Materials analogue of zero-stiffness structures Arun Kumar a ; Anandh Subramaniam a a

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1 This article was downloaded by: [Indian Institute of Technology] On: 14 June 2011 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Philosophical Magazine Letters Publication details, including instructions for authors and subscription information: Materials analogue of zero-stiffness structures Arun Kumar a ; Anandh Subramaniam a a Department of Materials Science and Engineering, Indian Institute of Technology, Kanpur , India First published on: 08 February 2011 To cite this Article Kumar, Arun and Subramaniam, Anandh(2011) 'Materials analogue of zero-stiffness structures', Philosophical Magazine Letters, 91: 4, , First published on: 08 February 2011 (ifirst) To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Philosophical Magazine Letters Vol. 91, No. 4, April 2011, Materials analogue of zero-stiffness structures Arun Kumar and Anandh Subramaniam* Department of Materials Science and Engineering, Indian Institute of Technology, Kanpur , India (Received 31 July 2010; final version received 5 January 2011) Anglepoise lamps and certain tensegrities are examples of zero-stiffness structures. These structures are in a state of neutral equilibrium with respect to changes in configuration of the system. Using Eshelby s example of an edge dislocation in a thin plate that can bend, we report the discovery of a non-trivial new class of material structures as an analogue to zerostiffness structures. For extended positions of the edge dislocation in these structures, the dislocation experiences a zero image force. Salient features of these material structures along with the key differences from conventional zero-stiffness structures are pointed out. Keywords: zero-stiffness structures; edge dislocation; finite-element method 1. Introduction Anglepoise lamps and certain other spring and lever balancing mechanisms, along with some tensegrities, have been identified as zero-stiffness structures [1 3]. The number of such distinct structures known so far remains small [4]. These structures are in equilibrium along a continuous path of configurations and thus exhibit mechanism-like properties (i.e. unlike usual structures that do not undergo large deformations, these can exist in a wide range of configurations that are equal in energy) [3]. Structures with negative stiffness have also been reported [5,6] and, in this context, a composite of materials with negative and positive stiffness can result in a material of zero stiffness [7 10]. Negative-stiffness elements (buckling beams), in combination with those having positive stiffness, have been used to achieve quasi-zero-stiffness structures [11]. In this letter, we try to answer the following questions: (i) Are there special circumstances wherein the image force can be zero for extended positions of an edge dislocation? (ii) Can non-trivial cases of zero-stiffness material structures (ZSMS) that are analogues of zero-stiffness structures exist? (iii) How are these new classes of material structures similar to the usual zero-stiffness structures and what are the key differences? One can imagine a myriad of situations wherein a material can exist in a series of configurations that are identical in energy; e.g. a vacancy at one of the many lattice *Corresponding author. anandh@iitk.ac.in ISSN print/issn online ß 2011 Taylor & Francis DOI: /

3 Philosophical Magazine Letters 273 positions. These represent trivial cases and the challenge is to find an example where the energy might be expected to change with configuration, but does not do so. Towards this end, let us consider a thin plate of a material (say aluminium), which bends in the presence of an edge dislocation (the Eshelby bend [12]). This problem itself is very old and has been considered before from experimental [13] and theoretical points of view [12,14,15]. When one is interested in solving a broader class of dislocation-related problems, it is seen that a numerical technique such as the finite-element method (FEM) is better suited to the task as compared to purely analytical techniques [16,17], which can often get very cumbersome (see, e.g. the comment by Eshelby in [12]). Additionally, it has been shown that standard analytical formulae prove inadequate in the case of nanocrystals [17]. Keeping this in mind, the FEM has been used in the current work to simulate an edge dislocation in a thin plate. 2. Finite-element methodology A brief outline of the simulation methodology is presented here, the details of which can be found elsewhere [18]. An edge dislocation in aluminium (a 0 ¼ 4.04 A, slip system: h110i{111}, Burgers vector (b) ¼ p 2a 0 /2 ¼ 2.86 A, G ¼ GPa, and ¼ [19]) is simulated by imposing stress-free (eigen-) strains corresponding to the introduction of a half-plane of atoms as in Figure 1. Isotropic plane strain conditions are considered and it is assumed that bulk material properties can be used at the length scales of the simulation. Using standard ABAQUS software, the total energy of the system, along with that of the deformed configuration, is computed for various positions of the edge dislocation (varying x in Figure 1) along the domain. 3. Results and discussion If the length L of the plate is sufficiently small (along with the thickness), then the energy of the dislocation will decrease as the dislocation is positioned closer to a free surface (the free lateral surface will be considered in the current work). This reduction in energy (as shown in Figure 2a) implies that the dislocation is attracted towards the free surface and is the origin of the image force (the gradient of the energy distance curve gives the image force). Now, if the domain is free to bend (model shown in Figure 1b), the energy landscape completely changes and becomes nearly flat (Figure 2b). This implies that as the system goes through a series of configurations (Figure 3) there is virtually no change in energy (within a numerical accuracy of less than 0.1%) and hence, the system behaves like a ZSMS. The range of dislocation positions over which the system shows zero stiffness is also marked in Figure 2. Two noteworthy points can be seen from a comparison of the two plots in Figure 2: (i) the domain that can bend has a lower energy (34% for the dislocation in the centre of the plate) along with a modified state of stress and (ii) the energy landscape has become flat. The first point has been noted before [12] and is obvious, but the second point is surprising and implies that the effect of the free surface has vanished. This aspect is also reflected in by the restoration of left right mirror symmetry in the x stress contour plot close to the position of the dislocation

