Topology design for maximization of fundamental frequency of couple-stress continuum

Abstract

In this paper, the topology optimization formulation of couple-stress continuum is investigated for maximizing the fundamental frequency. A modified bound formulation is used to prevent the order switching and the eigenvalue repeating during the optimization procedure. Also, a modified stiffness interpolation with respect to the element density is employed to avoid the localized mode. Numerical examples are carried out to verify the effectiveness of the present formulation. The results show that the fundamental frequencies increase with the increasing bending modulus but decrease with the increasing inertia of micro-rotation for couple-stress continuum. In addition, the bending modulus contributes dominantly in enlarging the fundamental frequencies. Moreover, the results show that the characteristic length is a factor to determine whether the couple-stress effect is important. The optimal topology and fundamental frequency are both quite different from those of classical continuum when the characteristic length is large and vice versa.

Appendix: Definitions of engineering constants of isotropic micropolar material

In the case of isotropic micropolar material, which neglects the thermal effects, the constitutive equations take the form:

$$ \begin{array}{l}{\sigma}_{kl}=\lambda {\varepsilon}_{mm}{\delta}_{kl}+\left(\mu +\kappa \right){\varepsilon}_{kl}+\mu {\varepsilon}_{lk}\hfill \\ {}{m}_{kl}=\alpha {\kappa}_{mm}{\delta}_{kl}+\beta {\kappa}_{kl}+\gamma {\kappa}_{kl}\hfill \end{array} $$

(23)

where λ, μ, κ, α, β and γ are the material moduli.

The Young’s modulus E, shear modulus G and coupling parameter N are defined as follows (Eringen 1999):

$$ E\equiv \frac{\left(2\mu +\kappa \right)\left(3\lambda +2\mu +\kappa \right)}{2\lambda +2\mu +\kappa } $$

(24)

$$ G\equiv \mu +\frac{\kappa }{2} $$

(25)

$$ N\equiv {\left[\frac{\kappa }{2\left(\mu +\kappa \right)}\right]}^{\frac{1}{2}} $$

(26)