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Fig. 7. Yield strength of selected high density two-dimensional lattices with three hierarchical levels. A shear macroscopic stress is applied at the third hierarchical level. In contrast to the stiffness (Fig. 6), for strength the regions occupied by each hierarchical level are quite distinct; the points relative to higher hierarchical levels are always located below the points representing lower hierarchical levels. Thus, the performance of the lattice decreases with higher levels of hierarchy. stiffness of the lattice has been first determined at each hierarchical level; then a unitary pure shear macroscopic load has been applied at the third hierarchical level and the macroscopic stress distribution, shown in the first col- umn of the plots in Fig. 6, has been obtained. Here, the ele- ment with the highest effective stress has been identified (circles in the plots) and the component of its strain state has been used as boundary condition to determine the stress distribution in the lattice at level 2. With an equiva- lent procedure, we calculate the stress of the lattice mate- rial at level 1. We note that at each level the stiffness of the lattice depends on both the cell topology at that level and the macroscopic stiffness of the lattice at the lower level of the hierarchy. In addition, the microscopic stress applied at a location of the lattice in a given level is controlled by the macroscopic stress acting on the lattice at the upper level. For this reason, we can conclude that the problem is completely coupled. At each hierarchical level, the stress Fig. 7 maps the design space of the yield strength for lattices with more than one hierarchical level. In the figure, we plot the ratio o,/¢,;, which represents the load that must be applied at the highest hierarchical level to produce a unitary effective stress in the most stressed location of the solid material. We observe that the existence of multi- ple hierarchical levels is not beneficial for the lattice topol- ogies under investigation. The points correspond lattices with three levels of hierarchy are located ing to below the points corresponding to two hierarchical level lattices, which are below the solid line representing the attices with one hierarchical level. We also note that the yield strength of the lattices, at each hierarchical level, approximately with the first power of the relative d scales ensity, as expected in a first order approximation. This occurs be- cause for a given macroscopic load, the stress scales with

Figure 7 Yield strength of selected high density two-dimensional lattices with three hierarchical levels. A shear macroscopic stress is applied at the third hierarchical level. In contrast to the stiffness (Fig. 6), for strength the regions occupied by each hierarchical level are quite distinct; the points relative to higher hierarchical levels are always located below the points representing lower hierarchical levels. Thus, the performance of the lattice decreases with higher levels of hierarchy. stiffness of the lattice has been first determined at each hierarchical level; then a unitary pure shear macroscopic load has been applied at the third hierarchical level and the macroscopic stress distribution, shown in the first col- umn of the plots in Fig. 6, has been obtained. Here, the ele- ment with the highest effective stress has been identified (circles in the plots) and the component of its strain state has been used as boundary condition to determine the stress distribution in the lattice at level 2. With an equiva- lent procedure, we calculate the stress of the lattice mate- rial at level 1. We note that at each level the stiffness of the lattice depends on both the cell topology at that level and the macroscopic stiffness of the lattice at the lower level of the hierarchy. In addition, the microscopic stress applied at a location of the lattice in a given level is controlled by the macroscopic stress acting on the lattice at the upper level. For this reason, we can conclude that the problem is completely coupled. At each hierarchical level, the stress Fig. 7 maps the design space of the yield strength for lattices with more than one hierarchical level. In the figure, we plot the ratio o,/¢,;, which represents the load that must be applied at the highest hierarchical level to produce a unitary effective stress in the most stressed location of the solid material. We observe that the existence of multi- ple hierarchical levels is not beneficial for the lattice topol- ogies under investigation. The points correspond lattices with three levels of hierarchy are located ing to below the points corresponding to two hierarchical level lattices, which are below the solid line representing the attices with one hierarchical level. We also note that the yield strength of the lattices, at each hierarchical level, approximately with the first power of the relative d scales ensity, as expected in a first order approximation. This occurs be- cause for a given macroscopic load, the stress scales with

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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Provided for non-commercial research and education use. Not for reproduction, distribution or commercial use.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Provided for non-commercial research and education use. Not for reproduction, distribution or commercial use.
Fig. 1. Microlattice with multiple hierarchical levels.
Fig. 1. Microlattice with multiple hierarchical levels.
Fig. 2. Schematic protocol to recursively calculate the material stiffness of a hierarchical material.
Fig. 2. Schematic protocol to recursively calculate the material stiffness of a hierarchical material.
