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Small superimposed radial oscillations for a class of damaged limited elastic tubes

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Abstract

Small superimposed radial oscillations for a class of damaged limited elastic, incompressible, isotropic and homogeneous thick-walled circular cylindrical tube are examined. Both the damage function and the considered limited elastic class depend only on the first invariant \( I_{1} \) of the left Cauchy–Green deformation tensor. The present work puts some light in the context of the surgical procedure of angioplasty, used especially to treat cardiovascular diseases. Our simplified isotropic assumption allows for a much faster and analytical computation of comparably accurate results. The examples of three limited elastic material models are presented, namely the Arruda–Boyce, Gent, and Demiray. The radial oscillations are further investigated for a damaged limited elastic thin membrane, and the examples above are explored. Universal relations are found for the three material membranes. The obtained results are matched with the clinical data of the existing literature. It appears that the pressure ratio for the virgin and damaged arterial tissue is higher before angioplasty and vice versa. A similar trend for the frequency ratio also follows.

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Notes

  1. A careful observation of Figure 5 of [17] will assert that the arterial tissue is having the limiting extensibility stretch of \( \lambda _{m}=\lambda _{1M}=1.7827 \). Thus, the limiting value of the invariant is \( I_{m}=I_{1M}=\lambda ^{2}+\dfrac{2}{\lambda }=4.3 \).

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai, India

    Kriti Arya

  2. Indian Institute of Technology Kharagpur, Kharagpur, India

    Ranjan Bhattacharyya

  3. Indian Institute of Technology Patna, Patna, India

    Somnath Sarangi

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  1. Kriti Arya
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  2. Ranjan Bhattacharyya
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  3. Somnath Sarangi
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Appendix 1. Detailed derivation of Eq. (26)

Appendix 1. Detailed derivation of Eq. (26)

For the forced radial oscillations presented here, the respective Knowles’ equation (17) incorporating the damage function is detailed below:

The deformation gradient for the considered tube problem is given by

$$\begin{aligned} \mathbf{F} =\dfrac{R}{r}{} \mathbf{e} _\mathbf{rr }+\dfrac{r}{R}{} \mathbf{e} _{\varvec{\theta \theta }}+\mathbf{e} _\mathbf{zz }, \end{aligned}$$
(70)

and so the left Cauchy–Green deformation tensor \(\mathbf{B} =\mathbf{F} {} \mathbf{F} ^{T}\) is obtained as

$$\begin{aligned} \mathbf{B} =\dfrac{R^{2}}{r^{2}}{} \mathbf{e} _\mathbf{rr }+\dfrac{r^{2}}{R^{2}}{} \mathbf{e} _{\varvec{\theta \theta }}+\mathbf{e} _\mathbf{zz }. \end{aligned}$$
(71)

The respective principal invariants of \(\mathbf{B} \) are

$$\begin{aligned} \textit{I}_1=\textit{I}_2=1+\dfrac{r^2}{R^2}+\dfrac{R^2}{r^2}=1+u+\dfrac{1}{u}, and \ \textit{I}_3=1. \end{aligned}$$
(72)

Consequently, from the constitutive relation (15) we obtain the principal components of the Cauchy stress \( \mathbf{T} \) as

$$\begin{aligned} T_{rr}=-\textit{q}+\frac{R^2}{r^2}\beta _1\ , \ T_{{\theta }{\theta }}=-\textit{q}+\frac{r^2}{R^2}\beta _1\ , \ T_{zz}=-\textit{q}+\beta _1. \end{aligned}$$
(73)

Thus, use of the equilibrium equation

$$\begin{aligned} Div (\mathbf{T} )=\rho \ddot{r}, \end{aligned}$$
(74)

and the boundary conditions \( T_{rr}=-p_{1}(t) \) and \( T_{rr}=-p_{2}(t) \) at \( r=r_{1} \) and \( r=r_{2} \), respectively, provides the Knowles’ equation (17) for the damaged case.

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Arya, K., Bhattacharyya, R. & Sarangi, S. Small superimposed radial oscillations for a class of damaged limited elastic tubes. Acta Mech 232, 2765–2780 (2021). https://doi.org/10.1007/s00707-021-02980-z

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