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Analysis of Different Homogenization Methods for the Calculation of Deformations in 3D Printed Circuit Boards
Keywords: Homogenization, Finite element modelling, Thermal strains
Printed Circuit Boards (PCBs) connect electronic components by circuit traces on multiple layers. The width of those traces is only some micrometres, but the dimension of a whole PCB can be several centimetres. Analysing strains and stresses in PCBs with finite element models thus imposes a great challenge, because important features for the deformation can be by a factor of 10000 smaller than the whole model. In order to describe these problems with sufficient accuracy and within reasonable computational times, so-called “homogenization” techniques are usually employed. Since the traces in a PCB are normally not distributed in a periodic manner, homogenizing a PCB structure is not as simple as for typical composites. In homogenization of PCBs, the whole geometry is divided into small cells with homogenized elastic properties. The properties of each cell can be determined by the following approaches: 1. Assigning the material properties of the predominant material in the cell (Voxel-approach) 2. Analytical computation of the material combination using (i) uniform strain or (ii) uniform stress assumption (Grenestedt und Hutapea 2003) 3. Using periodic boundary conditions to extract properties for the non-uniform cells 4. Employing empirical equations for the stiffness components extracted with approach 3 (Ubachs et al. 2006; Hutapea et al. 2006) In the referred papers, no fundamental study of the accuracy of the results obtained from those different methods is presented. Ubachs et al. (Ubachs et al. 2006) state that their maximum error in the stiffness components for a homogenized structure is 10% for the shear stiffness, but it is not clear whether this is due to the fitted stiffness components of the cells (which are dependent on the copper volume content and anisotropy ratio) or the periodic boundary conditions that were used for cells embedded into a non-periodic surrounding. In this work, the above approaches are analysed utilizing a simple but realistic PCB structure and are compared to a model with full geometry in terms of the PCB’s stiffness and coefficients of thermal expansion. The structure represents 5 layers with copper traces and vias. The non-symmetric build-up of the structure causes warpage under temperature loading. The global model discretizes the PCB structure with different numbers of partitions in the plane: 2x3, 4x5, and so on. Over the thickness, 5 partitions are made (one for each layer). The stiffness and Coefficients of Thermal Expansion (CTE) for each cell are computed with a) the voxel method, b) the uniform strain method, c) the uniform stress method, d) conventional periodic boundary conditions and e) modified periodic boundary conditions. The latter attempts to improve the accuracy of homogenized model results. Since the boundary conditions already impose some simplification (and thus potential source of error) the approach 4 is not studied. The direction dependent stiffness and CTE values of the total structure are calculated depending on the number of discretizations in the plane and the homogenization approach. Those stiffness and CTE values are compared to those from the model structure with fully discretized geometry, which represents the reference case. Naturally, by increasing the number of discretization cells, the results of the different approaches get closer to the results of the reference case. The cell numbers at which the different approaches yield acceptable results are shown for the PCB structure and the possibility of improvement for the widely used periodic boundray conditions are discussed.
Martin Pletz,
Designing Plastics and Composite Materials, Department of Polymer Engineering and Science, Montanuniversitaet Leoben
Leoben, Styria

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