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Analysis of the Yield Condition of Porous Bone-implant Fixation
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 177 (2017) 358 – 362
XXI International Polish-Slovak Conference “Machine Modeling and Simulations 2016”
Analysis of the yield condition of porous bone-implant fixation Maciej Berdychowskia*, Janusz Mielniczukb, Mirosław Matyjaszczykc Chair of Basics of Machine Design, Poznan University of Technology ul. Piotrowo 3, 60-965 Poznań, Poland b Rail Vehicles Institute “TABOR”, ul. Warszawska181, 61-055 Poznań, Poland c University of Zielona Gora, ul. Licealna 9, 65-417 Zielona Góra, Poland
a
Abstract In this paper a theoretical analysis of porous bone-implant coupling is performed. The cancellous bone is described as a hollow circular cylinder formed of isotropic poroelastic material filled with viscous intraosseous fluid. The conical metallic implant, axially compressed, is assumed to be undeformable in comparison to cancellous bone. The Biot’s formulation of theory of poroelasticity is used and by means of the modified Huber-Mises yield criterion the new yield condition (constitutive equation) has been determined as a function of strain. From this criterion it is possible the maximum permissible displacement of implant seating in bone to determine. Authors.Published Publishedby byElsevier ElsevierLtd. Ltd.This is an open access article under the CC BY-NC-ND license © 2017 The Authors. © Peer-review under responsibility of the organizing committee of MMS 2016. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MMS 2016 Keywords: cancellous bone; porous; implant; poroelasticity; Huber-Mises yield criterion; critical pressure load;
1. The problem of porous bone – implant fixation In the article [1] there was considered the fixation of the porous cortical bone of long bone diaphysis with the medullar cylindrical metallic porous coated implant. Now we consider the fixation of the porous bone and a new type of endoprothesis with needle palisade fixation system. The idea of the new type of endoprothesis is shown in Fig. 1. The implant has a very characteristic system of needles put on the spherical surface to mount implant in to the porous cancellous bone. [2, 3]
* Corresponding author. Tel.: 48 61 224 45 14. E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MMS 2016
doi:10.1016/j.proeng.2017.02.207
Maciej Berdychowski et al. / Procedia Engineering 177 (2017) 358 – 362
359
Fig. 1. The idea of the new type of endoprothesis: (a) needles penetrate into cancellous bone; (b) model of the endoprothesis with needle palisade fixation system based on the European Patent EP072418 B1.
We assume that the implant with needle palisade fixation system is axially compressed with the quasi-static axial load P in the porous cancellous bone, which is described as a hollow circular cylinder formed of isotropic poroelastic material filled with viscous intraosseous fluid (Fig. 2). The hollow isotropic cylinder is subjected to an internal radial pressure load pa.
Fig. 2. The metallic cone needle axially compressed in the hollow circular isotropic poroelastic cylinder which is subjected to internal pressure load pa .
It is known that the cancellous bone is anisotropic material with relatively small anisotropy [4, 5]. However, if cancellous bone acts under longitudinal load (compression or tension without bending and/or torsion) a good approximation can be obtained from assuming cancellous bone to be isotropic. Most metals used for endoprostheses stems have higher (2÷3 range) values of Young‘s moduli of elasticity in comparison with those of cortical or cancellous bone [4]. Thus it can be assumed that strains in metallic needle of the considered implant are relatively small in comparison to the elastic strains of cortical or cancellous bone. Then we will assume that implant is practically undeformable in comparison to bone.
