Research Paper
Contents |
Finite Element Analysis of the stress distribution near damage Si particle clusters in cast Al-Si alloy
Ken Gall^{a}, Mark Horstemeyer^{a}, David L. McDowell^{b}, Jinghong Fan^{b}
^{a}Materials and Engineering Sciences Center, Solid and Material Mechanics Department, Sandia National Laboratories, 7011 East Avenue, MS 9405, Livermore, CA 94550, USA
^{b}GWW School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
Received 12 April 1999; received in revised form 24 September 1999
Abstract
The objective of this research is to study the influence of morphology on fracture and debonding of silicon particles embedded in an Al-1%Si matrix. Fracture and deboning is caused by applied tensile and compressive cyclic loading conditions. The finite element method is used for this study to accurately represent particle geometry, particle interactions, and the stress-strain behavior of the aluminum matrix. For this study, a cluster of 4 to 8 silicon particle inclusion is chosen over infinite array of inclusion or single isolated inclusion. Silicon particles are modeled with linear elastic constitutive relationship and matrix material using internal state variable cyclic plasticity model. A two level design of experiments (DOE) method is used to test 16 sets of combination made with 7 variables; relative particle size, shape, spacing, configuration, alignment, grouping and matrix microporosity. Results of the study demonstrated the dominance of shape and alignment during initial phases fracturing and debonding and spacing during later phases. Local intensification of stresses in induced by particle debonding in Al-1%Si matrix. This intensification of stresses is higher than that of particle fracture. Enhancement is spacing due to consecutive fracturing in the cluster becomes a dominant factor due to large local intensification of stresses as mentioned above
Methodology
Two level design of experiments (DOE) method is used to study the effects of seven parameters at their chosen range of conditions which are shown in Fig.1. A total of 16 significant combinations of the 7 morphological parameters at their extreme conditions are chosen for the study, shown in the following Table 1. The extremes for these parameters are based on micro-graphical observations from an A365 aluminum alloy study which constitutes silicon particles. Finite element cases were created for all 16 combinations with following assumptions;
- Traces of other element are not considered in the model though which are generally present in the alloy to promote hardening and other casting properties.
- The silicon particles are assumed to behave in an isotropic linear elastic manner. The Al matrix material is described using an internal state variable plasticity model with coupled micro void growth.
- Temperature and strain rate dependence on the plasticity of the model were not considered. Experimental data regression is used to generate constants needed for the calculations to complete the model.
In Fig 2, stress-strain model output is compared to experimental output for cycles 1, 2, and 10. A point of saturation is attained at the end of 10 cycles in both cases. Maximum tensile principal stress is an important study parameter which is the proposed cause of fracture in Si particles. Debonding is studied on the basis of hydrostatic stresses on particle matrix interface. The screening of the two significant parameters is listed under table 2. Schematic from mesh # 10 is shown as an example in Fig. 4 along with finite element fine mesh region near the silicon particles in mesh # 10. In these cases, main intention is to quantify the pattern of fracture and debonding in adjacent particles and their effect on neighboring particles.
Fig. 1. Schematic demonstrating the different parameters considered in the present finite element study. The parameter ranges were determined by examining actual micrographs of a modified cast A356 aluminum alloy | Fig 2 : Experimental and model stress-strain response for the Al-1%Si matrix material in the finite element simulations. | Fig 3: (a) Schematic of mesh 10 with the silicon particles enlarged to highlight their distribution. The actual radii are given in mesh units on the lower left corner of the mesh. (b) The finite element fine mesh region near the silicon particles in mesh 10. |
Table 1: List of all 16 mesh combinations of the 7 morphological parameters shown in Fig 1 | Table 2: Maximum principal stress within a fully bonded silicon particle & Maximum hydrostatic stress at the Si-Al interface of a bonded Si particle | ||
Data from table 2 shows that shape and alignment plays a significant role in fracture and debonding of the Si particles that are highly intact. Spacing between the particles becomes a significant factor in later phases when several bonds are broken in the matrix. Spacing and configuration accelerates the rate of fracture and debonding in the cluster towards saturation. Some particles are fractured in the beginning in order to study stress distribution on the neighboring particles. Fig 4, 5 and 6 represent successive phases from bonded state towards cracked and debonded state. Contours in Fig 7-12 represent several combination and phases in terms of maximum principal stresses and maximum hydrostatic stresses.
