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Stiffness and strength design of composite bone plates
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 65 (2005) 73–85 www.elsevier.com/locate/compscitech
Stiffness and strength design of composite bone plates Zheng-Ming Huang
a,*
, K. Fujihara
b
a
b
School of Aeronautics, Astronautics & Mechanics, Tongji University, 1239 Siping Road, Shanghai 200092, PR China Division of Bioengineering, Biomaterials Laboratory, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 22 January 2004; received in revised form 14 June 2004; accepted 14 June 2004 Available online 19 August 2004
Abstract Although the potential of composites used for bone plate implantation has been realized for a long period of time, not much literature has addressed optimal design of composite bone plates upon variation in reinforcing fiber structures, plate thicknesses, and so on. In this paper, we present a preliminary but critical design procedure for composite bone plate with target on both its stiffness and its ultimate strength. While the material system used is the same, i.e. carbon fibers and PEEK matrix, three kinds of fibrous preforms namely unidirectional (UD) prepregs, braided fabrics, and knitted fabrics are considered as reinforcements, constituting six different candidate laminated plates. They are UD [0]ns, angle plied [±20]ns, multidirectional [0/±20/0]ns, two braiding angle diamond braid fabrics [5]ns and [15]ns, and plain weft knitted fabric laminates. By varying the number of layers (n) in the lamination, different plate thicknesses can be attained. The structure–property relationship of the composite plates is realized through the use of a micromechanics bridging model. A clinically used stainless steel plate is taken as a design benchmark. It has been found that the composite bone plates with all the considered lamination patterns except for the knitted fabric reinforcement and having a thickness near to that of the stainless steel plate can exhibit required properties of bending rigidity and maximum bending moment. The plate thickness is more controlled by strength rather than by stiffness requirement. However, the 5 braided fabric composite bone plate shows an overall superiority as compared to the other lamination forms, and is thus recommended as the first choice. 2004 Elsevier Ltd. All rights reserved. Keywords: Bone plate; B. Strength
1. Introduction In the surgery field, patients with diaphyseal fractures are generally treated using compression bone plates made of stainless-steel, Cr–Co and Ti alloys [1–4] as shown in Fig. 1. Although exhibiting reasonable fatigue strength, the stiffness of these metals (in between 110 and 220 GPa) is much higher than that of human cortical bone (around 20 GPa). As a result, the majority of
*
Corresponding author. Tel.: +86-216-598-5373; fax: +86-216-5982914. E-mail address: [email protected] (Z.-M. Huang). 0266-3538/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2004.06.006
the load is carried by the plate rather than by the underlying bone. Callus formation, ossification, and bone union at fractured part are refrained after the implant operation, and the whole bone structure, not only at the fractured part, becomes osteoporosis [5,6]. The bone mass can be decreased by 20% [6] and in some cases the bone re-fracture due to stress concentration around the bone screws can be induced after the removal of the plate [7,8]. These phenomena are widely recognized as ‘‘stress shielding’’ effect, which is a main drawback for the use of metal bone plates. Another drawback is that metal plates can generate considerable artifacts (Fig. 2) under an X-ray, which make the interpretation of radiographs difficult.
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Fig. 1. Schematic of a compression bone plate with a tension device. A compression plate is attached on tensile side of the fragmented bone. Tension device is pulling a compression plate and compression force is accordingly generated at the damaged bones [1].
Fig. 2. Comparison of artifacts between metal and composite under X-ray.
In order to tackle aforementioned two drawbacks, polymer based composite materials, which have less stiffness, high fatigue strength, and good radiolucency, have been proposed for bone plate applications. Most composite plates developed so far have used UD (unidirectional) laminates [9–17] and discontinuous short fibers [18] as reinforcement. Recently, textile preforms such as braided structures have also been adopted in bone plate development [19,20]. As we know, composites are inherently anisotropic. Many parameters can influence the mechanical performance of a resulting composite bone plate such as constituent materials, their contents, reinforcing structure, plate thickness, etc. By varying all or some of them, an optimal composite bone plate can be expected. However, the majority of the existing composite bone plates has been essentially developed based on some trial and error method as a consequence of no relevant standard available at the present. Limited reports have been found in the open literature addressing actually optimal designs for composite bone plates upon varying the aforementioned parameters especially the fiber reinforcing structures. As mentioned before, too high stiffness of a bone plate will result in ‘‘stress shielding’’ effect, which is of an obstacle to the bone healing. On the other hand, too low stiffness may give rise to micro-movements of the fractured bone segments beneath the plate, and hence
also hinders the bone healing. This means that a proper amount of stiffness is necessary to be targeted by a plate design. Moreover, bone plate is a primary load-carrying element. It works in a very severe environment of human body, loaded with significant cycling forces. Thus, the plate load-carrying ability should be targeted as high as possible whereas the plate weight (or the volume) should be reduced to minimum to facilitate patientsÕ carriage. These targets can be fully achieved only when the microstructure–property relationship of the composite bone plate is thoroughly understood. In this paper, we present a methodology for a preliminary but critical design of composite bone plates with targets at plate rigidity and flexural strength, by varying reinforcing fiber structures and plate thicknesses. In this preliminary work, the influence of screw holes and curvature on the plate performance is neglected. However, the maximum load-carrying ability has been explored. Essentially three different fibrous preforms, i.e. UD prepregs, braided fabrics, and knitted fabrics, have been employed, which are arranged in UD, angle-plied, and multidirectional laminates, and multilayer braided and knitted fabric composite plates. The structure–property relationship of these composite plates is realized by using the micromechanics bridging model [21]. The bending rigidity and the maximum bending moment of a clinically used stainless steel plate are taken as reference basis in our design. The bending rigidities and ultimate bending strengths of various fibrous preform composite plates are plotted against their thickness, from which simple relationships between the bending properties of different composite structure plates and the plate thickness have been obtained. It has been shown that the composite bone plates with all the considered lamination patterns except for the knitted fabric reinforcement and having a thickness near to that of the stainless steel plate can exhibit required properties of bending rigidity and maximum bending moment. The plate thickness is more controlled by strength rather than by stiffness requirement. However, the 5 braided fabric composite bone plate shows an overall superiority as compared to the other lamination forms, and is thus recommended as the first choice in a new composite plate development.
2. Sample plate development 2.1. Material selection Possible toxic effects of non-reacted monomers and non-suitability to re-shape at the time of surgery make thermoset polymer based composites not suitable for bone plate application. In this work, we choose to use thermoplastic carbon/PEEK (poly-ether-ether-ketone) materials system, because both are well biocompatible
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materials. Composite bone plates will be developed using a commingled yarn method [19,20]. A microbraiding technique is used to generate the commingled yarns in which the reinforcement Toray carbon fibers (diameter = 7 lm, 1000 filaments, Toray Co., Ltd., Japan), comparable in properties to T300 carbon fibers, are uniformly arranged within PEEK matrix fibers (230 dTEX, 30 filaments, ZYEX Ltd., UK). These microbraiding yarns are then fabricated into different fibrous preforms, which are later fabricated into composite bone plates using a hot press machine. 2.2. Preform of micro-braiding yarn A challenge problem in using thermoplastic polymers as matrix materials is their high viscosity. However, due to a uniform combination of reinforcing and matrix fibers, the commingled yarn method can result in good impregnation of the matrix into the fibers once the yarn is hot pressed. In this work, a micro-braiding technique is applied to make such type of commingled yarns. An attractive feature of this yarn is that the reinforcing and the matrix fibers are easily mixed using a simple tubular braiding technique [19]. In a tubular braided fabric, fiber yarns can be inserted as a middle-end-fiber or axial fiber (Fig. 3(a)). Moreover, a desired yarn size can be flexibly designed by choosing suitable tow sizes of the reinforcing and matrix fiber yarns. In the sample fabrication, three carbon fiber yarns, all as middle-endfibers, were inserted in the tubular braided fabric of 10 PEEK fiber yarns, as shown in Fig. 3(b). The microbraiding yarn, serving as a commingled yarn, is then used to make flat fibrous preforms, such as a braid (Fig. 3(c)), for bone plate fabrication.
75
2.3. Fabrication of composite bone plate Required layers of the braided preforms were placed in a stainless steel mould (Fig. 4) which was then put into the hot press machine for melting and compressing treatments, with an averaged holding pressure of 4.5 MPa and a heating temperature of 380 C for 20 min. A detailed diagram showing the temperature versus time cycle is plotted in Fig. 5. The bone plate was achieved when the mould was cooled down to room temperature and removed, with an averaged fiber volume fraction of 53% (measured through a burning method) and a void content of less than 1%. A typical micrograph for the cross-section of the developed bone plate is shown in Fig. 6. Good impregnation of the PEEK matrix into carbon fiber braided fabrics has been recognized. As screws are necessary to fix the plate to human bone, the bone plate must be fabricated with screw
Fig. 4. Schematic drawing of (a) a molding and (b) fiber architecture of a flat braided fabric around a pin.
Fig. 3. (a) Schematic drawing of a tubular braided fabric with middle-end-fibers and an axial fiber, (b) photograph of a micro-braiding commingled yarn, and (c) photograph of a flat braided fabric using the commingled yarns.
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plate (DCP) of AO Institute, which is used widely in surgery, was also tested in order to make a comparison for the properties of the composite plates developed. Details can refer to [20].
3. Theoretical background
Fig. 5. Temperature vs time cycle in the fabrication of braided CF/ PEEK composite bone plate.
