On the mechanical modeling of the extreme softening/stiffening response of axially loaded tensegrity prisms
Graphical abstract
Introduction
The category of ‘Extremal Materials’ has been introduced in Milton and Cherkaev (1995) to define materials that at the same time show very soft and very stiff deformation modes (unimode, bimode, trimode, quadramode and pentamode materials, depending on the number of soft modes). Such a definition applies to a special class of mechanical metamaterials, i.e. composite materials, structural foams, pin-jointed trusses; cellular materials with re-entrant cells, rigid rotational elements: chiral lattices, etc., which feature special mechanical properties. The latter may include, e.g., auxetic deformation modes, negative compressibility, negative stiffness phases, high composite stiffness and damping, to name just a few examples (cf. Lakes, 1987, Milton, 1992, Milton, 2002, Kadic et al., 2012, Spadoni and Ruzzene, 2012, Nicolaou and Motter, 2012, Milton, 2013, Kochmann, 2014, and references therein). Extremal materials are well suited to manufacture composites with enhanced toughness and shear strength (auxetic fiber reinforced composite), artificial blood vessels, energy absorption tools, and intelligent materials (cf. Liu, 2006). Rapid prototyping techniques for the manufacturing of materials with nearly pentamode behavior, and bistable elements with negative stiffness have been recently presented in Kadic et al. (2012) and Kashdan et al. (2012), respectively.
Structural lattices are nowadays employed to manufacture phononic crystals and acoustic metamaterials, i.e., periodic arrays of particles/units, freestanding or embedded in fluid or solid matrices with contrast in mass density and/or elastic moduli, eventually engineered with local resonant inclusions (Lu et al., 2009). Such artificial materials are designed to gain a variety of unusual acoustic behaviors, such as, e.g., phononic band-gaps, sound control, negative effective mass density, negative effective bulk modulus, negative effective refraction index, and wave steering and directional behavior (cf. Liu et al., 2000, Li and Chan, 2004, Ruzzene and Scarpa, 2005, Daraio et al., 2006, Engheta and Ziolkowski, 2006, Fang et al., 2006, Gonella and Ruzzene, 2008, Lu et al., 2009, Zhang et al., 2009, Zhang, 2010, Bigoni et al., 2013, Casadei and Rimoli, 2013, and the references therein). Particularly interesting is the use of geometrical nonlinearities for the in situ tuning of phononic crystals (Bertoldi and Boyce, 2008, Wang et al., 2013), pattern transformation by elastic instability (Lee et al., 2012), as well as the optimal design of auxetic composites, and soft metamaterials incorporating fluids, gels and soft solid phases (Kochmann and Venturini, 2013, Brunet et al., 2013). Nonlinear metamaterials may support very compact compression solitary waves, in correspondence with a stiffening elastic response of the unit cells; or alternatively rarefaction pulses, when instead the unit cells exhibit a softening-type elastic behavior (cf. Friesecke and Matthies, 2002, Fraternali et al., 2012, Nesterenko, 2001, Herbold and Nesterenko, 2013).
This paper presents a mechanical study of the axial response of tensegrity prisms featuring large displacements, varying aspect ratios, prestress states, and material properties. We focus on the response of such structures under uniform axial loading, showing that they can exhibit extreme stiffening or, alternatively, extreme softening behavior, depending on suitable design variables. Interestingly, such a variegated mechanical response is a consequence of purely geometric nonlinearities. By extending the tensegrity prism models already in the literature (Oppenheim and Williams, 2000, Fraternali et al., 2012), we assume that the bases and bars of the tensegrity prism may show either elastic or rigid behavior. The presented models lead us to recover the extreme stiffening-type response in the presence of rigid bases already studied in Oppenheim and Williams (2000) and Fraternali et al. (2012). In addition, we discover a new, extreme softening-type response. The latter is associated with a snap buckling phenomenon eventually leading to the complete axial collapse of the structure. We validate our theoretical and numerical results through comparisons with an experimental study on the quasi-static compression of physical models (Amendola et al., 2014). The extreme hard/soft behaviors of tensegrity prisms can be usefully exploited to manufacture metamaterials supporting special types of solitary waves, and 2D or 3D highly anisotropic systems including soft and hard units (Fraternali et al., 2012, Herbold and Nesterenko, 2013, Ruzzene and Scarpa, 2005, Casadei and Rimoli, 2013). The structure of this paper is as follows: in Section 2, we formulate a geometrically nonlinear model of a regular minimal tensegrity prism. Next, we present a collection of numerical results referring to tensegirity prisms with different aspect ratios, prestress states, and material properties (Section 3). In Section 4, we validate such results against compression tests on physical models. We end in Section 5 by drawing the main conclusions of the present study, and discussing future applications of tensegrity structures for the manufacture of innovative periodic lattices and metamaterials.
Section snippets
Geometrically nonlinear model of an axially loaded tensegrity prism
Let us consider an arbitrary configuration of a regular minimal tensegrity prism (Skelton and de Oliveira, 2010), which consists of two sets of horizontal strings: 1–2–3 (bottom strings) and 4–5–6 (top strings); three cross strings: 1–6, 2–4, and 3–5; and three bars: 1–4, 2–5, and 3–6 (Fig. 1). The horizontal strings form two equilateral triangles with side length ℓ, which are rotated with respect to each other by an arbitrary angle of twist φ. On introducing the Cartesian frame
Numerical results
The current section presents a collection of numerical results aimed to illustrate the main features of the mechanical models presented in Section 2. We examine the mechanical response of tensegrity prisms having the same features as the physical models studied in Amendola et al. (2014). Such prisms are equipped with M8 threaded bars made out of white zinc plated grade 8.8 steel (DIN 976-1), and strings consisting of braided fibers with 0.76 mm diameter (commercialized by
Experimental validation
The present section deals with an experimental validation of the models presented in 2 Geometrically nonlinear model of an axially loaded tensegrity prism, 3 Numerical results, against the results of quasi-static compression tests on physical samples (Amendola et al., 2014) (cf. also Section 3). We first examine the experimental responses of the thick prism specimens described in Table 4, where denotes the axial force carried by the cross-strings in correspondence with the reference
Concluding remarks
We have presented a fully elastic model of axially loaded tensegrity prisms, which generalizes previous models available in the literature (Oppenheim and Williams, 2000, Fraternali et al., 2012). The mechanical theory presented in Section 2.1 assumes that all the elements of a tensegrity prism respond as elastic springs, and relaxes the rigidity constraints introduced in Oppenheim and Williams (2000). On adopting the equilibrium approach to tensegrity systems described in Skelton and de
Acknowledgments
Support for this work was received from the Italian Ministry of Foreign Affairs, Grant no. 00173/2014, Italy-USA Scientific and Technological Cooperation 2014-2015 (‘Lavoro realizzato con il contributo del Ministero degli Affari Esteri, Direzione Generale per la Promozione del Sistema Paese’). The authors would like to thank the anonymous referees for helpful comments, Robert Skelton and Mauricio de Oliveira (University of California, San Diego) for many useful discussions and suggestions, and
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2023, International Journal of Mechanical SciencesCitation Excerpt :For these applications, symmetric configurations are the preferred prototypes because of their excellent assembly capabilities and computability. Typical symmetric tensegrities include polygonal tensegrities [7–9], prismatic tensegrities [10–12], truncated polyhedral tensegrities [13,14], and other special tensegrities [15–18]. As a well-known type of them, truncated regular polyhedral (TRP) tensegrities can be constructed following truncated regular polyhedrons from one-bar elementary cells [19].