Evaluating the snappability of bar-joint frameworks
It is well-known that there exist bar-joint frameworks (without continuous flexions) whose physical models can snap between different realizations due to non-destructive elastic deformations of material. We present a method to measure these snapping capability – shortly called snappability – based on the total elastic strain energy of the framework by computing the deformation of all bars using Hook’s law. The presented theoretical results give further connections between shakiness and snapping beside the well-known technique of averaging and deaveraging.
Keywords:Snapping framework, multistability, model flexor, elastic deformation
We consider a framework in the Euclidean -dimensional space which consists of a knot set
and an abstract graph on fixing the combinatorial structure. We denote the edge connecting to with by
with and collect all indices of knots edge-connected to in the knot neighborhood .
Moreover we denote the number of edges in the graph by and assign
By materializing all edges by straight bars and linking them by -joints
In the following we consider the configuration of knots , where denotes the -dimensional coordinate vector of the knot , which together with the graph implies the framework’s realization . In the rigidity community (e.g. connelly_book ()) each edge is assigned with a stress (coefficient) . If in every knot the so-called equilibrium condition
is fulfilled then the -dimensional vector is refereed as
self-stress (or equilibrium stress).
According to Gluck gluck () and Roth roth () the existence of a non-zero self-stress
Shakiness (of order one
Before we plunge in medias res we provide a short review on snapping (also called multistable; cf. goldberg ()) structures. During the last years the interest in these structures has increased due to practical applications (e.g. rafsanjani (); haghpanah (); shang ()).
It is pointed out in stachel_between () that there is a direct connection between shakiness and snapping through the technique of averaging and deaveraging, respectively (cf. (schulze, , page 1604) and ivan ()). The latter allows to construct snapping frameworks in any dimension. Moreover for snapping bipartite frameworks in an explicit result in terms of confocal hyperquadrics is known (cf. (stachel_between, , page 112) under consideration of stachel_palermo ()). Most results are known for the dimension , which are summarized next.
There is a series of papers of Walter Wunderlich on snapping spatial structures
(octaeder, Bennett mechanisms, antiprisms, icosaeder, dodecaeder),
which are reviewed in stachel_wunderlich (). In this context also the paper goldberg ()
should be cited, where buckling polyhedral surfaces and Siamese dipyramids are introduced.
Snapping structures are also related to so-called model flexors
2 Physical model of deformation
First we consider a single bar and apply equal but opposite directed forces to it’s ends pointing outwards/inwards the bar,
which imply a tensile/compression stress leading to an expansion/decrease of the bar.
According to Hooke’s law, which can be applied due to the elastic deformation during the process of snapping,
the tensile/compression stress in a uniform bar equals the product of the modulus of elasticity
where is the deformed length (stressed length) of the bar , while is the original length (unstressed length) of the bar . Note that corresponds to a stretching and to a compression. Taking Eq. (2) into account the well-known relation , where is the cross-sectional area of the bar , can be rewritten as
This force acts on the end points and of the involved bar by the force vectors
where denotes the standard norm.
Moreover, the elastic strain energy
according to (mittemeijer, , page 512), where is the axial stiffness of the bar . As a consequence the total deformation energy of the framework reads as
Expressing in dependence of the knots and the system of partial derivatives
where is the coordinate vector of , equals the condition for . With respect to our physical model the meaning of the real values in Eq. (1) is , which is the so-called force density with respect to the stressed length tibert (). Moreover it should be pointed out that each critical point of the total elastic strain-energy corresponds to a self-stressed framework realization , which is called deformed for and undeformed for .
The system of equations was also obtained by Linkwitz and Schek (schek, , page 149ff.), who studied the form finding problem for cable networks. Within their proposed force density method tibert () values are assigned to the mentioned force densities rendering these equations linear in the knot coordinates.
2.1 Metric interpretation of
This physical model implies in the -dimensional space of bar lengths the following scalar product : with and
as the involved diagonal matrix is positive definite. Consequently this scalar product induces a norm : with and a metric (distance function) : with . Thus of Eq. (6) can be seen as squared distance between the deformed framework and the initial one ; i.e. . It should be pointed out that this distance only depends on the intrinsic metric of the framework and is therefore independent of the actual realization.
In order to reduce the distance measure to its geometric core we set and for all bars in the remainder of the article.
