A microarchitecture design methodology to achieve extreme isotropic elastic properties of composites based on crystal symmetries

Abstract

The present contribution describes an optimization-based design technique of elastic isotropic periodic microarchitectures with crystal symmetries aiming at the realization of composites with extreme properties. To achieve this goal, three consecutive procedures are followed: (i) a series of inverse homogenization problems with symmetry constraints, (ii) a correlation analysis between symmetries and effective elastic properties of the attained microarchitectures, and (iii) the pattern resemblance recognition of these topologies and their redesign, by adopting microstructures with two length-scales, through optimized parametric geometries. This paper is devoted to assessing the third procedure because the first two procedures have been evaluated in previous works of the authors, and here they are only summarized. By applying the methodology, two plane group symmetries are assessed to define two families of 2D periodic parameterized microarchitecture. Once the parameters have been optimized, the resulting composites achieve elastic isotropic properties close to the whole range of the theoretically estimated bounds. Particularly, an unprecedented microstructure attaining the theoretical maximum stiffness is reported. Starting from these parameterized topologies, simple, one-length scale, and easily manufacturable geometries are defined. One of the so-designed microarchitectures has been manufactured and tested, displaying an effective Poisson’s ratio of − 0.90 simultaneously with a high shear modulus.

Appendix. Equations used in the multi-objective optimization problems

In the mathematical formulation of the multi-objective problem (10), the volume fraction constraints are written as follows.

In the case of p31m microstructure, replacing (11) into (10):

$$ \gamma \ {\varOmega}_{\text{lam}}(\alpha,t) + {\varOmega}_{1}(\alpha,t) - {f_{1}^{t}} \ {\varOmega}_{\mu} = 0 , $$

(13)

where:

$$ \begin{array}{@{}rcl@{}} {\varOmega}_{\text{lam}} &=& 6 \times (\frac{\sqrt{3}/2}{\sin(120^{\circ} - \alpha/2) + \sin(\alpha/2)} \times L \times t - \frac{\sin(\beta_{1}+60^{\circ}-\alpha/2)}{\sin(\beta_{1}+90^{\circ}-\alpha/2)} \\ &&\times t^{2} - \frac{\sin(\beta_{2}-60^{\circ}+\alpha/2)}{\sin(\beta_{2}-30^{\circ}+\alpha/2)} \times t^{2}) , \\ &&{\varOmega}_{1} = \frac{3}{2} \sin(30^{\circ} + \alpha/2) \times (\frac{1}{\sin(\alpha/2)} + \frac{1}{\cos(30^{\circ} - \alpha/2)}) \times t^{2} ,\\ &&{\varOmega}_{\mu} = \frac{\sqrt{3}}{2} L^{2} . \end{array} $$

(14)

The angles β₁ and β₂ depend on α and are calculated from the following expressions:

$$ \begin{array}{@{}rcl@{}} \tan(\beta_{1}) = \frac{3 - 2\times \sin(30^{\circ} + \alpha)}{\sqrt{3} - 2 \times \cos(30^{\circ} + \alpha)} , \\ \tan(\beta_{2}) = \frac{3 + 2\times \cos(\alpha)}{\sqrt{3} + 2 \times \sin(\alpha)} . \end{array} $$

(15)

In the case of the p3m1, by replacing (12) into (10):

$$ \gamma_{1} \ {\varOmega}_{\text{lam}1}(d,t) + \gamma_{2} \ {\varOmega}_{\text{lam}2}(d,t) + \gamma_{3} \ {\varOmega}_{\text{lam}3}(d,t) + {\varOmega}_{1}(d,t) - {f_{1}^{t}} \ {\varOmega}_{\mu}= 0 , $$

(16)

where γ₁, γ₂, γ_3, and ${f_{1}^{t}}$ are parameters defined in Fig. 5 and:

$$ \begin{array}{@{}rcl@{}} {\varOmega}_{\text{lam}1} = {\varOmega}_{\text{lam}2} = 3 \times d \times L \times t - \sqrt{3} \times t^{2} ,\\ {\varOmega}_{\text{lam}3} = 3 \times (L - 2 \times d \times L)\times t - \sqrt{3} \times t^{2} ,\\ {\varOmega}_{1} = \frac{3 \sqrt{3}}{2} t^{2} ,\\ {\varOmega}_{\mu} = \frac{\sqrt{3}}{2} L^{2} . \end{array} $$

(17)

The d parameter may increase beyond the vanishing of the γ₃ laminate region (see for example microstructures P4 and P5 in Fig. 7). In this situation, the relation between the parameters to respect a given volume fraction comes from:

$$ \gamma_{1} \ {\varOmega}_{\text{lam}1}(d,t) + \gamma_{2} \ {\varOmega}_{\text{lam}2}(d,t) + {\varOmega}_{1}(d,t) - {f_{1}^{t}} \ {\varOmega}_{\mu} = 0 , $$

(18)

where:

$$ \begin{array}{@{}rcl@{}} {\varOmega}_{\text{lam}1} &=& {\varOmega}_{\text{lam}2} = 3 \times (L - d \times L)\times t - 2\times \sqrt{3} \times t^{2} ,\\ &&{\varOmega}_{1} = 2 \times \sqrt{3} \times t^{2} - 6 \times \sqrt{3} \times (L/2 - d \times L)^{2} . \end{array} $$

(19)