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Evaluation of Hyperelastic Material Properties of Sheep Anterior Cruciate Ligament by In-Vitro and 3D Finite Element Analysis

Year 2021, Volume: 11 Issue: 3, 2245 - 2254, 01.09.2021
https://doi.org/10.21597/jist.895137

Abstract

Modeling of ligament tissues in computer simulations in biomechanics is important for achieving the simulation anatomically. These are often described as hyperelastic materials in these types of studies. However, in order to be defined as a hyperelastic material, a mathematical material model created with data obtained from tests such as tension, compression, and creep is needed. There are many phenomenological models that can be used as material models. In this study, ligament tissue simulation was created according to Neo-Hookean, Ogden 2nd Order and Yeoh 2nd Order hyperelastic material models. In-vitro uniaxial tensile testing of sheep anterior cruciate ligament was performed for data on the stressstrain curve to be used by the models. According to the data obtained from the tensile test, the material constants required for the material models were calculated. As a result of the analysis, it was determined that Ogden 2nd Order and Yeoh 2nd Order models were close results and Neo-Hookean model gave results with different stress values. The fit of the stress-strain curves obtained from three models and in-vitro test was evaluated according to the Root Mean Square Error(RMSE) values. RMSE values of Neo-Hookean, Ogden 2nd Order and Yeoh 2nd Order hyperelastic material models were obtained as 4.9597, 1.9704 and 2.3644, respectively. As a result, the Ogden 2nd Order hyperelastic material model with a high number of material constants produced results closer to in-vitro test results compared to both normal and von-mises stress values and RMSE values. Among the three models analyzed for simulations of ligament tissue, it was determined that the use of this material model was more appropriate.

