Gecko and many insects (Fig. 1) have evolved fibrillar structures

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1 Shape insensitive optimal adhesion o nanoscale ibrillar structures Huajian Gao* and Haimin Yao Max Planck Institute or Metals Research, Heisenbergstrasse 3, D Stuttgart, Germany Edited by Jan D. Achenbach, Northwestern University, Evanston, IL, and approved April 8, 2004 (received or review February 3, 2004) Gecko and many insects have adopted nanoscale ibrillar structures on their eet as adhesion devices. Here, we consider adhesion between a single iber and a substrate by van der Waals or electrostatic interactions. For a given contact area A, the theoretical pull-o orce o the iber is th A where th is the theoretical strength o adhesion. We show that it is possible to design an optimal shape o the tip o the iber to achieve the theoretical pull-o orce. However, such design tends to be unreliable at the macroscopic scale because the pull-o orce is sensitive to small variations in the tip shape. We ind that a robust design o shape-insensitive optimal adhesion becomes possible only when the diameter o the iber is reduced to length scales on the order o 100 nm. In general, optimal adhesion could be achieved by a combination o size reduction and shape optimization. The smaller the size, the less important the shape. At large contact sizes, optimal adhesion could still be achieved i the shape can be manuactured to a suiciently high precision. The robust design o optimal adhesion at nanoscale provides a plausible explanation or the convergent evolution o hairy attachment systems in biology. Gecko and many insects (Fig. 1) have evolved ibrillar structures on their eet to achieve extraordinary adhesion on vertical walls and ceilings. These hairy biological attachment systems consist o inely structured protruding hairs with size ranging rom a ew hundred nanometers to a ew micrometers, depending on the animal species. The density o surace hairs increases with the body weight o the animal, and gecko has the highest density among all animal species that have been studied (1). Dierent mechanisms (2, 3) have been proposed in the past or biological attachment structures, but only until recently has strong evidence been presented (4, 5) that molecular adhesion by van der Waals interaction plays a dominant role in the attachment by ibrillar structures in biology. This discovery may appear somewhat surprising because it takes a much greater orce to pull a gecko away rom a ceiling than removing a human hand rom a table, even though the same van der Waals orce is expected to exist in both situations. A question thus arises: What determines the adhesion strength o a iber in adhesive contact with a substrate? The chemical nature o materials cannot explain why the same van der Waals orce results in strong adhesion in geckos but not in humans. Apparently, nature has evolved mechanisms to use weak van der Waals orces in animal species or which adhesion is crucial or survival. Dry adhesion between solid objects has been an active topic o research in contact mechanics. The Johnson Kendal Robert (JKR) (6) and Derjaguin Muller Toporov models (7) are particularly useul in modeling adhesive contact at two opposite extremes o a dimensionless parameter representing the ratio between the elastic deormation o the contacting suraces and the eective range o surace interaction orces. The transition between JKR and Derjaguin Muller Toporov theories is described by the Maugis Dugdale model (8) based on a cohesive description o the surace orces. More recent studies have expanded these theories to coupled normal and shear loading (9, 10) and to viscoelastic materials (11). Various mechanical models have been developed to model speciic iber-array structures (12, 13), and signiicant progress has been made in using the JKR model to show that the hairy attachment systems, in which a macroscopic contact area is split into Fig. 1. The nanoscale ibrillar structures in animals with hairy attachment pads. The density o surace hairs increases with the body weight o animal, and gecko has the highest density among all animal species. (Photograph courtesy o S. Gorb, Max Planck Institute or Metals Research. Adapted rom re. 14.) many smaller contact patches, could greatly enhance the adhesive strength (5, 14). However, little is known about why the characteristic size o the ibrillar ultrastructure o bioattachment systems alls in a narrow range between a ew hundred nanometers and a ew micrometers. In addition, no eort has been made to address the optimal conditions under which the theoretical pull-o orce th A, where th is the theoretical strength o molecular interaction and A is the cross-sectional area o the iber, can be achieved. A thorough understanding o these issues could also be o value to structural engineering and artiicial materials design (15). In conventional engineering, i