PaperlayerView exact revision on arXiv

Source-linked AI summary

Determination of Metamaterial Parameters by Means of a Homogenization Approach Based on Asymptotic Analysis

H. Yang, B. E. Abali, W. H. Müller, D. Timofeev

arXiv:1906.07556v1cs.CE

TL;DR

Predicting metamaterial behavior is difficult when the macroscopic structure is not much larger than its microscale substructure, a regime where classical homogenization has limitations. This paper uses asymptotic homogenization to connect first-order microscale elasticity with second-order strain-gradient macroscale theory and computes parameters by FEM. The approach yields parameters that vanish as material becomes homogeneous and are independent of RVE repetition.

  • Problem

    Classical homogenization has limitations when the macroscopic length scale is comparable to the microscale, where size effects are not captured by same-order theories.

    Source: §1 · p. 2

  • Method

    The approach derives macroscale strain-gradient parameters through asymptotic homogenization and energy equivalence, then solves the resulting boundary-value problems using FEM.

    Source: §1 · p. 3, §2.2 · p. 7, §3 · p. 10

  • Results

    The proposed method derives classical and strain-gradient stiffness tensors for a square lattice, with strain-gradient parameters vanishing as the material becomes homogeneous and independent of RVE repetition.

    Source: §6 · p. 19

  • Takeaways & Limitations

    The method addresses compatibility with homogeneous material and reliability across repeated-cell RVE choices for the computed strain-gradient parameters.

    Source: §6 · p. 19

    The microscale displacement decomposition assumes a smooth macroscopic displacement field and y-periodic local fluctuations with vanishing mean within each RVE.

    Source: §2.2 · p. 7

Abstract

from arXiv · show

Owing to additive manufacturing techniques, a structure at millimeter length scale (macroscale) can be produced by using a lattice substructure at micrometer length scale (microscale). Such a system is called a metamaterial at the macroscale as the mechanical characteristics deviate from the characteristics at the microscale. As a remedy, metamaterial is modeled by using additional parameters; we intend to determine them. A homogenization approach based on the asymptotic analysis establishes a connection between these different characteristics at micro- and macroscales. A linear elastic first order theory at the microscale is related to a linear elastic second order theory at the macroscale. Relation for parameters at the macroscale is derived by using the equivalence of energy at macro- and microscales within a so-called Representative Volume Element (RVE). Determination of parameters are succeeded by solving a boundary value problem with the Finite Element Method (FEM). The proposed approach guarantees that the additional parameters vanish if the material is purely homogeneous, in other words, it is fully compatible with conventional homogenization schemes based on spatial averaging techniques. Moreover, the proposed approach is reliable as it ensures that such resolved additional parameters are not sensitive to choices of RVE consisting in the repetition of smaller RVEs but depend upon the intrinsic size of the structure.

1 Introduction

Metamaterial response depends on both material and substructure morphology, while detailed FEM modeling is costly and classical homogenization has limits when macro- and microscales are comparable. The paper proposes an asymptotic homogenization approach using strain-gradient theory to determine macroscale parameters while addressing compatibility and reliability requirements.

1 Introduction

  • Metamaterial response depends on both the material and the morphology of its repetitive substructure.Periodic lattice structures are studied for properties including relatively low manufacturing costs and high specific stiffness.Source: §1 · p. 2
  • Detailed FEM modeling can be computationally costly, motivating homogenization to upscale microscale response through a macroscale constitutive equation.The mesh size must be at least one order smaller than the substructure's geometric size.Source: §1 · p. 2
  • Classical homogenization is limited when macro- and microscales are comparable, while higher-gradient terms are a natural way to include size effects.The proposed direction relates first-order microscale theory to second-order macroscale theory with additional parameters.Source: §1 · p. 2
  • The approach must ensure strain-gradient stiffness parameters vanish for homogeneous material and remain insensitive to repetition of the basic cell.Source: §1 · p. 3
  • The paper presents an alternative derivation of a correction term and applies FEM to determine 2D metamaterial parameters using a linear elastic microscale model.The method is applied to simple general 2D metamaterials and a square lattice.Source: §1 · p. 3

2 Connection of micro- and macroscale parameters

The section connects a heterogeneous microscale RVE to a homogeneous macroscale model by equating their stored energies, using first-order theory at the microscale and second-order theory at the macroscale. Asymptotic scale separation and the resulting periodic corrector problems provide the basis for deriving effective parameters.

