### Effects of stem length on the joint stresses

#### Von Mises stresses at bone cement

Figure 2 shows the variations on the maximum Von Mises stresses at the bone cement when using CoCrMo stems with different lengths. The model analysis of hip joint stresses shows that the greatest extreme values of stresses take place around the stem end (distal point). It can be remarked that the maximum Von Mises stresses at the bone cement increase significantly when increasing the stem length. For example, the stress at the bone cement increases by 50 % when a stem of 190 mm length is used instead of a 135 mm stem length. The significant increase in bone cement stress due to using long stems can be attributed to the increase in the distance between point of load application (at the femur head) and the hip joint fixation point (at the distal end). The longer distance between the loading point (proximal end) and the hip joint fixation point (distal end) induce larger bending stresses at the distal end. For CoCrMo stem, the values of maximum Von Mises stresses on the bone cement are 13.5 MPa, 20.3 MPa, 27 MPa and 35 MPa for 135 mm, 165 mm, 190 mm and 220 mm stem length respectively. These values are lower than the yield strength of bone cement (PMMA) which is around 39 MPa [29]. Similar trends have been also obtained for the variations of maximum Von Mises stresses at the bone cement when using Ti alloy, FG1 and FG2 stems with different lengths as indicated in Figure 3.

Figure 4 shows the distribution of maximum Von Mises stresses at bone cement along the cement length for 135 mm and 190 mm CoCrMo stems. The results of this figure indicate that the maximum Von Mises stress at bone cement takes place at the distal end (end of stem with bone cement). The values of these stresses increase as the stem length increases and their minimum values are observed at the proximal end of the hip joint. There is no change in the stresses’ distribution profile due to using different stem lengths, only the value of maximum Von Mises stresses increases when increasing the stem length.

### Shear stresses at bone cement/stem interface

Figure 5 shows the variations on the maximum shear stresses at bone cement and CoCrMo stem interface for different stem lengths. The results indicate that the stem length has remarkable effect on the resultant shear stresses at the bone cement/stem interface. These stresses decrease when increasing the stem length. The maximum values of stresses are located at the distal end of the prosthesis as observed in many previous results [13, 30]. The results indicate that the shear stress at the bone cement decreases by 38%, 23.3% and 13.3% when a stem of 220 mm,190 mm and 165 mm length is used instead of 135 mm stem length respectively. Similar trends have been also obtained for the variations of maximum shear stresses at the bone cement when using Ti alloy, FG1 and FG2 stems with different lengths as indicated in Figure 6. The values of maximum shear stresses on the bone cement are 21 MPa, 18.2 MPa, 16.1 MPa and 13.1 MPa for 135 mm, 165 mm, 190 mm and 220 mm CoCrMo stem length respectively. Similar results have been obtained by Sivasankar M. et al. [31] for the variation on shear stresses at the stem/cement interface with stem length. In this study, stem lengths of 145, 147.5, 150 and 155 mm are used in their finite element modeling. The shear stresses at the interface between bone cement and different stems are found to be related to the stem length. The maximum shear stresses at bone cement are found to be changed from 21 MPa to 17.1 MPa when 147.5 mm stem is used instead of 152.5 mm. This reduction can be attributed to the increase of contact area between bone cement and stem when using long stems.

### Von Mises stresses at bone

The variations of maximum Von Mises stresses at the bone for different CoCrMo stems are shown in Figure 7. The results indicate that the resultant Von Mises stress at bone slightly increase with increasing the stem length. The greatest extreme values of bone stresses take place around fixation point of the stem. The results show that the stresses at bone increase by 1.3%, 22% and 23.3 % when a CoCrMo stem of 165 mm, 190 mm and 220 mm length is used instead of 135 mm CoCrMo stem respectively. Similar trends have also been obtained for the variations of maximum Von Mises stresses at the bone when using Ti alloy, FG1 and FG2 stems with different lengths as indicated in Figure 8. The values of maximum Von Mises stresses on the bone are 111 MPa, 120 MPa, 120 MPa and 121 MPa for 135 mm, 165 mm, 190 mm and 220 mm FG1 stems respectively. Similar results have been obtained by Abdullah A.H. et al. [32] where the stresses at bone increased from 112 MPa to 204 MPa and to 278 MPa when a long stem is used instead of medium and short length stem. It is important to mention that in the present study the maximum stresses at bone are below the yield strength of bone that ranges from 130 MPa to 160 MPa even for long stem [11, 33].