two elastic bodies (Fig. 2a) are joined together by adhesion and then subject to an externally applied load, stress concentration is expected to occur near the edge o the joint (Fig. 2b). As the load increases, the intensity o stress concentration ultimately reaches a critical level to drive a crack to propagate and break the joint. Under this circumstance, the material in the joint is not being used most eiciently because only a small raction o material is highly stressed at any instant o time, and ailure occurs by incremental crack propagation. On the other hand, it is theoretically always possible to design the shape o the contact bodies so that at the instant o pull-o, the stress is uniormly distributed over the contact region with magnitude equal to the theoretical adhesion strength th (Fig. 2c). Assuming that the molecular adhesion orce between two contacting bodies is determined by the separation between them, a uniorm stress ield at pull-o requires a uniorm separation o the two contacting bodies over the entire contact region. In other words, the optimal adhesion occurs i the contacting suraces perectly conorm to each other at the critical moment o pull-o. The tip geometry o the iber, which gives rise to a uniorm stress This paper was submitted directly (Track II) to the PNAS oice. Abbreviation: JKR, Johnson Kendal Robert. *To whom correspondence should be addressed. hjgao@m.mpg.de. Stress concentration can still exist near the edge o the contact region beore pull-o occurs by The National Academy o Sciences o the USA APPLIED PHYSICAL SCIENCES cgi doi pnas PNAS May 25, 2004 vol. 101 no

2 Fig. 2. Shape sensitivity o the pull-o orce. (a) Two structures are bonded together and subject to an external load. (b) Stress concentration occurs at the edge and could initiate a crack to propagate and break the joint. The springs represent the molecular interaction orces between the contacting suraces. (c) The optimum distribution o stress at pull-o is a uniorm tensile stress equal to the theoretical strength o adhesion th. (d) The optimal shape proiles o a rigid iber in rictionless or nonslip contact with an elastic substrate. The solid line curve is the optimal shape proile or rictionless contact, and the dotted line curve is the optimal shape proile or the nonslip contact. Here the Poisson ratio is taken to be 0.3. ield at pull-o rom a substrate, can thus be calculated by the theory o elasticity (see Appendix). Why hasn t such an optimal shape been evolved in nature or used in engineering? One problem is that the optimal shape described here is sensitive to small variations in the shape o the tip o the iber. In act, only a small variation in geometry is needed to alter the tip shape to induce a singular stress ield like that near a crack. Here, in contrast to the optimal shape, we also introduce a deinition or the singular shape such that the stress distribution is equivalent to the singular stress distribution associated with a crack external to the connecting area A at pull-o. The dierence between the optimal shape and the singular shape is oten very small. This can be illustrated by considering the case o a rigid iber in contact with an elastic substrate. I riction is neglected, the singular shape is just a lat-ended iber that can be described by the shape unction crack r 0. r R [1] In contrast, the optimal shape giving rise to the theoretical pull-o orce is (see Appendix) th r thr 4 E(r R) 2, (r R) [2] E* where E is the complete elliptic integral o the second kind, R is the radius o the iber, and E* is deined as E* E (1 2 ), E being Young s modulus and Poisson s ratio o the substrate. Normally, the ratio th E* is quite small (1 2%) and so is the dierence between crack (r) and th (r). I riction is included with no slip along the interace between the iber and substrate, the optimal shape o Eq. 2 would be modiied by an additional term o th R 1 2 E* 1 r2 R 2 1. A derivation o these solutions is given in Appendix. The optimal shape or the case o nonslip contact diers only slightly rom that or the rictionless case, as plotted in Fig. 2d. More general riction cases are ound to lie in between these two limits. These results show that the eect o shear plays only a minor role in the optimal shape problem. The pull-o orce associated with the singular shape can be calculated according to the Griith condition or crack initiation (16) as P crack and that associated with the optimal shape is 2 R 8 E* 1 2, [3] R P th R 2 th, [4] where is the work o adhesion. For van der Waals interaction, usually ranges between 10 and 50 mj m 2. We shall assume 10 mj m 2 in this article. Taking the eective interaction range o van der Waals orce to be 0.5 nm, we estimate th to be 20 MPa. In general, it is diicult to determine the singular and optimal shapes in closed orm solutions, although they can be calculated by numerical methods such as the inite element method. In