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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°

RTD prediction, modelling and measurement of gas flow in reactor

Thyn J., Ha J. J., Strasak P., Zitny R.

Nukleonika

1998

Vol. 43, nr 1

95-114

Problems of the measurement and evaluation of RTD of gas phase in chemical reactors are presented and discussed. The measurement of RTD in an industry is mostly used in trouble shooting or for process intensification or optimization. Special PC-programs have been prepared for selection of an RTD model and for its verification by parametrical analysis. An example of the procedure used in trouble shooting is demonstrated on basis of data of radiotracer impulse response of gas flow in a heat exchanger. The results of methods for identification (deconvolution) with regularization, from so called nonparametrical analysis are presented. A combined model described by a set of differential equations has been suggested for gas flow in the reactor with baffles. The model identification by nonlinear regression yields important process parameters, the relative dead volume and mass exchange coefficient e.g. The prediction of RTD for other steady state conditions (e.g. for another flow rate) can be done on the basis of dimensionless impulse response. However the similarity assumption should be verified by other experiments, because the same flow pattern as well as the same relative active volume must be assured at different steady conditions. Another possibility is the prediction of a change of RTD (change of the values of RTD model parameters) on the basis of numerical solution of transport equations for fluid dynamics using particle tracking method or transient analysis of temperature pulse spreading. The results of the RTD prediction for a broad range of Reynolds numbers using software Fluent are presented.

Omówiono problemy pomiaru i wyznaczania rozkładu czasu przebywania (RTD) fazy gazowej w reaktorach chemicznych. Pomiar RTD w przemyśle stosuje się najczęściej przy ocenie niezawodności lub w celu intensyfikacji lub optymalizacji procesu. Opracowano specjalne programy komputerowe umożliwiające wybranie optymalnego modelu RTD i jego weryfikację na drodzeprocedury optymalizacyjnej. Podano przykład postępowania przy badaniu niesprawności wymiennika ciepła oparty na analizie odpowiedzi impulsowej otrzymanej w eksperymencie radioznacznikowym dla fazy gazowej. Przedstawiono wyniki uzyskane z zastosowaniem procedury identyfikacyjnej (dekonwolucji) z regularyzacją tzw. metodą analizy nieparametrycznej. Dla przepływu gazu w reaktorze z przegrodami zaproponowano złożony model opisany układem równań różniczkowych. Identyfikacja modelu na drodze regresji nieliniowej pozwala na wyznaczenie podstawowych parametrów procesu, np. względnej objętości martwej i współczynnika wymiany masy. Określenie RTD dla innych stanów stacjonarnych (np. innego natężenia przepływu) można przeprowadzić w oparciu o analizę bezwymiarowej odpowiedzi impulsowej. Jednakże należy sprawdzić założone podobieństwo na drodze innych doświadczeń, ponieważ należy się upewnić, że w różnych warunkach stacjonarnych struktura przepływu i względna objętność czynna są takie same. Można również przewidzieć zmiany RTD (zmiany wartości parametrów modelu RTD) na podstawie numerycznego rozwiązania równań transportu dla dynamiki cieczy, stosując metodę "cząstek znakowanych" lub analizę stanów przejściowych dyspersji impulsu temperaturowego. Przedstawiono wyniki wyznaczania RTD dla szerokiego zakresu liczb Reynoldsa stosując oprogramowanie Fluent.