Figure 9 shows the distribution of Von Mises stresses at bone along the bone length for 135 mm and 190 mm CoCrMo stems. The results of this figure indicate that the resultant Von Mises stresses at bone increase when increasing the stem length. These stresses decrease from their maximum value at the femur fixation point to their minimum values at the proximal end. According to the present FE results, the use of longer stems is believed to give higher stresses at femur and then be capable of reducing stress shielding problems while the use of shorter stems decreases the load transfer to the cortical bone and encourages the stress shielding, bone resorption problems and then prosthesis fails.

### Effects of stem material on the joint stresses

#### Von Mises stresses at bone cement

The effects of stem stiffness on the variations of maximum Von Mises stresses at the bone cement are shown in Figure 10. The results indicate that the maximum value of stresses occurs at the stem end/cement interface. From this figure, it can be noticed that the resultant Von Mises stresses at the bone cement decrease significantly when using FG materials instead of CoCrMo or Ti ally. For 135 mm stem, the Von Mises stresses at the bone cement decrease by 30%, 49% and 58.5% when using FG1 instead of FG2, Ti alloy and CoCrMo stem respectively. For 190 mm stem length, the Von Mises stresses at the bone cement decrease by 47%, 60% and 66.7% when using FG1 material instead of FG2, Ti alloy and CoCrMo stem respectively. For 220 mm stem length, the maximum Von Mises stresses at bone cement decrease from 35 MPa to 14 MPa when using FG1 stems instead of CoCrMo stem.

From the above results, it can be concluded that the proposed FG stem results in significant reduction in the resultant Von Mises stresses at bone cement compared with Ti and CoCrMo stems. The significant decrease in bone cement stress due to using FG stems can be attributed to the fact that low stiffness FG material allows for more deformation and can absorb more strain energy compared to high stiffness ones. The resultant maximum Von Mises stresses induced in bone cement when using FG1 (14 MPa) is much lower than the yield strength of bone cement (PMMA), which is around 39 MPa [29]. Similar trends are obtained by Senalp et al.[10], where the resultant stresses on the artificial hip joint decrease when using Ti alloy stem instead of CoCrMo stems under static and dynamic loading conditions. Also, the results obtained by Sabatani et al. [11] indicate that the use of Ti6Al4V stem exhibits lower stresses than those obtained when using CoCrMo and SS stems. Finally, Rawal et al. [33] indicated that the use of stem material with lower modulus (E=117 GPa) would be more beneficial for artificial hip joint performance than the use of stiffer material such as steel (E=210 GPa).

Figure 11 shows the distribution of maximum Von Mises stresses at bone cement along the cement length for 135 mm FG1, FG2, Ti alloy and CoCrMo stems. The results of this figure indicate that the maximum Von Mises stress at bone cement takes place at the joint distal end. From the results, it can be concluded that the use of proposed FG1 as stem successfully reduces the predicted stresses at bone cement. Also, the stresses’ distribution along the bone cement length when using FG1 is more uniform along the whole bone cement compared to other stem materials. The more uniform predicted stresses at the bone cement that induced when using FG1 will help in the reduction of the artificial hip joint loosening rate and improving its short and long term performance.

### Shear stresses at bone cement/stem interface

Figure 12 shows the variations on the maximum shear stresses at the bone cement/stem interface with stem stiffness. The results indicate that the maximum value of stresses occurs at the distal end. It can also be noticed that the resultant shear stresses at the bone cement decrease significantly when using FG1 material instead of CoCrMo, Ti ally and FG2 even for long stems. The decrease in shear stress at bone cement/stem interface can be attributed to the low stiffness of FG stem compared with other stems. For example, for 135 mm stem, the shear stresses at the bone cement decrease from 13.1 MPa, 12.8 MPa, 8.2 MPa to 6.7 MPa when using FG1 material instead of CoCrMo, Ti alloy and FG2stem respectively. For 190 mm stem length, the shear stresses at the bone cement decrease by 51%, 44% and 12% when using FG1 material instead of CoCrMo, FG2 and Ti alloy stem respectively. For 220 mm stem length, the maximum shear stresses at bone cement decrease from 21 MPa to 10.2 MPa when using FG1 stems instead of CoCrMo stem. The significant decrease in bone cement shear stress due to using the proposed FG material even for long stems will improve the short and long term performance of the total hip joint replacement. It can be concluded that in order to obtain acceptable and uniform stresses at the bone cement, a stem of suitable length (not too long or too short) and low stiffness is required.

