Size-dependent microstructure design for maximal fundamental frequencies of structures

Abstract

Topology optimization on a unit cell is a common technique to improve the fundamental frequencies of periodic cellular solid structures. During this procedure, the effective properties of cellular solids are primarily computed by the homogenization method. This homogenization method is based on the classic continuum theory under the assumption that the unit cell is infinitely small. Hence, this classic strategy is inadequate to interpret the size dependence of the optimal results. The aim of this study was to describe and examine size dependence in relation to the topology design of the unit cell to achieve maximization of the structural fundamental frequencies. For this purpose, we determined the effective properties of the cellular solids and constructed the optimization formulation based on the couple-stress theory rather than the classic theory. A modified bound formulation of the objective and constraint functions was used to avoid the non-differentiability of repeated frequencies. Although the existing theory does not reflect size dependence, our optimization formulation was able to identify the size dependence of both the microstructural topologies and the fundamental frequencies. The size-dependent results are achieved by varying of the mechanisms to achieve the maximal fundamental frequencies in response to cell size variation. The present formulation is suitable for the unit cell design of cellular solid structures that possess local dimensions comparable to the cell size, and this novel formulation has expanded the application scope of the classic microstructural design problem for periodic materials.

Appendix calculations for the effective couple-stress constitutive constants

Four tests are constructed to determine the components of stiffness matrix C and another two tests are constructed to determine the components of stiffness matrix D for the homogenization of a cellular solid unit cell to homogeneous couple-stress continuum (Fig. 12). These six computations are based on the equivalent strain energy. In addition, a computation that is based on the geometry analysis is applied to determine the density ρ. The other computation that is based on the equivalent rotational kinetic energy is applied to determine the micro-rotational inertia Θ of the effective couple-stress continuum.

• 1) Horizontal uniaxial extension test for C₁₁: by applying the unit strain to the unit cell

$$ {\varepsilon}_x=1,\kern0.33em {\varepsilon}_y={\gamma}_{xy}=0,\kern0.33em {\kappa}_{xz}={\kappa}_{yz}=0,\kern0.66em \mathrm{in}\kern0.33em \varOmega $$

(A1)

The corresponding boundary conditions are

$$ u=x,\kern0.33em v=0,\kern0.33em \mathrm{on}\kern0.33em \partial \varOmega $$

(A2)

The deformation of the unit cell is shown in Fig. 13(a).

Then it follows that

$$ {C}_{11}=2{U}_{disc}^{(1)}/V $$

(A3)

where ${U}_{disc}^{(1)}$ is the strain energy of the unit cell with boundary conditions Eq.(A2) and V is the volume of the unite cell.

• 2) Vertical uniaxial extension test for C₂₂: by applying the unit strain to the unit cell

$$ {\varepsilon}_y=1,\kern0.33em {\varepsilon}_x={\gamma}_{xy}=0,\kern0.33em {\kappa}_{xz}={\kappa}_{yz}=0,\kern0.66em \mathrm{in}\kern0.33em \varOmega $$

(A4)

The corresponding boundary conditions are

$$ u=0,\kern0.33em v=y,\kern0.33em \mathrm{on}\kern0.33em \partial \varOmega $$

(A5)

The deformation of the unit cell is shown in Fig. 13(b).

Fig. 12

Homogenization of a cellular solid unit cell to homogeneous couple-stress continuum

Fig. 13

Sketches for deformations of the unit cell to compute C₁₁~D₂₂, where solid lines denote the undeformed cells and dashed lines denote the deformed cells (a) C₁₁, (b) C₂₂, (c) C₁₂, (d) C₆₆, (e) D₁₁, and (f) D₂₂

Then it follows that

$$ {C}_{22}=2{U}_{disc}^{(2)}/V $$

(A6)

• 3) Biaxial extension test for C₁₂: by applying the unit strain to the unit cell

$$ {\varepsilon}_x={\varepsilon}_y=1,\kern0.33em {\gamma}_{xy}=0,\kern0.33em {\kappa}_{xz}={\kappa}_{yz}=0,\kern0.66em \mathrm{in}\kern0.33em \varOmega $$

(A7)

The corresponding boundary conditions are

$$ u=x,\kern0.33em v=y,\kern0.33em \mathrm{on}\kern0.33em \partial \varOmega $$

(A8)

The deformation of the unit cell is shown in Fig. 13(c).

