Yielding of Porous Ductile Materials
Today new and advanced materials are constantly being developed to optimise various properties such as strength, weight, thermal/electrical conductivity and cost. These are often composite or porous materials. Examples include metal and polymer foams, ceramic-metal composites, porous fuel-cell electrodes, alloys, polymer blends, aerogels and filters. When engineers design objects using these materials, one of the material properties they need to know is the yield criterion. As well as quantifying when a porous object is going to "yield", the yield criterion of porous materials plays an important role in the fracture of ductile solid materials, since ductile fracture involves the nucleation and growth of microscopic pores.
When small stresses are applied to most objects, they behave elastically - that is, when the stress is removed, the object returns to its initial state. If too much stress is applied to objects, they yield and start to deform plastically - when the stress is removed the object does not return to its initial state. The yield criterion for an object is a constraint on the stresses which states when the object will begin to deform plastically.
Dr Andy Wilkins of the School of Physical Sciences at the University of Queensland is using computational simulations and modelling to determine the yield criterion of porous materials. The porous material is modelled as a solid ductile "matrix", with spherical voids representing the pores in the material. Dr Wilkins has so far been looking at symmetrical packings of voids within the matrix, such as simple-cubic, body-centred-cubic, and face-centred-cubic arrays (see figure 1), similar to the configurations of atoms in many crystals. While these configurations may not be entirely realistic - industrially-produced materials would have randomly shaped voids with irregular packing - they provide a nice starting point, as they are easy to code into the computer and provide some insight into the non-regular situations.
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Figure 1: Models of porous materials with (from left) simple-cubic, body-centred-cubic and face-centred-cubic arrays of voids (spheres) in a (transparent) ductile matrix. |
In order to determine the yield criterion, Dr Wilkins first chooses a void volume fraction, up to the percolation threshold (at which point the matrix is no longer connected). He then applies a stress state to the material, increasing the stress until the object yields. Varying the way in which the stress is applied allows the yield point to be determined for many points in "stress space" - for instance the front and back faces of the cube being modelled may be pulled while the sides, top and bottom are kept still. Alternately you may find the yield point by twisting two opposing faces in opposite directions. This data is accrued for many different stress states, allowing the yield criterion for that void volume fraction to be interpolated. This process is then repeated for different void volume fractions.
Pioneering work done by Gurson in 1977 resulted in the derivation of an upper bound on the yield criterion, known as Gurson's bound. This means that if any stress state outside of Gurson's bound were applied to the material it would definitely yield. However not much was known about states inside this bound. Dr Wilkins has been studying these states, and his simulations have shown that Gurson's bound can overestimate the yield criterion by as much as 50%. For example, figure 2 shows a comparison between Gurson's bound (shown as the red line) and the numerically determined true yield curve (mapped out by the data points shown as black crosses), for a void volume fraction of 25%. The equivalent stress is a measure of the sheer stress applied to the object, while hydrostatic stress is the average pressure put on the object. The interpretation of this plot is that any stress state inside the black crosses will cause the object to behave elastically, while any stress state outside the black crosses will cause plastic deformation of the object.
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Figure 2: The upper bound on the yield criterion determined by Gurson (shown in red) compared to the experimentally-determined yield curve (shown by the black crosses). Any stress state inside the black crosses causes the object to behave elastically, while any stress state outside the crosses causes the object to behave plastically. |
Figure 3: A Quicktime movie showing the plastic deformation of an (almost) rigid unit cell of material with a pore at its centre. The graph shows the stress level, while the red pixels indicate the parts of the material behaving elastically. (Right click to download 14MB MOV file). |
Figure 3 above shows one frame of a Quicktime movie (generated by Mark Barry of QUT), which shows the transition from elastic to plastic behaviour as the front and back faces of an (almost) rigid unit cell (with one pore at its centre) are pulled in opposite directions. As the stress on the object increases, more and more of the material begins to flow plastically (represented by the increasing number of red pixels), until finally, when the maximum stress is applied, the material yields - the rigid parts of the cell are no longer connected.
Fracture of brittle heterogeneous materials
Dr Wilkins is also investigating fracture processes in heterogeneous materials such as bone. Before looking at complicated three-dimensional problems, Dr Wilkins is first studying the 2D case, trying to predict the crack path in a sheet of perspex containing randomly drilled holes. Figure 4(b) shows a comparison of the numerically-determined and actual crack path in a sheet of perspex - the numerically determined path is shown in red while the actual crack path is shown in black.
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| a) Sheet of perspex containing randomly drilled holes and a saw kerf to initiate cracking. | b) Comparison of the simulation and actual crack path in a sheet of perspex with randomly drilled holes. |
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These experimental results are very difficult to reproduce - minor changes in hole sizes or perspex size can dramatically affect the crack path.
Success in the 2D case has led to the submission of a paper to the Journal of Computational Physics3. Dr Wilkins now hopes to apply these methods to three dimensional problems.
Computational Methods
The numerical methods used to carry out these calculations are implemented as Fortran90 code on QCIF supercomputers at UQ's HPC facility. The code was parallelised in 2005 by summer research student Tim O'Sullivan, which dramatically decreased the time required to carry out calculations. Dr Wilkins is starting to move to the engineering software package ABAQUS, already in use on the QCIF computers. ABAQUS will allow Dr Wilkins to verify past results, and will also allow more challenging problems to be studied.
Contact
Dr Andy
Wilkins, Dr Tony Roberts
Department of Mathematics, University
of Queensland
Publications
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D. L. S. McElwain, A. P. Roberts, A. H. Wilkins, "Yield functions for porous materials with cubic symmetry using different definitions of yield" (invited paper), Advanced Engineering Materials (in press, 2006).
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D. L. S. McElwain, A. P. Roberts, A. H. Wilkins, "Yield criterion of porous materials subjected to complex stress states", Acta Materialia, 54 (2006) 1995-2002.
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A. H. Wilkins, A. P. Roberts, "A simple, versatile finite element method for fracture in heterogeneous materials", Submitted to Journal of Computational Physics, June 2006.
Reports
Summer Internship 2006 report - Timothy Sullivan (88 KB pdf)
Written by Dr A. Wilkins and T. Curtis, September 2006.