Fracture Behavior of the Cement Mantle of Reconstructed Acetabulum in the Presence of a Microcrack Emanating from a Microvoid

In this work, the fin ite element method is used to analyze the behavior of the crack emanating from a microvoid in acetabular cement mantle by computing the stress intensity factor. A simple 2D multilayer model developed by Benbarek et al.[1] to reproduce the stress distributions in the cement mantle has been used. To provide the place of birth of the crac k, the stress distribution around the microvoid is determined in several positions for three different loads. The effect of axial an d radial d isplacement of the microvoid in the cement is highlighted. The results indicate that the stress distribution , xx  yy  and xy  induced in the cement around the microvoid are not homogeneous and this, whatever its position. In addit ion, there is a large birth risk of cracks in several radial directions depending on the position of the microvoid in the cement mantle. The crack can be triggered in several d irect ions in mode I or mode II, while the mixed mode is dominant. The KI and KII SIF varies according to the position of the microcrack and the microvoid in the cement. They increase proportionally with the increase of the weight of the patient. It should be noted that the KI SIF are two t imes higher than the SIF KII. The maxima of the KI SIF are obtained for the position of the microvoid α = 100° and θ = 45° of the microcrack and the risk of the propagation of the microcrack is very important for this orientation.


Introduction
Although the Polymethylmethacrylate has long been known as a fixat ive in orthopedics dental prostheses, its first use in hip arthroplasty in 1962 [2]. Despite the various disadvantages of PMMA, improved techniques of preparation and implementation of cement and implementation methods contributes to the survival of cemented arthroplasties. In addition, the function of fixing the implant, the bone cement is responsible for transferring the loads of the joint to the bone. Face loads transmitted, which can reach in some circu mstances eight times the weight of the patient [3,4], bio-co mpetence cement mus t be good [5].
Thus, the mechanical and physical properties of cement are determin ing in the service life of the imp lant [6,7]. These properties are strongly affected by the size and number of pores in cement [8]. Indeed, the porosity can cause crack initiat ion by fat igue, by creating irregular areas [9,10]. Thus, surgeons tend to reduce the porosity to ensure greater resistance to fatigue.
Go ld, that this trend is directly related to the chosen method of mixing during the preparation of cement [11]. For example, the conventional method of mixing leads to a porosity ranging from 5 to 16% depending on the type of cement, while the method of "vacuum mixing" generates a porosity of 0.1 to 1% [12,13]. So me authors assume that the latter method, increases the mechanical properties largely due to the decrease in micropores and macropores [14,15], thus improving the life of the cement [16,17].
The effect of the position and orientation of a crack in the cement in three loads using the finite element method has been studied by Serier et al. [18] and Bachir Bouiad jra et al. [19]. They indicate that, fo r the third case load, the risk of crack p ropagation is higher when the crack is in the horizontal position for both failure modes. Achour et al. [20] presented a study on the mechanical behavior of the damage (failure) of the interface between the cement / bone and cement / stem in total h ip prosthesis. They conclude that interfacial crack (cement / bone) in the distal region can spread by opening and shear; it can cause a risk of brutal fracture if the crack length exceeds 0.6 mm. The risk of failure of the interface cement/bone or cement/stem in the pro ximal area is less important compared with medial and distal areas. Flitti et al. [21] studied the effect of the position of a microcrack on mechanical behavior out of a total hip prosthesis under the effect of a 90kg patient's weight. They concluded that the initiat ion of a crack in the cement area distal femu r gro ws in mixed mode, unlike init iated in the pro ximal zone which can propagate in mode II. Bouziane et al. [22] examined the behavior of microvoids located in the cement of a model of the hip prosthesis simp lified three -dimensional. They show that when the microvoid is located at the proximal and distal areas, the static charge causes a higher stress field that the dynamic load. Un like the work of Benbarak et al. [1] and [18][19][20] (microcrack constant), wh ich showed that the effect of the position of the microcrack constant emanating from the microvoid ; in this paper we have shown the variation K I and K II factors as a function of the length of the microcrack emanating fro m the microvoid and for a plurality o f positions in the cement. These positions are chosen according to the critical amplification Von Mises determined fro m the microvoid on along the circu mference and on the depth of the thickness of the cement (P1-P9). To co mp lete this study, we evaluated the principal stresses at the two interfaces of the cement (upper and lower). Also, the presence of two microcracks fro m of the microvoid is highlighted.
The objective of this study are expected to shed light on the influence of the presence of microvoid and a crack emanating fro m the microvoid on the fracture behavior of bone cement, by using finite element method. The effect of the position of the microvoid in cement and effect of the size of the microcrack on the fracture behavior are h ighlighted. The stress intensity factor to the microcrack-tip is used like criterion of rupture. The analysis of the distribution of the Von Mises stresses in the various components of the acetabular part and the implant is made to a zero angle between the necks of the imp lant relat ive to the axis of the cup. We are required to develop a fin ite element model to analyze the p resence of a microvoid on the behavior and strength of bone cement

