Micropolar homogenization constitutive modeling and size effect analysis of lattice materials

doi:10.1007/s10483-026-3338-9

Abstract

Abstract:

Lattice materials have demonstrated promising potential in engineering applications owing to their exceptional lightweight, high specific strength, and tunable mechanical properties. However, the traditional homogenization methods based on the classical elasticity theory struggle to accurately describe the non-classical mechanical behaviors of lattice materials, especially when dealing with complex unit-cell geometries featured by non-symmetric configurations or non-single central node connections. In response to this limitation, this study establishes a generalized homogenization model based on the micropolar theory framework, employing Hill’s boundary conditions to precisely predict the equivalent moduli of complex lattice materials. By introducing the independent rotational degree of freedom (DOF) characteristic of the micropolar theory, the proposed model successfully overcomes the limitation of conventional methods in accurately describing the asymmetric deformation and scale effects. We initially calculate the constitutive relations of two-dimensional (2D) cross-shaped multi-node chiral lattices and subsequently extend the method to three-dimensional (3D) lattices, successfully predicting the mechanical properties of both traditional and eccentric body-centered cubic (BCC) lattices. The theoretical model is validated through the finite element numerical verification which shows excellent consistency with the theoretical predictions. A further parametric study investigates the influence of geometric parameters, revealing the underlying size-effect mechanism. This paper provides a reliable theoretical tool for the design and property optimization of complex lattice materials.

Key words: lattice material, size effect, micropolar theory, homogenization method, constitutive relationship

2010 MSC Number:

O343.7

Tingrui CHEN, Fan YANG, Jingchun ZHANG, Dong HAN, Qingcheng YANG. Micropolar homogenization constitutive modeling and size effect analysis of lattice materials. Applied Mathematics and Mechanics (English Edition), 2026, 47(1): 39-60.

TrendMD

Figures/Tables 12

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 1

Constitutive parameters for the zigzag chiral lattice with θ=30∘"

$L$ /mm	$A 1111$ /MPa	$A 1112$ /MPa	$A 1212$ /MPa	$H 1313$ /MPa^-1
40	3.024 7	$− 0.387 4$	0.203 9	$2.030 3 × 10 − 2$
4	3.024 7	$− 0.387 4$	0.203 9	$2.030 3 × 102$
0.4	3.024 7	$− 0.387 4$	0.203 9	$2.030 3 × 106$
0.04	3.024 7	$− 0.387 4$	0.203 9	$2.030 3 × 1010$
0.004	3.024 7	$− 0.387 4$	0.203 9	$2.030 3 × 1014$

Table 1

Fig. 7

Fig. 8

Fig. 9

Table 2

Relationship between the constitutive constants and geometrical parameters for BCC lattice structures"

$d / l$	$l$ /mm	$A 1111$ /MPa	$A 1122$ /MPa	$A 1212$ /MPa	$H 1111$ /MPa^-1	$H 1122$ /MPa^-1	$H 1212$ /MPa^-1
0.1	0.001	5.235	5.196	5.215	$6.314 × 107$	$3.754 × 107$	$6.333 × 107$
0.1	0.1	5.235	5.196	5.215	$6.314 × 103$	$3.754 × 103$	$6.333 × 103$
0.1	10	5.235	5.196	5.215	$6.314 × 10 − 1$	$3.754 × 10 − 1$	$6.333 × 10 − 1$
0.2	0.001	21.25	20.63	20.94	$3.946 × 106$	$2.346 × 106$	$3.993 × 106$
0.2	0.1	21.25	20.63	20.94	$3.946 × 102$	$2.346 × 102$	$3.993 × 102$
0.2	10	21.25	20.63	20.94	$3.946 × 10 − 2$	$2.346 × 10 − 2$	$3.993 × 10 − 2$
0.4	0.001	90.01	80.01	85.01	$2.466 × 105$	$1.467 × 105$	$2.578 × 105$
0.4	0.1	90.01	80.01	85.01	$2.466 × 101$	$1.467 × 101$	$2.578 × 101$
0.4	10	90.01	80.01	85.01	$2.466 × 10 − 3$	$1.467 × 10 − 3$	$2.578 × 10 − 3$

Table 2

Fig. 10

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