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Survey of some constitutive models for arterial walls under uniaxial and biaxial loading

Jankowska M. A., Będziński R.

Vibrations in Physical Systems

2012

Vol. 25

177--184

The paper deals with some constitutive equations for arterial walls subjected to uniaxial and biaxial extension tests. In spite of a great number of various approaches to development of the strain-stress relations, the models selected for presentation seems to be good representatives of the most characteristic and fundamental ones.

Limiting fiber extensibility model for arterial wall

Horny L., Zitny R., Chlup H., Konvickova S.

Engineering of Biomaterials

2008

Vol. 11, no. 81-84

112--113

Arterial walls exhibit anisotropic, nonlinear and inelastic response to external loads. Moreover arterial wall is non–homogenous material with complicated internal structure. These facts make the question about the best material model for arterial wall still unanswered. Nowadays approach to building constitutive models is characterized by incorporating structural information when considering e.g. layers, fibers, fiber orientation or waviness. The most frequent method how to incorporate structural information is to regard arterial wall as a fiber reinforced composite. Considerations about preferred directions are subsequently implemented into the framework of continuum mechanics. Constitutive models are usually based on the theory of hyperelastic materials. Thus mechanical response of an arterial wall is supposed to be governed by a strain energy (or free energy) density function like in (1). The theory of hyperelastic materials is widely applied and studied in details in polymer science. Due to some phenomenological and structural similarities between rubber–like materials and biological tissues, methods of polymer physics are frequently applied in biomechanics, see Holzapfel [1]. Gent [2] suggested the new isotropic model for strain energy density function which was based on an assumption of limiting chain extensibility in polymer materials. The Gent model expresses strain energy y as a function of first invariant I1 of the right Cauchy-Green strain tensor as follows [formula]. In equation (1) μ denotes stress–like parameter, so–called infinitesimal shear modulus. Jm denotes limiting value of I 1 -3. The domain of logarithm requires [formula]. Thus, Jm can be interpreted as limiting value for macromolecular chains stretch. Horgan and Saccomandi in [3] suggested its anisotropic extension. They recently published modification based on usual concept of anisotropy related to fiber reinforcement, see paper [4]. Horgan and Saccomandi use rational approximations to relate the strain energy expression to Cauchy stress representation formula. We adopted this term with small modification as follows [formula] In (2) μ denote shear modulus. J m is the material parameter related to limiting extensibility of fibers. The similar definitional inequality like in (1) must be hold for logarithm in (2). Thus I 4 must satisfy [formula] denotes so called fourth pseudo–invariant of the right Cauchy-Green strain tensor which arises from the existence of preferred direction in continuum. It is worth to note that total number of invariants of the strain tensor is five in the case of transversely isotropic material and nine in the case of orthotropy. Details can be found in e.g. Holzapfel [5]. Model (2) presumes two preferred directions in continuum which are mechanically equivalent. Due to cylindrical shape of an artery we can imagine it as helices with same helix angel but with antisymmetric rientation. This is illustrated in the FIG. 1 I 4 can be expressed in the form given in (3) [formula] Stretched configuration of the tube is characterized by λ t , what denotes circumferential stretch and λ z what denotes axial stretch, respectively. Model (2) contains three material parameters. Above described μ, J m and β. The third material parameter β has the meaning of angle between fiber direction and circumferential axis. There are two families of fibers with angle ±β, however, I 4 is symmetric with respect to ±β. In order to verify capability of (2) to govern multi–axial mechanical response of an artery regression analysis based on previously published experimental data was performed. Details of experimental method and specimen can be found in Horny et al. [6]. Briefly we resume basic facts. Male 54–year–old sample of thoracic aorta underwent inflation test under constant axial stretch. The tubular sample was 6 times pressurized in the range 0kPa–18kPa–0kPa under axial pre–stretch λ z =1.3 and 3 times in the pressure range 0kPa–20kPa–0kPa under λ z =1.42, respectively. The opening angle was measured in order to account residual strains. Radial displacements were photographed and evaluated by image analysis. Regression analysis based on least square method gave the estimations for material parameters μ, Jm and β. The vessel was modeled as thick–walled tube with residual strains. The material was supposed to be hyperelastic and incompressible. No shear strains were considered. Fitting of material model was based on comparison of model predicted and measured values of internal pressure. Results are illustrated in FIG. 2. We can conclude that proposed material model fits experimental data successfully. Thus strain energy given in (2) is suitable to govern arterial response during its inflation and extension. Estimated values of parameters for material model (2) are as follows: μ =26kPa; J m =1.044; β=37.2°