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Numerical analysis of the crack growth path in the cement mantle of the reconstructed acetabulum Smaïl Benbarek ⁎, Bel Abbes Bachir Bouiadjra, Bouziane Mohamed El Mokhtar, Tarik Achour, Boualem Serier Mechanics and Physics of Materials Laboratory, Djillali Liabes University of Sidi Bel-Abbes, BP89 cité Larbi Ben M'hidi, Sidi Bel-Abbes, Algeria

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Article history: Received 16 November 2011 Received in revised form 7 September 2012 Accepted 28 September 2012 Available online 6 October 2012 Keywords: Biomechanics Orthopedic cement Crack propagation Abaqus python scripting

a b s t r a c t In this study, we use the ﬁnite element method to analyze the propagation's path of the crack in the orthopedic cement of the total hip replacement. In fact, a small python statement was incorporated with the Abaqus software to do in loop the following operations: extracting the crack propagation direction from the previous study using the maximal circumferential stresses criterion, drawing the new path, meshing and calculating again (stresses and fracture parameters). The loop is broken when the user's desired crack length is reached (number of propagations) or the value of the mode I stress intensity factor is negative. Results show that the crack propagation's path can be inﬂuenced by human body posture. The existing of a cavity in the vicinity of the crack can change its propagation path or can absolutely attract it enough to meet it. Crack can propagate in the outward direction (toward the acetabulum bone) and cannot propagate in the opposite direction, the mode I stress intensity factor increases with the crack length and that of mode II vanishes. © 2012 Elsevier B.V. All rights reserved.

1. Introduction In the clinical loosening of implants in total hip replacement surgery, fracture of the cement mantle is the main indicated reason [1]. Cracks initiate from micro-voids in cement and propagate due to the cyclic loading of the human body weight during the walking gate cycles [2]. Crack growth analysis is very important to improve the total hip life span. In literature; there has been a little amount of researches carried out in to the crack's growth path in the orthopedic cement, there has been some studies dealing with the fatigue life of orthopedic cement, but not following the crack's growth path [3–6]. Byeongsoo et al. [7] analyzed the fracture parameters (KI, KII and Keff) in a cross section of the femur part of the total hip joint. The focus of this study is to follow the path of a crack initiated from a cavity, and study the interaction between the crack path and cavity in order to highlight to the implants designer the most dangerous positions of the crack in order to predict the life span of the cement mantle using the fatigue laws. The Abaqus python programming language [8] permits us to implement a parametric study easily, but when making crack propagation, we must redraw the whole model and added the new crack's propagating increment as a geometry entities (edges, lines, surfaces, focused mesh…etc.) according to the computed crack deviation. This work was done using a python program loop incorporated with the Abaqus main

⁎ Corresponding author. Tel.: +213 554388759. E-mail addresses: [email protected] (S. Benbarek), [email protected] (B.A. Bachir Bouiadjra), [email protected] (B.M. El Mokhtar), [email protected] (T. Achour), [email protected] (B. Serier). 0928-4931/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msec.2012.09.029

software. A schematic drawing is showed in Fig. 1 to explain the main operations of the program. The behavior of a crack emanating from cavity localized at 20° and 100° according to the cylindrical coordinate system given in Fig. 2 is numerically analyzed. The choice of the ﬁrst position is based on the stress distribution in the cement (higher stresses are located on this position) and the analysis of the second cavity position is done to be compared to the ﬁrst one. 2. Geometric deﬁnition Fig. 3 shows the geometrical model. It was taken from our previous study [9]. In this model, one choose a 200 μm circular cavity. Foucat [10] found that the size of the cavities existing in the cement is between a few micro-millimeters to 1 mm. The chosen size in this work represents the most frequent case. The analysis of the crack propagation can be done in any plane across the acetabulum bone; we have to change only the boundary conditions, the considered plane and compute the crack propagation's path. 3. Material's deﬁnition Fig. 4 shows the assignment of different materials constants to their respective geometries. Table 1 contains materials' properties of cement, cup and all subregions of the acetabulum bone. 4. Loading model Two selected load cases corresponding to the stem position equivalent respectively to 0° and 50° with an average body's weight of 70 kg

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Importing the initial geometry with the inclined initial crack.

