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

International Journal for Numerical Methods in Engineering

Volumn 79, Issue 4, 2009, Pages 474-504

  Alastrue V ^a,b   Martinez M A ^a,b   Menzel, A ^c,d   Doblare M ^a,b  

^a UNIVERSITY OF ZARAGOZA   (Spain)

^b BIOMATERIALES Y NANOMEDICINA CIBER BBN   (Spain)

^c TU DORTMUND UNIVERSITY   (Germany)

^d LUND UNIVERSITY   (Sweden)

Author keywords

Anisotropy; Collagen fibres; Integration of unit spheres; Microsphere; Non linear transformation; Soft tissue

Indexed keywords

COLLAGEN FIBRES; COMPUTATIONAL COSTS; CONSTITUTIVE LAW; DISCRETIZATIONS; FINITE ELEMENT; HYPER ELASTIC; INHOMOGENEOUS DEFORMATION; INTEGRATION POINTS; INTEGRATION PROCESS; INTEGRATION SCHEME; INTERNAL VARIABLES; ISOTROPIC MATERIALS; MACROSCOPIC RESPONSE; MECHANICAL BEHAVIOUR; MECHANICAL FEATURE; MICRO-STRUCTURAL; MODELLING TOOLS; NON-LINEAR TRANSFORMATION; NON-LINEAR TRANSFORMATIONS; NUMERICAL TECHNIQUES; ORIENTATION DISTRIBUTION FUNCTION; SIMULATION RESULT; SOFT BIOLOGICAL TISSUE; SOFT TISSUE; SPHERE DISCRETIZATIONS; SYMMETRIC FUNCTIONS; VON MISES;

ANISOTROPY; COLLAGEN; COMPUTATIONAL EFFICIENCY; CONSTITUTIVE MODELS; DISTRIBUTION FUNCTIONS; HISTOLOGY; MATHEMATICAL TRANSFORMATIONS; MICROSPHERES; SPHERES; STRESS ANALYSIS; TISSUE;

INTEGRATION;

EID: 68649106680     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2577     Document Type: Article

References (38)

