A Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials Using the Micropolar Elasticity Theory

2021 ◽  
pp. 113594
Author(s):  
A.I. Gad ◽  
X.L. Gao ◽  
K. Li
Keyword(s):  
Elasticity Theory ◽  
Strain Energy ◽  
Homogenization Method ◽  
Cellular Materials ◽  
Micropolar Elasticity

scholarly journals A Generalized Strain Energy-Based Homogenization Method for 2-D and 3-D Cellular Materials with and without Periodicity Constraints

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1870
Author(s):  
Ahmad I. Gad ◽  
Xin-Lin Gao
Keyword(s):  
Strain Energy ◽  
Homogenization Method ◽  
Cellular Materials ◽  
Cellular Material ◽  
Equilibrium Equations ◽  
New Approach ◽  
Cubic Symmetry ◽  
Representative Cell ◽  
The Matrix ◽  
Hill’S Lemma

A generalized strain energy-based homogenization method for 2-D and 3-D cellular materials with and without periodicity constraints is proposed using Hill’s Lemma and the matrix method for spatial frames. In this new approach, the equilibrium equations are enforced at all boundary and interior nodes and each interior node is allowed to translate and rotate freely, which differ from existing methods where the equilibrium conditions are imposed only at the boundary nodes. The newly formulated homogenization method can be applied to cellular materials with or without symmetry. To illustrate the new method, four examples are studied: two for a 2-D cellular material and two for a 3-D pentamode metamaterial, with and without periodic constraints in each group. For the 2-D cellular material, an asymmetric microstructure with or without periodicity constraints is analyzed, and closed-form expressions of the effective stiffness components are obtained in both cases. For the 3-D pentamode metamaterial, a primitive diamond-shaped unit cell with or without periodicity constraints is considered. In each of these 3-D cases, two different representative cells in two orientations are examined. The homogenization analysis reveals that the pentamode metamaterial exhibits the cubic symmetry based on one representative cell, with the effective Poisson’s ratio v¯ being nearly 0.5. Moreover, it is revealed that the pentamode metamaterial with the cubic symmetry can be tailored to be a rubber-like material (with v¯ ≅0.5) or an auxetic material (with v¯< 0).

Effects of Shear Deformation of Struts in Hexagonal Lattices on their Effective In-Plane Material Properties

2021 ◽  
Vol 1034 ◽  
pp. 193-198
Author(s):  
Pana Suttakul ◽  
Thongchai Fongsamootr ◽  
Duy Vo ◽  
Pruettha Nanakorn
Keyword(s):  
Shear Deformation ◽  
Material Properties ◽  
Strain Energy ◽  
Equivalent Strain ◽  
Homogenization Method ◽  
Two Dimensional ◽  
Engineering Applications ◽  
Large Numbers ◽  
Effective Material Properties ◽  
Unit Cells

Two-dimensional lattices are widely used in many engineering applications. If 2D lattices have large numbers of unit cells, they can be accurately modeled as 2D homogeneous solids having effective material properties. When the slenderness ratios of struts in these 2D lattices are low, the effects of shear deformation on the values of the effective material properties can be significant. This study aims to investigate the effects of shear deformation on the effective material properties of 2D lattices with hexagonal unit cells, by using the homogenization method based on equivalent strain energy. Several topologies of hexagonal unit cells and several slenderness ratios of struts are considered. The effects of struts’ shear deformation on the effective material properties are examined by comparing the results of the present study, in which shear deformation is neglected, with those from the literature, in which shear deformation is included.

Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations

2016 ◽  
Vol 22 (2) ◽  
pp. 224-242 ◽  
Author(s):  
Soroosh Hassanpour ◽  
Glenn R. Heppler
Keyword(s):  
Elasticity Theory ◽  
Elastic Constants ◽  
Couple Stress ◽  
Elastic Moduli ◽  
Experimental Results ◽  
Experimental Investigations ◽  
Micropolar Elasticity ◽  
Classical Elasticity ◽  
Material Constants ◽  
Isotropic Theory

This paper is devoted to a review of the linear isotropic theory of micropolar elasticity and its development with a focus on the notation used to represent the micropolar elastic moduli and the experimental efforts taken to measure them. Notation, not only the selected symbols but also the approaches used for denoting the material elastic constants involved in the model, can play an important role in the micropolar elasticity theory especially in the context of investigating its relationship with the couple-stress and classical elasticity theories. Two categories of notation, one with coupled classical and micropolar elastic moduli and one with decoupled classical and micropolar elastic moduli, are examined and the consequences of each are addressed. The misleading nature of the former category is also discussed. Experimental investigations on the micropolar elasticity and material constants are also reviewed where one can note the questionable nature and limitations of the experimental results reported on the micropolar elasticity theory.