2 Connection of micro- and macroscale parameters

  • The method identifies macroscale parameters by equating stored energies for a heterogeneous microscale RVE and its homogeneous macroscale counterpart, using first- and second-order theories respectively.The microscale material properties are known, while the macroscale parameters are sought.Source: §2 · p. 4

2.1 The macroscale energy for an RVE

  • A quadratic Taylor approximation of macroscale displacement accounts for strain-gradient effects and expresses the RVE energy in terms of macroscopic deformation gradients.The resulting macroscale energy is used to connect parameters to the microscale energy.Source: §2.1 · pp. 5–6

2.2 The microscale energy for an RVE

  • Asymptotic homogenization separates global displacement variation from local fluctuations using global coordinates X and local coordinates y.The RVE size in local coordinates can be varied by the homothetic ratio while its global size remains fixed; the macroscopic field is smooth and local fluctuations are y-periodic with vanishing mean.Source: §2.2 · pp. 6–7
  • Periodic corrector fields ϕ and ψ solve partial differential equations, and the resulting microscale energy is compared with the macroscale expression to generate an effective-parameter algorithm.For centro-symmetric materials, the rank-5 tensor vanishes.Source: §2.2 · pp. 8–10

3 Numerical solution of strain gradient homogenization problems

The authors solve the microscale problems on a two-dimensional RVE using Galerkin FEM with periodic boundary conditions, then use the solutions to determine classical and strain-gradient stiffness tensors.

3 Numerical solution of strain gradient homogenization problems

  • Galerkin FEM solves Eqs. (20) and (21) on a two-dimensional RVE with periodic Dirichlet boundary conditions.The calculations are restricted to indices in {1, 2}.Source: §3 · p. 10
  • Solutions for the classical coefficients yield ϕ, and solutions for the strain-gradient coefficients yield ψ, which are used to determine CM and DM.The procedure solves for ab = 11, 12, 21, 22 and for the eight listed abc combinations.Source: §3 · p. 10
  • Voigt’s notation is provided for the 2D strain tensors and strain-gradient tensors.Source: Table 1 · p. 11, Table 2 · p. 11
  • The weak-form systems are solved with GMRES and a Jacobi preconditioner, using relative tolerance 10^-5 and absolute tolerance 10^-10.Source: §3 · p. 11

4 Identification of the classical and strain gradient stiffness tensors

The identified strain-gradient parameters vanish for a homogeneous material and vary with the square lattice’s basic-cell size, while its classical stiffness remains unchanged across the tested cell sizes.

4 Identification of the classical and strain gradient stiffness tensors

  • For homogeneous material, DM components are 10^-6 N or smaller, consistent with the strain-gradient parameters vanishing in the homogeneous limit.The test uses Young’s modulus 100 MPa and Poisson’s ratio 0.3.Source: §4 · p. 11

4.1 Parameter determination for the square lattice structure

  • The square-lattice study uses polymer-like rods and much softer inclusions, with RVEs made by repeating one, four, or nine basic cells.The inclusion volume fraction is 81%.Source: §4.1 · p. 12
  • For 1×1 cells, D221221 is 4 times its value for 0.5×0.5 cells and 25 times its value for 0.2×0.2 cells; all nonzero strain-gradient parameters approach zero as cells shrink.The classical stiffness tensors are identical across these cell sizes; the reported strain-gradient parameters are compiled in Table 5.Source: §4.1 · pp. 13–14, Table 5 · p. 13

5 Computational validation of determined parameters

The validation compares microscale simulations with classical and strain-gradient macroscale models across lattice size ratios. Strain-gradient predictions match the microscale across the tested cell sizes, while classical elasticity approaches it as scale separation increases.

5 Computational validation of determined parameters

  • Validation compares microscale, classical-elasticity, and strain-gradient simulations using deformation energy, with isogeometric FEM providing the required C1 continuity.The simulations use a clamped left edge and prescribed rotation on the right edge.Source: §5 · p. 14

5.1 Computations for square lattices with the same basic cell sizes and varied macro sizes

  • With basic cells unchanged, increasing L/l makes classical-elasticity computations approach microscale results; classical elasticity is adequate for accuracy when L/l > 10, whereas comparable macro- and internal-substructure scales make strain-gradient effects significant, a size effect.Source: §5.1 · p. 16

5.2 Computations for square lattices with varied basic cell sizes and the same macro sizes

  • For three 4 mm × 4 mm square lattices divided into 16, 64, and 400 basic cells, respectively, microscale strain energy at 0.2 radian shows an increasing trend with basic-cell size.Source: §5.2 · p. 16
  • Across the tested cell sizes, strain-gradient computations match the microscale quantitatively and resolve the lattice size effect.Figure 10 compares the microscale, classical-elasticity macroscale, and strain-gradient macroscale computations.Source: Figure 10 · p. 18, §5.2 · p. 19

6 Conclusions

The paper develops and validates an asymptotic-analysis homogenization methodology for determining strain-gradient parameters in metamaterials.

6 Conclusions

  • The methodology derives classical and strain-gradient stiffness tensors and uses FEM to solve the homogenization-generated partial differential equations.The square-lattice material parameters are explicitly computed.Source: §6 · p. 19
  • The determined strain-gradient parameters vanish as the material becomes homogeneous and are independent of RVE repetition but depend on intrinsic material size.Source: §6 · p. 19
  • Numerical computations show that the determined strain-gradient parameters accurately capture lattice size effects, and the methodology applies to metamaterials with an RVE.Source: §6 · p. 19

Appendix: Asymptotic solution for the displacement field

The appendix derives an asymptotic solution for the displacement field using successive orders and cell-dependent fields.

Appendix: Asymptotic solution for the displacement field

Loading 1906.07556v1…