Abstract
An anisotropic damage model for soft fibered tissue is presented in this paper, using a multi-scale scheme and focusing on the directionally dependent behavior of these materials. For this purpose, a micro-structural or, more precisely, a microsphere-based approach is used to model the contribution of the fibers. The link between micro-structural contribution and macroscopic response is achieved by means of computational homogenization, involving numerical integration over the surface of the unit sphere. In order to deal with the distribution of the fibrils within the fiber, a von Mises probability function is incorporated, and the mechanical (phenomenological) behavior of the fibrils is defined by an exponential-type model. We will restrict ourselves to affine deformations of the network, neglecting any cross-link between fibrils and sliding between fibers and the surrounding ground matrix. Damage in the fiber bundles is introduced through a thermodynamic formulation, which is directly included in the hyperelastic model. When the fibers are stretched far from their natural state, they become damaged. The damage increases gradually due to the progressive failure of the fibrils that make up such a structure. This model has been implemented in a finite element code, and different boundary value problems are solved and discussed herein in order to test the model features. Finally, a clinical application with the material behavior obtained from actual experimental data is also presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alastrué V, Peña E, Martinez MA, Doblare M (2007a) Assessing the use of the “opening angle method” to enforce residual stresses in patient-specific arteries. Ann Biomed Eng 35(10): 1821–1837
Alastrué V, Rodriguez J, Calvo B, Doblare M (2007b) Structural damage models for fibrous biological soft tissues. Int J Solids Struct 44(18–19): 5894–5911
Alastrué V, Martinez MA, Doblare M, Menzel A (2009a) Anisotropic micro-sphere-based finite elasticity applied to blood vessel modelling. J Mech Phys Solids 57(1): 178–203
Alastrué V, Martinez MA, Menzel A, Doblare M (2009b) On the use of non-linear transformations for the evaluation of anisotropic rotationally symmetric directional integrals. Application to the stress analysis in fibred soft tissues. Intl J Numer Meth Eng 79(4): 474–504
Alastrue V, Saez P, Martinez MA, Doblare M (2010) On the use of the bingham statistical distribution in microsphere-based constitutive models for arterial tissue. Mech Res Commun 37(8): 700–706
Ateshian GA (2007) On the theory of reactive mixtures for modeling biological growth. Biomech Model Mech 6(6): 423–445
Balzani D, Schroder J, Gross D (2006) Simulation of discontinuous damage incorporating residual stresses in circumferentially overstretched atherosclerotic arteries. Acta Biomater 2(6): 609–618
Bažant P, Oh BH (1986) Efficient numerical integration on the surface of a sphere. ZAMM-Z Angew Math Comput Mech 66(1): 37–49
Boehler JP (1987) Applications of tensor functions in solid mechanics. CISM courses and lectures. Springer, Berlin
Buehler MJ (2008) Nanomechanics of collagen fibrils under varying cross-link densities: atomistic and continuum studies. J Mech Behav Biomed Mater 1(1): 59–67
Calvo B, Peña E, Martinez MA, Doblaré M (2007) An uncoupled directional damage model for fibred biological soft tissues. Formulation and computational aspects. Int J Numer Meth Eng 69(10): 2036–2057
Caner FC, Carol I (2006) Microplane constitutive model and computational framework for blood vessel tissue. J Biomech Eng 128(3): 419–427
Carew TE, Vaishnav RN, Patel DJ (1968) Compressibility of the arterial wall. Circ Res 23(1): 61–68
Chuong CJ, Fung YC (1984) Compressibility and constitutive equation of arterial wall in radial compression experiments. J Biomech 17(1): 35–40
Dal H, Kaliske M (2009) A micro-continuum-mechanical material model for failure of rubber-like materials: application to ageing-induced fracturing. J Mech Phys Solids 57(8): 1340–1356
Demiray H, Weizsacker HW, Pascale K, Erbay H (1988) A stress-strain relation for a rat abdominal aorta. J Biomech 21(5): 369–374
Ehret AE, Itskov M (2009) Modeling of anisotropic softening phenomena: application to soft biological tissues. Int J Plasticity 25(5): 901–919
Ehret AE, Itskov M, Schmid H (2010) Numerical integration on the sphere and its effect on the material symmetry of constitutive equations—a comparative study. Intl J Numer Meth Eng 81(2): 189–206
Flory PJ (1961) Thermodynamic relations for high elastic materials. T Faraday Soc 57: 829–838
Fung YC (1990) Biomechanics: mechanical properties of living tissues. Springer, Berlin
Gasser TC, Holzapfel GA (2007) Finite element modeling of balloon angioplasty by considering overstretch of remnant non-diseased tissues in lesions. Comput Mech 40(1): 47–60
Gasser TC, Ogden RW, Holzapfel GA (2006) Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J Roy Soc Interface 3: 15–35
Göktepe S, Miehe C (2005) A micro-macro approach to rubber-like materials. Part iii: the micro-sphere model of anisotropic mullins-type damage. J Mech Phys Solids 53(10): 2259–2283
Hardin RH, Sloane NJA (1996) Mclaren’s improved snub cube and other new spherical designs in three dimensions. Discrete Comput Biol Med Geom 15(4): 429–441
Head DA, Levine AJ, MacKintosh FC (2003) Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletal and semiflexible polymer networks. Phys Rev E 68(6): 061907
Heo S, Xu Y (2001) Constructing fully symmetric cubature formulae for the sphere. Math Comput 70: 269–279