The design purpose is to obtain suitable mechanical specifically bending properties of various composite bone plates upon their different fibrous preforms and plate thicknesses. Based on these results, one can choose a suitable plate for the final development. In order to do so, we must understand the structure–property relationships of these composites. In this work, three kinds of fibrous preforms, i.e. UD fiber pregregs, braided fabrics, and knitted fabrics using the same carbon/PEEK materials system, are considered as candidate reinforcements. Relevant equations and modeling procedure for predicting the bending properties of the composite plates with these three kinds of reinforcements are outlined below. 3.1. Bending analysis of laminated plate
Fig. 6. A typical cross-section micrograph of the composite bone plate.
holes. It has been recognized that drilling a hole after the composite has been fabricated can reduce its load-carrying capacity to a remarkable extent due to the breakage of fiber continuity. On the other hand, if the hole is achieved before matrix impregnation and without significant damage of the fibers, the situation can be improved considerably [22]. In this regard, the fabrication mould had six inserted pins with its diameter equal to that of the screws (Fig. 4). The female mould consisted of three parts, which are connected through screw bolts. The pins were carefully penetrated through the fabrics, without breaking the continuity of yarns (see in Fig. 4(b)). 2.4. Mechanical characterization Bending stiffness and ultimate bending strength of the bone plate is the most critical mechanical property from the application viewpoint, and is generally evaluated based on the maximum bending moment (P/2 · L) and bending stiffness (moment/2h calculated from the initial linear moment against the bending angle at a support (see Fig. 7(c)). Static 4-point bending tests have been conducted for the composite bone plate samples, following ISO 9585 standard. The testing results indicated that the sample bone plates have sufficiently high bending properties [20]. A stainless-steel dynamic compression
Use of different coordinate systems is important for a laminate analysis. Let us choose (X, Y, Z), (x, y, z), and (x1, x2, x3) to represent the global, ply, and local systems, respectively (Fig. 7). The coordinates Z and z parallel to each other. The origin of Z is at the middle surface of the laminate. Suppose that the laminate consists of a number of laminas subjected to a bending load. Here, a lamina can be a UD preform, a single layer braided fabric reinforced composite, or a single layer knitted fabric reinforced composite as shown in Figs. 7(d.1), (d.2) and (d.3), respectively. For UD and braided fabric laminas, their global and ply coordinate systems are coincided each other. The analysis of the laminate is based on the classical lamination theory [21], with stress and strain increments, {dr}G = {drXX, drYY, drXY}T and {de}G = {deXX, deYY, 2deXY}T, correlated by [21] G
G
1
T
G
fdrgk ¼ ½ðC Gij Þk fdegk ¼ ð½T c Þk ð½Sk Þ ð½T c Þk fdegk ;
G
fdegk ¼
ð1:1Þ de0XX þ
þ
Z k þ Z k1 0 djXX ; de0YY : 2
Z k þ Z k1 0 djYY ; 2de0XY þ ðZ k þ Z k1 Þdj0XY 2
T ; ð1:2Þ
where Zk1 and Zk are the Z-coordinates of the bottom and top surfaces of the kth ply. [S]k is the instantaneous compliance matrix of the kth lamina in its ply system.
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77
Fig. 7. Analysis procedure of laminated beams with different fiber preform reinforcements.
[T]c is a coordinate transformation matrix between the ply and the global systems, whose detail can be found in [21]. Suppose that the overall applied bending moment increment on a unit length (width) of a beam cross-section is dMYY. For instance, on the central cross-section of the beam (Fig. 7(c)), we have dMYY = a dP/(2b), where a is the distance between a support and a nearer loading point and b is the beam width. The middle surface strain and curvature increments are solved from the following equations [21]: 8 0 9 9 2 I 8 II II 3 > dexx > 0 > Q11 QI12 QI16 QII > 11 Q12 Q16 > > > > 0 > > > > 6 QI QI QI QII QII QII 7> > > > deyy > > > > > 0 > > > > > 7 6 12 22 26 12 22 26 > > > > > 6 I > > 7> 0 = = < < I I II II II 6 Q16 Q26 Q66 Q16 Q26 Q66 7 2dexy 0 7 ¼6 6 QII QII QII QIII QIII QIII 7> dj0 >: > dM YY > > > 6 11 12 16 11 xx > 12 16 7> > > > > > > > > 6 II > > > II III III III 7 0 > > > > > 5 4 Q12 QII 0 dj Q Q Q Q > > > yy > 22 26 12 22 26 > > > > > ; > : ; : II II III III III 0 QII 2dj0xy 16 Q26 Q66 Q16 Q26 Q66 ð2Þ The in-plane modulus of the laminate in axial direction 2 is given by EXX ¼ ½1 ðQI12 Þ =ðQI11 QI22 ÞQI11 =h, where h is the beam thickness. The beam deflection increment, dw0 can be integrated from the equation o2 ðdw0 Þ ¼ dj0XX ; ð3Þ oX 2 together with boundary conditions dw0(0) = dw0(l) = 0, where l is the support span, and the total deflection is updated through w0 = w0 + dw0. With the increase of applied load, some lamina must fail first before others. Suppose that the k0th lamina ply has failed. Then, the overall stiffness elements of the remaining laminate are defined by
QIij ¼
N X
ðC Gij Þk ðzk zk1 Þ;
k¼1 k6¼k 0
QII ij ¼
N 1 X ðC Gij Þk ðz2k z2k1 Þ; 2 k¼1 k6¼k 0
QIII ij ¼
ð4Þ
N 1 X ðC Gij Þk ðz3k z3k1 Þ: 3 k¼1 k6¼k 0
In this way, a layer-by-layer failure analysis can be continued until the ultimate failure occurs. It should be pointed out that the ultimate failure of the laminate under a bending generally does not correspond to its last ply failure. Instead, it corresponds to an intermediate ply failure [23,24]. In order to completely determine this specific ply, we need an additional controlling parameter, i.e. a measured critical deflection or curvature of the beam at which the beam attains its maximum load [23,24]. For a preliminary design, however, some earlierply such as the first or the second-ply failure can be regarded the ultimate failure for the controlling parameter. The design thus obtained may be somewhat conservative. 3.2. Analysis of a UD lamina The purpose of a lamina analysis is to provide its compliance matrix [S]k required in Eq. (1.1) and to check its failure status. This will be accomplished by
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the bridging model [21], because this model can give an instantaneous rather than an initial constant compliance matrix. Moreover, the stress states in the fiber and matrix materials of the lamina at every load level can be explicitly determined with this model so that the lamina failure can be detected only against the constituent strengths. This latter feature is very useful because otherwise all the three kinds of laminas, UD, braided fabric, and knitted fabric reinforced composite laminas, have to be fabricated and tested. The lamina analysis is generally carried out in the local coordinate system (Fig. 8), with its local instantaneous compliance matrix given by [21] ½S ðLÞ ¼ ðV f ½S f þ V m ½S m ½AÞðV f ½I þ V m ½AÞ1 ¼ ðV f ½S f þ V m ½S m ½AÞ½B;
ð5Þ
where ‘‘L’’ refers to the local system, Vf and Vm are the fiber and matrix volume fractions, [Sf] and [Sm] are the instantaneous compliance matrices of the fiber and matrix materials, [I] is a unit matrix, [B] = (Vf[I] + Vm[A])1 and [A] is a bridging matrix. Elements of all these matrices can be found in [21]. The required compliance matrix in the ply system is obtained from Eq. (5) through ðLÞ
T
½S k ¼ ð½T s Þk ½S ð½T s Þk ;
ð6Þ
where [Ts] is another coordinate transformation matrix between the local and the ply systems [21]. The stress increments in the fiber and matrix materials denoted in the local system are given by fdrf g
ðLÞ
¼ ½Bfdrg
ðLÞ
ð7:1Þ
The total stresses in the fiber and matrix are simply updated through frf g
ðLÞ
frm g
ðLÞ
¼ frf g
ðLÞ
¼ frm g
ðLÞ
þ fdrf g ;
ðLÞ
ðLÞ
þ fdrm g :
ð8Þ
The next step is to check whether fiber or matrix has attained its failure status. If so, the lamina will be considered to have failed. Otherwise, an additional load increment can be applied. Evidently, the lamina failure (ultimate strength) should correspond to the constituent fracture (ultimate) failure (rather than, e.g. a yielding failure). Thus, the maximum normal stress failure criterion is best applicable. In other words, the lamina failure is attained as long as any of the following conditions: rf11 þ rf22 1 þ 2 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðrf11 rf22 Þ þ 4ðrf12 Þ P rfu ;
ð9:1Þ
rf11 þ rf22 1 2 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðrf11 rf22 Þ þ 4ðrf12 Þ 6 rfu;c ;
ð9:2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m 2 m 2 ðrm 11 r22 Þ þ 4ðr12 Þ P ru ;
ð9:3Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m 2 m 2 ðrm 11 r22 Þ þ 4ðr12 Þ 6 ru;c ;
ð9:4Þ
m rm 1 11 þ r22 þ 2 2 m rm 1 11 þ r22 2 2
m is met, where rfu ; rfu;c and rm u ; ru;c are the ultimate tensile and compressive strengths of the fiber and the matrix materials, respectively. For the fiber material, rfu and rfu;c should be obtained along its axial direction.
and fdrm g
ðLÞ
ðLÞ
¼ ½A½Bfdrg ;
ð7:2Þ
with fdrg
ðLÞ
¼
T G ½T s fdrg :
ð7:3Þ
Fig. 8. Analysis of a UD lamina: (a) the lamina taken from the laminate (Fig. 7(d.1)) and (b) the lamina analysis in the local coordinate system.