References

  • Bashkuev M, Reitmaier S, Schmidt H, 2020. Relationship between intervertebral disc and facet joint degeneration: A probabilistic finite element model study. Journal of Biomechanics, 102: 109518.
  • Benítez, JM, Montáns FJ, 2017. The mechanical behavior of skin: Structures and models for the finite element analysis. Computers & Structures, 190, 75-107.
  • Bermel E.A, Thakral S, Claeson AA, Ellingson AM, Barocas VH, 2020. Asymmetric in-plane shear behavior of isolated cadaveric lumbar facet capsular ligaments: Implications for subject specific biomechanical models, Journal of Biomechanics, 105: 109814.
  • Bijalwan A, Patel BP, Marieswaran M, Kalyanasundaram D, 2018. Volumetric locking free 3D finite element for modelling of anisotropic visco-hyperelastic behaviour of anterior cruciate ligament. Journal of Biomechanics, 73: 1–8.
  • Brandão S, Parente M, Mascarenhas T, da Silva ARG, Ramos I, Jorge RN, 2015. Biomechanical study on the bladder neck and urethral positions: Simulation of impairment of the pelvic ligaments. Journal of Biomechanics, 48(2): 217–223.
  • Caragiuli M, Mandolini M, Landi D, Bruno G, De Stefani A, Gracco A, Toniolo I, 2021. A finite element analysis for evaluating mandibular advancement devices. Journal of Biomechanics, 119: 110298.
  • Chen ZW, Joli P, Feng ZQ, Rahim M, Pirro N, Bellemare ME, 2015. Female patient-specific finite element modeling of pelvic organ prolapse (POP), Journal of Biomechanics, 48(2): 238-245.
  • Cifuentes-De la Portilla C, Pasapula C, Larrainzar-Garijo R, Bayod J, 2020. Finite element analysis of secondary effect of midfoot fusions on the spring ligament in the management of adult acquired flatfoot. Clinical Biomechanics, 76: 105018.
  • Damlar İ, Özyilmaz E, Altan A, Özyilmaz E, 2014. Üç Boyutlu Sonlu Eleman Analiz Yöntemiyle İki Ticari İmplant Sisteminin Gerilme Dağılımlarının İncelenmesi. Mühendislik Bilimleri ve Tasarım Dergisi, 2(3): 175–180.
  • Gan RZ, Yang F, Zhang X, Nakmali D, 2011. Mechanical properties of stapedial annular ligament. Medical Engineering & Physics, 33(3): 330–339.
  • Henninger HB, Ellis BJ, Scott SA, Weiss JA, 2019. Contributions of elastic fibers, collagen, and extracellular matrix to the multiaxial mechanics of ligament. Journal of the Mechanical Behavior of Biomedical Materials, 99: 118–126.
  • Hexter AT, Shahbazi S, Thangarajah T, Kalaskar D, Haddad FS, Blunn G. 2020. "Characterisation of the tensile properties of Demineralised Cortical Bone when used as an anterior cruciate ligament allograft". Journal of the Mechanical Behavior of Biomedical Materials, 110, 103981.
  • Horgan CO, Smayda MG, 2012. The importance of the second strain invariant in the constitutive modeling of elastomers and soft biomaterials. Mechanics of Materials, 51: 43–52.
  • Jannesar S, Allen M, Mills S, Gibbons A, Bresnahan JC, Salegio EA, Sparrey CJ, 2018. Compressive mechanical characterization of non-human primate spinal cord white matter. Acta Biomaterialia, 74: 260–269.
  • Kara A, 2019. Uzun-Kısa Süreli Bellek Ağı Kullanarak Global Güneş Işınımı Zaman Serileri Tahmini. Gazi Üniversitesi Fen Bilimleri Dergisi Part C: Tasarım ve Teknoloji, 7(4): 882–892.
  • Karimi A, Razaghi R, Biglari H, Rahmati SM, Sandbothe A, Hasani M, 2020. Finite element modeling of the periodontal ligament under a realistic kinetic loading of the jaw system. The Saudi Dental Journal, 32(7): 349–356.
  • Koh Y-G, Park K-M, Kang K, Kim PS, Lee YH, Park KK, Kang K-T, 2021. Finite element analysis of the influence of the posterior tibial slope on mobile-bearing unicompartmental knee arthroplasty. The Knee, 29: 116–125.
  • Korhonen RK, Saarakkala S, 2011. Biomechanics and Modeling of Skeletal Soft Tissues. Theoretical Biomechanics, 6.
  • Łagan, SD, Liber-Kneć A. 2017. Experimental testing and constitutive modeling of the mechanical properties of the swine skin tissue. Acta of Bioengineering and Biomechanics, 19(2), 93-102.
  • Mallett KF, Arruda EM, 2017. Digital image correlation-aided mechanical characterization of the anteromedial and posterolateral bundles of the anterior cruciate ligament. Acta Biomaterialia, Gradients in Biomaterials, 56: 44–57.
  • Martins PALS, Natal Jorge RM, Ferreira AJM, 2006. A Comparative Study of Several Material Models for Prediction of Hyperelastic Properties: Application to Silicone-Rubber and Soft Tissues. Strain, 42(3): 135–147.
  • Morales-Orcajo E, Becerro de Bengoa Vallejo R, Losa Iglesias M, Bayod J, 2016. Structural and material properties of human foot tendons. Clinical Biomechanics, 37: 1–6.
  • Naghibi Beidokhti H, Janssen D, van de Groes S, Hazrati J, Van den Boogaard T, Verdonschot N, 2017. The influence of ligament modelling strategies on the predictive capability of finite element models of the human knee joint. Journal of Biomechanics, 65: 1–11.
  • Nikkhoo M, Hassani K, Tavakoli Golpaygani A, Karimi A, 2020. Biomechanical role of posterior cruciate ligament in total knee arthroplasty: A finite element analysis. Computer Methods and Programs in Biomedicine, 183: 105109.
  • Park H-S, Ahn C, Fung DT, Ren Y, Zhang L-Q, 2010. A knee-specific finite element analysis of the human anterior cruciate ligament impingement against the femoral intercondylar notch. Journal of Biomechanics, 43(10): 2039–2042.
  • Polak-Kraśna K, Robak-Nawrocka S, Szotek S, Czyż M, Gheek D, Pezowicz C, 2019. The denticulate ligament – Tensile characterisation and finite element micro-scale model of the structure stabilising spinal cord. Journal of the Mechanical Behavior of Biomedical Materials, 91: 10-17.
  • Ristaniemi A, Stenroth L, Mikkonen S, Korhonen RK, 2018. Comparison of elastic, viscoelastic and failure tensile material properties of knee ligaments and patellar tendon. Journal of Biomechanics, 79: 31–38.
  • Sarma PA, Pidaparti RM, Moulik PN, Meiss RA, 2003. Non-linear material models for tracheal smooth muscle tissue. Bio-medical Materials and Engineering, 13(3): 235-45.
  • Song Y-L, Lee C-F 2012. Computer-aided modeling of sound transmission of the human middle ear and its otological applications using finite element analysis. Tzu Chi Medical Journal, 24(4), 178-180.
  • Sudres P, Evin M, Wagnac E, Bailly N, Diotalevi L, Melot A, Arnoux P-J, Petit Y., 2021. Tensile mechanical properties of the cervical, thoracic and lumbar porcine spinal meninges. Journal of the Mechanical Behavior of Biomedical Materials, 115: 104280.
  • Sugerman GP, Kakaletsis S, Thakkar P, Chokshi A, Parekh SH, Rausch MK 2021. A whole blood thrombus mimic: Constitutive behavior under simple shear. Journal of the Mechanical Behavior of Biomedical Materials, 115, 104216.
  • Sun L, Wu B, Tian M, Liu B, Luo Y. 2013. Comparison of graft healing in anterior cruciate ligament reconstruction with and without a preserved remnant in rabbits. The Knee, 20(6), 537-544.
  • Weiler A, Peine R, Pashmineh-Azar A, Abel C, Südkamp NP, Hoffmann, RFG. 2002. Tendon healing in a bone tunnel. Part I: Biomechanical results after biodegradable interference fit fixation in a model of anterior cruciate ligament reconstruction in sheep. Arthroscopy: The Journal of Arthroscopic & Related Surgery, 18(2), 113-123.
  • Yahyaiee Bavil A, Rouhi G, 2020. The biomechanical performance of the night-time Providence brace: experimental and finite element investigations. Heliyon, 6(10): e05210.
  • Yucel I, Karaca E, Ozturan K, Yıldırım Ü, Duman S, Degirmenci E. 2009. Biomechanical and histological effects of intra-articular hyaluronic acid on anterior cruciate ligament in rats. Clinical Biomechanics, 24(7), 571-576.
  • Zhou L, Lin J, Wang B, Gan W, Huang A, Lin Y, 2019. Biomechanical effect of anterior talofibular ligament injury in Weber B lateral malleolus fractures after lateral plate fixation: A finite element analysis. Foot and Ankle Surgery, 26(8): 871-875.

