anisotropic blistering) but dips sharply at the point of blistering before increasing again. The linear regime is well described by the known result for the indentation of an ellipsoidal shell Eq. (6). Experimental results are shown for shells with t=b ¼ 0.0134 0.0025 (△) and t=b ¼ 0.0165 0.0025 (□); two runs were performed for each scenario (open and closed symbols) and indicative error bars (resulting from the heterogeneity in shell thickness) are shown.
l ∼ ðatÞ1=2 , as for the first blister [23]. Balancing the stretching or compression and bending energy densities, we find that the number of blisters n and the indentation depth at which the nth blister first appears, Z0n , are related by 1=2 a n−1∼ ðZ0n − Z0c Þ1=2 : (5) b This scaling is consistent with the experimental and numerical results shown in Fig. 3(b). We also consider the load P required to impose a given displacement. For sufficiently small indentation depths, the zone of contact between the plate and the shell is sufficiently small that the problem is analogous to the case of the point indentation of an ellipsoidal shell, which has been studied previously [14,15]. Based on our previous theoretical analysis [14], we expect the force to increase linearly with displacement; in particular, we expect the force and displacement to be related by P¼
4 Et2 4 Et3 0 Z ¼ Z; ½3ð1 − ν2 Þ1=2 ðRx Ry Þ1=2 ½3ð1 − ν2 Þ1=2 a
(6)
with ν the Poisson ratio. This result is borne out (prior to the onset of anisotropic blistering) by the numerical and
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experimental results shown in Fig. 4. A key feature of this data is that at the point of blistering, the force decreases dramatically, as has also been observed in the formation of localized dimples [4]. The force then increases rapidly, although a logarithmic plot reveals that the force law is not necessarily quadratic as first appears. Finally, we note that the precise force-displacement relationship that is observed depends sensitively on the thickness of the shell, though the indentation depth at which the force drop associated with blistering is observed is a robust feature of the instability. In conclusion, we have explored a novel instability of ellipsoidal shells deformed by contact with a horizontal plate: a series of rectangular blisters form aligned with the long axis of the ellipsoid. We found that the number of blisters formed grows with increasing indentation although the width of each blister remains approximately constant during indentation [see Fig. 1(c)]. This ability to form a variable number of blisters by controlling the indentation depth has potential application in a range of fields. For example, the blisters could be used as fluid channels in a microelectromechanical system giving a fluid carrying capacity that depends on the indentation in much the same way as the deformation of a single blister has been used to control such flows recently [24]. We also showed that the onset and form of the anisotropic blistering instability is sensitive to the shell thickness. Conventional AFM techniques [10,14] to measure the stiffness of an ellipsoidal object in indentation yield the combination Et2 =a. While the lateral dimension a is easy to measure directly, the combination Et2 is difficult to disentangle into its constituents E and t and is sensitive to heterogeneity in the thickness, as shown in the large error bars shown in Fig. 4. However, even in the absence of any force measurements, measuring the characteristic size of any blisters that form, l, and using the scaling law Eq. (4) would yield an estimate for the thickness of the shell, t. If force and/or indentation depth measurements are available then a more accurate estimate of thickness t may be determined by measuring the indentation depth Zc at which delamination first occurs, since this corresponds to a sudden dip in the applied force (see Fig. 4); from Fig. 2 and Eq. (3), we see that t ≈ Zc b=ð1.78aÞ. Such a method could prove useful for measurements of long thin cells that might be modeled as ellipsoids with aspect ratio a=b ≳ 2 (and, hence, be subject to the anisotropic blistering instability described here); this includes elongated epidermal cells as in the shoot or the root of plants [25] as well as fission yeast [26,27]. We are grateful to Pedro Reis and his group for valuable input into the experimental protocols used here. A. A., H. E., andA. V. arethankfulfor the U.S. AirForceOffice ofScientific Research under AFOSR YIP Grant No. FA 9550-10-1-0145 under the technical supervision of Dr. Joycelyn Harrison. D. V. is partially supported by a Leverhulme Trust Research Fellowship. H. E. and A. A. contributed equally to this work.
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PHYSICAL REVIEW LETTERS
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