International Journal of Mechanics and Materials in Design (2004) 1: 329–346 DOI 10.1007/s10999-005-3307-4
Springer 2005
A method of material optimization of cementless stem through functionally graded material H. S. HEDIA*, M. A. N. SHABARA, T. T. EL-MIDANY and N. FOUDA Department of Faculty of Engineering, Prod. Eng. & M/C Design, Mansoura University, Mansoura, Egypt *Author for correspondence (E-mail:
[email protected]) Received 5 August 2003; accepted in revised form 2 February 2004 Abstract. Ideally, a bone implant should be such that it exhibits an identical response to loading as real bone and is also biocompatible with existing tissue. A stiff stem, which is usually made of titanium, shields the proximal bone from mechanical loading (stress shielding). On the other hand, decreasing the stem stiffness increases the proximal interface shear stress and the risk of proximal interface failure. Therefore the purpose of this study is to solve these conflicting requirements in order to have more uniform interface shear stress distribution and less stress shielding through the concept of functionally graded material (FGM). FGM is a kind of advanced composite materials, which changes its composition and structure gradually over one or two directions of its volume, resulting in corresponding changes in the properties of the material. This study is divided into two parts; in the first part, the finite element analysis and optimization technique are used to design the stem as one-dimensional FGM, while in the second part, the stem is designed as two-dimensional functionally graded material. The aim of both designs is to overcome the above mentioned problems. In the case of part one (one-dimensional FGM), the gradation of elastic modulus is changed along the vertical direction (model 1) and along the horizontal direction (model 2), in order to find the optimal gradation direction. It is found that the optimal design is to change the elastic modulus gradually from 110 GPa (Hydroxyapatite) at the top of the stem to 1GPa (Collagen) at the bottom (model 1). This optimal gradation decreases stress shielding by 83%, while reduces the maximum interface shear stress by 32% compared to homogenous titanium stem. However, in the second part (twodimensional FGM, model 3) the materials of optimal design are found to be hydroxyapatite, Bioglass, and collagen. This design leads to the same stress shielding reduction as in model 1, while at the same time, the maximum interface shear stress is reduced by 45% and 63% compared to the optimal one-dimensional FGM design and homogenous titanium stem, respectively. Key words: Material optimization, functionally graded material, cementless stem, von Mises stress, interface shear stress.
1 Introduction It has been well documented that bone adaptations around femoral hip stems are related to mechanical conditions. The natural stress distribution in the femur is significantly altered after total hip arthroplasty. When an implant is introduced, it will carry a portion of the load, causing a reduction of stress in some regions of the remaining bone. This phenomenon is commonly known as stress shielding. In response to the changed mechanical environment the shielded bone will remodel, resulting in a loss of bone mass through a biological process called resorption. Resorption can, in turn, cause or contribute to loosening of the prosthesis. However, the bone– prosthesis interface is a major source of problems associated with cementless total hip replacement. Finite element analysis of bone–prosthesis configuration almost invariably shows that high interface stress peaks appear at the interface edges whereas the remaining part of the interface is virtually unloaded. These stress concentrations are likely to cause problems such as loosening and pain (Kuiper and Huiskes, 1997). These two main problems can be avoided by
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controlling the stem stiffness. As the stem stiffness is a function of its shape and material, the problem can be alleviated by controlling these two factors. (Kuiper and Huiskes, 1992) developed a numerical optimization method to determine the optimal stiffness characteristics of a cementless hip stem, in order to minimize the probability for interface failure while limiting the amount of bone resorption. The method is based on finite element analysis in combination with an optimization procedure. The parameters describing a non-homogenous elastic modulus distribution were considered as design variables. The method was applied to a 2-dimensional finite element model of a femoral hip replacement. A prosthesis, which was relatively stiff at the proximal end, and whose stiffness gradually reduced towards the distal end was found to satisfy the requirements and to be much better than a stem made out of a homogenous material. They also applied a mathematical optimization of elastic modulus to a simplified model of a cementless hip stem (Kuiper and Huiskes, 1997), assuming long-term bone loss. They concluded that a maximum interface stress can be reduced by over 50% compared to a homogenous flexible stem. On applying the concept of shape optimization, (Huiskes and Boeklagen, 1989) introduced a method for shape optimization applied in a simplified model of a cemented femoral stem fixation. The results showed that 30–70% cement and interface stress reductions can be obtained using a model with proximal and distal tapers, and a belly-shaped middle. (Hedia et al., 1996) dimensioned the high values of stress in the cement in a cemented prosthesis, while increased the fatigue notch factor in proximal medial bone in order to reduce stress shielding in this region by using the concept of shape optimization. Again they described the effect of changes in the elastic modulus of the stem material for both the original Charnley stem and the optimized shape (Hedia et al., 1997). They concluded that a composite