1. Bažant ZP, Oh BH. Microplane model for progressive fracture of concrete and rock. Journal of Structural Engineering (ASCE) 1985; 111:559-582.
2. Bažant ZP, Prat PC. Microplane model for brittle-plastic material: I. Journal of Structural Engineering (ASCE) 1988; 114(10):1672-1688.
3. Bažant ZP, Prat PC. Microplane model for brittle-plastic material: II verification. Journal of Structural Engineering (ASCE) 1988; 114(10):1689-1702.
4. Kuhl E, Ramm E. Microplane modelling of cohesive frictional materials. European Journal of Mechanics- A/Solids 2000; 19(special issue):S121-S143.
5. Kuhl E, D'Addetta GA, Herrmann HJ, Ramm E. A comparison of discrete granular material models with continuous microplane formulations. Granular Matter 2000; 2(3):113-121.
6. Bažant ZP, Adley MD, Carol I, Jirásek M, Akers SA, Rohani B, Cargile JD, Caner FC. Large-strain generalization of microplane constitutive model for concrete and application. Journal of Structural Engineering (ASCE) 2000; 126(9):971-980.
7. Caner FC, Guo Z, Moran B, Bažant ZP, Carol I. Hyperelastic anisotropic microplane constitutive model for annulus fibrosus. Journal of Biomechanical Engineering 2007; 129(5):632-641.
8. Miehe C, Göktepe S, Lulei F. A micro-macro approach to rubber-like materials-part I: the non-affine micro-sphere model of rubber elasticity. Journal of the Mechanics and Physics of Solids 2004; 52(11):2617-2660.
9. Miehe C, Göktepe S. A micro-macro approach to rubber-like materials-part II: the micro-sphere model of finite rubber viscoelasticity. Journal of the Mechanics and Physics of Solids 2005; 53(10):2231-2258.
10. Göktepe S, Miehe C. A micro-macro approach to rubber-like materials-part III: the micro-sphere model of anisotropic mullins-type damage. Journal of the Mechanics and Physics of Solids 2005; 53(10):2259-2283.
11. Alastrué V, Martínez MA, Doblaré M, Menzel A. Anisotropic micro-sphere-based finite elasticity applied to blood vessel modelling. Journal of the Mechanics and Physics of Solids 2009; 57(1):178-203.
12. Stroud A. Approximate Calculation of Multiple Integrals. Prentice-Hall: Englewood Cliffs, NJ, 1971.
13. Heo S, Xu Y. Constructing fully symmetric cubature formulae for the sphere. Mathematics of Computation 2001; 70:269-279.
14. Bažant ZP, Oh BH. Efficient numerical integration on the surface of a sphere. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 1986; 66(1):37-49.
15. Baant ZP, Caner FC, Carol I, Adley MD, Akers SA. Microplane model M4 for concrete: I. formulation with work-conjugate deviatoric stress. Journal of Structural Engineering (ASCE) 2000; 126(9):944-953.
16. Telles JCF. A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals. International Journal for Numerical Methods in Engineering 1987; 24(5):959-973.
17. Elliott D. Sigmoidal transformations and the trapezoidal rule. Journal of the Australian Mathematical Society, Series B-Applied Mathematics 1998; 40(E):E77-E137.
18. Yun BI, Kim P. A new sigmoidal transformation for weakly singular integrals in the boundary element method. SIAM Journal on Scientific Computing 2003; 24(4):1203-1217.
19. Doblaré M, Gracia L. On non-linear transformations for the integration of weakly-singular and cauchy principal value integrals. International Journal for Numerical Methods in Engineering 1997; 40(18):3325-3358.
20. Singh KM, Tanaka M. On non-linear transformations for accurate numerical evaluation of weakly singular boundary integrals. International Journal for Numerical Methods in Engineering 2001; 50(8):2007-2030.
21. Yun BI. An extended sigmoidal transformation technique for evaluating weakly singular integrals without splitting the integration interval. SIAM Journal on Scientific Computing 2003; 25(1):284-301.
22. Elliott D, Johnston PR. The iterated sinh transformation. International Journal for Numerical Methods in Engineering 2008; 75(1):43-57.
23. Johnston BM, Johnston PR, Elliott D. A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals. International Journal for Numerical Methods in Engineering 2007; 69(7):1460-1479.
24. Yun BI. A generalized non-linear transformation for evaluating singular integrals. International Journal for Numerical Methods in Engineering 2006; 65(12):1947-1969.
25. Fisher R. Dispersion on a sphere. Proceedings of the Royal Society of London, Series A-Mathematical and Physical Sciences 1953; 217:295-305.
26. Gasser TC, Ogden RW, Holzapfel GA. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. Journal of the Royal Society Interface 2006; 3:15-35.
27. Weisstein EW. 'Erfi.' From MathWorld-a Wolfram web resource, 2004. http://mathworld.wolfram.com/Erfi.html.
28. Flory PJ. Thermodynamic relations for high elastic materials. Transactions of the Faraday Society 1961; 57: 829-838.
29. Arruda EM, Boyce MC. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids 1993; 41(2):389-412.
30. Bischoff JE, Arruda EA, Grosh K. A microstructurally based orthotropic hyperelastic constitutive law. Journal of Applied Mechanics 2002; 69(5):570-579.
31. Kuhl E, Garikipati K, Arruda EM, Grosh K. Remodeling of biological tissue: mechanically induced reorientation of a transversely isotropic chain network. Journal of the Mechanics and Physics of Solids 2005; 53(7):1552-1573.
32. Alastrué V, Rodríguez JF, Calvo B, Doblaré M. Structural damage models for fibrous biological soft tissues. International Journal of Solids and Structures 2007; 44(18-19):5894-5911.
33. Kuhl E, Menzel A, Garikipati K. On the convexity of transversely isotropic chain network models. Philosophical Magazine 2006; 86:3241-3258.
34. Marko JF, Siggia ED. Stretching DNA. Macromolecules 1995; 28(26):8759-8770.
35. Bustamante C, Bryant Z, Smith SB. Ten years of tension: single-molecule DNA mechanics. Nature 2003; 421(6921):423-427.
36. Holzapfel GA, Sommer G, Gasser CT, Regitnig P. Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling. American Journal of Physiology-Heart and Circulatory Physiology 2005; 289(5):H2048-H2058.
37. Holzapfel GA, Gasser TC, Ogden RW. A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal of Elasticity 2000; V61(1):1-48.
38. Bingham C. An antipodally symmetric distribution on the sphere. Annals of Statistics 1974; 2(6):1201-1225.