Modelowanie konstytutywne tkanki kostnej i materiałów poliuretanowych do zastosowań w chirurgii ortopedycznej

• Autorzy: Pawlikowski M.

Warianty tytułu

EN

Constitutive modelling of bone tissue and polyurethane materials for orthopaedical application

• Języki publikacji: PL

Abstrakty

PL

Niniejsza monografia dotyczy modelowania konstytutywnego korowej tkanki kostnej oraz dwóch materiałów poliuretanowych. Przez modelowanie konstytutywne należy tutaj rozumieć formułowanie, dla danego materiału, równania definiującego zależność między naprężeniem a odkształceniem. Nieliniowy lepkosprężysty model konstytutywny wspomnianych materiałów tworzony był na podstawie zapostulowanej lub dobranej funkcji energii odkształcenia. Stałe materiałowe, występujące w modelu, zostały zidentyfikowane w oparciu o wyniki testów relaksacji i testów monotonicznego ściskania przeprowadzonych z trzema prędkościami deformacji. Należy podkreślić, że liczba czasów relaksacji nie została z góry narzucona. Liczba czasów relaksacji, jak również ich wartości, zostały określone na podstawie testów relaksacji. Testy monotonicznego ściskania obejmowały fazę obciążania i odciążania. Sformułowany model konstytutywny uwzględniał więc również inną cechę materiałów lepko sprężystych, tj. pętlę histerezy. Wyprowadzone równania konstytutywne zostały zastosowane w prostych symulacjach numerycznych wykonanych za pomocą metody elementów skończonych z wykorzystaniem komercyjnego systemu Abaqus®. W analizach tych zasymulowano różne warianty obciążenia, tj. ściskanie, rozciąganie, obciążenie cykliczne. W pracy przedstawiono także zaawansowaną symulację numeryczną implantowanego segmentu kręgosłupa lędźwiowego. Model numeryczny segmentu obejmował, oprócz dwóch kręgów lędźwiowych, endoprotezę krążka międzykręgowego składającą się z dwóch metalowych płytek i poliuretanowej wkładki. Ten ostatni element endoprotezy został zamodelowany za pomocą jednego z sformułowanych równań konstytutywnych dla materiału poliuretanowego.

EN

The monograph deals with constitutive modelling of cortical bone tissue and two polyurethane materials. The term „constitutive modelling” denotes formulation of an equation relating stress and strain. Non-linear visco-elastic constitutive models of the mentioned materials were formulated on the basis of the postulated or selected strain energy function. The material constants were identified on the basis of relaxation tests and monotonic compression tests performed at three strain rates. It has to be emphasized that the number of relaxation times was not assumed a priori. It was determined, together with the values of the relaxation times, on the basis of relaxation tests. The monotonic compression tests comprised the phase of loading and that of unloading. The constitutive laws are, then, able to describe another viscoelastic property, i.e. hysteresis loop. The formulated constitutive models were implemented into the finite element method system Abaqus®, which was utilised to perform some simple numerical simulations. In the monograph, an advanced numerical simulation of the implanted lumbar segment was also presented. The numerical model of the segment comprised, apart from the vertebrae, the three-element prosthesis of an intervertebral disc. One of the prosthesis elements, i.e. the polyurethane inlay, was simulated by means of the constitutive model derived for one of the polyurethane materials.

Słowa kluczowe

PL

równanie konstytutywne; funkcja energii odkształcenia; lepkosprężystość nieliniowa; materiały poliuretanowe; tensor sztywności

EN

constitutive equation; strain energy function; nonlinear viscoelasticity; polyurethane materials; stiffness tensor

• Wydawca: Oficyna Wydawnicza Politechniki Warszawskiej
• Czasopismo: Prace Naukowe Politechniki Warszawskiej. Mechanika
• Rocznik: 2013
• Tom: z. 260
• Strony: 3--147
• Opis fizyczny: Bibliogr. 171 poz., rys., tab., wykr.

Twórcy

autor

Pawlikowski M.