3 Snappability of realizations
If we have a realization with non-zero self-stress and the corresponding total elastic energy is not at a local minimum, then small perturbations would deform the framework according to the minimum total potential energy principle. Therefore we are interested in the set of stable realizations, i.e. realizations which are at a local minimum of the total elastic strain-energy. Note that the set is not empty as it contains at least the undeformed framework realizations.
3.1 Computation of the set
For the computation of the set for a given framework with a given intrinsic metric the following approach is used. We introduce new variables fulfilling the side condition with and make the Lagrange ansatz
with the -dimensional vectors and , where the latter is composed of the Lagrange multipliers . By taking the partial derivatives of with respect to the variables we obtain a system of equations, which is of algebraic nature. Therefore we can use homotopy continuation method (e.g. Bertini; cf. bates ()), as other approaches (e.g. Gröbner base, resultant based elimination) are not promising due to the number of unknowns and degree of equations. First of all we can restrict to the obtained real critical points of with all as only these correspond to realizations. This resulting set of realizations is split into a set and its absolute complement , where the elements of correspond to local minima of . They can be identified by the so-called second derivative test; i.e. all eigenvalues of the Hessian matrix of the function are positive. Finally the desired set can be obtained as the quotient SE(), where SE() denotes the group of direct isometries of . In the same way we define the set SE() of unstable realizations, which is needed later on.
3.2 Measuring the snappability
The evaluation of the snappability has to be based on the intrinsic metric of the framework, as minor changes of this metric can heavily effect its spatial shape (cf. examples of the four-horn schwabe () and Siamese dipyramids gorkavyy ()). Our intrinsic metric approach towards the determination of the snappability of a stable realization is based on the following theorem:
If a frameworks snaps out of a given stable realization by applying the minimum energy needed to it, then the corresponding deformation of the realization has to pass a shaky realization at the maximum state of deformation.
We think of as a graph function over the space of knots ; i.e. the ordered pair . In order to get out of the valley of the local minimum , which corresponds to the given stable realization , with a minimum of energy needed, one has to pass a saddle point of the graph, which corresponds to a realization . As all realizations of are self-stressed and deformed ( no zero self-stress) they are infinitesimally flexible.
Note that the concrete curve on the graph connecting the local minimum and the saddle point does not play a role as long as the deformation energy of each bar is monotonic increasing with respect to the curve parameter . This is due to the fact that along each curve of this possible set of curves the same amount of mechanical work (namely the minimum work needed) is performed on the framework to reach the saddle point.
Based on the pseudometric : with the snappability index of can be quantified as follows
where and are the bar lengths of the and , respectively, using the notation of the above proof and denotes the framework’s total length. Note that due to the division by the framework’s volume , which is constant according to our assumed Poisson’s ratio , the snappability index can be interpreted as the change of the elastic strain energy density . Therefore is invariant with respect to scaling (taking into account ; cf. Sec. 2.1) and enables the comparison of frameworks, which differ in the number of knots, the combinatorial structure and intrinsic metric. Note that the minimum of the obtained snappability indices over all undeformed realizations can be seen as the snappability index of the
The algorithm to compute reads as follows: Let us assume that yields the minimal positive value for where is a stable realization. We consider the simplest possible path in , namely the straight line segment from to and parametrize it with respect to the time yielding . This path corresponds to different 1-parametric deformations of realizations in . If among these a deformation with the property
exists ( ), then the stable realization can be left over the unstable realization and we get a value for . Computationally the property (12) can be checked e.g. by a parameter homotopy approach (e.g. Bertini; cf. (bates, , Sec. 6)). If such a deformation does not exist then we redefine as and run again the procedure explained in this paragraph. This loop either breaks down if we get a value for or if one cannot snap out from the realizations over a realizations of . In this case we set .
Note that our computational approach using Bertini does not recognize if the tracked path between the real starting point and real endpoint of the homotopy is entirely real. More generally it is an open problem to check if there exists at least one real curve , which corresponds to such a continuous real deformation.
After applying the mechanical minimum work needed to deform into the unstable realization , the framework will relax according to the minimum total potential energy principle. Therefore this self-acting deformation will end up either in a stable realization or it get stuck on the way to such a local minimum by reasons of reality. Note that a realization at the boarder of reality has to be infinitesimal flexible, as a real solution of an algebraic set of equations can only change over into a complex one through a double root. This results in the following theorem:
A snap of a framework described in Theorem 1 ends up in a realization which is either undeformed or a deformed one with a shakiness.
Note that is independent of the final snapping realization .