3.3. Analysis of a braided fabric lamina In this work, a diamond braid (Fig. 7(a.2)) is used as reinforcement. Through the previous laminate analysis, the stresses shared by a specific braid lamina can be determined (Fig. 7(d.2)). Let us now perform an analysis for the braid lamina. As before, there are three purposes for this analysis. One is to update the lamina instantaneous compliance matrix required in Eq. (1.1), another is to determine the internal stress increments in the fiber, and the last to determine those in the matrix. Essentially, the analysis for a complex fiber structure composite such as a braid lamina consists of three steps. In the first step, the lamina is discretized into a series of UD composites. The second step deals with the analysis of the UD composites as precedingly shown. An assemblage of all the UD composites is performed in the third step to accomplish the purposes of the lamina analysis. A schematic diagram to show the discretization of a diamond braid lamina into UD composites is given in Fig. 9. The whole fabric structure (Fig. 7(d.2) or
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79
Fig. 9. Schematic diagram to show analysis procedure for a braided fabric lamina.
Fig. 9(a)) can be constructed by repeating some unit cell (Fig. 9(b)), which can be further divided into four identical or symmetrical sub-cells. One such sub-cell together with its surrounding matrix is taken as a representative volume element (RVE) for the lamina. Thus, the analysis for the braided fabric lamina can be achieved through that for the RVE. The RVE is sub-divided in the fabric plane into sub-elements, as shown in Fig. 9(c). Each sub-element, Fig. 9(d), can have at most four material layers, i.e. the braider yarn 1, the braider yarn 2, and the top and bottom pure matrix layers. These material layers are considered as UD composites in their respective local coordinate systems (both the pure matrix layers can be regarded as a UD composite with a zero fiber volume fraction), as indicated in Figs. 9(e)–(g). The mechanical responses (the internal stresses in the fiber and matrix materials and the instantaneous compliance matrix of the composite) of these UD composites are determined by using the bridging model. Then, an assemblage of the three UD composites gives the responses of the sub-element, where the lamina (RVE) properties are obtained by assembling the contribution of all the sub-elements. In order to perform the subdivision and then assemblage, the yarn orientations in the fabric must be identified. An analytical model has been proposed to describe the diamond braid geometry using only parameters of the braiding angle, yarn width, yarn thickness, and inter-yarn gap. Formulation of this model and detailed analysis procedure for the braid lamina can be found in [25].
3.4. Analysis of a knitted fabric lamina A plain weft knit has been chosen as the knitting structure in this work (Fig. 7(a.3)). Through laminate analysis, the stress increments sustained by a knitted fabric lamina can be determined, which are shown in Fig. 7(d.3). Similarly, we must then perform a lamina analysis to achieve the three purposes. Schematic for the analysis of a knitted fabric lamina is shown in Fig. 10. The stress state of Fig. 7(d.3) (i.e. Fig. 10(a)) given in the global system is first transformed into the ply system (Fig. 10(b)). It is seen from Fig. 10(b) that the entire fabric can be constructed using a repeating unit cell shown in Fig. 10(c), which can be further divided into four symmetrical or identical sub-cells. One such sub-cell is taken as the RVE for the knitted fabric lamina, as indicated in Fig. 10(d). There are two yarns in the RVE, whose geometric positions are critical. Fortunately, for the present plain weft knitted fabric structure, the Leaf and Glaskin model [26] can be employed to identify the yarn orientations. Only three parameters, i.e. the wale number (W), the course number (C), and the yarn diameter (d), are required by the model to completely specify the fabric geometry [27,28]. The RVE in Fig. 10(d) is subdivided into even small elements along the x-direction. Owing to the small subdivision, the yarn segment together with its surrounding matrix can be regarded as a UD composite having an inclined angle with the x-direction (Fig. 10(e)). The response of the UD composite in its local coordinate system (Fig. 10(f)) is determined through
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Fig. 10. Schematic diagram to show analysis procedure for a knitted fabric lamina.
the bridging model. The RVE properties, i.e. the internal stresses in the constituent fiber and matrix materials and the instantaneous compliance/stiffness matrix of the RVE, can be obtained through assembling the contributions of all the sub-elements. More details can be found in [27,28].
No standard is presently available for the design of a composite bone plate. Thus, the mechanical behavior of the clinically used stainless steel plate can be taken as a reference. For the stainless steel bone plate that has 3.8 mm thickness (h) and 210 GPa modulus, its unit rigidity (the rigidity per unit width) is given by k ¼ ðEI=bÞ ¼ Eh3 =12 ¼ 960:3
4. Design philosophy 4.1. Design in rigidity From a mechanical point of view, there are two critical quantities that must be taken into account in the design of a composite bone plate. One is its rigidity. As mentioned before, a high rigidity bone plate can result in ‘‘stress shielding’’. On the other hand, a low rigidity plate may adequately appose two fracture fragments but allow motion in excess of that needed for fracture healing without callous formation. Thus, a proper choice of the bone plate rigidity is necessary. Rigidity is interpreted in different ways: axial, flexural, and torsional. For a bone plate, the flexural rigidity is the most important, which is proportional to the material stiffness and the cube of the plate thickness.
ðN mÞ;
ð10Þ
where b is the plate width and the effect of plate holes has been ignored for the sake of simplicity in the present preliminary design. The design is begun with the specification of constituent material properties. The T300 carbon fibers are transversely isotropic and linearly elastic until rupture, whose properties are taken from [29] and are summarised in Table 1. It is noted that the fiber elastic properties at tension and compression are considered as the same and are directly taken from [29]. However, the fiber ultimate strengths are back calculated upon the composite strengths provided in [29], see [24] for detail. Uniaxial tensile tests have been performed for the PEEK matrix material using bulk specimens. Two typical stress–strain curves are plotted in Fig. 11. The curves are then approximated using multiple linear segments, whose representative parameters are summarised in
Table 1 Mechanical properties of T300 carbon fibers Ef11 (GPa)
mf12
Ef22 (GPa)
Gf12 (GPa)
mf23
rfu (MPa)
rfu;c (MPa)
230
0.2
15
15
0.071
2468
1470
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81
Fig. 11. Measured stress–strain curve of PEEK material.
Table 2 Mechanical properties of PEEK matrix (mm = 0.38) i
Tensile properties 1
(rm Y )i (Em T )i
(MPa) (MPa) ðEm T Þi ,
50 3400 ðrm Y Þi1
Note: Em ¼ when a Ultimate strength value.
Compressive properties
2
3
80.5 2130 6
rm e
6
ðrm Y Þi ,
89.4 900 with
ðrm Y Þ0
4
1 a
91.5 270
50 3400
2 82.5 2130
3
4
92 900
a
270
¼ 0.
Table 2. It is noted that the matrix properties at compression have been adjusted according to a bending test. The composite bone plate must be fabricated from a laminated structure. Changing the laminate layers will vary the plate rigidity. Each layer is a fiber preform reinforced PEEK matrix lamina. Three different preforms, i.e. UD prepreg, braided fabric, and knitted fabric, are used as reinforcement. However, the unidirectional prepreg for a different layer can be arranged in a different direction. More over, two different braiding angles, 5 and 15, are used. The candidate composite bone plates are supposed to be fabricated essentially following the procedure as described in Section 2. Namely, the carbon and the PEEK fibers are commingled into yarns that are fabricated as the fiber preforms. These preforms are then made into bone plates using fabrication moulds similar to that shown in Fig. 4 under a hot press machine. In this preliminary design, following composite laminates are considered: unidirectional laminates ([0]ns), angle-plied laminates ([±20]ns), multidirectional laminates ([0/±20/0]ns), braided fabric (with a diamond braid pattern) laminates ([5]ns and [15]ns), and knitted (plain weft knit) fabric laminates, where the subscript n refers to the repeated number. The braided fabric laminates are assumed to have a volume fraction of 48% with each layer being 0.35 mm thick [19], whereas the knitted fabric laminates have a volume fraction of 40% with
each layer being 0.2 mm thick. The other laminates constructed from the unidirectional prepregs are considered to have a fiber volume fraction of 50%, with each layer being 0.17 mm thick. Furthermore, the yarn width is 0.791 mm in the 5 braid and 1.02 mm in the 15 braid. The yarn thickness is the same, equal to 0.35/2 = 0.175 mm, in both the braids. The geometric parameters for the knitted fabrics are: W(wale number) = 1.5(loops/ cm), C(course number) = 3.5(loops/cm), and d(yarn diameter) = 0.404 mm. The preliminary design results for the rigidity of the composite bone plates are shown in Fig. 12. The rigidity of the stainless steel plate, given by Eq. (10), is also shown in the figure. It is seen that to achieve a rigidity comparable to that of the stainless steel plate, the composite plates must be significantly thicker. However, as mentioned before, the rigidity of the stainless steel plate is much more than necessity. In fact, a research carried out by Bradley et al. indicated that a composite bone plate with a stiffness in 10–25% of that of the stainless steel plate could be applicable [10]. In this regard, the required thickness can be significantly reduced into around the stainless steel plate thickness. Although an evaluation (see the previous section) for the rigidity of a composite laminate plate is much more complicated than for that of an isotropic plate (beam), a simple relation similar to Eq. (10) exists, which can sufficiently well interpret the response of the laminated
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Fig. 12. Unit rigidity of laminated plates varied with plate thickness.
