Introduction

The topic of homogenization has come up several times in this technical blog.

(2) Adidas 3DP Shoes Explanation: Design of Mechanical Metamaterials that Convert Compressive Forces into Shear Deformations

Homogenizing Mechanical Metamaterial

This article introduces the method of homogenization of lattice structures by numerical experiments and visualization of the results in terms of elastic bodies and fluids (channels).

This article is the first part and introduces homogenization of elastic bodies.

The second part is here: (2) Homogenization of Lattice Structures

Note that while this article introduces homogenization of lattices as an Additive Manufacturing prerequisite,
In client work, it is common to utilize metamaterial structural design techniques while assisting in the development of designs that can be mass-produced at the client's manufacturing facility.

Homogenization of lattice structures into elastic matrices by numerical experiments

There is a homogenization technique that incorporates the behavior of the lattice structure during small deformations into the elasticity matrix.

A numerical experimental approach is often used to obtain elastic matrices equivalent to lattice structures.

Here is an example of one of these numerical experiments.

By Hooke's law, the elastic matrix \(C\) is written as a matrix linking the strain \(\epsilon_{ij}\) and the stress \(\epsilon_{k\ell}\).

\(\begin{align*}
        C
        =
       \left(
            \begin{array}{ccc}
                 c_{xxxx} & c_{xxyy} & c_{xxzz} & c_{xxyz} & c_{xxzx} & c_{xxxy}\\
                 c_{yyxx} & c_{yyyy} & c_{yyzz} & c_{yyyz} & c_{yyzx} & c_{yyxy}\\
                 c_{zzxx} & c_{zzyy} & c_{zzzz} & c_{zzyz} & c_{zzzx} & c_{zzxy}\\
                 c_{yzxx} & c_{yzyy} & c_{yzzz} & c_{yzyz} & c_{yzzx} & c_{yzxy}\\
                 c_{zxxx} & c_{zxyy} & c_{zxzz} & c_{zxyz} & c_{zxzx} & c_{zxxy}\\
                 c_{xyxx} & c_{xyyy} & c_{xyzz} & c_{xyyz} & c_{xyzx} & c_{xyxy}
            \end{array}
        \right)
    \end{align*}\)

\(\begin{align*}
        \left(
            \begin{array}{ccc}
                 \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz} \\ \sigma_{yz} \\ \sigma_{zx} \\ \sigma_{xy}
            \end{array}
        \right)
        =
       \left(
            \begin{array}{ccc}
                 c_{xxxx} & c_{xxyy} & c_{xxzz} & c_{xxyz} & c_{xxzx} & c_{xxxy}\\
                 c_{yyxx} & c_{yyyy} & c_{yyzz} & c_{yyyz} & c_{yyzx} & c_{yyxy}\\
                 c_{zzxx} & c_{zzyy} & c_{zzzz} & c_{zzyz} & c_{zzzx} & c_{zzxy}\\
                 c_{yzxx} & c_{yzyy} & c_{yzzz} & c_{yzyz} & c_{yzzx} & c_{yzxy}\\
                 c_{zxxx} & c_{zxyy} & c_{zxzz} & c_{zxyz} & c_{zxzx} & c_{zxxy}\\
                 c_{xyxx} & c_{xyyy} & c_{xyzz} & c_{xyyz} & c_{xyzx} & c_{xyxy}
            \end{array}
        \right)
       \left(
            \begin{array}{ccc}
                 \epsilon_{xx} \\ \epsilon_{yy} \\ \epsilon_{zz} \\ 2\epsilon_{yz} \\ 2\epsilon_{zx} \\ 2\epsilon_{xy}
            \end{array}
        \right)
    \end{align*}\)

The elasticity matrix \(C\) is originally defined for each material, but the behavior of the lattice structure during elastic deformation is included in the \(C\) matrix based on the idea that the lattice structure itself can be considered a homogeneous material when viewed on a macro scale sufficiently larger than the unit size of the lattice structure.

To obtain the elasticity matrix \(C\) homogenizing the Lattice structure, we take the unit structure of the Lattice structure and perform numerical experiments.

After setting the cyclic boundaries of stress and strain at the boundary of the unit structure, the analysis is performed by applying six types of strain as follows.