the case o an elastic iber in contact with a rigid substrate (a more realistic model o bioattachment systems), the lat-ended iber becomes the optimal shape. In this case, the singular shape cannot be determined in closed orm. Fortunately, detailed solutions o the singular and optimal shapes are not necessary to calculate the associated pull-o orces. In act, it can be shown that the same results in Eqs. 3 and 4 apply to an elastic iber on a rigid substrate when E* is interpreted as the modulus o the iber. More generally, or an elastic iber with Young s modulus E and Poisson s ratio in contact with an elastic substrate with Young s modulus E s and Poisson s ratio s, the pull-o orce associated with the singular shape would still be given by Eq. 3 i E* is properly generalized according to cgi doi pnas Gao and Yao

3 Fig. 3. Saturation o adhesion strength or the singular shape o a iber. (a) The variation o the normalized pull-o orce with the nondimensional parameter (E* R 2 th ) 1/2.(b) The distribution o the adhesive stress along the contact radius or dierent iber radii. (c) The shape sensitivity o adhesion strength. As E* R 2 th increases, or the iber radius decreases, the dierence between the adhesive strength o the singular shape and that o the optimal shape decreases, eventually vanishing at a critical length on the order o 100 nm. 1 E* s. [5] E E s The pull-o orce associated with the optimal shape is always deined by Eq. 4. Fig. 3a plots the normalized adhesive strength P ( R 2 th )asa unction o the nondimensional parameter (E* R 2 th ) 1/2. The predictions rom Eqs. 3 and 4 are plotted as two straight solid lines. We have also developed a numerical method to determine the pull-o orce associated with the singular shape o a rigid iber on an elastic substrate with molecular interaction characterized by the Lennard Jones potential (17) h h h 9, [6] where is the equilibrium distance at zero interaction and h is the separation between two interactive suraces. The numerically calculated values o the pull-o orce or the singular shape, shown as illed squares in Fig. 3a, agree with the prediction o Eq. 3 based on Griith condition or large iber sizes but asymptotically approach the theoretical adhesion strength as the iber size decreases. This trend occurs because the theoretical strength is the upper limit o adhesion. The Griith condition assumes that ailure always occurs by crack propagation, i.e., a crack is assumed to nucleate at the edge o the contact and propagates to break the joint. This assumption breaks down or very small ibers. The results o Fig. 3a show that the theoretical adhesion strength can be achieved in two ways; the irst is by accurately adopting the optimal shape o the tip o the iber, and the second is by reducing the diameter o the iber. I theoretical strength is achieved, ailure occurs no longer by crack propagation but rather by a uniorm detachment over the entire contact region. For very small sizes, the condition or crack propagation cannot be satisied beore the theoretical strength is reached. The calculated pull-o orce associated with the singular shape based on the Lennard Jones potential shows a smooth transition between two ailure modes, crack propagation at large sizes and uniorm detachment at small sizes. Fig. 3b shows the distribution o adhesive traction at pull-o or the singular shape with dierent iber sizes. Note that the adhesive traction becomes more and more uniorm as the iber size decreases, eventually becoming uniorm at a critical size. At macroscopic sizes, a small variation in the tip shape o the iber results in large changes in pull-o orce. For example, taking 10 mj m 2, th 20 MPa, and E* 1 GPa (the case o a keratinous iber in contact with a rigid substrate) and iber radius equal to 1 mm, the pull-o orce or the optimal shape is estimated to be 62.8 N and that or the singular shape is only 0.5 N. In other words, small variations in geometry result in large changes in pull-o orce. The design o optimal shape, although theoretically easible, is unrealizable in practice at the macroscopic scale. Fig. 3c shows that the dierence between the adhesive strength o the optimal shape and that o the singular shape decreases as the size o the iber is reduced. At the critical size R cr 8 E* 2, [7] the strength o the singular shape predicted by Eq. 3 becomes equal to that o the optimal shape predicted by Eq. 4. Taking 10 mj m 2, th 20 MPa, and E* 1 GPa, we estimate R cr 64 nm. From these results we can conclude that the sensitivity o adhesion strength to tip geometry o the iber decreases as the iber diameter is reduced, and a robust design o optimal adhesion becomes possible around a critical size at which the pull-o orce is no longer sensitive to variations in tip geometry. This length scale is 100 nm and indicates that the nanometer size o the ibrillar ultrastructure (spatula) o gecko and many insects may be the result o