- Email: [email protected]
PEEK biocomposite
ARTICLE IN PRESS
Biomaterials 25 (2004) 5363–5373
Influence of interphase layer on the overall elasto-plastic behaviors of HA/PEEK biocomposite J.P. Fana, C.P. Tsuia,*, C.Y. Tanga, C.L. Chowb a
Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Room DE404, ISE Research Office, Hung Hom, Kowloon, Hong Kong, China b Department of Mechanical Engineering, The University of Michigan-Dearborn, Dearborn, MI, USA Received 19 September 2003; accepted 19 December 2003
Abstract A three-dimensional finite element unit cell model has been designed and constructed for studying mechanical properties of hydroxyapatite (HA) reinforced polyetheretherketone (PEEK) biocomposite. The model consists of an elastic-brittle HA spherical particle, an elasto-plastic matrix and an interphase layer between the particle and the matrix. The interphase layers with four different kinds of material behaviors have been taken into consideration to examine their effects on the overall properties of the composite. The damage evolution in the matrix and the interphase layer, and the interface failure, were also taken into account. Some other factors, such as mesh sensitivity, loading velocity and mass scale scheme, were also discussed in this investigation. A general-purpose finite element software package, ABAQUS, incorporated with a user-defined material subroutine, was used to perform the analysis. The predicted results were compared with the experimental data obtained from existing literatures. The results predicted by using the cell model with consideration of the matrix degradation and the effects of the damage and failure on the interphase layer are in good agreement with the experimental ones. Hence, the suitability of our proposed cell model incorporated with an appropriate type of the interphase layer for modeling the mechanical properties of the particulate biocomposite could be verified. r 2004 Elsevier Ltd. All rights reserved. Keywords: Polyetheretherketone; Finite element analysis; Degradation; Interface
1. Introduction Traditionally, metallic materials, such as stainless steel and cobalt–chromium alloys, have been used as bone implant in orthopedic areas for many years. However, these materials can experience stress-shielding problem due to their high elastic modulus [1]. In order to minimize the stress-shielding effect and the formation of callus during healing of fractures, a polymer matrixbased material should be used. Bioactive second-phase reinforcing particles can be incorporated into the polymer matrix to enhance the mechanical properties of the composite for satisfying the load-bearing requirement of a bone implant. The concept of using bioactive hydroxyapatite (HA) particles reinforced polymer composite as an implant material for bone replacement was introduced by Bonfield [2] in the early 1980s, because the *Corresponding author. Tel.: +852-2766-6635. E-mail address: [email protected] (C.P. Tsui). 0142-9612/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.biomaterials.2003.12.050
HA particle is bioactive with a calcium-to-phosphorus ratio similar to that in the natural bone. Since then, particulate reinforced polymer-based biocomposites have been widely studied for bone tissue replacement [3–5]. Composites with HA and a polymer such as polyethylene, polyethyl ester or polyphosphasone and polyetheretherketone (PEEK) [6–9] have been used as potential materials for bone tissue replacement. Special attention has been focused on the particulate biocomposites with the bio-inert PEEK matrix, because PEEK has superior mechanical properties due to its highly aromatic structure [9]. Besides, it possesses additional benefits including good chemical and fatigue resistance, hightemperature durability, high wear properties, high ease of processibility, its capacity to be repeatedly sterilized and shaped readily by machining and heat contouring to fit the shape of bone [10]. All these benefits have rendered particulate reinforced PEEK to become attractive materials for biomedical applications such as major load-bearing orthopaedic implant materials [11].
ARTICLE IN PRESS 5364
J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
Under excessive loading, particle–matrix debonding and microdamage of the matrix can occur in particulate biocomposites [12,13]. In addition, the microdamage of the matrix may be induced during fabrication of the composite. The damage can be associated with microcracks initiating and growing from the imperfections within a matrix and along the particle–matrix interface [14]. Clustering of particles in the composite may also be originated from the fabrication processes [15]. Microcracks may initiate in the region of the particle clustering due to a high stress concentration. However, literatures on the modeling of the effects of microdamage on the mechanical properties of particulate biocomposites are very limited. Numerical techniques, such as finite element methods (FEMs), have been used for evaluating the micromechanical behaviors of particulate-reinforced composites on the basis of a unit cell model. Guild adopted the cylindrical cell model and Lee used the plane strain model as representative volume element [16–18]. Although using two-dimensional numerical models can reduce computation time as compared with the threedimensional ones, there exist gaps between the exact cell and its approximation. Recently, three-dimensional models for predicting the mechanical properties of hydroxyapatite-polyethylene and hydroxyapatite-poly l-lactide composites have been developed [19,20]. These models mainly focused on prediction of the elastic properties of the biocomposites. Good prediction for the elastic properties of the biocomposites could be achieved. However, the prediction for the failure processes of the biocomposites using a linear elastic model was found to have a low accuracy. Hence, the more complicated process of a nonlinear analysis including plastic deformation and damage should be employed. In addition, particulate filled composites possess heterogeneous structures, in which each phase is separated by an interfacial phase. This interphase forms spontaneously on the surface of the filler during the process of material fabrication due to the adsorption of the polymer in a thermoplastic composite [21] and because of surface treatment on the filler or modified curing chemistry in a thermoset matrix. The mechanical characteristics of the interphase are entirely different from those of the matrix and the reinforcing filler. The presence of the interlayer can modify stress distribution around the inclusions, micromechanical deformation processes and the final properties of the composites [22,23]. However, limited investigation has been performed about the influence of the interphase properties on the mechanical properties of the composites, especially the particulate reinforced biocomposites. In this study, a three-dimensional finite element unit cell model was established for studying the influence of four different material behaviors of the interphase layer
on the micromechanical properties of HA reinforced PEEK biocomposites. In addition to the different kinds of the interphase layer, a ductile damage evolution in the matrix was taken into account to describe the degradation behavior of the matrix. The model was applied to predict Young’s modulus, tensile strength, stress–strain relation and stress distribution of the HA/PEEK biocomposites with different material properties of the interphase layers. The simulation results were compared with experimental ones.
2. Properties of the constituent materials PEEK with a basic formula of (–C6H4–O–C6H4–O– C6H4–CO–)n is a polyaromatic semicrystalline thermoplastic polymer. It has a melting temperature of 343 C, a crystallization peak at 343 C and a glass transition temperature of 145 C. The high-temperature performance makes it become a stable material in human body [24]. In addition, its superior combination of strength, stiffness and toughness, as well as outstanding chemical, hydrolysis and wear resistance, together with its extensive biocompatibility, have enabled it to be suitable for in vitro load-bearing medical device applications [25,26]. Moreover, PEEK is non-cytotoxic and can be repeatedly sterilized using conventional steam, gamma and ethylene oxide processes without evident degradation on its mechanical properties [26,27]. Thus, this renders it to become long-term medical implants in orthopaedic, cardiovascular and dental markets. Experimental measurement demonstrated that the matrix of PEEK experienced elasto-plastic deformation [9,11]. Therefore, in this investigation, the von Mises yield surface for the PEEK matrix was assumed to have an associated flow rule and exhibit isotropic hardening. HA, with a chemical formula of (Ca10(PO4)6(OH)2), has a chemical and crystallographic structure being similar to that of bone material [28]. Moreover, HA is biocompatible with hard tissues of human beings and possesses osteoconductive properties [29]. Thus, HA has been widely used as a bone-replacement material in restorative dental and orthopaedic implants [28]. The high-modulus HA particle in biocomposites usually undergoes elastic deformation and seldom suffers crack failure during a load-bearing process [19]. Therefore, the deformation behavior of the HA reinforcement can be characterized by a purely elastic constitutive law. The generalized Hooke’s law [30] is satisfactory to reflect the deformation behavior of the HA particle in the biocomposite. The mechanical properties of the interphase layer are hard to measure [31], so that it was firstly assumed to have the same material properties as the matrix. In addition, four different material behaviors of the interphase layers, including a perfectly adhesive interface, a weaker stiffness, a degradable elasto-plastic,
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
and a combination of both damage and failure behaviors, would be taken into account and explained in detail in Section 4.
3. Explicit integral technique Unlike the implicit integral technique, explicit dynamic formulation has a series of advantages, such as low memory requirement, no iteration-convergent difficulty, easy treatment of contact condition and the presence of vanishing element treatment. All these advantages render it to become a rather powerful tool in actual application. The general form of the principle of virtual work is described as balance of internal work, inertia work, work done by body force and work exerted by surface force [32]. The discretization of the body into finite elements and the assembly of all element contributions into global matrices and vectors lead to the following matrix–vector equation . þ K U ¼ F; M U
ð1Þ
where M is the mass matrix, K is the stiffness matrix, F is the vector of the prescribed forces, and U ¼ UðtÞ is the nodal displacement vector in global coordinate. To solve this second-order differential equation, it requires discretization of the time interval into finite steps of Dt and calculation of the approximate solution on the time tn ¼ n Dt (for n ¼ 0; 1, 2, y). The central difference method was applied so that the first-order and second’ and U . of the global displacement order derivatives U vector were then approximately replaced by the displacement vectors Uðtn1 Þ; Uðtn Þ and Uðtnþ1 Þ: The truncation errors in these approximations are of the order OððDtÞ2 Þ: When the time step tends to be zero such that Dt-0; the central difference method approaches an accurate solution. Convergence of the solution is now assured when the algorithm satisfies the stability condition, i.e., when small numerical errors are not amplified by taking one time step. The central difference method is conditionally stable, meaning that the time step must be sufficiently small. The following restriction is required [33]: DtpDtmax ¼
2 ; oh
ð2Þ
where oh is the maximum natural frequency of system vibration represented by Eq. (1). This frequency depends on element size h and dilatational wave speed cd ; and is proportional to cd =h: The proportional constant is related to the element type. Thus, it is clear that the time step Dt is restricted by the smallest element in the entire element division.
