The aim of this work was to simulate the behaviour of hip
prostheses under mechanical shocks. When hip joint is replaced by
prosthesis, during the swing phase of the leg, a microseparation
between the prosthetic head and the cup could occur. Two different
sizes of femoral heads were studied: 28 and 32 mm diameter,
made, respectively, in alumina and zirconia. The shock-induced
stress was determined numerically using finite element analysis
(FEA), Abaqus software. The influence of inclination, force,
material, and microseparation was studied. In addition, an
algorithm was developed from a probabilistic model, Todinov's
approach, to predict lifetime of head and cup. Simulations showed
maximum tensile stresses were reached on the cup's surfaces
near to rim. The worst case was the cup-head mounted at
Ceramics were investigated in orthopaedics since many years because of their good wear-resistance and loading capacity [
Nevertheless, retrieved heads and cups have sometimes shown wear stripes [
Components for total hip replacement are shown in Figure
Scheme hip articulation [
Clearly, the shock resistance issue requires a better understanding of the biomechanics of gait cycle. Hip joint forces have been estimated indirectly using inverse dynamics and analytical methods or directly measured with implanted transducers. In general, the force at the hip joint reaches an initial peak in early stance phase and a second peak in late stance phase. This pattern was described in the initial work of Paul [
Because of the increase of life expectancy, new materials or designs are needed, with a significantly improved lifetime and better mechanical properties. The goal of this study was to develop a model combining finite element analysis and crack growth simulation. The influence of several parameters was then assessed to determine the most significant parameter in relation with shock resistance.
The degradation of two types of materials, alumina and zirconia, Table
Physical and mechanical properties of alumina and zirconia.
Density (g/cm3) | Young modulus (GPa) | Poisson’s ratio | |
---|---|---|---|
Al2O3 | 4.00 | 400 | 0.23 |
ZrO2 | 6.05 | 200 | 0.23 |
(a) Shock machine. (b) Assembly head and cup at the standard anatomic position.
The FE simulations were implemented using commercial software (ABAQUS 6.8, Dassault Systèmes, 2008).
Materials were assumed to be perfectly elastic, so no plastic deformation was possible. The modelling was carried out in static conditions.
The shock-induced stress was considered constant. Finally, the friction between head and cup was considered negligible.
Friction coefficient for contact ceramic/ceramic is the weakest compared to other couples of materials. As the friction behaviour, in terms of finite elements analysis (FEA), is not well known, the sliding movement was not considered in the contact. As the friction coefficient is weak, one might suggest that friction forces are negligible compared to contact forces and friction constant was zero.
However, a particular attention was paid on elastic contact and rebound, mechanical energy restored after a shock. An algorithm was developed using a bouncing ball made from zirconia diameter 32 mm. Head-taper fit and head cup were modeled with the same contact algorithm penalty. This contact with the option “hard contact” allows modelling elastic contacts. This algorithm was developed using a ball made of zirconia which falls from a height of 10 mm and impact. Coefficient of restitution was then calculated
Even though is possible to input a friction model in Abaqus, the model is unknown. That is why we said that there is not sliding in the contact. Moreover, the cup was constrained in all degree of freedoms to reproduce the shocks device.
In addition, symmetry conditions were imposed on the vertical axis. A load of 9 kN was defined for the contact head cup at the upper rim. Load was applied at the centre of the cone and increased linearly from zero to its maximum value at 11 ms which are the same conditions that experimental shock tests. Boundary conditions and definition of angles are shown in Figures
(a) Boundary conditions: degrees of freedom and vertical symmetry. (b) Definition of angles from the center of cup and head.
Angles will be called in following work by words, theta and phi, instead of Greek letters,
Two different sizes were studied, 28 and 32 mm in diameter. The first model consisted of a 28 mm femoral head and a cup, both made of zirconia and a cone made of Ti-6Al-4V. Young’s modulus and Poisson’s ratio for zirconia were set to 200 GPa and 0.23, respectively; see Table
(a) Assembly of the cup, head and cone for the finite element simulation. (b) Refined mesh of the assembly, head with 4800 nodes and cup with 2534 nodes.