prosthesis with a layer of modulus 31 GPa added to the optimized stainless steel stem in the proximal region only was found to significantly increase the stresses in the proximal bone and reduce fatigue notch factor in the cement. The design of a controlled stiffness composite femoral prosthesis is described by (Simoes et al., 2000). The prosthesis is composed of a cobalt–chrome core (E=200 GPa) surrounded by a flexible composite outer layer (E=12 GPa), using braided hybrid carbon-glass fiber performs and epoxy resin. By varying the composite layer thickness it was possible to control the prosthesis stiffness. A 3-dimensional material optimization of femoral components of hip prostheses was described by (Katoozian et al., 2001). Two objective functions were defined based on interface failure criteria and bone adaptive remodeling to avoid interface disruption and to reduce the risk of bone loss. The results demonstrate the potential of fiber-reinforced composites for improved design. Akay and Aslan (1996) designed carbon fiber/polyetheretherketone (CF/PEEK) composite stems which possess a mechanical behavior similar to that of the femur. Finite element analysis indicated that compared to conventional metallic stems more favorable stresses and deformations could be generated in the femur using composite material. Hydroxyapatite/Collagen (HAP/Col) composite having a bone-like nanostructure was designed by (Itoh et al., 2002) to develop an artificial vertebra system using this novel implant for anterior fusion of the cervical spine, they tested this implant in dogs. The histological and radiographical analysis suggested that this novel biomaterial is a suitable replacement for anterior fusion of the cervical spine. (HAP/Col) composite also has a self-organized character similar to that of natural bone (Itoh et al., 2000).
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Recently, divers serious activities can be observed in materials science to develop a new generation of biomaterials for bone replacement which combines bone-like mechanical properties with characteristic feature of the living bone and the capability of remodeling by cellular activity. These two goals open the avenue for application of basic concepts of functionally graded materials to new bone replacement material (Pompe et al., 2001). Functional gradation is one characteristic feature of living tissue, in which a compositional change from surface to surface optimizes the overall performance of the component. Dental implants with functionally graded structures composed of titanium (Ti) and ceramic hydroxyapatite (HAP) were fabricated by (Miyao et al., 2001), to satisfy both mechanical and biocompatible property requirements. The implant was functionally graded in the longitudinal direction with more titanium in the upper part and more apatite in the lower part. They found that the biocompatibility tests were successfully performed using miniature FGM specimens, and no inflammation was observed through the implantation period of 1–4 weeks. FGMs composed of polymer-ceramics and metal-ceramics were fabricated for biomedical applications and the effect of gradient structure was investigated by (Watari et al., 2003). They concluded that functionally graded core and post made of composite resin with the filler content decreasing from core part to the apex of post showed gradual decrease in elastic modulus and contributed to stress relaxation as evaluated by stress analysis using finite element method. They also found that Ti/HAP functionally graded implant accelerated the maturation of newly formed bone in HAP-rich region from the earlier state. The gradient functions in both mechanical properties with stress relaxation and biochemical affinity to osteogenesis could attain efficient biocompatibility. Titanium has reasonable stiffness and strength while hydroxyapatite has low stiffness, low strength and high ability to reach full integration with living bone. Therefore, a design optimization of functionally graded dental implant was carried out by (Hedia and Nemat-Alla, 2004) to achieve the advantages of both titanium and hydroxyapatite, using finite element method. The investigation has shown that the maximum stress in the bone for hydroxyapatite/ titanium FGM implant decreases by about 22% and 28% compared to currently used titanium and stainless steel implants, respectively. From the foregoing review of literature, one can see that several works have been carried out to control the stiffness of hip implants by changing the shape and material, using composite femur. However, the concept of FGM was usually carried out on dental implants. Therefore, the application of FGMs on the human artificial hip joints is deemed to be new. So, the aim of the current investigation is to use the concept of FGM in order to find an optimal stem material, made from FGM instead of that made from homogenous materials. In this paper the concept of FGM will be used to overcome the two conflicting problems of stress shielding at the proximal femur and increased maximum shear stresses at the edges of the stem/bone interface. FGM changes its composition and structure gradually over one or two directions of its volume. This study is divided into two parts; in the first part the stem is designed as one-dimensional FGM, while in the second part, the stem is designed as two-dimensional functionally graded material. In the case of part one (one-dimensional FGM), the gradation of elastic modulus is changed along the vertical direction (model 1) and along the horizontal direction (model 2), in order to find the optimal gradation direction. However, in the second part a two-dimensional FGM (model 3) is designed. The rules of mixtures and volume fraction relations which are applied to represent one and two-dimensional FGM of the stem are used through the finite element method and optimization technique, utilizing the ANSYS finite element program.