\(　　　　\begin{align*}
       \left(
            \begin{array}{ccc}
                 \epsilon_{xx} \\ \epsilon_{yy} \\ \epsilon_{zz} \\ 2\epsilon_{yz} \\ 2\epsilon_{zx} \\ 2\epsilon_{xy}
            \end{array}
        \right)
=
       \left(
            \begin{array}{ccc}
                 1 \\ 0\\  0\\ 0 \\ 0 \\ 0
            \end{array}
        \right)
,
       \left(
            \begin{array}{ccc}
                 0 \\ 1\\  0\\ 0 \\ 0 \\ 0
            \end{array}
        \right)
,
       \left(
            \begin{array}{ccc}
                 0 \\ 0\\  1\\ 0 \\ 0 \\ 0
            \end{array}
        \right)
,
       \left(
            \begin{array}{ccc}
                 0 \\ 0\\  0\\ 1 \\ 0 \\ 0
            \end{array}
        \right)
,
       \left(
            \begin{array}{ccc}
                 0 \\ 0\\  0\\ 0 \\ 1 \\ 0
            \end{array}
        \right)
,
       \left(
            \begin{array}{ccc}
                 0 \\ 0\\  0\\ 0 \\ 0 \\ 1
            \end{array}
        \right)
    \end{align*}\)

Calculate the average stress \(\bar{\sigma}^{(n)}_{i,j}\) in each direction, including the void space, for the six types of strain analysis.

Based on the calculated mean stress \(\bar{\sigma}^{(n)}_{i,j}\), a temporary elastic matrix is obtained as follows.

\(\begin{align*}
        \left(
            \begin{array}{ccc}
                 \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz} \\ \sigma_{yz} \\ \sigma_{zx} \\ \sigma_{xy}
            \end{array}
        \right)
        =
       \left(
            \begin{array}{ccc}
                 \bar{\sigma}^{(1)}_{xx} & \bar{\sigma}^{(2)}_{xx} & \bar{\sigma}^{(3)}_{xx} & \bar{\sigma}^{(4)}_{xx} & \bar{\sigma}^{(5)}_{xx} & \bar{\sigma}^{(6)}_{xx}\\
\bar{\sigma}^{(1)}_{yy} & \bar{\sigma}^{(2)}_{yy} & \bar{\sigma}^{(3)}_{yy} & \bar{\sigma}^{(4)}_{yy} & \bar{\sigma}^{(5)}_{yy} & \bar{\sigma}^{(6)}_{yy}\\
\bar{\sigma}^{(1)}_{zz} & \bar{\sigma}^{(2)}_{zz} & \bar{\sigma}^{(3)}_{zz} & \bar{\sigma}^{(4)}_{zz} & \bar{\sigma}^{(5)}_{zz} & \bar{\sigma}^{(6)}_{zz}\\
\bar{\sigma}^{(1)}_{yz} & \bar{\sigma}^{(2)}_{yz} & \bar{\sigma}^{(3)}_{yz} & \bar{\sigma}^{(4)}_{yz} & \bar{\sigma}^{(5)}_{yz} & \bar{\sigma}^{(6)}_{yz}\\
\bar{\sigma}^{(1)}_{zx} & \bar{\sigma}^{(2)}_{zx} & \bar{\sigma}^{(3)}_{zx} & \bar{\sigma}^{(4)}_{zx} & \bar{\sigma}^{(5)}_{zx} & \bar{\sigma}^{(6)}_{zx}\\
\bar{\sigma}^{(1)}_{xy} & \bar{\sigma}^{(2)}_{xy} & \bar{\sigma}^{(3)}_{xy} & \bar{\sigma}^{(4)}_{xy} & \bar{\sigma}^{(5)}_{xy} & \bar{\sigma}^{(6)}_{xy}\\
            \end{array}
        \right)
       \left(
            \begin{array}{ccc}
                 \epsilon_{xx} \\ \epsilon_{yy} \\ \epsilon_{zz} \\ 2\epsilon_{yz} \\ 2\epsilon_{zx} \\ 2\epsilon_{xy}
            \end{array}
        \right)
    \end{align*}\)

The elasticity matrix must be symmetric. Even if an element should be zero due to the symmetry of the structure, it will not be zero due to numerical error.