Both FEA models were three-dimensional and meshed using tetrahedral solid elements, ideal for meshing complexes geometries. In order to obtain more accurate results, the mesh in the hypothetical contact zones between head and cup were refined, especially at the rim of the cup. Meshing of head, 4799 nodes, cup, 2534 nodes, and cone, 145 nodes, are shown in Figure
Several parameters were changed in order to determinate their influence on the stress. These parameters are summarized in Table
Studied parameters: materials, force, microseparation, and inclination.
Parameter | Material | Force | Microseparation (mm) | Inclination |
---|---|---|---|---|
Force | Zirconia | 2–9 kN | 1.3 | 45° |
Microseparation | Zirconia | 9 kN | 0, 0.7, 1.0, 1.3, 1.6, 1.9 | 45° |
Inclination | Zirconia | 9 kN | 1.3 | 30°, 45°, 60° |
Material | zirconia alumina | 9 kN | 1.3 | 45° |
An algorithm implemented in Matlab (Matlab 7.7 The MathWorks Inc.) was used to interpolate stresses on the nodes obtained with ABAQUS. This step is necessary for increasing the number of nodes in order to predict the structure behaviour. For each element, stresses in the nodes were interpolated into a cube; see Figures
3D Interpolation of stresses on the cup. (a) A finite element. (b) Interpolation cube including the element.
Thus, stresses on the cup were calculated in about 3 millions of points instead of a few thousands. Number of points was multiplied by 1000.
Stress intensity factor,
Only tensile stresses were considered. Stress is supposed to be maximum and constant during the 11 ms of the simulation. Therefore,
For alumina, the same ratio between
Parameters for
Al2O3 | 40.0 | 4.2 | 2.1 | |
ZrO2 | 20.0 | 5.2 | 2.7 |
Curves
The configuration is considered unstable, and the associated flaw will grow while
Algorithm for determining the number of unstable configurations.
For each shock
Algorithm for crack growth simulation.
Following this algorithm, Figure
Given that size and location of the flaws are randomly determined, test simulations were investigated in order to assess the optimal number of configurations
Determination of optimal number of configurations. (a) Average percentage of growth as function of the simulated number of configurations. Red dots represent minimum and maximum of percentage for every number of configurations. (b) Standard deviation of average percentage of growth. (c) Average calculation time, minutes, for detecting unstable flaws (according to the number of configurations).
The configurations number would be a compromise between the calculation time versus the growth percentage and the standard deviation. According to the results in Figure
The previous algorithm, described in part 2, allows getting the
Moreover, thanks to probabilistic model developed by Todinov [
By making the hypothesis that there is no flaws coalescence and they evolve independently, the probability of failure,
Determination of flaws density according to porosity and average flaw size.
Various configurations will be considered in this part. One factor changes, and the others are constant: force, microseparation, angle, and materials. In this part, the flaws increasing will not be mentioned. One focuses the attention on the influence on the experimental parameters to better understand the influence of each one.
In order to validate the crack growth model from experimental results, some modellings were investigated for a cup tilted at 45°, with former microseparation from 0.0 to 1.3 mm and a force between 2 and 9 kN. First, a contact between the head and the upper rim of the cup takes place. Then, the head impacts the upper rim of the cup, rebounds, and impacts the lower rim at the end of simulation, and the duration of this movement is 11 ms.
On the cup, the maximum stresses were always located near the rim; see Figure
Influence of applied force on stresses on the cup, microseparation of 1.3 mm, and angle of 45°. (a) Location of tensile stresses on the cup under a force of 6, 8, and 9 kN. (b) Maximum stresses as function of the applied force.
For higher forces, stresses are also observed on the lower rim of the cup. However, stresses on the upper rim are slightly greater than those on the lower rim; see Figure
This phenomenon, probably related to rebounds at the lower and the upper rims, is likely to appear in the experimental tests, since two wear stripes were located on the head. These results corroborate exactly some explants studies [
The influence of the microseparation was investigated with a cup position of 45° and a force of 9 kN. Three vertical displacements were 0, 0.5, 0.75, 1.0, 1.25, and 1.5 mm, which correspond respectively to these microseparations: 0, 0.7, 1.0, 1.3, 1.6, and 1.9 mm. As mentioned above, stresses are located on the upper rim and the lower rim of the cup, Figure
Influence of microseparation on the surface stresses of the cup; tensile stresses on the cup for 0, 0.7, 1.0, 1.3, 1.6, and 1.9 mm.