Computing the crack parameters (SIF and the direction of propagation).

Drawing a new path with the user’s desired length according to the propagation’s direction. Fig. 3. Geometrical model.

Re-meshing and re-computing the crack parameters(SIF and direction of propagation).

(Fig. 5). Fig. 6 shows boundary conditions acting on the model. The boundary conditions imposed on the structure are taken from previous work [9]. 5. Results and analysis 5.1. Cavity position effect on the path of the crack propagation

Y

N SIF>0

The program gives a text file including crack parameters(SIF and crack length). Fig. 1. A ﬂowchart of the main operations which make the crack propagations.

Fig. 7 shows a crack propagation created by the program for an initially inclined crack. This ﬁgure shows clearly the propagation of a crack emanating from micro-void, and initially inclined with 60° under a static loading. This illustrates the efﬁciency of the developed schedule to predict the path propagation. The analysis of this ﬁgure shows that the initially inclined crack has a tendency to redress itself during the ﬁrsts propagations' steps (2 to 3 steps) until getting a direction of about 40° according to the horizontal plane. This behavior is due to stress ﬁeld nature around the crack tip. The mesh around the crack tip is very reﬁned before and during propagation which gives a good simulation of the stress ﬁeld close to and in the vicinity of the crack tip (Fig. 8). A deep zoom of the frame in the vicinity of the crack shows clearly its deviation and opening (Fig. 9). During its development the initially inclined crack tends to vanishes its inclination and propagate according to the direction that makes an angle of 40° with the horizontal plane. 5.2. Crack path according to the initial crack inclination This analysis was done for a crack initiated around the cavity with a step of 10° for the ﬁrst stem position. Fig. 10 shows the distribution of 36 cracks initiated around the micro-void (each crack was initiated and propagated alone). This ﬁgure analysis shows clearly that the cracks which are inclined between 0° and 90° turn and propagate

Fig. 2. Cylindrical coordinate system (r,θ).

Fig. 4. Composition of a reconstructed acetabulum.

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Table 1 Materials' properties. Materials

Young modulus E (MPa)

Poisson ratio (ν)

Cortical bone Sub-chondral bone Spongious bone 1 Spongious bone 2 Spongious bone 3 Cup (UHMWPE) Cement (PMMA) Metallic implant

17,000 2000 132 70 2 690 2300 210,000

0.30 0.30 0.20 0.20 0.20 0.35 0.30 0.30

according to the direction that makes 40° with the horizontal plane. The other cracks' propagation seems to stop. This shows clearly that the cracks emanating from micro-void cannot propagate if they are not favorably oriented (according to the radial direction). We notice that the geometry of the cement is spherical. In our previous parametric studies; one noticed that the crack is dangerous when its inclination is radial [9,11]. The effect of the second stem position on the crack instability around the cavity is given in Fig. 11. The implant orientation leads to a weak crack rotation. In this case, the cracks change their orientation slightly and propagate in the same direction. This behavior can be explained by the fact that the stem position induces a more wide uniform circumferential stress ﬁeld in the vicinity of the crack tip. Only cracks initially inclined between − 10° and 90° may propagate. These favorably inclined cracks grow progressively according to the radial direction of the cement mantle. Cracks initially inclined in the opposite direction propagate but they stop propagating after a few steps because the schedule stops their propagation if the mode one stress intensity factor is negative. Results from this study show that the loading nature deﬁnes the crack instability and its growth path. For more information about the loading effect on the crack path, three inclinations (0°, 30° and 60°) of an emanated crack from a micro-void at the position of 100° are analyzed (Figs. 12–14). These ﬁgures show that the propagation's path depends on the stem position. Under the ﬁrst position loading, the propagation path is directed slightly to the left side comparing to that given with the second implant position. This behavior is noticed for the three analyzed crack positions.

