A generalized statistical approach for modeling fiber-reinforced materials.

*(English)*Zbl 1408.74036Summary: We present a generalized statistical approach for the description of fully three-dimensional fiber-reinforced materials, resulting from the composition of two independent probability distribution functions of two spherical angles. We discuss the consequences of the proposed formulation on the constitutive behavior of fibrous materials. Upon suitable assumptions, the generalized formulation recovers existing alternative models, based on averaged structure tensors both at first- and second-order approximations. We demonstrate that the generalized formulation embeds standard behaviors of fiber-reinforced materials such as planar isotropy and transverse isotropy, while any intermediate behavior is easily obtained through the calibration of two material parameters. We illustrate the performance of the model by means of uniaxial and biaxial tests. For uniaxial loading, we introduce a preliminary discussion concerning the generalized tension-compression switch procedure.

##### Keywords:

fiber-reinforced materials; fourth pseudo-invariant; isochoric anisotropic hyperelasticity; statistical fiber distribution
Full Text:
DOI

##### References:

[1] | Volokh K Y (2016) Mechanics of soft materials. Springer, Singapore |

[2] | Pandolfi, A; Manganiello, F, A model for the human cornea: constitutive formulation and numerical analysis, Biomech Model Mechanobiol, 5, 237-246, (2006) |

[3] | Hurtado, DE; Villaroel, N; Retamal, J; Bugedo, G; Bruhn, A, Improving the accuracy of registration-based biomechanical analysis: a finite element approach to lung regional strain quantification, IEEE Trans Med Imaging, 35, 580-588, (2016) |

[4] | Cyron CJ, Müller KW, Bausch AR, Wall WA (2013) Micromechanical simulations of biopolymer networks with finite elements. J Comput Phys 244:236-251 · Zbl 1377.92010 |

[5] | Gizzi, A; Vasta, M; Pandolfi, A, Modeling collagen recruitment in hyperelastic bio-material models with statistical distribution of the fiber orientation, Int J Eng Sci, 78, 48-60, (2014) · Zbl 1423.74616 |

[6] | Cyron, CJ; Aydin, RC; Humphrey, JD, A homogenized constrained mixture (and mechanical analog) model for growth and remodeling of soft tissue, Biomech Model Mechanobiol, 15, 1389-1403, (2016) |

[7] | Wu, JZ; Herzog, W; Federico, S, Finite element modeling of finite deformable, biphasic biological tissues with transversely isotropic statistically distributed fibers: toward a practical solution, Zeitschrift für angewandte Mathematik und Physik, 67, 26, (2016) · Zbl 1338.74015 |

[8] | Sacks, MS, Incorporation of experimentally-derived fiber orientation into a structural constitutive model for planar collagenous tissues, J Biomech Eng, 125, 280-287, (2003) |

[9] | Federico, S; Gasser, TC, Nonlinear elasticity of biological tissues with statistical fibre orientation, J R Soc Interface, 7, 955-966, (2010) |

[10] | Lanir, Y, Constitutive equations for fibrous connective tissues, J Biomech, 16, 1-12, (1983) |

[11] | Gasser, TC; Ogden, RW; Holzapfel, GA, Hyperelastic modeling of arterial layers with distributed collagen fibre orientations, J R Soc Interface, 3, 15-35, (2006) |

[12] | Alastrué, V; Saez, P; Martinez, MA; Doblaré, M, On the use of the Bingham statistical distribution in microsphere-based constitutive models for arterial tissue, Mech Res Commun, 37, 700-706, (2010) · Zbl 1272.74470 |

[13] | Spencer AJM (1989) Continuum mechanics. Longman Group Ltd, London · Zbl 0427.73001 |

[14] | Saccomandi G, Ogden RW (eds) (2004) Mechanics and thermomechanics of rubberlike solids, vol. 452, Springer |

[15] | Merodio, J; Ogden, RW, Mechanical response of fiber-reinforced incompressible non-linearly elastic solids, Int J Non-Linear Mech, 40, 213-227, (2005) · Zbl 1349.74057 |

[16] | Horgan, CO; Saccomandi, G, A new constitutive theory for fiber-reinforced incompressible nonlinearly elastic solids, J Mech Phys Solids, 53, 1985-2015, (2005) · Zbl 1176.74026 |

[17] | Federico, S; Herzog, W, Towards an analytical model of soft tissues, J Biomech, 41, 3309-3313, (2008) |

[18] | Pandolfi, A; Vasta, M, Fiber distributed hyperelastic modeling of biological tissues, Mech Mater, 44, 151-162, (2012) |

[19] | Holzapfel, GA; Niestrawska, JA; Ogden, RW; Reinisch, AJ; Schriefl, AJ, Modelling non-symmetric collagen fibre dispersion in arterial walls, J R Soc Interface, 12, 20150188, (2015) |

[20] | Vasta, M; Gizzi, A; Pandolfi, A, On three- and two-dimensional fiber distributed models of biological tissues, Probab Eng Mech, 37, 170-179, (2014) |

[21] | Holzapfel, GA; Ogden, RW, On the tension-compression switch in soft fibrous solids, Eur J Mech A, 49, 561-569, (2015) · Zbl 1406.74496 |

