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A Multiscale Approach to Assess the Effect of Multilevel Structuring on the Properties of Hierarchical Lattice Materials

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A Multiscale Approach to Assess the Effect of Multilevel Structuring on the Properties of Hierarchical Lattice Materials.
Journal:
2011 MRS Fall Meeting
Manuscript ID: 1160628 Manuscript Type: Symposium OO Date Submitted by the Author: 15-Nov-2011 Complete List of Authors: Vigliotti, Andrea; Mc Gill University, Mechanical Engineering Pasini, Damiano; Mc Gill University, Mechanical Engineering Keywords: cellular (material type), microstructure, biomimetic (assembly)
A Multiscale Approach to Assess the Effect of Multilevel Structuring on the Properties of Hierarchical Lattice Materials
Andrea Vigliotti
1
and Damiano Pasini
1
1
Department of Mechanical Engineering - McGill University 817 Sherbrooke Street West, Montreal, QC
ABSTRACT
Natural materials have often a defined multilevel hierarchy which governs their macroscopic properties and mechanical efficiency. The structural features at a given length scale impart a specific mechanical contribution. One of these features is embodied by cellular patterns, which can be revealed at one level of the structural hierarchy, or can appear, in certain instances, repeated times at multiple levels. Cellular arrangements, either random, or periodic, or a mix of the two, yield a major contribution to the overall performance of the material. Furthermore, when they are nested at multiple length scale, they confer a remarkable cumulative effect on the mechanical properties. This paper presents a multiscale approach to the analysis of a hierarchical lattice component which exhibits nested levels of lattice. A number of three-dimensional topologies as well as the effect of lattice geometry parameters have been investigated. The results of the analysis are plotted onto material property charts. The visualization of the properties helps gain insight into the contribution that each hierarchical layer imparts to the overall properties of a component hierarchically structured with lattice materials.
INTRODUCTION
Materials with defined levels of structural hierarchy are widespread in Nature. Bone, wood, seashells and muscular tissues, are only a few examples of biological materials displaying multiple layers of structural organization. At the macroscopic scale, several of them exhibit cellular patterns, characterised by a large abundance of voids and channels that allows them to perform multifunctional tasks. Their cell walls, at a closer look, might present further cellular structure, which can be nested at more than one level. Wood, for example, is a cellular solid consisting mainly of hollow tubes; the cell wall in turn has its own structure, which can be regarded as a fibre composite made of cellulose microfibrils embedded into a matrix of hemicelluloses reinforced with lignin [1]. Certain bones, or parts of bones, display at higher levels of hierarchy a spongy network of trabeculae, whose material itself might consist of porous hollow fibres; at lower levels of hierarchy, the cell wall is a composite of collagen and mineral nanoparticles made of carbonated hydroxyapatite [2, 3]. Nacre, the material of seashells, is made
of an ordered multi-layered arrangement of mineral calcium carbonate tablets, embedded in a soft organic matrix. Although the constituents are structurally poor, their peculiar arrangement provides exceptional toughness properties [4]. In general, the superposition of multiple levels of structural hierarchy can impart multifunctional properties to the material. This effect has been proved to be remarkable. Ortiz [5] showed that the microstructure of several natural materials confer global properties that exceed those of their constituents by order/s of magnitude. Dipanjan [6] showed that a significant improvement in toughness and crack propagation resistance can be
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obtained from a brittle material solely by stacking levels with defined structural organization, with no need to introduce other materials. This paper presents a procedure for the analysis of hierarchical materials with nested layers of lattice structure. Works available in literature mainly used simplified models to analyse the features of hierarchical lattices, and estimate the properties of a hierarchical structure as the product of the properties at each hierarchical level [2].In this work, we present a multiscale approach to model the micromechanics of each level and assess its impact at the global level. The mechanical properties of a component are obtained from the parameters of the lattice, which can be defined arbitrarily at a given hierarchical layer. Starting from the uniform solid material, at level 0, the properties of the lattice are evaluated at level 1 via a multiscale approach, which is then reiterated in cascade at the following levels.
THE MULTISCALE APPROACH
Consider a lattice structure made of a uniform material. As explained in [7], its macroscopic stiffness can be obtained following the steps shown in Figure 1. The procedure is based on the following assumptions: (i) the scale of the microscopic structure is much smaller than the scale of the component, so as to locally approximate the behaviour of the lattice as infinite; (ii) the scale of the lattice is sufficiently larger than the molecular scale of the solid material so as to use continuum mechanics.