Fig. 6. Boundary conditions.

Fig. 7. Crack propagation path.

The ﬁrst position of the stem is the most common human body position which simulates the standing position. Under this position the path is slightly shifted to the left side. This is the mostly noticed path. Previous studies [12,13] found the same behavior of the crack with respect of the stem position. 5.3. Parameters of the cracks propagation To get a better information of the mechanical fracture behavior of the orthopedic cement, the stress intensity factor analysis was carried out with respect to the propagating crack length (cracks which are

Fig. 5. Stem positions.

Fig. 8. Mesh model.

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Fig. 9. An open propagated crack. Fig. 12. Stem's position effect on crack path for a 0° crack inclination.

In mode II, the stress intensity factor behavior in terms of the crack length and the stem position is completely different with that in mode I (Fig. 15). In fact, shorts cracks give the maximum stress intensity factors. For long cracks, the stress intensity factor in mode II vanishes. This results show that during its growth, the crack rotates itself toward a favorable orientation, which lead to the vanishing of stress intensity factor in mode II. In Fig. 16, the mode II factor seems to be independent to implant position. Thus, the opening fracture mode is the preponderant one. 5.4. Case of a cavity at 20°

able to propagate) and the stem orientation. For this case, the stress intensity factor was analyzed for a crack inclined with 60° emanated from a cavity at 100°. The obtained results are plotted in Fig. 15. This ﬁgure shows that the 1st stem position lead to a growth of the stress intensity factor in mode I. this growth is very important for long cracks. Thus under this loading conditions, cracks which are initially favorably oriented can propagate and can lead to the collapse of the orthopedic cement with the opening fracture mode. One can notice that the second stem's position can lead to a signiﬁcant decrease of the stress intensity factor level. The mode I SIF seems to be little sensible to the second stem's position.

To show the cement cavity position effect, the behavior of an initiated crack from a cavity positioned at 20° in the (r,θ) coordinate system was analyzed. To do a comparative study with the precedent one, cracks are initiated around this cavity and propagated with the same steps. Fig. 17 shows the 36 crack propagation paths for the ﬁrst stem position. The analysis of this ﬁgure shows clearly that the stem's position lead to the propagation of a great amount of cracks. Theses cracks propagate mainly in two directions making an angle of − 20° and 110° with respect to the horizontal plane. The second stem position (Fig. 18) solicits differently in the cement. Cracks that may propagate are those initiated from this cavity and are initially oriented between −20° and 70°. These cracks propagate in a preferential direction which is 25° with the horizontal plane. Cracks which are initiated in the range between 190° and 240° grow weakly and stop propagating after a few steps. Fig. 19 shows the effect of the stem position on the crack propagation's path for the same initiated crack. Under the act of the ﬁrst implant position, the crack propagation's path is directed

Fig. 11. Cracks path vs. crack initiation inclination for the second stem position.

Fig. 13. Stem's position effect on crack path for a 30° crack inclination.

Fig. 10. Cracks path vs. crack initiation inclination for the ﬁrst stem position.

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Fig. 17. Cracks path along crack initiation inclination for the ﬁrst stem position. Fig. 14. Stem's position effect on crack path for a 60° crack inclination.

0,10

KI [MPa(mm)1/2]

0,09 0,08

load1 load2

0,07 0,06 0,05 0,04 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

Fig. 18. Crack path along crack initiation inclination for the second stem position.

Crack length [mm] Fig. 15. Mode I SIF along crack length.

according to the radial direction but under the second position, the growth path is shifted by 25° from the ﬁrst path. Fig. 20 shows the inﬂuence of both implant position and crack length on the mode I stress intensity factor. The SIF increases linearly with respect to the crack length whatever the human posture is. This growth is more important when moving from the ﬁrst stem position to the second one. The difference between both SIF is constant whatever the crack length. The propagation risk is higher when the man is in standing position. 5.5. Interaction between cavity and crack path

are given in Fig. 21. This ﬁgure shows the propagation direction of the crack emanated from cavity at 100° close to the cement–cup interface. This crack was initiated inclined with 30°. This ﬁgure shows that this crack propagates following a linear path toward the radial direction of the cement mantle. Secondly, a second cavity was created near the crack path to shows its effect on the path shapes. In the vicinity of the second cavity, this crack was deviated by the stress ﬁeld around the cavity (Fig. 22a). In fact this ﬁeld exerts an attraction effort on the crack leading to its deviation to the cavity. After being deviated, the crack reorients itself according the previous path (radial direction). Far from the cavity, this factor cannot have any effect on the crack path. A photo taken by

