On the viscoelastic dynamic beam modelling

Abstract

There is a renewed interest in the development of underwater robotic fish due to the advancement in control systems. The cross section of real fish broadly consists of solid bone at the core followed by viscoelastic matrix (tissue mass). The precise material development for such systems is still a challenge, as the desired motion requires viscoelastic dynamic response with probably varying axial stiffness. It is thus important to develop physically accurate material model of fish that can be employed in fluid–structure interaction codes. The primary objective of the present work is to develop viscoelastic dynamic material model, where the viscous response is obtained by Kelvin–Voigt model. The detailed formulation of the model is presented followed by its numerical implementation by finite element (space discretization) and Newmark (time discretization) methods. The correctness of the developed MATLAB code is successfully demonstrated by solving several problems having known analytical solutions. The entire code is made available in the supplementary material file, which can be readily deployed in the design and development of robotic fish systems.

Appendix

1.1 Two KV elements in series

A creep compliance function, for two KV elements connected in series, is derived in this section. This expression is employed to represent the variation of E and ${\upeta }$, as shown in Fig. 2, with respect to the time parameter.

Let us consider two KV elements, having $(E_1, {\upeta }_1)$ and $(E_2, {\upeta }_2)$ material constants, connected in the series. A constant stress ${\sigma }= {\sigma }_0\,\text{ H }(t)$ is applied across this arrangement, where $\text{ H }(t)$ is usual Heaviside function (delta function) implying an application of ${\sigma }_0$ at $t \ge 0^{+}$. An equilibrium of individual KV element is given as

$$\begin{aligned} {\sigma }_1 = E_1\,{\varepsilon }_{1} + {\upeta }_1\,{\dot{\varepsilon }}_{1},\,\,\,\,{\sigma }_2 = E_2\,{\varepsilon }_{2} + {\upeta }_2\,{\dot{\varepsilon }}_{2} \end{aligned}$$

(A.1)

where the subscripts $(\bullet )_1$ and $(\bullet )_2$ imply first and second KV element, respectively, and the stress and strain equilibrium result in ${\sigma }= {\sigma }_1 = {\sigma }_2$ and $\varepsilon = \varepsilon _1 + \varepsilon _2$, respectively. The Laplace transform of Eq. (A.1) results in

$$\begin{aligned} \overline{{\sigma }}_1 = (E_1 + {\upeta }_1\,p)\,\overline{\varepsilon }_1,\,\,\,\,\overline{{\sigma }}_2 = (E_2 + {\upeta }_2\,p)\,\overline{\varepsilon }_2 \end{aligned}$$

(A.2)

where $(\overline{\bullet })$ represents a quantity in Laplace transform domain, and p is Laplace variable. Eq. (A.2) is substituted in the strain equilibrium to obtain

$$\begin{aligned} \overline{\varepsilon } = {\sigma }_0\, \left[ \left( \frac{1}{p\,(E_1 + p\,{\upeta }_1)}\right) + \left( \frac{1}{p\,(E_2 + p\,{\upeta }_2)}\right) \right] \end{aligned}$$

(A.3)

Eq. (A.3) is simplified by partial fraction as

$$\begin{aligned} \overline{\varepsilon } = \frac{{\sigma }_0}{E_1}\, \left[ \frac{1}{p} - \frac{1}{\left( \frac{E_1}{{\upeta }_1} + p\right) } \right] + \frac{{\sigma }_0}{E_2}\, \left[ \frac{1}{p} - \frac{1}{\left( \frac{E_2}{{\upeta }_2} + p\right) } \right] \end{aligned}$$

(A.4)

Eq. (A.4) is transferred to the time domain by inverse Laplace transform to get

$$\begin{aligned} \varepsilon (t) = {\sigma }_0 \underbrace{\left\{ \frac{1}{E_1}\,\left[ 1 - \text{ e}^{(-t/{\uptau }_1)} \right] + \frac{1}{E_2}\,\left[ 1 - \text{ e}^{(-t/{\uptau }_2)} \right] \right\} }_{S(t)} \end{aligned}$$

(A.5)

where ${\uptau }_1 = ({\upeta }_1 / E_1)$ and ${\uptau }_2 = ({\upeta }_2/E_2)$ is substituted, and S(t) is the creep compliance function. The $E(t) = (1/S(t))$ and ${\upeta }(t) = {\upeta }_1 \,\text{ e}^{(-t/{\uptau }_1)} + {\upeta }_2 \,\text{ e}^{(-t/{\uptau }_2)}$ are used in VE dynamic beam deflection solution due to earthquake type loading.