A methodology for achieving the maximum bulk or shear modulus in an elastic composite composed of two isotropic phases with distinct Poisson’s ratios is proposed. A topology optimization algorithm is developed which is capable of finding microstructures with extreme properties very close to theoretical upper bounds. The effective mechanical properties of the designed composite are determined by a numerical homogenization technique. The sensitivities with respect to design variables are derived by simultaneously interpolating Young’s modulus and Poisson’s ratio using different parameters. The so-called solid isotropic material with penalization method is developed to establish the optimization formulation. Maximum bulk or shear modulus is considered as the objective function, and the volume fraction of constituent phases is taken as constraints. The method of moving asymptotes is applied to update the design variables. Several 3D numerical examples are presented to demonstrate the effectiveness of the proposed structural optimization method. The effects of key parameters such as Poisson’s ratios and volume fractions of constituent phase on the final designs are investigated. A series of novel microstructures are obtained from the proposed approach. It is found that the optimized bulk and shear moduli of all the studied composites are very close to the Hashin-Shtrikman-Walpole bounds.

Methods to achieve optimal engineering designs have been intensively investigated for decades. Various topology optimization approaches have been proposed and developed in the field of structural designs [

The bulk modulus and shear modulus are generally devoted to describe the stiffness of an isotropic elastic material. And earlier work in topology optimization of materials normally involves maximizing the bulk modulus or shear modulus of a 2D or 3D material. Except for the existing composites such as composite sphere, finite rank laminates and Vigdergauz microstructures, Sigmund [

In comparison to traditional cellular materials composed of a solid phase and a void phase, composites made of two or more materials are more preferable to practical applications due to their synthetic mechanical properties. Conventional topological design of composite microstructure with maximized stiffness always involves constituent phases with different Young’s moduli and the same Poisson’s ratios. However, Liu et al. [

The rest of the paper is organized as follows. The homogenized effective elasticity matrix is calculated within periodic base cell (PBC) in Section 2. Also, the sensitivity of the elastic matrix is derived from the interpolation of both Young’s modulus and Poisson’s ratio. Section 3 establishes the topological optimization formulation the microstructure. Section 4 provides several typical numerical examples to validate the effectiveness of the proposed approach. Finally, the main conclusion of this paper is drawn in Section 5.

For a two-phase periodic heterogeneous material, the macroscopic elasticity tensor

where _{pqrs}

Here

Based on

The constituent phases are assumed to be isotropic in this study. To derive the sensitivities, a SIMP function for both the Young’s modulus and Possion’s ratio is adopted to interpolate between two phases in each element [

where _{j}_{e}_{e}_{e}

We assume the materials used to build the composite are isotropic, the elastic matrix for the element

where

Here for simplicity, the subscript e is omitted in

From

Using the chain rule, the sensitivity of the elastic matrix for the element _{e}

Thus, the macroscopic elasticity tensor with respect to density variable can be derived by using the adjoint method, which is

where

The bulk modulus

To obtain optimal design of composites with homogenized properties, the optimization problem is defined through maximizing bulk modulus or shear modulus subjected to volume fraction constraints of phase 1.

where _{1} (_{0} are the volume of phase 1 material and the volume of design domain,

The bounds of physical properties are of significant importance in composite design. HS bound is one of the most popular estimations, which defines tight theoretical limits of bulk and shear moduli for isotropic materials, in terms of volume fraction of two-phase composites based on variational principles. Such bounds are achievable when the microstructures and volume fraction of individual phase are determined [_{2} − _{1}) (_{2} − _{1}) > 0. Walpole [_{2} − _{1}) (_{2} − _{1}) < 0).

The bulk and shear moduli of each phase are defined as

For two-phase composites with 3D microstructure, the HSW upper limits on effective bulk modulus ^{U} and shear modulus ^{U} can be found by

where

The HSW upper bound is attainable if the following conditions are satisfied.

Here in it is assumed that

In this section, several numerical examples are presented to demonstrate the effectiveness and capabilities of the proposed optimization method. It is assumed that in all cases the PBC, which represents the microstructure of a composite, is discretized to

In this study, we simply define two different initial designs. In initial design 1, the element density increases proportionally to the distance from the element center to the base cell center, as shown in _{e}

For easier identification, the phase 1 in the microstructure is displayed in red, while phase 2 in blue.

The objective of topology optimization of this example is to obtain microstructure with maximum bulk modulus. In this example, the material properties are defined as: Young’s modulus, _{1} = 2.5 for phase 1 and _{2} = 1.0 for phase 2; Poisson’s ratio, _{1} = 0.8333 and _{2} = 0.1190 for the two phases, respectively. The prescribed volume fraction of phase 1 is 60%. The obtained optimal microstructures and their effective matrices are given in

It can be seen that the obtained microstructures are cubic symmetry, thus the two-phase composites are orthotropic. The bulk moduli can be calculated from the obtained microstructures as 0.5148 and 0.5128, starting from initial designs 1 and 2, respectively. These results are slightly below the HSW upper bound ^{U} = 0.5159 according to ^{U} = 1.1432 according to

The stiffness of resulting composites in one or more directions could be enhanced, as the benefits of using phase with significantly different Poisson’s ratios. For the purpose of comparison, the upper bound of effective Young’s modulus estimated by Voigt theory can be calculated as

To better understand the influence of constituent phase on the topology of optimal composites, optimization process on design domain of merely phase 1 is performed. The two optimal microstructures and the corresponding elasticity matrices of the composite are presented in

The microstructure obtained from initial design 1 in

To investigate the effect of the Poisson’s ratio on the maximum bulk modulus and the HSW bounds, the change in Poisson’s ratios of phase 2 are manually defined, ranging from −0.95 to 0.45 with two ends, i.e., −0.99 and 0.49, while phase 1 has a fixed Poisson’s ratio. Meanwhile, the prescribed volume fraction of phase 1 is set to be 50%. Other parameters are the same as those in Example I.