beam. Namely, the unit rigidity of the laminated beam can be well approximately expressed as k ¼ kðh=h30 Þ; where h0 is any but fixed thickness at which the laminate rigidity is known. Choosing a standard thickness (e.g. h0 = 1), the unit rigidity of the laminated beam can be rewritten as k ¼ jh3 :
ð11Þ
From the responses shown in Fig. 12, the rigidity parameters j of various laminates are summarized in Table 3. In light of BradleyÕ research and the data of Table 3, it might be concluded that except for the knitted fabric reinforcement the carbon/PEEK composite plates reinforced with all the other considered fiber preforms and with the same as the stainless steel plate thickness would be rigidity suitable for applications. 4.2. Design in strength The second important quantity of the bone plate is its ultimate bending strength, or, equivalently, its maximum bending moment. Similarly as done in the above, we can make a preliminary design for the unit maximum bending moments (the moment per unit width) of the laminated bone plates. Taking the ultimate bending strength of the stainless steel as rb = 760 MPa, its unit maximum bending moment is given by
m ¼ rb ðh2 =6Þ ¼ 1829:1 ðNÞ:
ð12Þ
As understood from Section 3.1, to obtain the ultimate bending strength of a composite plate, we need an additional governing parameter, the plate critical deflection or curvature. However, the exact value of this parameter is not available since the plate is still under design and has not been developed yet. Thus, we have to assume such additional parameter in priori for the strength design purpose. In [24,30] where the braided and knitted T300 carbon fiber reinforced epoxy composite laminates under bending were theoretically and experimentally analysed, respectively. The braded fabric laminates had eight whereas the knitted fabric laminates had six layers of laminas. Those laminates were recognised to mostly assume their ultimate failure at the third-ply failure. Considering that the present composite bone plates may most probably have more than eight lamina layers, we assume that the failure moment corresponding to the third-ply failure is defined as the ultimate bending moment of the designed composite plates. Then, the unit maximum bending moments of the laminated plates under consideration varied with the plate thicknesses can be estimated, and are plotted in Fig. 13. The figure demonstrates that to achieve the same amount of the maximum bending moment as the stainless steel plate, the thicknesses of the composite plates with all the other fibrous structure reinforcements except for the knitted fabric need to be marginally thicker. Even so, the result-
Table 3 Rigidity parameters of various T300/PEEK laminates Material 3
j (N m/mm ) a b c
Stainless steel
UDa [0]ns
MDb [0/±20/0]ns
Braid [5]ns
AP c [±20]ns
Braid [15]ns
Knitted laminate
17.5
4.175
3.406
3.304
2.738
2.359
1.148
Unidirectional. Multidirectional. Angle plied.
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Fig. 13. Maximum unit bending moments of various laminate plates varied with plate thickness.
ing composite plate will be much lighter in weight than the stainless steel plate. This is because the carbon/ PEEK composite material has a mass density less than 2 g/cm3 (as a consequence of 1.79 g/cm3 density for the carbon fibers and 1.49 g/cm3 density for the PEEK matrix), much smaller than 7.8 g/cm3, the stainless steel plate density. Considering the comment on the bending rigidity design given above, we clearly understand that as long as the composite plate thickness is chosen in such a way that the plate fulfils the ultimate bending moment requirement it will also satisfy the bending stiffness requirement. In other words, the ultimate bending moment is a key parameter to control the composite bone plate design. Furthermore, from Fig. 13, a relation similar to Eq. (12) is obtained as m ¼ gh2 :
ð13Þ
The moment parameters g of the various laminates are listed in Table 4. 4.3. Discussion Figs. 12 and 13 show that we can always choose a suitable plate thickness to make the bending moment and rigidity of any composite bone plate comparable
to those of the stainless steel plate. On the other hand, when the strength and rigidity are sufficient and without considering other designing issues, the thinner the thickness of a bone plate the better it would be to the patient. In this regard, both the figures together with Tables 3 and 4 demonstrate that the unidirectional laminate plate displays the maximum bending rigidity and bending strength among all the candidate composite plates of a fixed plate thickness. However, a significant limitation exists for the choice of the unidirectional laminate bone plate. Namely, it is very poor in sustaining the load component transverse to the fiber direction. As the bone plate generally possesses a curvature in its cross-section, the stresses near the plate holes transmitted by screws can easily cause the unidirectional laminate plate to split along the axial direction (Fig. 14). In the bending test of such a plate specimen with holes, a splitting load has been found to be much lower than that given in Fig. 13. Due to this reason, an angle-plied or multidirectional tape laminate would be necessary. Nevertheless, more fabrication defects using the angle-plied or multidirectional tape laminate than using that shown in Fig. 4 may result. For instance, the plate holes may have to be drilled if the unidirectional prepregs are used in the fabrication. These drilled holes will break down the continuity of some fibers and hence will cause a reduction in
Table 4 Moment parameters of various T300/PEEK laminates Material 2
g (N/mm ) a b c
Stainless steel
UDa [0]ns
MDb [0/±20/0]ns
Braid [5]ns
APc [±20]ns
Braid [15]ns
Knitted laminate
126.46
124.76
106.78
110.54
110.73
108.29
48.52
Unidirectional. Multidirectional. Angle plied.
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Fig. 14. Schematic diagram of a bending load applied to a bone plate with a curvature. There is a moment component acting in the transverse direction of the plate so that a unidirectional laminate plate can be easily splitted.
the plate load-carrying ability. Considering that the bending rigidity and bending strength of a braided fabric laminate plate are comparable to those of the multidirectional/angle-plied laminate plate (Figs. 12 and 13) and that the plate holes can be easily generated during the fabrication of the braid laminate plate (see Fig. 4), the braided fabric preforms (e.g. with a 5 braiding angle) is recommended as the first candidate for the development of a new composite bone plate. It is worth mentioning that the designed properties (i.e. stiffness and strength) of the composite bone plates also depend on their constituent fiber and matrix behaviors. Different fiber and matrix properties will vary accordingly the designed bending modulus and ultimate bending strength. This means that a scatter in the measured fiber and/or matrix properties (e.g. that shown in Fig. 11) will generally result in a scatter in the designed unit rigidity and maximum unit bending moment for the composite. The relationship, however, between the constituent properties and the overall properties of the composite bone plate is more complicated. As understood, an epoxy matrix modulus has negligible influence on the longitudinal modulus of a carbon fiber composite but predominant influence on its transversely modulus. Thus, a scatter in matrix modulus will have a negligibly influence on the bending stiffness of a small angle braided composite bone plate. But in other cases, usually we will have to perform the design procedure as described above to understand exactly the influence amount.
5. Conclusion A theoretical designing procedure is presented in this paper for an optimal design of composite bone plates reinforced with various fiber structures and varied with plate thicknesses. Not only the plate rigidity but also its ultimate bending moment has been targeted in the design. For the same carbon/PEEK materials system, the numerical experiments indicated that ideally the composite bone plates made of unidirectional, angle-plied, multidirectional, and braided fabric laminates and hav-
ing a thickness comparable to that of a clinically used stainless steel plate may all satisfy the stiffness and strength requirements for applications. The unidirectional and the braided fabric laminate plates exhibit better mechanical performances. Considering a possible reduction in the load-carrying abilities of the unidirectional plate due to a possible splitting along the plate axial direction and a breakage of fiber continuity at screw holes, the composite bone plate made of a small angle braided fabric laminas shows overall the best advantage in terms of mechanical properties. Future researches should incorporate the influence of screw holes and plate curvature on the mechanical behavior of a composite bone plate.
References [1] Thakur AJ. The elements of fracture fixation. New York: Churchill Livingstone; 1997. [2] Burwell HN, Charnley AD. Treatment of forearm fractures in adults with particular reference to plate fixation. J Bone Joint Surg 1964;46-B(3):404–25. [3] Dodge S, Cady GW. Treatment of fractures of the radius and ulna with compression tests. J Bone Joint Surg 1970;54A(6):1167–76. [4] Anderson LD, Sisk TD, Tooms RE, Park WI. Compression-plate fixation in acute diapheseal fractures of the radius and ulna. J Bone Joint Surg 1975;57-A(3):287–96. [5] Tonino AJ, Davidson CL, Klopper PJ, Linclau LA. Protection from stress in bone and its effect. J Bone Joint Surg 1976;58B(1):107–13. [6] Schwyzer HK, Cordey J, Brum S, Matter P, Perren SM. Bone less after internal fixation using plates-determination in humans using computed topography. In: Perren SM, Schneider E, editors. Biomechanics; 1984. p. 191–5. [7] Hidaka S, Gustilo RB. Refracture of bones of the forearm after plate removal. J Bone Joint Surg 1984;64-A(8):1241–3. [8] Deluca PA, Lindsey RW, Ruwe PA. Refracture of bones of the forearm after the removal of compression plates. J Bone Joint Surg 1988;70-A(9):1372–6. [9] Woo SL-Y, Akeson WH, Levenetz B, Coutts RD, Matthews JV, Amiel D. Potential application of graphite fiber and methyl methacrylate resin composites as internal fixation plates. J Biomed Mater Res 1974;8:321–8. [10] Bradley GW, Mckenna GB, Dunn HK, Daniels AU, Statton WO. Effects of flexural rigidity of plates on bone healing. J Bone Joint Surg 1979;61A(6):866–72.