4 274 A. Kumar and A. Subramaniam Figure 1. Finite-element model with boundary conditions (schematic not to scale) to simulate an edge dislocation in a thin plate: (a) plate constrained from bending and (b) plate free to bend. In (a) the entire bottom surface (AC) is constrained in the Y-direction and additionally the point B is constrained in the X-direction. In (b) only point B is constrained in the X- and Y-directions. Figure 2. Plot of energy of the system as a function of the position of the dislocation in the domain: (a) for the model considered in Figure 1a (constrained domain) and (b) for the model considered in Figure 1b. Scaled-up view (inset) shows that, even between 0b and 10b, curve (a) is sloping down.

5 Philosophical Magazine Letters 275 Figure 3. Series of configurations as the dislocation is positioned at various places in the plate (model in Figure 1b): (a) x ¼ 0b, (b) x ¼ 10b, (c) x ¼ 20b and (d) x ¼ 30b. Contours correspond to x stresses and the deformation scale factor is set to three for better visualisation. Figure 4. Plot of x stress contours for a dislocation at 30b from the centre of the domain, for the model shown in: (a) Figure 1a and (b) Figure 1b. Dimensions correspond to the undistorted domain. Deformation scale factor is one. (comparison shown in Figure 4), which makes the domain behave as an infinite body in the X-direction. Some points worthy of comment are as follows. (i) The system does not exhibit zero stiffness when the dislocation is positioned very close to the free surface. This is similar to the range of configurations for some zero-stiffness structures, exhibiting zero stiffness; for example, in tensegrities the zero stiffness property exists only for a limited (albeit wide) range of configurations [20]. (ii) The range of configurations is not truly continuous as in the case of a zero-stiffness structure as the dislocation has a minimum change in position corresponding to the Burgers vector. (iii) The dislocation resides in a local energy minimum and has to overcome a Peierls barrier to cross over to the next position (a Burgers vector apart). These Peierls oscillations (maxima and minima of energy) have not been considered

6 276 A. Kumar and A. Subramaniam here owing to the previous point. (iv) If the plate is very long then there will be a region in the centre of the domain that will exhibit a flat energy landscape of a trivial kind (as the free surfaces are far away). (v) The bending is significant only when the thickness of the plate is on the nanoscale. This is unlike zero-stiffness structures, which are on the macroscale. (vi) Ignoring the core energy does not affect our conclusions, as the core energy is a constant additive term to the energy, except for a position of the dislocation very close to the surface, i.e. few Burgers vectors, which do not affect the slope of the curves in Figure 2. In any case, for positions of the dislocation very close to the surface, the material does not display zero stiffness. (vii) The range of edge dislocation positions over which the system displays zero stiffness can be increased by using a thin harder material in conjunction with the original plate [21]. (viii) Since zero stiffness is being considered with respect to a range of configurations of the system, these ZSMS are inherently different from the zero-stiffness composites. (ix) The thickness of the plate plays an important role in ZSMS. Thicker plates, as compared to the 20b thick plate considered so far, have a reduced configurational range over which zero stiffness behaviour is seen, while thinner plates represent trivial cases (i.e. for most configurations the dislocation does not experience an image force) [21]. (x) To emphasise, the phrase zero stiffness has been used in the precise sense as discussed before and not in the sense of resistance to deformations to external loads. Further points on this comment follow later. (xi) In this first report, we have considered some aspects leading to ZSMS. Future work should be directed towards gaining a physical understanding of the results and the general principles involved, along with addressing issues like the effects of screw and mixed dislocations in thin finite plates, and of the geometry of the domain, etc. Typically, when describing a property, the geometrical contribution is separated from the material contribution with these two aspects remaining distinct. For example, elastic modulus is a material property and stiffness is a combination of geometry along with the elastic modulus. Exceptions to this rule do exist, such as plane strain fracture toughness, wherein the geometrical role is included in the definition of the material property [22]. The case of a plate with an edge dislocation on the one hand has a material side to it as the defect is crystallographic in origin; on the other hand it also has a bent geometry which undergoes a series of configurations (i.e. configurations with respect to the position of the dislocation and the resultant bending of the plate). Hence, this new class of material structures falls right between materials and structures. A pertinent question that can be asked at this juncture (and which arises from point (x) above) is how does a beam with a dislocation compare with a beam without a dislocation with respect to bending deformations? To answer this question, two models of cantilever beams shown in Figures 5a and b are compared for deformations (D) plotted against force/energy of the system. It is seen that the model with the dislocation can accommodate a range of D values without change in energy as compared to the beam without the dislocation (Figure 5c). The energy of the simple cantilever beam (without the dislocation) increases with vertical displacement of point P (Figure 5c). A point to be noted is that the dislocation can move from one position to another only if the shear stresses can overcome the Peierls stresses [23]. An accurate determination of the Peierls stress is the ongoing effort of many research groups

7 Philosophical Magazine Letters 277 Figure 5. (a) Model of a beam with a dislocation, (b) beam without a dislocation and (c) variation of the energy and force (at point P) as a function of vertical displacement D of point P in the downward direction for the two systems (models (a) and (b)). In the models, the surface DE is constrained in the X- and Y-directions and against rotations. The forces plotted refer to the force required to cause displacement of point P. It is to be noted that energies and forces are plotted on two different scales.