Fig. 3. Schematic protocol to recursively calculate the material strength of a hierarchical material. A. Vigliotti, D. Pasini/ Mechanics of Materials 62 (2013) 32-43 the neighbourhood of cracks, domain boundaries and where external loads are applied. The extent of this region depends ont he topology of t he lattice. For example, for tri- angulated and hexagonal lattices, the boundary region is limited to a up to 10 cel few strut lengt s. The lattice d hs; for the Kagome, it can be eformation returns then uni- form in the material domain outside the boundary layer. On the base least one order of magnitud scale of two levels in the structural hierarchy. We note that the effects o f length scale s of this observation, we can conclude that at e should separate the length eparation among hierarchical orders is not considered in this paper; thus the results are valid in the hypothesis of asymptotic separation. Yet, the findings provide insight into the effect of multiple hier- archical levels on the stress distribution, and into the exis- tence of stress concentration in hierarchical lattices. Table 1 Properties of selected low density two-dimensional lattices. In the above expressions, / = t/L is the slenderness ratio, where t is the edge thickness and Lis the length. E; is the Young’s modulus of the solid material and v the Poisson’s ratio. K is the macroscopic stiffness tensor of the lattice that yields o = Ke, with € = [€, €, 7], and O = [Ox, Gy, Tx]. p* is the relative density, 4; are the eigenvalues of the stiffness tensor of the lattice. The expressions are valid for 2 < 1/20. For comparative purposes, the corre- sponding expressions are also given for a uniform isotropic material.
Fig. 3. Schematic protocol to recursively calculate the material strength of a hierarchical material. A. Vigliotti, D. Pasini/ Mechanics of Materials 62 (2013) 32-43 the neighbourhood of cracks, domain boundaries and where external loads are applied. The extent of this region depends ont he topology of t he lattice. For example, for tri- angulated and hexagonal lattices, the boundary region is limited to a up to 10 cel few strut lengt s. The lattice d hs; for the Kagome, it can be eformation returns then uni- form in the material domain outside the boundary layer. On the base least one order of magnitud scale of two levels in the structural hierarchy. We note that the effects o f length scale s of this observation, we can conclude that at e should separate the length eparation among hierarchical orders is not considered in this paper; thus the results are valid in the hypothesis of asymptotic separation. Yet, the findings provide insight into the effect of multiple hier- archical levels on the stress distribution, and into the exis- tence of stress concentration in hierarchical lattices. Table 1 Properties of selected low density two-dimensional lattices. In the above expressions, / = t/L is the slenderness ratio, where t is the edge thickness and Lis the length. E; is the Young’s modulus of the solid material and v the Poisson’s ratio. K is the macroscopic stiffness tensor of the lattice that yields o = Ke, with € = [€, €, 7], and O = [Ox, Gy, Tx]. p* is the relative density, 4; are the eigenvalues of the stiffness tensor of the lattice. The expressions are valid for 2 < 1/20. For comparative purposes, the corre- sponding expressions are also given for a uniform isotropic material.
Fig. 4. FE meshes of selected two-dimensional lattices for decreasing values p*. For high density lattices, the stubby elements of the unit cell should be modelled with continuous plane stress elements. This choice enables not only to account for the actual stress distribution in the edges if the hypothesis of slenderness is not met, but also to accurately determine the joint stiffness and relative density. In contrast in a dis- crete model, the cross-section properties are assumed to be concentrated in the centre of the element cross-section, and overlapping volumes at the joints are not correctly ac- counted. Fig. 4 shows the finite element meshes of the topologies under investigation, for three alternative values Let us now consider the properties of lattices with two and three hierarchical levels. In Fig. 5, the points relative to lattices with two hierarchical levels are shown with an empty square marker, while the points relative to three
Fig. 4. FE meshes of selected two-dimensional lattices for decreasing values p*. For high density lattices, the stubby elements of the unit cell should be modelled with continuous plane stress elements. This choice enables not only to account for the actual stress distribution in the edges if the hypothesis of slenderness is not met, but also to accurately determine the joint stiffness and relative density. In contrast in a dis- crete model, the cross-section properties are assumed to be concentrated in the centre of the element cross-section, and overlapping volumes at the joints are not correctly ac- counted. Fig. 4 shows the finite element meshes of the topologies under investigation, for three alternative values Let us now consider the properties of lattices with two and three hierarchical levels. In Fig. 5, the points relative to lattices with two hierarchical levels are shown with an empty square marker, while the points relative to three
Eigenvalues and eigenvectors of the three-dimensional lattices analysed in the paper. For comparative purposes, the first column reports the eigen- values of an isotropic homogeneous material. Table 2
Eigenvalues and eigenvectors of the three-dimensional lattices analysed in the paper. For comparative purposes, the first column reports the eigen- values of an isotropic homogeneous material. Table 2
Fig. 5. Stiffness of selected high density two-dimensional lattices with three hierarchical levels. A solid line identifies the stiffness of a lattice with on hierarchical level; the empty square markers denote the stiffness of lattices with two hierarchical levels; the solid diamond marker refers to a lattice wit! three hierarchical levels. The green colour refers to the stiffness for equibiaxial stress, while the red corresponds to the shear stiffness. The Poisson’s ratio o the solid material is v = 0.3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.
Fig. 5. Stiffness of selected high density two-dimensional lattices with three hierarchical levels. A solid line identifies the stiffness of a lattice with on hierarchical level; the empty square markers denote the stiffness of lattices with two hierarchical levels; the solid diamond marker refers to a lattice wit! three hierarchical levels. The green colour refers to the stiffness for equibiaxial stress, while the red corresponds to the shear stiffness. The Poisson’s ratio o the solid material is v = 0.3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.
