Design of Frame-Like Periodic Solids for Isotropic Symmetry by Member Sizing

2016 ◽  
Vol 33 (1) ◽  
pp. 41-54 ◽  
Author(s):  
K. Theerakittayakorn ◽  
P. Suttakul ◽  
P. Sam ◽  
P. Nanakorn
Keyword(s):  
Closed Form ◽  
Elastic Constants ◽  
Strain Energy ◽  
Equivalent Strain ◽  
Homogenization Method ◽  
Unit Cell ◽  
Cubic Symmetry ◽  
Effective Elastic Constants ◽  
Common Unit ◽  
2D And 3D

AbstractIn this study, a methodology to design frame-like periodic solids for isotropic symmetry by appropriate sizing of unit-cell struts is presented. The methodology utilizes the closed-form effective elastic constants of 2D frame-like periodic solids with square symmetry and 3D frame-like periodic solids with cubic symmetry, derived using the homogenization method based on equivalent strain energy. By using the closed-form effective elastic constants, an equation to enforce isotropic symmetry can be analytically constructed. Thereafter, the equation can be used to determine relative unit-cell strut sizes that are required for isotropic symmetry. The methodology is tested with 2D and 3D frame-like periodic solids with some common unit-cell topologies. Satisfactory results are observed.

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.

scholarly journals A closed-form orthotropic constitutive model for fused filament fabrication materials

2019 ◽  
Author(s):  
Ruiqi Chen ◽  
Debbie G. Senesky
Keyword(s):  
Constitutive Model ◽  
Closed Form ◽  
Elastic Constants ◽  
Complex Variable ◽  
Shape Parameter ◽  
Antiplane Shear ◽  
Fused Filament Fabrication ◽  
Effective Elastic Constants ◽  
Out Of Plane ◽  
Simulation Results

We model two common fused filament fabrication mesostructures, square and hexagonal, using an orthotropic constitutive model and derive closed-form expressions for all nine effective elastic constants. The periodic void shapes are modeled using three and four point hypotrochoid curves with a single shape parameter that controls the sharpness of the points. Using the complex variable method of elasticity, we derive the in-plane elastic constants (Exx, Eyy, Gxy, nuxy) as well as out-of-plane antiplane shear constants (Gzx and Gzy). The remaining out-of-plane elastic constants (Ezz, nuzx, nuzy) are derived by directly solving the linearelasticity equations. We compare our results by conducting unit cell simulations on both mesostructures and at various porosity values. The simulations match the closed-form expressions exactly for Ezz, nuzx, and nuzy. For the remaining elastic constants, the simulation results match the closed-form expressions better for the square mesostructure than the hexagonal mesostructure. Differences between simulation and closed-form expressions are less than 10% for porosity values less than 6% (hexagonal mesostructure) and 10% (square mesostructure) for any of the nine elastic constants.

Effective elastic constants and acoustic properties of single-crystal TiN/NbN superlattices

1992 ◽  
Vol 7 (8) ◽  
pp. 2248-2256 ◽  
Author(s):  
Jin O. Kim ◽  
Jan D. Achenbach ◽  
Meenam Shinn ◽  
Scott A. Barnett
Keyword(s):  
Single Crystal ◽  
Elastic Constants ◽  
Acoustic Waves ◽  
Mass Density ◽  
Surface Acoustic Waves ◽  
Dispersion Curves ◽  
Acoustic Properties ◽  
Tetragonal Symmetry ◽  
Cubic Symmetry ◽  
Effective Elastic Constants

Using the measured elastic constants of TiN and NbN single crystals with cubic symmetry, the effective elastic constants of single-crystal TiN/NbN superlattice films with tetragonal symmetry, namely c11, c12, c13, c33, c44, and c66 have been calculated for various thickness ratios of the layers. Using a line-focus acoustic microscope, measurements of surface acoustic waves (SAWs) have been carried out on single-crystal TiN/NbN superlattice films grown on the (001) plane of cubic crystal MgO substrates. The phase velocities measured as functions of the angle of propagation display the expected anisotropic nature of cubic crystals. Dispersion curves of SAWs propagating along the symmetry axes have been obtained by measuring wave velocities for various film thicknesses and frequencies. The SAW dispersion curves calculated from the effectiveelastic constants and the effective mass density of the superlattice films show very good agreement with experimental results. The results of this paper exhibit no anomalous dependence of the elastic constants on the superlattice period of TiN/NbN superlattices.

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).

Effective Elastic Constants of Fiber-Eutectics and Transformation Particles Composite Ceramic

2010 ◽  
Vol 177 ◽  
pp. 182-185 ◽  
Author(s):  
Bao Feng Li ◽  
Jian Zheng ◽  
Xin Hua Ni ◽  
Ying Chen Ma ◽  
Jing Zhang
Keyword(s):  
Elastic Modulus ◽  
Elastic Constant ◽  
Effective Stress ◽  
Elastic Constants ◽  
Transverse Isotropy ◽  
Analytical Formula ◽  
Composite Ceramic ◽  
Phase Model ◽  
Composite Ceramics ◽  
Effective Elastic Constants

The composite ceramics is composed of fiber-eutectics, transformation particles and matrix particles. First, the recessive expression between the effective stress in fiber-eutectic and the flexibility increment tensor is obtained according to the four-phase model. Second, the analytical formula which contains elastic constant of the fiber-eutectic is obtained applying Taylor’s formula. The eutectic is transverse isotropy, so there are five elastic constants. Third, the effective elastic constants of composite ceramics are predicted. The result shows that the elastic modulus of composite ceramic is reduced with the increase of fibers fraction and fibers diameter.

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