Analysis of the cavity effect on the propagation path of a crack emanated from micro-void was performed. Firstly, a crack propagation analysis in pure cement was carried out. The obtained results

0,020

load 1 load 2

KII [MPa(mm)1/2]

0,015 0,010 0,005 0,000 -0,005 -0,010 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

Crack length [mm] Fig. 16. Mode II SIF along crack length.

Fig. 19. Stem's position effect on crack path.

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S. Benbarek et al. / Materials Science and Engineering C 33 (2013) 543–549 Crack little far from cavity Crack close to cavity Crack in pure cement

0,040 0,035

0,12

Load 1 Load 2

KI[MPa(mm)1/2]

KI [MPa(mm)1/2]

0,030 0,025 0,020 0,015 0,010

0,10

0,08

0,06

0,005

0,04 0,000

0,0 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

crack length1 [mm]

Crack length [mm]

Fig. 23. Stress intensity factor along the crack length. Fig. 20. Mode I SIF along crack length.

tip and lead to an intensiﬁcation of the stress intensity factor. Close to the cavity, this factor strongly drops due to stress concentration around the cavity. Far from the cavity, this effect disappears and the SIF values rise again (line+ square graph). If the crack is very close to the cavity, the cavity strongly attracts the crack till meeting it, which explains the shortness of the third crack length (line + circle graph of Fig. 23). 6. Conclusion The propagation study of a crack emanating from a micro-void in the cement of a reconstructed acetabulum is carried out using the ﬁnite element method associated with a schedule which does the propagation. The obtained results are:

Fig. 21. Crack path in pure cement.

Nikolaus [14] during a compression test is given in Fig. 22b, it shows the great similitude with our numerical results. Fig. 23 shows the variation of the stress intensity factor (SIF) according to the crack size and the crack–cavity distance. This ﬁgure shows that when the crack is close to the cavity, the interaction becomes important; the stresses' interaction between the crack tip and the cavity increase the mechanical energy in the vicinity of the crack

• For a cavity at a position of 100°, the crack emanating from micro-void and initially inclined between 0° and 90° can propagate by reorienting itself toward a 40° direction and then follows a linear path; the other cracks are closed. A slight shifting of the stem reduces the angle of reorientation, but the propagation direction still the same. • The stress intensity factor in mode I, increases with respect to the crack length; that in mode II vanishes. The position of the stem changes the values of the stress intensity factor KI. • For a cavity at 20°, most cracks cans propagate and follow one of both directions −20° or 110°. For a slightly oriented stem, only cracks inclined between −20° and 70° can propagate. • If a crack is emanated from a cavity in pure cement (without any defect around), it propagates linearly in the radial direction until the

Fig. 22. a. Crack path close to cavity (numerical), b. Crack path close to cavity (experimental).

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bone–cement interface. If another cavity exist around crack path, this last behaves in two manner: ﬁrstly, the crack close to the second cavity will deviate from its initial path and then takes back its path; secondly, when the cavity is very close the crack path, it will completely attract the crack and they meet. • According to this results, a reinforcement of the cement by any procedure (ex: nano-particles) in the direction of propagation can stop or retard this propagation and consequently improve the life span of the hip prosthesis. • Using the results of this study (crack length and SIF), one can predict the fatigue life of the orthopedic cement and consequently, the life span of the prosthesis. Acknowledgment The authors extend their appreciation to the Deanship of Scientiﬁc Research at King Saud University for funding the work through the research group no. RGP-VPP-035.

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