Geometrical Model
The geometrical model is generated fro m a roentgenogram of a 4mm slice normal to the acetabulum through the pubic and iliu m. The cup has an outer diameter of 54 mm and an inner diameter of 28 mm. It is sealed with the bone cement mantle to uniform thickness of 2 mm [23]. The inside diameter of the UHMWPE cup is 54 mm. The interfaces between the cup-cement and cement-subchondral bone are assumed to be fully bonded. In this work two cases were analyzed: the first is to take the presence of a microvoid in different positions in cement. The stress concentrations are determined. In the second case we assume init ially the propagation of a microcrack emanating fro m the microvoid in the determined position and characterized by a high stress concentration gradient; and another time it is assumed that the microcrack emanating fro m the microvoid in different positions. The stress intensity factors are evaluated. The model was div ided in seven different reg ions ( Figure 1) according to the different elastic constants with isotropic properties considered in each region. The main areas are: cortical bone, subchondral bone and spongious bone [24][25][26][27][28]. The femo ral head was modeled as a spherical surface that was attached to the spherical acetabular cavity. The acetabular cavity is located on the outside of the hip bone at the junction of its three co mponents ( Figure 1): iliu m, ischium and pubic bone. Table 1 su mmarizes the material properties of cement mantle, cup and all sub-regions of acetabulum bone.

Finite Element Modelling
The acetabulum was modeled using the finite element code Abaqus 6.11.1 [30]. To simp lify the study, the 2D model of the acetabulum was considered. This representation was used to be representative of a section taken through the transverse plane of the acetabulum. Berg mann [25] found that the variation of the resultant forces acting on the acetabulum is larger in the transverse plane. A very fine discretizat ion was used to represent all possible, and to be closer to reality, and special mesh type ¼ was used near a microcrack tip. Figure 2 shows the mesh of the geometrical model. The geometrical model consists of 20611 elements in total, 13564 quadratic elements of type CPS4R and 7047 triangular elements of type CPS3. We opted for an orientation defined by an angle of 0° between the implant neck and the axis of the cup. The latter reflects a posture of the human body.   . For reasons to be in the worst case, we chose a zero inclination angle between the neck of the imp lant and the axis of the cup (see Fig. 3) which was used by Benbarek et al. [1] whose they indicate that they present more stress concentration. The considered Body weight is 70, 140 and 210 kg. The sacroiliac jo int was co mpletely stationary while the pubic joint was free in the sagittal plane. The boundary conditions considered are shown in the configuration of Figure 3, pubic nodes are blocked in all directions, on the wing o f the iliu m the nodes are blocked along the x axis and a uniformly distributed load applied on the implant. The contact between the bone and cement and between the cement and the cup was taken as fully bound, and between femoral head and the cup was assumed to be without frict ion under small slip.

