# Stochastic modelling of lattice structures

Stochastic modelling of lattice structures for additive layer manufacturing process

Additive layer manufacturing improvises the complexity involved in current subtractive manufacturing techniques. The process facilitates to build any complex geometry with less lead time and minimises the requirement of assembling procedure.  The technique enables to construct the material from mesoscale. This entails the product to be build up in the form of periodic meso structural unit cells, which consist of struts. However, in case of metallic structures, the strut nodes (i.e. the locations where several struts are connected) are relatively weak, because voids occur here due to the fact that strut nodes are heated several times.  Also layering of struts involving multiple interfaces, causes defects in the final product. Therefore, the strut nodes are accurately represented to obtain a truly predictive model.

3D finite element model of unit cell consisting of strut node

General aim of the work:

To quantify the variation of material parameters for products manufactured using additive layer manufacturing process.

Summary of obtained results:

The finite element model of unit cell is discretised into 3D tetrahedrons and solved for compressible, isotropic, linear elastic material using infinitesimal strain theory and for isotropic compressible materials using hyper elastic material model defined with respect  to relation between stored energy and deformation. These unit cells consisting of strut nodes are coupled with beam elements to obtain a complete lattice structure. Therefore, the displacement of nodes present at the boundaries of unit cell has to be in accordance with the deformation of beam element. For this, a constrained minimization problem is solved using Lagrange multiplier method.  It is to be noted that the 3D mesh is point wise symmetric and periodic boundary conditions are applied in order to efficiently manage the computational effort.

Localisation of 3D unit cell

2D unit cell for which constraints are incorporated to boundary nodes

Future work:

The finite element model of single unit cell will be developed further to include damage mechanics. The material model considering large deformations are non linear, consequently, it is computationally expensive. Therefore, a Proper Orthogonal Decomposition (POD) based model order reduction strategy will be applied to reduce the number of degrees of freedom. A hyper reduction strategy will be used to integrate the modes of POD, which additionally reduces the computational cost.

Acknowledgement:

We thank the financial support of the Fonds National de la Recherche Luxembourg FNRS-FNR grant INTER/ FNRS/15/11019432/EnLightenIt to support the research.