3.3 Pinned frameworks
From the mechanical point of view it makes sense to fix a system to the ground by pinning a subset of the knot set . The results of this paper also hold for this scenario of so-called pinned graphs (or grounded graphs) due to their fundamental properties summarized in (nixon, , Sec. 2.1). One only has to keep in mind that bars between pinned knots cannot be deformed and that the equilibrium condition (1) only has to hold for knots as pinned knots can counterbalance any force. This becomes clear by studying a trivial example (comparison of a pinned and unpinned triangular framework) given in the Appendix (cf. Sec. 5.1), where the computation of the snappability index is also demonstrated for a more sophisticated framework (pinned 3-legged planar parallel manipulator; cf. Sec. 5.2).
4 Conclusion and open problems
The total elastic strain energy of the framework (based on a physical model for the deformation of bars using Hook’s law) serves as base for the presented snappability index and the theoretical results of Theorem 3.1 and 3.2, which give further connections between shakiness and snapping beside the technique of averaging and deaveraging.
Note that our approach neglect the possibility of collision of bars during the frameworks deformation. A further open problem is mentioned in Remark 3.
Acknowledgements.The research is supported by Grant No. P 30855-N32 of the Austrian Science Fund FWF. Thanks to Hellmuth Stachel for constructive feedback on the final draft.
5 Appendix of examples
5.1 Pinned and unpinned triangular framework
We study the triangle with vertices and edge lengths , and . In the first approach we consider an unpinned framework where we attach the fixed frame in the following way to the framework: The origin coincide with and the positive -axis points into direction of ; i.e.
In the second approach we pin and yielding
Unpinned case: According to the computation of Sec. 3.1, the set consists of the following two undeformed realizations:
and the set of the following three unstable realizations:
The two undeformed realizations and the trajectories of under the snapping deformations between them passing through the unstable realizations are illustrated in Fig. 1(left) and Fig. 2, respectively. Note that the snappability index equals , which corresponds to the elastic strain energy density of the green unstable realization. The corresponding values for the red and violet unstable realizations are and ,
Pinned case: We obtain the same set as in the unpinned case (cf. Eq. (15)), but the set consists only of the following two realizations:
which are illustrated in Fig. 1(right) together with the trajectories of under the two corresponding snapping deformations. For this trivial example the two sattle points of the graph of the elastic strain energy density function can even be visualized (see Fig. 3). The function values of the red and green sattle points are and , respectively, where the latter one equals the snappability index of the pinned triangular framework.
5.2 Pinned 3-legged planar parallel manipulator
This more sophisticated framework (compared to the triangular one studied before) consists of six knots , where the first three knots are pinned to the ground possessing the following coordinates:
Each of these knots is connected by a bar, the so-called -th leg, with one of the remaining three knots (for ) with
which form a joint-bar triangle (cf. Fig. 4). The intrinsic metric of the framework is given by:
and implies that the joint-bar triangle degenerates as holds. According to the computation of Sec. 3.1, the set consists of the following three stable realizations:
where the first (blue) and second (cyan) are undeformed realizations and the third (magenta) is a deformed one (cf. Fig. 4(left)). The elastic strain energy density of the latter realization equals .
The set contains unstable realizations where the one with the smallest elastic strain energy density value of is given by:
The critical points of the elastic strain energy function where computed by Bertini based on the splitting of the variables into the following two groups:
which resulted in 59136 paths. In contrast the full homotopy yields 262144 paths.
The framework cannot snap out of the magenta realization by passing the green one, as the elastic strain energy density of the latter realization is lower. Therefore we consider in the space of edge lengths the straight line segment between the point given in Eq. (23) and the corresponding point of the green realization given by
One of the corresponding 1-parametric deformations has the property that it connects the blue realization with the green realization (cf. Eq. (12)). A further corresponding deformation also ends up in the green realization. This second deformation does not start at the cyan realization, but in the following complex solution for an undeformed realization of the framework:
Therefore the framework will relax from the green realization towards this complex solution. The realization, where this 1-parametric deformation hits the boarder of reality, is illustrated by the red shaky realization in Fig. 4(right).
- email: firstname.lastname@example.org
- This assignment corresponds to the definition of the intrinsic metric of the framework.
- denotes the spherical joint, which enables the group of spherical motions SO() of . Note that a -joint equals a rotational joint (R-joint).
- differs from the -dimensional zero vector.