Himpel G, Menzel A, Kuhl E, Steinmann P (2008) Time-dependent fibre reorientation of transversely isotropic continua. Finite element formulation and consistent linearization. Intl J Numer Meth Eng 73(10): 1413–1433
Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley, London
Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61(1): 1–48
Holzapfel GA, Sommer G, Gasser CT, Regitnig P (2005) Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling. Am J Physiol Heart Circ Physiol 289(5): H2048–H2058
Humphrey JD (1995) Mechanics of the arterial wall: review and directions. Crit Rev Bio Eng 23(1–2): 1–162
Humphrey JD, Rajagopal KR (2002) A constrained mixture model for growth and remodeling of soft tissues. Math Models Methods Appl Sci 12(3): 407–430
Kuhl E, Askes H, Steinmann P (2006) An illustration of the equivalence of the loss of ellipticity conditions in spatial and material settings of hyperelasticity. Eur J Mech A Solids 25(2): 199–214
Kuhl E, Garikipati K, Arruda EM, Grosh K (2005) Remodeling of biological tissue: mechanically induced reorientation of a transversely isotropic chain network. J Mech Phys Solids 53(7): 1552–1573
Kuhl E, Holzapfel G (2007) A continuum model for remodeling in living structures. J Mater Sci 42(21): 8811–8823
Kuhl E, Ramm E (1999) Simulation of strain localization with gradient enhanced damage models. Comput Mater Sci 16(1–4): 176–185
Landuyt M (2006) Structural quantification of collagen fibers in abdominal aortic aneurysms. Master’s thesis, Royal Institute of Technology in Stockholm, Department of Solid Mechanics and Ghent University, Department of Civil Engineering
Marsden JE, Hughes TJR (1994) Mathematical foundations of elasticity. Dover, New York
Menzel A, Harrysson M, Ristinmaa M (2008) Towards an orientation-distribution-based multi-scale approach for remodelling biological tissues. Comput Meth Biomech Biomed Eng 11(5): 505–524
Menzel A, Steinmann P (2003) A view on anisotropic finite hyper-elasticity. Eur J Mech A Solids 22(1): 71–87
Menzel A, Waffenschmidt T (2009) A microsphere-based remodelling formulation for anisotropic biological tissues. Phil Trans R Soc A 367(1902): 3499–3523
Miehe C (1995) Discontinuous and continuous damage evolution in ogden-type large-strain elastic-materials. Eur J Mech A Solids 14(5): 697–720
Miehe C, Göktepe S (2005) A micro-macro approach to rubber-like materials. Part ii: the micro-sphere model of finite rubber viscoelasticity. J Mech Phys Solids 53(10): 2231–2258
Miehe C, Göktepe S, Lulei F (2004) A micro-macro approach to rubber-like materials—part i: the non-affine micro-sphere model of rubber elasticity. J Mech Phys Solids 52(11): 2617–2660
Natali AN, Pavan PG, Carniel EL, Lucisano ME, Taglialavoro G (2005) Anisotropic elasto-damage constitutive model for the biomechanical analysis of tendons. Med Eng Phys 27(3): 209–214
Oktay H (1994) Continuum damage mechanics of ballon angioplasty. In Internal Report. UMI
Peña E (2011) A rate dependent directional damage model for fibred materials. Application to soft biological tissues. Comput Mech, page Accepted. doi:10.1007/s00466-011-0594-5
Peña E, Alastrue V, Laborda A, Martinez MA, Doblare M (2010) A constitutive formulation of vascular tissue mechanics including viscoelasticity and softening behaviour. J Biomech 43(5): 984–989
Peña E, Doblaré M (2009) An anisotropic pseudo-elastic approach for modelling mullins effect in fibrous biological materials. Mech Res Commun 36(7): 784–790
Peerlings RHJ, Geers MGD, de Borst R, Brekelmans WAM (2001) A critical comparison of nonlocal and gradient-enhanced softening continua. Int J Solids Struct 38(44–45): 7723–7746
Pijaudier-Cabot G, Bazant ZP (1987) Nonlocal damage theory. J Eng Mech 113(10): 1512–1533
Rhodin JAG (1980) Handbook of Physiology, The Cardiovascular System, volume 2, chapter Architecture of the vessel wall, pp 1–31. American Physiological Society, Bethesda, Maryland
Simo JC (1987) On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput Method Appl M 60(2): 153–173
Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, Berlin
Souza Neto EAD, Peric D, Owen DRJ (1998) Continuum modelling and numerical simulation of material damage at finite strains. Arch Comput Meth Eng 5(4): 311–384
Spencer AJM (1971) Theory of invariants. In: Eringen AC (ed) Continuum physiscs. Academic Press, New York, pp 239–253
Steinmann P (1999) Formulation and computation of geometrically non-linear gradient damage. Intl J Numer Meth Eng 46(5): 757–779
Taber LA (1998) A model for aortic growth based on fluid shear and fiber stresses. J Biomech Eng-T ASME 120(3): 348–354
Tang Y, Ballarini R, Buehler MJ, Eppell SJ (2010) Deformation micromechanisms of collagen fibrils under uniaxial tension. J R Soc Interface 7(46): 839–850
Truesdell C, Noll W (2004) The non-linear field theories of mechanics, 3rd edn. Springer, Berlin
Weisstein EW (2004) Erfi. From MathWorld—a wolfram web resource. http://mathworld.wolfram.com/Erfi.html
Author information
Authors and Affiliations
Corresponding author
Electronic Supplementary Material
The Below is the Electronic Supplementary Material.
ESM 1 (MPG 1722 kb)
ESM 2 (MPG 1396 kb)
ESM 3 (MPG 1620 kb)
Rights and permissions
About this article
Cite this article
Sáez, P., Alastrué, V., Peña, E. et al. Anisotropic microsphere-based approach to damage in soft fibered tissue. Biomech Model Mechanobiol 11, 595–608 (2012). https://doi.org/10.1007/s10237-011-0336-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10237-011-0336-9