5365
4. The damage evolution 4.1. Basic consideration of damage The term damage is used to indicate the deterioration of the material load-carrying capability. From a general viewpoint, damage develops in the material microstructure when non-reversible phenomena take place, such as microvoid or microcrack formation in the polymer matrix. A representative volume element in a damaged body is selected, as shown in Fig. 1. Let A0 be the nominal intersection area, and Aeff be the effective resisting section area in which the area occupied by microdefects has been removed. Thus, a damage variable is defined as [34] Aeff D¼1 : ð3Þ A0 This definition means that an isotropic damage is developed, such that microcracks and microvoids with orientation distribute uniformly in all directions. The damage value does not depend on the direction, so that it can be completely characterized by the scalar value of D: 4.2. Matrix damage evolution In this study, the matrix was considered to be a ductile material, which undergoes elasto-plastic deformation. Thus, a ductile damage is likely to take place in the matrix and a damage potential function j is taken as follows [34]: S Y 2 j¼ pHðp pD Þ: ð4Þ ’ 2 S where S is a material-dependent constant, p’ is an equivalent plastic strain rate, Y is the damage strain energy and HðUÞ is a Heaviside step function.
A0
A eff
Fig. 1. Representative volume element for a damaged body.
ARTICLE IN PRESS 5366
J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
In order to implement a finite element analysis, an incremental form of the damage evolution is derived by applying the backward Euler method: seq Rv DD ¼ ð5Þ DpHðp pD Þ; 2ESð1 DÞ2 where Rv is a stress triaxiality function, which depends on the rate of hydrostatic stress to von Mises equivalent stress. Obviously, the damage evolution depends upon the accumulated plastic strain, the stress triaxiality and the von Mises equivalent stress. 4.3. Damage evolution equation for the interphase layer There are three types of criteria proposed for the particle–matrix debonding: stress-based, strain-based and energy-based criteria. The stress-based criterion is generally more appropriate to be incorporated into the interphase layer for modelling the particle–matrix debonding of the ceramic particle reinforced polymer composite [35]. At a certain level of loading, a damage will be developed in the interphase element. The damage variable D used in this investigation is defined as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 2ffi sn 2 st1 st2 D¼ þ c þ c ; ð6Þ scn st st
geometry of the constituent phases, and interfacial properties among the phases [37]. The thermal spraying experiment [9] illustrated that the HA particle appears to be spherical in shape with a glassy smooth surface, as shown in Fig. 2. For convenience of studies, these spherical particles are assumed to be evenly dispersed in the polymer composite in this investigation, so that it could be represented by a periodically repeated threedimensional cubic array, as shown in Fig. 3(a). Because of the assumed symmetry for the packing of the spherical particles, only one-eighth of the sphere embedded in a cube as shown in Fig. 3(b) is needed for numerical analysis. Particle volume fraction (PVF), Vp is given by pr1 3 ; ð7Þ Vp ¼ 6 r while the interphase volume fraction, Vi is expressed as p r2 3 r1 3 Vi ¼ ; ð8Þ 6 r r whereas the thickness of the interphase layer h is given by h ¼ r2 r1 ; in which r1 is the particle radius and r2 is the outer radius of the interphase layer depicted in Fig. 3(b). Here r is the cell length, which is set to be
where sn is the normal stress, st1 and st2 are two tangential stresses within the interphase layer. scn and sct are the normal and tangential critical stress values, respectively. The damage variable D varies in the range of [0, 1]. When D value in the interphase element is greater than zero, this means that a damage has been developed in this element. When D value in any interphase element reaches one, this element would fail and be removed using a vanishing element technique [36]. In order to perform the analysis, the commercial software package ABAQUS/Explicit was used. However, the composite with our required material behaviors is not available in the material library of our finite element software. In order to take into account the elastic deformation of the particle, the elasto-plasticdamage behavior of polymer matrix, damage-coupled material behaviors and failure criterion of the interphase layer, a user-defined material subroutine named ‘‘VUMAT’’ has been established and incorporated into the ABAQUS finite element code.
Fig. 2. Morphology of flame spheroidized HA particles.
H
G
z F
E
y
r2
5. Finite element modeling
The mechanical behavior of the composite is dependent on physical properties, volume fraction, packing
C
r1 A
5.1. Formation of a unit cell model
D
(a)
B
x
(b)
Fig. 3. Geometry of particle-reinforced composite: (a) packing of spherical particles in the composite; (b) three-dimensional unit cell model.
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
equal to one. The thickness of the interphase layer is set to be 1% of the cell length [38]. Perfect bonding was initially assumed between the particle and the interphase layer, and between the interphase layer and the polymer matrix. These cell models were then converted into finite element cell models constructed by three-dimensional brick eight-node linear elements. The finite element mesh of the cell model containing 20% PVF is shown in Fig. 4, which clearly illustrates that the model is composed of the particle, the interphase layer and the polymer matrix. In the present finite element analysis, the particle shape and the thickness of the interphase layer were kept fixed, while the PVF was varied up to 40%. The mechanical properties of the constituent materials are summarized in Table 1. 5.2. Boundary conditions External force and influence of adjacent material deformation were taken into account by using appropriate boundary conditions. Due to the symmetry of the unit cell, the normal displacements on the surfaces of ABCD, ABFE and ADHE shown in Fig. 3(b) are constrained such that ux ¼ 0 on x ¼ 0;
5367
surface. On the other hand, to ensure compatibility among all periodic representative cells, the faces BCGF, DCGH and EFGH of the unit cell are required to remain plane and parallel to their initial state after deformation. Thus, the normal force acting on the both faces of BCGF and DCGH must be set to zero such that Z ð10Þ sn dA ¼ 0 on x ¼ r and y ¼ r and all shear stresses on these boundaries should be set to zero. To satisfy the parallel plane requirement during deformation, an arbitrary point P; which does not belong to any parts of the model and is allowed to freely move in any directions, was created and related to all nodes on the line x ¼ r and y ¼ r: Thus, displacement of all element nodes on the face x ¼ r and y ¼ r in the x-direction and y-direction, respectively, are the same and equal to that of the point P: With this method, no external force is added onto these boundaries. Hence, the boundary conditions set in Eq. (10) could be satisfied. The velocity boundary condition was applied on the upper surface at z ¼ r:
6. Results and discussions
uy ¼ 0 on y ¼ 0; uz ¼ 0 on z ¼ 0:
ð9Þ
These boundary conditions can be easily enforced by restraining the nodal displacement on the corresponding
6.1. Young’s modulus For the estimation of Young’s modulus of the biocomposite, the interphase layer was taken to be the same material as the matrix. Each constituent underwent elastic deformation, and no damage and failure happened. In order to verify the accuracy of the numerical prediction, the Halpin–Tsai model, which is a well-known model for composite materials [39], was applied to predict the variation of Young’s modulus of the HA/PEEK biocomposite with PVF. The semiempirical equation of the Halpin-Tsai model for predicting Young’s modulus E of the composite is given by E ¼ Em
Fig. 4. Three-dimensional finite element mesh of unit cell model.
Table 1 Material properties Materials
E (GPa)
n
r (kg/m3)
PEEK HA
3.2 85
0.42 0.3
1291 3160
ð1 þ zZVp Þ Ep =Em 1 where Z ¼ ; ð1 ZVp Þ Ep =Em þ z
ð11Þ
where Vp is PVF, z is a measure of particle geometry, packing the geometry and loading conditions of the particle. The predicted results generated from our finite element model and the analytical Halpin–Tsai model are compared with the experimental results obtained from some literatures [9,11] shown in Fig. 5. The predicted Young’s modulus increases correspondingly with increasing HA content. The results acquired from the numerical prediction are closer to the experimental data than those from the analytical Halpin–Tsai formula. On the other hand, the composite with 40%
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
5368
14
9
Experiment 10 6 Stress (MPa)
Young's modulus (GPa)
12
8 6
3
4 Halpin-Tsai 2
Finite element Prediction
0 0
10
20
30
0
40
0
HA Volume (%) Fig. 5. Young’s modulus of the composite.
1.5
2
Fig. 6. Stress–strain relation of the interphase with degradable elastoplastic behavior.
100 40% PVF 80
Stress (MPa)
The stress–strain relation of the HA reinforced PEEK composites have been predicted by using four different kinds of interphase layers. (a) perfectly adhesive interface: this interphase layer was taken to be the same material as the matrix, and no damage and failure were assumed to take place; (b) interphase with a weaker stiffness: the interphase layer with an elastic modulus only one-thousandth of that of the matrix was assumed to undergo elastic deformation only; (c) interphase with degradable elasto-plastic behavior: the stress–strain relation of this kind of interphase layer is shown in Fig. 6; (d) interphase with damage and failure behaviors: this interphase layer was assumed to undergo damage deterioration after a certain threshold strain is reached. When the damage value D in the elements of this interphase layer approaches one, the failure will happen and these elements will be removed from the finite element mesh. Fig. 7 illustrates the predicted stress–strain relation of the composite with the perfectly adhesive interface. Unfilled PEEK exhibits a ductile-like behavior similar to semicrystalline polymer. It firstly underwent elastic deformation, followed by a non-linear stress–strain behavior prior to its yield point, and then deforms plastically at a more or less constant load before rupture. For the composite with higher PVF, the elastic modulus enhancement could be seen from the steeper
1 Strain (%)
of HA particles has the Young’s modulus 3.5 times greater than the unfilled PEEK. The Young’s modulus of cortical bone tissue is about 7–30 GPa [9]. Therefore, the stiffness of HA/PEEK with 40% PVF could satisfy the stiffness requirement of the cortical bone. 6.2. The stress–strain relation
0.5
30% PVF 20% PVF
60
Pure PEEK
10% PVF
40
20
0 0
2
4 6 Strain (%)
8
10
Fig. 7. Predicted stress–strain relation with perfectly adhesive interface.