optimization or reliable and optimal adhesion. It has recently been shown (18, 19) that the nanometer size o mineral particles in bone-like biological materials may have been selected to ensure optimum racture strength and maximum tolerance o crack-like laws. The present study shows that the nanoscale dimension may play a crucial role in achieving reliable adhesion in a ibrillar structure. The optimal adhesion strength described here can be compared with the JKR solution or a iber having radius o curvature R at the tip in contact with a lat surace. The ratio o the optimal solution to that o JKR is P th P JKR 2 3 th R th. [8] For 10 mj m 2, th 20 MPa, this ratio is calculated to be as large as 10 6 or a iber with a radius o 1 mm and 10 3 or that 1 m. Thereore, a huge magniication o pull-o orce can be achieved by modiication o the tip shape (shape optimization). Depending on animal species and convenience, optimal adhesion could be achieved by a combination o size reduction and shape optimization. The smaller the size, the less important the shape. At large sizes, the optimal adhesion could still be achieved i the shape could be manuactured to suiciently high precision. APPLIED PHYSICAL SCIENCES Gao and Yao PNAS May 25, 2004 vol. 101 no

4 Fig. 4. Variation o adhesive orce as the optimally shaped rigid iber approaches and withdraws rom an elastic substrate. (a) Schematic illustration o the problem. (b e) Dierent behaviors o ibers with radius equal to R 0.13E* 2 th (b), R 1.3E* 2 th (c), R 11.3E* 2 th (d), and R 13.7E* 2 th (e). is the separation between the substrate and the center o the tip surace o the iber, and is the equilibrium distance at zero interaction. The discussion so ar has been ocused on how to achieve the theoretical pull-o orce by size reduction or shape optimization. It is also interesting to investigate the variation in adhesive orce as a iber approaches or withdraws rom a substrate. For this purpose, we have calculated the quasi-static process o an optimally shaped (Eq. 2) rigid iber approaches an elastic substrate (Fig. 4a). O special interest is how the interaction orce changes as the iber size decreases toward the critical size or shape insensitivity. Figs. 4 b e display the variation o the normalized adhesive orce P R 2 th asaunctiono, where is the separation between the substrate and the center o the tip surace o the iber. Fibers o dierent sizes approaching and withdrawing rom a substrate are considered. Fig. 4b shows the behavior o a iber with radius R 0.13 E* 2 th (3.2 nm or 10 mj m 2, th 20 MPa and E* 1 GPa). For this iber, the approach and withdrawal ollow the same path with no hysteresis. As the iber approaches the substrate, the adhesive orce is always attractive and the iber is drawn to the substrate until an equilibrium contact position is reached with zero interactive orce. As the iber is withdrawn rom the equilibrium position, a maximum adhesive orce exists equal to the theoretical pull-o orce, conirming that the tip shape is indeed optimized. 2 Fig. 4c shows the behavior o a larger iber with R 1.3E* th (32.5 nm or 10 mj m 2, th 20 MPa, and E* 1 GPa). Now the approach and withdrawal ollow dierent paths, giving rise to a hysteresis. During approaching, the adhesive orce is always positive and the iber is drawn toward the substrate until equilibrium. The approaching process is unstable. At a critical distance rom substrate, there is a sudden increase in adhesive orce and the iber jerks toward the substrate. During withdrawal, the adhesive orce increases until reaching the theoretical pull-o orce and then abruptly drops to a small value. For a even larger iber with R 13.7E* 2 th (342.5 nm or 10 mj m 2, th 20 MPa, and E* 1 GPa) shown in Fig. 4e, the theoretical adhesive strength cannot be achieved spontaneously. As the iber approaches the substrate, the adhesive orce vanishes at two equilibrium positions. Only ater the second equilibrium position can the iber achieve the theoretical pull-o orce on withdrawal. I the iber is pulled back near the irst equilibrium position, ull contact between the iber and substrate has not been achieved and the withdrawal goes back along the approaching path BA with a tiny hysteresis and small pull-o orce. Between the two equilibrium positions, the adhesive orce is actually negative, implying that an externally applied pressure is needed to reach ull contact. In this case, the theoretical pull-o orce cannot be achieved unless ull contact is achieved by pressing the iber hard enough against the substrate beore withdrawal. The preceding analysis indicates that a threshold radius exists or the spontaneous approaching o substrate, which is calculated to be R spon 11.3E* 2 th or a rigid iber interacting with an elastic substrate. The behavior o such a iber at threshold radius is shown in Fig. 4d. Taking 10 mj m 2, E* 1 GPa, and th 20 MPa, R spon is estimated to be nm. Fibers with radius smaller than R spon can spontaneously achieve ull contact without externally applied pressure and attain the theoretical pull-o orce. The same concept is expected to hold or the more general case o an optimally shaped elastic iber approaching an elastic substrate cgi doi pnas Gao and Yao