• Wydział Inżynierii Produkcji

Bibliografia

• [1] Ahrens J., Shelokov A.P., Carver J.L., Normal joint mobility is maintained with an artificial disc prosthesis. New York: North American Spine Society, 1997.
• [2] Akeson W.H., Woo S.L., Taylor T.K., Ghosh P., Bushell G.R., Biomechanics and biochemistry of the intervertebral disks: the need for correlation studies. ClinOrthop; 129:133-40, 1977.
• [3] Aklonis J.J., MacKnight W.J., Introduction to polymer viscoelasticity. Second Edition. New York: John Wiley, 1983.
• [4] Alfrey T. Jr., Mechanical behaviour of high polymers Interscience Publishers, New York, 1948.
• [5] Ali A., Hosseini M., Sahari B.B., A Review of Constitutive Models for Rubber-Like Materials, American J. of Engineering and Applied Sciences 3 (1): 232-239, 2010.
• [6] Arcan M., Hashin Z., Voloshin A., A method to produce uniform plane-stress states with applications to fiber-reinforced materials, Exp. Mech., 18, 141-146, 1978.
• [7] Arruda E.M., Boyce M.C., A three-dimensional constitutive model for the large stretch behavior of elastomers. J. Mech. Phys. Solids 41, 389-412, 1993.
• [8] Ascenzi A., Biomechanics and Galileo Galilei, J. Biomechanics, 26, 95-100, 1993.
• [9] Ashman R.B., Ultrasonic Determination of the Elastic Properties of Cortical Bone: Techniques and Limitations, Ph.D. thesis; Tulane University, New Orleans, LA, 1982.
• [10] Bartkowiak-Jowsa M., Będziński R., Szaraniec B., Chłopek J., Mechanical, biological, and microstructural properties of biodegradable models of polymeric stents made of PLLA and alginate fibers, Acta of Bioengineering and Biomechanics, 13(4), 21-28, 2011.
• [11] Bayraktar H.H., Keaveny T.M., Mechanisms of uniformity of yield strains for trabecular bone, J. Biomech 37:1671-1678, 2004.
• [12] Będziński R., Biomechanika inżynierska. Zagadnienia wybrane, Oficyna Wydawnicza Politechniki Wrocławskiej, Wrocław, 1997.
• [13] Bertagnoli R., Schonmayr R. Surgical and clinical results with the PDN prosthetic disc-nucleus device. Eur Spine J; 11 (Suppl 2):S143-8, 2002.
• [14] Bonet J., Large strain viscoelastic constitutive models, Int. J. Solid Struct. 38, 2953-2968, 2001.
• [15] Bonfield W., Li C.H., The temperature dependence of the deformation of bone, J. Biomech., 1, 323-329, 1968.
• [16] Bonfield W., Tully A.E., Ultrasonic analysis of the Young's modulus of cortical bone, J. Biomed. Eng., 4, 23-27, 1982.
• [17] Borkowski P., Kędzior K., Krzesiński G., Skalski K.R., Wymysłowski P., Zagrajek T., Numerical Investigation of a New Type of Artificial Lumbar Disc, J. Theor. Appl. Mech., 42, 2, 253-268, 2004.
• [18] Broz J.J., Simske S.J., Greenherg A.R., Luttges M.W., Effects of rehydration state on the flexural properties of whole mouse long bones, J. Biomech. Eng., 115, 447-449, 1993.
• [19] Caler W.E., Carter D.R., Bone creep-fatigue damage accumulation, J. Biomech., 22, 625-635, 1989.
• [20] Carter D.R., Caler W.C., A cumulative damage model for bone fracture, J. Orthop. Res., 3, 84-90, 1985.
• [21] Carter D.R., Hayes W.C., Compact bone fatigue damage. A microscopic examination, Clin. Orthop. Relat. Res., 127, 265-274, 1977.
• [22] Carter D.R., Hayes W.C., Fatigue life of compact bone - I. Effects of stress amplitude, temperature and density, J. Biomech., 9, 27-34, 1976.
• [23] Carter D.R., Hayes W.C., The compressive behavior of bone as a two-phase porous structure, J. Bone Joint Surg., 59-A, 954-962, 1977.
• [24] Caspi I., Levinkopf M., Nerubay J., Results of lumbar disk prosthesis after a follow-up period of 48 months. Isr. Med. Assoc.; 5:9-11, 2003.