8 278 A. Kumar and A. Subramaniam (the reader may consult Wang [24], Lubarda and Markenscoff [25]), and it has even been suggested that the stress might be negligible for pure metals [26]. In the case of the beam with a dislocation, the deformations (D) arise because of the presence of the dislocation and should be classified as plastic deformation (as it can exist in the absence of external loading). Additionally, these plastic deformations can be termed (curiously) as reversible plastic deformations due to elasticity. The term reversible implies that the system is in equilibrium at positions of the dislocation under consideration. Shear stresses of opposite sign can drive the dislocation in the opposite direction (without change in energy of the system in the zero-stiffness regime). The term elastic implies that these deformations arise due to elastic stress fields of the edge dislocation. It is to be noted that usual plastic deformations by slip occur by dislocations leaving the crystal surface or the grain boundary (leading to the formation of a step) and in that case, the energy of the system changes with continued plastic deformation (this process is not reversible because opposite stresses of the same magnitude cannot create a dislocation). 4. Summary It has been shown that for extended positions of a dislocation in a thin finite plate (of specific geometry) that can bend (the Eshelby bend) the system is in a state of neutral equilibrium (i.e. does not change its energy with respect to the position of the dislocation and thus implying zero image force). This serves as the first (non-trivial) example of a materials analogue of zero-stiffness structures, which we term as ZSMS. Acknowledgements The authors acknowledge financial assistance from the Department of Science and Technology, Government of India (SR/S3/ME/048/2006). They thank Professor Sumit Basu and Professor Madhav Ranganathan for invaluable discussions. References [1] G. Carwardine, Improvements in Elastic Force Mechanisms, UK Patent , filed 21 March 1931, granted 22 August [2] M.J. French and M.B. Widden, Proc. Inst. Mech. Eng. C 214 (2000) p.501. [3] M. Schenk, S.D. Guest and J.L. Herder, Int. J. Solids Struct. 44 (2007) p [4] T. Tarnai, Int. J. Mech. Sci. 45 (2003) p.425. [5] C.M. Lee, V.N. Goverdovskiy and A.I. Temnikov, J. Sound Vib. 302 (2007) p.865. [6] Y. Estrin, A.V. Dyskin, E. Pasternak, S. Schaare, S. Stanchits and A.J. Kanel-Belov, Scr. Mater. 50 (2004) p.291. [7] R.S. Lakes, Phys. Rev. Lett. 86 (2001) p [8] W.J. Drugan, Phys. Rev. Lett. 98 (2007) p [9] Y.C. Wang and R.S. Lakes, Am. J. Phys. 72 (2004) p.40. [10] R.S. Lakes, T. Lee, A. Bersie and Y.C. Wang, Nature 410 (2001) p.565. [11] D.L. Platus, Proc. SPIE Int. Soc. Opt. Eng (1999) p.98. [12] J.D. Eshelby and A.N. Stroh, Phil. Mag. 42 (1951) p.1401.

9 Philosophical Magazine Letters 279 [13] S. Amelinckx, The direct observation of dislocations, in Solid State Physics, F. Seitz and D. Turnbull, eds., Academic Press, London, 1964, p.405. [14] F. Kroupa, Appl. Math. 4 (1959) p.239. [15] V.L. Indenbom, Sov. Phys.-Dokl. 4 (1960) p [16] T. Belytschko and R. Gracie, Int. J. Plast. 23 (2007) p [17] P. Khanikar, A. Kumar and A. Subramaniam, Phil. Mag. 91 (2011) p.730. [18] P. Khanikar and A. Subramaniam, J. Nano Res. 10 (2010) p.93. [19] E.A. Brandes, Smithells Metals Reference Book, 6th ed., Butterworths, London, [20] M. Schenk, J.L. Herder and S.D. Guest, in Proceedings of DETC/CIE 2006, ASME, Philadelphia, [21] Unpublished work. [22] D. Broek, Elementary Engineering Fracture Mechanics, Kluwar Academic Publishers, Dordrecht, [23] D. Hull and D.J. Bacon, Introduction to Dislocations, Butter-Heinemann, Oxford, [24] J.N. Wang, Mater. Sci. Eng. A 206 (1996) p.259. [25] V.A. Lubarda and X. Markenscoff, Arch. Appl. Mech. 77 (2007) p.147. [26] J.J. Gilman, Phil. Mag. 87 (2007) p.5601.