slope of its stress–strain curve in the linear segment. The tensile strength is enhanced as PVF is increasing. Thus, the particle plays a role of reinforcement in both the stiffness and the strength of the composite with the perfectly adhesive interface. The stress–strain relation of the composites with 20% PVF and different interphase properties is predicted in Fig. 8. The degradable interphase is found to have a little effect on the stiffness of the composite, but have a weakening effect on the tensile strength of the composite. On the other hand, the interphase with a weaker stiffness has an adverse effect on both the strength and stiffness of the composite. The weakening effect for this case on the tensile strength of the composite is greater than that for the degradable
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373 80
80 perfect adhesive
70
70 Degradable
60
Weakening Damage
50
Stress (MPa)
Stress (MPa)
5369
40 30
Pure PEEK
60
10%
50
20%
40 30% 30
20
40%
20 10
10 0 2
4 6 Strain (%)
8
10
Fig. 8. Predicted stress–strain relation of the composites with 20% PVF and different interphase behaviors.
interphase. For the interphase layer with damage and failure effects, after a first element failure and stress redistribution, a series of successive failure takes place in the interphase layer. This phenomenon can be seen in the curve through a small wave at a region near the end of elastic deformation. Both the strength and stiffness were heavily influenced after the interphase layer had undergone damage and failure. Interfacial bond strength of the actual fabricated composite may not be perfect. By using the unit cell model incorporated with different material behaviors of the interphase layers, stress–strain relation of the composites with the bond strength ranging from the perfect to the weaker one due to the damage and failure effects could be simulated. Hence, the safety limit of a structure made of the HA/PEEK composite for a load-bearing application could be estimated by using the predicted range. When a ductile damage evolution in the matrix was also considered in addition to the damage and failure effects on the interphase layer, the stress–strain relation of the composites with different PVFs is as predicted in Fig. 9. The results are entirely opposite to those of the perfectly adhesive case. The tensile strength of the composites decreases as PVF is increasing. The effect of particle reinforcement fails due to the damage and failure of the interphase layer. The predicted stress– strain relation of the composite with 10% PVF is extracted from Fig. 9 and compared with the experimental data obtained by axial tensile test [9,11], as shown in Fig. 10. Good agreement between the predicted and experimental results is obtained both for the elastic and plastic regions. The results indicate that the cell model with consideration of the damage evolution in the matrix and the interphase layer, and the interface failure is well suitable for predicting the
0
0
2
4
6
8
10
Strain (%)
Fig. 9. Predicted stress–strain relation of the composite with matrix damage, and interphase damage and failure effects.
70 60 50
Stress (MPa)
0
40 Finite element Prediction
30 20
Experiment 10 0
0
2
4
6
Strain (%) Fig. 10. Comparison between predicted and experimental stress–strain relation of the composite with matrix damage, and interphase damage and failure effects of 10% PVF.
mechanical properties of the particulate reinforced biocomposite. 6.3. Tensile strength It has been known from our previous results that the ultimate tensile strength of the composite with perfectly adhesive interface can be enhanced by addition of particle. This is usually true especially for the metallicmatrix composite [40]. Unfortunately, this advantage fails to appear in a real biocomposite due to its poor interfacial bond. A simple empirical model that assumes that no interfacial adhesion takes place between the
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
5370
matrix and the particle is expressed as [41] sc ¼ sm ð1 1:21Vp2=3 Þ;
ð12Þ
where sc and sm are the tensile strengths of the composite and the matrix, respectively. The ultimate tensile strength of the composite with consideration of the damage and failure effects on the interphase layer and the matrix damage was predicted in comparison with the results from the empirical model and the experimental results as shown in Fig. 11. The ultimate tensile strength decreases with increasing PVF because of failure of the interphase layer. When the composite is strained beyond a certain threshold strain, 80
Finite element prediction Tensile strength (MPa)
60
40 Empirial model 20 Experimental data 0 0
10
20
30
40
HA Volume
Fig. 11. Ultimate tensile strength of the composite with matrix damage, and interphase damage and failure effects at 10% PVF.
the filled particles will debond from the matrix. Thus, the stress transfer between the particle and the matrix is greatly hindered and reinforcing effect of the particles disappears. Like a form of cavity, the load can only be carried by the remaining matrix, thus inducing a decrease in the strength of the composite. It can be observed from Fig. 11 that our predicted results using our cell model are in good agreement with the experimental ones.
6.4. Stress field distribution The von Mises stress distribution in the unit cell model is shown in Fig. 12. For the composite with the perfectly adhesive interface, the particle can bear most of the loading, and therefore the stress transfer between the particle and the matrix can reach a maximum. Thus, the particle serves as a reinforcement in the composite for both the strength and stiffness. For the composite with the degradable interphase layer, the stiffness of the interphase decreases with an increase in the load after a certain strain, thus making part of the loading unable to transfer to the particle. Therefore, the particle can bear a lesser part of the loading. It can be observed from Fig. 12(b) that the load-bearing region around the centre of the particle in the model with the degradable interphase layer is much smaller than that in the model with the perfectly adhesive interface, as shown in Fig. 12(a). However, the maximum von Mises stress is almost the same as the case for the perfectly adhesive interphase. For the composite containing the interphase layer with the damage and failure effects, the particle hardly helps to bear the loading due to occurrence of interfacial
Fig. 12. von Mises stress distribution: (a) perfect adhesive interface; (b) degradable interfacial material; (c) interfacial damage and failure.
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
6.5. Influence of other factors on the simulation results 6.5.1. Mesh sensitivity The amount of finite elements used affects the data storage space and calculating time. For an experienced user, as fewer as possible number of elements should be adopted, without losing appropriate calculating accuracy. In order to investigate the mesh sensitivity of the numerical results, three different meshes have been considered for our simulations. Mesh number 1, 2 and 3 have 20,000, 6000 and 1000 elements, respectively. Fig. 13 illustrates the effect of different element amounts on the stress-strain relation of the composite. For mesh number 3, a total of 1000 elements were adopted. There is a great influence at the stage of successive failure in the interphase elements because each element in this mesh has a larger size than the other two meshes. Beyond this stage, no great wave appears, especially when the amount of elements increases. The stress– strain curve of the composite was obtained through an averaging technique. It was found that mesh number 2 containing 6000 elements is enough for obtaining desirable results. 6.5.2. The loading velocity The explicit dynamic algorithm has several significant advantages over the conventional implicit static 60
50
Stress (MPa)
40
mesh number 2
20 mesh number 3 10
0
0
2
4 6 Strain (%)
8
50
40
30 v=0.1
v=1
20 v=10
10
0
0
2
4 6 Strain (%)
8
10
Fig. 14. Loading velocity test of stress–strain curves of the composites with 20% PVF.
algorithm. In the explicit method, there is no banded equation solver like the Newton–Raphson method. Consequently, the computational cost of a solution does not grow quadratically with the problem size. In general, the computational cost is linearly proportional to the problem size in the explicit dynamic procedure. The major disadvantage of the explicit dynamic procedure is that it is a time- and rate-dependent dynamic procedure, which sometimes loses the static stability of a solution. Generally, some sort of artificial time scales are required to introduce into the analysis to achieve an economical solution. When loading velocity is great, the inertia effect heavily affects the calculation results. On the other hand, if loading velocity is too small, a large amount of computation time is required to achieve the approximation of quasi-static solutions. Fig. 14 illustrates the predicted stress–strain curves of composites with 20% PVF under three loading velocities, v ¼ 0:1; 1, 10. For v ¼ 10; the curve is waved around its quasi-static case. For v ¼ 1 or smaller, a desirable result is achieved. 6.5.3. Mass scale scheme A mass scaling scheme was introduced because the time scale of the analysis is of particular importance in dynamic analysis. The dilatational wave speed cd is determined by pffiffiffiffiffiffiffiffiffiffi ð13Þ cd ¼ Et =r;
mesh number 1
30
60
Stress (MPa)
debonding in the composite. Thus, the maximum von Mises stress in this case is lower than that in the case of the perfectly adhesive interface, and is mainly located in the matrix. With the contour plot of the stress field distribution in the composite, the effects of the varying interphase properties on the strength of the composite can be visualized.
5371
10
Fig. 13. Mesh sensitivity of stress–strain curves of the composites with 20% PVF.
where Et is the tangent modulus and r is the material density. The maximum incremental time in one loading step is pffiffiffi 2 2h 2h r p ffiffiffiffiffi : Dtmax ¼ ¼ ¼ ð14Þ oh cd Et
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
5372
7. Conclusion
60
50
Stress (MPa)
40 10
30 100
20
scaling factor=1 10
0
0
2
4 6 Strain (%)
8
10
Fig. 15. Mass scaling test of stress–strain curves of the composites with 20% PVF.
The wave speed of the material is inversely proportional to the square root of the material density. Increasing the density by a factor of 100 will increase the stable time increment by a factor of 10. For an economic analysis, the mass scaling method was adopted. The simulation results are shown in Fig. 15 with three mass scaling factors, 1, 10, and 100. The curve demonstrates that there exists a great deviation between the scaling value and the real one during the stage of successive failure in the interphase layer. Moreover, the mass scaling calculation provides the more accurate results. 6.6. Biological response of HA/PEEK composites as implant materials For the biological response and tissue in-growth of the HA/PEEK composites as implant materials at an early implantation period, in vivo study has been performed by Abu Bakar et al. [11]. The HA/PEEK implants with varying porosity, pore size distribution and HA particle volume fraction were inserted into 35 kg pigs up to 24 months. For the period of 6 weeks of implantation, the presence of fibro-vascular tissue grew within the pores of the implant. For the period of 16 weeks, mature bone was formed within the pores of the implant. These results showed that the HA/PEEK implants had favorable bioactivity and biocompatibility. Histological study of animal model revealed the appearance of fibroblast cells, which promote vascularization evidently during early stages of implantation. Furthermore, osteoblast has been activated in the formation of osteoid and osteocytes within lamellar bone in developing mature bone at longer implantation periods. Thus, these findings have promised the use of HA/PEEK composites for high load-bearing medical implants and devices.
