Periodic Trusses Research Articles

Structural Analysis of Periodic Trusses and Lattice Materials: States of Self-Stress, Mechanisms, and Mechanical Properties

Abstract In the realm of low relative density, a rigid-jointed lattice material exhibits a mechanical response analogous to that of a pin-jointed periodic truss with an identical micro-architecture. This correspondence simplifies the evaluation of the lattice's structural performance. While established matrix methods in linear algebra are capable of determining the states of self-stress, infinitesimal mechanisms, and mechanical properties of pin-jointed periodic trusses, they lack a systematic approach to deriving closed-form expressions for these properties. This paper presents a straightforward approach to address this gap. Furthermore, we introduce an inventive finite element framework that directly yields self-stress states and infinitesimal mechanisms in periodic trusses subjected to specific uniform macroscopic loading conditions. This framework allows for the determination of mechanical properties for periodic trusses by solving straightforward boundary value problems, such as uniaxial tension/compression or simple shear. The developed finite element framework offers greater simplicity compared to the matrix methods, with its benefits becoming more pronounced for complex micro-architectures. To demonstrate the effectiveness of the two newly developed methods, we conduct structural analyses on five different lattice materials, achieving successful outcomes.

Rainbow trapping of out-of-plane mechanical waves in spatially variant beam lattices

We numerically and experimentally investigate the propagation of mechanical waves in two-dimensional periodic and spatially graded elastic beam lattices. Experiments on metallic lattices admit the characterization of the linear elastic wave dispersion over a wide range of frequencies, resulting in complete, experimentally-constructed dispersion surfaces in excellent agreement with predictions obtained from finite element-based Bloch wave analysis. While Timoshenko beam theory is shown to be sufficiently accurate for predicting the lowest modes, experiments prove that solid finite elements are required to capture the dispersion relations at higher frequencies as well as when mode coupling occurs. Based on an improved numerical procedure, group velocity maps further highlight the directionality of wave dispersion and allow for the simple identification of bandgaps. In addition to classically studied periodic trusses, we extend the framework to spatially graded structures and demonstrate acoustic rainbow trapping in beam lattices undergoing out-of-plane vibrations. Our experiments confirm broadband vibration attenuation of the typical meta-wedge type previously observed only in optics and few mechanical studies. Results further show convincing agreement between Bloch theory-based predictions, finite element simulations, and experimental measurements. Such spatially-variant architected lattices show great promise for steering the motion of elastic waves in applications from wave guiding and wave shielding to energy harvesting.

Predicting the influence of geometric imperfections on the mechanical response of 2D and 3D periodic trusses

Although architected materials based on truss networks have been shown to possess advantageous or extreme mechanical properties, those can be highly affected by tolerances and uncertainties in the manufacturing process, which are usually neglected during the design phase. Deterministic computational tools typically design structures with the assumption of perfect, defect-free architectures, while experiments have confirmed the inevitable presence of imperfections and their possibly detrimental impact on the effective properties. Information about the nature and expected magnitude of geometric defects that emerge from the additive manufacturing processes would allow for new designs that aim to mitigate (or at least account for) the effects of defects and to reduce the uncertainty in the effective properties. To this end, we here investigate the effects of four most commonly found types of geometric imperfections in trusses, applied to eleven representative truss topologies in two and three dimensions. Through our study, we (i) quantify the impact of imperfections on the effective stiffness through computational homogenization, (ii) examine the sensitivity of the various truss topologies with respect to those imperfections, (iii) demonstrate the applicability of the model through experiments on 3D-printed trusses, and (iv) present a machine learning framework to predict the presence of defects in a given truss architecture based merely on its mechanical response.

Equivalent plate dynamic modeling of space periodic truss structures

Antenna structures used in spaceflight always contain nonlinear joints distributed along their cross section; hence, an equivalent beam model (EBM) that uses a point to describe the cross section is not ideally suited to depicting connection nonlinearity in the structure. To address this difficulty, a novel orthotropic equivalent plate model (EPM) of a space periodic truss structure is proposed. Firstly, utilizing the Mindlin plate hypothesis, strain in the periodic truss element is expanded into a binary high-order Taylor series, and corrections are introduced to normal strain. Secondly, employing the principle of energy equivalence, the equivalent elastic and inertial matrices of the EPM are calculated for in-plane and out-of-plane vibration. Finally, the static condensation method is utilized to reduce the dimensions of the equivalent model to establish an orthotropic equivalent Mindlin plate dynamic model, and its inherent characteristics are analyzed. The simulations show that when in-plane shear strain is expanded to the third order, the EPM can more accurately represent torsional vibration of the periodic truss structure. By introducing normal strain correction coefficients in the longitudinal direction, lateral bending vibrations of the periodic truss structure can be simulated more precisely. Compared with the EBM, the EPM can more accurately depict mechanical characteristics along the cross section. The EPM broadens the scope of continuum equivalent modeling of periodic trusses, and is more useful in engineering practice. In addition, the frequencies and mode shapes of the analytical EPM are consistent with a model using the finite element method, which supports the accuracy and effectiveness of the EPM. The analytical EPM provides a convenient way to study the nonlinear connections and vibration control of antenna structures with multiple joints along their cross section.

