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# A nonlinear mechanics model of bio-inspired hierarchical lattice materials consisting of horseshoe microstructures.

^{1}, Cheng H

^{2}, Jang KI

^{3}, Luan H

^{4}, Hwang KC

^{1}, Rogers JA

^{3}, Huang Y

^{4}, Zhang Y

^{1}.

### Author information

- 1
- Center for Mechanics and Materials, AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P.R. China.
- 2
- Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA.
- 3
- Department of Materials Science and Engineering and Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.
- 4
- Department of Civil and Environmental Engineering; Department of Mechanical Engineering; Department of Materials Science and Engineering; Center for Engineering and Health; Skin Disease Research Center; Northwestern University, Evanston, IL 60208, USA.

### Abstract

Development of advanced synthetic materials that can mimic the mechanical properties of non-mineralized soft biological materials has important implications in a wide range of technologies. Hierarchical lattice materials constructed with horseshoe microstructures belong to this class of bio-inspired synthetic materials, where the mechanical responses can be tailored to match the nonlinear J-shaped stress-strain curves of human skins. The underlying relations between the J-shaped stress-strain curves and their microstructure geometry are essential in designing such systems for targeted applications. Here, a theoretical model of this type of hierarchical lattice material is developed by combining a finite deformation constitutive relation of the building block (i.e., horseshoe microstructure), with the analyses of equilibrium and deformation compatibility in the periodical lattices. The nonlinear J-shaped stress-strain curves and Poisson ratios predicted by this model agree very well with results of finite element analyses (FEA) and experiment. Based on this model, analytic solutions were obtained for some key mechanical quantities, e.g., elastic modulus, Poisson ratio, peak modulus, and critical strain around which the tangent modulus increases rapidly. A negative Poisson effect is revealed in the hierarchical lattice with triangular topology, as opposed to a positive Poisson effect in hierarchical lattices with Kagome and honeycomb topologies. The lattice topology is also found to have a strong influence on the stress-strain curve. For the three isotropic lattice topologies (triangular, Kagome and honeycomb), the hierarchical triangular lattice material renders the sharpest transition in the stress-strain curve and relative high stretchability, given the same porosity and arc angle of horseshoe microstructure. Furthermore, a demonstrative example illustrates the utility of the developed model in the rapid optimization of hierarchical lattice materials for reproducing the desired stress-strain curves of human skins. This study provides theoretical guidelines for future designs of soft bio-mimetic materials with hierarchical lattice constructions.

#### KEYWORDS:

Bio-inspired materials; Finite deformation; Hierarchical design; Horseshoe microstructure; Lattice materials; Stress-strain curves

- PMID:
- 27087704
- PMCID:
- PMC4831080
- DOI:
- 10.1016/j.jmps.2016.02.012