Mechanics of non-euclidean plates

Transcription

1 Oddelek za fiziko Seminar - 4. letnik Mechanics of non-euclidean plates Author: Nina Rogelj Adviser: doc. dr. P. Ziherl Ljubljana, April, 2011 Summary The purpose of this seminar is to describe the mechanics of non-euclidean plates. In the first part we focus on definition of thin plates. Later we discuss some examples of non-euclidean plates in nature and end with man-made ones.

2 CONTENTS 1 Contents 1 Introduction 1 2 Mechanical instabilities Elastic energy and t 0 limit Ordinary buckling and wrinkling Metrics and Gauss theorem Experiments Leaves, flowers and garbage bags Responsive NEP Conclusion 12 5 References 13 1 Introduction Crocheting a flat thin disk can be really hard if you are a beginner. An expert creates a flat disk where all the stitches are the same size. On the other hand, beginner s stitches are far from being the same size. The difference in size results in buckling of the disk. One typically produces a non-euclidean plate [1]. Euclidean plates are flat plates and when we crumple, stretch or tear them, we get curves on the surface and geometry that is not Euclidean. Thin elastic sheets that form non-euclidean plates (NEPs) can be easily found in nature; edges of leaves, flowers, torn plastic sheets, growing tissues (Fig. 1, Fig. 2)... All of them are determined by elastic instabilities. On the other hand, artificial NEPs are not common at all mainly because of two main problems. First, we have great difficulties formulating a practical theoretical model of these instabilities. The second problem is implementation. If we knew the distribution that would turn a disk into an ornamental bowl, how would we achieve it? Therefore many of the experiments involving NEPs are quite recent [2]. NEPs are characterized by complex patterns as seen in examples shown in Figs. 1 and 2. These patterns are made by rolling, crumpling, stretching or tearing thin sheets. When describing NEPs we use mechanical instabilities like buckling and wrinkling. We will discuss this in the next section.

3 2 MECHANICAL INSTABILITIES 2 Figure 1: Living tissues are a great example of non-euclidean surfaces with their wavy edges. On the left side of the picture is a lichen (Sticta limbata) and on the right is an orchid (Schomborgkia beysiana) [3]. Figure 2: Examples of non-euclidean plates found in nature. A potato chip that adopts a saddle shape during frying (top left). Acetabularia schenckii: A green algae where its top is a representative of NEP (top right). A dead leaf (bottom left). A leaf infected by cotton leaf crumple virus (bottom right) [4]. 2 Mechanical instabilities 2.1 Elastic energy and t 0 limit The selection of a NEPs shape is determined by the minimum of the elastic energy. We use an approximation of the energy composed from a stretching and a bending term The two terms of the energy are given by E b = E = E s + E b. (1) E s = Y 24(1 ν 2 ) t3 B (2) Y 2(1 ν 2 ts (3) ) where ν is Poisson ratio, S is a function of local strain and B is a function of the mean curvature of the surface [2]. We get a competition between the two terms and the winner is

4 2 MECHANICAL INSTABILITIES 3 determined by the thickness of the surface. When we decrease the thickness, the bending term decreases faster then the stretching term. Bending becomes energetically favorable and thus in the limit of t 0 we get a stretch-free configuration. To explain the limit we will consider ordinary buckling and wrinkling. 2.2 Ordinary buckling and wrinkling We will start with ordinary buckling. Consider a ruler of length L and thickness t that is compressed at both ends. Compression results in a displacement of 2δ as seen in Fig. 3. The elastic energy of such compressed system is ( ) δ 2 E S t. (4) L If we allow a symmetry break, we can consider a second solution, bending the ruler to form an arc. The latter solution is buckling and the energy is E B t 3 k 2, (5) where k is the curvature of the ruler [2]. Total energy of a thin sheet is composed from stretching and bending term as expressed in (1). The stretching term is linearly dependent on thickness t, whereas the bending term scales as t 3. When determining the shape of NEP, the minimum of the total energy is sought. The stretching term decreases faster then the bending term when t is large. In this case, the equilibrium state corresponds to k = 0 and we obtain a bend-free solution. On the other hand, when t is small the bending term decreases faster than the stretching term and the equilibrium state is a stretch free solution, i.e., δ = 0. Figure 3: Bending. A ruler of length L and thickness t is compressed (top). A pure compressed configuration is flat (center). Its mid plane (dashed line) is compressed by 2δ. In the pure bending configuration the ruler is bent into an arc (bottom). Its mid plane preserves the length L [2].