Then it follows that

$$ {C}_{12}=\left(2{U}_{disc}^{(3)}/V-{C}_{11}-{C}_{22}\right)/2 $$

(A9)

• 4) Shearing test for C₆₆: by applying the unit strain to the unit cell

$$ {\varepsilon}_x={\varepsilon}_y=0,\kern0.33em {\gamma}_{xy}=1,\kern0.33em {\kappa}_{xz}={\kappa}_{yz}=0,\kern0.66em \mathrm{in}\kern0.33em \varOmega $$

(A10)

The corresponding boundary conditions are

$$ u=y/2,\kern0.33em v=x/2,\kern0.33em \mathrm{on}\kern0.33em \partial \varOmega $$

(A11)

The deformation of the unit cell is shown in Fig. 13(d).

Then it follows that

$$ {C}_{66}=2{U}_{disc}^{(4)}/V $$

(A12)

• 5) Bending test for D₁₁: by applying the mixed field of strain and stress to the unit cell

$$ {\varepsilon}_x=-y,{\sigma}_y=0,\kern0.33em {\gamma}_{xy}=0,\kern0.33em {\kappa}_{xz}=1,{\kappa}_{yz}=0,\kern0.66em \mathrm{in}\kern0.33em \varOmega $$

(A13)

The corresponding boundary conditions are

$$ {\left.u\right|}_{\partial \varOmega }=- xy,\kern0.33em {\left.v\right|}_{y=0}={x}^2/2\kern0.33em $$

(A14)

The deformation of the unit cell is shown in Fig. 13(e).

Then it follows that

$$ {D}_{11}=\left(2{U}_{disc}^{(5)}-{\int}_{\varOmega }{E}_x{y}^2 dV\right)/V $$

(A15)

• 6) Bending test for D₂₂: by applying the mixed field of strain and stress to the unit cell

$$ {\sigma}_x=0,{\varepsilon}_y=x,,\kern0.33em {\gamma}_{xy}=0,\kern0.33em {\kappa}_{xz}=0,{\kappa}_{yz}=1,\kern0.66em \mathrm{in}\kern0.33em \varOmega $$

(A16)

The corresponding boundary conditions are

$$ {\left.u\right|}_y=0=-{y}^2/2\kern0.33em ,\kern0.33em {\left.v\right|}_{\partial \varOmega }= xy $$

(A17)

The deformation of the unit cell is shown in Fig. 13(f).

Then it follows that

$$ {D}_{22}=\left(2{U}_{disc}^{(6)}-{\int}_{\varOmega }{E}_y{x}^2 dV\right)/V $$

(A18)

More detailed introduction and examples should be found in reference (Liu and Su 2009).

ρ and Θ are computed in the following:

• 1) Computing ρ: the effective density ρ is exactly the mean density of the base cell

$$ \rho ={\int}_{\varOmega_s}{\rho}_sd{\varOmega}_s/V $$

(A19)

where ρ_s denotes the density of the solid material of cellular solids.

• 2) Computing Θ: assuming the unit cell rotates about its centroid at a given angular velocity, then based on the equivalent kinetic energy, Θ is computed as

$$ \varTheta =\left({\rho}_s{\int}_{\varOmega_s}{r}^2d{\varOmega}_s-\rho {\int}_{\varOmega }{r}^2 d\varOmega \right)/V $$

(A20)

Detail derivations are given in reference (Su and Liu 2014).