مدل‌سازی رفتار ناهمسانگرد بافت‌های نرم با روش بدون شبکه هیدرودینامیک هموار ذرات

نوع مقاله : مقاله پژوهشی

نویسندگان

گروه مهندسی مکانیک، دانشگاه صنعتی خاتم الانبیاء بهبهان، بهبهان، خوزستان، ایران

چکیده

مدل‌سازی رفتار مکانیکی بسیاری از بافت‌های نرم و برخی مواد مصنوعی ساخته‌شده توسط انسان نیازمند نمایش دقیق طبیعت پیچیده و ناهمسانگرد آن‌ها است. در این پژوهش، مدل هایپرالاستیک ناهمسانگرد هولتزاپفل-گاسر-اوگدن (HGO) با استفاده از روش بدون شبکه هیدرودینامیک هموار ذرات (SPH) پیاده‌سازی شده است. برای غلبه بر چالش‌های ناشی از تراکم‌ناپذیری در این دسته از مواد و دشواری‌های عددی مرتبط، از روابط ترکیبی استفاده شده است. بر این اساس، مدل SPH در چارچوب تمام لاگرانژی توسعه داده شده که در آن از قوانین بقای مرتبه اول برای مومنتوم خطی، تانسور گرادیان تغییر شکل، تانسور نگاشت سطح و نگاشت حجم بهره گرفته شده است. دقت این پیاده‌سازی با حل چندین مثال استاندارد در رفتار دینامیکی جامدات تحت تغییر شکل‌های بزرگ بررسی شده است. همچنین، نتایج مدل هایپرالاستیک با نتایج حاصل از نرم‌افزار اجزای محدود آباکوس مقایسه شده است. علاوه بر این، رفتار پوست انسان به‌عنوان یک بافت نرم ناهمسانگرد تقویت‌شده با الیاف‌های کلاژن شبیه‌سازی و با نتایج تجربی موجود مقایسه شده است. مقایسه‌ها نشان می‌دهد که مدل SPH به‌طور مؤثری رفتار ناهمسانگرد بافت‌های نرم تحت تغییر شکل‌های بزرگ را شبیه‌سازی می‌کند. یافته‌های این پژوهش نشان می‌دهد که روش بدون شبکه SPH می‌تواند به‌عنوان یک چارچوب برای مدل‌سازی رفتار بافت‌های نرم با تغییر شکل‌های پیچیده مورد استفاده قرار گیرد.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Modeling Anisotropic Hyperelastic Behavior of Soft Tissues Using the Smoothed Particle Hydrodynamics Method

نویسندگان [English]

  • Hojjat Badnava
  • Ali Nikpour
Behbahan Khatam Alanbia University of Technology, Behbahan, Khuzestan, 63616-63973, Iran
چکیده [English]

Modeling the mechanical behavior of many soft tissues and certain man-made synthetic materials requires accurate representation of their complex and anisotropic nature. In this research, we implement the anisotropic hyperelastic Holzapfel-Gasser-Ogden (HGO) model using the meshless smoothed particle hydrodynamics (SPH) method. To overcome the challenges of incompressibility, inherent in this class of materials and the associated difficulties of the numerical solution, mixed formulations are employed. Accordingly, the SPH model is developed within a total Lagrangian framework, utilizing first-order conservation laws for linear momentum, the deformation gradient tensor, surface mapping and volume mapping. The accuracy of the implementation is demonstrated by solving several well-known benchmarks in the dynamic behavior of solids undergoing large deformations. Additionally, the results of the hyperelastic model are compared with those obtained from the finite element software Abaqus. Furthermore, the behavior of human skin as an anisotropic soft tissue reinforced with collagen fibers is simulated and compared with available experimental results. The comparisons show that the SPH model effectively simulates the anisotropic behavior of soft tissues under large deformations. The findings of this research indicate that the meshless SPH method can be utilized as a framework for modeling the behavior of soft tissues with complex deformations.