5 2 MECHANICAL INSTABILITIES 4 On the other hand, wrinkling can be represented as a ruler supported by a thick substrate (Fig. 4). Again, we compress the system by 2δ. In this case, the ruler keeps its original length L by bending. The number of waves of the ruler is determined by its thickness. As the thickness is decreased, the wavelength λ and the amplitude A of wrinkles is decreased too [2]. Figure 4: Wrinkling. A stiff ruler of length L and thickness t is attached to a soft thick substrate (top). The system is compressed by 2δ and the ruler preserves its length L. For large thicknesses of the ruler (center) the system wrinkles with large wavelength and amplitude. For small thicknesses (bottom)the system wrinkles with small wavelength and amplitude [2]. When we consider ordinary buckling and wrinkling, we have external force that deforms our object. On the other hand, flowers do not experience an external force and still remain buckled. We will show on a simple example of a disk that we get buckling or wrinkling-like instabilities because of internal stresses. We take a thin disk of radius R and thickness t (Fig. 5). All the material within radius ρ < R grows uniformly and isotropically by a factor η. As with the ruler there exist two options. One is a flat configuration with no bending. The inner part is radially and azimuthally compressed, whereas the outer is radially compressed and azimuthally stretched. The second option is buckling out of plane, which eliminates stretching [2]. With the decreasing thickness of the disk, the latter option is energetically preferable because t 3 decreases faster than t when t Metrics and Gauss theorem For an easier description of NEPs geometry we will introduce metric and Gauss theorem. Metric provides the information that we need to determine NEPs surface. We define it by comparing the surface in two different states. Imagine a flat sheet in its reference state and draw a grid to form a coordinate system. Now stretch or compress the sheet. With deformation we change the distances between the lines. We define metric g from square of this distance dl 2 = g ij dx i dx j. (6)

6 3 EXPERIMENTS 5 Figure 5: Buckling because of internal stresses. We start with a flat disk of radius R and we let all the material within radius ρ < R undergo a growth. The bottom left panel shows a flat configuration that is under stress. Blue color indicates compression and red indicates tension. The bottom right shape is a 3D configuration where only bending is present [2]. For a stress-free flat elastic disk of radius R and thickness t we express distance between two neighboring points in polar coordinate system dl 2 = dr 2 + r 2 dθ 2 (7) with corresponding 2D metric tensor g = ( r 2 ) (8) [2]. We define two different metric tensors. The actual metric is obtained from the real configuration of the system, i.e. from the shape that the system is in after the deformation. The target metric is a metric that we would obtain from an equilibrium state of a system where every spring that connects points in the system is at its rest length. The difference between the two metrics is the local strain. In case of NEPs we want the strain to be as small as possible which leads to a shape that is curved without applied force [3]. Gauss theorem shows that metric of the surface determines the local Gaussian curvature. If the surface is completely flat, the Gaussian curvature at every point is K = 0. When K < 0 we have a hyperbolic surface and we get an elliptic surface when K > 0. For a metric of form ( ) 1 0 g = 0 φ 2 (9) (x) the Gaussian curvature is [2] 3 Experiments K = 1 2 φ φ x 2. (10) In this seminar we will take a look at two experiments involving NEPs. The first one is the edge of a torn plastic sheet. The results obtained from it will be used to explain wavy leaves and flowers. In the second experiment we will look at a responsive NEP that changes its form according to external stimulation.

7 3 EXPERIMENTS Leaves, flowers and garbage bags This is an experiment easily done at home. We take a garbage bag, cut a square and tear it apart. Then we take a look at the edge of a torn plastic (Fig. 6) [4]. Figure 6: The edge of a torn plastic sheet. We can observe a complex structure, waves upon waves. This property of pattern is called a fractal [4]. Experiment has been performed where a plastic sheet was ripped apart in a controlled laboratory setting. It was pulled at a uniform rate on opposite sites as seen in Fig. 7. Figure 7: Puling plastic apart under controlled settings. By tearing regular wave pattern is generated at the edge of the tear. Lines on the scale in the upper left corner are 1 mm apart [5]. Upon taking a closer look at the edge, we observe waves upon waves. Fig. 8(a) shows us different magnifications of the edge. We take a part of the edge, magnify it by a factor of 3.2 and we get almost identical image as the original. We repeat the process 5 times before we loose the wave pattern. This property of the pattern is called a fractal. The amplitude A of a specific wave is related to the wavelength by equation A = 0.15λ. The linear relation of amplitude and wavelength is illustrated in Fig. 8(b) [6]. The reason for this complex shape of the edge is its metric. The change in metric of the surface means the change in distances on the surface. But the change is not the same all over the surface. The further away from the edge, the smaller the change and the less stretching of the plastic sheet. The relation of the change in the metric from the distance form the edge is plotted in Fig. 8(c). The edge of the plastic is now too long and that is reflected in a new metric. Because of increase of the length of the edge the edge buckles, like the thin ruler that we mentioned before. We saw that buckling is energetically favorable when thickness is decreased. This is the reason why plastic is so highly susceptible to buckling. The Gaussian curvature is not 0, which means that the metric is not flat. The sheet chooses to bend buckle out of plane generating saddle points everywhere. This is energetically favorable compared to compression needed to accommodate the metric.