• [25] Charalambides M.N., Wanigasooriya L., Williams J.G., Goh S.M., Chakrabarti S., Large deformation extensional rheology of bread dough, Rheol. Acta 46: 239-248, 2006.
• [26] Charlebios M., Jirásek M., Zysset P.K., A nonlocal constitutive model for trabecular bone softening in compression, Biomech Model Mechanobiol, 9:597-611, 2010.
• [27] Christensen R.M., Theory of viscoelasticity. Second Edition London: Academic Press, 1982.
• [28] Ciambella J., Paolone A., Vidoli S., A comparison of nonlinear integral-based viscoelastic models through compression tests on filled rubber, Mech. Mat. 42, 932-944, 2010.
• [29] Cinotti G., David T., Postacchini F., Results of disc prosthesis after a minimum follow-up period of 2 years. Spine; 21:995-1000, 1996.
• [30] Coleman B.D., Foundation of Linear Viscoelasticity, Rev. of Modern Physics, Vol. 33, No. 2, 239-240, 1961.
• [31] Coleman B.D., Thermodynamics of Materials with Memory. Arch. Rat. Mech. Anal. 17, 3-46, 1964.
• [32] Cowin S.C., Bones have ears. Acta of Bioengineering and Biotmechanics. Vol. 4, Suppl. 1, 29-32, 2002.
• [33] Cowin S.C., He Q.-C., Tensile and compressive stress yield criteria for cancellous bone. J. Biomech; 38:141-4, 2005.
• [34] Cowin S.C., Moss M.L., Mechanosensory mechanisms in bone, in Textbook in Tissue Engineering, (2nd edition) Lanza R., Langer R., Chick W. eds., Academic Press, San Diego, 723-738, 2000.
• [35] Currey J.D., The effects of drying and re-wetting on some mechanical properties of cortical bone, J. Biomech., 21, 439-441, 1988.
• [36] Currcy J., Tensile yield in compact bone is determined by strain. post-yield behavior by mineral content, J. Biomech., 37:549-56, 2004.
• [37] Da Silva Soares J.F., Constitutive modeling for biodegradable polymers for application in endovascular stents, Texas A&M University, 2008.
• [38] Datta J., Leszkowski K., Investigation of chemical stability of ether-urethane prepolymers, Polymers (Polimery), 53, 115, 2008.
• [39] de Groot J.H., de Vrijer R., Pennings A.J., Klompmaker J., Veth R.P.H., Jansen H.W.B., Use of porous polyurethanes for meniscal reconstruction and meniscal prostheses. Biomater. 17, 163-173, 1996.
• [40] Dempster W.T., Liddicoat R., Compact bone as a non-isotropic material, Am. J. Anat., 91, 331-362, 1952.
• [41] Dibbets J.M.H., One century of Wolff’s law, in Bone Biodynamics in Orthodontic and Orthopaedic Treatment (eds. D.S. Carlson and S.A. Goldstein), Craniofacial Growth Series (Vol. 27), University of Michigan, Ann Arbor, 1-13, 1992.
• [42] Dietrich L., Lekszycki T., Turski K., Problems of identification of mechanical characteristics of viscoelastic composites, Acta Mechanica, 126, 153-167, 1998.
• [43] Doll S., Schweizehof K., On the development of volumetric strain-energy functions, J. Appl. Mech. 67, 17-21, 2000.
• [44] Evans F.G., Lebow M., Regional differences in some of the physical properties of the human femur, J. Appl. Physiol., 3, 563-572, 1951.
• [45] Evans F.G., Mechanical Properties of Bone, Charles C. Thomas, Springfield, IL, 1973.
• [46] Findley W.N., Lai J.S., Onaran K., Creep and relaxation of nonlinear viscoelastic materials, Amsterdam: North-Holland, 1976.
• [47] Fondrk M.T., Bahniuk E.H., Davy D.T., Inelastic strain accumulation in cortical bone during rapid transient tensile loading, J. Biomech. Eng. 121:616-621, 1999.
• [48] Forwood M.R., Parker A.W., Microdamage in response to repetitive torsional loading in the rat tibia, Calcif. Tissue Int., 45, 47-53, 1989.
• [49] Fung Y.C., Biomechanics. Mechanical Properties of Living Tissues, Springer, 1993.
• [50] Fung Y.C., Podstawy mechaniki ciała stałego, PWN, Warszawa, 1969.
