Abstract
The elasticity tensor providing the power-conjugation of the Green-Naghdi rate of the Kirchhoff stress and the deformation rate is required, e.g. by the commercially available Finite Element package ABAQUS/Standard for the material user subroutine UMAT, used to input material behaviours other than those included in the libraries of the package. This elasticity tensor had been studied in the literature, but its symmetries have only been briefly discussed, and only its component form in Cartesian coordinates was known. In this work, we derived a covariant, component-free expression of this elasticity tensor and thoroughly studied its symmetries. We found that, although symmetry on both pair of feet (indices) has been deemed to be desirable in the literature, the expression of the tensor available to-date in fact possesses only symmetry on the first pair of feet (indices), whereas the second pair lacks symmetry, and therefore carries a skew-symmetric contribution. This contribution is unnecessary, as it is automatically filtered in the contraction of the elasticity tensor with the symmetric deformation rate tensor. In order to avoid carrying this unnecessary skew-symmetric contribution in the computations, we employ a tensor identity that naturally symmetrises the second pair of feet of the elasticity tensor. We demonstrated the validity and robustness of the implementation of the user-defined material based on this tensor representation by simulating a benchmark problem consisting in biaxial tests of porcine and human atrial tissue, with material properties taken from previously performed experiments. We compared the results obtained by means of our user-defined material and those obtained through an equivalent built-in material, and obtained identical results.
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ABAQUS: Abaqus v6.9 Documentation. Simulia, Dassault Systémes, Providence (2009)
Altenbach H., Eremeyev V.A.: On the effective stiffness of plates made of hyperelastic materials with initial stresses. Int. J. Non-Linear Mech. 45, 976–981 (2010)
Altenbach, H., Eremeyev, V.A.: Vibration analysis of non-linear 6-parameter prestressed shell. Meccanica, doi:10.1007/s11012-013-9845-1
Andreaus, U., Giorgio, I., Madeo, A.: Modeling of the interaction between bone tissue and resorbable biomaterial as linear elastic materials with voids. ZAMP—Zeitschrift für Angewandte Mathematik und Physik (J. Appl. Math. Phys.), (in press)
Ateshian G.A., Weiss J.A.: Anisotropic hydraulic permeability under finite deformation. J. Biomech. Eng. 132, 111004 (2010)
Atluri S.N., Cazzani A.: Rotations in computational solid mechanics. Arch. Comput. Methods Eng. 2, 49–138 (1995)
Auffray, N.: On anisotropic polynomial relations for the elasticity tensor. Math. Mech. Solids. doi:10.1177/1081286513507941
Auffray N., Kolev B., Petitot M.: On the algebraic structure of isotropic generalized elasticity theories. J. Elast. 115, 77–103 (2014)
Bellini C., Di Martino E.S.: A mechanical characterization of the porcine atria at the healthy stage and after ventricular tachypacing. J. Biomech. Eng. 134, 021008 (2012)
Bellini C., Di Martino E.S., Federico S.: Mechanical behaviour of the human atria. Ann. Biomed. Eng. 41, 1478–1490 (2013)
Bonet J., Wood R.D.: Nonlinear Continuum Mechanics for Finite Element Analysis, 2nd edn. Cambridge University Press, Cambridge (2008)
Cazzani A., Atluri S.N.: Four-noded mixed finite elements, using unsymmetric stresses, for linear analysis of membranes. Comput. Mech. 11, 229–251 (1993)
Cazzani A., Lovadina C.: On some mixed finite element methods for plane membrane problems. Comput. Mech. 20, 560–572 (1997)
Cuomo M., Contrafatto L.: Stress rate formulation for elastoplastic models with internal variables based on augmented lagrangian regularisation. Int. J. Solids Struct. 37, 3935–3964 (2000)
Curnier A., He Q.-C., Zysset P.: Conewise linear elastic materials. J. Elast. 37, 1–38 (1995)
