Dynamic modeling for silicone beams using higher-order ANCF beam elements and experiment investigation

Abstract

The present paper mainly concentrates on dynamic modeling and analysis of the isotropic, hyperelastic and nonlinear nearly incompressible silicone beams adopting higher-order beam elements based on absolute nodal coordinate formulation (ANCF), and the combined use of higher-order beam elements with nonlinear incompressible material model is developed. Different from the previously proposed conventional lower-order fully parameterized ANCF beam element for modeling of rubber-like beams, in which linear interpolation in the transverse directions is used, the higher-order beam element which covers quadratic interpolation in the transverse directions is employed to investigate dynamic behaviors of silicone beams within the ANCF framework. The utilization of the higher-order beam element cannot only avoid the stiffening problem caused by volumetric locking, but also improve precision in simulating large bending deformation compared with the modified lower-order beam element based on selective reduced integration method for the particular silicone beams considering damping. By introducing the volumetric energy penalty function, the nonlinear elastic force and its derivative formulations of the Arruda–Boyce model are deduced. The availability and accuracy of the higher-order beam element are verified and compared firstly through static equilibrium experiment and ANSYS simulation. Subsequently, the dynamic experimental investigation that captures free-falling motion of the silicone cantilever beam under the action of gravitational force is executed, then corresponding dynamic simulations are implemented employing three types of different beam elements in consideration of damping effect, and the simulation results are compared with experimental data. The validity, simulation accuracy of different beam elements and diverse nonlinear constitutive models in the dynamic analysis are discussed. In addition, the superiority of the higher-order beam element relative to two other types of lower-order beam elements can be further examined through physical experiment.

Appendix

In this appendix, the derivations of some identities used in this present investigation to obtain the elastic forces associated with the nonlinear elastic constitutive models in the absolute nodal coordinate formulation are presented. Recall that the deformation gradient tensor $\mathbf{J}$ is given by

$$ \mathbf{J} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} \dfrac{\partial \mathbf{r}}{\partial x} & \dfrac{\partial \mathbf{r}}{\partial y} & \dfrac{\partial \mathbf{r}}{\partial z} \end{array}\displaystyle \right ] = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} \dfrac{\partial \mathbf{S}}{\partial x}\mathbf{q}_{e} & \dfrac{ \partial \mathbf{S}}{\partial y}\mathbf{q}_{e} & \dfrac{\partial \mathbf{S}}{\partial z}\mathbf{q}_{e} \end{array}\displaystyle \right ] $$

(25)

Defining

$$ \begin{aligned} &\frac{\partial \mathbf{r}}{\partial x} = \mathbf{r}_{x} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} r_{x1} & r_{x2} & r_{x3} \end{array}\displaystyle \right ]^{\mathrm{T}} \qquad \frac{\partial \mathbf{r}}{\partial y} = \mathbf{r}_{y} = \left [ \displaystyle\textstyle\begin{array}{c@{\quad }c@{\quad }c} r_{y1} & r_{y2} & r_{y3} \end{array}\displaystyle \right ]^{\mathrm{T}} \\ &\frac{\partial \mathbf{r}}{\partial z} = \mathbf{r}_{z} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} r_{z1} & r_{z2} & r_{z3} \end{array}\displaystyle \right ] \end{aligned} $$

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$$ \frac{\partial \mathbf{S}}{\partial x} = \mathbf{S}_{x} \qquad \frac{\partial \mathbf{S}}{\partial y} = \mathbf{S}_{y} \qquad \frac{\partial \mathbf{S}}{\partial z} = \mathbf{S}_{z} $$

(27)

The right Cauchy–Green deformation tensor is

$$ \mathbf{C} = \mathbf{J}^{\mathrm{T}}\mathbf{J} = \left [ \renewcommand{\arraystretch}{1.2}\textstyle\begin{array}{c@{\quad }c@{\quad }c} \mathbf{r}_{x}^{\mathrm{T}}\mathbf{r}_{x} & \mathbf{r} _{x}^{\mathrm{T}}\mathbf{r}_{y} & \mathbf{r}_{x}^{\mathrm{T}} \mathbf{r}_{z} \\ \mathbf{r}_{y}^{\mathrm{T}}\mathbf{r}_{x} & \mathbf{r} _{y}^{\mathrm{T}}\mathbf{r}_{y} & \mathbf{r}_{y}^{\mathrm{T}} \mathbf{r}_{z} \\ \mathbf{r}_{z}^{\mathrm{T}}\mathbf{r}_{x} & \mathbf{r} _{z}^{\mathrm{T}}\mathbf{r}_{y} & \mathbf{r}_{z}^{\mathrm{T}} \mathbf{r}_{z} \end{array}\displaystyle \right ] $$