• [51] Fung Y.C., Tong P., Classical and Computational Solid Mechanics, World Scientific Publishing, 2001.
• [52] Fung Y.C.B., Elasticity of soft tissues in sample elongation, American Journal of Physiology, vol. 213, 1532-1544, 1967.
• [53] Gamradt S.C., Wang J.C., Lumbar disc arthroplasty. The Spine Journal; 5: 95-103, 2005.
• [54] Garbarski J., Opis zjawisk lepkosprężystości liniowej polimerów konstrukcyjnych, Wydawnictwa Politechniki Warszawskiej, Warszawa, 1990.
• [55] Garcia D., Zysset P.K., Charlebois M., Curnier A., A three-dimensional elastic plastic damage constitutive law for bone tissue, Biomech Model Mechanobiol, 8:149-165, 2009.
• [56] Gardiner J.C., Weiss J.A., Simple Shear Testing of Parallelfibered, Planar Soft Tissue, ASME J. Biomech. Eng., 123, 170-175, 2001.
• [57] Gent A.N., A new constitutive relation for rubber, Rubber chemistry and technology, Vol. 69, 59-61, 1996.
• [58] Gibson L.J., Ashby M.F., Cellular solids: structure and properties, 2. Cambridge University Press, Cambridge, 1999.
• [59] Gosh S.M., Charalambides M.N., Williams J.G., Determination of the Constitutive Constants of Non-Linear Viscoelastic Materials, Mech. Time-Depend. Mater. 8, 255-268, 2004.
• [60] Gordon J., Dmitriev A.E., Hu N., Mcaffee P.C., Biomechanical evalualion of total disc arthroplasty: an in-vitro human cadaveric model. New Orleans, LA: American Academy of Orthopaedic Surgeons, 2003.
• [61] Goulet R.W., Goldstein S.A., Ciarelli M.J., Kuhn J.L., Brown M.B., Feldkamp L.A., The relationship between the structural and orthogonal compressive properties of trabecular bone. J. Biomech. 27, 375-389, 1994.
• [62] Green A.E., Naghdi P.M., A general theory of an elastic-plastic continuum, Springer, Berlin, 1965.
• [63] Green M.S., Tobolsky A.V., A new approach to the theory of relaxing polymeric media. J. Phys. Chem. 14, 80-92, 1946.
• [64] Gupta A., Bayraktar H.H., Fox J.C., Keaveny T.M., Papadopoulos P., Constitutive modeling and algorithmic implementation of a plasticity-like model for trabecular bone structures, Comput. Mech. 40: 61-72, 2007.
• [65] Harrigan T.P., Jasty M., Mann R.W., Harris W.H., Limitations of the continuum assumption in cancellous bone, J. Biomech., 21, 269-275, 1988.
• [66] Harris R.F., Molecular weight advancement of poly(ethylene ether carbonate) polyols, J. Appl. Polym. Sci. 38, 463-476, 1989.
• [67] Hartmann S., Neff P., Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. Int. J. Solids Struct. 40, 2767-2791, 2003.
• [68] Holzapfel G.A., Nonlinear Solid Mechanics. A Continuum Approach for Engineering, John Wiley & Sons, LTD, 2000.
• [69] Hoppen H.J., Leenslag J.W., Pennings A.J., van der Lei B., Robinson P.H., Two-ply biodegradable nerve guide: basic aspects of design, construction and biological performance, Biomater. 11, 286-290, 1990.
• [70] Huber N., Tsakmakis C., Discussion of finite deformation viscoelasticity laws with reference to torsion loading. Continuum Mech. Thermodyn. 12, 303-323, 2000b.
• [71] Hubner N., Tsakmakis C., Finite deformation viscoelasticity laws, Mech. Mater. 32, 1-18, 2000a.
• [72] Humphrey J.D., Determination of a constitutive relation for passive myocardium: I: A new functional form, Journal of Biomechanical Engineering 112, 333-339, 1990.
• [73] Hvid I., Jensen J., Cancellous bone strength at the proximal human tibia. Eng. Med., 13, 21-25, 1984.
• [74] Iosipescu N., New accurate procedure for single shear testing of metals, J. Mater., 2, 537-566, 1967.
• [75] Jemioło S., Studium hipersprężystych własności materiałów izotropowych, Oficyna Wydawnicza Politechniki Warszawskiej, Warszawa, 2002.