Figure 1 The multiscale mechanics scheme
Given the macroscopic displacements at the lattice level, (1,2) the macroscopic strain is determined from the displacement gradient; (3) by means of proper kinematic relations, the microscopic displacements of the unit cell are evaluated as a function of the macroscopic strain; (4) the microscopic stress is then determined for the uniform solid material; (5) the specific strain energy is evaluated as shown in Figure 1; (6) the components of the macroscopic stress are calculated by differentiating the microscopic strain energy density with respect to the macroscopic strain components; (7,8) element forces are finally assessed by means of the virtual work principle at the macroscopic level. The procedure described above can be reiterated to evaluate the stiffness and the strength of hierarchical lattice structure, as illustrated in Figure 2. At a given level, the macroscopic stiffness of the material is only a function of the lattice topology at that level and of the stiffness of the lattice at the previous level. Starting from the solid material at level 0, the density,
n
M
ρ
, and the macroscopic stiffness,
n
M
K
, of a component can be obtained in cascade by calculating those at the intermediate levels. Once the macroscopic stiffness at the component level has been found, the structural problem can be solved and the macroscopic displacement,
n
M
u
, stress,
n
M
σ
, and strain,
n
M
ε
, are obtained. These values can be used to determine the status of the
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microstructure at each layer. Given the macroscopic stress at each level, the occurrence of local buckling in the lattice members can be verified as well as yielding of the solid material.
Figure 2 Multiscale analysis of a hierarchical lattice structure
ANALYSIS OF LOW DENSITY THREE DIMENSIONAL LATTICES
Figure 3 shows selected lattice topologies, whose properties are here examined. The body centred cube (BCC) is a typical crystallographic arrangement of the austenitic form of steel and other common metals and alloys, characterised by high stiffness and strength. The regular octet is an optimal truss topology extensively studied in the literature [8, 9]. The cubocathedron is the
only stretching dominated Archimedean polyhedron. The truncated dodecahedron is the only regular polyhedron besides the cube, capable of regularly tessellating the space with an unitary packing factor; it is of particular interest for metallic foams since it is the shape under which foams self-arrange [10].
Body centred cube Regular octet Cubocathedron Truncated dodecahedron
Figure 3 Three dimensional topologies under investigation
As cases in point, a lattice structure with one, two and three levels of structural organization are here examined. For each cell topology with both closed and open configuration, the stiffness and the strength are determined at each hierarchic level (h-level) as a function of the relative density. Properties charts can be produced for any arbitrary stress state to capture the general behaviour of a lattice. So far, we have investigated the effect of multiple hierarchies on buckling and yield strength. In this work, charts of shear states are presented, since they are generally the weakest stress states of the material. The edges and the walls of the selected lattices have been modelled respectively as Euler-Bernoulli beams and Kirchhoff plates. To satisfy the hypothesis of slenderness, the lattice geometric parameters are constrained to the values: 20
≥
d L
00
≤≤
d t
( 1 ) where L is the edge length; d is the side of the edge cross section, which has been assumed square; t is the thickness of the wall. Open cell lattices correspond to zero wall thickness and they yield the lowest relative density. The dimensionless parameters in ( 1 ) can fully determine the geometry of the lattice cell and they conveniently allow to plot the property space of a 3D lattice as a 2D region of a material chart. The limits in equation ( 1 ) define also the feasible range of relative density, which is different for each lattice topology. Figure 4 shows the shear stiffness as a function of the relative density for one, two and three level lattices. The effect of nesting various layers depends on the cell-topology. For the BCC, the regular octet and the cuboctahedron, that are all stretching dominated, the lattice
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stiffness at higher h-levels generally scales with the same power of the relative density as the lattices with one h-level. For the truncated octahedron, on the other hand, the shear stiffness rapidly decreases as the thickness of the walls approaches zero and its properties at all h-levels decrease at a faster rate. The reason for this is to attribute to the bending dominated deformation of its open cell configuration. For a closed cell, on the other hand, the lattice remains stretching dominated at all h-levels. Hence, since the closed cell of the truncated octahedron is stretching dominated at any relative density, the nesting of h-levels with closed cell lattice has the effect of reducing its density without penalizing its stiffness.
Figure 4 Shear stiffness for one, two and three hierarchic levels.The properties at h-level 2 (green points) are obtained from selected points on theline multisegment (blue) representing one h-level lattice. These points are calculated by letting the relativity density span in the range allowed by equation ( 1 ). The procedure has been recoursivlely applied to the points of h-level 2 to obtain the properties (red) at h-level 3
Figure 5 groups the strength charts for the shear stress state. As observed, the lattice strength tends to improve significantly at higher h-levels. For 2-level hierarchies, the properties are located above the solid line corresponding to the single level lattice. Thus for a given density, a two level lattice can provide higher strength compared to a single level lattice. The effect of the multiple hierarchies is to maximise the efficiency of the structure by an optimal use of the solid material. In multiple h-level lattices, the structural mass is optimised at each level; therefore, adding h-levels to reduce the mass of a lattice preserves its strength more than reducing the cross section of its members. Figure 6 shows the charts for the shear yield strength. Also in this case, nesting multiple hierarchies improves the yield strength. However, although significant, the improvement of the yield strength is smaller than the improvement of the buckling strength (Fig. 6). This can be explained as follow. For a single hierarchy lattice, the buckling strength, which scales as , decreases considerably faster than the yield strength that scales as Adding hierarchical layers on a higher density lattice allows to reduce the relative density while
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