The homogenized elasticity matrix is calculated after adding corrections such as taking the average of symmetric elements.

In the lattice consisting of the beams illustrated above, the actual homogenized elastic matrix for a material with Young's modulus \(7×10^{10}\) Pa and Poisson's ratio \(0.33\) is as follows

\(\begin{align*}
       \left(
            \begin{array}{ccc}
                 4.89×10^8 & 4.73×10^8 & 4.73×10^8 & 0 & 0 & 0\\
                 4.73×10^8 & 4.89×10^8 & 4.73×10^8 & 0 & 0 & 0\\
                 4.73×10^8 & 4.73×10^8 & 4.89×10^8 & 0 & 0 & 0\\
                 0 & 0 & 0 & 4.41×10^8 & 0 & 0\\
                 0 & 0 & 0 & 0 & 4.41×10^8 & 0\\
                 0 & 0 & 0 & 0 & 0 & 4.41×10^8\\
            \end{array}
        \right) .
    \end{align*}\)

Using this elastic matrix, the lattice structure portion of a cantilever beam with a \(5×5\) lattice structure was homogenized and analyzed, results are shown below. (The displacement scaling of boundary conditions and visualization is identical for results with and without homogenization.)

It is clear that the macro deformations are well reproduced.

On the other hand, a plot of von Mises stresses shows the following, indicating that microscopic stresses can no longer be simply estimated due to homogenization.

By performing localized analysis in conjunction with homogenization analysis, microscopic stresses can also be estimated to some extent.

It should also be noted that homogenization assumes a situation where the unit structure of the lattice structure is repeated a sufficiently large number of times, so that homogenization will not be valid if the number of repetitions is small.

Use homogenization with a full understanding of its characteristics.

Visualization of elastic matrices

Next we will show you how to visualize the homogenized elastic matrix.

Since it is difficult to visualize all the information in an elasticity matrix, we consider here the visualization of Young's modulus effectively working in each direction.

As shown in the figure below, the effective Young's modulus in each direction is represented by a point in 2-D or 3-D space that is separated from the origin by a distance corresponding to the Young's modulus in each direction.

Connecting those points to form a line or plane visualizes the effective Young's modulus in each direction.

The effective Young's modulus is defined as the inverse of the diagonal components (\(S_{xx},S_{yy},S_{zz}\)) of the compliance matrix \(S\) (the inverse of the elastic matrix \(C\)) for each direction.

For directions other than `\(x,y,z\), the elasticity matrix is calculated by transforming it according to the coordinate transformations.

This value is Young's modulus itself for isotropic materials

Below we present visualizations of elasticity matrices (Young's modulus) in isotropic and orthotropic materials and in several lattice structures.

Isotropic materials

In isotropic materials, Young's modulus is naturally the same in all directions, and when visualized in three dimensions, the material becomes a sphere.

Orthotropic materials

For orthotropic materials (with \(E_x, E_y, E_z=20, 18, 15\rm{GPa}\),\(\nu_{xy}, \nu_{xy}, \nu_{xz}=0.12,0.10,0.14\) and \(G_{xy}, G_{xy}, G_{xz}=9, 8, 6\rm{GPa}\)) shows that the effective Young's modulus is different in each direction.

Lattice 1

It can be inferred that this lattice structure has a direction in which the beams are aligned, and Young's modulus is greater in that direction.

Lattice 2

truss lined cross section (see the third figure below, parallel to the y=z plane), and it can be inferred that the effective Young's modulus is higher in the vicinity of the y=z direction

Lattice 3

We can infer that the effective Young's modulus is larger in the vicinity of the z direction because of the truss lined cross section (parallel to the x=0 plane and perpendicular to the y=0 plane).

Finally

This article introduced the homogenization of lattice structures by elastic matrices and how to visualize elastic matrices.

Homogenization allows the analysis of complex lattice structures at a relatively low computational cost.

If you are interested in doing an analysis of lattice structures, please use it.

The second part of the presentation will focus on homogenization of channels using a porous model.