[22] | Vergori, L; Destrade, M; McGarry, P; Ogden, RW, On anisotropic elasticity and questions concerning its finite element implementation, Comput Mech, 52, 1185-1197, (2013) · Zbl 1388.74015 |

[23] | Nolan, DR; Gower, AL; Destrade, M; Ogden, RW; McGarry, JP, A robust anisotropic hyperelastic formulation for the modelling of soft tissue, J Mech Behav Biomed Mater, 39, 48-60, (2014) |

[24] | Petsche, SJ; Pinsky, PM, The role of 3-d collagen organization in stromal elasticity: a model based on x-ray diffraction data and second harmonic-generated images, Biomech Model Mechanobiol, 12, 1101-1113, (2013) |

[25] | Abass, A; Hayes, S; White, N; Sorensen, T; Meek, KM, Transverse depth-dependent changes in corneal collagen lamellar orientation and distribution, J R Soc Interface, 12, 20140717, (2015) |

[26] | Advani, SG; Rucker, CLI, The use of tensors to describe and predict fiber orientation in short fiber composites, J Rheol, 31, 751-784, (1987) |

[27] | Hashlamoun, K; Grillo, A; Federico, S, Efficient evaluation of the material response of tissues reinforced by statistically oriented fibres, Zeitschrift für angewandte Mathematik und Physik, 67, 113, (2016) · Zbl 1432.74152 |

[28] | Latorre, M; Montáns, FJ, Material-symmetries congruency in transversely isotropic and orthotropic hyperelastic materials, Eur J Mech A, 53, 99-106, (2015) · Zbl 1406.74154 |

[29] | Gizzi, A; Pandolfi, A; Vasta, M, Statistical characterization of the anisotropic strain energy in soft materials with distributed fibers, Mech Mater, 92, 119-138, (2016) |

[30] | Tomic, A; Grillo, A; Federico, S, Poroelastic materials reinforced by statistically oriented fibres—numerical implementation and application to articular cartilage, IMA J Appl Math, 79, 1027-1059, (2014) · Zbl 1299.74171 |

[31] | Hashlamoun, K; Federico, S, Transversely isotropic higher-order averaged structure tensors, ZAMP, 68, 88, (2017) · Zbl 06816892 |

[32] | Volokh KY (2017) On arterial fiber dispersion and auxetic effect. J Biomech (in press) |

[33] | Aydin RC, Brandstaeter S, Braeu FA, Steinberger M, Marcus RP, Nikoloau K, Notohamiprodjo M, Cyron CJ (2017) Experimental characterization of the biaxial mechanical properties of porcine gastric tissue. J Mech Behav Biomed Mater (in press) |

[34] | Latorre M, Montáns FJ (2016) On the tension-compression switch of the Gasser-Ogden-Holzapfel model: analysis and a new pre-integrated proposal. J Mech Behav Biomed Mater 57:175-189 |

[35] | Pisano, AA; Fuschi, P; Domenico, D, Failure modes prediction of multi-pin joints FRP laminates by limit analysis, Composites B, 46, 197-206, (2013) |

[36] | Pisano, AA; Fuschi, P; Domenico, D, Peak load prediction of multi-pin joints FRP laminates by limit analysis, Compos Struct, 96, 763-772, (2013) |

[37] | Slesarenko, V; Volokh, KY; Aboudi, J; Rudykh, S, Understanding the strength of bioinspired soft composites, Int J Mech Sci, 131-132, 171-178, (2017) |

[38] | Fuschi, P; Pisano, AA; Domenico, D, Plane stress problems in nonlocal elasticity: finite element solutions with a strain-difference-based formulation, J Math Anal Appl, 431, 714-736, (2015) · Zbl 1328.74015 |

[39] | Maceri, F; Marino, M; Vairo, G, A unified multiscale mechanical model for soft collagenous tissues with regular fiber arrangement, J Biomech, 43, 355-363, (2010) |

[40] | Marino, M; Vairo, G, Stress and strain localization in stretched collagenous tissues via a multiscale modelling approach, Comput Methods Biomech Biomed Eng, 17, 11-30, (2014) |

[41] | Gizzi, A; Cherubini, C; Pomella, N; Persichetti, P; Vasta, M; Filippi, S, Computational modeling and stress analysis of columellar biomechanics, J Mech Behav Biomed Mater, 15, 46-58, (2012) |

[42] | Gizzi, A; Pandolfi, A; Vasta, M, Viscoelectromechanics modeling of intestine wall hyperelasticity, Int J Comput Methods Eng Sci Mech, 17, 143-155, (2016) |

[43] | Pandolfi, A; Gizzi, A; Vasta, M, Coupled electro-mechanical models of fiber-distributed active tissues, J Biomech, 49, 2436-2444, (2016) · Zbl 1394.74126 |

[44] | Cyron, CJ; Aydin, RC, Mechanobiological free energy: a variational approach to tensional homeostasis in tissue equivalents, ZAMM, 97, 1011-1019, (2017) |

[45] | Pandolfi A, Gizzi A, Vasta M (2017) Visco-electro-elastic models of fiber-distributed active tissues. Meccanica 52:3399-3415 · Zbl 1394.74126 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.