Z.-M. Huang, K. Fujihara / Composites Science and Technology 65 (2005) 73–85 [11] Bradley JS, Hastings GW, Johnson-Nurse C. Carbon fibre reinforced epoxy as a high strength, low modulus material for internal fixation plates. Biomaterials 1980;1:38–40. [12] Mckenna GB, Bradley GW, Dunn HK, Statton WO. Mechanical properties of some fibre reinforced polymer composites after implantation as fracture fixation plates. Biomaterials 1980;1:189–92. [13] Williams DF, Gore LF, Clark GCF. Quantitative microradiography of cortical bone in disuse osteoporosis following fracture fixation. Biomaterials 1983;4:285–8. [14] Tayton K, Bradley JS. How stiff should semi-rigid fixation of the human tibia be?. J Bone Joint Surg 1983;65-B:312–5. [15] Zimmerman M, Parsons JR, Alexander H. The design and analysis of a laminated partially degradable composite bone plate for fracture fixation. J Biomed Mater Res 1987;21(3):345–61. [16] Skirving AP, MacDonald W, McLaren R. Carbon fiber reinforced plastic (CFRP) plates versus stainless steel dynamic compression plates in the treatment of fractures of the tibiae in dogs. Clin Orthop Relat Res 1987;224:117–24. [17] Ali MS, French TA, Hastings GW, Rae T, Rushton N, Ross ERS, et al. Carbon fibre composite bone plates. J Bone Joint Surg 1990;72B(4):586–91. [18] Gillett N, Brown SA, Dumbleton JH, Pool RP. The use of short carbon fibre reinforced thermoplastic plates for fracture fixation. Biomaterials 1985;6:113–21. [19] Fujihara K, Huang ZM, Ramakrishna S, Satkunanantham K, Hamada H. Development of braided carbon/PEEK composite bone plates. Adv Compos Lett 2000;110(1):13–20. [20] Fujihara K, Huang ZM, Ramakrishna S, Satknanantham K, Hamada H. Performance study of braided carbon/PEEK composite compression bone plates. Biomaterials 2003;24:2661–7.
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[21] Huang ZM. Simulation of the mechanical properties of fibrous composites by the bridging micromechanics model. Composites A 2001;32(2):143–72. [22] Maekawa Z, Hamada H, Yokoyama A, Ueda S. Tensile behavior of braided flat bar with a circular hole. J Jpn Soc Compos Mater 1988;14(3):116–23. [23] Huang ZM, Teng XC, Ramakrishna S. Bending behavior of laminated knitted fabric reinforced beams. Adv Compos Lett 2001;10(5):211–8. [24] Huang ZM, Fujihara K, Ramakrishna S. Flexural failure behavior of laminated composites reinforced with braided fabrics. AIAA J 2002;40(7):1415–20. [25] Huang ZM. The mechanical properties of composites reinforced with woven and braided fabrics. Compos Sci Technol 2000;60(4):479–98. [26] Leaf GAV, Glaskin A. The geometry of a plain knitted loop. J Textile Inst 1955;45:T587–605. [27] Huang ZM, Ramakrishna S, Tay AAO. A micromechanical approach to the tensile strength of a knitted fabric composite. J Compos Mater 1999;33(19):1758–91. [28] Huang ZM, Ramakrishna S. Micromechanical modelling approaches for the stiffness and strength of knitted fabric composites: a review & comparative study. Composites A 2000;31(5):479–501. [29] Soden PD, Hinton MJ, Kaddour AS. Lamina properties, lay-up configurations and loading conditions for a range of fiberreinforced composite laminates. Compos Sci Technol 1998;58:1011–22. [30] Huang ZM, Teng XC, Ramakrishna. Progressive failure analysis of laminated knitted fabric composites under bending. J Thermoplast Compos Mater 2001;14(6):499–522.
Stiffness and strength design of composite bone plates Zheng-Ming Huang
a,*
, K. Fujihara
b
a
b
School of Aeronautics, Astronautics & Mechanics, Tongji University, 1239 Siping Road, Shanghai 200092, PR China Division of Bioengineering, Biomaterials Laboratory, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 22 January 2004; received in revised form 14 June 2004; accepted 14 June 2004 Available online 19 August 2004
Abstract Although the potential of composites used for bone plate implantation has been realized for a long period of time, not much literature has addressed optimal design of composite bone plates upon variation in reinforcing fiber structures, plate thicknesses, and so on. In this paper, we present a preliminary but critical design procedure for composite bone plate with target on both its stiffness and its ultimate strength. While the material system used is the same, i.e. carbon fibers and PEEK matrix, three kinds of fibrous preforms namely unidirectional (UD) prepregs, braided fabrics, and knitted fabrics are considered as reinforcements, constituting six different candidate laminated plates. They are UD [0]ns, angle plied [±20]ns, multidirectional [0/±20/0]ns, two braiding angle diamond braid fabrics [5]ns and [15]ns, and plain weft knitted fabric laminates. By varying the number of layers (n) in the lamination, different plate thicknesses can be attained. The structure–property relationship of the composite plates is realized through the use of a micromechanics bridging model. A clinically used stainless steel plate is taken as a design benchmark. It has been found that the composite bone plates with all the considered lamination patterns except for the knitted fabric reinforcement and having a thickness near to that of the stainless steel plate can exhibit required properties of bending rigidity and maximum bending moment. The plate thickness is more controlled by strength rather than by stiffness requirement. However, the 5 braided fabric composite bone plate shows an overall superiority as compared to the other lamination forms, and is thus recommended as the first choice. 2004 Elsevier Ltd. All rights reserved. Keywords: Bone plate; B. Strength
1. Introduction In the surgery field, patients with diaphyseal fractures are generally treated using compression bone plates made of stainless-steel, Cr–Co and Ti alloys [1–4] as shown in Fig. 1. Although exhibiting reasonable fatigue strength, the stiffness of these metals (in between 110 and 220 GPa) is much higher than that of human cortical bone (around 20 GPa). As a result, the majority of
*
Corresponding author. Tel.: +86-216-598-5373; fax: +86-216-5982914. E-mail address: [email protected] (Z.-M. Huang). 0266-3538/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2004.06.006
the load is carried by the plate rather than by the underlying bone. Callus formation, ossification, and bone union at fractured part are refrained after the implant operation, and the whole bone structure, not only at the fractured part, becomes osteoporosis [5,6]. The bone mass can be decreased by 20% [6] and in some cases the bone re-fracture due to stress concentration around the bone screws can be induced after the removal of the plate [7,8]. These phenomena are widely recognized as ‘‘stress shielding’’ effect, which is a main drawback for the use of metal bone plates. Another drawback is that metal plates can generate considerable artifacts (Fig. 2) under an X-ray, which make the interpretation of radiographs difficult.
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Fig. 1. Schematic of a compression bone plate with a tension device. A compression plate is attached on tensile side of the fragmented bone. Tension device is pulling a compression plate and compression force is accordingly generated at the damaged bones [1].
Fig. 2. Comparison of artifacts between metal and composite under X-ray.
In order to tackle aforementioned two drawbacks, polymer based composite materials, which have less stiffness, high fatigue strength, and good radiolucency, have been proposed for bone plate applications. Most composite plates developed so far have used UD (unidirectional) laminates [9–17] and discontinuous short fibers [18] as reinforcement. Recently, textile preforms such as braided structures have also been adopted in bone plate development [19,20]. As we know, composites are inherently anisotropic. Many parameters can influence the mechanical performance of a resulting composite bone plate such as constituent materials, their contents, reinforcing structure, plate thickness, etc. By varying all or some of them, an optimal composite bone plate can be expected. However, the majority of the existing composite bone plates has been essentially developed based on some trial and error method as a consequence of no relevant standard available at the present. Limited reports have been found in the open literature addressing actually optimal designs for composite bone plates upon varying the aforementioned parameters especially the fiber reinforcing structures. As mentioned before, too high stiffness of a bone plate will result in ‘‘stress shielding’’ effect, which is of an obstacle to the bone healing. On the other hand, too low stiffness may give rise to micro-movements of the fractured bone segments beneath the plate, and hence
also hinders the bone healing. This means that a proper amount of stiffness is necessary to be targeted by a plate design. Moreover, bone plate is a primary load-carrying element. It works in a very severe environment of human body, loaded with significant cycling forces. Thus, the plate load-carrying ability should be targeted as high as possible whereas the plate weight (or the volume) should be reduced to minimum to facilitate patientsÕ carriage. These targets can be fully achieved only when the microstructure–property relationship of the composite bone plate is thoroughly understood. In this paper, we present a methodology for a preliminary but critical design of composite bone plates with targets at plate rigidity and flexural strength, by varying reinforcing fiber structures and plate thicknesses. In this preliminary work, the influence of screw holes and curvature on the plate performance is neglected. However, the maximum load-carrying ability has been explored. Essentially three different fibrous preforms, i.e. UD prepregs, braided fabrics, and knitted fabrics, have been employed, which are arranged in UD, angle-plied, and multidirectional laminates, and multilayer braided and knitted fabric composite plates. The structure–property relationship of these composite plates is realized by using the micromechanics bridging model [21]. The bending rigidity and the maximum bending moment of a clinically used stainless steel plate are taken as reference basis in our design. The bending rigidities and ultimate bending strengths of various fibrous preform composite plates are plotted against their thickness, from which simple relationships between the bending properties of different composite structure plates and the plate thickness have been obtained. It has been shown that the composite bone plates with all the considered lamination patterns except for the knitted fabric reinforcement and having a thickness near to that of the stainless steel plate can exhibit required properties of bending rigidity and maximum bending moment. The plate thickness is more controlled by strength rather than by stiffness requirement. However, the 5 braided fabric composite bone plate shows an overall superiority as compared to the other lamination forms, and is thus recommended as the first choice in a new composite plate development.