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2. Method of analysis 2.1. VOLUME FRACTIONS AND RULES OF MIXTURE OF ONE-DIMENSIONAL FGM A simplified model of cementless hip prosthesis, which was used before by many investigators (Kuiper and Huiskes, 1997; Gross and Abdel, 2001; Jeffers et al., 2003) will be applied in this analysis. Therefore the stem will be considered as a plate of FGM with porosity p that is functionally graded from ceramic and metal. The volume fraction of metal (Vm) and of ceramic (Vc), are distributed over y-direction in model 1 and over x-direction in model 2, as shown in Figure 1, according to the following relations (Nemat-Allah, 2003; Hedia and Nemat-Alla, 2004). For model 1: Vm ¼ ðy=lÞm
ð1Þ
For model 2: Vm ¼ ðx=tÞm
ð2Þ
For both models: Vc ¼ ð1 Vm Þ
ð3Þ
where: l is the total length of stem, t is the stem thickness and m is a non-homogenous parameter that controls the composition variations through the length in model 1 and through the thickness in model 2. The composition is rich in metal when m1. The porosity p of the FGM is represented for model 1 as: yn h yz i p¼A 1 ð4Þ l l While for model 2, xn h xz i p¼A 1 t t Where
Figure 1. (a) Gradation in y-direction. (model 1). (b) Gradation in x-direction (model 2).
ð5Þ
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ððn þ zÞ=nÞn A 0; 1 ðn=ðn þ zÞÞz A, n and z are arbitrary parameters that control the porosity. The effective values of the material properties for FGM, with porosity and continuously graded profile, are determined by employing the suspended spherical grain model. It was derived based on the assumption that the granular phase is in a matrix phase. The following relations give the rules of mixture for the elastic modulus (Nemat-Allah, 2003). E¼
EO ð1 pÞ 1 þ pð5 þ 8mÞð37 8mÞ=f8ð1 þ mÞð23 þ 8mÞg
ð7Þ
where " EO ¼ Ec
#
2=3
Ec þ ðEm Ec ÞVm
ð8Þ
2=3
Ec þ ðEm Ec ÞðVm Vm Þ
2.2. VOLUME FRACTIONS AND RULES OF MIXTURE OF TWO-DIMENSIONAL FGM 2-D FGM is made of continuous gradation of three distinct material phases (model 3). It is fabricated in such a way that the volume fractions of the constituents are varied continuously in a predetermined composition profile. Now let us consider the stem which will be studied in model 3 as a plate which shown in Figure 2, and discuss the volume fractions and porosity of the 2-D FGM at any arbitrary point B on the 2-D FGM that consists of two volume fractions: VS which is a mixture of V1 and V2, and V3. The two volume fractions, VS and V3 can be proposed for model 3 as, ymy V3 ¼ ð9Þ l
VS ¼ 1
ymy l
ð10Þ
y t V2
V1
l
B
y V3
x
x
Figure 2. Coordinate system of volume fraction distribution for 2-D FGM stem.
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The porosity variation in y-direction can be written as: yny h yzy i py ¼ Ay 1 l l where
ny þzy ny
1
ð11Þ
ny
zy ny ny þzy
Ay 0
ð12Þ
where ny, zy, and Ay are arbitrary parameters that control the porosity in y-direction. Also, the volume fractions V1 and V2 can be obtained by considering the upper surface of the 2-D FGM plate as one-dimensional FGM, then V1, V2 and porosity px can be expressed as: y m y i h xmx i h V1 ¼ 1 1 ð13Þ l t ymy ixmx h V2 ¼ 1 l t
px ¼ Ax
xnx
"
t
yzy y2zy þ 12 l l
ð14Þ #" 1
xnx t
# ð15Þ
where Ax represents the amount of porosity in x direction in the mixture and can be calculated as follows: nx þzx nx
V2 þ V1
nx
nx nx þzx
zx A x 0
ð16Þ
The subscripts 1, 2 and 3 denote material 1, material 2 and material 3 of the basic constituents. Also, my and mx are non-homogenous parameters that represent the composition distributions of the basic constituents materials in y- and x-directions. nx, and zx are arbitrary parameters that control the porosity in x direction. From the proposed volume fractions at any point, the volume fractions of the three basic constituents materials on each boundary surface are: V1 ¼ 1; V2 ¼ 0; V3 ¼ 0; at y ¼ l; x ¼ 0 V1 ¼ 0; V2 ¼ 1; V3 ¼ 0 at y ¼ l; x ¼ t V1 ¼ 0; V2 ¼ 0; V3 ¼ 1 at y ¼ 0; x ¼ 0 ! t From the above volume fraction distribution, it is clear that the proposed volume fractions of the composition of 2-D FGM changes from 100% of V3, material 3, on the lower surface of the stem, to 100% of V1, material 1, on the left upper corner of the upper surface of the stem and 100% of V2, material 2, on the right corner of the upper surface of the stem.
