Asymptotic self-restabilization of a continuous elastic structure
A challenge in soft robotics and soft actuation is the determination of an elastic system which spontaneously recovers its trivial path during postcritical deformation after a bifurcation. The interest in this behaviour is that a displacement component spontaneously cycles around a null value, thus producing a cyclic soft mechanism. An example of such a system is theoretically proven through the solution of the Elastica and a stability analysis based on dynamic perturbations. It is shown that the asymptotic self-restabilization is driven by the development of a configurational force, of similar nature to the Peach-Koehler interaction between dislocations in crystals, which is derived from the principle of least action. A proof-of-concept prototype of the discovered elastic system is designed, realized, and tested, showing that this innovative behaviour can be obtained in a real mechanical apparatus.
Corresponding author: ]email@example.com; +39 0461 282507
The straight configuration of an axially compressed elastic rod becomes unstable at buckling, so that a bent configuration emerges and the rod usually does not recover its straight configuration during the post-bifurcation deformation Timoshenko-Gere-Libro:1961 (); Kuznetsov-Levyakov:2002 (). In the traditional mechanical design, the post-bifurcation behaviour, characterized by large deformations, is typically avoided to preserve mechanical integrity, but nowadays bio-inspired structures are often designed to work in a large displacement regime nature (); desimo (); Laschi (); Bigoni (); esse2 (); Yang (). Therefore, an interesting challenge for applications to soft-robotics and compliant mechanisms is to find a ‘self-restabilizing structure’, in which the straight configuration is spontaneously recovered during loading after buckling. Examples of these elastic systems have been provided, for which the instability region of the trivial path is bounded, becoming a sort of ‘island’. However, these restabilizations usually do not occur ‘spontaneously’, so that the systems have to be ‘externally’ moved back to the straight configuration (see the examples presented by Feodosyev Feodosyev-Libro:1977 () and by Bigoni et al. Bigoni-Bosi-DalCorso-Misseroni-1:2014 (), the former referred to a discrete system, the latter to a continuous elastic structure). For a two-degrees of freedom discrete system, spontaneous restabilization has been theoretically proven, but only in the presence of non-linear springs PotierFerry:1987 ().
The aim of the present article is to design, realize, and test a continuous elastic system which displays an ‘asymptotic self-restabilization’ in the following sense: although bifurcation does not occur, because the system is imperfect, the deflection initially grows and subsequently decays up to vanish during a monotonically increasing loading.
The continuous elastic system is a planar, slender and inextensible rod, modeled as the elastica sebastiano (); estrudimi (); magnetico (); basilio (). In particular, the developed system is an imperfect version of that analyzed in Bigoni-Bosi-DalCorso-Misseroni-1:2014 (), where an elastic rod (with bending stiffness ) penetrates into a frictionless sliding sleeve and is restrained with a linear spring (of stiffness ). The imperfection is the tilt angle , which is null in the perfect case, see Fig. 1. A dead load is applied at the upper end of the elastic rod. The rod has a total length , of which the length lies inside the sliding sleeve, with when .
The imperfect system displays two different behaviours. In the first one, self-restabilization does not occur (Fig. 2, lower part, on the right), so that the elastic rod is bent and, at increasing load, is progressively ejected from the sliding sleeve. In the second behaviour, asymptotic self-restabilization occurs (Fig. 2, upper part, on the right), so that the rod deflection initially increases and then progressively decreases until the rod is totally inserted into the sliding sleeve. As Fig. 2 shows, each of the two behaviours is connected to what is observed in the perfect system. In particular, the asymptotic restabilization of the imperfect system occurs when the perfect system does not buckle (Fig. 2, upper part, on the left) while, oppositely, it does not occur when the perfect system does buckle at increasing load (Fig. 2, lower part, on the left).