The interphase layer between the particle and the matrix plays a key role in high-performance composites such as particulate reinforced biocomposites. Investigation of the effect of the different interfacial behaviors on the overall mechanical properties of the composites can be beneficial to the material design and application. The three-dimensional cubic unit cell model has been successfully applied to predict the mechanical properties of HA/PEEK biocomposites, including the Young’s modulus, the stress–strain relation and the ultimate tensile strength. Four different kinds of the interphase layers, including the perfectly adhesive interphase layer, the interphase layer with a weaker stiffness, the interphase layer with a degradable elasto-plastic behavior, and the interphase layer with both the damage and failure effects, have been taken into account. The matrix degradation in the composite was also considered by using a matrix damage evolution equation. A userdefined VUMAT subroutine describing the damagedcoupled elasto-plastic constitutive behavior of the matrix material, and the damage evolution and the damage-induced failure criterion for the interphase layer have been successfully established and incorporated into the ABAQUS finite element code for prediction of the effects of different interphase behaviors on the micromechanical properties of the composties. Some other factors, such as mesh sensitivity, loading velocity and mass scale scheme were also found to be important on the prediction accuracy. The predicted Young’s modulus of the composite with the perfectly adhesive interface increases with increase in the HA content. The predicted tensile strength of the composite containing the interphase layer with both damage and failure effects, and the matrix with ductile damage, shows a decrease on increasing the particle concentration. The predicted results are in good agreement with the experimental ones. Thus, the threedimensional finite element unit cell model with appropriate material behaviors of the interphase layer and with consideration of the matrix degradation is successful in predicting not only the trend of the mechanical properties of the HA/PEEK composites in relation with the HA content, but also the non-linear constitutive relation.
Acknowledgements The authors would like to thank the Research Grant Council of Hong Kong for funding support of this project (PolyU5176/00E) and Prof. Khor for providing us the SEM pictures of the HA/PEEK biocomposite.
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
References [1] Sloten JV, Labey L, Audekeroke RV, Perre GV. Materials selection and design for orthopaedic implants with improved long-term performance. Biomaterials 1998;19:1455–9. [2] Bonfield W, et al. Hydroxyapatite reinforced polyethylene—a mechanically compatible implant material for bone replacement. Bioceramics 1981;2:185–6. [3] Bonfield W, Wang M, Tanner KE. Interfaces in analogue biomaterials. Acta Mater 1998;46:2509–18. [4] Wang M. Bioactive ceramic–polymer composites for bone replacement. Proceedings of the 13th International Conference on Composite Materials (ICCM-13), Beijing, China, 2001, Paper x1541. [5] Cheang P, Khor KA. Effect of particulate morphology on the tensile behaviour of polymer–hydroxyapatite composites. Mater Sci Eng 2003;A345:47–54. [6] Huang J, Di Silvio L, Wang M, Tanner K, Bonfield W. In vitro mechanical and biological assessment of hydroxyapatitereinforced polyethylene composite. J Mater Sci: Mater Med 1997;8:775–9. [7] Liu Q, De Wijn J, Van Blitterswijk C. Nano-apatite/polymer composites: mechanical and physicochemical characteristics. Biomaterials 1997;18:1263–70. [8] Reed C, TenHuisen K, Brown P, Allcock H. Thermal stability and compressive strength of calcium-deficient hydroxyapatitepoly (bis(carboxylatophenoxy) phosphazane) composites. Chem Mater 1996;8:440–7. [9] Abu Bakar MS, Cheang P, Khor KA. Tensile properties and microstructural analysis of spheroidized hydroxyapatite-poly (etheretherketone) biocomposites. Mater Sci Eng A 2003;345: 55–63. [10] Barton AJ, Sagers RD, Pitt WG. Bacterial adhesion to orthopaedic implant polymers. J Biomed Mater Res 1996;30: 403–10. [11] Abu Bakar MS, Cheng MHW, Tang SM, Yu SC, Liao K, Tan CT, Khor KA, Cheang P. Tensile properties, tension–tension fatigue and biological response of polyetheretherketone–hydroxyapatite composites for load-bearing orthopedic implants. Biomaterials 2003;24:2245–50. [12] Santos C, Charke RL, Braden M, Guitian F, Davy KWM. Water absorption characteristics of dental composites incorporating hydroxyapatite filler. Biomaterials 2002;23:1897–904. [13] Soderholm KJM, Zigan M, Ragan M, Fischlschweiger W, Bergman M. Hydrolytic degradation of dental composites. J Dent Res 1984;63:1248–54. [14] Matous K. Damage evolution in particulate composite materials. Intl J Solids Struct 2003;40:1489–503. [15] Tszeng TC. The effects of particle clustering on the mechanical behavior of particle reinforced composites. Composer: Part B 1998;29B:299–308. [16] Guild FJ, Young RJ. A predictive model for particulate filled composite materials. J Mater Sci 1989;24:298–306. [17] Guild FJ, Bonfield W. Predictive modeling of hydroxyapatite– polyethylene composite. Biomaterials 1993;14:985–93. [18] Lee ST, Chiang HC, Lin CT, Huang HM, Dong DR. Finite element analysis of thermo-debonding mechanism in dental composites. Biomaterials 2000;21:1315–26. [19] Guild FJ, Bonfield W. Predictive modelling of the mechanical properties and failure processes in hydroxyapatite–polyethylene (Hapext) composite. J Mater Sci: Mater Med 1998;9:497–502.
5373
[20] Balac I, Uskokovic PS, Ignjatovic N, Aleksic R, Uskokovic D. Stress analysis in hydroxyapatite/poly-l-lactide composite biomaterials. Comput Mater Sci 2001;20:275–83. [21] Voros G, Pukanszky B. Effect of a soft interlayer with changing properties on the stress distribution around inclusions and yielding of composites. Composer: Part A 2001;32:343–52. [22] Lauke B, Schuller T, Beckert W. Calculation of adhesion strength at the interface of a coated particle embedded within matrix under multiaxial load. Comput Mater Sci 2000;18:362–80. [23] Ismar H, Schroter F, Streicher F. Effects of interfacial debonding on the transverse loading behaviour of continuous fiber-reinforced metal matrix composites. Comp Struct 2001;79:1713–22. [24] Wang C, Ma J, Cheng W. Formation of polyetheretherketone polymer coating by electrophoretic deposition method. Surf Coat Tech 2003;171:271–5. [25] Abu Bakar MS, Cheang P, Khor KA. Thermal processing of hydroxyapatite reinforced polyetheretherketone composites. J Mater Process Tech 1999;89–90:462–6. [26] Abu Bakar MS, Cheang P, Khor KA. Mechanical properties of injection molded hydroxyapatite–polyetheretherketone biocomposites. Compos Sci Tech 2003;63:421–5. [27] Andres TE, Griffiths I. Proceedings of the 1998 SPE Annual West Regional Meeting, Anaheim, CA Soc Pet Eng, Richardson, TX, 1998. p. 203. [28] Ning CQ, Zhou Y. In vitro bioactivity of a biocomposite fabricated from HA and Ti powders by powder metallurgy method. Biomaterials 2002;23:2909–15. [29] Jarcho M. Calcium phosphate ceramics as hard tissue prosthetics. Clin Orthop Relat Res 1981;157:259–78. [30] Haddadi H, Teodosiu C. 3D-analysis of the effect of interfacial debonding on the plastic behaviour of two-phase composites. Comput Mater Sci 1999;16:315–22. [31] Lee SY, Chiang HC, Lin CT, Huang HM, Dong D-R. Finite element analysis of thermo-debonding mechanism in dental composites. Biomaterials 2000;21:1315–26. [32] Bathe KL. Finite element procedure in engineering analysis. New Jersey: Prentice-Hall; 1982. [33] Hughes TJR. The finite element method, linear static and dynamic finite element analysis. New Jersey: Prentice-Hall; 1987. [34] Lemaitre J. Continuum damage mechanics theory and applications. Berlin: Spinger; 1987. [35] Haddadi H, Teodosiu C. 3D-analysis of the effect of interfacial debonding on the plastic behaviour of two-phase composites. Comput Mater Sci 1916;1–4:315–22. [36] Li G, Ling X, Shen H. On the mechanism of void growth and the effect of straining mode in ductile materials. Int J Plast 1916;1: 39–57. [37] Wu Y, Dong ZF. Three-dimensional finite element analysis of composites with coated spherical inclusions. Mater Sci Eng 1995; 203:314–23. [38] Wu W, Sadeghipour K, Boberick K, Baran G. Predictive modeling of elastic properties of particulate-reinforced composites. Mater Sci Eng A 2002;332:362–70. [39] Halpin JC. Primer on composite materials analysis, 2nd ed. New Holland, Lancaster, PA: Technomic Pub. Co.; 1992. [40] Farrissey L, Schmauder S, Dong M, Soppa E, Poech MH, McHugh P. Investigation of the strengthening of particulate reinforced composites using different analytical and finite element models. Comput Mater Sci 1999;15:1–10. [41] Levita G, Marchetti A, Lazzeri A. Fracture of ultrafine calcium carbonate/polypropylene composites. Polym Compos 1989;10:39–43.