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The rules of mixture for the 2-D FGM with porosity can be obtained using the same way used to obtain the porosity of one dimensional FGM with some mathematical manipulation. For any point on the 2-D FGM plate with volume fractions V1, V2 and V3 as shown in Figure 2, using Eqs. (3) to (16), the rules of mixture for the different mechanical properties may be obtained as: (Nemat-Allah, 2003; Hedia and Nemat-Alla, 2004). For PoissonÔs ratio: m ¼ m1 V1 þ m2 V2 þ m3 V3
ð17Þ
For modulus of elasticity E¼
Eoy ð1 py Þ 1 þ py ð5 þ 8mÞð37 8mÞ=f8ð1 þ mÞð23 þ 8mÞg
ð18Þ
where " Eoy ¼ Ex
Ex ¼
2=3
Ex þ ðE3 Ex ÞV3
#
2=3
Ex þ ðE3 Ex ÞðV3 1 þ Vx Þ
Eox ð1 px Þ 1 þ px ð5 þ 8mx Þð37 8mx Þ=f8ð1 þ mx Þð23 þ 8mx Þg "
Eox ¼
2=3
E1 þ ðE2 E1 ÞV2 2=3
E1 þ ðE2 E1 ÞðV2 V2 Þ
mx ¼ m1 V1 þ m2 V2
ð19Þ
ð20Þ
# ð21Þ
ð22Þ
2.3. FATIGUE NOTCH FACTOR The fatigue life of mechanically loaded parts consists of the crack initiation time and crack propagation time. The crack initiation time is determined by the fatigue notch factor Kf, which describes the ratio of the fatigue strength (alternating stress range to cause failure in a given number of cycles) of a specimen with no stress concentration to the fatigue strength of the corresponding specimen with a stress concentration. There are many hypotheses which give formulas for Kf calculation. In this analysis, the statistical hypothesis (weakest-link-model) is considered, which leads to the following formula (Hedia et al., 1996). (R )1=k A2 ðg1 ðx; y; zÞÞk dA2 kf ¼ kt ð23Þ A1 where g1 ðx; y; zÞ ¼
rðx; y; zÞ rmax
ð24Þ
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A1 = surface area of the smooth specimen, A2 = surface area of the notched specimen and k = statistical parameter (not a material constant) which can vary from 1 to 24. A low value of k implies that the volume of material under high stress is a more important parameter while a high value implies that it is the magnitude of the stress which is more important. In the present work a minimum value of k is taken, (k=1), thus the optimization based on a low value of k which reduces the stress along the whole of the interface will be more effective in reducing the probability of fatigue failure than one based on a high value of k which will only decrease the peak stresses. The statistical hypothesis for axisymmetrical FE model leads to the following equation (Hedia, 2000; Hedia et al., 2000). Z 1=k ð2pÞ1=k k Kf ¼ ð r ð x; y Þ Þ rdr ð25Þ 1=k rn A1 where rn is the nominal stress calculated from applied loading, r is the radius of each node along the boundary of interest. A linear distribution of von Mises stress between adjacent finite element nodes on each surface is assumed (Hedia, 2000; Hedia et al., 2000). The above integral along a line of nodes is carried out analytically using the following equation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 X ðriþ1 þ ri Þðriþ1 þ ri Þ I¼ ðXiþ1 Xi Þ2 þðYiþ1 Yi Þ2 ð26Þ 2 i¼1 where r is the von Mises equivalent stress and X,Y are the coordinates of each node. In this case r=X. Thus Kf can be calculated as follows: Kf ¼
2p1=k 1=K
rn A1
I¼CI
ð27Þ
where C is a constant for a given loading situation. 2.4. FINITE ELEMENT IDEALIZATION AND OPTIMIZATION TECHNIQUE An axisymmetric finite element model of a simplified cementless hip prosthesis is constructed and analyzed using ANSYS finite element program. The dimensions of simplified cementless hip joint are taken as used before by (Kuiper and Huiskes, 1997). The prosthetic stem length, l, is taken equal 91.3 mm, the outer radius of bone is15 mm, and the inner radius of bone (radius of stem) is 10 mm, as shown in Figure 3. The distal end of the model was rigidly fixed and a 3000 N point load was applied at 11 to the vertical at the proximal end of the stem as used before by Gross and Abdel, (2001). The interface between the stem and bone