Vibration Analysis of Periodic Trusses with Rigid Joints Using an Energy Equivalent Method

Vibration Analysis of Periodic Trusses with Rigid Joints Using an Energy Equivalent Method

Continuum models for stretching- and bending-dominated periodic trusses undergoing finite deformations

Advances in additive manufacturing across scales have enabled the creation of random, periodic, or hierarchical truss networks containing millions and more of individual truss members. In order to significantly reduce computational costs while accurately capturing the dominant deformation mechanisms, we introduce a simple yet powerful homogenized continuum description of truss lattices, which is based on applying a multi-lattice Cauchy–Born rule to a representative unit cell (RUC). Beam theory applied at the level of the RUC introduces rotational degrees of freedom and leads to a generalized continuum model that depends on the effective deformation gradients, rotation, and curvature on the macroscale. While affinely deforming the RUC is shown to produce excellent results for simple Bravais lattices, a multi-lattice extension is required for general and especially bending-dominated lattices, which cannot be described by a pure Taylor expansion of the RUC deformation; the importance of the multi-lattice concept is demonstrated through analytical examples. The resulting method is a beneficial compromise between inefficient FE2 techniques and micropolar theories with limited applicability. By implementing the model within a finite element framework, we solve and report several benchmark tests in 2D to illustrate the accuracy and efficiency of the model, which comes with only a small fraction of the computational costs associated with the full, discrete truss calculation. By using a corotational beam description, we also capture finite beam rotations. We further demonstrate that a second-gradient homogenization formulation is beneficial in examples involving localization, providing higher local accuracy at the RUC level than a first-gradient scheme, while affecting the global response only marginally.

Defect Sensitivity of Truss Strength

Periodic trusses with missing or defective struts may exhibit reduced strength relative to those of otherwise pristine trusses. The principal goal of the present study is to determine the extent to which individual strut defects and free surfaces, both separately and together, elevate strains in neighboring struts and, in turn, the effects of strain elevations on truss strength, especially in trusses made from elastic-brittle materials. The goals are pursued through finite element analyses of three stretch-dominated truss structures with low relative density under uniaxial compressive loading. In all cases considered, strain elevations due to bulk defects (distant from free surfaces) are comparable to or lower than those associated with the surfaces themselves. Although defects located at truss corners and truss edges cause the highest elevations in strut strains, their effects on truss strength are small (5–25%). Guidelines are presented for the minimal tensile fracture strain of the constituent material required to achieve the full strength potential of the truss, dictated by large-scale strut buckling.

Effects of nodal fillets and external boundaries on compressive response of an octet truss

Additive manufacturing has enabled fabrication of weight-efficient periodic trusses over a wide range of length scales. The principal objective of the present study is to address two coupled aspects of truss design and performance: (i) the extent to which circular nodal fillets enhance node stiffness and alleviate stress concentrations, and (ii) the extent to which external boundaries affect local strut strains. To this end, octet trusses with and without filleted nodes were fabricated out of a hard thermoplastic and tested in compression, with digital image correlation used to monitor axial and bending strut strains and nodal rotations. Complementary finite element simulations were also performed. Macroscopic compressive failure occurs through bending and buckling of certain inclined struts followed by tensile rupture of struts oriented perpendicular to the loading direction. The struts experiencing the highest tensile strain and most prone to rupture are those located along the truss mid-plane and that intersect external edges. Additional boundary effects are manifested in exacerbated nodal rotations and bending of inclined struts intersecting the truss corners. Although buckling begins in the affected struts, large-scale buckling is preceded by tensile rupture. The implications for design and failure prediction of finite truss structures are discussed.

Periodic truss structures

Despite the recognition of the enormous potential of periodic trusses for use in a broad range of technologies, there are no widely-accepted descriptors of their structure. The terminology has been based loosely either on geometry of polyhedra or of point lattices: neither of which, on its own, has an appropriate structure to fully define periodic trusses. The present article lays out a system for classification of truss structure types. The system employs concepts from crystallography and geometry to describe nodal locations and connectivity of struts. Through a series of illustrative examples of progressively increasing complexity, a rational taxonomy of truss structure is developed. Its conceptual evolution begins with elementary cubic trusses, increasing in complexity with non-cubic and compound trusses as well as supertrusses, and, finally, with complex trusses. The conventions and terminology adopted to define truss structure yield concise yet unambiguous descriptions of structure types and of specific (finite) trusses. The utility of the taxonomy is demonstrated by bringing into alignment a disparate set of ad hoc and incomplete truss designations previously employed in a broad range of science and engineering fields. Additionally, the merits of a particular compound truss (comprising two interpenetrating elementary trusses) is shown to be superior to the octet truss for applications requiring high stiffness and elastic isotropy. By systematically stepping through and analyzing the finite number of structure types identified through the present classification system, optimal structures for prescribed mechanical and functional requirements are expected to be ascertained in an expeditious manner.