Biomaterials 25 (2004) 5363–5373
Influence of interphase layer on the overall elasto-plastic behaviors of HA/PEEK biocomposite J.P. Fana, C.P. Tsuia,*, C.Y. Tanga, C.L. Chowb a
Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Room DE404, ISE Research Office, Hung Hom, Kowloon, Hong Kong, China b Department of Mechanical Engineering, The University of Michigan-Dearborn, Dearborn, MI, USA Received 19 September 2003; accepted 19 December 2003
Abstract A three-dimensional finite element unit cell model has been designed and constructed for studying mechanical properties of hydroxyapatite (HA) reinforced polyetheretherketone (PEEK) biocomposite. The model consists of an elastic-brittle HA spherical particle, an elasto-plastic matrix and an interphase layer between the particle and the matrix. The interphase layers with four different kinds of material behaviors have been taken into consideration to examine their effects on the overall properties of the composite. The damage evolution in the matrix and the interphase layer, and the interface failure, were also taken into account. Some other factors, such as mesh sensitivity, loading velocity and mass scale scheme, were also discussed in this investigation. A general-purpose finite element software package, ABAQUS, incorporated with a user-defined material subroutine, was used to perform the analysis. The predicted results were compared with the experimental data obtained from existing literatures. The results predicted by using the cell model with consideration of the matrix degradation and the effects of the damage and failure on the interphase layer are in good agreement with the experimental ones. Hence, the suitability of our proposed cell model incorporated with an appropriate type of the interphase layer for modeling the mechanical properties of the particulate biocomposite could be verified. r 2004 Elsevier Ltd. All rights reserved. Keywords: Polyetheretherketone; Finite element analysis; Degradation; Interface
1. Introduction Traditionally, metallic materials, such as stainless steel and cobalt–chromium alloys, have been used as bone implant in orthopedic areas for many years. However, these materials can experience stress-shielding problem due to their high elastic modulus [1]. In order to minimize the stress-shielding effect and the formation of callus during healing of fractures, a polymer matrixbased material should be used. Bioactive second-phase reinforcing particles can be incorporated into the polymer matrix to enhance the mechanical properties of the composite for satisfying the load-bearing requirement of a bone implant. The concept of using bioactive hydroxyapatite (HA) particles reinforced polymer composite as an implant material for bone replacement was introduced by Bonfield [2] in the early 1980s, because the *Corresponding author. Tel.: +852-2766-6635. E-mail address: [email protected] (C.P. Tsui). 0142-9612/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.biomaterials.2003.12.050
HA particle is bioactive with a calcium-to-phosphorus ratio similar to that in the natural bone. Since then, particulate reinforced polymer-based biocomposites have been widely studied for bone tissue replacement [3–5]. Composites with HA and a polymer such as polyethylene, polyethyl ester or polyphosphasone and polyetheretherketone (PEEK) [6–9] have been used as potential materials for bone tissue replacement. Special attention has been focused on the particulate biocomposites with the bio-inert PEEK matrix, because PEEK has superior mechanical properties due to its highly aromatic structure [9]. Besides, it possesses additional benefits including good chemical and fatigue resistance, hightemperature durability, high wear properties, high ease of processibility, its capacity to be repeatedly sterilized and shaped readily by machining and heat contouring to fit the shape of bone [10]. All these benefits have rendered particulate reinforced PEEK to become attractive materials for biomedical applications such as major load-bearing orthopaedic implant materials [11].
ARTICLE IN PRESS 5364
J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
Under excessive loading, particle–matrix debonding and microdamage of the matrix can occur in particulate biocomposites [12,13]. In addition, the microdamage of the matrix may be induced during fabrication of the composite. The damage can be associated with microcracks initiating and growing from the imperfections within a matrix and along the particle–matrix interface [14]. Clustering of particles in the composite may also be originated from the fabrication processes [15]. Microcracks may initiate in the region of the particle clustering due to a high stress concentration. However, literatures on the modeling of the effects of microdamage on the mechanical properties of particulate biocomposites are very limited. Numerical techniques, such as finite element methods (FEMs), have been used for evaluating the micromechanical behaviors of particulate-reinforced composites on the basis of a unit cell model. Guild adopted the cylindrical cell model and Lee used the plane strain model as representative volume element [16–18]. Although using two-dimensional numerical models can reduce computation time as compared with the threedimensional ones, there exist gaps between the exact cell and its approximation. Recently, three-dimensional models for predicting the mechanical properties of hydroxyapatite-polyethylene and hydroxyapatite-poly l-lactide composites have been developed [19,20]. These models mainly focused on prediction of the elastic properties of the biocomposites. Good prediction for the elastic properties of the biocomposites could be achieved. However, the prediction for the failure processes of the biocomposites using a linear elastic model was found to have a low accuracy. Hence, the more complicated process of a nonlinear analysis including plastic deformation and damage should be employed. In addition, particulate filled composites possess heterogeneous structures, in which each phase is separated by an interfacial phase. This interphase forms spontaneously on the surface of the filler during the process of material fabrication due to the adsorption of the polymer in a thermoplastic composite [21] and because of surface treatment on the filler or modified curing chemistry in a thermoset matrix. The mechanical characteristics of the interphase are entirely different from those of the matrix and the reinforcing filler. The presence of the interlayer can modify stress distribution around the inclusions, micromechanical deformation processes and the final properties of the composites [22,23]. However, limited investigation has been performed about the influence of the interphase properties on the mechanical properties of the composites, especially the particulate reinforced biocomposites. In this study, a three-dimensional finite element unit cell model was established for studying the influence of four different material behaviors of the interphase layer
on the micromechanical properties of HA reinforced PEEK biocomposites. In addition to the different kinds of the interphase layer, a ductile damage evolution in the matrix was taken into account to describe the degradation behavior of the matrix. The model was applied to predict Young’s modulus, tensile strength, stress–strain relation and stress distribution of the HA/PEEK biocomposites with different material properties of the interphase layers. The simulation results were compared with experimental ones.
2. Properties of the constituent materials PEEK with a basic formula of (–C6H4–O–C6H4–O– C6H4–CO–)n is a polyaromatic semicrystalline thermoplastic polymer. It has a melting temperature of 343 C, a crystallization peak at 343 C and a glass transition temperature of 145 C. The high-temperature performance makes it become a stable material in human body [24]. In addition, its superior combination of strength, stiffness and toughness, as well as outstanding chemical, hydrolysis and wear resistance, together with its extensive biocompatibility, have enabled it to be suitable for in vitro load-bearing medical device applications [25,26]. Moreover, PEEK is non-cytotoxic and can be repeatedly sterilized using conventional steam, gamma and ethylene oxide processes without evident degradation on its mechanical properties [26,27]. Thus, this renders it to become long-term medical implants in orthopaedic, cardiovascular and dental markets. Experimental measurement demonstrated that the matrix of PEEK experienced elasto-plastic deformation [9,11]. Therefore, in this investigation, the von Mises yield surface for the PEEK matrix was assumed to have an associated flow rule and exhibit isotropic hardening. HA, with a chemical formula of (Ca10(PO4)6(OH)2), has a chemical and crystallographic structure being similar to that of bone material [28]. Moreover, HA is biocompatible with hard tissues of human beings and possesses osteoconductive properties [29]. Thus, HA has been widely used as a bone-replacement material in restorative dental and orthopaedic implants [28]. The high-modulus HA particle in biocomposites usually undergoes elastic deformation and seldom suffers crack failure during a load-bearing process [19]. Therefore, the deformation behavior of the HA reinforcement can be characterized by a purely elastic constitutive law. The generalized Hooke’s law [30] is satisfactory to reflect the deformation behavior of the HA particle in the biocomposite. The mechanical properties of the interphase layer are hard to measure [31], so that it was firstly assumed to have the same material properties as the matrix. In addition, four different material behaviors of the interphase layers, including a perfectly adhesive interface, a weaker stiffness, a degradable elasto-plastic,
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
and a combination of both damage and failure behaviors, would be taken into account and explained in detail in Section 4.
3. Explicit integral technique Unlike the implicit integral technique, explicit dynamic formulation has a series of advantages, such as low memory requirement, no iteration-convergent difficulty, easy treatment of contact condition and the presence of vanishing element treatment. All these advantages render it to become a rather powerful tool in actual application. The general form of the principle of virtual work is described as balance of internal work, inertia work, work done by body force and work exerted by surface force [32]. The discretization of the body into finite elements and the assembly of all element contributions into global matrices and vectors lead to the following matrix–vector equation . þ K U ¼ F; M U
ð1Þ
where M is the mass matrix, K is the stiffness matrix, F is the vector of the prescribed forces, and U ¼ UðtÞ is the nodal displacement vector in global coordinate. To solve this second-order differential equation, it requires discretization of the time interval into finite steps of Dt and calculation of the approximate solution on the time tn ¼ n Dt (for n ¼ 0; 1, 2, y). The central difference method was applied so that the first-order and second’ and U . of the global displacement order derivatives U vector were then approximately replaced by the displacement vectors Uðtn1 Þ; Uðtn Þ and Uðtnþ1 Þ: The truncation errors in these approximations are of the order OððDtÞ2 Þ: When the time step tends to be zero such that Dt-0; the central difference method approaches an accurate solution. Convergence of the solution is now assured when the algorithm satisfies the stability condition, i.e., when small numerical errors are not amplified by taking one time step. The central difference method is conditionally stable, meaning that the time step must be sufficiently small. The following restriction is required [33]: DtpDtmax ¼
2 ; oh
ð2Þ
where oh is the maximum natural frequency of system vibration represented by Eq. (1). This frequency depends on element size h and dilatational wave speed cd ; and is proportional to cd =h: The proportional constant is related to the element type. Thus, it is clear that the time step Dt is restricted by the smallest element in the entire element division.
