Viscoelastic truss metamaterials as time-dependent generalized continua

Raphaël N. Glaesener, Jan-Hendrik Bastek, Frederick Gonon, Vignesh Kannan, Bastian Telgen, Ben Spöttling, Stephan Steiner, Dennis M. Kochmann
2021 Journal of the mechanics and physics of solids  
Mechanical metamaterials provide tailorable functionality based on a careful combination of base material and structural architecture. Truss-based metamaterials, e.g., exploit structural topology and beam geometry to achieve beneficial mechanical and physical properties from stiffness and wave dispersion to strength and toughness. While the focus to date has been primarily on static metamaterial properties or linear wave motion, 3D-printed polymeric base materials naturally come with
more » ... viscoelasticity, making the effective truss response time-and rate-dependent. Here, we report a theoretical-numerical-experimental study which (i) deploys a linear viscoelastic corotational beam description (capturing finite rotations at small strains), (ii) implements the latter in a finite element framework, (iii) calibrates a generalized Maxwell model based on viscoelastic experiments on 3D-printed polymer samples, (iv) validates the theory and implementation through experimental truss benchmark tests, and (v) introduces a generalized continuum formulation for the efficient simulation of viscoelastic truss metamaterials containing large numbers of structural members. We show that the viscoelastic beam approach, calibrated via tension tests on individual strut samples, performs well when applied to complex truss lattices undergoing time-dependent stress relaxation -as verified by the effective mechanical response and full-field deformation maps. The resulting variational generalized continuum framework uses on-the-fly periodic homogenization based on a representative unit cell and is extended to dynamics by including inertial effects. By comparison to discrete numerical simulations we demonstrate the accuracy of the continuum approach, which is promising for modeling and optimizing 3D-printed truss metamaterials for engineering applications from shock-absorbing structures to rate-dependent architected materials and soft robotics. J o u r n a l P r e -p r o o f Journal Pre-proof with application-dependent static (Surjadi et al., 2019) and dynamic optimized properties, the latter including shock mitigating by impact energy absorption (Schaedler et al., 2014; Frenzel et al., 2016) and wave guiding (Zelhofer and Kochmann, 2017). As additive manufacturing is the prime means of fabrication, versatile 3D printing materials have been used as base materials for metamaterials, which requires detailed understanding of their mechanical behavior to accurately design, model, and optimize the engineered metamaterial response. While for additively manufactured trusses made of metallic (Zadi-Maad et al., 2018) and ceramic base materials (Zocca et al., 2015) the assumption of linear elasticity is oftentimes sufficient, compliant polymers require rate-dependent models to describe the complex time-dependent behavior and history dependence (Gibson et al., 2010; Wang et al., 2017). Although being a technical complication for modeling, the viscoelastic relaxation, creep, frequency-dependent harmonic or generally time-dependent response of such architectures facilitates fascinating new features and applications (Gomez et al., 2019) ranging from rate-dependent buckling patterns and bistable beams (Dykstra et al., 2019; Janbaz et al., 2020) to advanced vibration damping (Wang and J., 2018) and frequency control and dispersion tuning in soft phononic crystals (Frazier and Hussein, 2015; Parnell and De Pascalis, 2019). Seizing the opportunities provided by viscoelastic metamaterials requires accurate and efficient models for the design space exploration and property optimization. While the classical theory of viscoelasticity was laid out long time ago (see e.g. Schapery (1969); Taylor et al. (1970); Christensen (1982); Lakes (1999)), the modeling of viscoelastic structural members is a more recent topic, including, e.g., finite element (FE) models for viscoelastic plates (Marques and Creus, 1994; Yi and Hilton, 1994), beams (Hilton, 2009), and sandwich structures (Galucio et al., 2004). Rate-dependent trusses were studied, among others, by Ghayesh et al. (2016) who studied the viscoelastic response of a single Euler-Bernoulli beam, whereas Hamed (2012) focused on a 2D Timoshenko beam, and Bottoni et al. (2008) modeled linear viscoelastic thin-walled beams. Recently, Ananthapadmanabhan and Saravanan (2020) reported numerical techniques to calculate the response of nonlinear viscoelastic truss networks (yet ignoring bending strains). In addition, Lestringant et al. (2020) and Lestringant and Kochmann (2020) introduced a general formulation of slender, geometrically exact beams made of viscoelastic base materials, which has not yet been applied to trusses. With the objective of providing a simple and accurate truss description and an