Since the effect of the imperfection is found to be crucial in the restabilization process, the mechanical behaviour of the system is investigated through both a theoretical and an experimental approach, when the inclination angle is varied. In particular, the experiment reported in Fig. 3 and performed with an inclination shows (see the snapshots taken at =N) the asymptotic self-restabilization during the application of an increasing load at the rod’s free edge. Indeed, at increasing loads, the lateral deflection of the rod initially increases (from snapshot A to C), but later decreases (from snapshot C to E) and eventually vanishes when the rod is completely inserted into the sliding sleeve, which occurs when .
The restabilization phenomenon is related to the action of a configurational force in the system, developed at the frictionless sliding sleeve (), whenever the rod curvature does not vanish there. Configurational forces have been introduced by Eshelby Eshelby:1951 () Eshelby:1956 () to describe the motion of inhomogeneities within solids and have been recently interpreted as the resultant of Newtonian contact forces on a moving inhomogeneity Royer-Ballarini:2016 (). Essentially, a configurational force is generated whenever an elastic system can change its configuration through a release of potential energy. For example, when a dislocation is present within a stressed crystal, it tends to move within it and this tendency can be quantified with a decrease of potential energy. For a dislocation loop represented by a closed curve of unit tangent (at the point singled-out by the arc-length ), Burgers vector , and stressed by , the configurational force driving the dislocation motion is given by the Peach-Koehler relation
where is the alternating Levi-Civita tensor. Recently, configurational (or ‘Eshelby-like’) forces acting on elastic structures have been theoretically and experimentally proven Bigoni-Bosi-DalCorso-Misseroni-2:2014 (); inietto (), exploited for the realization of self-encapsulation incapsulo () and for the design of innovative devices, such as the ‘elastica arm scale’ Bosi-DalCorso-Misseroni-Bigoni:2014 () and the ‘torsional gun’ torsionallocomotion ().
The present analysis showing self-restabilization is based on a variational approach which provides the equations governing the dynamics of the system and gives full evidence to the configurational force. Asymptotic self-restabilization is demonstrated through the determination of the equilibrium paths of the structure under quasi-static conditions. Finally, the stability of the equilibrium paths is verified through the investigation of the response of the structure to dynamic perturbations. The theoretical predictions are fully validated by experimental tests performed on a proof-of-concept structure.
Ii The dynamics of the system from the principle of least action
The equations governing the dynamics of the elastic system sketched in Fig. 1 are derived below by means of the principle of the least action. An inextensible elastic rod of length , straight in the undeformed configuration, and with a linear elastic behaviour relating the bending moment and the curvature through a constant bending stiffness , is constrained by a frictionless sliding sleeve ending with a linear spring of stiffness . The sliding sleeve is tilted by an angle from the vertical. Denoting by the curvilinear coordinate along the rod, , the exit point of the sliding sleeve lies at for all time . Therefore, the rod is partially inserted into the sleeve with the region from to lying inside the sleeve, so that the region from to remains outside the sleeve. The origin of the fixed frame is placed at the exit point of the sliding sleeve (). A vertical dead load is hanged at the rod’s edge , so that . In the unloaded state (), the rod is straight and lies along the axis, , so that the region from to is free-standing.
Introducing the positions and of the rod in the reference frame , and the rotation between the tangent to the rod and the axis, the following boundary conditions hold:
We look now at the dynamics of the rod. Neglecting rotational inertia for the rod, we consider the kinetic energy of the rod given by
where is the constant linear mass density of the rod and . The potential energy of the system comprises the work of the dead load , , the extensional strain energy of the spring , and the bending strain energy of the rod , where . As the part of the rod which lies inside the sleeve is straight
the curvature term for this region is zero and the total potential energy reads
The Lagrangian to be considered is provided by the difference , under the inextensibility constraints and
where we introduce the Lagrangian multipliers and , which are to be identified with the and components of the internal force of the rod. The dynamics of the rod is consequently given by first-order conditions to minimize the action
Following this minimization procedure (reported in the appendix), the equations governing the dynamics of the region of the rod outside the sliding sleeve, , are obtained as
complemented by the following boundary conditions
is disclosed from the equilibrium equation along the -axis (8b), recalling that .