• [76] Jemioło S., Telega J.J., Fabric tensor in bone mechanics, Eng. Trans., 46, 3-26, 1998.
• [77] Keaveny T.M., Guo X.E., Wachtel E.F., McMahon T.A., Hayes W.C., Trabecular bone exhibits fully linear elastic behavior and yields at low strains, J. Biomech., 27, 1127-1136, 1994.
• [78] Keaveny T.M., Hayes W.C., A 20-year perspective on the mechanical properties of trabecular bone, J. Biomech. Eng., 115, 534-542, 1993.
• [79] Keaveny T.M., Wachtel E.F., Kopperdahl D.L., Mechanical behavior of human trabecular bone after overloading, J. Orthop. Res. 17:346-353, 1999.
• [80] Kilian H.G., Equation of state of real networks, Polymer, Vol. 22, 209-217, 1981.
• [81] Klara P.M., Ray C.D. Artificial nucleus replacement: clinical experience. Spine: 27:1374-7, 2002.
• [82] Kleiber M., Wykłady z nieliniowej termo-mechaniki ciał odkształcalnych, Warszawa, 2000.
• [83] Knowles J.K., The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids, International Journal of Fracture, Vol.13, 611-639, 1977.
• [84] Kobayashi M., Inoue S., Tsuruta T., Copolymerization of Carbon Dioxide and Epoxide by the Dialkylzinc Carboxylic Acid System, J. Polym. Sci.: Polym. Chem. Ed. 11. 2383-85, 1973.
• [85] Koerner H., Liu W., Alexander M., Mirau P., Dowty H., Vaia R.A., Deformation-morphology correlations in electrically conductive carbon nanotube-thermoplastic polyurethane nanocomposites, Polymer, 46, 4405-4420, 2005.
• [86] Korochkina T.V., Claypole T.C., Gethin D.T., Choosing Constitutive Models for Elastomers used in Printing Processes. In: Constitutive models for Rubber IV, Austrell, P.E. and L. Kari (Eds.). AA Balkema Publishers, UK., ISBN: 10: 0415383463, pp: 431-435, 2005.
• [87] Krag M.H., Seroussi R.E., Wilder D.G., Pope M.H., Internal displacement distribution from in vitro loading of humam thoracic and lumbar spinal motion segments: experimental results and theoretical predictions. Spine; 12:1001-7, 1987.
• [88] Laiarinandrasana L., Piques R., Robisson A., Visco-hyperelastic model with internal state variable coupled with discontinuous damage concept under total Lagrangian, Int. J. Plasticity 19, 977-1000, 2003.
• [89] Lakes R.S., Katz J.L., Sternstein S.S., Viscoelastic properties of wet cortical bone - I. Torsional and biaxial studies, J. Biomech., 12, 657-678, 1979.
• [90] Lakes R.S., Saha S., Cement line motion in bone, Science, 204, 501-503, 1979.
• [91] Lelah M.D., Cooper S.L., Polyurethanes in Medicine, CRC Press, Boca Raton, Florida, 1986.
• [92] Lemaitre J., Chaboche J.L., Mécanique des matériauxsolides, 2eédn. Dunod, Paris, 2001.
• [93] Limbert G., Middleton J., A constitutive model of the posterior cruciate ligament, Medical Engineering & Physics 28, 99-113, 2006.
• [94] Linde F., Hvid I., Madsen F., The effect of specimen geometry on the mechanical behaviour of trabecular bone specimens, J. Biomech. 25, 359-368, 1992.
• [95] Linde F., Hvid I., The effect of constraint on the mechanical behavior of trabecular bone specimens, J. Biomech., 22, 485-490, 1989.
• [96] Linde F., Sørensen H.C., The effect of different storage methods on the mechanical properties of trabecular bone, Journal of Biomechanics, 26(10), 1249-1252, 1993.
• [97] Link H.D., History, design and biomechanics of the LINK SB Charité artificial disc. Eur Spine J. 11(Suppl 2):S98-S105, 2002.
• [98] Lion A., A physically based method to represent the thermo-mechanical behavior of elastomers, Acta Mech. 123, 1-25, 1997.
• [99] Lubliner J., A model of rubber viscoelasticity, Mech. Res. Comm. 12, 93-99, 1985.
• [100] Macosko C.W., Rheology: principles, measurement and applications. VCH Publishers, ISBN 1-56081-579-5, 1994.