Koyun Ön Çapraz Bağının Hiperelastik Malzeme Özelliklerinin In-Vitro ve 3 Boyutlu Sonlu Elemanlar Analizi ile Değerlendirilmesi

Year 2021, Volume: 11 Issue: 3, 2245 - 2254, 01.09.2021
https://doi.org/10.21597/jist.895137

Abstract

Biyomekanik alanındaki bilgisayar simülasyonlarında bağ dokularının modellenmesi simülasyonun anatomiye uygun olarak başarılması için önemlidir. Bağ dokuları, bu tip çalışmalarda çoğunlukla hiperelastik malzeme olarak tanımlanırlar. Ancak hiperelastik malzeme olarak tanımlanması için çekme, basma, sürünme vb. gibi testlerden elde edilmiş verilerle oluşturulan bir matematiksel malzeme modeline ihtiyaç vardır. Malzeme modeli olarak kullanılabilecek birçok fenomenolojik model bulunmaktadır. Bu çalışmada, bağ dokusunun simülasyonu, Neo-Hookean, Ogden 2. Derece ve Yeoh 2. Derece hiperelastik malzeme modellerine göre oluşturulmuştur. Modellerin kullanacağı gerilme-birim şekil değiştirme eğrisi verileri için koyun ön çapraz bağının in-vitro olarak tek eksenli çekme deneyi yapılmıştır. Deneyden elde edilen verilere göre malzeme modelleri için gerekli olan malzeme sabitleri hesaplanmıştır. Yapılan analizler sonucunda bağ dokusunda, Ogden 2. Derece ve Yeoh 2. Derece modellerinin birbirine yakın, Neo-Hookean modelinin ise farklı gerilme değerlerine sahip sonuçlar verdiği belirlenmiştir. Üç modelden ve in-vitro testten elde edilen gerilme-birim şekil değiştirme eğrilerinin uyumu Kök Ortalama Kare Hatası(RMSE) değerlerine göre değerlendirilmiştir. Neo-Hookean, Ogden 2. Derece ve Yeoh 2. Derece hiperelastik malzeme modellerinin RMSE değerleri sırasıyla 4.9597, 1.9704 ve 2.3644 olarak elde edilmiştir. Sonuç olarak, malzeme sabiti sayısı fazla olan Ogden 2. Derece hiperelastik malzeme modeli hem normal ve vonmises gerilme değerlerine hem de RMSE değerlerine göre in-vitro test sonuçlarına daha yakın sonuçlar üretmiştir. Bağ dokusunu simülasyonları için analiz edilen üç model arasında bu malzeme modelinin kullanımının daha uygun olduğu belirlenmiştir.