5 Fig. 5. Methodology or determining the optimal and singular shapes or dry adhesion between two solid objects. (a) Deormed coniguration o the optimal shape at pull-o. At this state, a uniorm stress equal to the theoretical adhesion strength th is distributed over the contact region. (b) Reverse o the pull-o coniguration by superposition o a uniorm pressure equal to th over the contact region, which results in the undeormed coniguration o the optimal shape depicted in c. (d) Deormed coniguration o the singular shape at pull-o. At this state, a singular stress equivalent to that associated with a crack external to the contact region is assumed to be distributed over the contact interace. (e) Reverse o the pull-o coniguration by superposition o a singular pressure distribution over the contact region, giving the undeormed coniguration o the singular shape depicted in. APPLIED PHYSICAL SCIENCES The concepts described in this article are not just limited to adhesion o ibrillar structures. The concept o a critical length or robust optimal adhesion can be generalized to a pair o arbitrary contacting bodies through the ollowing deinitions and theorems. Deinition 1. The optimal shape o two objects in adhesive contact over a surace area A is deined as such that the stress distribution is uniorm and equal to the theoretical strength th o adhesion at pull-o. The optimal shape can be calculated rom the coniguration at pull-o by elasticity. I desired, the optimal shape can be determined by numerical methods such as the ine element method. Deinition 2. The singular shape o two contacting objects over a surace area A is deined as such that the stress distribution is equivalent to the singular stress distribution associated with a crack external to the connecting area A at pull-o. The singular shape can be calculated rom the coniguration at pull-o by elasticity. Theorem 1. As the size o the contact area decreases, the adhesive strength associated with the singular shape increases until it reaches the theoretical strength o adhesion near the critical size deined by Eq. 7. At the critical size, the strength o adhesion is no longer sensitive to the local geometry o the contacting objects. The optimal shape and the singular shape give rise to the same pull-o orce. Theorem 2. Optimal adhesion at theoretical strength could be achieved by a combination o size reduction and shape optimization. The smaller the size, the less important the shape optimization becomes. At large contact sizes, optimal adhesion could still be achieved i the shape can be manuactured to a suiciently high precision. Because these concepts are based on the classical subjects o elasticity and linear elastic racture mechanics, or which the basic existence and uniqueness theorems have long been proven (20), we do not ind it necessary to go beyond setting up the boundary value problems or the singular and optimal shapes. Fig. 5 a c shows schematically the linear elastic boundary value problems or the optimal shape and Fig. 5 d shows those or the singular shape. The conclusions o this article are generally applicable as long as the elastic boundary problems depicted in Fig. 5 can be solved to obtain the optimal and singular shapes. Further discussions are made in Appendix, where we also show how to obtain the analytical solution or the optimal shape in the special case o a rigid iber in contact with an elastic substrate. The concepts developed in this article should be o general value in the understanding o biological attachment systems and the design o engineering systems. In the present study, we have implicitly assumed that the ibrillar structure is sot enough to achieve conormal contact with a rough surace. The eect o surace roughness should be more systematically investigated in the uture. Also, we have not considered many important aspects o optimal design or dry adhesion. Open questions still exist that should be subjected to urther investigation. For example, the more general cases involving coupled tension, shear, and bending loads on a ibrillar structure can be investigated under similar questions with respect to optimized adhesion or riction. It will be especially interesting to consider these issues in conjunction with the eects o the aspect ratio, distribution, and collective behaviors o a ibrillar structure. Appendix Fig. 5a shows that the optimal shape o a iber is deined as such that a uniorm distribution o normal stress equal to the theoretical adhesion strength th is achieved at pull-o. Fig. 5c shows that the substrate is assumed to be lat in its undeormed coniguration. We can use these conditions to calculate the undeormed coniguration o the optimal shape o the iber. To determine the optimal shape, a uniorm pressure with magnitude equal to th is separately applied over the contact region o the two objects which will deorm as depicted in Fig. 5b. Superposing Fig. 5a and Fig. 5b should yield the optimal shape o the iber and the undisturbed lat surace o the substrate (Fig. 5c). Here, in Fig. 5, one o the contacting objects (substrate) has been assumed to be a lat surace, and we ocus on inding the optimal shape o Gao and Yao PNAS May 25, 2004 vol. 101 no