Fig. 6. Effective stress amplification factor at each hierarchical level. Colours indicate the dimensionless amplification factor between the applied macroscopic stress and the resulting microscopic stress in the unit cell. At the third hierarchical level, a unitary shear stress is applied, the element with the highest effective stress is identified and the relative stress is applied at the second level. The same procedure is applied at a lower level of the hierarchy to obtain the stress regime. (For interpretation of the references to colour in this figure legend, the reader is referred to the electronic version of this article.) Similar results were also obtained for other stress states. To obtain the data plotted in Fig. 6, the macroscopic We now consider the strength of two-dimensional hier- archical lattices. Since this section focuses on high density lattices (A > 1/20), we only consider as a failure mode
Fig. 6. Effective stress amplification factor at each hierarchical level. Colours indicate the dimensionless amplification factor between the applied macroscopic stress and the resulting microscopic stress in the unit cell. At the third hierarchical level, a unitary shear stress is applied, the element with the highest effective stress is identified and the relative stress is applied at the second level. The same procedure is applied at a lower level of the hierarchy to obtain the stress regime. (For interpretation of the references to colour in this figure legend, the reader is referred to the electronic version of this article.) Similar results were also obtained for other stress states. To obtain the data plotted in Fig. 6, the macroscopic We now consider the strength of two-dimensional hier- archical lattices. Since this section focuses on high density lattices (A > 1/20), we only consider as a failure mode
Fig. 9. Shear stiffness of three-dimensional topologies. The solid line refers to lattices with a single hierarchical level; the dots to lattice with multip hierarchical levels.
Fig. 9. Shear stiffness of three-dimensional topologies. The solid line refers to lattices with a single hierarchical level; the dots to lattice with multip hierarchical levels.
Fig. 8. Three-dimensional topologies under investigation. Fig. 8 shows the three-dimensional topologies exam- ined in this work: the body centred cube (BCC); the regular octet, an optimal truss topology that has been extensively studied in the literature (Deshpande et al., 2001b; Wallach and Gibson, 2001); the cuboctahedron, the only stretching dominated Archimedean polyhedron; and the truncated octahedron, which is the only Archimedean polyhedron, capable of regularly tessellating the space with a unitary packing factor (Torquato and Jiao, 2009). The truncated octahedron is also of special interest since it is very similar to the tetradecahedron, the polyhedron that minimizes the surface to volume ratio. It differs from the tetrakaidecahe- dron only for a slight curvature of the faces, and approxi- mates the shape under which foams self-arrange (Gong et al., 2005). Among the lattices shown in Fig. 8, the trun-
Fig. 8. Three-dimensional topologies under investigation. Fig. 8 shows the three-dimensional topologies exam- ined in this work: the body centred cube (BCC); the regular octet, an optimal truss topology that has been extensively studied in the literature (Deshpande et al., 2001b; Wallach and Gibson, 2001); the cuboctahedron, the only stretching dominated Archimedean polyhedron; and the truncated octahedron, which is the only Archimedean polyhedron, capable of regularly tessellating the space with a unitary packing factor (Torquato and Jiao, 2009). The truncated octahedron is also of special interest since it is very similar to the tetradecahedron, the polyhedron that minimizes the surface to volume ratio. It differs from the tetrakaidecahe- dron only for a slight curvature of the faces, and approxi- mates the shape under which foams self-arrange (Gong et al., 2005). Among the lattices shown in Fig. 8, the trun-
Fig. 10. Shear buckling strength for lattices with one, two and three hierarchical levels. where the expressions for a, f and y depend on the topol- ogy and on the properties of the solid material (Vigliotti and Pasini, 2012a). The eigenvalues and the eigenvectors of Kmat are reported in Table 2. As can be observed, the eigenspace of the stiffness matrix of the lattices examined in this section is similar to that of the lattices analysed in the previous section. In particular, the largest eigenvalue, where t¢ is the thickness of each beam, represented by the thick lines in Fig. 9, which we assume have a square cross section, L is the edge length and d is the thickness of the walls. The first inequality guarantees that the Euler—Ber- noulli assumption for the beams is fulfilled; the second is to prevent an overestimation of the cell stiffness due to the overlapping portion of adjacent cell elements. To com- pare the lattice performance for different configurations, we fix the slenderness ratio as 4 = 20 and let £ vary in the
Fig. 10. Shear buckling strength for lattices with one, two and three hierarchical levels. where the expressions for a, f and y depend on the topol- ogy and on the properties of the solid material (Vigliotti and Pasini, 2012a). The eigenvalues and the eigenvectors of Kmat are reported in Table 2. As can be observed, the eigenspace of the stiffness matrix of the lattices examined in this section is similar to that of the lattices analysed in the previous section. In particular, the largest eigenvalue, where t¢ is the thickness of each beam, represented by the thick lines in Fig. 9, which we assume have a square cross section, L is the edge length and d is the thickness of the walls. The first inequality guarantees that the Euler—Ber- noulli assumption for the beams is fulfilled; the second is to prevent an overestimation of the cell stiffness due to the overlapping portion of adjacent cell elements. To com- pare the lattice performance for different configurations, we fix the slenderness ratio as 4 = 20 and let £ vary in the
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