References

  • Bashkuev M, Reitmaier S, Schmidt H, 2020. Relationship between intervertebral disc and facet joint degeneration: A probabilistic finite element model study. Journal of Biomechanics, 102: 109518.
  • Benítez, JM, Montáns FJ, 2017. The mechanical behavior of skin: Structures and models for the finite element analysis. Computers & Structures, 190, 75-107.
  • Bermel E.A, Thakral S, Claeson AA, Ellingson AM, Barocas VH, 2020. Asymmetric in-plane shear behavior of isolated cadaveric lumbar facet capsular ligaments: Implications for subject specific biomechanical models, Journal of Biomechanics, 105: 109814.
  • Bijalwan A, Patel BP, Marieswaran M, Kalyanasundaram D, 2018. Volumetric locking free 3D finite element for modelling of anisotropic visco-hyperelastic behaviour of anterior cruciate ligament. Journal of Biomechanics, 73: 1–8.
  • Brandão S, Parente M, Mascarenhas T, da Silva ARG, Ramos I, Jorge RN, 2015. Biomechanical study on the bladder neck and urethral positions: Simulation of impairment of the pelvic ligaments. Journal of Biomechanics, 48(2): 217–223.
  • Caragiuli M, Mandolini M, Landi D, Bruno G, De Stefani A, Gracco A, Toniolo I, 2021. A finite element analysis for evaluating mandibular advancement devices. Journal of Biomechanics, 119: 110298.
  • Chen ZW, Joli P, Feng ZQ, Rahim M, Pirro N, Bellemare ME, 2015. Female patient-specific finite element modeling of pelvic organ prolapse (POP), Journal of Biomechanics, 48(2): 238-245.
  • Cifuentes-De la Portilla C, Pasapula C, Larrainzar-Garijo R, Bayod J, 2020. Finite element analysis of secondary effect of midfoot fusions on the spring ligament in the management of adult acquired flatfoot. Clinical Biomechanics, 76: 105018.
  • Damlar İ, Özyilmaz E, Altan A, Özyilmaz E, 2014. Üç Boyutlu Sonlu Eleman Analiz Yöntemiyle İki Ticari İmplant Sisteminin Gerilme Dağılımlarının İncelenmesi. Mühendislik Bilimleri ve Tasarım Dergisi, 2(3): 175–180.
  • Gan RZ, Yang F, Zhang X, Nakmali D, 2011. Mechanical properties of stapedial annular ligament. Medical Engineering & Physics, 33(3): 330–339.
  • Henninger HB, Ellis BJ, Scott SA, Weiss JA, 2019. Contributions of elastic fibers, collagen, and extracellular matrix to the multiaxial mechanics of ligament. Journal of the Mechanical Behavior of Biomedical Materials, 99: 118–126.
  • Hexter AT, Shahbazi S, Thangarajah T, Kalaskar D, Haddad FS, Blunn G. 2020. "Characterisation of the tensile properties of Demineralised Cortical Bone when used as an anterior cruciate ligament allograft". Journal of the Mechanical Behavior of Biomedical Materials, 110, 103981.
  • Horgan CO, Smayda MG, 2012. The importance of the second strain invariant in the constitutive modeling of elastomers and soft biomaterials. Mechanics of Materials, 51: 43–52.
  • Jannesar S, Allen M, Mills S, Gibbons A, Bresnahan JC, Salegio EA, Sparrey CJ, 2018. Compressive mechanical characterization of non-human primate spinal cord white matter. Acta Biomaterialia, 74: 260–269.
  • Kara A, 2019. Uzun-Kısa Süreli Bellek Ağı Kullanarak Global Güneş Işınımı Zaman Serileri Tahmini. Gazi Üniversitesi Fen Bilimleri Dergisi Part C: Tasarım ve Teknoloji, 7(4): 882–892.
  • Karimi A, Razaghi R, Biglari H, Rahmati SM, Sandbothe A, Hasani M, 2020. Finite element modeling of the periodontal ligament under a realistic kinetic loading of the jaw system. The Saudi Dental Journal, 32(7): 349–356.
  • Koh Y-G, Park K-M, Kang K, Kim PS, Lee YH, Park KK, Kang K-T, 2021. Finite element analysis of the influence of the posterior tibial slope on mobile-bearing unicompartmental knee arthroplasty. The Knee, 29: 116–125.
  • Korhonen RK, Saarakkala S, 2011. Biomechanics and Modeling of Skeletal Soft Tissues. Theoretical Biomechanics, 6.
  • Łagan, SD, Liber-Kneć A. 2017. Experimental testing and constitutive modeling of the mechanical properties of the swine skin tissue. Acta of Bioengineering and Biomechanics, 19(2), 93-102.
