Research Paper

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AbstractMethodologyMaterial ModelInput DataResultsAcknowledgmentsReferences


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. The fracture and deboning is caused by applied tensile and compressive cyclic loading conditions. Finite element method is used to study these effects to accurately represent particle geometry, particle interactions, and the stress-strain behavior of the aluminum matrix. A cluster of 4 to 8 silicon particle inclusion is chosen for the study 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 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 demonstrates 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.

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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 (click on the image to enlarge).
Figure 2. This is a plot of the accessibility of each grain boundary structure for an asymmetric tilt grain boundary. For complex boundaries, a large number of boundaries may need to be sampled to find the "global" minimum energy boundary. (click on the image to enlarge).


Two level design of experiments (DOE) method is used to study the effects of seven parameters at their chosen range of conditions is 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;

  1. Traces of other element are not considered in the model though that are generally present in the alloy to promote hardening and other casting properties.
  2. The silicon particles are assumed to behave in an isotropic linear elastic. The Al matrix material is described using an internal state variable plasticity model with coupled micro void growth.
  3. Temperature and strain rate dependence on the plasticity of the model were not considered. Experimental data regression is thoroughly used to generate constants for 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. 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.

Material Model

Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)

Input Data

See LAMMPS Input Deck for Grain boundary generation


This methodology has resulted in structures that agree with HRTEM images and grain boundary energies that agree with experimentally-measured grain boundary energies. Figure 3 shows an example plot of the grain boundary energy versus inclination angle for a complex grain boundaries in Cu (Sigma 3 asymmetric grain boundaries). This system of boundaries displays a phase transformation at the boundary to the orthorhombic 9R phase for inclination angles of 70-90 degrees. The methodology used above agrees nicely with experimental results and previous atomistic calculations of Wolf and coworkers.
Al <100> symmetric tilt grain boundaries.
Al <110> symmetric tilt grain boundaries.
Cu <100> symmetric tilt grain boundaries.
Cu <110> symmetric tilt grain boundaries.
Figure 3. Plot of the grain boundary energy as a function of inclination angle for asymmetric tilt grain boundaries in Cu.


M.A. Tschopp would like to acknowledge funding provided under an NSF graduate fellowship for the initial work. Continued funding for investigating structure-property relationships in grain boundaries under the NEAMS (Nuclear Energy Advanced Modeling and Simulation) program is also acknowledged.


The initial methodology was used in the following papers:

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