Introduction
The purpose of this project was to study the effects that contact stresses have on brittle
materials. Specifically, this project is an analysis of the contact stresses developed when a
tungsten steel ball is pressed into the surface of a Dicor/MGC plate. We are interested in the
stresses that result from that pressure and how the plate reacts to it.
Background
Another name for this analysis is the Hertzian Indentation. In this project, a steel ball is pressed into a plate surface. We want to obtain the distribution of the contact stresses. To solve this problem without a computer, theories would have to be employed and the solution would be limited to elastic materials only. Also, the type of load applications for these theories were limited to point loads and distributed loads. However, in the modern world, we can use Finite Element Analysis (FEA) packages to obtained the stress distribution for elastic-plastic materials with a variety of loadings and constraints. Furthermore, the finite element solution can be much more accurate than the theoretical solution, if setup properly. One popular FEA software package is ANSYS. ANSYS allows us to import geometries or to make them from within ANSYS, apply loading and constraints, and then solve for a solution. Then the desired quantities (stress, strain, DOF) can be plotted from the solution.
For this project, the solution was done in stages. The analysis of the sphere and plate can be done in two-dimensions or three-dimensions. Also, ANSYS has the ability to show crack initiation and propagation for brittle materials. Based on what ANSYS can do, the stages of the project were broken up as follows:
After the stages have been identified, the following procedures were implemented.
Procedures
The FEA model for the 1st Stage and 2nd Stage analysis is simplified due to the fact that symmetry exists in two axes. Furthermore, the material properties used for the sphere and plate are as follows:
| Young's Modulus (Gpa) | Yield Stress (GPa) | Poisson's Ratio | |
| Sphere - Tungsten Steel | 614 | 6 | 0.22 |
| Plate - Dicor/MGC | 70 | 1.63 | 0.26 |
Create Element Mesh for the two areas
Create the Contact Elements that link the two areas together
Set Material Properties
Tungsten Steel - Sphere
Dicor/MGC - Plate
Apply Loadings
Apply Constraints taking symmetry in account
Run Analysis
Plot Results and Create Animation
1 Create Areas in ANSYS - Circle and Plate
2. Create Element Mesh for the two areas
Create the Contact Elements that link the two areas together
Set Material Properties
Tungsten Steel - Sphere
Dicor/MGC - Plate
Apply Loadings
Apply Constraints taking symmetry in account
Run Analysis
Remove element with highest stress
Repeat steps 7 and 8 to grow crack until the crack arrests
Plot Results and Create Animations
Create Volumes in Pro/Engineer - Sphere and Plate
Apply Loadings
Apply Constraints - for cutaway view, symmetry must be taken into account
Create Element Mesh
Export to ANSYS
Set Material Properties
Tungsten Steel - Sphere
Dicor/MGC - Plate
Run Analysis
Plot Results and Create Animations
Create Volumes in Pro/Engineer - Sphere and Plate
Apply Loadings
Apply Constraints - for cutaway view, symmetry must be taken into account
Create Element Mesh
Export to ANSYS
Set Material Properties
Tungsten Steel - Sphere
Dicor/MGC - Plate
Run Analysis
Remove element with highest stress
Repeat steps 7 and 8 to grow crack until the crack arrests
Plot Results and Create Animations
After running the procedures for the four stages of the project, the final solution plots and animations are shown in the Results section.
Discussion
Note that the Results section displays the animation of each Stage frame by frame (with the exception of the 4th Stage that will be discussed later). There are several points that need to be noted in the results from the 1st and 3rd Stages:
Compressive stresses at the contact point between the sphere and plate
High tensile stresses on plate surface
For brittle materials (Dicor/MGC), the tensile stresses act as initiation site for crack propagation (Crack propagation can be seen in 2nd Stage Results)
For the 2nd Stage results the crack is shown initiating and then propagating downward into the plate at about a 20o angle with the horizontal. Also as the crack propagates, the stresses in at the crack tip dissipate. Another thing to note is that the mesh for the 2D analysis is finer that the 3D analysis. This is because the student version of ANSYS is limited to 10,000 elements. Therefore, because the 3D model requires more elements, they are space out more to stay below 10,000 elements total. This results in an solution that is not as accurate as the 2D analysis, however it is close enough to notice the similarity between the 2D and 3D analysis.
The 4th Stage was not implemented due to the lack of resources and time. However, the
result would have been a crack in the shape of a cone centered around the contact point. This
crack would be traveling downward into the plate. The desired result was to make a transparent
3D model with the crack surfaces being dyed. This way, a person could visually see inside the
plate with the conic crack.
Applications of Analysis
The model used in this project can be used in many other types of design applications. For example, in the field of dentistry this type of FEA has already been implemented (Figure 1). Specifically, ANSYS has been used to study the Occlusal Contact. In a more specific case, the Occlusal Contact is a simulation of the biting action between an all-ceramic crown and an opposing object (the steel ball). Using the steel ball to represent the opposing object to the crown has numerous advantages over the previous methods of analysis. One advantage is that the surface radius of the ball represents more accurately the surface, which will exert forces on the crown. Prior methods of analysis, as mentioned earlier, were limited to point loads or simple variants of distributed loading. The radius of the steel ball in this case can be changed to account for varying conditions. From the Hertzian Indentation type of FEA analysis with ANSYS, various types of stresses and strains as well as locations of failure initiation can be solved for and visually estimated.
The fractures of brittle material that occur because of the tensile stresses are a topic of another type of application that deals with the effects of machining on brittle materials such as ceramics. Imagine the ball in our project as a cutting tool and the plate as a piece of brittle material, such as Dicor/MGC, that is being machined. When the cutting tool is introduced to the material the stresses that occur can cause the same type of conical fracture as demonstrated by our model analysis. These fractures can penetrate into the newly cut surface creating a surface finish where crack initiation has already begun. This type of analysis is important because once cracks are initiated, it is almost certain that failure will occur at that location.
Another application of the Hertzian Indentation model, more closely related to model we
analyzed, is in the design of ceramic floor tiles. Ceramic floor tiles are very likely to be
subjected to the same type of stress concentrations as the plate in our analysis. Objects that may
impart the forces that cause these stresses are ball roller feet on chairs or carts, tables with balled
feet, or wheels of any type. In many of these cases the same type of symmetry can be assumed
which will simplify the solution greatly. The smaller model size will allow for a finer mesh and
this in turn will render more accurate results. The stress concentrations and crack initiation for
different types of ceramic tiles are important to a person choosing a tile to achieve a particular
level of performance. These characteristics of the tiles are also important when attempting to
determine a cause for failure or predicting the general location in which a failure might occur.
Conclusion
In final summary, the Hertzian Indentation Model is useful because it predicts crack initiation and propagation for brittle materials subjected to contact stresses. Furthermore, this model can be applied to real world engineering problems in an attempt to avoid cracking. The solution for the Hertzian Indentation model is simplified due to the many Finite Element Analysis packages available such as ANSYS. In this project, a plate being subject to a contact load from a sphere, results in tensile stresses on the surface of the plate surrounding the contact point. Furthermore, these tensile stresses act as the site for crack initiation and propagation which lead to failure of the plate.