Variation of Von Mises
Before analyzing the stress intensity factor at the microcrack tip, it is necessary to analyze the stress distribution around the microvoid to predict the microcrack initiat ion. In It is clear that the stress distribution is not uniform around the microvoid. We note several peaks in each rad ial position of the microvoid. All these stresses are due to the compression effect produced by the weight of the patient. At the radial position corresponds to   0  the maximu m stress at the interface is cup-cement of the order o f 20M Pa and the bone-cement interface subchondral is of the order of 35M Pa. The first interface to the second interface stress changes from single to double, this shows that when the microvoid is close to the bone-cement interface subchondral interaction effect is much larger than when it is close to the interface head cup-cement.
The maximu m stresses in the microvoid near the interface cement / bone sub-chonral into position   0  are of the order of 35M Pa, 70MPa and 140MPa, respectively for the weight of 70kg, 140kg and 210kg. Th is shows the effect of the interaction between the microvoid and the interface. In these three cases the maximu m stress exceeds the tensile failure, which shows the severity of the defect position in the cement. In addition, depending on the axial position of the microvoid, the constraints become important. Fro m the position P1 where the cavity is close to the cup-cement interface, the maximu m stress increases progressively to approach the interface cement-subchondral bone. This finding is significant regardless of the radial position of the microvoid. The stress levels at the radial position of microvoid    is about four times lower of the compression fracture limit, while the traction is three times lower, wh ich shows that they are relatively low. By against, a weight of 140kg and the position of the microvoid to 100°, the constraints tend to the tensile strength limit to angles 30° and 210°. The stress yy  greatly exceeds the strength in tension and co mpression. In this case, the cement is almost frag mented in tension or compression depending on the position of the microvoid in the binder. and in the interval varying fro m 90° to 120° in both interfaces.

Stress Variation
The first peak is obtained at 0° and the second at 100° for the two interfaces of the cement. In this case, the Von Mises stresses are almost three times less to tensile strength stress. It should be noted that if a microvoid is in these two areas of peak stress, the defect will quadruplicate the stress and therefore present a high risk of microcrack init iation, and the likelihood of its spread is high. The Von Mises stresses are higher in the cup/cement interface that in the cement interface-subchondral bone and it exp lains that the cement is a stress absorber. If a cav ity is close to the interface, the stresses in the interface and the cavity will be increased as a result of interaction and therefore the risk of damage is major. This behavior shows that the existence of the microvoid is a source of increasing stress concentrations and consequently the risk of loosening of the prosthesis

Variation of SIF of Microcrack Emanating from the Microvoid
In this section we have studied the evolution of the stress intensity factor K I and K II as a function of the length of the microcrack emanating fro m the microvoid located in the bone cement. This latter is taken in the most unfavorable radial positions previously established. Three patient' s weight loads are considered, 70kg, 210kg and 140kg. According to figures 9.1-9.6, we find that the stress intensity factors K I and K II will vary as a function of the increase in the length of the microcrack emanating fro m the microvoid. This variat ion is more marked with increasing of patient weight. The stress intensity factors KI for the positions of the microvoid 40° are positive and for positions 0° are negative. While K II SIF are negatives whatever the microvoid position. We note that the SIF K I and K II obtained for the position   0  of the microvoid are much larger in absolute value compared to other positions, showing that the birth of a microcrack emanating fro m a cavity at an angle constitute a high risk of rupture co mpared with other positions. This is due to the edge effect. The K II SIF is almost ten times s maller than the KI except for the case of load 70kg, where it is almost neglig ible for large microcracks. In position   100


, the K I SIF shows significant positive values that can cause rupture of the cement easily. This microvoid position affects significantly the bone-cement fracture toughness, which controls the failure process at the interfaces.
In Figure 10 we present the Von M ises stress levels for four different orientations . The same behavior has been marked when the microvoid is at the position α = 120°. If the microvoid is at the position α = 40°, the microcrack is susceptible to propagate in pure mode I at θ = 135° or at θ = 335° or pure mode II at θ = 20° or θ = 170°. And if it is at 0°, the SIF K I reaches its maximu m negative at 0 ° and 335 ° for the SIF K II .

Conclusions
This study was conducted to analyze the fracture behavior of bone cement in the presence of a microvoid and a microcrack emanating fro m the microvoid. Results emerge the following findings: →The distribution induced by stresses