Iii Asymptotic self-restabilization
subject to the following boundary conditions
with the moving boundary , defined by the axial equilibrium at the exit of the sliding sleeve,
where the load parameter has been introduced and the rotation field is subject to the boundary conditions and .
iii.1 Equilibrium paths
The solution at equilibrium can be obtained through an analytical manipulation of equations (III), (13), based on change of variables and integrations. Defining as the rotation at the free end () at equilibrium, , the dimensionless load and the dimensionless stiffness ratio as
the equilibrium configuration (restricted to the first deformation mode) is provided at varying load and tilt angle as the solution and to the following system of nonlinear equations
In equation (15), and are the complete and incomplete elliptic integral of the first kind respectively and the parameters and depend on the angles and as
Once the nonlinear system (15) is solved, the kinematical fields can be evaluated through
where , are the axial and transverse displacement fields, the functions am, cn and sn denote the Jacobi amplitude, Jacobi cosine amplitude and Jacobi sine amplitude functions, while is the incomplete elliptic integral of the second kind of modulus . Equations (17) and (17), evaluated at the rod’s loaded end, provide the following expressions for the axial and transversal displacements of the free edge of the rod
The solution of the cubic equation (15) provides three values of associated to the same triad given by the free end rotation , the stiffness ratio and the angle , namely
with being always real, while and taking real or complex values depending on the positive or negative sign of the discriminant ,
When the discriminant is positive, the roots are all real and can be expressed through the following trigonometric relation Cox2012Galois-Theory ()
where , respectively for the root .
Equilibrium paths for the system with stiffness ratio and are reported in Fig. 4 (upper and lower parts, respectively) for three different tilt angles . The roots , , and , given by equation (21), of the cubic equation (15) are plotted in Fig. 4 (left) as functions of the free end rotation . Stable and unstable paths (see next section for the details) are reported as continuous and dashed lines, respectively. The limit values of the ranges where the roots , , and prevail are marked in the figure with circular and square spots, the former indicating the transition from solution to , the latter from to .
Using equation (III.1), the stable equilibrium paths (departing from the undeformed state) are reported in Fig. 4 (right) in terms of the dimensionless load as a function of the dimensionless displacement components, namely, the axial displacement and the deflection .
It is clear from Fig. 4 that the equilibrium paths are strongly affected by the tilt angle and, depending on this parameter, the asymptotic self-restabilization may or may not occur. In particular, during a monotonic loading, whenever the tilt angle is smaller or equal than the maximum tilt angle for restabilization, , the rod initially leaves the undeformed straight configuration and then ‘spontaneously’ and smoothly returns back to this configuration, until the limit is approached of complete penetration of the rod into the sliding sleeve. Therefore the rod deflection initially increases, but later decreases until it vanishes, in the limit , so that asymptotic self-restabilization occurs. When this does not occur, , the complete penetration of the rod into the sliding sleeve does never realize. Therefore, in the cases considered in Fig. 4, asymptotic self-restabilization occurs for (Fig. 4, upper part) and for (Fig. 4, lower part).
The restabilization phenomenon is displayed whenever the discriminant , (III.1), takes non-positive values for a set of the end rotations . Through this criterion, the values of the stiffness ratio and the tilt angle for which the asymptotic self-restabilization occurs have been numerically obtained and reported in Fig. 5. It is observed that the maximum tilt angle (reported as red continuous line in Fig. 5) can be approximately described through the following equation
which is represented as the blue dashed line in Fig. 5.
We finally note that asymptotic self-restabilization may occur only for , which is the condition for which the bifurcation does not occur in the case when the tilt angle is null () Bigoni-Bosi-DalCorso-Misseroni-1:2014 (). Therefore, asymptotic self-restabilization is strictly related to the fact that the perfect system does not display buckling, but only a pure penetration of the straight rod into the sliding sleeve at increasing load. As a further evidence of this concept, trajectories of the loaded end, together with intermediate deformed configurations, obtained during a monotonic loading are shown in Fig. 2 for =0.3 (upper part) and for =1.2 (lower part) in the perfect (, left) and imperfect case (, right).
