An anisotropic visco-hyperelastic model for ligaments at finite strains. Formulation and computational aspects

Abstract

In this paper we present a fully three-dimensional finite strain anisotropic visco-hyperelastic model for ligaments and tendons. The structural model is formulated within the framework of non-linear continuum mechanics and is well-suited for its finite element implementation. This model is based on a local additive decomposition of the stress tensor into initial and non-equilibrium parts as resulted from the assumed structure of the free-energy density function that generalizes Kelvin-Voigt linear viscous models. Also, we use a local multiplicative decomposition of the deformation gradient into volume-preserving and dilatational parts that permits to model the incompressible properties of soft biological tissues. To simulate the viscoelastic properties of this kind of tissues, we consider different viscoelastic behaviours for the matrix and the different families of fibers. A second-order accurate numerical integration procedure is used, established entirely in the reference configuration. Expressions for the stress and elasticity tensors in the spatial description are also presented.Of all soft tissues, we have focused in ligaments due to the importance of their viscoelastic properties in the clinical practise. In order to show clearly the performance of the constitutive model, we present 3D simulations of the behaviour of the anterior cruciate ligament and patellar tendon graft. The model was also tested for various multi-axial loading situations. The relaxation and creep responses and the strain rate dependent behaviour of anterior cruciate ligament and patellar tendon graft were accurately predicted.

Mathematical and computational modelling of skin biophysics: a review

“…Using a multi-modal Maxwell element rheological analogy, this type of formulation was applied by Holzapfel & Gasser [152] to model the viscoelastic orthotropic behaviour of arteries. Peña et al [153,154] developed a similar approach to model the ligaments of the knee but conceptually used a combination of Kelvin-Voigt elements. Ehret et al [155] proposed a microstructurally motivated finite strain viscoelastic model for soft tissues which is very relevant for skin mechanics.…”

A Structural Approach Including the Behavior of Collagen Cross-Links to Model Patient-Specific Human Carotid Arteries

“…What may be described as the normal behavior of materials, as a purely mechanical response to a stimulus, is only a small part of the biological tissue behavior. They undergo damage 11 and viscoelastic effects 32 , like many conventional materials (rubbers, steel, aluminum etc.). But they also constitute a very active component where chemical reactions, cell synthesis and apoptosis, growth 23,27,28 , remodeling 22,29,7 , creation of atherome plaque 25 etc., are involved in their response.…”

A microstructurally motivated anisotropic viscoelastic model for soft tissues

“…In order to model this behaviour, many different approaches have been proposed such as hereditary integral formulations (see [1] and references therein), the theory of internal variables (e.g. [2][3][4]) or formulations based on invariants of strain rate tensors (e.g. [5,6]).…”