2. Sample plate development 2.1. Material selection Possible toxic effects of non-reacted monomers and non-suitability to re-shape at the time of surgery make thermoset polymer based composites not suitable for bone plate application. In this work, we choose to use thermoplastic carbon/PEEK (poly-ether-ether-ketone) materials system, because both are well biocompatible
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materials. Composite bone plates will be developed using a commingled yarn method [19,20]. A microbraiding technique is used to generate the commingled yarns in which the reinforcement Toray carbon fibers (diameter = 7 lm, 1000 filaments, Toray Co., Ltd., Japan), comparable in properties to T300 carbon fibers, are uniformly arranged within PEEK matrix fibers (230 dTEX, 30 filaments, ZYEX Ltd., UK). These microbraiding yarns are then fabricated into different fibrous preforms, which are later fabricated into composite bone plates using a hot press machine. 2.2. Preform of micro-braiding yarn A challenge problem in using thermoplastic polymers as matrix materials is their high viscosity. However, due to a uniform combination of reinforcing and matrix fibers, the commingled yarn method can result in good impregnation of the matrix into the fibers once the yarn is hot pressed. In this work, a micro-braiding technique is applied to make such type of commingled yarns. An attractive feature of this yarn is that the reinforcing and the matrix fibers are easily mixed using a simple tubular braiding technique [19]. In a tubular braided fabric, fiber yarns can be inserted as a middle-end-fiber or axial fiber (Fig. 3(a)). Moreover, a desired yarn size can be flexibly designed by choosing suitable tow sizes of the reinforcing and matrix fiber yarns. In the sample fabrication, three carbon fiber yarns, all as middle-endfibers, were inserted in the tubular braided fabric of 10 PEEK fiber yarns, as shown in Fig. 3(b). The microbraiding yarn, serving as a commingled yarn, is then used to make flat fibrous preforms, such as a braid (Fig. 3(c)), for bone plate fabrication.
75
2.3. Fabrication of composite bone plate Required layers of the braided preforms were placed in a stainless steel mould (Fig. 4) which was then put into the hot press machine for melting and compressing treatments, with an averaged holding pressure of 4.5 MPa and a heating temperature of 380 C for 20 min. A detailed diagram showing the temperature versus time cycle is plotted in Fig. 5. The bone plate was achieved when the mould was cooled down to room temperature and removed, with an averaged fiber volume fraction of 53% (measured through a burning method) and a void content of less than 1%. A typical micrograph for the cross-section of the developed bone plate is shown in Fig. 6. Good impregnation of the PEEK matrix into carbon fiber braided fabrics has been recognized. As screws are necessary to fix the plate to human bone, the bone plate must be fabricated with screw
Fig. 4. Schematic drawing of (a) a molding and (b) fiber architecture of a flat braided fabric around a pin.
Fig. 3. (a) Schematic drawing of a tubular braided fabric with middle-end-fibers and an axial fiber, (b) photograph of a micro-braiding commingled yarn, and (c) photograph of a flat braided fabric using the commingled yarns.
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plate (DCP) of AO Institute, which is used widely in surgery, was also tested in order to make a comparison for the properties of the composite plates developed. Details can refer to [20].
3. Theoretical background
Fig. 5. Temperature vs time cycle in the fabrication of braided CF/ PEEK composite bone plate.
The design purpose is to obtain suitable mechanical specifically bending properties of various composite bone plates upon their different fibrous preforms and plate thicknesses. Based on these results, one can choose a suitable plate for the final development. In order to do so, we must understand the structure–property relationships of these composites. In this work, three kinds of fibrous preforms, i.e. UD fiber pregregs, braided fabrics, and knitted fabrics using the same carbon/PEEK materials system, are considered as candidate reinforcements. Relevant equations and modeling procedure for predicting the bending properties of the composite plates with these three kinds of reinforcements are outlined below. 3.1. Bending analysis of laminated plate
Fig. 6. A typical cross-section micrograph of the composite bone plate.
holes. It has been recognized that drilling a hole after the composite has been fabricated can reduce its load-carrying capacity to a remarkable extent due to the breakage of fiber continuity. On the other hand, if the hole is achieved before matrix impregnation and without significant damage of the fibers, the situation can be improved considerably [22]. In this regard, the fabrication mould had six inserted pins with its diameter equal to that of the screws (Fig. 4). The female mould consisted of three parts, which are connected through screw bolts. The pins were carefully penetrated through the fabrics, without breaking the continuity of yarns (see in Fig. 4(b)). 2.4. Mechanical characterization Bending stiffness and ultimate bending strength of the bone plate is the most critical mechanical property from the application viewpoint, and is generally evaluated based on the maximum bending moment (P/2 · L) and bending stiffness (moment/2h calculated from the initial linear moment against the bending angle at a support (see Fig. 7(c)). Static 4-point bending tests have been conducted for the composite bone plate samples, following ISO 9585 standard. The testing results indicated that the sample bone plates have sufficiently high bending properties [20]. A stainless-steel dynamic compression
Use of different coordinate systems is important for a laminate analysis. Let us choose (X, Y, Z), (x, y, z), and (x1, x2, x3) to represent the global, ply, and local systems, respectively (Fig. 7). The coordinates Z and z parallel to each other. The origin of Z is at the middle surface of the laminate. Suppose that the laminate consists of a number of laminas subjected to a bending load. Here, a lamina can be a UD preform, a single layer braided fabric reinforced composite, or a single layer knitted fabric reinforced composite as shown in Figs. 7(d.1), (d.2) and (d.3), respectively. For UD and braided fabric laminas, their global and ply coordinate systems are coincided each other. The analysis of the laminate is based on the classical lamination theory [21], with stress and strain increments, {dr}G = {drXX, drYY, drXY}T and {de}G = {deXX, deYY, 2deXY}T, correlated by [21] G
G
1
T
G
fdrgk ¼ ½ðC Gij Þk fdegk ¼ ð½T c Þk ð½Sk Þ ð½T c Þk fdegk ;
G
fdegk ¼
ð1:1Þ de0XX þ
þ
Z k þ Z k1 0 djXX ; de0YY : 2
Z k þ Z k1 0 djYY ; 2de0XY þ ðZ k þ Z k1 Þdj0XY 2
T ; ð1:2Þ
where Zk1 and Zk are the Z-coordinates of the bottom and top surfaces of the kth ply. [S]k is the instantaneous compliance matrix of the kth lamina in its ply system.
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77
Fig. 7. Analysis procedure of laminated beams with different fiber preform reinforcements.
[T]c is a coordinate transformation matrix between the ply and the global systems, whose detail can be found in [21]. Suppose that the overall applied bending moment increment on a unit length (width) of a beam cross-section is dMYY. For instance, on the central cross-section of the beam (Fig. 7(c)), we have dMYY = a dP/(2b), where a is the distance between a support and a nearer loading point and b is the beam width. The middle surface strain and curvature increments are solved from the following equations [21]: 8 0 9 9 2 I 8 II II 3 > dexx > 0 > Q11 QI12 QI16 QII > 11 Q12 Q16 > > > > 0 > > > > 6 QI QI QI QII QII QII 7> > > > deyy > > > > > 0 > > > > > 7 6 12 22 26 12 22 26 > > > > > 6 I > > 7> 0 = = < < I I II II II 6 Q16 Q26 Q66 Q16 Q26 Q66 7 2dexy 0 7 ¼6 6 QII QII QII QIII QIII QIII 7> dj0 >: > dM YY > > > 6 11 12 16 11 xx > 12 16 7> > > > > > > > > 6 II > > > II III III III 7 0 > > > > > 5 4 Q12 QII 0 dj Q Q Q Q > > > yy > 22 26 12 22 26 > > > > > ; > : ; : II II III III III 0 QII 2dj0xy 16 Q26 Q66 Q16 Q26 Q66 ð2Þ The in-plane modulus of the laminate in axial direction 2 is given by EXX ¼ ½1 ðQI12 Þ =ðQI11 QI22 ÞQI11 =h, where h is the beam thickness. The beam deflection increment, dw0 can be integrated from the equation o2 ðdw0 Þ ¼ dj0XX ; ð3Þ oX 2 together with boundary conditions dw0(0) = dw0(l) = 0, where l is the support span, and the total deflection is updated through w0 = w0 + dw0. With the increase of applied load, some lamina must fail first before others. Suppose that the k0th lamina ply has failed. Then, the overall stiffness elements of the remaining laminate are defined by
QIij ¼
N X
ðC Gij Þk ðzk zk1 Þ;
k¼1 k6¼k 0
QII ij ¼
N 1 X ðC Gij Þk ðz2k z2k1 Þ; 2 k¼1 k6¼k 0
QIII ij ¼
ð4Þ
N 1 X ðC Gij Þk ðz3k z3k1 Þ: 3 k¼1 k6¼k 0
In this way, a layer-by-layer failure analysis can be continued until the ultimate failure occurs. It should be pointed out that the ultimate failure of the laminate under a bending generally does not correspond to its last ply failure. Instead, it corresponds to an intermediate ply failure [23,24]. In order to completely determine this specific ply, we need an additional controlling parameter, i.e. a measured critical deflection or curvature of the beam at which the beam attains its maximum load [23,24]. For a preliminary design, however, some earlierply such as the first or the second-ply failure can be regarded the ultimate failure for the controlling parameter. The design thus obtained may be somewhat conservative. 3.2. Analysis of a UD lamina The purpose of a lamina analysis is to provide its compliance matrix [S]k required in Eq. (1.1) and to check its failure status. This will be accomplished by
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the bridging model [21], because this model can give an instantaneous rather than an initial constant compliance matrix. Moreover, the stress states in the fiber and matrix materials of the lamina at every load level can be explicitly determined with this model so that the lamina failure can be detected only against the constituent strengths. This latter feature is very useful because otherwise all the three kinds of laminas, UD, braided fabric, and knitted fabric reinforced composite laminas, have to be fabricated and tested. The lamina analysis is generally carried out in the local coordinate system (Fig. 8), with its local instantaneous compliance matrix given by [21] ½S ðLÞ ¼ ðV f ½S f þ V m ½S m ½AÞðV f ½I þ V m ½AÞ1 ¼ ðV f ½S f þ V m ½S m ½AÞ½B;
ð5Þ
where ‘‘L’’ refers to the local system, Vf and Vm are the fiber and matrix volume fractions, [Sf] and [Sm] are the instantaneous compliance matrices of the fiber and matrix materials, [I] is a unit matrix, [B] = (Vf[I] + Vm[A])1 and [A] is a bridging matrix. Elements of all these matrices can be found in [21]. The required compliance matrix in the ply system is obtained from Eq. (5) through ðLÞ
T
½S k ¼ ð½T s Þk ½S ð½T s Þk ;
ð6Þ
where [Ts] is another coordinate transformation matrix between the local and the ply systems [21]. The stress increments in the fiber and matrix materials denoted in the local system are given by fdrf g
ðLÞ
¼ ½Bfdrg
ðLÞ
ð7:1Þ
The total stresses in the fiber and matrix are simply updated through frf g
ðLÞ
frm g
ðLÞ
¼ frf g
ðLÞ
¼ frm g
ðLÞ
þ fdrf g ;
ðLÞ
ðLÞ
þ fdrm g :
ð8Þ
The next step is to check whether fiber or matrix has attained its failure status. If so, the lamina will be considered to have failed. Otherwise, an additional load increment can be applied. Evidently, the lamina failure (ultimate strength) should correspond to the constituent fracture (ultimate) failure (rather than, e.g. a yielding failure). Thus, the maximum normal stress failure criterion is best applicable. In other words, the lamina failure is attained as long as any of the following conditions: rf11 þ rf22 1 þ 2 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðrf11 rf22 Þ þ 4ðrf12 Þ P rfu ;
ð9:1Þ
rf11 þ rf22 1 2 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðrf11 rf22 Þ þ 4ðrf12 Þ 6 rfu;c ;
ð9:2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m 2 m 2 ðrm 11 r22 Þ þ 4ðr12 Þ P ru ;
ð9:3Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m 2 m 2 ðrm 11 r22 Þ þ 4ðr12 Þ 6 ru;c ;
ð9:4Þ
m rm 1 11 þ r22 þ 2 2 m rm 1 11 þ r22 2 2
m is met, where rfu ; rfu;c and rm u ; ru;c are the ultimate tensile and compressive strengths of the fiber and the matrix materials, respectively. For the fiber material, rfu and rfu;c should be obtained along its axial direction.