is assumed to be perfectly bonded. The YoungÕs modulus of the bone is assumed equal 20 GPa and PoissonÔs ratio is assumed equal 0.3. The aim of the investigation is to find the optimal material of the stem, in order to overcome the previously mentioned stress shielding and stress concentration problems. The concept of volume fraction and rule of mixtures of one and two-dimensional FGM are carried out using a numerical optimization procedure in combination with the axisymmetrical finite element model. A computer program was developed using ANSYS parametric design language (APDL) to calculate the modulus of elasticity and PoissonÔs ratio, using Eqs. from (1) to (23) at different locations of the stem. Also a computer program is developed to calculate the fatigue notch factor using Eqs. (25), (26) and (27).
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bone
stem bone
91.2 mm
Material optimization of cementless stem
20 mm 30 mm
Figure 3. Shape and dimensions of a simplified cementless finite element model.
The concept of FGM was carried out through three models: Model 1: By changing the material of the stem through the vertical direction from y=0 to the full length of the stem at y=l; part 1 of the study Model 2: By changing the material of the stem through horizontal direction from x=0 to the radius of the stem at x=t; part 1 of the study Model 3: By changing three distinct materials continuously through the stem as a 2-D FGM which has been discussed previously in Sections 2.2 and 2.3; part 2 of this study The first two models are studied comparatively under the optimization procedure in order to find the best gradation direction. These two models are also compared with a homogenous flexible ‘‘iso-elastic’’ stem with a YoungÔs modulus equals 20 GPa (equivalent to cortical bone which is taken as a reference for stress shielding) and a stiff titanium stem with YoungÔs modulus equals 110 GPa. The design objective is to minimize the maximum shear stress in bone at the bone/stem interface to reduce the stress concentrations at the edges. The state variable is to maximize the stresses at the proximal bone by maximizing the fatigue notch factor, the state variable, Kf. The design variables for this problem are the elastic modulii of the basic constituents of the FGM stem (Em and Ec) and the composition variations Vm and Vc, which are described by the constant m. Therefore, the constraints for this optimization problem are: a. To maximize the stresses at the proximal bone in order to reduce stress shielding at this region. In other words, to maintain the state variable in bone at the proximal region at or above its initial value Kfi£ Kf£ any arbitrary value less than or equal to the value when using stem material as a bone. The Kfi = fatigue notch factor in bone at the proximal interface using titanium stem, b. To maintain the values of the design variables m within limits used in literature, i.e. 0£ m£ 10, and c. To maintain the values of the design variables within a wide range of biomaterials and bioceramics which are used in the literature and have the ranges, 1 GPa £ Em and Ec £ 210 GPa Finally, the optimal gradation model is compared with the third model of 2-D FGM using three distinct materials with three different elastic moduli, E1, E2, E3, and three volume fractions, V1, V2, V3, varied continuously as described before in section 2.3. This 2-D FGM stem
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consists of three main constituents 100% E1 at the upper left stem corner, 100% E2 at the upper right stem corner, and 100% E3 at the lower surface of the stem. Finally, a comparative study is carried out between the optimal one-dimensional FGM and the 2-D FGM model. The design optimization procedure can not be used in the third model (2-D FGM) due to the requirement of high capacity memory, and only a comparative study is used.
