Numerical modeling of the thickness dependence of zinc die-cast materials

Abstract

Zinc die casting alloys show varying material properties over the thickness in their final solid state, which causes a change in the mechanical response for specimens with different thicknesses. In this article, we propose a modeling concept to account for the varying porosity in the constitutive modeling. The material properties are effectively incorporated by combining a partial differential equation describing the distribution of the pores by a structural parameter with the Mori–Tanaka approach for linear elasticity. The distribution of the porosity is determined by polished cut images, for which the procedure is explained in detail. Finite element simulations of the coupled system incorporating the thickness dependence show the applicability of this approach.

Appendix: Porosity determination

In Fig. 3a, one can recognize a non-homogeneous distribution of the porosity through the thickness of the specimen. If we define a cylindrical coordinate system with the center in the revolution axis of the cylindrical specimen, we can express this dependence as $\varPhi = {\hat{\varPhi }}(r)$. For the porosity evaluation, a script in Matlab is used. At first, the micro-graph is converted into a gray-level image with the resolution $m \times n$ and a pixel-density of 6000 ppi. This image is processed as a matrix ${{\mathbf {\mathsf{{{P}}}}}}_{\text {GL}}\in \mathbb {N}^{m \times n}$, where the (x, y)-position and the gray-value $p_{ij} = {0,\ldots ,255}$ of the pixel is stored. In the next step, the position of every pixel in the $(r,\varphi )$-coordinate system is determined. Then, the pixels of one image are evaluated as “pores” ($p_{ij} > 175$) or “not pores” ($p_{ij} < 175$) in intervals $\varDelta r = {4.233}{\upmu }$m over the discretized radial coordinate $r_{k+1} = r_k + \varDelta r$. The radial coordinate has the limits $r_i$ and $r_o$, and we evaluate the pores within the angles $\varphi _{\text {min}}$ and $\varphi _{\text {max}}$. For each coordinate position, the porosity can be calculated with Eq. (2), where $A_H(r_k)$ is equal to the number of pixels considered as pores in a certain radial interval $r_k$ multiplied with the area of a pixel. Finally, the average value of the porosity of all the sectors for every radial interval is calculated. Figure 3b represents the development of the porosity through the thickness of the specimen.