Figure 1
3D PTM. (a) unit cell (
Muñoz-Rojas et al., 2010
Muñoz-Rojas, P.A., Carniel, T.A., Silva, E.C.N., Öchsner, A. (2010). Optimization of a unit periodic cell in lattice block materials aimed at thermo-mechanical applications, in A. Öchsner and G.E. Murch (eds.), Heat Transfer in Multi-Phase Materials, Adv Struct Mater 2, Springer-Verlag Berlin Heidelberg.
); (b) Corresponding material (
Guth et al., 2012
Guth, D.C, Luersen, M.A., Muñoz-Rojas, P.A. (2012). Optimization of periodic truss materials including constitutive symmetry constraints, Mat.-wiss. u.Werkstofftech 43(5): 447-456.
).
Figure 2
A continuum domain with microscopic heterogeneities modeled by asymptotic homogeneization. The unit cell is composed of a truss-like structure.
Figure 3
Periodic boundary conditions used for AH.
Figure 4
Boolean matrix
for the condition when the value of the degree of freedom J is constrained to equal the value of the degree of freedom I.
Figure 5
(a) generic cellular structure containing both macro and micro scales; (b) microscopic truss-like unit cell with its global stiffness matrix; (c) EMsFEM equivalent finite element; (d) macroscopic model with considerably less degrees of freedon compared to th model in (a).
Figure 6
Scheme for construction EMsFEM using LBC.
Figure 7
Numerical interpolation function obtained for the PBC scheme.
Figure 8
A particular element “e” connecting nodes “p” and “q”.
Figure 9
Initial material design. (a) unit cell, (b) macroscopic material and (c) corresponding elastic homogenized tensor (
Guth et al., 2012
Guth, D.C, Luersen, M.A., Muñoz-Rojas, P.A. (2012). Optimization of periodic truss materials including constitutive symmetry constraints, Mat.-wiss. u.Werkstofftech 43(5): 447-456.
).
Figure 10
Polar plot of the component rotated in the plane XY and normalized by(Guth et al., 2015Guth, D.C, Luersen, M.A, Muñoz-Rojas, P.A. (2015). Optimization of three-dimensional truss-like periodic materials considering isotropy constraints, Struct Multidisc Optim 52:889-901.).
Figure 11
Polar plots for the elastic tensor (Guth 2012Guth, D.C, Luersen, M.A., Muñoz-Rojas, P.A. (2012). Optimization of periodic truss materials including constitutive symmetry constraints, Mat.-wiss. u.Werkstofftech 43(5): 447-456.): (a)component rotated and normalized with respect to; (b)component rotated and normalized with respect to.
Figure 12
Initial material design. (a) unit cell, (b) macroscopic material and (c) corresponding elastic homogenized tensor (Guth 2012Guth, D.C, Luersen, M.A., Muñoz-Rojas, P.A. (2012). Optimization of periodic truss materials including constitutive symmetry constraints, Mat.-wiss. u.Werkstofftech 43(5): 447-456.).
Figure 13
Polar plots for the elastic tensor (Guth 2012Guth, D.C, Luersen, M.A., Muñoz-Rojas, P.A. (2012). Optimization of periodic truss materials including constitutive symmetry constraints, Mat.-wiss. u.Werkstofftech 43(5): 447-456.): (a) component rotated and normalized with respect to; (b) component rotated and normalized with respect to.
Figure 14
Initial material design. (a) unit cell, (b) macroscopic material and (c) corresponding elastic homogenized tensor (Guth et al., 2012Guth, D.C, Luersen, M.A., Muñoz-Rojas, P.A. (2012). Optimization of periodic truss materials including constitutive symmetry constraints, Mat.-wiss. u.Werkstofftech 43(5): 447-456.).
Figure 15
Polar plots for the elastic tensor (Guth 2012Guth, D.C, Luersen, M.A., Muñoz-Rojas, P.A. (2012). Optimization of periodic truss materials including constitutive symmetry constraints, Mat.-wiss. u.Werkstofftech 43(5): 447-456.): (a) component rotated and normalized with respect to; (b)component rotated and normalized with respect to.
Figure 16
Initial material design. (a) unit cell and (b) macroscopic material, (c) corresponding elastic homogenized tensor (Guth 2012Guth, D.C, Luersen, M.A., Muñoz-Rojas, P.A. (2012). Optimization of periodic truss materials including constitutive symmetry constraints, Mat.-wiss. u.Werkstofftech 43(5): 447-456.).
Figure 17
Polar plots for the elastic tensor (Guth 2012Guth, D.C, Luersen, M.A., Muñoz-Rojas, P.A. (2012). Optimization of periodic truss materials including constitutive symmetry constraints, Mat.-wiss. u.Werkstofftech 43(5): 447-456.): (a) component rotated and normalized with respect to; (b) component rotated and normalized with respect to.
Figure 18
Initial material design. (a) unit cell, (b) macroscopic material and (c) corresponding elastic homogenized tensor (Guth 2012Guth, D.C, Luersen, M.A., Muñoz-Rojas, P.A. (2012). Optimization of periodic truss materials including constitutive symmetry constraints, Mat.-wiss. u.Werkstofftech 43(5): 447-456.).
Figure 19
Polar plots for the elastic tensor (Guth 2012Guth, D.C, Luersen, M.A., Muñoz-Rojas, P.A. (2012). Optimization of periodic truss materials including constitutive symmetry constraints, Mat.-wiss. u.Werkstofftech 43(5): 447-456.): (a) component rotated and normalized with respect to; (b) component rotated and normalized with respect to.
Figure 20
Numerical interpolation functions obtained for LBC.
Figure 21
Numerical interpolation functions obtained for PBC.
Figure 22
Numerical interpolation functions obtained for LBC.
Figure 23
Numerical interpolation functions obtained for PBC.
Figure 24
Numerical interpolation functions obtained for LBC.
Figure 25
Numerical interpolation functions obtained for PBC.
Figure 26
Force decomposition showing large axial resultants on nearly collinear bars.
Figure 27
Numerical interpolation functions obtained for LBC.
Figure 28
Numerical interpolation functions obtained for PBC.
Figure 29
Numerical interpolation functions obtained for LBC.
Figure 30
Numerical interpolation functions obtained for PBC.
Figure 31
FEM discretization macro and micro scale problem.
Figure 32
Poisson ratio evaluated using EMsFEM - initial cell plane stress FEM fully integrated linear quadrilateral element (AH) -
0.23532.
Figure 33
Max. Poisson ratio evaluated using EMsFEM - only areas l plane stress FEM fully integrated linear quadrilateral element (AH) -0.99975.
Figure 34
Max. Poisson ratio evaluated using EMsFEM - areas and coordinates plane stress FEM fully integrated linear quad. element (AH) -.
Figure 35
Min. Poisson ratio evaluated using EMsFEM - only areas plane stress FEM fully integrated linear quad. element (AH) -.
Figure 36
Min. Poisson ratio evaluated using EMsFEM - areas and coordinates plane stress FEM fully integrated linear quad. element (AH) -
.
Figure 37
Evaluation of Poisson ratio for each cell rotated 90 clockwise for isotropy check. Although three cells are extremely non symmetric, the same Poisson value is confirmed in two directions.