Finally, as a quantitative measure of the self-restabilization effect, the maximum deflection attained during loading has been evaluated for a tilt angle corresponding to the maximum tilt angle, . This quantity is reported as a function of the stiffness ratio in the inset of Fig. 5. It is observed that the largest deflection which may occur in the system during the asymptotic self-restabilization is about of the initial external length and is attained for , which corresponds to as maximum tilt angle.
iii.2 Stability from vibrations
where is the frequency and . Inserting the representation (24) into the dynamics equations, using equations (10) and retaining only the linear terms in , yields the following non-autonomous linear system of differential equations for the modes
The stability of the equilibrium solution is therefore related to the sign of , with corresponding to a stable configuration, and to an unstable configuration. We solve the linear boundary value problem (25)-(26) using a shooting method approach: at only three quantities are unknown , the other being given by equations (26a)-(26). Once the integration of the linear system (25) is performed, the boundary conditions (26c) depend linearly on , so that writing this linear problem can be expressed as
Requiring to be non-zero imposes the matrix to be singular. Defining , the mode frequencies can be obtained as the roots of , namely, solving the equation . As equations (25)-(26) can be recast as a Sturm-Liouville problem Bigoni-Bosi-DalCorso-Misseroni-1:2014 (), we know that the lowest eigenvalue is finite Boyce1959Vibrations-Of-Twisted (), Rao2007Vibration-of-continuous (). For each equilibrium solution, we plot as function of and record the smallest eigenvalue . The equilibrium is then stable if the lowest eigenvalue is strictly positive, and unstable if it is strictly negative. Such analysis allowed the definition of stability for the equilibrium paths reported in Figs. 4. As an example, the typical behaviour of the function for a stable and an unstable equilibrium configuration is reported in Figure 6, showing respectively a positive and a negative value for the lowest root of the determinant . The two considered equilibrium configurations are both characterized by the same values of , , and , while they correspond to the two different free end rotations, namely, (stable configuration) and (unstable configuration).
Experiments have been performed at the ‘Instabilities Lab’ of the University of Trento on the prototype reported in Fig. 7 as a sketch and a photo, where the linear elastic axial spring in Fig. 1 has been realized by hanging a highly-stiff bar (to which the elastic rod is clamped and orthogonal) to two carbon steel (EN 10270-1 SH) springs (D19100, 1.25 mm wire diameter and 8 mm mean coil diameter, N/m, purchased from D.I.M.). The stiff bar can only rigidly translate as constrained by two linear bushings (LHFRD12, Misumi Europe) parallel to the rod in its undeformed state. The tilt angle has been provided by simply inclining the prototype, through a lifting the right support (with the movable crosshead of a MIDI 10 load frame, from Messphysik). The penetration length of the rod has been obtained by measuring the displacement of the lower edge of the rod through a magnetic non-contact displacement transducer GC-MK5 (from Gemac). The data have been acquired with a NI CompactDAQ system, interfaced with Labview 8.5.1 (National Instruments). All the photos were taken with a Sony NEX 5N digital camera, equipped with 3.5-5.6/18-55 lens (optical steady shot from Sony Corporation).
A first experiment showing the asymptotic self-restabilization is reported in Fig. 3, where five snapshots of the system for a tilt angle are reported at increasing applied loading, N, on a C62 Carbon-steel rod (25 mm x 2 mm cross section, bending stiffness Nm) of external length in the unloaded configuration m. It can be noted that after an initial increase in the deflection (from snapshot A to C), a further increase of load realizes a decrease in the deflection (from snapshot C to E). The experimental values of the deflection (normalized through division of the initial external length), measured from image post-processing, are also reported dotted, as function of the dimensionless load and compared with the theoretical equilibrium path.