8 3 EXPERIMENTS 7 Figure 8: Buckling cascades. (a) The edge of a torn plastic at different magnifications. The dotted boxed region on the left is 3.2-times magnified and showed under the previous picture. The width of the picture is shown under it. (b) Relation between amplitude and wavelength of the cascade. (c) The length of a sheet normalized by its original length as a function of distance from the edge. Dashed line indicates original size. (d) Top panel is side and front view of a plastic sheet. Bottom panel is beet leaf in both views. We can note the resemblance between the two [6]. The results from this experiment can be expanded to leaves and flowers. Till now it was thought that the waviness of the leaves is generated by a complex genetic code. But after observing the edge of a torn plastic, we see that waves can be produced only by changing the metric of the sheet. This means that the plant has to control only the growth of the cells at the edge. To support this theory, an experiment was carried out using a flat leaf. With growth-regulating hormone that was applied to the edge of a leaf, they created a wavy leaf. This transformation can be seen in Fig. 9. Because the edge of the leaf grew more than the center, it started to buckle. The amplitude of the waves increased with time and waves upon waves were seen just like in the plastic sheet [4]. Another demonstration of the symmetry breaking of the leaf-edge is comparing a naturally flat and naturally wavy leaf. Both leaves were cut into thin strips along its edges and placed between two glass plates. The diameter of a flat leaf increases as you move outward. This is illustrated in Fig. 10. On the other hand the diameter of a wavy leaf decreases as we approach the edge. The closer to the edge of a wavy leaf, the longer and the more buckled the arc [4]. In Fig. 2 (top left) we see that thermal expansion turns a flat slice of potato into a potato chip which is non-euclidean. Desiccation can also lead to bending and crumpling (Fig. 2 - bottom left). In living tissues viruses can modify the growth process and infected plants exhibit curled or crumpled leaves (Fig. 2 - bottom right) [7]. Another example is our skin. When it is exposed to water for a longer period of time, the skin wrinkles [8].

9 3 EXPERIMENTS 8 Figure 9: From a flat to a wavy leaf. The edge of a normally flat leaf of an eggplant was treated with growth hormone. This caused an enhanced growth and therefore a negative Gaussian curvature. The edge of a leaf became similar to torn plastic sheet. After 10 days the waves have developed. After 14 days waves grew bigger and waves upon waves became visible [4]. Figure 10: Comparing a flat and a wavy leaf. Strips are cut from both leaves and placed between glass plates. Strips from a flat leaf shown on the left have an increasing radius of the arcs. Strips from a wavy leaf shown on the right side have a decreasing radius of the arcs. Geometry of a wavy leaf therefore requires negative Gaussian curvature. Both leaves were collected from the same bay leaf bush [4].

10 3 EXPERIMENTS 9 We can expand the same principle to flowers. The only difference is that in this case we do not have a flat sheet to start with, but a cylinder. Its diameter grows exponentially and this increase leads us to a trumpet-like shape. If the diameter continues to increase, it results in a symmetry breaking and the edge of the flower buckles (Fig. 11) [4]. Figure 11: Modeling a flower. (a) We start with a ring of cells. (b) The number of cells in each line increases. (c) We obtain a trumpet-like shape. (d) If we keep increasing the number of cells in each row, we get a daffodil-like shape. [2] The form of a daffodil was made in an experimental study using thin tubes made of polyacrylamide gel. Metric of the gel was changed by water and acetone (Fig. 12). The tube was first dipped into acetone which shrank it uniformly. One end was then dipped into water so the water diffused into the tube. The gel reacts to water by swelling, therefore the edge of the tube grew bigger and it had to buckle. Depending on the transition between water and acetone, two different shapes were obtained. A trumpet-like shape was obtained where the transition was over a longer distance and a daffodil shape was obtained where transition took place over a shorter distance. The shorter transition produced a steeper change in metric and thus a wavier edge [4]. Figure 12: Modeling a daffodil using thin tube made of gel, which swells in water and shrinks in acetone. (a) The tube was dipped in acetone and then the edge in the water. A trumpet-like shape was obtained. (b) A computer simulation of a trumpet-like shape. (c) Transition water-to-acetone took place on a shorter distance. Metric of the cylinder had a steeper increase and that forced the tube to break symmetry by producing wavy edge. (d) A computer simulation of a daffodil-like shape. On the right panel:a real daffodil. [4]

