Finite Element Analysis
The Finite Element Analysis Method using ANSYS Workbench Platform
Finite element analysis (FEA) is a ‘fast, cheap and non-destructive way to analyse the stresses within anvil cells’ (Adams et al, 1992), thus, to date a great deal of research has been done on the FEA of DACs. There are a number of different programs which can be used for this type of analysis, each with differing benefits and limitations; for this study ANSYS Workbench Platform will be used. This program has the ability to analyse axially symmetric structures by the finite element method. The method uses the principal of Von Mises stress distribution;
where σz is the axial stress, σr is the radial stress, σθ is the hoop stress and τrz is the shear stress (for more information click here). ‘In any three-dimensional system two of the principal stresses are the largest compressive and largest tensile stresses at any point and the values of all other direct stresses lie between them’. The maximum shear stresses act on planes at 45° to the principal planes. The chosen FEA program uses a fine ‘mesh’ which creates significantly smaller elements over the area under question. The force equations are then solved by convergence to give an equilibrium state solution giving in turn the stress values and distribution throughout the problem. For accuracy, the mesh has increased density in regions of interest where stresses are likely to be large, such as, the anvil’s working face.
Stress distribution has been well understood for working DACs through FEA which has allowed important optimisations to be made to their design. ANSYS Workbench Platform is a very powerful program which will hopefully be able to provide accurate results into the working conditions of sapphire anvils. Some FEA research on sapphires has been done to date but there must be a great deal more in order for sapphire anvils to become as reliable and efficient as DACs.
Stress distribution has been well understood for working DACs through FEA which has allowed important optimisations to be made to their design. ANSYS Workbench Platform is a very powerful program which will hopefully be able to provide accurate results into the working conditions of sapphire anvils. Some FEA research on sapphires has been done to date but there must be a great deal more in order for sapphire anvils to become as reliable and efficient as DACs.
Modelling & Meshing
For simplicity and to save simulation running time the anvil set up was modelling in 2D. The model was also 'cut' into quarters down its two lines of symmetry (y-axis & x-axis) thus meaning only a quarter of the full set-up was required to be modelled. The symmetry edges were defined as simply supported to allow realistic analysis. The parts where then individually meshed with sufficiently small element sizes (0.1-0.05mm). Refinements to the mesh were made on areas of particular interest such as the sample or anvil's working face. The mesh on the quarter 2D set-up can be seen in the images.
Pressure was applied via the back surface of the support (i.e. top of image a above). This allowed Ansys to calculate the stress distribution across the quarter anvil at a particular pressure applied to the support, allowing the value and position of the maximum shear, tensile and compressive stresses to be obtained. The sample pressure was also able to be found through the results obtained by the program model. Systematic increase of the pressure to the back of the anvil made it then possible to find the abilty of the anvil geometry - i.e. the pressure within the sapphire at which the stresses within the anvil exceed critical values.
Results of the Finite Element Analysis
It was found that the 5mm diameter spherical sapphire anvil with 1mm diameter working face would fail under compression while applying a sample pressure of 2.16GPa. This was very realistic value according to previous studies and also considering limitations of the modelling program and accuracy of the assumptions made for the set-up.
Plots of the stress distribution at critical pressure can be seen below, click on each image to enlarge it for clearer examination.
Plots of the stress distribution at critical pressure can be seen below, click on each image to enlarge it for clearer examination.
Pressure Distribution Across the Sample
Principal Stress Plot (tensile & compressive)
Stress concentrations at 'corner' of anvil face due to its angular quality as can be seen by the principle stress contour plot. The red colour is in tension ranging through to the blue which is in compression and will be the initial site of failure.
At a sample pressure of 2.16GPa, the maximum compressive stress was found here to be just over 2.95GPa indicating failure. |
Shear Stress Plot
Stress concentrations once again at corner of anvil face with sizeable 'plum' above. This was most likely created by the flowing out of the gasket material during pressurisation.
At a sample pressure of 2.16GPa, the maximum shear stress was found here to be 2.30GPa. This is below the critical shear stress for sapphire and thus, the anvil will fail first in compression. |
Validation of Simulation & Normalisation of Errors
The simulation of the experiment was validated by experimental research on spherical sapphire obtaining similar results. Further information on this is proved here.
The simulation often gave unrealistic results. These are explained here, where methods for the normalisation of these errors has been presented.
The simulation often gave unrealistic results. These are explained here, where methods for the normalisation of these errors has been presented.