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Maciej Berdychowski et al. / Procedia Engineering 177 (2017) 358 – 362
2. Constitutive equations for cancellous bone on the base of Biot’s theory of poroelasticity The bone is treated as porous isotropic solid representing the matrix structure of bone that is filled by viscous compressible extracellular fluid including the bone cells and blood flowing through the matrix structure. The physical relations for such defined material can be given using Biot’s theory of poroelasticity [6-8]: ߪ݆݅ ݏൌ ʹܰߝ݆݅ ሺ ߝܣ ܳ߆ሻߜ݆݅
(1)
ߪ ݂ ൌ ܳߝ ܴ߆ where: σijs, i, j = r, θ, z are the stress applied to the solid, σf represents the total stress applied to the fluid, εij, i, j = r, θ, z, are the strain components of solid, ε = εrr + εθθ + εzz and Θ = Θrr + Θθθ + Θzz are the dilatations of the solid and fluid respectively. N, A, Q, R are the poroelastic constants of the material, in accordance with Biot’s formulation [6-8] and δij is the Kronecker’s symbol. The physical relations for an isotropic poroelastic material have in cylindrical coordinates the following form: ݏ ߪݎݎ ൌ ʹܰߝ ݎݎ ܣሺߝ ݎݎ ߝߠߠ ߝ ݖݖሻ ܳሺ߆ ݎݎ ߆ߠߠ ߆ ݖݖሻ ݏ ߪߠߠ ൌ ʹܰߝߠߠ ܣሺߝ ݎݎ ߝߠߠ ߝ ݖݖሻ ܳሺ߆ ݎݎ ߆ߠߠ ߆ ݖݖሻ ݏ ߪݖݖ ൌ ʹܰߝ ݖݖ ܣሺߝ ݎݎ ߝߠߠ ߝ ݖݖሻ ܳሺ߆ ݎݎ ߆ߠߠ ߆ ݖݖሻ ݏ ߪߠݎ ൌ ܰߝߠݎ
(2)
ݏ ߪߠݖ ൌ ܰߝߠݖ ݏ ߪݎݖ ൌ ܰߝݎݖ
ߪ ݂ ൌ ܳሺߝ ݎݎ ߝߠߠ ߝ ݖݖሻ ܴሺ߆ ݎݎ ߆ߠߠ ߆ ݖݖሻ 3. Boundary conditions and modified Huber-Mises yield criterion The boundary condition (3) reflects the radial internal pressure exerted on the solid part of the cylindrical surface of the medullary canal by forced fit of the medullared implant, whereas (4) refer to the pressure exerted by the environment around the bone on the outer surface of the hollow cylinder. For the simplicity there have been assumed, that at the boundary r = a and r = b the fluid is allowed to flow out of the bone matrix. Then the fluid phase do not transmit the stresses, σf = 0. The pressure outside the hollow cylinder is taken PO = 0. The boundary conditions at the inner and outer surfaces of the cylinder are: ݏ ݏ ݏ at r=a, ߪ ݏݎൌ െ ܽǡ ߪߠݎ ൌ ߪݎݖ ൌ ߪߠݖ ൌͲ (3) ݏ ݏ ݏ at r=b, ߪ ݏݎൌ Ͳǡ ߪߠݎ ൌ ߪݎݖ ൌ ߪߠݖ ൌͲ (4) where: a and b denote the inner and outer radius of the cylinder. For the simplicity the double indexation can be replaced by singular, e.g. σrr = σr. Then, the physical relations (2) become: ߪ ݏݎൌ ʹܰߝ ݎ ܣሺߝ ݎ ߝߠ ሻ ܳሺ߆ ݎ ߆ߠ ሻ ߪߠ ݏൌ ʹܰߝߠ ܣሺߝ ݎ ߝߠ ሻ ܳሺ߆ ݎ ߆ߠ ሻ
(5)
ܳሺߝ ݎ ߝߠ ሻ ܴሺ߆ ݎ ߆ߠ ሻ ൌ Ͳ ݏ ݏ ݏ ߪߠݎ ൌ ߪߠݖ ൌ ߪݎݖ ൌͲ
The equations (5) formulate the relationship between stress and dilatation solid and fluid. In a situation where the structure of the outer cortical bone is affected and will expose cancellous bone, the liquid ceases to fulfill its role which is to carry the load. In this case compounds constitutive of the porous material can be written as: ߪ ݏݎൌ ʹܰߝ ݎ ܣሺߝ ݎ ߝߠ ሻ ߪߠ ݏൌ ʹܰߝߠ ܣሺߝ ݎ ߝߠ ሻ
(6)
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Maciej Berdychowski et al. / Procedia Engineering 177 (2017) 358 – 362
In order to determine the critical pressure load of bone-implant fixation we will use the modified Huber-Mises yield criterion for porous material [9]; for the considered system this criterion takes the form: ߪ ʹݎെ ߪ ߠߪ ݎ ߪߠʹ ൌ ሺܴ ݔܽ݉ݎሻʹ
(7)