6 the iber. This limitation can be easily removed by generalizing the methodology to two arbitrary contacting objects. We can always use two conorming suraces with a uniorm distribution o theoretical adhesion strength th to determine the undeormed conigurations o the two objects by reverse elasticity. The solution is unique only when the deormed coniguration o the contact interace is prescribed or known. I the deormed coniguration o the conorming interace is not known, we can in principle ind an ininite number o solutions or the optimally shaped contacting suraces which give rise to the theoretical pull-o orce. In the problem depicted in Fig. 5, the deormed coniguration o the substrate is assumed to be a lat surace and this condition should lead to a unique solution or the deormed coniguration o the conorming contact interace and the optimal shape o the iber. Figs. 5 d describes similar relationships that can be used to determine the singular shapes or the contacting objects. We emphasize again that the pull-o orces associated with the optimal and singular shapes given by Eqs. 3 5 do not require solutions to these shapes. A critical step in the methodology to determine the optimal and singular shapes or general adhesive joints is the boundary value problem deined in Fig. 5 b and e which involves compressive loading on the two objects. Assuming the deormation o the objects is small (linear elastic) at pull-o, the existence and uniqueness theorems o elasticity (20) guarantee that the solution can be uniquely determined. For very compliant structures, the condition o small deormation at pull-o may be violated and the methodology may breakdown. In those cases, the two theorems given in the article may cease to be valid. As illustration, in the ollowing text, we show how the optimal shape o a rigid cylindrical iber in contact with an elastic hal-space is determined. In the case o rictionless contact, the normal surace displacement caused by a uniorm normal traction th applied over a circular region o an initially perectly lat hal-space is given by (21) w r 4 thr E r R, r R [A1] E* where E is the complete elliptic integral o the second kind, E* is deined as E* E (1 2 ), E being Young s modulus and being Poisson s ratio o the hal-space. According to Fig. 5a, the optimal adhesion occurs i the contacting suraces should perectly conorm to each other at the critical moment o pull-o. Thereore, the optimal shape or the rigid iber tip is simply given by th r thr 4 E r R 2, r R [A2] E* where a constant has been added such that th (0) 0. Similarly, the optimal shape o a rigid cylindrical iber in nonslip contact with an elastic hal-space can be determined. In this case, the relative tangential displacements along the contact interace are ully constrained and the eects o radial shear tractions over the contact region must be taken into account. The normal surace displacement caused by a uniorm normal traction th with ully constrained tangential displacement over a circular region o an initially perectly lat hal-space is given by w r thr 4 2 E r R 1 E* 1 2 th R 2 E* 1 r2 R 2. r R [A3] Letting the tip shape o the rigid iber conorm to this unction yields the optimal shape E r R 2 th NS r thr E* th R 2 E* 1 r2 R 2 1. r R [A4] where a constant has been added to make th NS (0) 0. Here, NS stands or no slip. Eqs. A2 A4 also apply to the case o an elastic cylindrical iber within the approximation o linear elasticity. This work has been supported by the Max Planck Society, the National Science Foundation o China, and the Chang Jiang Scholar program through Tsinghua University. 1. Scherge, M. & Gorb, S. (2001) Biological Micro and Nano-Tribology (Springer, New York). 2. Gillett, J. D. & Wigglesworth, V. B. (1932) Proc. R. Soc. London Ser. B 111, Stork, N. E. (1980) J. Exp. Biol. 88, Autumn, K., Liang Y. A., Hsieh, S. 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Mater. 2, Griith, A. A. (1921) Philos. Trans. R. Soc. London A 221, Greenwood, J. A. (1997) Proc. R. Soc. London Ser. A 453, Gao, H., Ji, B., Jäger, I. L., Arzt, E. & Fratzl, P. (2003) Proc. Natl. Acad. Sci. USA 100, Gao, H.& Ji, B. (2003) Eng. Fract. Mech. 70, Timoshenko, S. P. & Goodier, J. N. (1985) Theory o Elasticity (McGraw Hill, Auckland). 21. Johnson, K. L. (1985) Contact Mechanics (Cambridge Univ. Press, Cambridge, U.K.) cgi doi pnas Gao and Yao