# Multiscale and Failure Analysis of Periodic Lattice Structures
## 3 Downscaling Technique  
The solution of the homogenized structure does not provide stresses in the actual lattice members. The stresses in the homogenized body are mathematically smeared stresses, but not the actual stresses experienced by lattice members. Thus, the downscaling process is conducted to compute the actual stresses in the lattice members because failure would occur in the overstressed lattice members. The lattice members may fail by tensile  
Figure 4: Unit cell of a rectangular lattice with diagonal members.  
Figure 5: Displacement of a diagonal member by horizontal elongation of the unit cell in Figure 3.  
failure or buckling under compression. The downscaling analysis is conducted using the same unit cell used for the upscaling process.  
The lattice members are modeled using beam elements, while the homogenized body is modeled using a continuum element, like plane stress or 3-D solid elements. Therefore, there is a mismatch in the degrees of freedom between the beam elements and continuum elements. The continuum elements do not have rotational degrees of freedom. The downscaling process is to resolve this issue and to compute the stresses in the lattice beam members.  
Let us use the rectangular unit cell in Figures 3 and 4 to explain the downscale process. The finite element matrix equation is developed using beam elements for the unit cell. The unit cell may or may not have internal members like the diagonal members, as shown in Figures 3 and 4, respectively. The unit-cell model has four nodes at the corners for Figure 3 and one additional center node for Figure 4. Every node has three degrees of freedom. All the degrees of freedom in the unit cell are classified into two groups. The first group contains the displacement degrees of the corner nodes, which are compatible with the continuum elements. The second group contains all the rotational degrees of freedom and the degrees of freedom other than at the corner nodes. This group is condensed out in the matrix equation as described below.  
The stiffness matrix equation of the unit cell is broken down into the following submatrices:  
\[\begin{bmatrix}K_{11}&K_{12}\\ K_{21}&K_{22}\end{bmatrix}\begin{pmatrix}U_{1}\\ U_{2}\end{pmatrix}=\begin{pmatrix}F_{1}\\ F_{2}\end{pmatrix} \tag{8}\]  
where \(U_{1}\) contains the degrees of freedom of the first group, while \(U_{2}\) Includes the degrees of freedom of the second group. The second row of Equation (8) is written as  
\[K_{21}U_{1}+K_{22}U_{2}=F_{2} \tag{9}\]  
Solving Equation (9) for \(U_{2}\) results in the following relationship with the first group \(U_{1}\):  
\[U_{2}=K_{22}^{-1}(F_{2}-K_{21}U_{1}) \tag{10}\]  
We compute the displacements \(U_{1}\) at the four corners of the unit cell from the solution of the homogenized model, where we want to calculate the stresses in the lattice members. Next, we compute the values of the condensed degrees of freedom \(U_{2}\) using Equation (10). Finally, both \(U_{1}\) and \(U_{2}\) are substituted into Equation (8) to compute the forces and moments at nodal locations. Lattice stresses are determined by the forces and moments using  
\[\sigma=\frac{F}{A}\pm\frac{My}{I} \tag{11}\]  
Equation (11) gives normal stress to the lattice member. Likewise, shear stress can be computed from the shear load on the lattice member. Equation (11) is for a 2-D lattice element. A similar equation is used for a 3-D lattice element.