INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2012; 89:1691–1706 Published online 2 December 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3308 A variational framework for fiber-reinforced viscoelastic soft tissues J. M. Vassoler * ,† , L. Reips and E. A. Fancello Department of Mechanical Engineering, Federal University of Santa Catarina, Florianópolis, SC, Brazil SUMMARY The mechanical properties of soft biological tissues vary depending on how the internal structure is orga- nized. Classical examples of tissues are ligaments, tendons, skin, arteries, and annulus fibrous. The main element of such tissues is the fibers which are responsible for the tissue resistance and the main mechanical characteristic is their viscoelastic anisotropic behavior. The objective of this paper is to extend an existing model for isotropic viscoelastic materials in order to include anisotropy provided by fiber reinforcement. The incorporation of the fiber allows the mechanical behavior of these tissues to be simulated. The model is based on a variational framework in which its mechanical behavior is described by a free energy incre- mental potential whose local minimization provides the constraints for the internal variable updates for each load increment. The main advantage of this variational approach is the ability to represent different mate- rial models depending on the choice of suitable potential functions. Finally, the model is implemented in a finite-element code in order to perform numerical tests to show the ability of the proposed model to repre- sent fiber-reinforced materials. The material parameters used in the tests were obtained through parameter identification using experimental data available in the literature. Copyright © 2011 John Wiley & Sons, Ltd. Received 28 October 2010; Revised 14 July 2011; Accepted 16 August 2011 KEY WORDS: biomechanics; anisotropy; nonlinear viscoelasticity 1. INTRODUCTION Soft biological tissues, such as skin, ligaments and tendons, are very important for the mechanical functioning of the body. They are responsible, respectively, for the protection of the body, and the transfer of loads between bones and between muscles and bones [1]. Soft biological tissues are formed mainly of elastin and collagen. Elastin, like rubber, is comprised of long flexible molecules that form three-dimensional networks by cross-linking and is responsible for the elasticity of the tissues. Collagen is the most important structural element of soft and hard tissues in animals. The particular arrangement of the collagen proteins - three left-handed helices twisted together into a right-handed triple helix - provides this structure with a high stretch resis- tance to traction [2]. These molecules can be aggregated to form different structures depending on the tissue. Firstly, the collagen molecules can wrap around themselves to form a collagen fibril which varies according to the animal species and tissue. Subsequently, bundles of collagen fibrils can be organized into collagen fibers and the fibers into tissues, where the packaging of collagen fibers has many different modes of organization that vary depending on the tissue. Connective tissues, such as ligaments and tendons, have a pronounced anisotropic mechanical behavior due to their internal structure, consisting of regular parallel collagen fibers, which it is the simplest tissue structure. The fiber structure of tendons has a largely parallel organization, and *Correspondence to: J. M. Vassoler, Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Rua Sarmento Leite, 425 - Cidade Baixa, 90050-170, Porto Alegre, RS, Brazil. † E-mail: vassoler@mecanica.ufrgs.br Copyright © 2011 John Wiley & Sons, Ltd.