efficient finite element treatment, we here extend the corotational beam element of Crisfield (1990) to linear viscoelastic beams, which accounts for rate dependence in the axial, flexural and torsional stress-strain relations (based on a general one-dimensional linear viscoelastic constitutive law) and accounts for finite rotations. Implementation of the model in a FE framework allows us to simulate the linear viscoelastic response of complex truss networks in two and three dimensions (2D and 3D, respectively). Validation is realized by comparison to relaxation experiments on 3D-printed 2D hexagonal trusses, whose generalized Maxwell model parameters are extracted from tensile tests. The comparison between simulations and experiments on the strained hexagon lattice shows excellent agreement, which confirms the applicability of our viscoelastic corotational beam formulation to polymeric trusses. Even though we present an efficient truss description, the cost of solving boundary problems involving thousands to millions of struts inside a truss becomes prohibitive, so multiscale techniques become beneficial (Kochmann et al., 2 J o u r n a l P r e -p r o o f Journal Pre-proof 2019). Here, we incorporate the linear viscoelastic beam into a generalized continuum formulation, which was previously introduced for linear elastic beam networks (Glaesener et al., 2019 (Glaesener et al., , 2020 . The discrete truss is replaced by a continuous body, whose effective mechanical constitutive behavior is obtained from on-the-fly numerical homogenization of a representative unit cell, and whose deformation is solved by a finite element calculation on the macroscale (comparable to FE 2 ). This two-scale model captures finite strains on the macroscale (accommodated by finite rotations of truss members on the microscale), and it applies to stretching-and bending-dominated truss topologies. Unlike Vigliotti et al. (2014); Pal et al. (2016) , who pursued similar approaches in 2D but with rotational degrees of freedom condensed out on the microscale, we introduce both translational and rotational degrees of freedom on the macroscale and pass those to the representative unit cell. We compare simulation results of discrete numerical calculations to those of the two-scale generalized continuum approximation for several 3D relaxation and vibration benchmarks, using a selection of bending-and stretching-dominated truss topologies, which demonstrate the homogenization scheme's accuracy and efficiency as well as its applicability. In addition, our results highlight the exciting opportunities for exploiting viscoelastic base materials for the design of time-dependent metamaterial properties. The remainder is structured as follows. Section 2 lays out the linear viscoelastic beam theory, introduces its FE implementation through corotational beam elements, and summarizes the formulation of the two-scale generalized continuum representation of viscoelastic trusses. Section 3 validates the viscoelastic truss model by comparing simulated results to experimental relaxation tests on 2D hexagonal trusses. 3D quasistatic stress relaxation and dynamic vibration examples in Section 4 demonstrate good agreement between fully resolved discrete numerical calculations and efficient FE simulations based on the two-scale representation, before Section 5 concludes our study. Linear viscoelastic corotational beams We consider slender beams made of a linear viscoelastic material in 3D, which we describe by an extension of the corotational beam formulation originally introduced by Crisfield (1990) for linear elastic Euler-Bernoulli beams. Owing to the linear constitutive relations and the resulting applicability of Boltzmann's superposition principle, the formulation and numerical implementation of our viscoelastic beams follows that of their linear elastic counterparts and, particularly, admits the decoupling of stretching, flexural and torsional stress and strain components. Viscoelastic beam theory We assume our corotational beam to be sufficiently slender, so it experiences a linear combination of axial strains due to stretching and bending, on the one hand, and shear strains due to torsion, on the other hand, while we neglect all further strain components. Let the x-axis be locally aligned with the neutral axis of the beam (Figure1(a) ), so that ε xx (x, t) = ε ax. (x, t) − y κ z (x, t) + z κ y (x, t). (1) for every point x = (x, y, z) T on the beam at time t. We describe the displacement of the beam's center-line by the displacement field u(x, t) = u x (x, t), u y (x, t), u z (x, t) , so that ε ax. (x, t) = u x (x, t) is an axial strain in the beam that 3 J o u r n a l P r e -p r o o f Journal Pre-proof        . (12) 5 J o u r n a l P r e -p r o o f Journal Pre-proof Despite those limitations, the presented model has provides an accurate and efficient reduced-order representation of complex viscoelastic trusses and truss-based architected materials.
doi:10.1016/j.jmps.2021.104569 fatcat:kssbz7yarve7tavomlrt4qxfze