  • Mallett KF, Arruda EM, 2017. Digital image correlation-aided mechanical characterization of the anteromedial and posterolateral bundles of the anterior cruciate ligament. Acta Biomaterialia, Gradients in Biomaterials, 56: 44–57.
  • Martins PALS, Natal Jorge RM, Ferreira AJM, 2006. A Comparative Study of Several Material Models for Prediction of Hyperelastic Properties: Application to Silicone-Rubber and Soft Tissues. Strain, 42(3): 135–147.
  • Morales-Orcajo E, Becerro de Bengoa Vallejo R, Losa Iglesias M, Bayod J, 2016. Structural and material properties of human foot tendons. Clinical Biomechanics, 37: 1–6.
  • Naghibi Beidokhti H, Janssen D, van de Groes S, Hazrati J, Van den Boogaard T, Verdonschot N, 2017. The influence of ligament modelling strategies on the predictive capability of finite element models of the human knee joint. Journal of Biomechanics, 65: 1–11.
  • Nikkhoo M, Hassani K, Tavakoli Golpaygani A, Karimi A, 2020. Biomechanical role of posterior cruciate ligament in total knee arthroplasty: A finite element analysis. Computer Methods and Programs in Biomedicine, 183: 105109.
  • Park H-S, Ahn C, Fung DT, Ren Y, Zhang L-Q, 2010. A knee-specific finite element analysis of the human anterior cruciate ligament impingement against the femoral intercondylar notch. Journal of Biomechanics, 43(10): 2039–2042.
  • Polak-Kraśna K, Robak-Nawrocka S, Szotek S, Czyż M, Gheek D, Pezowicz C, 2019. The denticulate ligament – Tensile characterisation and finite element micro-scale model of the structure stabilising spinal cord. Journal of the Mechanical Behavior of Biomedical Materials, 91: 10-17.
  • Ristaniemi A, Stenroth L, Mikkonen S, Korhonen RK, 2018. Comparison of elastic, viscoelastic and failure tensile material properties of knee ligaments and patellar tendon. Journal of Biomechanics, 79: 31–38.
  • Sarma PA, Pidaparti RM, Moulik PN, Meiss RA, 2003. Non-linear material models for tracheal smooth muscle tissue. Bio-medical Materials and Engineering, 13(3): 235-45.
  • Song Y-L, Lee C-F 2012. Computer-aided modeling of sound transmission of the human middle ear and its otological applications using finite element analysis. Tzu Chi Medical Journal, 24(4), 178-180.
  • Sudres P, Evin M, Wagnac E, Bailly N, Diotalevi L, Melot A, Arnoux P-J, Petit Y., 2021. Tensile mechanical properties of the cervical, thoracic and lumbar porcine spinal meninges. Journal of the Mechanical Behavior of Biomedical Materials, 115: 104280.
  • Sugerman GP, Kakaletsis S, Thakkar P, Chokshi A, Parekh SH, Rausch MK 2021. A whole blood thrombus mimic: Constitutive behavior under simple shear. Journal of the Mechanical Behavior of Biomedical Materials, 115, 104216.
  • Sun L, Wu B, Tian M, Liu B, Luo Y. 2013. Comparison of graft healing in anterior cruciate ligament reconstruction with and without a preserved remnant in rabbits. The Knee, 20(6), 537-544.
  • Weiler A, Peine R, Pashmineh-Azar A, Abel C, Südkamp NP, Hoffmann, RFG. 2002. Tendon healing in a bone tunnel. Part I: Biomechanical results after biodegradable interference fit fixation in a model of anterior cruciate ligament reconstruction in sheep. Arthroscopy: The Journal of Arthroscopic & Related Surgery, 18(2), 113-123.
  • Yahyaiee Bavil A, Rouhi G, 2020. The biomechanical performance of the night-time Providence brace: experimental and finite element investigations. Heliyon, 6(10): e05210.
  • Yucel I, Karaca E, Ozturan K, Yıldırım Ü, Duman S, Degirmenci E. 2009. Biomechanical and histological effects of intra-articular hyaluronic acid on anterior cruciate ligament in rats. Clinical Biomechanics, 24(7), 571-576.
  • Zhou L, Lin J, Wang B, Gan W, Huang A, Lin Y, 2019. Biomechanical effect of anterior talofibular ligament injury in Weber B lateral malleolus fractures after lateral plate fixation: A finite element analysis. Foot and Ankle Surgery, 26(8): 871-875.
There are 36 citations in total.