where ܴ – ݔܽ݉ݎthe yield stress for radial compression of bone. Thus, finally, we get the boundary condition in the form: Ͷܰ ʹ ሺߝ ʹݎെ ߝ ߠߝ ݎ ߝߠʹ ሻ ሺߝ ݎ ߝߠ ሻʹ ሺʹܰ ܣ ʹܣሻ ൌ ሺܴ ݔܽ݉ݎሻʹ
(8)
It is a yield condition expressed in strains. Special arrangement allows you to specify limits displacement of the implant in the bone. His transgression leads to the permanent destruction of the bone structure. Now we can determine the deep we can press the implant into the bone before we reach the yield point. To do it we need to define a dependence between strains and geometry of needles (fig. 3). We can write: ߝ ݎൌ ߝߠ ൌ
߂ܴ ܴ ߂ܥ ܥ
ൌ െͳ ʹ
ൌ
ܴʹ െܴͳ ሺ݈െʹܴͳ ሻ
(9)
ܴʹ െܴͳ ܴͳ
where R1, R2 – radius of the needle at the beginning and at the end of insertion, l is the distance between adjacent axis of the needle and Co denote circumference before deformation.
Fig. 3. The metallic implant with needle palisade fixation system pressed into cancellous bone.
It is important that R1 is a radius of the needle at the beginning of insertion and should be equal the average radius of pores of material in which we press the implant. The last we need it is equation for taper convergence to describe a geometry of the needles.
362
Maciej Berdychowski et al. / Procedia Engineering 177 (2017) 358 – 362 ܴʹ െܴͳ ݄
ൌ ݊ܽݐ
ߙ
(10)
ʹ
where α – is a cone angle, and h is the depth of penetration. Now we can describe a strain as a function of the depth of penetration. ߝ ݎൌ െ ͳ ʹ
ߝߠ ൌ
݄ ݊ܽݐ
ߙ ʹ
(11)
ሺ݈െʹܴͳ ሻ
݄ ݊ܽݐ
ߙ ʹ
ܴͳ
Finally using equation (8) and (11) we can formulate function which let us determine a maximum depth of penetration ݄݉ܽݔ during pressing needle into the cancellous bone. ݄݉ܽ ݔൌ ට
ሺܴ ݔܽ݉ݎሻʹ
Ͷܰ ʹ ܨʹܰ ܭܣʹܣܭ
(12)
where N, A are the poroelastic constants of the material in accordance with Biot’s formulation [6, 7, 8] and K, F are constants depending from geometric of needle. ߙ
ܭൌ
ܨൌ
ͳ
ʹ ʹ ݊ܽݐቀܴͳʹ െܴͳ ሺ݈െʹܴͳ ሻͶ ሺ݈െʹܴͳ ሻʹ ቁ ͳ ʹ ܴ ሺ݈െʹܴͳ ሻʹ Ͷ ͳ
ߙ ʹ
(13)
ͳ ͳ ʹ Ͷ ͳ ʹ ʹ ܴ ሺ݈െʹܴͳ ሻ Ͷ ͳ
ʹ ݊ܽݐቀܴͳʹ ܴͳ ሺ݈െʹܴͳ ሻ ሺ݈െʹܴͳ ሻʹ ቁ
The presented theoretical mechanical analysis of porous cancellous bone - metallic implant fixation makes possible to determine the stress state and strains in cancellous bone and determine a maximum depth of penetration ݄݉ܽ ݔduring pressing needle into the cancellous bone. If we assume that ܴ ݔܽ݉ݎǡ ܰǡ ܣare constant (for example for human femur), then ݄݉ܽ ݔis function only geometric parameters of needle. Thus, we can analyze and find the best design for needles. Calculation of ݄݉ܽ ݔis important also for surgeon. If we calculate ݄݉ܽ ݔ, surgeon will know how deep he should press implant into the bone. References [1] J. Mielniczuk, R. Uklejewski, M. Berdychowski, M. Winiecki, Mathematical model for determining critical load of porous cortical bone-implant fixation theoretical analysis. Trans VŠB-TU Ostrava, Metallurgical Series, 51, 1 (2008) 76-81. [2] J. Mielniczuk, P. Rogala, R. Uklejewski, M. Winiecki, G. Jokś, A. Auguściński, M. Berdychowski, Modelling of the needle-palisade fixation system for the total hip resurfacing arthroplasty endoprosthesis. Trans VŠB-TU Ostrava, Metallurgical Series, 5, 1 (2008) 160-166. [3] P. Rogala, R. Uklejewski, J. Mielniczuk, M. Winiecki, Prototyp małoinwazyjnej endoprotezy powierzchniowej stawu biodrowego wytworzony metodą SLM, TS