کلیدواژه‌ها [English]

  • Anisotropic hyperelasticity
  • SPH
  • Meshfree
  • Soft tissue
  • Incompressibility
[1] Schröder, J., and Neff, P., ‟Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions”, International Journal of Solids and Structures, Vol. 40, No. 2, pp. 401-445, 2003.
[2] Balzani, D., Neff, P., Schröder, J., and Holzapfel, G. A., ‟A polyconvex framework for soft biological tissues. Adjustment to experimental data”, International Journal of Solids and Structures, Vol. 43, No. 20, pp. 6052-6070, 2006.
[3] Schröder, J., Neff, P., and Ebbing, V., ‟Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors”, Journal of the Mechanics and Physics of Solids, Vol. 56, No. 12, pp. 3486-3506, 2008.
[4] Itskov, M. and Aksel, N., "A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function", International Journal of Solids and Structures, Vol. 41, No. 14, pp. 3833-3848, 2004.
[5] Schröder, J., Neff, P., and Balzani, D., "A variational approach for materially stable anisotropic hyperelasticity", International Journal of Solids and Structures, Vol. 42, No. 15, pp. 4352-4371, 2005.
[6] Holzapfel, G. A., "Arterial Tissue in Health and Disease: Experimental Data, Collagen-Based Modeling and Simulation, Including Aortic Dissection," in Biomechanical Modelling at the Molecular, Cellular and Tissue Levels, G. A. Holzapfel and R. W. Ogden Eds. Vienna: Springer Vienna, 2009, pp. 259-344.
[7] Gasser, T. C., Ogden, R. W., and Holzapfel, G. A., "Hyperelastic modelling of arterial layers with distributed collagen fibre orientations", Journal of The Royal Society Interface, Vol. 3, No. 6, pp. 15-35, 2006.
[32] Ren, J. S., "Effects of dispersion of fiber orientation on the mechanical property of the arterial wall", J Theor Biol, Vol. 301, pp. 153-60, 2012.
[8] Grytz, R. and Meschke, G., "Constitutive modeling of crimped collagen fibrils in soft tissues", Journal of the Mechanical Behavior of Biomedical Materials, Vol. 2, No. 5, pp. 522-533, 2009.
[10] Nolan, D. R., Gower, A. L., Destrade, M., Ogden, R. W., and McGarry, J. P., "A robust anisotropic hyperelastic formulation for the modelling of soft tissue", J Mech Behav Biomed Mater, Vol. 39, pp. 48-60, 2014.
[11] Skacel, P. and Bursa, J., "Comparison of constitutive models of arterial layers with distributed collagen fibre orientations", Acta Bioeng Biomech, Vol. 16, No. 3, pp. 47-58, 2014.
[12] Weisbecker, H., Unterberger, M. J., and Holzapfel, G. A., "Constitutive modelling of arteries considering fibre recruitment and three-dimensional fibre distribution", J R Soc Interface, Vol. 12, No. 105, 2015.
[13] Pranavi, D., Steinmann, P., and Rajagopal, A., "A unifying finite strain modeling framework for anisotropic mixed-mode fracture in soft materials", Computational Mechanics, Vol. 73, No. 1, pp. 123-137, 2024.
[14] Chittajallu, S., Richhariya, A., Tse, K. M., and Chinthapenta, V., "A Review on Damage and Rupture Modelling for Soft Tissues", Bioengineering (Basel), Vol. 9, No. 1, 2022.
[15] Gil, A. J. and Ortigosa, R., "A new framework for large strain electromechanics based on convex multi-variable strain energies: Variational formulation and material characterisation", Computer Methods in Applied Mechanics and Engineering, Vol. 302, pp. 293-328, 2016.
[16] Doll, S., Schweizerhof, K., Hauptmann, R., and Freischläger, C., "On Volumetric locking of low‐order solid and solid‐shell elements for finite elastoviscoplastic deformations and selective reduced integration", Engineering Computations, Vol. 17, No. 7, pp. 874-902, 2000.
[17] Simo, J. C. and Rifai, M. S., "A class of mixed assumed strain methods and the method of incompatible modes", International Journal for Numerical Methods in Engineering, Vol. 29, No. 8, pp. 1595-1638, 1990.