• [101] Maksimov R.D., Ivanova T., Kalnins M., Zicans J., Mechanical properties of high-density polyethylene/chlorinated polyethylene blends, Mech. Comp. Mat. 40, 331-340, 2004.
• [102] Marckmann G., Verron E., Comparison of hyperelastic models for rubberlike materials, Rubber Chemistry and Technology, 79(5), 835-858, 2006.
• [103] Marlow R.S., A General First-Invariant Hyperelastic Constitutive Model, In: Constitutive Models for Rubber III, Busfield, J. (Ed.). AA Balkema Publishers, UK., ISBN: 10: 9058095665, pp: 157-160, 2003.
• [104] Mathur A.B., Collier T.O., Kao W.J., Wiggins M., Schubert M.A., Hiltner A., Anderson J.M. In vivo biocompatibility and biostability of modified polyurethanes. J Biomed Mater Res 36(2):246-57, 1997.
• [105] Matsuura M., Eckstein F., Lochmüller E.-M., Zysset P.K., The role of fabric in the quasi-static compressive mechanical properties of human trabecular bone from various anatomical locations, Biomech Model Mechanobiol 7(1):27-42, 2008.
• [106] McNally D.S., Adams M.A., Internal intervertebral disc mechanics as revealed by stress profilometry. Spine;17:66-73, 1992.
• [107] Melnis A.E., Ozola B.O., Moorlat P.A., Comparative characteristics of the creep properties of human compact bone tissue under various specimen storage and testing conditions, Mechanics of Composite Materials, 17, 3, 355-360, 1981.
• [108] Meyer G.H., Die Architektur der Spongiosa, ArchisfurAnatomie, Physiologie und wissenschaftliche Medizin, Reichert und DuBois-Reymonds Archiv, 34, 615-628, 1867.
• [109] Morgan E.F., Keaveny T.M., Dependence of yield strain of human trabecular bone on anatomic site, J Biomech 34:569-577, 2001.
• [110] Morman K.N., Rubber viscoelasticity - a review of current understanding. Dearborn: Ford Motor Company, 1985.
• [111] Nachemson A., Towards a better understanding of low-back pain. A review of the mechanics of the lumbar disc. Rheumatology and Rehabilitation, 14, 129-143, 1975.
• [112] Natali A.N., Carniel E.L., Pavan P.G., Constitutive modelling of inelastic behaviour of cortical bone, Medical Engineering & Physics 30, 905-912, 2008.
• [113] Nowacki W., Teoria sprężystości, PWN, Warszawa, 1970.
• [114] Ogden R., Saccomandi G., Sgura l., Fittinghyperelastic models to experimental data, Comput. Mech. 34, 484-502, 2004.
• [115] Ogden R.W., Nonlinear Elastic Deformations, Ellis Horwood, Chichester, U.K., 1984.
• [116] Ogdez R.W., Large Deformation Isotropic Elasticity - On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids. Proceedings of the Royal Society of London. Series A, Math Phys Sci, 326: 565-584, 1972.
• [117] Osti OL, Vernon-Roberts B, Moore R, Fraser RD. Annular tears and disc degeneration in the lumbar spine. A post-mortem study of 135 discs. J Bone Joint Surg Br; 74:678-82, 1992.
• [118] Oxnard C.E., Yang H.C., Beyond biometrics: studies of complex biological patterns, Symposia Zool. Soc. Lond., 46, 127-167, 1981.
• [119] Park H.C., Lakes R.S., Cosserat micromechanics of human bone: strain redistribution by a hydration sensitive constituent, J. Biomech., 19. 385-397, 1986.
• [120] Patzák B., Bittnar Z., Design of object oriented finite element code, Adv Eng Softw 32(10-11): 759-767, 2001.
• [121] Pawlikowski M., Cortical Bone Tissue Viscoelastic Properties and Its Constitutive Equation - Preliminary Studies. Arch. Mech. Eng. LIX, 31-52, 2012.
• [122] Pawlikowski M., Non-linear approach in visco-hyperelastic constitutive modelling of polyurethane nanocomposite, Mech Time Dependent Mat, DOI: 10.1007/s11043-013-9208-2, 2013.
• [123] Pawlikowski M., Klasztorny M., Skalski K., Studies on constitutive equation that models bone tissue, Acta of Bioengineering and Biomechanics, 10, 39-47, 2008.