Raport, 48 (2008) 6-7 (in Polish). [4] J.D. Currey, Bones: Structure and Mechanics, Princeton: Princetown University Press, 2002. [5] A.N. Natali, E.A. Meroi, A review of the mechanical properties of bone as a material, J. Biomed Engng 11 (1989) 266–276. [6] M.A. Biot, General theory of three-dimensional consolidation, J Appl Phys, 12 (1941) 155–165. [7] M.A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid, J Appl Phys 26(1955) 182–185. [8] M.A. Biot, D.G. Willis, Elastic coefficients of the theory of consolidation, J Appl Mech, 79 (1957) 594–601. [9] J. Mielniczuk, Plasticity of Porous Materials: Theory and the Limit Load Capacity, Wyd. Politechniki Poznańskiej, Poznań, 2000.
ScienceDirect Procedia Engineering 177 (2017) 358 – 362
XXI International Polish-Slovak Conference “Machine Modeling and Simulations 2016”
Analysis of the yield condition of porous bone-implant fixation Maciej Berdychowskia*, Janusz Mielniczukb, Mirosław Matyjaszczykc Chair of Basics of Machine Design, Poznan University of Technology ul. Piotrowo 3, 60-965 Poznań, Poland b Rail Vehicles Institute “TABOR”, ul. Warszawska181, 61-055 Poznań, Poland c University of Zielona Gora, ul. Licealna 9, 65-417 Zielona Góra, Poland
a
Abstract In this paper a theoretical analysis of porous bone-implant coupling is performed. The cancellous bone is described as a hollow circular cylinder formed of isotropic poroelastic material filled with viscous intraosseous fluid. The conical metallic implant, axially compressed, is assumed to be undeformable in comparison to cancellous bone. The Biot’s formulation of theory of poroelasticity is used and by means of the modified Huber-Mises yield criterion the new yield condition (constitutive equation) has been determined as a function of strain. From this criterion it is possible the maximum permissible displacement of implant seating in bone to determine. Authors.Published Publishedby byElsevier ElsevierLtd. Ltd.This is an open access article under the CC BY-NC-ND license © 2017 The Authors. © Peer-review under responsibility of the organizing committee of MMS 2016. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MMS 2016 Keywords: cancellous bone; porous; implant; poroelasticity; Huber-Mises yield criterion; critical pressure load;
1. The problem of porous bone – implant fixation In the article [1] there was considered the fixation of the porous cortical bone of long bone diaphysis with the medullar cylindrical metallic porous coated implant. Now we consider the fixation of the porous bone and a new type of endoprothesis with needle palisade fixation system. The idea of the new type of endoprothesis is shown in Fig. 1. The implant has a very characteristic system of needles put on the spherical surface to mount implant in to the porous cancellous bone. [2, 3]
* Corresponding author. Tel.: 48 61 224 45 14. E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MMS 2016
doi:10.1016/j.proeng.2017.02.207
Maciej Berdychowski et al. / Procedia Engineering 177 (2017) 358 – 362
359
Fig. 1. The idea of the new type of endoprothesis: (a) needles penetrate into cancellous bone; (b) model of the endoprothesis with needle palisade fixation system based on the European Patent EP072418 B1.