Tensile stress of 400 MPa in Figure
The influence of the cup angle, the inclination, was studied for 30°, 45°, and 60°; see Figure
Influence of inclination on the surface stresses of the cup. (a) Tested inclinations. (b) Stresses on the surface of the cup for each inclination and a microseparation of 1.3 mm.
An alumina-alumina prosthesis of 28 mm was considered, mounted at 45°, with a force of 9 kN. Stresses on the cup were 30% higher than stresses on the zirconia one. Results are shown on Figure
Influence of material on the surface stresses on the cup, with a load of 9 kN, inclination of 45°, and microseparation of 1.3 mm.
In this part, the Todinov’s approach is investigated for predicting the crack growth according to a flaws distribution. Positions of the flaws are shown in Figure
Location of simulated flaws, cup inclined at 45°, load of 9 kN and microseparation of 1.3 mm. (a) Location of critical flaws for a flaw size between 19 and 35
There was no critical flaw deeper than 3 mm. Moreover, these critical flaws seemed to be distributed on almost all the surface of the cup, Figure
This approach is macroscopic. Any investigation was carried out at the microscale, and this means that taking into account the grain structure is not considered in this study. However, the deepest flaws are related to the fracture zone. From the point of view of manufacturing, if defects are in the bulk alumina and if they are higher than 10
As mentioned before, material flaws density directly depends on the material porosity (
Evolution of crack probabilities versus the number of shocks, according to flaw size (from 19 to 40
The modelling parameters are as follows: zirconia-zirconia and alumina-alumina, head diameter of 28 mm, inclination of 45°, force of 9 kN and microseparation of 1.3 mm. The minimum flaws sizes likely to growth in alumina are smaller than those for zirconia: 4
Influence of material on critical number of shocks.
The main aim of this work is to present if zirconia or alumina materials involve the highest shock number according to the flaw size. If the shocks number is weak and the flaw size is lower than 10
Finally, four modellings were investigated: head diameters of 28 mm and 32 mm for alumina and zirconia. The inclination was constant and equal to 45°, the load was of 9 kN, and the microseparation was of 1.3 mm. Alumina 32 mm with a thick cup appears to be more resistant to shocks than alumina diameter 28 mm; see Figure
Influence of geometry on critical number of shocks. Critical number of shocks as function of flaw size for (a) alumina and (b) zirconia. Dotted lines correspond to the interval of lifetime found experimentally.
The finite element analysis allowed highlighting the significant parameters for the crack growth modelling (stresses) and establishing the mechanical behaviour description. Concerning the values of stresses, it is worth noting that the head is subjected to compressive stresses, whereas the cup is mainly subjected to tensile stresses more harmful. In addition, a weak microseparation has a huge influence on location of stresses. Thus, microseparation produced a stresses concentration at the rim of cup, experimental tests exhibit, too, a fracture at the rim of the cup. Rebounds and conflicts between head and cup, when the hydraulic jack go down involved two wear zones, at the upper and the lower part of the cup, which matched with the two wear stripes during the experimental tests.
Moreover, the stress maximum only depended on the head velocity before impacting the cup. In fact, this maximum is also influenced by the mechanical behaviour of the assembly.
Three inclinations were then tested: 30°, 45°, and 60°. When the cup is mounted at 60° the smaller maximum of stress is obtained regardless of the microseparation.
This simulation also highlighted the differences between the two types of tested bioceramics, zirconia and alumina. Alumina is submitted to higher compressive stresses than zirconia, since its Young’s modulus is two times higher. Finally, the worst case would be described in the following conditions: load of 9 kN; microseparation of 1.3 mm, inclination of 30° and alumina material. The inclination depends on the anatomical factors. The usual surgical inclination is within the range from 35° to 60° [
Nevertheless, severe modelling hypotheses were considered, and these results (in terms of lifetime and location of critical flaws) are in good agreement with the experimental tests; see Figure
The validation of results was made by comparing theoretical and experimental lifetime for zirconia-zirconia contact. The order of magnitude, that is, the lifetime, was similar regardless of the fact that crack growth laws were taken from literature [
The studied parameters were: material, head diameter (consequently cup diameter), cup geometry applied force, inclination and microseparation. Excluding the applied force which has an obvious influence on the lifetime of the prosthetic elements, microseparation and probably inclination of the cup seem to be both significant parameters.
