Application of Comsol Multiphysics 3.2 to finite strain ... - Humusoft

Holzapfel [3] and used by Holzapfel & Gasser [5] to calculate the viscoelastic deformation of fiberreinforced composite material undergoing finite strains.The model is based on the theory of compressible hyperelasticity with the decoupledrepresentation of the Helmholtz free energy function with the internal variables (Holzapfel [1], p. 283) :C , 1,..... , m= ∞VOL J ∞ISOmC ∑ C , , C= J −2/3 C . (1)The first two terms in (1) characterize the equilibrium state and describe the volumetric elastic responseand the isochoric elastic response as t ∞ , respectively. The third term is the dissipative potentialresponsible for the viscoelastic contribution. The derivation of the 2 nd Piola-Kirchhoff stress withvolumetric and isochoric terms:∞S VOL∞S ISOS=2 ∂ C , 1 , ..... , m ∂C=1=S ∞VOLS ∞ISOm∑=1Q (2)where and is the volumetric and the isochoric stress response respectively and theoverstress Q is stress of 2 nd Piola-Kirchhoff type.S ∞ISOS ∞VOL=J d ∞VOLd J J C −1 ,=J −2/ 3 Dev [2 ∂ ∞ISO∂ CC ] ,Dev .=.−1/3[ .:C ]C −1 . (5)(3)(4)where Dev . is the deviatoric operator in the Lagrangian description. Motivated by the generalizedMaxwell rheological model (Fig. 1), the evolution equation for the internal variable Q takes on theform (6).Fig. 1. Maxwell rheological modelṠ ∞ISOQ˙ Q =Ṡ ISO,Ṡ ISO= ∞Ṡ ∞ ISOC ,whereC = ∂ S ∞ISO∂ C Ċ =C Ė ISOC ISO=2 ∂ S ∞ISO∞∂ C =2 ∂ S ISO∂C∂C∂C(6)(7)(8)(9) ∞ ∈0,∞ in the expression (7) is the nondimensional strain energy factor [2,4], is therelaxation time, C ISO is isochoric contribution of the tangent elasticity tensor and Ė is the materialstrain rate tensor