• [124] Pioletti D.P., Rakotomanana, L.R., Non-linear viscoelastic laws for soft biological tissues, Eur. J. Mech. A/Solids 19, 749-759, 2000.
• [125] Pioletti D.P., Viscoelastic properties of soft tissues: application to knee ligaments and tendons, Thesis, Ecole Polytechnique Fédéralede Lausanne.
• [126] Pokharkar V., Sivaram S., Poly(alkylene carbonate)s by the Carbonate Interchange Reaction of Aliphatic Diols with Dimethyl Carbonate: Synthesis and Characterization, Polymer 36, 4851-54, 1995.
• [127] Pope M.H., Frymoyer J.W., Lehmann T.R., Structure and function of the lumbar spine. In: Pope MH, Anderson GBJ, Frymoyer JW, Chaffin DB, eds. Occupational low back pain: assessment, treatment, and prevention. St. Louis, MO: CV Mosby, 95-113, 1991.
• [128] Pope M.H., Wilder D.G., Matteri R.E., Frymoyer J.W., Experimental measurements of vertebral motion under load. OrthopClin North Am; 8:155-67 1977.
• [129] Pucci E., Saccomandi G., A note on the Gent model for rubber-like materials, Rubber chemistry and technology, vol. 75, 839-851, 2002.
• [130] Reese S., Govindjee S., A theory of finite viscoelasticity and numerical aspects, Int. J. Solid Struct. 35, 3455-3482, 1998.
• [131] Reilly D.T., Burstein A.H., Frankel V.H., The elastic modulus for bone, Biomech., 7, 271-275, 1974.
• [132] Rincón-Kohli L., Zysset P., Multi-axial mechanical properties of human trabecular bone, Biomech Model Mechanobiol 8(3): 195-208, 2009.
• [133] Rivlin R.S., Saunders D.W., Large elastic deformations of isotropic materials, VII. Experiments on the deformation of rubber, Phil. Trans. R. Soc. A-243, 251-288, 1951.
• [134] Rivlin R.S.. Some applications of elasticity theory to rubber engineering. In Collected Papers of R.S. Rivlin. 1 and 2 Springer, 1997.
• [135] Robinson P.H., van der Lei B., Knol K.E., Pennings A.J., Patency and long-term biological fate of a two-ply biodegradable microarterial prosthesis in the rat, Br. J. Plast. Surg. 42, 544-549, 1989.
• [136] Roesler H., Some historical remarks on the theory of cancellous bone structure (Wolff's Law), Mechanical Properties of Bone, S.C. Cowin, New York, American Society of Mechanical Engineers, 27-42, 1981.
• [137] Rokicki G., Kowalczyk T., Synthesis of oligocarbonate diols and their characterization by MALDI-TOF spectrometry, Polymer 41, 9013-9031, 2000.
• [138] Rokicki G., Piotrowska A., Kowalczyk T., Kozakiewicz J., Cykliczne węglany w syntezie oligowęglanodioli metodą polireakcji stopniowej, Polimery, 46, 483-493, 2001.
• [139] Rolander S.D., Motion of the lumbar spine with special reference to the stabilizing effect of posterior fusion. An experimental study on autopsy specimens. Acta Orthop Scand; (Suppl 90), 1-144, 1966.
• [140] Roux W., Beitragezur Morphologie der funktionellen Anpassung, Arch. Anat. Physiol. Abt., 120-185, 1885a.
• [141] Rubin C.T., Lanyon L.E., Limb mechanics as a function of speed and gait: a study of functional strains in the radius and tibia of horse and dog, J. Exp. Biol., 101, 187-211, 1982.
• [142] Ryszkowska J.L., Jurczyk-Kowalska M., Szymborski T., Kurzydłowski K.J., Dispersion of carbon nanotubes in polyurethane matrix, Physica E 39, 124-127, 2007.
• [143] Schaffler M.B., Radin E.L., Burr D.B., Mechanical and morphologlcal effects of strain rate on fatigue of compact bone, Bone, 10, 207-214, 1989.
• [144] Schnell H., Chemistry and physics of polycarbonates, New York: Wiley, 1964.
• [145] Schulz M.J., Kelkar A.D., Sundaresan M.J., Nanoengineering of Structural Functional and Smart Materials, CRC, Taylor & Francis, New York, 285-326, 2006.