We assume that the implant with needle palisade fixation system is axially compressed with the quasi-static axial load P in the porous cancellous bone, which is described as a hollow circular cylinder formed of isotropic poroelastic material filled with viscous intraosseous fluid (Fig. 2). The hollow isotropic cylinder is subjected to an internal radial pressure load pa.
Fig. 2. The metallic cone needle axially compressed in the hollow circular isotropic poroelastic cylinder which is subjected to internal pressure load pa .
It is known that the cancellous bone is anisotropic material with relatively small anisotropy [4, 5]. However, if cancellous bone acts under longitudinal load (compression or tension without bending and/or torsion) a good approximation can be obtained from assuming cancellous bone to be isotropic. Most metals used for endoprostheses stems have higher (2÷3 range) values of Young‘s moduli of elasticity in comparison with those of cortical or cancellous bone [4]. Thus it can be assumed that strains in metallic needle of the considered implant are relatively small in comparison to the elastic strains of cortical or cancellous bone. Then we will assume that implant is practically undeformable in comparison to bone.
360
Maciej Berdychowski et al. / Procedia Engineering 177 (2017) 358 – 362
2. Constitutive equations for cancellous bone on the base of Biot’s theory of poroelasticity The bone is treated as porous isotropic solid representing the matrix structure of bone that is filled by viscous compressible extracellular fluid including the bone cells and blood flowing through the matrix structure. The physical relations for such defined material can be given using Biot’s theory of poroelasticity [6-8]: ߪ݆݅ ݏൌ ʹܰߝ݆݅ ሺ ߝܣ ܳ߆ሻߜ݆݅
(1)
ߪ ݂ ൌ ܳߝ ܴ߆ where: σijs, i, j = r, θ, z are the stress applied to the solid, σf represents the total stress applied to the fluid, εij, i, j = r, θ, z, are the strain components of solid, ε = εrr + εθθ + εzz and Θ = Θrr + Θθθ + Θzz are the dilatations of the solid and fluid respectively. N, A, Q, R are the poroelastic constants of the material, in accordance with Biot’s formulation [6-8] and δij is the Kronecker’s symbol. The physical relations for an isotropic poroelastic material have in cylindrical coordinates the following form: ݏ ߪݎݎ ൌ ʹܰߝ ݎݎ ܣሺߝ ݎݎ ߝߠߠ ߝ ݖݖሻ ܳሺ߆ ݎݎ ߆ߠߠ ߆ ݖݖሻ ݏ ߪߠߠ ൌ ʹܰߝߠߠ ܣሺߝ ݎݎ ߝߠߠ ߝ ݖݖሻ ܳሺ߆ ݎݎ ߆ߠߠ ߆ ݖݖሻ ݏ ߪݖݖ ൌ ʹܰߝ ݖݖ ܣሺߝ ݎݎ ߝߠߠ ߝ ݖݖሻ ܳሺ߆ ݎݎ ߆ߠߠ ߆ ݖݖሻ ݏ ߪߠݎ ൌ ܰߝߠݎ
(2)
ݏ ߪߠݖ ൌ ܰߝߠݖ ݏ ߪݎݖ ൌ ܰߝݎݖ
