Linear multiscale analysis and finite element validation of stretching and bending dominated lattice materials

Abstract

The paper presents a multiscale procedure for the linear analysis of components made of lattice materials. The method allows the analysis of both pin-jointed and rigid-jointed microtruss materials with arbitrary topology of the unit cell. At the macroscopic level, the procedure enables to determine the lattice stiffness, while at the microscopic level the internal forces in the lattice elements are expressed in terms of the macroscopic strain applied to the lattice component. A numeric validation of the method is described. The procedure is completely automated and can be easily used within an optimization framework to find the optimal geometric parameters of a given lattice material.

Introduction

Cellular materials are a broad range of natural and artificial materials characterized by an abundance of microvoids confined in cells. The macroscopic characteristics of a cellular material depend not only on the shape and volume of the voids, but also on the material and cross section of the cell walls. As a subset of cellular materials, lattice materials are characterised by an ordered periodic microstructures obtained by replicating a unit cell along independent tessellation vectors. For a given density, lattice materials are tenfold stiffer and threefold stronger than foams, which, due to their stochastic arrangement of cells, lie below the lattices in the matreial charts (Ashby, 2005).

Recent developments in additive manufacturing enable to build lattice materials with a high level of quality at affordable cost (Yang et al., 2002, Stampfl et al., 2004). Such techniques provide material designers with a superior degree of control on the material properties and allow them to tailor the material performance to meet prescribed multifunctional requirements. For instance, desired macroscopic stiffness, strength, and collapse mode can be attained in given directions by properly selecting the geometric parameters of the microstructure. Unusual mechanical behaviour, such as negative macroscopic Poisson’s ratio, can be obtained by selecting auxetic topologies of the lattice (Lakes, 1987). In the aerospace sector, lattice materials can be applied for the design of morphing wings for next generation aircrafts (Spadoni, 2007, Alderson and Alderson, 2007, Gonella and Ruzzene, 2008). In the biomedical field, lattice materials have been proposed for advanced bone-replacement prosthesis, where the microtruss can be designed to resemble the inner architecture of trabecular bones, allowing seamless bone-implant integration, with reduced stress-shielding and bone resorption (Murr et al., 2010).

Reliable constitutive models are necessary to accurately predict the properties of lattice materials and exploit fully their potential. If the microscopic dimensions of the lattice are small compared to the macroscopic dimensions of the component, the number of degrees of freedom of a detailed model becomes extremely large and a direct approach involving the individual modelling of each cell is not practical.

An abundance of literature exists about constitutive models for cellular and lattice materials. In a work discussing alternative approaches for the analysis of large periodic structures, Noor (1988) emphasized that modelling the discrete structure as an equivalent continuum is the most promising strategy. He also outlined a method to evaluate the elastic constants of the surrogate continuum based on the isolation of the repeating cell and the use of the Taylor series expansion to approximate the displacement field inside the cell. His conclusion is that the Cauchy strain tensor can be used for the analysis of pin-jointed lattices, while for rigid jointed lattices the micropolar strain theory should be adopted.

In their comprehensive work on cellular materials, Gibson and Ashby (1988) estimated the stiffness and the strength of hexagonal and cubic lattices considering only bending in the cell walls. Their analysis focuses on a single cell under uni-axial load conditions and models the cell walls as either beams or plates. Zhu et al. (1997) applied a similar approach to model the tetrakaidecahedral topology, the cell shape usually assumed by foams, and obtained the Young’s and shear moduli as a function of the relative density.

Wang et al. (2005) analysed the behaviour of extruded beams with cellular cross section, subjected to combined in-plane and out-of plane loadings. The in-plane macroscopic stiffness of the beam cross section was derived for a number of bidimensional lattices, considering a single cell subjected to shear and compression along different axes. The elastic constants of the lattices were determined through a detailed analysis of each case, the pertinent loads were applied to the unit cell, and the lattice stiffness was calculated from the resulting nodal displacements (Wang and McDowell, 2004). Kumar and McDowell (2004), on the other hand, used the micropolar theory to estimate the stiffness of rigid-jointed lattices. The rotational components of the micropolar field were used to account for nodal rotations. The displacements and rotations within the unit cell were expressed by a second order Taylor expansion about the cell centroid; then, the micropolar constitutive constants were determined by equating the expressions of the deformation energy of the micropolar continuum and of the discrete lattice. The analysis was limited to unit cell topologies that included a single internal node only. Lately, Gonella and Ruzzene (2008) analysed the wave propagation in repetitive lattices by considering an equivalent continuous media; the generalized displacements field of the unit cell was expressed by a Taylor series expansion around a reference node; the equivalent elastic properties were obtained by direct comparison of the wave equations of the homogenized model, and of a uniform plate under plane-stress. The method was illustrated with specific reference to the regular hexagonal and re-entrant honeycombs. The same authors, in a more recent paper (Gonella and Ruzzene, 2010), noted that the order of the Taylor series expansion is limited by the number of boundary conditions that can be imposed on the unit cell, and limits the accuracy of the continuous model; the authors, thus, proposed an alternative approach using multiple cells as repeating units to improve the capability of the continuous model in capturing local deformation modes. Another approach was recently presented by Hutchinson and Fleck (2006), who resorted to the Bloch theorem and the Cauchy–Born rule to analyse pin-jointed lattice materials with nodes only on the boundary of the unit cell. Elsayed and Pasini (2010a) expanded this method introducing the dummy node rule, for the analysis of pin-jointed lattices with elements intersecting the unit cell envelope. The same authors used this approach for the analysis of the compressive strength of columns made of lattice materials (Elsayed and Pasini, 2010b).

This paper presents an alternative method for the analysis of both pin-jointed and rigid-jointed lattices. The procedure is based on a multiscale approach, where the macroscopic properties of the lattice are determined by expressing the microscopic deformation work as a function of the macroscopic strain field. In contrast to previous approaches relying on the Taylor series or the Cauchy–Born rule for the approximation of the displacements within the repeating cell, this method do not make any kinematic assumption on the internal points, but only on the boundary points of the cell. In addition, our approach does not resort to micropolar theory for the determination of the lattice nodal rotations; rather the rotational degrees of freedom (DoFs) of the cell nodes are evaluated enforcing periodic equilibrium conditions on the unit cell. At the microscopic level, after expressing the nodal DoFs of the unit cell as a function of the components of the macro-strain field, the internal forces in the lattice members are determined to verify whether the solid material of the cells fails. The procedure is illustrated with reference to three bidimensional topologies, namely the triangular, the hexagonal and the Kagome lattice. The method is vaildated by comparing the displacements of a finite lattice to those of an equivalent continuous model for prescribed geometry of the component, applied loads and boundary conditions.