Flexibility mechanisms in ideal zeolite frameworks

Zeolites are microporous crystalline aluminosilicate materials whose atomic structures can be usefully modelled in purely mechanical terms as stress-free periodic trusses constructed from rigid corner-connected SiO4 and AlO4 tetrahedra. When modelled this way, all of the known synthesized zeolite frameworks exhibit a range of densities, known as the flexibility window, over which they satisfy the framework mechanical constraints. Within the flexibility window internal stresses are accommodated by force-free coordinated rotations of the tetrahedra about their apices (oxygen atoms). We use rigidity theory to explore the folding mechanisms within the flexibility window, and derive an expression for the configurational entropic density throughout the flexibility window. By comparison with the structures of pure silica zeolite materials, we conclude that configurational entropy associated with the flexibility modes is not a dominant thermodynamic term in most bulk zeolite crystals. Nevertheless, the presence of a flexibility window in an idealized hypothetical tetrahedral framework may be thermodynamically important at the nucleation stage of zeolite formation, suggesting that flexibility is a strong indicator that the topology is realizable as a zeolite. Only a small fraction of the vast number of hypothetical zeolites that are known exhibit flexibility. The absence of a flexibility window may explain why so few hypothetical frameworks are realized in nature.

Microarchitectured cellular solids – the hunt for statically determinate periodic trusses

AbstractThe mechanical properties of open‐cell metallic foams and periodic lattice materials are explained by a structural analysis of their parent pin‐jointed truss structures. A matrix method of analysis is presented in order to elucidate whether a periodic truss structure can collapse by macroscopic strain producing mechanisms, or by periodic collapse mechanisms which induce zero macroscopic strain to first order. It is shown that the planar Kagome truss is of the latter type, and is consequently stiff. The in‐plane stiffness of the 2D Kagome grid and the bending stiffness of the 3D Kagome double‐layer grid (KDLG) are determined for elastic members. An alternative, statically determinate, double‐layer grid is derived from the octet truss, and is referred to as the modified octet truss (MOT). This structure has faces comprising a hexagonal grid and a semi‐regular triangular grid, and is much less stiff than the Kagome double‐layer grid. Both structures have morphing capability due to their static and kinematic determinacy.

Continuum modeling of lattice structures in large displacement applications to buckling analysis

We consider very large periodic trusses called lattice structures. In classical calculus, the periodic truss character and its global geometry are forgotten. With the homogenization method that we have developed, the lattice is replaced by a continuum model which approaches, in a certain sense, the real structure. This is true when the number of constitutive cells becomes large. The homogenization method has been developed in linear static cases and in free vibration. This paper generalises it for the large displacement assumption. The truss equilibrium is written on the unknown deformed configuration.

Vibration mode localization in disordered large planar lattice trusses

Vibration mode localization in large randomly disordered planar lattice trusses is studied in this paper. The equations of motion of a general n s-storey and n b-bay truss are formulated using a finite element method. The localization factor, which characterizes the exponential rate of growth or decay of amplitudes of vibration of a randomly disordered periodic structure, is related to the smallest positive Lyapunov exponent for the corresponding random discrete dynamical system. A numerical algorithm for determining all the Lyapunov exponents for a discrete random dynamical system is presented. As a second approach, Green's function formulation for a block tridiagonal matrix system is applied to determine the localization factors for disordered periodic trusses. Numerical results of vibration mode localization for single-bay and double-bay trusses are obtained by using the method of transfer matrix and the method of Green's function.

Vibration of periodic truss structures

Recent trends in underwater vehicle design suggest the use of truss-like structures, connected to the hull by a limited number of attachment points, to support vibrating machinery. This work investigates the dynamic behavior of a set of spatially periodic trusses of practical interest. As expected, experimental and numerical modeling shows that this periodicity causes the trusses to behave as mechanical comb filters in frequency. However, the multiple periodicities at work in the truss and impedance discontinuities at the truss joints cause the resonant peaks at low frequencies to widen significantly. The consequence of this effect suggests that trusses with varied strut lengths tend to a low-pass characteristic. This is confirmed by data from a truss with several strut lengths. [Research supported by ARPA and ONR.]

The localization of buckling modes in nearly periodic trusses

The global buckling problem of nearly periodic trusses, including intermediate weak members and disordered panels, is considered. The possibility of local buckling or plastification of individual bars is neglected. The problem is solved numerically using a finite element program. The results show the buckling load factor as well as the buckled mode shapes as functions of a disorder related parameter. These relations are found to be of a nonlinear nature for both the critical loads and the normalized displacements of selected nodes of the studied structures for a given mode shape. Thus, small changes in the disorder related parameter result in a sharp decrease of normalized displacements everywhere but at a certain region of the trusses presented. These results suggest that the phenomenon of buckling mode localization may be utilized in order to passively control structural elastic instability.