In a second experiment, the dead load at the free end of the elastic rod of length m (25 mm x 2 mm cross section, bending stiffness Nm) was imposed by filling two containers with water at a constant rate of 10 g/s, in order to obtain a slow and continuous increase in the applied load, which was measured with two miniaturized Leane XFTC301 (R.C. 500N) load cells. The experimental results, expressed in terms of applied dimensionless load as a function of both the dimensionless deflection (left) and amount of the rod inserted into the sliding sleeve, , (right) are reported in Fig. 8 for a dimensionless stiffness parameter , together with the theoretically predicted behaviour. Results reported as pink curves refer to a tilt angle , for which asymptotic self-restabilization occurs, whereas results reported as blue curves refer to a tilt angle , for which asymptotic self-restabilization does not occur. Experimental results are reported with continuous lines (from continuous measures) and dots or squares (from measures extracted from a video of the experiments through image post-processing, for and respectively), together with the theoretical prediction (dashed line), showing a nice agreement between theory and experiments.
A movie showing the experiments considered in Fig. 8 is available as electronic supplementary material.
A structure which self-restabilizes, namely, capable of recovering its initial trivial path after a post-bifurcation deformation, would be useful for applications in soft robotics and deformable mechanisms. One example of such a structure has been found in an asymptotic sense, so that a structural system has been shown to exhibit a deflection initially increasing from zero and later decreasing until vanishing. The asymptotic self-restabilization is determined by the effect of a configurational (or ‘Eshelby-like’) force, which has been theoretically deduced from the principle of least action. The mechanical behaviour of the structure and the stability of the equilibrium paths have been theoretically solved and experimentally confirmed, so that the presented results may open new perspectives for technological applications. The achievement of (non-asymptotic) self-restabilization in the presence of linear elastic constraints still remains a challenge to be addressed.
Acknowledgments The authors gratefully acknowledge financial support from the ERC, Advanced Grant ‘Instabilities and nonlocal multiscale modelling of materials’ ERC-2013-AdG-340561-INSTABILITIES (2014-2019). S.N. also acknowledges funding from the french ‘Agence Nationale de la Recherche’ (ANR-14-CE07-0023-01).
Appendix A Lagrangian first variation
Introducing the vector , collecting the functions describing the kinematics of the system, we consider the conditions for a state to minimize the action , under the boundary conditions (1) and fixed , , , and , at the two instants and . Calculus of variations shows that a necessary condition for configuration to be a solution is given by
where is a variation of the configuration such that
We perform integrations by parts with regard to time for the kinetic energy and remark that the boundary terms vanish due to the condition (A). We then perform integrations by parts with regard to the arc-length for the total potential energy and finally obtain
where the possibility of having a jump at the sliding sleeve exit point in the internal force and has been taken into account. Note that for simplicity of presentation, the term in equation (33) and in the following, refers to the non-null value of the function at this point, that is just outside the sliding sleeve, , so that we have . Using conditions (31), a further manipulation of equation (33) leads to
which annihilation, for every variation , yields the equations (7) governing the dynamics of the elastic system and the related boundary conditions (8). In equation (34) the symbol denotes the jump of the relevant argument evaluated at a specific point, namely
together with the boundary condition
On the other hand, imposing equation (32) to be zero for every variation in the displacement fields and , and for , yields the dynamic equations for the system along the and directions
as well as the translational equilibrium at specific points: at the loaded end
at the end attached to the spring
and at the sliding sleeve exit
Equation (42) discloses that a non-null jump is present in the axial internal force at the exit of the sliding sleeve and that is provided by the presence of an outward configurational force developed there,
Due to the linear elastic behaviour of the rod, the bending moment is related to curvature through , and the configurational force can be rewritten as equation (9) similarly to Bigoni-Bosi-DalCorso-Misseroni-2:2014 (), inietto (), incapsulo (), Bosi-DalCorso-Misseroni-Bigoni:2014 (). We note that there is also a jump in the component of the internal force,