• [146] Sidoroff F., Un modele viscoelastique non lineaire avec configuration intermediaire, J. MCc. 13, 679-713, 1974.
• [147] Simo J.C., Taylor R.L., A three dimensional finite deformation viscoelastic model accounting for damage effects, Rept. No. UCBiSESMiG-2. Department of Civil Engineering, University of California, Berkeley, CA, 1985.
• [148] Simo J.C., Taylor R.L., Pister K.S., Variational and projection methods for the volume constraint in finite deformation elastoplasticity, Comput. Meths. Appl. Mech. Engrg. 51, 177-208, 1985.
• [149] Simon S.R., Kinesiology. In: Simon SR, ed. Orthopaedic basic science. Rosemont, IL: American Academy of Orthopaedic Surgeons, 519-622, 1994.
• [150] Sorvari J., Malinen M., On the direct estimation of creep and relaxation functions, Mech Time-Depend Mater 11: 143-157, 2007.
• [151] Stokes K., McVenes R., Anderson J., Polyurethane elastomer bio-stability, J. Biomater. Appl. 9, 321-54, 1995.
• [152] Stokes K., Polyether Polyurethanes: Biostable or Not?, J Biomater Appl; 3:228-259, 1988.
• [153] Sun W., Yuan Y-X., Optimization theory and methods. Nonlinear Programming. Springer, 2006.
• [154] Taylor R.L., Pister K.S., Goudreau G.L., Thermomechanical Analysis of Viscoelastic Solids, International Journal for Numerical Methods in Engineering, 2, 45-59 1970.
• [155] Thiel W., The preservation of the whole corpse with natural color. Ann Anat; 174, 185-95, 2002.
• [156] Treloar L.R.G., Stress-strain data for vulcanized rubber under various types of deformation, Trans. Faraday Soc. 40, 59-70, 1944.
• [157] Tropiano P., Huang R.C., Girardi F.P., Marnay T., Lumbar disc replacement: preliminary results with ProDisc II after a minimum follow-up period of 1 year. J Spinal Disord Tech; 16:362-8, 2003.
• [158] Truesdell C., Noll W., The Non-linear Field Theories. Springer, 1992.
• [159] Turner C.H., Yield behavior of cancellous bone, J. Biomech. Eng., 111, 1-5, 1989.
• [160] Ün K., Bevilla G, Keaveny T.M., The effects of side-artifacts on the elastic modulus of trabecular bone, J. Biomech., 39, 1955-1963, 2006.
• [161] Unger S., Blauth M., Schmoelz W., Effects of three different preservation methods on the mechanical properties of human and bovine cortical bone, Bone, 47, 1048-1053, 2010.
• [162] Volterra V., Sulleequazioniintegro-differenziali della teoria dell'estatica, Attidella Reale Accademiadei Lincei, 18, 295, 1909.
• [163] Walrath D.E., Adams D.F., The Iosipescu shear test as applied to composite materials, Exp. Mech., 23, 105-110, 1983.
• [164] Walton A.G., Formation and structure of calcified tissue, in: Advances in Bioengineering. Chem. Eng. Progr. Symp., 1974.
• [165] Wei G., Ma P.X., Structure and properties of nano-hydroxyapatite/polymer composite scaffolds for bone tissue engineering, Biomater. 25, 4749-4757, 2004.
• [166] Williams D.F., On the mechanisms of biocompatibility, Biomaterials, 29, 2941-2953, 2009.
• [167] Wolff J., The Law of Bone Remodelling, Springer-Verlag, Berlin, 1986.
• [168] Yeoh O.H., Characterization of elastic properties of carbon black filled rubber vulcanizates, Rubber chemistry and technology, Vol. 63, 792-805, 1990.
• [169] Yeoh O.H., Some forms of the strain energy function for rubber, Rubber chemistry and technology, Vol. 66, 754-771, 1993.
• [170] Zysset P.K., Curnier A., An implicit projection algorithm for simultaneous flow of plasticity and damage in standard generalized materials. Int J Numer Methods Eng 39:3065-3082, 1996.
• [171] Zysset P.K., A review of morphology-elasticity relationships in human trabecular bone: theories and experiments, J. Biomech. 36(10):1469-1485, 2003.