[18] Hughes, T. J. R., "Generalization of selective integration procedures to anisotropic and nonlinear media", International Journal for Numerical Methods in Engineering, Vol. 15, No. 9, pp. 1413-1418, 1980.
[19] Sussman, T. and Bathe, K.-J., "A finite element formulation for nonlinear incompressible elastic and inelastic analysis", Computers & Structures, Vol. 26, No. 1, pp. 357-409, 1987.
[20] Bathe, K.-J. and Dvorkin, E. N., "A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation", International Journal for Numerical Methods in Engineering, Vol. 21, No. 2, pp. 367-383, 1985.
[21] Neto, E. A. d. S., Pires, F. M. A., and Owen, D. R. J., "F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I: formulation and benchmarking", International Journal for Numerical Methods in Engineering, Vol. 62, No. 3, pp. 353-383, 2005.
[22] de Souza Neto, E. A., Peric, D., Owen, D. R. J., "Finite Strain Hyperelasticity," in Computational Methods for Plasticity, 2008, pp. 517-571.
[23] Bonet J, Wood R., "HYPERELASTICITY," in Nonlinear Continuum Mechanics for Finite Element Analysis, J. Bonet and R. D. Wood Eds., 2 ed. Cambridge: Cambridge University Press, 2008, pp. 155-187.
[24] Schröder, J., Wriggers, P., and Balzani, D., "A new mixed finite element based on different approximations of the minors of deformation tensors", Computer Methods in Applied Mechanics and Engineering, Vol. 200, No. 49, pp. 3583-3600, 2011.
[25] Bonet, J., Gil, A. J., Lee, C. H., Aguirre, M., and Ortigosa, R., "A first order hyperbolic framework for large strain computational solid dynamics. Part I: Total Lagrangian isothermal elasticity", Computer Methods in Applied Mechanics and Engineering, Vol. 283, pp. 689-732, 2015.
[26] Hesch, C. et al., "A framework for polyconvex large strain phase-field methods to fracture", Computer Methods in Applied Mechanics and Engineering, Vol. 317, pp. 649-683, 2017.
[27] Gil, A. J., Lee, C. H., Bonet, J., and Ortigosa, R., "A first order hyperbolic framework for large strain computational solid dynamics. Part II: Total Lagrangian compressible, nearly incompressible and truly incompressible elasticity", Computer Methods in Applied Mechanics and Engineering, Vol. 300, pp. 146-181, 2016.
[28] Haider, J., Lee, C. H., Gil, A. J., and Bonet, J., "A first-order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total Lagrangian scheme", International Journal for Numerical Methods in Engineering, Vol. 109, No. 3, pp. 407-456, 2017.
[29] Vidal, Y., Bonet, J., and Huerta, A., "Stabilized updated Lagrangian corrected SPH for explicit dynamic problems", International Journal for Numerical Methods in Engineering, Vol. 69, No. 13, pp. 2687-2710, 2007.
[30] Lee, C. H., Gil, A. J., and Bonet, J., "Development of a cell centred upwind finite Volume algorithm for a new conservation law formulation in structural dynamics", Computers & Structures, Vol. 118, pp. 13-38, 2013.
[31] Lee, C. H., Gil, A. J., and Bonet, J., "Development of a stabilised Petrov–Galerkin formulation for conservation laws in Lagrangian fast solid dynamics", Computer Methods in Applied Mechanics and Engineering, Vol. 268, pp. 40-64, 2014.
[32] Lee, C. H., Gil, A. J., Greto, G., Kulasegaram, S., and Bonet, J., "A new Jameson–Schmidt–Turkel Smooth Particle Hydrodynamics algorithm for large strain explicit fast dynamics", Computer Methods in Applied Mechanics and Engineering, Vol. 311, pp. 71-111, 2016.
[33] Gil, A. J., Lee, C. H., Bonet, J., and Aguirre, M., "A stabilised Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics", Computer Methods in Applied Mechanics and Engineering, Vol. 276, pp. 659-690, 2014.
[34] Lee, C. H., Gil, A. J., Hassan, O. I., Bonet, J., and Kulasegaram, S., "A variationally consistent Streamline Upwind Petrov–Galerkin Smooth Particle Hydrodynamics algorithm for large strain solid dynamics", Computer Methods in Applied Mechanics and Engineering, Vol. 318, pp. 514-536, 2017.