- Each additional coinciding realization raises the order of the infinitesimal flexibility by one wohlhart ().
- Mathematically these structures do not posses a continuous flexibility but due to free bendings without visible distortions of materials their physical models flex.
- In this paper we assume as for conventional structural material is positive.
- Equals in this case the engineering normal strain of a material line segment .
- Note that elastic strain energy is a form of potential energy.
- Bates, D.J., Hauenstein J.D., Sommese, A.J., Wampler C.W.: Numerically Solving Polynomial Systems with Bertini. SIAM Philadelphia (2013)
- Connelly, R.: Rigidity. Handbook of Convex Geometry (P.M. Gruber, J.M. Wills eds.), pages 223–271, Elsevier (1993)
- Connelly, R., Whiteley, W.: The Stability of Tensegrity Frameworks. International Journal of Space Structures 7(2) 153–163 (1992)
- Gluck, H.: Almost all simply connected closed surfaces are rigid. Geometric Topology (L.C. Glaser, T.B. Rushing eds.), pages 225–239, Springer (1975)
- Goldberg, M.: Unstable Polyhedral Structures. Mathematics Magazine 51(3) 165–170 (1978)
- Gorkavyy, V., Fesenko, I.: On the model flexibility of Siamese dipyramids. Journal of Geometry 110:7 (2019)
- Haghpanah, B., Salari-Sharif, L., Pourrajab, P., Hopkins, J., Valdevit, L.: Multistable Shape-Reconfigurable Architected Materials. Advanced Materials 28(36) 7915–7920 (2016)
- Izmestiev, I.: Statics and kinematics of frameworks in Euclidean and non-Euclidean geometry. Eighteen Essays in Non-Euclidean Geometry (V. Alberge, A. Papadopoulos eds.), pages 191–233, EMS Publishing House (2019)
- Kötter, E.: Über die Möglichkeit, Punkte in der Ebene oder im Raume durch weniger als oder Stäbe von ganz unveränderlicher Länge unverschieblich miteinander zu verbinden. Festschrift Heinrich Müller-Breslau (H. Boost et al eds.), pages 61–80, Alfred Kröner Verlag Leipzig (1912)
- Linkwitz, K., Schek, H.-J.: Einige Bemerkungen zur Berechnung von vorgespannten Seilnetzkonstruktionen. Ingenieur-Archiev 40(3) 145–158 (1971)
- Milka, A.D.: Linear bendings of right convex polyhedra. Matematicheskaya fizika, anliz, geometriya 1(1) 116–130 (1994)
- Mittemeijer, E. J.: Fundamentals of Materials Science. Springer (2011)
- Nixon, A., Schulze, B., Sljoka, A., Whiteley, W.: Symmetry Adapted Assur Decomposition. Symmetry 6(3) 516–550 (2014)
- Rafsanjani, A., Akbarzadeh, A., Pasini, D.: Snapping Mechanical Metamaterials under Tension. Advanced Materials 27(39) 5931–5935 (2015)
- Roth, B.: Rigid and Flexible Frameworks. The American Mathematical Monthly 88(1) 6–21 (1981)
- Schulze, B., Whiteley, W.: Rigidity and scene analysis. Handbook of Discrete and Computational Geometry (J.E. Goodman et al eds.), pages 1593–1632, 3rd edition, CRC Press (2017)
- Shang, X., Liu, L., Rafsanjani, A., Pasini, D.: Durable bistable auxetics made of rigid solids. Journal of Materials Research 33(3) 300–308 (2018)
- Stachel, H.: W. Wunderlichs Beiträge zur Wackeligkeit. Technical Report No. 22, Institute of Geometry, TU Wien (1995)
- Stachel, H.: Configuration theorems on bipartite frameworks. Rendiconti del Circolo Matematico di Palermo (Series 2) 70(II) 335–351 (2002)
- Stachel, H.: What lies between rigidity and flexibility of structures. Serbian Architectural Journal 3(2) 102–115 (2011)
- Tibert, A.G., Pellegrino, S.: Review of Form-Finding Methods for Tensegrity Structures. International Journal of Space Structures 26(3) 241–255 (2011)
- Wohlhart, K.: Degrees of shakiness. Mechanism and Machine Theory 34(7) 1103–1126 (1999)
- Wunderlich, W., Schwabe, C.: Eine Familie von geschlossenen gleichflächigen Polyedern, die fast beweglich sind. Elemente der Mathematik 41(4) 88–93 (1986)