scholarly journals Finite Element Analysis of Bend Test of Sandwich Structures Using Strain Energy Based Homogenization Method

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Hassan Ijaz ◽  
Waqas Saleem ◽  
Muhammad Zain-ul-Abdein ◽  
Tarek Mabrouki ◽  
Saeed Rubaiee ◽  
...  
Keyword(s):  
Finite Element ◽  
Strain Energy ◽  
Effective Properties ◽  
Sandwich Structures ◽  
Sandwich Beam ◽  
Energy Criterion ◽  
Homogenization Method ◽  
Fe Analysis ◽  
Bend Test ◽  
Element Analysis

The purpose of this article is to present a simplified methodology for analysis of sandwich structures using the homogenization method. This methodology is based upon the strain energy criterion. Normally, sandwich structures are composed of hexagonal core and face sheets and a complete and complex hexagonal core is modeled for finite element (FE) structural analysis. In the present work, the hexagonal core is replaced by a simple equivalent volume for FE analysis. The properties of an equivalent volume were calculated by taking a single representative cell for the entire core structure and the analysis was performed to determine the effective elastic orthotropic modulus of the equivalent volume. Since each elemental cell of the hexagonal core repeats itself within the in-plane direction, periodic boundary conditions were applied to the single cell to obtain the more realistic values of effective modulus. A sandwich beam was then modeled using determined effective properties. 3D FE analysis of Three- and Four-Point Bend Tests (3PBT and 4PBT) for sandwich structures having an equivalent polypropylene honeycomb core and Glass Fiber Reinforced Plastic (GFRP) composite face sheets are performed in the present study. The authenticity of the proposed methodology has been verified by comparing the simulation results with the experimental bend test results on hexagonal core sandwich beams.

Step-by-Step Simplification of the Micropolar Elasticity Theory to the Couple-Stress and Classical Elasticity Theories

2014 ◽  
Author(s):  
Soroosh Hassanpour ◽  
G. R. Heppler
Keyword(s):  
Elasticity Theory ◽  
Linear Theory ◽  
Elastic Constants ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Material Model ◽  
Micropolar Elasticity ◽  
Classical Elasticity ◽  
Stress Theory ◽  
Special Case

The micropolar elasticity theory provides a useful material model for dealing with fibrous, coarse granular, and large molecule materials. Though being a well-known and well-developed elasticity model, the linear theory of micropolar elasticity is not without controversy. Specially simplification of the microppolar elasticity theory to the couple-stress and classical elasticity theories and the required conditions on the material elastic constants for this simplification have not been discussed consistently. In this paper the linear theory of micropolar elasticity is reviewed first. Then the correct approach for a consistent and step-by-step simplification of the micropolar elasticity model with six elastic constants to the couple-stress elasticity model with four elastic constants and the classical elasticity model with two elastic constants is presented. It is shown that the classical elasticity is a special case of the couple-stress theory which itself is a special case of the micropolar elasticity theory.

A 149 Line Homogenization Code for Three-Dimensional Cellular Materials Written in matlab

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
Guoying Dong ◽  
Yunlong Tang ◽  
Yaoyao Fiona Zhao
Keyword(s):  
Material Properties ◽  
Three Dimensional ◽  
Homogenization Method ◽  
Cellular Materials ◽  
Cellular Structures ◽  
Numerical Homogenization ◽  
Matlab Code ◽  
Voxel Model ◽  
Anisotropic Elastic ◽  
Constitutive Matrix

Cellular architectures are promising in a variety of engineering applications due to attractive material properties. Additive manufacturing has reduced the difficulty in the fabrication of three-dimensional (3D) cellular materials. In this paper, the numerical homogenization method for 3D cellular materials is provided based on a short, self-contained matlab code. It is an educational description that shows how the homogenized constitutive matrix is computed by a voxel model with one material to be void and another material to be solid. A voxel generation algorithm is proposed to generate the voxel model easily by the wireframe scripts of unit cell topologies. The format of the wireframe script is defined so that the topology can be customized. The homogenization code is then extended to multimaterial cellular structures and thermal conductivity problems. The result of the numerical homogenization shows that different topologies exhibit anisotropic elastic properties to a different extent. It is also found that the anisotropy of cellular materials can be controlled by adjusting the combination of materials.

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