[35] Lee, C. H., Gil, A. J., Ghavamian, A., and Bonet, J., "A Total Lagrangian upwind Smooth Particle Hydrodynamics algorithm for large strain explicit solid dynamics", Computer Methods in Applied Mechanics and Engineering, Vol. 344, pp. 209-250, 2019.
[36] Badnava, H., Lee, C. H., Nourbakhsh, S. H., and Refachinho de Campos, P. R., "A stabilised Total Lagrangian Element-Free Galerkin method for transient nonlinear solid dynamics", Computational Mechanics, 2024.
[37] de Campos, P. R. R., Gil, A. J., Lee, C. H., Giacomini, M., and Bonet, J., "A New Updated Reference Lagrangian Smooth Particle Hydrodynamics algorithm for isothermal elasticity and elasto-plasticity", Computer Methods in Applied Mechanics and Engineering, Vol. 392, p. 114680, 2022.
[38] Ghavamian, A., Lee, C. H., Gil, A. J., Bonet, J., Heuzé, T., and Stainier, L., "An entropy-stable Smooth Particle Hydrodynamics algorithm for large strain thermo-elasticity", Computer Methods in Applied Mechanics and Engineering, Vol. 379, p. 113736, 2021.
[39] Badnava, H., "Modeling metal forming processes using an Updated Lagrangian smoothed particle hydrodynamics method", Journal of Manufacturing Innovations, Vol. 1, No. 2, pp. 17-27, 2023.
[40] Ghadampour, Z., Hashemi, M. R., taleb beydokhti, N., Hossein Nikseresht, A., “Comparison of two projection methods in SPH for modeling flow under a gate”, Journal of Computational Methods in Engineering, Vol. 31, No. 2, pp. 79-97, 2013. (In Farsi).
[41] Edalati, H.,  Soltani, B., “Analysis of Thin Isotropic and Orthotropic Plates with Element-Free Galerkin Method and Various Geometric Shapes”, Journal of Computational Methods in Engineering, Vol. 34, No. 2, pp. 143-157, 2016. (In Farsi).
[42] Heidargheitaghi, F. F., Ghadiri Rad, M. H., Kazemi, M., “Buckling Analysis of Non-Prismatic Columns Subjected to Non-Uniform Loading Using the Meshless Local Petrov-Galerkin Method”, Journal of Computational Methods in Engineering, Vol. 40, No. 2, pp 39-56, 2022. (In Farsi).
[43] Hassan, O. I., Ghavamian, A., Lee, C. H., Gil, A. J., Bonet, J., and Auricchio, F., "An upwind vertex centred finite Volume algorithm for nearly and truly incompressible explicit fast solid dynamic applications: Total and Updated Lagrangian formulations", Journal of Computational Physics: X, Vol. 3, p. 100025, 2019.
[44] Bonet, J. and Kulasegaram, S., "Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations", International Journal for Numerical Methods in Engineering, Vol. 47, No. 6, pp. 1189-1214, 2000.
[45] Abd. Karim, I., Hean Lee, C., J. Gil, A., and Bonet, J., "A two-step Taylor-Galerkin formulation for fast dynamics", Engineering Computations, Vol. 31, No. 3, pp. 366-387, 2014.
[46] Holzapfel, G. A., Niestrawska, J. A., Ogden, R. W., Reinisch, A. J., and Schriefl, A. J., "Modelling non-symmetric collagen fibre dispersion in arterial walls", J R Soc Interface, Vol. 12, No. 106, 2015.
[47] Holzapfel, G. A., Ogden, R. W., and Sherifova, S., "On fibre dispersion modelling of soft biological tissues: a review", Proc Math Phys Eng Sci, Vol. 475, No. 2224, p. 20180736, 2019.
[48] Ní Annaidh, A., Bruyère, K., Destrade, M., Gilchrist, M. D., and Otténio, M., "Characterization of the anisotropic mechanical properties of excised human skin", Journal of the Mechanical Behavior of Biomedical Materials, Vol. 5, No. 1, pp. 139-148, 2012.
[49] Ní Annaidh, A. et al., "Automated Estimation of Collagen Fibre Dispersion in the Dermis and its Contribution to the Anisotropic Behaviour of Skin", Annals of Biomedical Engineering, Vol. 40, No. 8, pp. 1666-1678, 2012.

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