dell’Isola F., Seppecher P., Madeo A.: How contact interactions may depend on the shape of Cauchy cuts in n-th gradient continua: approach à la D’Alembert. ZAMP—Zeitschrift für Angewandte Mathematik und Physik (J. Appl. Math. Phys.) 63, 1119–1141 (2012)
Epstein M.: The Geometrical Language of Continuum Mechanics. Cambridge University Press, Cambridge (2010)
Eremeyev V.A., Pietraszkiewiecz W.: Local symmetry group in the general theory of elastic shells. J. Elast. 85, 125–152 (2006)
Eremeyev V.A., Pietraszkiewiecz W.: Material symmetry group of the non-linear polar-elastic continuum. Int. J. Solids Struct. 49, 1993–2005 (2012)
Eringen A.C.: Mechanics of Continua. Robert E. Krieger Publishing Company, Huntington (1980)
Federico S.: Covariant formulation of the tensor algebra of non-linear elasticity. Int. J. Non-Linear Mech. 47(2), 273–284 (2012)
Federico, S.: Some remarks on metric and deformation. Math. Mech. Solids. doi:10.1177/1081286513506432 (2013)
Federico S., Grillo A., Giaquinta G., Herzog W.: Convex Fung-type potentials for biological tissues. Meccanica 43, 279–288 (2008)
Fung Y.C.: Elasticity of soft tissues in simple elongation. Am. J. Physiol. 213, 1532–1544 (1967)
Fung Y.C.: Biomechanics: Mechanical Properties of Living Tissue. Springer, New York (1981)
Grillo A., Federico S., Wittum G.: Growth, mass transfer, and remodeling in fiber-reinforced, multi-constituent materials. Int. J. Non-Linear Mech. 47, 388–401 (2012)
Grillo A., Federico S., Wittum G., Imatani S., Giaquinta G., Mićunović M.V.: Evolution of a fibre-reinforced growing mixture. Nuovo Cimento C 32, 97–119 (2009)
Grillo, A., Wittum, G.: Growth and mass transfer in multi-constituent biological materials. In: Simos, T.E., Psihoyios, G., Tsitouras, Ch. (eds.), ICNAAM 2010, International Conference on Numerical Analysis and Applied Mathematics, in AIP Conference Proceedings, vol. 1281, pp. 261–283, Rhodes, Greece, 19–25 September 2010. doi:10.1063/1.3498474
Grillo, A., Wittum, G., Tomic, A., Federico, S.: Remodelling in statistically oriented fibre-reinforced materials and biological tissues. Math. Mech. Solids. doi:10.1177/1081286513515265
Hughes T.J.R., Marsden J.E.: Some applications of geometry in continuum mechanics. Rep. Math. Phys. 12, 35–44 (1977)
Jog C.S.: A concise proof of the representation theorem for fourth-order isotropic tensors. J. Elast. 85, 119–124 (2006)
Lekszycki T., dell’Isola F.: A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials. ZAMM—Zeitschrift für Angewandte Mathematik und Mechanik (J. Appl. Math. Mech.) 92, 426–444 (2012)
Madeo A., Lekszycki T., dell’Isola F.: A continuum model for the bio-mechanical interactions between living tissue and bio-resorbable graft after bone reconstructive surgery. Comptes Rendus de l’Académie des Sciences II B 339, 625–640 (2011)
Marsden J.E., Hughes T.J.R.: The Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliff (1983)
Mehrabadi M.M., Nemat-Nasser S.: Some basic kinematical relations for finite deformations of continua. Mech. Mater. 87, 127–138 (1987)
Prot V., Skallerud B., Holzapfel G.A.: Transversely isotropic membrane shells with application to mitral valve mechanics. constitutive modelling and finite element implementation. Int. J. Numer. Methods Eng. 71(8), 987–1008 (2007)
Segev, R.: Notes on metric independent analysis of classical fields. Math. Methods Appl. Sci. 36(5), 497–566 (2013)
Simo J.C., Hughes T.J.R.: Computational Inelasticity. Springer, New York (1998)
Sun W., Chaikof E.L., Levenston M.E.: Numerical approximation of tangent moduli for Finite Element implementations of nonlinear hyperelastic material models. J. Biomech. Eng. 130, 061003 (2008)
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Bellini, C., Federico, S. Green-Naghdi rate of the Kirchhoff stress and deformation rate: the elasticity tensor. Z. Angew. Math. Phys. 66, 1143–1163 (2015). https://doi.org/10.1007/s00033-014-0421-x
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DOI: https://doi.org/10.1007/s00033-014-0421-x