Theory and finite element implementation of orthotropic and transversely isotropic incompressible hyperelastic membrane Jarraya Abdessalem, Imen Kammoun Kallel and Dammak Fakhreddine Research Unit of Mechanical, Modelisation and Manufacturing Unit, National School of Engineers of Sfax, Sfax, Tunisia Abstract Purpose – The purpose of this paper is to describe a general theoretical and finite element implementation framework for the constitutive modelling of biological soft tissues. Design/methodology/approach – The model is based on continuum fibers reinforced composites in finite strains. As an extension of the isotropic hyperelasticity, it is assumed that the strain energy function is decomposed into a fully isotropic component and an anisotropic component. Closed form expressions of the stress tensor and elasticity tensor are first established in the general case of fully incompressible plane stress which orthotropic and transversely isotropic hyperelasticity. The incompressibility is satisfied exactly. Findings – Numerical examples are presented to illustrate the model’s performance. Originality/value – The paper presents a constitutive model for incompressible plane stress transversely isotropic and orthotropic hyperelastic materials. Keywords Modelling, Histology, Biological soft tissues, Hyperelasticity, Transversely isotropic, Orthotropic, Elasticity tensor Paper type Research paper 1. Introduction Almost all biological tissues are anisotropic, viscoelastic, incompressible and undergo large deformations. Theoretical models have the ability to allow investigation into the complexity of these complex systems and avoid the costly experimental studies. However, the numerical studies of these biological materials require the knowledge of their accurate constitutive equations. Many of these tissues are assumed to be an isotropic solid matrix reinforced by one family of fibers. Such material anisotropy is called transversely isotropic. One can use this approach to simulate the behavior of ligaments and tendons. However, if the tissue is made with branching and interwoven collagen fibers, it is relevant to consider two families of fibers. If these two families of fibers are orthogonal, such material anisotropy is called orthotropic. The constitutive equations to model anisotropic, orthotropic and transversely isotropic materials in small strain are well established. When materials undergo large The current issue and full text archive of this journal is available at www.emeraldinsight.com/1573-6105.htm MMMS 7,4 424 Received 30 March 2011 Revised 23 June 2011 Accepted 24 June 2011 Multidiscipline Modeling in Materials and Structures Vol. 7 No. 4, 2011 pp. 424-439 q Emerald Group Publishing Limited 1573-6105 DOI 10.1108/15736101111185298