3. Results 3.1. PART 1: ONE-DIMENSIONAL FUNCTIONALLY GRADED MATERIAL The finite element analysis and optimization procedure are carried out to find the optimal material gradation in the stem for both FGM models. The stresses in bone for FGM models are studied and compared with the stresses using homogenous titanium stem and flexible iso-elastic stem with a YoungÔs modulus equivalent to cortical bone which are taken as a reference for stress shielding. Starting from titanium stem as an initial material design and using the previously discussed objective function, design variables, and state variables, it is found that for the first model, the optimal elastic modulii Em and Ec are 110 GPa which represented by hydroxyapatite at the top of the stem and 1 GPa which represented by collagen at the bottom, respectively. For the second model it is found that Em and Ec are 30 GPa which represented by bioglass at the outmost layer of stem and 110 GPa (hydroxyapatite) at the stem core, respectively. For both models, the optimal composition variations parameter is found to be 0.1, which means that the stem is rich in hydroxyapatite (110 GPa) in the first model, while it is rich in bioglass (30 GPa) in the second model. In order to check the reliability of the introduced optimal design of the FGM implant a comparisons of von Mises and shear stresses for the titanium (initial design) and optimal design (FGM) are carried out as follows: 1. In studying the von Misses stress in the proximal lateral bone side, the bone length is divided into 3 regions (proximal, middle, and distal) as by (Weinans et al., 2000). As shown in Figure 4, using FGM model 1, the stress shielding in bone at the proximal region is decreased by 83%. However, when using FGM model 2 the stress shielding is found to be reduced by 67%. 2. As shown in Figure 5, using the flexible iso-elastic homogenous stem, as a bone, it is found that the maximum interface shear stress is concentrated at the proximal zone, while using the homogenous titanium stem concentrates the maximum interface shear stress distally. Using FGM model 1, it is found that the maximum interface shear stress is concentrated proximally but it is reduced by 32% compared to titanium stem. For FGM model 2, the maximum interface shear is concentrated proximally but it is reduced by 22% compared to titanium stem. In both FGM models, the maximum interface shear stress is reduced, however the stresses are not distributed uniformly. The poorly distributed load transfer from stem to bone is a point of concern for many investigators to avoid the risk of interface failure (Kuiper and Huiskes, 1997, 1992). Therefore, the 2-D FGM is introduced in the second part in order to obtain uniform interface shear stress distribution. However, it may be recommended from Figures 4 and 5 to take model 1 as the optimal model for this part of the study. The gradation of the elastic modulus for model 1 as a function of the composition variation parameter, m, is shown in Figure 6. From this figure, it is shown that the FGM is rich in metal when m is less than unity, while it is rich in ceramic when m is bigger than unity. This
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4.5
von Mises stress (MPa)
4 3.5 3 2.5 2 1.5 1 0.5
Bone Titanium FGM model (1) FGM model (2)
0 38 41 44 47 50 53 56 59 62 65 68 71 74 distance at the proximal bone (mm)
Figure 4. von Misses stresses in the proximal lateral bone.
relation between the elastic modulus gradation and the composition variation parameter is not changed using model. 3.2. PART 2: TWO-DIMENSIONAL FUNCTIONALLY GRADED MATERIAL In this part of study the optimization technique can not be applied to the 2-D FGM, because this needs a very large capacity memory. Therefore, depending on the optimal materials obtained from the first part of this study, it can be suggested the materials which can be used to design the stem with 2-D FGM. From one-dimensional FGM, it is concluded to use stiff stem at
16
interface shear stress (MPa)
14
Bone Titanium FGM model (1) FGM model (2)
12 10 8 6 4 2 0 0
4
8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76
distance from distal to proximal interface (mm) Figure 5. The interface shear stress distribution at stem/bone interface.
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H.S. Hedia et al.
Elastic modulus (MPa)
101000
m=0.1 Opt. m=0.2 m=0.5 m=1 m=2 m=5 m=10
81000
61000
41000
21000
1000
0
8
16
24
32
40
48
56
64
72
80
88
stem length from distalto proximal (mm) Figure 6. The gradation of the elastic modulus for model 1 as a function of the composition variation parameter, m.