Poisson’s ratio ( |
Resulting bulk modulus | HSW upper bound (bulk modulus) | Relative difference (%) |
---|---|---|---|

−0.99 | 0.4674 | 0.4707 | 0.698 |

−0.95 | 0.4566 | 0.4648 | 1.764 |

−0.85 | 0.4415 | 0.4528 | 2.489 |

−0.75 | 0.4369 | 0.4444 | 1.692 |

−0.65 | 0.4373 | 0.4396 | 0.528 |

−0.55 | 0.4424 | 0.4434 | 0.233 |

−0.45 | 0.4505 | 0.4545 | 0.888 |

−0.35 | 0.4624 | 0.4682 | 1.235 |

−0.25 | 0.4783 | 0.4852 | 1.426 |

−0.15 | 0.5007 | 0.5072 | 1.298 |

−0.05 | 0.5306 | 0.5367 | 1.145 |

0.05 | 0.5732 | 0.5782 | 0.876 |

0.15 | 0.6374 | 0.6410 | 0.565 |

0.25 | 0.7452 | 0.7471 | 0.254 |

0.35 | 0.9639 | 0.9649 | 0.238 |

0.45 | 1.6654 | 1.6667 | 0.258 |

0.49 | 2.7430 | 2.7333 | 0.355 |

In this example, the effect of various volume fractions of constitute phase on resulting bulk modulus is investigated. The Young’s moduli of two constituent materials are _{1} = 2.5 and _{2} = 1, respectively, and the Poisson’s ratio are

Volume fraction of phase 1 (%) | Resulting bulk modulus | HSW upper bound (bulk modulus) | Relative difference (%) |
---|---|---|---|

15 | 0.2396 | 0.2431 | 1.440 |

25 | 0.2944 | 0.2976 | 1.071 |

35 | 0.3519 | 0.3554 | 0.971 |

45 | 0.4136 | 0.4167 | 0.733 |

55 | 0.4795 | 0.4818 | 0.466 |

65 | 0.5509 | 0.5511 | 0.030 |

75 | 0.6249 | 0.6250 | 0.017 |

85 | 0.7039 | 0.7040 | 0.013 |

The objective of this example is to obtain the optimal microstructure with maximum shear modulus via the proposed optimization method. The Young’s moduli of constituent phases are selected as _{1} = 5.5 and _{2} = 1. The Poisson’s ratios are assumed to be _{1} = 2.75 and _{2} = 1.00, respectively. The prescribed volume fraction of phase 1 is 60%. According to

From the effective matrices, the shear moduli of the two designs are 1.8668 and 1.8677, respectively. As discussed previously, the two types of topologies differ significantly due to the distinct initial designs. However, the shear moduli are almost identical and slightly below the respective HSW bounds for shear modulus.

The objective of this example is to find the optimal microstructure with a maximum shear modulus. The Young’s modulus of constituent phase 1 is investigated and set to be ranging from 3.0 to 11.0. Other mechanical properties are the same as those in example IV. The prescribed volume fraction of phase 1 is defined as 50%.

Young’s modulus (_{2}) |
Resulting shear modulus | HSW upper bound (shear modulus) | Relative difference (%) |
---|---|---|---|

3.0 | 1.2189 | 1.2256 | 0.544 |

4.0 | 1.4094 | 1.4231 | 0.958 |

5.0 | 1.5932 | 1.6071 | 0.866 |

6.0 | 1.7695 | 1.7838 | 0.803 |

7.0 | 1.9416 | 1.9559 | 0.731 |

8.0 | 2.1089 | 2.1250 | 0.757 |

9.0 | 2.2800 | 2.2921 | 0.524 |

10.0 | 2.4426 | 2.4576 | 0.610 |

11.0 | 2.6048 | 2.6221 | 0.660 |

_{2}, the maximizing shear modulus and HSW upper bound rises. The microstructure of phase 2 is a Schwartz P structure when _{2} = 3.0. The Schwartz P structure and sphere inclusions for each phase coexist in the base cell when the Young’s modulus of phase 2 increases. The configurations of the microstructure are affected by the Young’s modulus of constitute phases. In all cases, the resulting shear moduli well agree with HSW upper bound calculated by

In this example, various volume fractions of the constituent phase on the final design are investigated. The mechanical properties and objective function are the same with those in example IV. The prescribed volume fraction of phase 1 ranges from15% to 85%.

Volume fraction of phase 1 (%) | Resulting shear modulus | HSW upper bound (shear modulus) | Relative difference (%) |
---|---|---|---|

15 | 1.1461 | 1.1827 | 3.095 |

25 | 1.2824 | 1.3158 | 2.540 |

35 | 1.4338 | 1.4592 | 1.739 |

45 | 1.5975 | 1.6140 | 1.020 |

55 | 1.7726 | 1.7818 | 0.514 |

65 | 1.9629 | 1.9641 | 0.065 |

75 | 2.1628 | 2.1631 | 0.014 |

85 | 2.3683 | 2.3811 | 0.538 |

From

In this paper, a topology optimization methodology is presented to design microstructures of materials that are composed of two isotropic phases with distinct Poisson’s ratios. The objective is to maximize the bulk or shear modulus of the composites. The effective elasticity matrix is determined by means of homogenization. Several 3D examples are presented to demonstrate the effectiveness of the proposed method. The examples show that the proposed approach is capable of generating composite microstructures with properties very close to the Hashin–Shtrikan–Walpole upper bounds. In the future research, several important directions furthered the proposed method includes the concurrent design of macrostructure and microstructure containing multiple phases [