and fdrm g
ðLÞ
ðLÞ
¼ ½A½Bfdrg ;
ð7:2Þ
with fdrg
ðLÞ
¼
T G ½T s fdrg :
ð7:3Þ
Fig. 8. Analysis of a UD lamina: (a) the lamina taken from the laminate (Fig. 7(d.1)) and (b) the lamina analysis in the local coordinate system.
3.3. Analysis of a braided fabric lamina In this work, a diamond braid (Fig. 7(a.2)) is used as reinforcement. Through the previous laminate analysis, the stresses shared by a specific braid lamina can be determined (Fig. 7(d.2)). Let us now perform an analysis for the braid lamina. As before, there are three purposes for this analysis. One is to update the lamina instantaneous compliance matrix required in Eq. (1.1), another is to determine the internal stress increments in the fiber, and the last to determine those in the matrix. Essentially, the analysis for a complex fiber structure composite such as a braid lamina consists of three steps. In the first step, the lamina is discretized into a series of UD composites. The second step deals with the analysis of the UD composites as precedingly shown. An assemblage of all the UD composites is performed in the third step to accomplish the purposes of the lamina analysis. A schematic diagram to show the discretization of a diamond braid lamina into UD composites is given in Fig. 9. The whole fabric structure (Fig. 7(d.2) or
Z.-M. Huang, K. Fujihara / Composites Science and Technology 65 (2005) 73–85
79
Fig. 9. Schematic diagram to show analysis procedure for a braided fabric lamina.
Fig. 9(a)) can be constructed by repeating some unit cell (Fig. 9(b)), which can be further divided into four identical or symmetrical sub-cells. One such sub-cell together with its surrounding matrix is taken as a representative volume element (RVE) for the lamina. Thus, the analysis for the braided fabric lamina can be achieved through that for the RVE. The RVE is sub-divided in the fabric plane into sub-elements, as shown in Fig. 9(c). Each sub-element, Fig. 9(d), can have at most four material layers, i.e. the braider yarn 1, the braider yarn 2, and the top and bottom pure matrix layers. These material layers are considered as UD composites in their respective local coordinate systems (both the pure matrix layers can be regarded as a UD composite with a zero fiber volume fraction), as indicated in Figs. 9(e)–(g). The mechanical responses (the internal stresses in the fiber and matrix materials and the instantaneous compliance matrix of the composite) of these UD composites are determined by using the bridging model. Then, an assemblage of the three UD composites gives the responses of the sub-element, where the lamina (RVE) properties are obtained by assembling the contribution of all the sub-elements. In order to perform the subdivision and then assemblage, the yarn orientations in the fabric must be identified. An analytical model has been proposed to describe the diamond braid geometry using only parameters of the braiding angle, yarn width, yarn thickness, and inter-yarn gap. Formulation of this model and detailed analysis procedure for the braid lamina can be found in [25].
3.4. Analysis of a knitted fabric lamina A plain weft knit has been chosen as the knitting structure in this work (Fig. 7(a.3)). Through laminate analysis, the stress increments sustained by a knitted fabric lamina can be determined, which are shown in Fig. 7(d.3). Similarly, we must then perform a lamina analysis to achieve the three purposes. Schematic for the analysis of a knitted fabric lamina is shown in Fig. 10. The stress state of Fig. 7(d.3) (i.e. Fig. 10(a)) given in the global system is first transformed into the ply system (Fig. 10(b)). It is seen from Fig. 10(b) that the entire fabric can be constructed using a repeating unit cell shown in Fig. 10(c), which can be further divided into four symmetrical or identical sub-cells. One such sub-cell is taken as the RVE for the knitted fabric lamina, as indicated in Fig. 10(d). There are two yarns in the RVE, whose geometric positions are critical. Fortunately, for the present plain weft knitted fabric structure, the Leaf and Glaskin model [26] can be employed to identify the yarn orientations. Only three parameters, i.e. the wale number (W), the course number (C), and the yarn diameter (d), are required by the model to completely specify the fabric geometry [27,28]. The RVE in Fig. 10(d) is subdivided into even small elements along the x-direction. Owing to the small subdivision, the yarn segment together with its surrounding matrix can be regarded as a UD composite having an inclined angle with the x-direction (Fig. 10(e)). The response of the UD composite in its local coordinate system (Fig. 10(f)) is determined through
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Fig. 10. Schematic diagram to show analysis procedure for a knitted fabric lamina.
the bridging model. The RVE properties, i.e. the internal stresses in the constituent fiber and matrix materials and the instantaneous compliance/stiffness matrix of the RVE, can be obtained through assembling the contributions of all the sub-elements. More details can be found in [27,28].
No standard is presently available for the design of a composite bone plate. Thus, the mechanical behavior of the clinically used stainless steel plate can be taken as a reference. For the stainless steel bone plate that has 3.8 mm thickness (h) and 210 GPa modulus, its unit rigidity (the rigidity per unit width) is given by k ¼ ðEI=bÞ ¼ Eh3 =12 ¼ 960:3
4. Design philosophy 4.1. Design in rigidity From a mechanical point of view, there are two critical quantities that must be taken into account in the design of a composite bone plate. One is its rigidity. As mentioned before, a high rigidity bone plate can result in ‘‘stress shielding’’. On the other hand, a low rigidity plate may adequately appose two fracture fragments but allow motion in excess of that needed for fracture healing without callous formation. Thus, a proper choice of the bone plate rigidity is necessary. Rigidity is interpreted in different ways: axial, flexural, and torsional. For a bone plate, the flexural rigidity is the most important, which is proportional to the material stiffness and the cube of the plate thickness.
ðN mÞ;
ð10Þ
where b is the plate width and the effect of plate holes has been ignored for the sake of simplicity in the present preliminary design. The design is begun with the specification of constituent material properties. The T300 carbon fibers are transversely isotropic and linearly elastic until rupture, whose properties are taken from [29] and are summarised in Table 1. It is noted that the fiber elastic properties at tension and compression are considered as the same and are directly taken from [29]. However, the fiber ultimate strengths are back calculated upon the composite strengths provided in [29], see [24] for detail. Uniaxial tensile tests have been performed for the PEEK matrix material using bulk specimens. Two typical stress–strain curves are plotted in Fig. 11. The curves are then approximated using multiple linear segments, whose representative parameters are summarised in
Table 1 Mechanical properties of T300 carbon fibers Ef11 (GPa)
mf12
Ef22 (GPa)
Gf12 (GPa)
mf23
rfu (MPa)
rfu;c (MPa)
230
0.2
15
15
0.071
2468
1470
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81
Fig. 11. Measured stress–strain curve of PEEK material.
Table 2 Mechanical properties of PEEK matrix (mm = 0.38) i
Tensile properties 1
(rm Y )i (Em T )i
(MPa) (MPa) ðEm T Þi ,
50 3400 ðrm Y Þi1
Note: Em ¼ when a Ultimate strength value.
Compressive properties
2
3
80.5 2130 6
rm e
6
ðrm Y Þi ,
89.4 900 with
ðrm Y Þ0
4
1 a
91.5 270
50 3400
2 82.5 2130
3
4
92 900
a
270
¼ 0.