5365
4. The damage evolution 4.1. Basic consideration of damage The term damage is used to indicate the deterioration of the material load-carrying capability. From a general viewpoint, damage develops in the material microstructure when non-reversible phenomena take place, such as microvoid or microcrack formation in the polymer matrix. A representative volume element in a damaged body is selected, as shown in Fig. 1. Let A0 be the nominal intersection area, and Aeff be the effective resisting section area in which the area occupied by microdefects has been removed. Thus, a damage variable is defined as [34] Aeff D¼1 : ð3Þ A0 This definition means that an isotropic damage is developed, such that microcracks and microvoids with orientation distribute uniformly in all directions. The damage value does not depend on the direction, so that it can be completely characterized by the scalar value of D: 4.2. Matrix damage evolution In this study, the matrix was considered to be a ductile material, which undergoes elasto-plastic deformation. Thus, a ductile damage is likely to take place in the matrix and a damage potential function j is taken as follows [34]: S Y 2 j¼ pHðp pD Þ: ð4Þ ’ 2 S where S is a material-dependent constant, p’ is an equivalent plastic strain rate, Y is the damage strain energy and HðUÞ is a Heaviside step function.
A0
A eff
Fig. 1. Representative volume element for a damaged body.
ARTICLE IN PRESS 5366
J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
In order to implement a finite element analysis, an incremental form of the damage evolution is derived by applying the backward Euler method: seq Rv DD ¼ ð5Þ DpHðp pD Þ; 2ESð1 DÞ2 where Rv is a stress triaxiality function, which depends on the rate of hydrostatic stress to von Mises equivalent stress. Obviously, the damage evolution depends upon the accumulated plastic strain, the stress triaxiality and the von Mises equivalent stress. 4.3. Damage evolution equation for the interphase layer There are three types of criteria proposed for the particle–matrix debonding: stress-based, strain-based and energy-based criteria. The stress-based criterion is generally more appropriate to be incorporated into the interphase layer for modelling the particle–matrix debonding of the ceramic particle reinforced polymer composite [35]. At a certain level of loading, a damage will be developed in the interphase element. The damage variable D used in this investigation is defined as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 2ffi sn 2 st1 st2 D¼ þ c þ c ; ð6Þ scn st st
geometry of the constituent phases, and interfacial properties among the phases [37]. The thermal spraying experiment [9] illustrated that the HA particle appears to be spherical in shape with a glassy smooth surface, as shown in Fig. 2. For convenience of studies, these spherical particles are assumed to be evenly dispersed in the polymer composite in this investigation, so that it could be represented by a periodically repeated threedimensional cubic array, as shown in Fig. 3(a). Because of the assumed symmetry for the packing of the spherical particles, only one-eighth of the sphere embedded in a cube as shown in Fig. 3(b) is needed for numerical analysis. Particle volume fraction (PVF), Vp is given by pr1 3 ; ð7Þ Vp ¼ 6 r while the interphase volume fraction, Vi is expressed as p r2 3 r1 3 Vi ¼ ; ð8Þ 6 r r whereas the thickness of the interphase layer h is given by h ¼ r2 r1 ; in which r1 is the particle radius and r2 is the outer radius of the interphase layer depicted in Fig. 3(b). Here r is the cell length, which is set to be
where sn is the normal stress, st1 and st2 are two tangential stresses within the interphase layer. scn and sct are the normal and tangential critical stress values, respectively. The damage variable D varies in the range of [0, 1]. When D value in the interphase element is greater than zero, this means that a damage has been developed in this element. When D value in any interphase element reaches one, this element would fail and be removed using a vanishing element technique [36]. In order to perform the analysis, the commercial software package ABAQUS/Explicit was used. However, the composite with our required material behaviors is not available in the material library of our finite element software. In order to take into account the elastic deformation of the particle, the elasto-plasticdamage behavior of polymer matrix, damage-coupled material behaviors and failure criterion of the interphase layer, a user-defined material subroutine named ‘‘VUMAT’’ has been established and incorporated into the ABAQUS finite element code.
Fig. 2. Morphology of flame spheroidized HA particles.
H
G
z F
E
y
r2
5. Finite element modeling
The mechanical behavior of the composite is dependent on physical properties, volume fraction, packing
C
r1 A
5.1. Formation of a unit cell model
D
(a)
B
x
(b)
Fig. 3. Geometry of particle-reinforced composite: (a) packing of spherical particles in the composite; (b) three-dimensional unit cell model.
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
equal to one. The thickness of the interphase layer is set to be 1% of the cell length [38]. Perfect bonding was initially assumed between the particle and the interphase layer, and between the interphase layer and the polymer matrix. These cell models were then converted into finite element cell models constructed by three-dimensional brick eight-node linear elements. The finite element mesh of the cell model containing 20% PVF is shown in Fig. 4, which clearly illustrates that the model is composed of the particle, the interphase layer and the polymer matrix. In the present finite element analysis, the particle shape and the thickness of the interphase layer were kept fixed, while the PVF was varied up to 40%. The mechanical properties of the constituent materials are summarized in Table 1. 5.2. Boundary conditions External force and influence of adjacent material deformation were taken into account by using appropriate boundary conditions. Due to the symmetry of the unit cell, the normal displacements on the surfaces of ABCD, ABFE and ADHE shown in Fig. 3(b) are constrained such that ux ¼ 0 on x ¼ 0;
5367
surface. On the other hand, to ensure compatibility among all periodic representative cells, the faces BCGF, DCGH and EFGH of the unit cell are required to remain plane and parallel to their initial state after deformation. Thus, the normal force acting on the both faces of BCGF and DCGH must be set to zero such that Z ð10Þ sn dA ¼ 0 on x ¼ r and y ¼ r and all shear stresses on these boundaries should be set to zero. To satisfy the parallel plane requirement during deformation, an arbitrary point P; which does not belong to any parts of the model and is allowed to freely move in any directions, was created and related to all nodes on the line x ¼ r and y ¼ r: Thus, displacement of all element nodes on the face x ¼ r and y ¼ r in the x-direction and y-direction, respectively, are the same and equal to that of the point P: With this method, no external force is added onto these boundaries. Hence, the boundary conditions set in Eq. (10) could be satisfied. The velocity boundary condition was applied on the upper surface at z ¼ r:
6. Results and discussions
uy ¼ 0 on y ¼ 0; uz ¼ 0 on z ¼ 0:
ð9Þ
These boundary conditions can be easily enforced by restraining the nodal displacement on the corresponding
6.1. Young’s modulus For the estimation of Young’s modulus of the biocomposite, the interphase layer was taken to be the same material as the matrix. Each constituent underwent elastic deformation, and no damage and failure happened. In order to verify the accuracy of the numerical prediction, the Halpin–Tsai model, which is a well-known model for composite materials [39], was applied to predict the variation of Young’s modulus of the HA/PEEK biocomposite with PVF. The semiempirical equation of the Halpin-Tsai model for predicting Young’s modulus E of the composite is given by E ¼ Em
Fig. 4. Three-dimensional finite element mesh of unit cell model.
Table 1 Material properties Materials
E (GPa)
n
r (kg/m3)
PEEK HA
3.2 85
0.42 0.3
1291 3160
ð1 þ zZVp Þ Ep =Em 1 where Z ¼ ; ð1 ZVp Þ Ep =Em þ z
ð11Þ
where Vp is PVF, z is a measure of particle geometry, packing the geometry and loading conditions of the particle. The predicted results generated from our finite element model and the analytical Halpin–Tsai model are compared with the experimental results obtained from some literatures [9,11] shown in Fig. 5. The predicted Young’s modulus increases correspondingly with increasing HA content. The results acquired from the numerical prediction are closer to the experimental data than those from the analytical Halpin–Tsai formula. On the other hand, the composite with 40%
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
5368
14
9
Experiment 10 6 Stress (MPa)
Young's modulus (GPa)
12
8 6
3
4 Halpin-Tsai 2
Finite element Prediction
0 0
10
20
30
0
40
0
HA Volume (%) Fig. 5. Young’s modulus of the composite.
1.5
2
Fig. 6. Stress–strain relation of the interphase with degradable elastoplastic behavior.
100 40% PVF 80
Stress (MPa)
The stress–strain relation of the HA reinforced PEEK composites have been predicted by using four different kinds of interphase layers. (a) perfectly adhesive interface: this interphase layer was taken to be the same material as the matrix, and no damage and failure were assumed to take place; (b) interphase with a weaker stiffness: the interphase layer with an elastic modulus only one-thousandth of that of the matrix was assumed to undergo elastic deformation only; (c) interphase with degradable elasto-plastic behavior: the stress–strain relation of this kind of interphase layer is shown in Fig. 6; (d) interphase with damage and failure behaviors: this interphase layer was assumed to undergo damage deterioration after a certain threshold strain is reached. When the damage value D in the elements of this interphase layer approaches one, the failure will happen and these elements will be removed from the finite element mesh. Fig. 7 illustrates the predicted stress–strain relation of the composite with the perfectly adhesive interface. Unfilled PEEK exhibits a ductile-like behavior similar to semicrystalline polymer. It firstly underwent elastic deformation, followed by a non-linear stress–strain behavior prior to its yield point, and then deforms plastically at a more or less constant load before rupture. For the composite with higher PVF, the elastic modulus enhancement could be seen from the steeper
1 Strain (%)
of HA particles has the Young’s modulus 3.5 times greater than the unfilled PEEK. The Young’s modulus of cortical bone tissue is about 7–30 GPa [9]. Therefore, the stiffness of HA/PEEK with 40% PVF could satisfy the stiffness requirement of the cortical bone. 6.2. The stress–strain relation
0.5
30% PVF 20% PVF
60
Pure PEEK
10% PVF
40
20
0 0
2
4 6 Strain (%)
8
10
Fig. 7. Predicted stress–strain relation with perfectly adhesive interface.
