Young's modulus for composite auxetic material

Question

If you look at the screenshot you will see a composite auxetic material. How would you calculate the E-modulus for it? Is it as simple as the rule of mixtures, E = V1 * E1 + V2 * E2? Or is it more complex because of its structure?

Answer 1

Rather than thinking about a Young's modulus directly, the problem becomes easier to grasp if you think in terms of a relationship between the load and displacement, and subsequently between stress and the strain.

If you consider the case where deformations are small, and just consider the stiff direction (tension/compression vertically and not horizontal shear in your first figure), you can apply a vertical load to the structure and compute the vertical deformation.

Let $P_y$ be the applied load and let $u_y$ be the resulting displacement. For the linear part of the load-displacement curve you will get a relation $$ P_y = K u_y $$ where $K$ is the slope of the curve.

To convert these into stresses and strains, you will need two quantities:

1. the area ($A_{xz}$) over which the load is applied
2. the initial length ($L_y$) of the unit cell.

The choice of these quantities, particularly the area, is not obvious and several options are possible. For example, the area might be:

1. just the area of the cylinder on which the load is applied
2. the area of the triangular domain of the unit cell
3. a square area that tiles space in the horizontal plane, etc.

You have to explicitly define what the area is and justify that choice satisfactorily. Once that choice has been made, you can compute the stress-strain relation using $$ \sigma_{yy} = \frac{P_y}{A_{xz}} = \frac{K L_y}{A_{xz}}\frac{u_y}{L_y} = \frac{K L_y}{A_{xz}} \varepsilon_{yy} =: E_{yy} \varepsilon_{yy} $$ where your definition of modulus is $$ E_{yy} = \frac{K L_y}{A_{xz}} $$ That's the preferred way of computing a modulus.

You can compare that with your mixture relation (or any other continuum theory) to check what the error in the rule of mixtures is.

Answer 2

Related: How do I calculate an estimate for the properties of a composite material?

In this case, we actually have an unusual case of the fourth method. We could look at each individual unit cell of the material and apply a unit force on the structure in the x, y, and z directions. Each application of unit force would correspond to a certain deflection. The net deflection divided by the bounding box dimensions of the unit cell would result in a strain, and the unit force divided by a stress would result in a stress.

However, this simple approach does not appreciate the 3D complexity of the problem. Since we are calculating the response to a force, we are finding the individual relations of the stiffness tensor. Note that for general anisotropic materials such as this one, we need to find 21 different relationships. However, afterwards, the stiffness tensor is not usable, we need to invert the matrix to find the compliance tensor. The elastic modulus would then be the inverse of the first term of the stiffness tensor, but the usable format would utilize the compliance tensor.