Table 2. It is noted that the matrix properties at compression have been adjusted according to a bending test. The composite bone plate must be fabricated from a laminated structure. Changing the laminate layers will vary the plate rigidity. Each layer is a fiber preform reinforced PEEK matrix lamina. Three different preforms, i.e. UD prepreg, braided fabric, and knitted fabric, are used as reinforcement. However, the unidirectional prepreg for a different layer can be arranged in a different direction. More over, two different braiding angles, 5 and 15, are used. The candidate composite bone plates are supposed to be fabricated essentially following the procedure as described in Section 2. Namely, the carbon and the PEEK fibers are commingled into yarns that are fabricated as the fiber preforms. These preforms are then made into bone plates using fabrication moulds similar to that shown in Fig. 4 under a hot press machine. In this preliminary design, following composite laminates are considered: unidirectional laminates ([0]ns), angle-plied laminates ([±20]ns), multidirectional laminates ([0/±20/0]ns), braided fabric (with a diamond braid pattern) laminates ([5]ns and [15]ns), and knitted (plain weft knit) fabric laminates, where the subscript n refers to the repeated number. The braided fabric laminates are assumed to have a volume fraction of 48% with each layer being 0.35 mm thick [19], whereas the knitted fabric laminates have a volume fraction of 40% with
each layer being 0.2 mm thick. The other laminates constructed from the unidirectional prepregs are considered to have a fiber volume fraction of 50%, with each layer being 0.17 mm thick. Furthermore, the yarn width is 0.791 mm in the 5 braid and 1.02 mm in the 15 braid. The yarn thickness is the same, equal to 0.35/2 = 0.175 mm, in both the braids. The geometric parameters for the knitted fabrics are: W(wale number) = 1.5(loops/ cm), C(course number) = 3.5(loops/cm), and d(yarn diameter) = 0.404 mm. The preliminary design results for the rigidity of the composite bone plates are shown in Fig. 12. The rigidity of the stainless steel plate, given by Eq. (10), is also shown in the figure. It is seen that to achieve a rigidity comparable to that of the stainless steel plate, the composite plates must be significantly thicker. However, as mentioned before, the rigidity of the stainless steel plate is much more than necessity. In fact, a research carried out by Bradley et al. indicated that a composite bone plate with a stiffness in 10–25% of that of the stainless steel plate could be applicable [10]. In this regard, the required thickness can be significantly reduced into around the stainless steel plate thickness. Although an evaluation (see the previous section) for the rigidity of a composite laminate plate is much more complicated than for that of an isotropic plate (beam), a simple relation similar to Eq. (10) exists, which can sufficiently well interpret the response of the laminated
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Fig. 12. Unit rigidity of laminated plates varied with plate thickness.
beam. Namely, the unit rigidity of the laminated beam can be well approximately expressed as k ¼ kðh=h30 Þ; where h0 is any but fixed thickness at which the laminate rigidity is known. Choosing a standard thickness (e.g. h0 = 1), the unit rigidity of the laminated beam can be rewritten as k ¼ jh3 :
ð11Þ
From the responses shown in Fig. 12, the rigidity parameters j of various laminates are summarized in Table 3. In light of BradleyÕ research and the data of Table 3, it might be concluded that except for the knitted fabric reinforcement the carbon/PEEK composite plates reinforced with all the other considered fiber preforms and with the same as the stainless steel plate thickness would be rigidity suitable for applications. 4.2. Design in strength The second important quantity of the bone plate is its ultimate bending strength, or, equivalently, its maximum bending moment. Similarly as done in the above, we can make a preliminary design for the unit maximum bending moments (the moment per unit width) of the laminated bone plates. Taking the ultimate bending strength of the stainless steel as rb = 760 MPa, its unit maximum bending moment is given by
m ¼ rb ðh2 =6Þ ¼ 1829:1 ðNÞ:
ð12Þ
As understood from Section 3.1, to obtain the ultimate bending strength of a composite plate, we need an additional governing parameter, the plate critical deflection or curvature. However, the exact value of this parameter is not available since the plate is still under design and has not been developed yet. Thus, we have to assume such additional parameter in priori for the strength design purpose. In [24,30] where the braided and knitted T300 carbon fiber reinforced epoxy composite laminates under bending were theoretically and experimentally analysed, respectively. The braded fabric laminates had eight whereas the knitted fabric laminates had six layers of laminas. Those laminates were recognised to mostly assume their ultimate failure at the third-ply failure. Considering that the present composite bone plates may most probably have more than eight lamina layers, we assume that the failure moment corresponding to the third-ply failure is defined as the ultimate bending moment of the designed composite plates. Then, the unit maximum bending moments of the laminated plates under consideration varied with the plate thicknesses can be estimated, and are plotted in Fig. 13. The figure demonstrates that to achieve the same amount of the maximum bending moment as the stainless steel plate, the thicknesses of the composite plates with all the other fibrous structure reinforcements except for the knitted fabric need to be marginally thicker. Even so, the result-
Table 3 Rigidity parameters of various T300/PEEK laminates Material 3
j (N m/mm ) a b c
Stainless steel
UDa [0]ns
MDb [0/±20/0]ns
Braid [5]ns
AP c [±20]ns
Braid [15]ns
Knitted laminate
17.5
4.175
3.406
3.304
2.738
2.359
1.148
Unidirectional. Multidirectional. Angle plied.
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83
Fig. 13. Maximum unit bending moments of various laminate plates varied with plate thickness.
ing composite plate will be much lighter in weight than the stainless steel plate. This is because the carbon/ PEEK composite material has a mass density less than 2 g/cm3 (as a consequence of 1.79 g/cm3 density for the carbon fibers and 1.49 g/cm3 density for the PEEK matrix), much smaller than 7.8 g/cm3, the stainless steel plate density. Considering the comment on the bending rigidity design given above, we clearly understand that as long as the composite plate thickness is chosen in such a way that the plate fulfils the ultimate bending moment requirement it will also satisfy the bending stiffness requirement. In other words, the ultimate bending moment is a key parameter to control the composite bone plate design. Furthermore, from Fig. 13, a relation similar to Eq. (12) is obtained as m ¼ gh2 :
ð13Þ
The moment parameters g of the various laminates are listed in Table 4. 4.3. Discussion Figs. 12 and 13 show that we can always choose a suitable plate thickness to make the bending moment and rigidity of any composite bone plate comparable
to those of the stainless steel plate. On the other hand, when the strength and rigidity are sufficient and without considering other designing issues, the thinner the thickness of a bone plate the better it would be to the patient. In this regard, both the figures together with Tables 3 and 4 demonstrate that the unidirectional laminate plate displays the maximum bending rigidity and bending strength among all the candidate composite plates of a fixed plate thickness. However, a significant limitation exists for the choice of the unidirectional laminate bone plate. Namely, it is very poor in sustaining the load component transverse to the fiber direction. As the bone plate generally possesses a curvature in its cross-section, the stresses near the plate holes transmitted by screws can easily cause the unidirectional laminate plate to split along the axial direction (Fig. 14). In the bending test of such a plate specimen with holes, a splitting load has been found to be much lower than that given in Fig. 13. Due to this reason, an angle-plied or multidirectional tape laminate would be necessary. Nevertheless, more fabrication defects using the angle-plied or multidirectional tape laminate than using that shown in Fig. 4 may result. For instance, the plate holes may have to be drilled if the unidirectional prepregs are used in the fabrication. These drilled holes will break down the continuity of some fibers and hence will cause a reduction in
Table 4 Moment parameters of various T300/PEEK laminates Material 2
g (N/mm ) a b c
Stainless steel
UDa [0]ns
MDb [0/±20/0]ns
Braid [5]ns
APc [±20]ns
Braid [15]ns
Knitted laminate
126.46
124.76
106.78
110.54
110.73
108.29
48.52
Unidirectional. Multidirectional. Angle plied.
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Fig. 14. Schematic diagram of a bending load applied to a bone plate with a curvature. There is a moment component acting in the transverse direction of the plate so that a unidirectional laminate plate can be easily splitted.
the plate load-carrying ability. Considering that the bending rigidity and bending strength of a braided fabric laminate plate are comparable to those of the multidirectional/angle-plied laminate plate (Figs. 12 and 13) and that the plate holes can be easily generated during the fabrication of the braid laminate plate (see Fig. 4), the braided fabric preforms (e.g. with a 5 braiding angle) is recommended as the first candidate for the development of a new composite bone plate. It is worth mentioning that the designed properties (i.e. stiffness and strength) of the composite bone plates also depend on their constituent fiber and matrix behaviors. Different fiber and matrix properties will vary accordingly the designed bending modulus and ultimate bending strength. This means that a scatter in the measured fiber and/or matrix properties (e.g. that shown in Fig. 11) will generally result in a scatter in the designed unit rigidity and maximum unit bending moment for the composite. The relationship, however, between the constituent properties and the overall properties of the composite bone plate is more complicated. As understood, an epoxy matrix modulus has negligible influence on the longitudinal modulus of a carbon fiber composite but predominant influence on its transversely modulus. Thus, a scatter in matrix modulus will have a negligibly influence on the bending stiffness of a small angle braided composite bone plate. But in other cases, usually we will have to perform the design procedure as described above to understand exactly the influence amount.
5. Conclusion A theoretical designing procedure is presented in this paper for an optimal design of composite bone plates reinforced with various fiber structures and varied with plate thicknesses. Not only the plate rigidity but also its ultimate bending moment has been targeted in the design. For the same carbon/PEEK materials system, the numerical experiments indicated that ideally the composite bone plates made of unidirectional, angle-plied, multidirectional, and braided fabric laminates and hav-
ing a thickness comparable to that of a clinically used stainless steel plate may all satisfy the stiffness and strength requirements for applications. The unidirectional and the braided fabric laminate plates exhibit better mechanical performances. Considering a possible reduction in the load-carrying abilities of the unidirectional plate due to a possible splitting along the plate axial direction and a breakage of fiber continuity at screw holes, the composite bone plate made of a small angle braided fabric laminas shows overall the best advantage in terms of mechanical properties. Future researches should incorporate the influence of screw holes and plate curvature on the mechanical behavior of a composite bone plate.
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