ߪ ݂ ൌ ܳሺߝ ݎݎ ߝߠߠ ߝ ݖݖሻ ܴሺ߆ ݎݎ ߆ߠߠ ߆ ݖݖሻ 3. Boundary conditions and modified Huber-Mises yield criterion The boundary condition (3) reflects the radial internal pressure exerted on the solid part of the cylindrical surface of the medullary canal by forced fit of the medullared implant, whereas (4) refer to the pressure exerted by the environment around the bone on the outer surface of the hollow cylinder. For the simplicity there have been assumed, that at the boundary r = a and r = b the fluid is allowed to flow out of the bone matrix. Then the fluid phase do not transmit the stresses, σf = 0. The pressure outside the hollow cylinder is taken PO = 0. The boundary conditions at the inner and outer surfaces of the cylinder are: ݏ ݏ ݏ at r=a, ߪ ݏݎൌ െ ܽǡ ߪߠݎ ൌ ߪݎݖ ൌ ߪߠݖ ൌͲ (3) ݏ ݏ ݏ at r=b, ߪ ݏݎൌ Ͳǡ ߪߠݎ ൌ ߪݎݖ ൌ ߪߠݖ ൌͲ (4) where: a and b denote the inner and outer radius of the cylinder. For the simplicity the double indexation can be replaced by singular, e.g. σrr = σr. Then, the physical relations (2) become: ߪ ݏݎൌ ʹܰߝ ݎ ܣሺߝ ݎ ߝߠ ሻ ܳሺ߆ ݎ ߆ߠ ሻ ߪߠ ݏൌ ʹܰߝߠ ܣሺߝ ݎ ߝߠ ሻ ܳሺ߆ ݎ ߆ߠ ሻ
(5)
ܳሺߝ ݎ ߝߠ ሻ ܴሺ߆ ݎ ߆ߠ ሻ ൌ Ͳ ݏ ݏ ݏ ߪߠݎ ൌ ߪߠݖ ൌ ߪݎݖ ൌͲ
The equations (5) formulate the relationship between stress and dilatation solid and fluid. In a situation where the structure of the outer cortical bone is affected and will expose cancellous bone, the liquid ceases to fulfill its role which is to carry the load. In this case compounds constitutive of the porous material can be written as: ߪ ݏݎൌ ʹܰߝ ݎ ܣሺߝ ݎ ߝߠ ሻ ߪߠ ݏൌ ʹܰߝߠ ܣሺߝ ݎ ߝߠ ሻ
(6)
361
Maciej Berdychowski et al. / Procedia Engineering 177 (2017) 358 – 362
In order to determine the critical pressure load of bone-implant fixation we will use the modified Huber-Mises yield criterion for porous material [9]; for the considered system this criterion takes the form: ߪ ʹݎെ ߪ ߠߪ ݎ ߪߠʹ ൌ ሺܴ ݔܽ݉ݎሻʹ
(7)
where ܴ – ݔܽ݉ݎthe yield stress for radial compression of bone. Thus, finally, we get the boundary condition in the form: Ͷܰ ʹ ሺߝ ʹݎെ ߝ ߠߝ ݎ ߝߠʹ ሻ ሺߝ ݎ ߝߠ ሻʹ ሺʹܰ ܣ ʹܣሻ ൌ ሺܴ ݔܽ݉ݎሻʹ
(8)
It is a yield condition expressed in strains. Special arrangement allows you to specify limits displacement of the implant in the bone. His transgression leads to the permanent destruction of the bone structure. Now we can determine the deep we can press the implant into the bone before we reach the yield point. To do it we need to define a dependence between strains and geometry of needles (fig. 3). We can write: ߝ ݎൌ ߝߠ ൌ
߂ܴ ܴ ߂ܥ ܥ
ൌ െͳ ʹ
ൌ
ܴʹ െܴͳ ሺ݈െʹܴͳ ሻ
(9)
ܴʹ െܴͳ ܴͳ
where R1, R2 – radius of the needle at the beginning and at the end of insertion, l is the distance between adjacent axis of the needle and Co denote circumference before deformation.
Fig. 3. The metallic implant with needle palisade fixation system pressed into cancellous bone.
It is important that R1 is a radius of the needle at the beginning of insertion and should be equal the average radius of pores of material in which we press the implant. The last we need it is equation for taper convergence to describe a geometry of the needles.
