In the paper, preliminary studies on formulation of a new constitutive equation of bone tissue are presented. A bone is modelled as a viscoelastic material. Thus, not only are elastic properties of the bone taken into account, but also both short-term and long-term viscoelastic properties are considered. A potential function is assumed for the bone, constant identification on the basis of experimental stress-strain curve fitting is completed and a preliminary constitutive equation is formulated. The experiments consisted of compressive tests performed on a cuboids-like bone sample of the following dimensions: 10x5x7.52 mm. The specimen was compressed along the highest dimension at the strain rates 0.016 s to the -1 and 0.00016 s to the -1. In addition to this, stress relaxation test was performed to identify long-term viscoelastic constants of bone. In the experiments, only displacement in the load direction was measured. The bone sample was extracted from a bovine femur. The form of the proposed potential function is such that it models a bone as a transversely isotropic material. For the sake of simplicity, it is assumed that the bone is incompressible. After the material constant identification the strain energy function proved to be adequate to describe bone behaviour under compressive load. Due to the fact that the function is convex, the results of the studies can be utilised in modelling of bone tissue in finite element analyses of an implant-bone system. Such analyses are very helpful in the process of a new prosthesis design as one can preoperatively verify the construction of the new implant and optimise its shape.
The article describes how to identify the boundary and yield surface for hypoplastic constitutive equations proposed by Wu, Gudehus and Bauer. It is shown how to identify and plot the surfaces for any equation in this class. Calculation errors are analyzed characteristic for applied set of numerical formulas. In the paper there are computer links to the source code prepared in the MATLAB system, based on instructions in the article. A sample consitutive domains are shown, plotted using the attached computer program.
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 performace of the developed code is examined using an exemplary problem.