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The first strain invariants of the tensor can be written as

$$\begin{aligned} I_{1} =& \operatorname{tr} ( \mathbf{C} ) = \mathbf{r}_{x}^{\mathrm{T}}\mathbf{r}_{x} + \mathbf{r} _{y}^{\mathrm{T}}\mathbf{r}_{y} + \mathbf{r}_{z}^{\mathrm{T}} \mathbf{r}_{z} \\ =& \mathbf{q}_{e}^{\mathrm{T}} \bigl( \mathbf{S}_{x}^{ \mathrm{T}}\mathbf{S}_{x} + \mathbf{S}_{y}^{\mathrm{T}} \mathbf{S}_{y} + \mathbf{S}_{z}^{\mathrm{T}}\mathbf{S} _{z} \bigr) \mathbf{q}_{e} \end{aligned}$$

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The derivative of $I_{1}$ with respect to the element absolute nodal coordinate $\mathbf{q}_{e}$ is deduced as

$$ \biggl( \frac{\partial I_{1}}{\partial \mathbf{q}_{e}} \biggr) ^{\mathrm{T}} = 2 \bigl( \mathbf{S}_{x}^{\mathrm{T}} \mathbf{S}_{x} + \mathbf{S}_{y}^{\mathrm{T}}\mathbf{S} _{y} + \mathbf{S}_{z}^{\mathrm{T}}\mathbf{S}_{z} \bigr) \mathbf{q}_{e} $$

(30)

Recall that the determinant of the deformation gradient tensor takes the form

$$ J = \det ( \mathbf{J} ) = \mathbf{r}_{x}^{ \mathrm{T}} ( \tilde{\mathbf{r}}_{y}\mathbf{r}_{z} ) = \mathbf{r}_{y}^{\mathrm{T}} ( \tilde{\mathbf{r}}_{z} \mathbf{r}_{x} ) = \mathbf{r}_{z}^{\mathrm{T}} ( \tilde{\mathbf{r}}_{x}\mathbf{r}_{y} ) $$

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where

$$ \begin{aligned} &\tilde{\mathbf{r}}_{x} = \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c} 0 & - r_{x3} & r_{x2} \\ r_{x3} & 0 & - r_{x1} \\ - r_{x2} & r_{x1} & 0 \end{array}\displaystyle \right ]\qquad \tilde{\mathbf{r}}_{y} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0 & - r_{y3} & r_{y2} \\ r_{y3} & 0 & - r_{y1} \\ - r_{y2} & r_{y1} & 0 \end{array}\displaystyle \right ] \\ & \tilde{\mathbf{r}}_{z} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0 & - r_{z3} & r_{z2} \\ r_{z3} & 0 & - r_{z1} \\ - r_{z2} & r_{z1} & 0 \end{array}\displaystyle \right ] \end{aligned} $$

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The derivative of the determinant of the deformation gradient tensor with respect to the element absolute nodal coordinate $\mathbf{q} _{e}$ is expressed as

$$\begin{aligned} \frac{\partial J}{\partial \mathbf{q}_{e}} =& ( \tilde{\mathbf{r}}_{y}\mathbf{r}_{z} )^{\mathrm{T}}\frac{ \partial \mathbf{r}_{x}}{\partial \mathbf{q}_{e}} + ( \tilde{\mathbf{r}}_{z}\mathbf{r}_{x} )^{\mathrm{T}}\frac{ \partial \mathbf{r}_{y}}{\partial \mathbf{q}_{e}} + ( \tilde{\mathbf{r}}_{x}\mathbf{r}_{y} )^{\mathrm{T}}\frac{ \partial \mathbf{r}_{z}}{\partial \mathbf{q}_{e}} \\ =& ( \tilde{\mathbf{r}}_{y}\mathbf{r}_{z}) ^{\mathrm{T}}\mathbf{S}_{x} + ( \tilde{\mathbf{r}}_{z} \mathbf{r}_{x} )^{\mathrm{T}}\mathbf{S}_{y} + ( \tilde{\mathbf{r}}_{x}\mathbf{r}_{y} )^{\mathrm{T}} \mathbf{S}_{z} \end{aligned}$$

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$$\begin{aligned} \biggl( \frac{\partial J}{\partial \mathbf{q}_{e}} \biggr)^{ \mathrm{T}} =& \mathbf{S}_{x}^{\mathrm{T}} ( \tilde{\mathbf{r}}_{y}\mathbf{r}_{z} ) + \mathbf{S}_{y}^{\mathrm{T}} ( \tilde{\mathbf{r}}_{z} \mathbf{r}_{x} ) + \mathbf{S}_{z}^{\mathrm{T}} ( \tilde{\mathbf{r}}_{x}\mathbf{r}_{y} ) \end{aligned}$$

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