A five-field finite element formulation for nearly inextensible and nearly incompressible finite hyperelasticity

A novel Hu-Washizu type 5-field virtual work principle for nearly inextensible and almost incompressible finite hyperelasticity is developed and implemented. The formulation is implemented in an hp-adaptive code providing the proper flexible environment for finite elements with variable order and mixed interpolation. The novel implementation includes residual based error estimation and mesh adaptivity. In the fully constrained limit the formulation provides the constraint manifold setting of hyperelasticity with the simple internal kinematic constraints of inextensibility and incompressibility. A study using a semi-inverse analytical solution and h-refinements and p-enrichments corroborates the convergence characteristics of the 5-field implementation. A new closed form solution for pure torsion of a circular cylindrical tube with inextensible fibres is derived and used for verification. As typical applications are found in e.g. soft tissue biomechanics, passive pressurisation of an ellipsoidal geometry resembling the left ventricle of a rabbit heart is analysed to demonstrate the capability of modelling transversely isotropic materials.

[1]  R. Ogden,et al.  A robust anisotropic hyperelastic formulation for the modelling of soft tissue. , 2014, Journal of the mechanical behavior of biomedical materials.

[2]  J. M. Zhang,et al.  Structural tensors for anisotropic solids , 1990 .

[3]  Gerhard A. Holzapfel,et al.  Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science , 2000 .

[4]  Z. Yosibash,et al.  p‐FEMs for hyperelastic anisotropic nearly incompressible materials under finite deformations with applications to arteries simulation , 2011 .

[5]  Hu Hai-chang,et al.  ON SOME VARIATIONAL PRINCIPLES IN THE THEORY OF ELASTICITY AND THE THEORY OF PLASTICITY , 1954 .

[6]  J. P. Boehler,et al.  Introduction to the Invariant Formulation of Anisotropic Constitutive Equations , 1987 .

[7]  M. Vogelius.  An analysis of thep-version of the finite element method for nearly incompressible materials , 1983 .

[8]  P. Flory,et al.  Thermodynamic relations for high elastic materials , 1961 .

[9]  Alain Goriely,et al.  Rotation, inversion and perversion in anisotropic elastic cylindrical tubes and membranes , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  Klaus-Jürgen Bathe,et al.  The inf–sup condition and its evaluation for mixed finite element methods , 2001 .

[11]  J. C. Simo,et al.  Variational and projection methods for the volume constraint in finite deformation elasto-plasticity , 1985 .

[12]  W. Rachowicz,et al.  A novel computational formulation for nearly incompressible and nearly inextensible finite hyperelasticity , 2014 .

[13]  K. Bathe,et al.  A finite element formulation for nonlinear incompressible elastic and inelastic analysis , 1987 .

[14]  E. Stein,et al.  On some mixed finite element methods for incompressible and nearly incompressible finite elasticity , 1996 .

[15]  Erwin Stein,et al.  Analysis, finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity , 2000 .

[16]  Gernot Plank,et al.  Influence of myocardial fiber/sheet orientations on left ventricular mechanical contraction , 2013 .

[17]  R. Ogden,et al.  A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models , 2000 .

[18]  R. Ogden,et al.  Mechanical response of fiber-reinforced incompressible non-linearly elastic solids , 2005 .

[19]  C. Horgan,et al.  Torsion of Incompressible Fiber-Reinforced Nonlinearly Elastic Circular Cylinders , 2011 .

[20]  Stefan Hartmann,et al.  Transversal isotropy based on a multiplicative decomposition of the deformation gradient within p‐version finite elements , 2015 .

[21]  Ernst Rank,et al.  p-FEM applied to finite isotropic hyperelastic bodies , 2003 .

[22]  Ray W. Ogden,et al.  On anisotropic elasticity and questions concerning its Finite Element implementation , 2013, 2009.00044.

[23]  D. Tortorelli,et al.  On hyperelasticity with internal constraints , 1996 .

[24]  Q. Zheng.  Theory of Representations for Tensor Functions—A Unified Invariant Approach to Constitutive Equations , 1994 .

[25]  M. Bercovier,et al.  Numerical Evaluation of Finite Element Methods for Rubber Parts , 1986 .

[26]  C. Truesdell,et al.  The Non-Linear Field Theories of Mechanics , 1965 .

[27]  Carlo Sansour,et al.  On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy , 2008 .