11 3 EXPERIMENTS 10 Another example of homemade NEPs is crocheting (Fig. 13). The metric is determined by controlling the number of stitches in a row much like in Fig. 11. Crocheting curved planes have recently become popular, creating coral reefs and flowers [1]. Figure 13: A crocheted annular hyperbolic plane [1] 3.2 Responsive NEP A new technique has been introduced in making responsive NEPs. In the making of the flowers they used a gel and changed its metric by dipping it in water and acetone. We can control the metric by different monomer concentration in the gel tube. Shrinkage of the gel is a function of monomer concentration as plotted in Fig. 14. To produce NEPs, gel disks were made using NIPA solutions. This gel undergoes a sharp, reversible shrinking transition at T c = 33. Flat disks were made with radial gradients in monomer concentration. The disk preparation protocol is shown in Fig. 15 [9]. Figure 14: Shrinkage of a NIPA gel as a function of monomer concentration. η is the ratio between the diameter of a warm disk and the diameter of the same disk when it is cold [2].

12 3 EXPERIMENTS 11 Figure 15: Making of NEP disks. High ( 30%) and low ( 10%) monomer concentration solution are mixed in a programmable mixer and injected between two plates. We obtain a disk with internal radial gradient in monomer concentration. On the right activation of the disk in a hot bath of temperature T > T c is shown. It shrinks differentially and the result is a 3D configuration [9]. Disks were then placed in a hot bath which induced shrinkage. Demonstration of shrinking can be seen in Ref. [10]. A radially decreasing monomer concentration produced a dome like shape because of a positive Gaussian curvature. Inverted concentration profile produced azimuthally oscillating shape with negative Gaussian curvature (Fig. 16). Experimenting with different thicknesses and monomer concentration leads to different shapes of disks. We see some examples in Fig. 17. In the same Figure we see that the initial shape is not restricted to disks; we can use tubes too [9]. Figure 16: Shaping of NEPs. A radially decreasing monomer concentration shown with red line prescribes a positive Gaussian curvature on the disk which results in a dome shape (lower image). Inverted concentration profile shown in blue line leads to an azimuthally oscillating shape (upper image). The initial thickness of disks is 0.5 mm [9].

13 4 CONCLUSION 12 Figure 17: Different structures made with NIPA gel. (a) A thick sheet (t = 0.75 mm) adopts a configuration of three waves. (b) If we take a thinner sheet (t = 0.3 mm) with larger gradient of monomer concentration, we get a configuration with two generation of waves. (c) Positive Gaussian curvature. (d) Combination of positive and negative Gaussian curvature leads us to a sombrero-like structure. (e) Axially symmetric metrics applied to a cylindrical sheet. Depending on the sheet thickness and metric profile we obtain tubes with (f) two, (g) four and (h) six waves [9]. 4 Conclusion In the seminar we have showed that non-euclidean plates are complex shapes produced by symmetry breaking. We considered only buckling but NEPs can be created by wrinkling as well. The physics behind the symmetry breaking is not simple. But the main challenge is to create NEPs from different materials. In the future new materials will be explored that produce NEPs which react to different external stimuli, such as light, ph, glucose level and other chemical signals. Such new NEPs would increase their applicative potential. They can be used whenever other production techniques for complex shapes are difficult. If we have the right material, creating desired shape is easy. One direction for NEPs is development of living tissue.

14 5 REFERENCES 13 5 References [1] D. W. Henderson and D. Taimina, The Mathematical Inteligencer 23, 17 (2001). [2] E. Sharon and E. Efrati, Soft Matter 6, 5693 (2010). [3] M. Marder, R. D. Deegan and E. Sharon, Phys. Today 6, 33 (2007). [4] E. Sharon, M. Marder and H. L. Swinney, American Scientist 92, 254 (2004). [5] E. Sharon, B. Roman and H. L. Swinney, Phys. Rev. E 75, (2007). [6] E. Sharon, B. Roman, M. Marder, G. Shin and H. L. Swinney, Nature 419, 579 (2002). [7] J. Dervaux and M. Ben Amar, Phys. Rev. Lett. 101, (2008). [8] E. Cerda and L. Mahadevan, Phys. Rev. Lett. 90, (2003). [9] Y. Klein, E. Efrati and E. Sharon, Science 315, 1116 (2007). [10] ( ).