### Von Mises stresses at bone

The effects of using the proposed FG stem on the variations of maximum Von Mises stresses at the bone compared to other stem materials are shown in Figure 13. The results indicate that using FG stem results in creating higher values of Von Mises stress at bone for all stem lengths. Also, the results indicate that the stresses at bone increase by 3.3%, and 23.3% when a 135 mm FG1 stem is used instead of 135 mm Ti alloy or CoCrMo stems. The maximum value of Von Mises stresses at bone when using the proposed FG material ranges from 111 MPa to 122 MPa. These stresses are below the yield stress of bone with factor of safety 1.6, indicating no risk of fracture [11, 33]. The increase in bone stresses due to using FG material can be attributed to high flexibility of hip joint structure when using FG1 stem. The flexibility of hip joint structure in bending is known to be mainly determined by the properties of femur bone (E= 17 GPa) and stem (10 GPa at distal end). This means that the low stiffness (FG material) stem will increase the stresses at the femur bone (reduce stress shielding) compared to high stiffness stems (Ti and CoCrMo).

Figure 14 shows the distribution of Von Mises stresses at bone along the bone length for 135 mm FG1, FG2, Ti alloy and CoCrMo stems. The results of this figure indicate that the level of Von Mises stresses at bone increases on the whole femur when using FG stems compared to Ti alloy and CoCrMo stems. These stresses decrease from their maximum values at the femur fixation point to minimum values at the proximal end. The minimum values of stresses at bone increase by 47% when using FG1 stem instead of CoCrMo stem. Therefore, according to the present FE results, the use of FG material is believed to give higher stresses at whole femur bone and then capable of reducing stress shielding problem.

### Stresses at stem

The effects of stem stiffness and length on the variation of maximum Von Mises stresses at stem are shown in Figure 15. The results indicate that the maximum Von Mises at stem increase with increasing the stem length. For example, for CoCrMo, the maximum Von Mises stresses at the increase from 88 MPa to 153 MPa, 190 MPa and 212 MPa when using 165 mm, 190 mm and 220 mm CoCrMo stems instead of 135 mm stem. The results show that the maximum Von Mises stress at the stems occurs at the distal end of the stem. This significant increase in the stem Von Mises stresses due to using long stems can be explained due to the increase in the distance between the point of load application (at the femur head) and the hip joint fixation point (at the distal end). The longer distance between the loading point (proximal end) and the hip joint fixation point (distal end) induces larger bending stresses on the whole joint at distal end. Previous results show that the resultant Von Mises stresses at similar Ti alloy stem range from 186 MPa to 253 MPa [7, 34]. The present results show that the possible maximum Von Mises stress at stem is 212 MPa, which occurs in case of 220 mm CoCrMo stem. This value is still below the yield strength of Ti ally or CoCrMo stems (800 MPa for Ti alloy and 720 MPa for CoCrMo) [10].

The results of Figure 15 also indicate that the maximum value of Von Mises stresses at stem decrease when using FG material stems instead of CoCrMo and Ti alloy stems at the same testing conditions. For example, for 135 mm stem, the stresses decrease from 88 MPa and 77 MPa to 43 MPa when FG1 stem is used instead of CoCrMo and Ti alloys stems respectively. For 220 mm stem, the stresses at the stem decrease by 57%,49 and 40% when FG1 stem is used instead of CoCrMo, Ti alloy and FG2 stems respectively. The decrease in stem stresses due to using low stiffness FG material can be attributed to the fact that a larger amount of load will be carried by the femur bone specially at the distal end (near fixation point). At this point, the FG stem has E=10 GPa, that is lower than the femur modulus (17 GPa). At case of Ti alloy or CoCrMo stems (that have very high modules compared to femur bone) larger amount of load will be carried by high stiffness stem resulting in a reduction of femur stresses and increases in stem stresses. For all kinds of stems, the maximum Von Mises stresses at stem take place at the distal end. There is no effect for stem length or material on the position of maximum Von Mises stresses at the stem or femur bone and also bone cement.