A Finite Element Implementation of Knowles Stored-Energy Function: Theory, Coding and Applications
This paper contains the full way of implementing a user-defined hyperelastic constitutive model into the finite element method (FEM) through defining an appropriate elasticity tensor. The Knowles stored-energy potential has been chosen to illustrate the implementation, as this particular potential function proved to be very effective in modeling nonlinear elasticity within moderate deformations. Thus, the Knowles stored-energy potential allows for appropriate modeling of thermoplastics, resins, polymeric composites and living tissues, such as bone for example. The decoupling of volumetric and isochoric behavior within a hyperelastic constitutive equation has been extensively discussed. An analytical elasticity tensor, corresponding to the Knowles stored-energy potential, has been derived. To the best of author's knowledge, this tensor has not been presented in the literature yet. The way of deriving analytical elasticity tensors for hyperelastic materials has been discussed in detail. The analytical elasticity tensor may be further used to develop visco-hyperelastic, nonlinear viscoelastic or viscoplastic constitutive models. A FORTRAN 77 code has been written in order to implement the Knowles hyperelastic model into a FEM system. The performance of the developed code is examined using an exemplary problem.
This work concerns mainly the finite element (FE) implementation of polyconvex incompressible hyperelastic models. A user material subroutine (UMAT) has been developed and can be utilized to define the aforementioned material behaviors in the FE system ABAQUS. The subroutine is written using a novel strategy in order to maximally simplify the relations for the analytical material Jacobian (MJ). The UMAT code is attached in the appendix. The developed subroutine allows to significantly decrease the time of computations and to avoid possible convergence difficulties. The structure of the code enables modifications which may lead to a rheological, damage or growth models, for instance.
In this study, a new viscoelastic-plastic constitutive model which has been formulated by utilizing the formalism of stress-like internal state variables is introduced. The developed constitutive equation allows for a good description of the inelastic material response of polymeric materials over a wide range of strain rates. An algorithm for numerical integration of the model equations has been derived. The FE implementation of the constitutive equation is widely discussed and the results of solving several exemplary problems are presented.
The present study is concerned with the finite element (FE) implementation of slightly compressible hyperelastic material models. A class of constitutive equations is considered where the isochoric potential functions are based on the first invariant of the right Cauchy-Green (C-G) deformation tensor. Special attention is paid to the most recently developed model formulations. The incremental form of hyperelasticity and its numerical implementation into both commercial and non-commercial FE software are discussed. A Fortran 77 UMAT code is attached which allows for a simple implementation of arbitrary first invariant-based constitutive models into Abaqus and Salome-Meca FE packages. Several exemplary problems are considered.
In the paper finite element (FE) analysis of implanted lumbar spine segment is presented. The segment model consists of two lumbar vertebrae L4 and L5 and the prosthesis. The model of the intervertebral disc prosthesis consists of two metallic plates and a polyurethane core. Bone tissue is modelled as a linear viscoelastic material. The prosthesis core is made of a polyurethane nanocomposite. It is modelled as a non-linear viscoelastic material. The constitutive law of the core, derived in one of the previous papers, is implemented into the FE software Abaqus . It was done by means of the User-supplied procedure UMAT. The metallic plates are elastic. The most important parts of the paper include: description of the prosthesis geometrical and numerical modelling, mathematical derivation of stiffness tensor and Kirchhoff stress and implementation of the constitutive model of the polyurethane core into Abaqus software. Two load cases were considered, i.e. compression and stress relaxation under constant displacement. The goal of the paper is to numerically validate the constitutive law, which was previously formulated, and to perform advanced FE analyses of the implanted L4-L5 spine segment in which non-standard constitutive law for one of the model materials, i.e. the prosthesis core, is implemented.
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