slope of its stress–strain curve in the linear segment. The tensile strength is enhanced as PVF is increasing. Thus, the particle plays a role of reinforcement in both the stiffness and the strength of the composite with the perfectly adhesive interface. The stress–strain relation of the composites with 20% PVF and different interphase properties is predicted in Fig. 8. The degradable interphase is found to have a little effect on the stiffness of the composite, but have a weakening effect on the tensile strength of the composite. On the other hand, the interphase with a weaker stiffness has an adverse effect on both the strength and stiffness of the composite. The weakening effect for this case on the tensile strength of the composite is greater than that for the degradable
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373 80
80 perfect adhesive
70
70 Degradable
60
Weakening Damage
50
Stress (MPa)
Stress (MPa)
5369
40 30
Pure PEEK
60
10%
50
20%
40 30% 30
20
40%
20 10
10 0 2
4 6 Strain (%)
8
10
Fig. 8. Predicted stress–strain relation of the composites with 20% PVF and different interphase behaviors.
interphase. For the interphase layer with damage and failure effects, after a first element failure and stress redistribution, a series of successive failure takes place in the interphase layer. This phenomenon can be seen in the curve through a small wave at a region near the end of elastic deformation. Both the strength and stiffness were heavily influenced after the interphase layer had undergone damage and failure. Interfacial bond strength of the actual fabricated composite may not be perfect. By using the unit cell model incorporated with different material behaviors of the interphase layers, stress–strain relation of the composites with the bond strength ranging from the perfect to the weaker one due to the damage and failure effects could be simulated. Hence, the safety limit of a structure made of the HA/PEEK composite for a load-bearing application could be estimated by using the predicted range. When a ductile damage evolution in the matrix was also considered in addition to the damage and failure effects on the interphase layer, the stress–strain relation of the composites with different PVFs is as predicted in Fig. 9. The results are entirely opposite to those of the perfectly adhesive case. The tensile strength of the composites decreases as PVF is increasing. The effect of particle reinforcement fails due to the damage and failure of the interphase layer. The predicted stress– strain relation of the composite with 10% PVF is extracted from Fig. 9 and compared with the experimental data obtained by axial tensile test [9,11], as shown in Fig. 10. Good agreement between the predicted and experimental results is obtained both for the elastic and plastic regions. The results indicate that the cell model with consideration of the damage evolution in the matrix and the interphase layer, and the interface failure is well suitable for predicting the
0
0
2
4
6
8
10
Strain (%)
Fig. 9. Predicted stress–strain relation of the composite with matrix damage, and interphase damage and failure effects.
70 60 50
Stress (MPa)
0
40 Finite element Prediction
30 20
Experiment 10 0
0
2
4
6
Strain (%) Fig. 10. Comparison between predicted and experimental stress–strain relation of the composite with matrix damage, and interphase damage and failure effects of 10% PVF.
mechanical properties of the particulate reinforced biocomposite. 6.3. Tensile strength It has been known from our previous results that the ultimate tensile strength of the composite with perfectly adhesive interface can be enhanced by addition of particle. This is usually true especially for the metallicmatrix composite [40]. Unfortunately, this advantage fails to appear in a real biocomposite due to its poor interfacial bond. A simple empirical model that assumes that no interfacial adhesion takes place between the
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
5370
matrix and the particle is expressed as [41] sc ¼ sm ð1 1:21Vp2=3 Þ;
ð12Þ
where sc and sm are the tensile strengths of the composite and the matrix, respectively. The ultimate tensile strength of the composite with consideration of the damage and failure effects on the interphase layer and the matrix damage was predicted in comparison with the results from the empirical model and the experimental results as shown in Fig. 11. The ultimate tensile strength decreases with increasing PVF because of failure of the interphase layer. When the composite is strained beyond a certain threshold strain, 80
Finite element prediction Tensile strength (MPa)
60
40 Empirial model 20 Experimental data 0 0
10
20
30
40
HA Volume
Fig. 11. Ultimate tensile strength of the composite with matrix damage, and interphase damage and failure effects at 10% PVF.
the filled particles will debond from the matrix. Thus, the stress transfer between the particle and the matrix is greatly hindered and reinforcing effect of the particles disappears. Like a form of cavity, the load can only be carried by the remaining matrix, thus inducing a decrease in the strength of the composite. It can be observed from Fig. 11 that our predicted results using our cell model are in good agreement with the experimental ones.
6.4. Stress field distribution The von Mises stress distribution in the unit cell model is shown in Fig. 12. For the composite with the perfectly adhesive interface, the particle can bear most of the loading, and therefore the stress transfer between the particle and the matrix can reach a maximum. Thus, the particle serves as a reinforcement in the composite for both the strength and stiffness. For the composite with the degradable interphase layer, the stiffness of the interphase decreases with an increase in the load after a certain strain, thus making part of the loading unable to transfer to the particle. Therefore, the particle can bear a lesser part of the loading. It can be observed from Fig. 12(b) that the load-bearing region around the centre of the particle in the model with the degradable interphase layer is much smaller than that in the model with the perfectly adhesive interface, as shown in Fig. 12(a). However, the maximum von Mises stress is almost the same as the case for the perfectly adhesive interphase. For the composite containing the interphase layer with the damage and failure effects, the particle hardly helps to bear the loading due to occurrence of interfacial
Fig. 12. von Mises stress distribution: (a) perfect adhesive interface; (b) degradable interfacial material; (c) interfacial damage and failure.
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
6.5. Influence of other factors on the simulation results 6.5.1. Mesh sensitivity The amount of finite elements used affects the data storage space and calculating time. For an experienced user, as fewer as possible number of elements should be adopted, without losing appropriate calculating accuracy. In order to investigate the mesh sensitivity of the numerical results, three different meshes have been considered for our simulations. Mesh number 1, 2 and 3 have 20,000, 6000 and 1000 elements, respectively. Fig. 13 illustrates the effect of different element amounts on the stress-strain relation of the composite. For mesh number 3, a total of 1000 elements were adopted. There is a great influence at the stage of successive failure in the interphase elements because each element in this mesh has a larger size than the other two meshes. Beyond this stage, no great wave appears, especially when the amount of elements increases. The stress– strain curve of the composite was obtained through an averaging technique. It was found that mesh number 2 containing 6000 elements is enough for obtaining desirable results. 6.5.2. The loading velocity The explicit dynamic algorithm has several significant advantages over the conventional implicit static 60
50
Stress (MPa)
40
mesh number 2
20 mesh number 3 10
0
0
2
4 6 Strain (%)
8
50
40
30 v=0.1
v=1
20 v=10
10
0
0
2
4 6 Strain (%)
8
10
Fig. 14. Loading velocity test of stress–strain curves of the composites with 20% PVF.
algorithm. In the explicit method, there is no banded equation solver like the Newton–Raphson method. Consequently, the computational cost of a solution does not grow quadratically with the problem size. In general, the computational cost is linearly proportional to the problem size in the explicit dynamic procedure. The major disadvantage of the explicit dynamic procedure is that it is a time- and rate-dependent dynamic procedure, which sometimes loses the static stability of a solution. Generally, some sort of artificial time scales are required to introduce into the analysis to achieve an economical solution. When loading velocity is great, the inertia effect heavily affects the calculation results. On the other hand, if loading velocity is too small, a large amount of computation time is required to achieve the approximation of quasi-static solutions. Fig. 14 illustrates the predicted stress–strain curves of composites with 20% PVF under three loading velocities, v ¼ 0:1; 1, 10. For v ¼ 10; the curve is waved around its quasi-static case. For v ¼ 1 or smaller, a desirable result is achieved. 6.5.3. Mass scale scheme A mass scaling scheme was introduced because the time scale of the analysis is of particular importance in dynamic analysis. The dilatational wave speed cd is determined by pffiffiffiffiffiffiffiffiffiffi ð13Þ cd ¼ Et =r;
mesh number 1
30
60
Stress (MPa)
debonding in the composite. Thus, the maximum von Mises stress in this case is lower than that in the case of the perfectly adhesive interface, and is mainly located in the matrix. With the contour plot of the stress field distribution in the composite, the effects of the varying interphase properties on the strength of the composite can be visualized.
5371
10
Fig. 13. Mesh sensitivity of stress–strain curves of the composites with 20% PVF.
where Et is the tangent modulus and r is the material density. The maximum incremental time in one loading step is pffiffiffi 2 2h 2h r p ffiffiffiffiffi : Dtmax ¼ ¼ ¼ ð14Þ oh cd Et
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
5372
7. Conclusion
60
50
Stress (MPa)
40 10
30 100
20
scaling factor=1 10
0
0
2
4 6 Strain (%)
8
10
Fig. 15. Mass scaling test of stress–strain curves of the composites with 20% PVF.
The wave speed of the material is inversely proportional to the square root of the material density. Increasing the density by a factor of 100 will increase the stable time increment by a factor of 10. For an economic analysis, the mass scaling method was adopted. The simulation results are shown in Fig. 15 with three mass scaling factors, 1, 10, and 100. The curve demonstrates that there exists a great deviation between the scaling value and the real one during the stage of successive failure in the interphase layer. Moreover, the mass scaling calculation provides the more accurate results. 6.6. Biological response of HA/PEEK composites as implant materials For the biological response and tissue in-growth of the HA/PEEK composites as implant materials at an early implantation period, in vivo study has been performed by Abu Bakar et al. [11]. The HA/PEEK implants with varying porosity, pore size distribution and HA particle volume fraction were inserted into 35 kg pigs up to 24 months. For the period of 6 weeks of implantation, the presence of fibro-vascular tissue grew within the pores of the implant. For the period of 16 weeks, mature bone was formed within the pores of the implant. These results showed that the HA/PEEK implants had favorable bioactivity and biocompatibility. Histological study of animal model revealed the appearance of fibroblast cells, which promote vascularization evidently during early stages of implantation. Furthermore, osteoblast has been activated in the formation of osteoid and osteocytes within lamellar bone in developing mature bone at longer implantation periods. Thus, these findings have promised the use of HA/PEEK composites for high load-bearing medical implants and devices.