the upper surface and flexible at the lower surface (model 1). However, it is recommended to use stiff stem at the stem core while using flexible stem at the outmost layer of the stem (model 2). Thus, it is recommended to use three distinct materials E1=110 GPa at the left upper corner of the stem with V1 equals 100% at this corner, E2=30 GPa at the right upper corner of stem with V2 equals 100% at this corner, and E3=1 GPa at the lower surface of the stem with V3 equals 100% at all of the lower stem surface, as illustrated in Figure 2. Then, the next step is to choose the optimal compositions variation parameter (mx) between V1 and V2, as well as the composition variation parameter (my) between VS (which is a mixture of V1 and V2) and V3. Therefore, mx and my are changed according to (0£ mx, my£ 10). The value of Kf at the proximal lateral bone surface and the value of the maximum interface shear stress at the proximal and distal parts are calculated at different values of mx, my. These trials are shown in details in Table 1. The maximum value of Kf must not exceed the value of Kf obtained using the flexible stem with the same YoungÔs modulus as cortical bone (Kf using stem with E=20 GPa equals 3960*C). Thus, from the Table 1, one can observe that the optimal values for Kf occurred at reasonable values for maximum interface shear stress are obtained using mx=0.1 and my=0.5. However, at these values of mx and my, it is found that the value of von Misses stresses at the proximal bone exceeds that obtained using the flexible stem. Therefore it is recommended to choose a value of mx=0.1 and my=1.0. Therefore the optimal 2-D FGM is recommended to be with E1=110 GPa, E2=30 GPa, and E3=1 GPa and with composition variations mx=0.1 and my=1.0. The distribution of 2-D FGM for elastic modulus gradation is illustrated in Figure 7. compared with the optimal one-dimensional FGM model (model 1) obtained from part 1 in this study. The results are shown below: 1. As shown in Figure 8, it is found that the von Misses stress along the proximal lateral bone side are approximately identical for both FGM models. The reduction in stress shielding is identical for both the optimal one-dimensional FGM (model 1) and 2-D FGM (model 3).
5490 4725 3659 3044 2721 2619 2615
4.2/6.8 4.3/5.8 4.5/4.5 4.7/3.9 5/3.7 5.4/3.7 5.7/3.7
(MPa)dis/pro*
5490 4710 3630 2994 2646 2520 2490
Kf/C
Kf/C
smax
0.2
0.1
mx
*dis/pro means distal/proximal
0.1 0.2 0.5 1.0 2.0 5.0 10
my
4.2/6.8 4.3/5.8 4.5/4.4 4.7/3.8 5/3.5 5.4/3.5 5.7/3.5
(MPa)dis/pro
smax
5460 4680 3563 2876 2470 2296 2277
Kf/C
0.5
4.2/6.7 4.3/5.7 4.5/4.2 4.7/3.5 5/3.2 5.4/3.1 5.7/3.1
(MPa)dis/pro
smax
5460 4650 3486 2745 2283 2061 2033
Kf/C
1.0
4.2/6.7 4.3/5.6 4.5/4 4.7/3.2 5/2.8 5.4/2.7 5.7/2.7
(MPa) dis/pro
smax
5448 4620 3399 2605 2087 1823 1788
Kf/C
2.0
4.2/6.6 4.3/5.4 4.5/3.8 4.7/2.9 5/2.5 5.5/2.4 5.8/2.4
(MPa) dis/pro
smax
5430 4578 3300 2448 1881 1586 1545
Kf/C
5.0
4.2/6.6 4.3/5.3 4.48/3.6 4.7/2.6 5/2.2 5.5/2 5.8/2
(MPa)dis/pro
smax
5423 4560 3251 2373 1785 1480 1437
Kf/C
10
4.2/6.5 4.3/5.3 4.5/3.5 4.7/2.5 5/2 5.5/1.9 5.8/1.9
smax (MPa) dis/pro
Table 1. The effect of composition variation parameters mx, my on the Kf value at the proximal lateral bone surface and the value of the maximum interface shear stress at the proximal and distal parts.
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120000
80000
60000
40000
20000
1
41 0
Figure 7. The distribution of 2-D FGM for elastic modulus gradation.
4 3.5 3 2.5 2 1.5 1
FGM model (1) Titanium 2D FGM
0.5 0 38
41
44
47
50
53
56
59
62
65
68
71
distance at the proximal bone (mm) Figure 8. von Misses stresses in the proximal lateral bone.
74
1
11
31
upper
21
41
51
71
61
91
stem
81
111
101
131
81 121
141
161
151
171
191
181
201
lower
von Mises stress (MPa)
221
61
outer c or e
21
211
Elastic modulus (MPa)
100000
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2. As shown in Figure 9, it is found that the shear stresses are uniformly distributed compared to both optimal one-dimensional FGM design and the homogenous titanium stem, the maximum interface shear stress is reduced by 45% and 63% compared to the optimal onedimensional FGM design and the homogenous titanium stem, respectively. To clarify the maximum interface shear stress values obtained using flexible stem, titanium, model 1, model 2, and model 3, Figure 10 illustrates these values clearly.