362
Maciej Berdychowski et al. / Procedia Engineering 177 (2017) 358 – 362 ܴʹ െܴͳ ݄
ൌ ݊ܽݐ
ߙ
(10)
ʹ
where α – is a cone angle, and h is the depth of penetration. Now we can describe a strain as a function of the depth of penetration. ߝ ݎൌ െ ͳ ʹ
ߝߠ ൌ
݄ ݊ܽݐ
ߙ ʹ
(11)
ሺ݈െʹܴͳ ሻ
݄ ݊ܽݐ
ߙ ʹ
ܴͳ
Finally using equation (8) and (11) we can formulate function which let us determine a maximum depth of penetration ݄݉ܽݔ during pressing needle into the cancellous bone. ݄݉ܽ ݔൌ ට
ሺܴ ݔܽ݉ݎሻʹ
Ͷܰ ʹ ܨʹܰ ܭܣʹܣܭ
(12)
where N, A are the poroelastic constants of the material in accordance with Biot’s formulation [6, 7, 8] and K, F are constants depending from geometric of needle. ߙ
ܭൌ
ܨൌ
ͳ
ʹ ʹ ݊ܽݐቀܴͳʹ െܴͳ ሺ݈െʹܴͳ ሻͶ ሺ݈െʹܴͳ ሻʹ ቁ ͳ ʹ ܴ ሺ݈െʹܴͳ ሻʹ Ͷ ͳ
ߙ ʹ
(13)
ͳ ͳ ʹ Ͷ ͳ ʹ ʹ ܴ ሺ݈െʹܴͳ ሻ Ͷ ͳ
ʹ ݊ܽݐቀܴͳʹ ܴͳ ሺ݈െʹܴͳ ሻ ሺ݈െʹܴͳ ሻʹ ቁ
The presented theoretical mechanical analysis of porous cancellous bone - metallic implant fixation makes possible to determine the stress state and strains in cancellous bone and determine a maximum depth of penetration ݄݉ܽ ݔduring pressing needle into the cancellous bone. If we assume that ܴ ݔܽ݉ݎǡ ܰǡ ܣare constant (for example for human femur), then ݄݉ܽ ݔis function only geometric parameters of needle. Thus, we can analyze and find the best design for needles. Calculation of ݄݉ܽ ݔis important also for surgeon. If we calculate ݄݉ܽ ݔ, surgeon will know how deep he should press implant into the bone. References [1] J. Mielniczuk, R. Uklejewski, M. Berdychowski, M. Winiecki, Mathematical model for determining critical load of porous cortical bone-implant fixation theoretical analysis. Trans VŠB-TU Ostrava, Metallurgical Series, 51, 1 (2008) 76-81. [2] J. Mielniczuk, P. Rogala, R. Uklejewski, M. Winiecki, G. Jokś, A. Auguściński, M. Berdychowski, Modelling of the needle-palisade fixation system for the total hip resurfacing arthroplasty endoprosthesis. Trans VŠB-TU Ostrava, Metallurgical Series, 5, 1 (2008) 160-166. [3] P. Rogala, R. Uklejewski, J. Mielniczuk, M. Winiecki, Prototyp małoinwazyjnej endoprotezy powierzchniowej stawu biodrowego wytworzony metodą SLM, TS Raport, 48 (2008) 6-7 (in Polish). [4] J.D. Currey, Bones: Structure and Mechanics, Princeton: Princetown University Press, 2002. [5] A.N. Natali, E.A. Meroi, A review of the mechanical properties of bone as a material, J. Biomed Engng 11 (1989) 266–276. [6] M.A. Biot, General theory of three-dimensional consolidation, J Appl Phys, 12 (1941) 155–165. [7] M.A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid, J Appl Phys 26(1955) 182–185. [8] M.A. Biot, D.G. Willis, Elastic coefficients of the theory of consolidation, J Appl Mech, 79 (1957) 594–601. [9] J. Mielniczuk, Plasticity of Porous Materials: Theory and the Limit Load Capacity, Wyd. Politechniki Poznańskiej, Poznań, 2000.