The interphase layer between the particle and the matrix plays a key role in high-performance composites such as particulate reinforced biocomposites. Investigation of the effect of the different interfacial behaviors on the overall mechanical properties of the composites can be beneficial to the material design and application. The three-dimensional cubic unit cell model has been successfully applied to predict the mechanical properties of HA/PEEK biocomposites, including the Young’s modulus, the stress–strain relation and the ultimate tensile strength. Four different kinds of the interphase layers, including the perfectly adhesive interphase layer, the interphase layer with a weaker stiffness, the interphase layer with a degradable elasto-plastic behavior, and the interphase layer with both the damage and failure effects, have been taken into account. The matrix degradation in the composite was also considered by using a matrix damage evolution equation. A userdefined VUMAT subroutine describing the damagedcoupled elasto-plastic constitutive behavior of the matrix material, and the damage evolution and the damage-induced failure criterion for the interphase layer have been successfully established and incorporated into the ABAQUS finite element code for prediction of the effects of different interphase behaviors on the micromechanical properties of the composties. Some other factors, such as mesh sensitivity, loading velocity and mass scale scheme were also found to be important on the prediction accuracy. The predicted Young’s modulus of the composite with the perfectly adhesive interface increases with increase in the HA content. The predicted tensile strength of the composite containing the interphase layer with both damage and failure effects, and the matrix with ductile damage, shows a decrease on increasing the particle concentration. The predicted results are in good agreement with the experimental ones. Thus, the threedimensional finite element unit cell model with appropriate material behaviors of the interphase layer and with consideration of the matrix degradation is successful in predicting not only the trend of the mechanical properties of the HA/PEEK composites in relation with the HA content, but also the non-linear constitutive relation.
Acknowledgements The authors would like to thank the Research Grant Council of Hong Kong for funding support of this project (PolyU5176/00E) and Prof. Khor for providing us the SEM pictures of the HA/PEEK biocomposite.
ARTICLE IN PRESS J.P. Fan et al. / Biomaterials 25 (2004) 5363–5373
References [1] Sloten JV, Labey L, Audekeroke RV, Perre GV. Materials selection and design for orthopaedic implants with improved long-term performance. Biomaterials 1998;19:1455–9. [2] Bonfield W, et al. Hydroxyapatite reinforced polyethylene—a mechanically compatible implant material for bone replacement. Bioceramics 1981;2:185–6. [3] Bonfield W, Wang M, Tanner KE. Interfaces in analogue biomaterials. Acta Mater 1998;46:2509–18. [4] Wang M. Bioactive ceramic–polymer composites for bone replacement. Proceedings of the 13th International Conference on Composite Materials (ICCM-13), Beijing, China, 2001, Paper x1541. [5] Cheang P, Khor KA. Effect of particulate morphology on the tensile behaviour of polymer–hydroxyapatite composites. Mater Sci Eng 2003;A345:47–54. [6] Huang J, Di Silvio L, Wang M, Tanner K, Bonfield W. In vitro mechanical and biological assessment of hydroxyapatitereinforced polyethylene composite. J Mater Sci: Mater Med 1997;8:775–9. [7] Liu Q, De Wijn J, Van Blitterswijk C. Nano-apatite/polymer composites: mechanical and physicochemical characteristics. Biomaterials 1997;18:1263–70. [8] Reed C, TenHuisen K, Brown P, Allcock H. Thermal stability and compressive strength of calcium-deficient hydroxyapatitepoly (bis(carboxylatophenoxy) phosphazane) composites. Chem Mater 1996;8:440–7. [9] Abu Bakar MS, Cheang P, Khor KA. Tensile properties and microstructural analysis of spheroidized hydroxyapatite-poly (etheretherketone) biocomposites. Mater Sci Eng A 2003;345: 55–63. [10] Barton AJ, Sagers RD, Pitt WG. Bacterial adhesion to orthopaedic implant polymers. J Biomed Mater Res 1996;30: 403–10. [11] Abu Bakar MS, Cheng MHW, Tang SM, Yu SC, Liao K, Tan CT, Khor KA, Cheang P. Tensile properties, tension–tension fatigue and biological response of polyetheretherketone–hydroxyapatite composites for load-bearing orthopedic implants. Biomaterials 2003;24:2245–50. [12] Santos C, Charke RL, Braden M, Guitian F, Davy KWM. Water absorption characteristics of dental composites incorporating hydroxyapatite filler. Biomaterials 2002;23:1897–904. [13] Soderholm KJM, Zigan M, Ragan M, Fischlschweiger W, Bergman M. Hydrolytic degradation of dental composites. J Dent Res 1984;63:1248–54. [14] Matous K. Damage evolution in particulate composite materials. Intl J Solids Struct 2003;40:1489–503. [15] Tszeng TC. The effects of particle clustering on the mechanical behavior of particle reinforced composites. Composer: Part B 1998;29B:299–308. [16] Guild FJ, Young RJ. A predictive model for particulate filled composite materials. J Mater Sci 1989;24:298–306. [17] Guild FJ, Bonfield W. Predictive modeling of hydroxyapatite– polyethylene composite. Biomaterials 1993;14:985–93. [18] Lee ST, Chiang HC, Lin CT, Huang HM, Dong DR. Finite element analysis of thermo-debonding mechanism in dental composites. Biomaterials 2000;21:1315–26. [19] Guild FJ, Bonfield W. Predictive modelling of the mechanical properties and failure processes in hydroxyapatite–polyethylene (Hapext) composite. J Mater Sci: Mater Med 1998;9:497–502.
5373
[20] Balac I, Uskokovic PS, Ignjatovic N, Aleksic R, Uskokovic D. Stress analysis in hydroxyapatite/poly-l-lactide composite biomaterials. Comput Mater Sci 2001;20:275–83. [21] Voros G, Pukanszky B. Effect of a soft interlayer with changing properties on the stress distribution around inclusions and yielding of composites. Composer: Part A 2001;32:343–52. [22] Lauke B, Schuller T, Beckert W. Calculation of adhesion strength at the interface of a coated particle embedded within matrix under multiaxial load. Comput Mater Sci 2000;18:362–80. [23] Ismar H, Schroter F, Streicher F. Effects of interfacial debonding on the transverse loading behaviour of continuous fiber-reinforced metal matrix composites. Comp Struct 2001;79:1713–22. [24] Wang C, Ma J, Cheng W. Formation of polyetheretherketone polymer coating by electrophoretic deposition method. Surf Coat Tech 2003;171:271–5. [25] Abu Bakar MS, Cheang P, Khor KA. Thermal processing of hydroxyapatite reinforced polyetheretherketone composites. J Mater Process Tech 1999;89–90:462–6. [26] Abu Bakar MS, Cheang P, Khor KA. Mechanical properties of injection molded hydroxyapatite–polyetheretherketone biocomposites. Compos Sci Tech 2003;63:421–5. [27] Andres TE, Griffiths I. Proceedings of the 1998 SPE Annual West Regional Meeting, Anaheim, CA Soc Pet Eng, Richardson, TX, 1998. p. 203. [28] Ning CQ, Zhou Y. In vitro bioactivity of a biocomposite fabricated from HA and Ti powders by powder metallurgy method. Biomaterials 2002;23:2909–15. [29] Jarcho M. Calcium phosphate ceramics as hard tissue prosthetics. Clin Orthop Relat Res 1981;157:259–78. [30] Haddadi H, Teodosiu C. 3D-analysis of the effect of interfacial debonding on the plastic behaviour of two-phase composites. Comput Mater Sci 1999;16:315–22. [31] Lee SY, Chiang HC, Lin CT, Huang HM, Dong D-R. Finite element analysis of thermo-debonding mechanism in dental composites. Biomaterials 2000;21:1315–26. [32] Bathe KL. Finite element procedure in engineering analysis. New Jersey: Prentice-Hall; 1982. [33] Hughes TJR. The finite element method, linear static and dynamic finite element analysis. New Jersey: Prentice-Hall; 1987. [34] Lemaitre J. Continuum damage mechanics theory and applications. Berlin: Spinger; 1987. [35] Haddadi H, Teodosiu C. 3D-analysis of the effect of interfacial debonding on the plastic behaviour of two-phase composites. Comput Mater Sci 1916;1–4:315–22. [36] Li G, Ling X, Shen H. On the mechanism of void growth and the effect of straining mode in ductile materials. Int J Plast 1916;1: 39–57. [37] Wu Y, Dong ZF. Three-dimensional finite element analysis of composites with coated spherical inclusions. Mater Sci Eng 1995; 203:314–23. [38] Wu W, Sadeghipour K, Boberick K, Baran G. Predictive modeling of elastic properties of particulate-reinforced composites. Mater Sci Eng A 2002;332:362–70. [39] Halpin JC. Primer on composite materials analysis, 2nd ed. New Holland, Lancaster, PA: Technomic Pub. Co.; 1992. [40] Farrissey L, Schmauder S, Dong M, Soppa E, Poech MH, McHugh P. Investigation of the strengthening of particulate reinforced composites using different analytical and finite element models. Comput Mater Sci 1999;15:1–10. [41] Levita G, Marchetti A, Lazzeri A. Fracture of ultrafine calcium carbonate/polypropylene composites. Polym Compos 1989;10:39–43.