4. Discussion Functionally graded material is an advanced composite material which is an important area in composites research. The main feature of a functionally graded composite is the almost continuously graded composition of the composite that results in two different materials at the two ends of the composite. This investigation recommended that the stem should be stiff at the upper surface, and at the lower surface it is recommended to be flexible, as it was previously concluded by (Kuiper and Huiskes, 1992) in their mathematical optimization of elastic properties applied on the cementless hip stem design. However, it is also recommended to use a stiff stem at the core while this stiffness is reduced gradually towards the outer surface of the stem, as it was studied previously by (Shairandami and Esat, 1990) when they used a stiff material at the stem core with elastic modulus equals 127 GPa and a stem shell with elastic modulus equals 68 GPa in their new design of hip prosthesis. However, the concept of FGM which is used in this investigation can be fabricated using many techniques such as spark plasma sintering (Miyao et al., 2001), or the approaches used by (Pompe et al., 2001) to fabricate the collagenhydroxyapatite materials. The cortical bone consists mainly of organic collagen fibers 16% and mineral hydroxyapatite 60% (Thompson and Hench, 2000). Thus using hydroxyapatite/collagen FGM seems to be a promising material. Hydroxyapatite/collagen has been previously used by many authors (Itoh et al., 2002, 2000). However, this composite material is formed in a simulated body fluid 14 FGM model (1)
interface shear stress (MPa)
12
Titanium 2D FGM
10 8 6 4 2 0 0
4
8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76
distance from distal to proximal interface (mm) Figure 9. The interface shear stress distribution at stem/bone interface.
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Figure 10. The maximum interface shear stress for homogenous and three FGM models.
environment by (Zhang et al., 2004), and they concluded that this composite material may be applicable for use as a bone substitute. In the first part of this analysis, it is concluded that the optimal composition variation parameter, m, equals 0.1. This means that the FGM in model 1 is rich in hydroxyapatite compared to collagen, which makes the stem more stiff proximally at the region of applied load. Some ceramics such as bioglass, sintered hydroxyapatite, and glassceramic, spontaneously bond to living bone (Kokubo et al., 2003). They are called bioactive materials which elicits a specific biological response at the interface of the material which results in the formation of a bond between the tissues and the material (Cao and Hench, 1996). Bioglass 45S5 is used in the FGM in this study which has an elastic modulus range 30–50 GPa (Thompson and Hench, 2000; Cao and Hench, 1996). A study using an extruded bioglass polysulfone composite in a sheep tibia indicates that the exposed glass has become active allowing bone bonding and some ingrowth into rough surface of the implant (Cao and Hench, 1996). The commercial bioactive glass bioglass 45S5 is chosen in this study, because it has the greatest bioactivity index (Ib). The bioactivity index is defined as the time taken for 50% of the interface between bone and an implant to chemically bond together (Ib for bioglass is 12.5, while for hydroxyapatite it is 3.2) (Gatti et al., 1998). The FGM in model 2 of this study is rich in bioglass compared to hydroxyapatite, which is reported as class A bioactive material as opposed to hydroxyapatite, class B, and can simulate osteoblast function faster than hydroxyapatite (Jones and Hench, 2001). In case of 2-D FGM the optimal composition variation parameter in x-direction between V1 and V2, mx, equals 0.1 which makes the FGM rich in hydroxyapatite at the upper surface of stem. However, the composition variation parameter between VS (which is a mixture of V1 and V2) and V3 equals 1.0 which means that the composition variation in y-direction is linear.
5. Conclusion Cementless hip prosthesis stem material can be optimized using one-dimensional FGM. The optimal one-dimensional FGM is expected to reduce the stress shielding at the proximal lateral bone by 83%, while the maximum interface shear stress at the stem/bone interface may be
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reduced by 32% compared to homogenous titanium stem. The optimal material gradation is changed from hydroxyapatite (E=110 GPa) at the upper surface of the stem, to collagen (E=1 GPa) at the lower surface. Using the optimal two-dimensional FGM stem is expected to reduce the maximum interface shear stress by 45% and 63% compared to the optimal one-dimensional FGM and the homogenous titanium stem, respectively. The shear stresses at the bone/stem interface become more uniformly distributed, which reduces the risk of failure and stem loosening. However, the reduction of the stress shielding at the proximal lateral bone side is identical to that obtained using one-dimensional FGM. The optimal two-dimensional FGM stem consists of hydroxyapatite with 100% volume fraction at the left upper stem corner, bioglass with 100% volume fraction at the right upper stem corner, and collagen with 100% volume fraction at the lower stem surface.
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