The inclination has an influence on the probability of failure. From previous investigations [
In the model, after the impact head cup, only the compressive stresses on the head were taken into account. Nevertheless, there could be important tensile stresses on the femoral head created by the taper fit. Further investigations should be made in order to confirm if these tensile stresses could lead to fracture on the femoral head as observed
The elastic recovery of the actuator of the cup was not taken into account. Additional investigations will be in progress for taking into account the mechanical impedance of the device. Concerning the FEA, the contact head cup was supposed frictionless, whereas during the tests, a wear stripe due to sliding movement was observed. The materials were assumed perfectly elastic, but at microscale, some plastic deformations could occur. A limited number of elements were chosen to have reliable results, only for the more solicited zones. Thus, it would be interesting to refine the mesh in the volume, with the aim of confirming that stresses are mostly located on the surface. Nevertheless, refining the mesh would increase the calculation time. In this study, the purpose was to obtain reliable results even by meshing roughly, since 2D results were interpolated with a refined mesh which multiplied the elements. Crack growth laws, crack rate as function of the stress intensity factor, were taken from literature. These values should be determined for the materials tested in the present study, but it would need a lot of time. Therefore, the crack growth law was developed only for a flaw supposed spherical. In addition, theoretical estimations of lifetime are in agreement with the experimental results for zirconia and alumina, which suggests that the crack growth curves used are not too different to the actual case.
The modelling confirmed the hypotheses concerning the assembly head cup. It has been demonstrated that the minor microseparation involves a huge influence. Moreover, the cup is subjected to tensile stresses, whereas the head is subjected to compressive stresses. The modelling also confirmed the location of high stressed zones at the rim of the cup, corresponding to location of wear stripes on the heads from the experimental tests. Crack growth algorithm is based on the stresses calculated by finite elements analysis. Despite the simplifying hypotheses and the crack growth curves taken from literature, results are on agreement with the experimental observations. In fact, all the simulated flaws likely to induce rupture are located on the cup surface corresponding to wear stripes on the head. Furthermore, the equation to calculate probabilities of failure takes into account the porosity. This model combining FEA with crack growth modellings confirmed the hypotheses made during experimental tests. In addition, this model served to theoretically testing parameters not studied experimentally. Thus, the inclination of the cup seems to play a significant role in the hip prostheses degradation. It has also been shown that microseparation produces wear stripes and flaws on the cup surface which might lead to the fracture. Given that wear stripes appear early on the head during
An interesting development concerning FEA could be refining the cup mesh in 3D. One should pay attention to calculating more accurate stresses and verifying that they are mainly located on the surface rather than the bulk cup. A submodel could be made with only the rim of the cup, since the bottom does not take part in the global kinetics of the head-cup assembly. Given that at the microscale some plastic deformation could occur in the cup, plastic properties should be set to alumina and zirconia. A multiscale approach should be developed for understanding the crack initiation and evolution from the microscopic, which is grain scale to macroscopic scale. Concerning the crack growth algorithm, the three-dimensional flaws should be investigated instead of only those on the surface and with a nonspherical shape but ellipsoidal. Thus, it would be necessary to give a specific orientation to each flaw. This study could be the base for developing new tests as closer as possible to
The authors do not have a direct financial relation with the commercial identity mentioned in this paper that might lead to a conflict of interests for any of the authors.
The authors declare that there is no conflict of interests.
The authors are grateful to: (i) ANR (Agence Nationale de la Recherche) for granting the project “Opt-Hip”; (ii) Region Rhône-Alpes provided funding for a Ph.D. grant about this work; and (iii) Nicolas Curt for his technical contribution about the shocks device. Their institution is responsible about the licences. Due to ethics rules, They had to notice the licence when they investigated works with specific software.