Fig 4: The quantitative main parameter effects on (a) cracking and (b) debonding. All of the adjacent particles are bonded. | Fig. 5. The quantitative main parameter effects on (a) cracking and (b) debonding. Some of the adjacent particles are bonded while others are cracked. | Fig. 6. The quantitative main parameter effects on (a) cracking and (b) debonding. Some of the adjacent particles are bonded while others are deboned. |
Fig. 7. Contours of the maximum principal stress σ_{I} (MPa) surrounding the silicon particles for mesh 10 at the maximum applied tensile strength in the third loading cycle. The results are symmetric about the left boundary of the plot. The mess has three different initial configurations: (a) all particle bonded, (b) two particle cracked and (c) two particles debonded. | Fig. 8. Contours of the maximum principal stress σ_{I} (MPa) surrounding the silicon particles for mesh 11 at the maximum applied tensile strength in the third loading cycle. The results are symmetric about the left boundary of the plot. The mess has three different initial configurations: (a) all particle bonded, (b) two particle cracked and (c) two particles debonded. | Fig. 9. Contours of the maximum principal stress σ_{I} (MPa) surrounding the silicon particles for mesh 12 at the maximum applied tensile strength in the third loading cycle. The results are symmetric about the left boundary of the plot. The mess has three different initial configurations: (a) all particle bonded, (b) two particle cracked and (c) two particles debonded. |
Fig. 10. Contours of the maximum principal stress σ_{H} (MPa, - in tension) surrounding the silicon particles for mesh 10 at the maximum applied tensile strength in the third loading cycle. The results are symmetric about the left boundary of the plot. The mess has three different initial configurations: (a) all particle bonded, (b) two particle cracked and (c) two particles debonded. | Fig. 11. Contours of the maximum principal stress σ_{H} (MPa, - in tension) surrounding the silicon particles for mesh 10 at the maximum applied tensile strength in the third loading cycle. The results are symmetric about the left boundary of the plot. The mess has three different initial configurations: (a) all particle bonded, (b) two particle cracked and (c) two particles debonded. | Fig. 12. Contours of the maximum principal stress σ_{H} (MPa, - in tension) surrounding the silicon particles for mesh 12 at the maximum applied tensile strength in the third loading cycle. The results are symmetric about the left boundary of the plot. The mess has three different initial configurations: (a) all particle bonded, (b) two particle cracked and (c) two particles debonded. |
Discussion
Results from the two level design of experiments shows that four out of seven parameters were the major contributors in the fracturing and debonding of silicon particles embedded in an Al-1%Si matrix. The contributing parameters are shape, alignment, spacing and configuration, whereas parameters like grouping, microporocity and size were insignificant. Two measured quantities (Table 2) i.e. maximum principle stress and maximum hydrostatic stress between Si particle and Al-1%Si matrix were used to determine the tendency of particle fracturing and debonding respectively.
In this study, shape and alignments were studies relative to pertinent microstructure parameters like spacing. The models aimed particle intactment for the pursuit of relative nature of the study. Elliptical particles oppose fracture and debonding when their major axis is perpendicular to direction of loading. Increase in tendency to fracture or debond is observed when particles are placed parallel to the direction of loading. Circular particles shows similar trend when bounded by elliptical particles. Matrix experiences small localization of stresses near centrally fractured Si particle which also induces crack.
Matrix experiences localization of stress near the fractured and debonded particles. Particles with a major axis perpendicular to the loading direction causees high stress in matrix and neighboring particles. Orientation wise, particles placed at a 45° angle, or along a direction perpendicular to the applied loading axis are major contributors to particle fracture and debonding.
References
[1] Ken Gall, Mark Horstemeyer, David L. McDowell, Jinghong Fan, Finite element analysis of the stress distributions near damaged Si particle clusters in cast Al–Si alloys, Mechanics of Materials, Volume 32, Issue 5, May 2000, Pages 277-301, ISSN 0167-6636, http://dx.doi.org/10.1016/S0167-6636(00)00003-X. (http://www.sciencedirect.com/science/article/pii/S016766360000003X)