Details

Primary Language Turkish
Subjects Mechanical Engineering
Journal Section Makina Mühendisliği / Mechanical Engineering
Authors

İsmail Hakkı Korkmaz 0000-0003-2440-0319

Publication Date September 1, 2021
Submission Date March 11, 2021
Acceptance Date May 7, 2021
Published in Issue Year 2021 Volume: 11 Issue: 3

Cite

APA Korkmaz, İ. H. (2021). Koyun Ön Çapraz Bağının Hiperelastik Malzeme Özelliklerinin In-Vitro ve 3 Boyutlu Sonlu Elemanlar Analizi ile Değerlendirilmesi. Journal of the Institute of Science and Technology, 11(3), 2245-2254. https://doi.org/10.21597/jist.895137
AMA Korkmaz İH. Koyun Ön Çapraz Bağının Hiperelastik Malzeme Özelliklerinin In-Vitro ve 3 Boyutlu Sonlu Elemanlar Analizi ile Değerlendirilmesi. J. Inst. Sci. and Tech. September 2021;11(3):2245-2254. doi:10.21597/jist.895137
Chicago Korkmaz, İsmail Hakkı. “Koyun Ön Çapraz Bağının Hiperelastik Malzeme Özelliklerinin In-Vitro Ve 3 Boyutlu Sonlu Elemanlar Analizi Ile Değerlendirilmesi”. Journal of the Institute of Science and Technology 11, no. 3 (September 2021): 2245-54. https://doi.org/10.21597/jist.895137.
EndNote Korkmaz İH (September 1, 2021) Koyun Ön Çapraz Bağının Hiperelastik Malzeme Özelliklerinin In-Vitro ve 3 Boyutlu Sonlu Elemanlar Analizi ile Değerlendirilmesi. Journal of the Institute of Science and Technology 11 3 2245–2254.
IEEE İ. H. Korkmaz, “Koyun Ön Çapraz Bağının Hiperelastik Malzeme Özelliklerinin In-Vitro ve 3 Boyutlu Sonlu Elemanlar Analizi ile Değerlendirilmesi”, J. Inst. Sci. and Tech., vol. 11, no. 3, pp. 2245–2254, 2021, doi: 10.21597/jist.895137.
ISNAD Korkmaz, İsmail Hakkı. “Koyun Ön Çapraz Bağının Hiperelastik Malzeme Özelliklerinin In-Vitro Ve 3 Boyutlu Sonlu Elemanlar Analizi Ile Değerlendirilmesi”. Journal of the Institute of Science and Technology 11/3 (September 2021), 2245-2254. https://doi.org/10.21597/jist.895137.
JAMA Korkmaz İH. Koyun Ön Çapraz Bağının Hiperelastik Malzeme Özelliklerinin In-Vitro ve 3 Boyutlu Sonlu Elemanlar Analizi ile Değerlendirilmesi. J. Inst. Sci. and Tech. 2021;11:2245–2254.
MLA Korkmaz, İsmail Hakkı. “Koyun Ön Çapraz Bağının Hiperelastik Malzeme Özelliklerinin In-Vitro Ve 3 Boyutlu Sonlu Elemanlar Analizi Ile Değerlendirilmesi”. Journal of the Institute of Science and Technology, vol. 11, no. 3, 2021, pp. 2245-54, doi:10.21597/jist.895137.
Vancouver Korkmaz İH. Koyun Ön Çapraz Bağının Hiperelastik Malzeme Özelliklerinin In-Vitro ve 3 Boyutlu Sonlu Elemanlar Analizi ile Değerlendirilmesi. J. Inst. Sci. and Tech. 2021;11(3):2245-54.