Generalization of the strain-split method and evaluation of the nonlinear ANCF finite elements

Abstract

This paper discusses the generalization of the strain-split method (SSM) for the locking alleviation of curved structures. The generalization is achieved by using proper definitions of the stress and strain tensors along the curved-coordinate lines using the matrix of position vector gradients in the reference configuration. This matrix, which accurately captures the element geometry at the integration points, allows using consistent gradient transformation in the calculation of the stress and strain tensors. The generalized SSM implementation is used to verify the results and evaluate the performance of the absolute nodal coordinate formulation (ANCF) finite elements (FE). The focus of this study is on the Poisson locking that characterizes fully parameterized ANCF elements that employ different orders of interpolation in different directions. ANCF beam and plate nonlinear problems are presented, and the obtained simulation results are compared with analytical solutions as well as results obtained using commercial FE computer programs. These results are also compared with the results obtained using ANCF beam and curved plate elements in the case of nonzero Poisson ratio in order to demonstrate the SSM effectiveness in alleviating the Poisson locking. It is shown that a much smaller number of ANCF plate elements is required to achieve approximately 0.9% difference from the results of commercial FE computer programs.

Appendices

Appendix A

Fully parameterized three-dimensional ANCF beam element

The fully parameterized three-dimensional ANCF beam element, proposed in [36], is a 2-node element with 24 degrees of freedom. The nodal coordinates $\mathbf{e}^{jk}$ at node k of the finite element j are defined as

$$\begin{aligned} \mathbf{e}^{jk}=\left[ {\mathbf{r}^{jk^{\mathrm{T}}} \;\mathbf{r}_x^{jk^{\mathrm{T}}} \;\mathbf{r}_y^{jk^{\mathrm{T}}} \;\mathbf{r}_z^{jk^{\mathrm{T}}} } \right] ^{\mathrm{T}} \qquad k=1,2, \end{aligned}$$

(A.1)

where $\mathbf{r}^{jk}$ is the absolute position vector at the node k of the element j and $\mathbf{r}_x^{jk}$, $\mathbf{r}_y^{jk}$, and $\mathbf{r}_z^{jk}$ are the position vector gradients obtained by differentiation with respect to the spatial coordinates x, y, and z, respectively. The shape function matrix can be written as $\mathbf{S}=\left[ {s_1 \mathbf{I}\hbox { }s_2 \mathbf{I}\hbox { }s_3 \mathbf{I}\hbox { }s_4 \mathbf{I}\hbox { }s_5 \mathbf{I}\hbox { }s_6 \mathbf{I}\hbox { }s_7 \mathbf{I}\hbox { }s_8 \mathbf{I}} \right]$, where

$$\begin{aligned} \left. {\begin{array}{l@{\quad }l} s_1 =1-3\xi ^{2}+2\xi ^{3} &{} s_2 =l\left( {\xi -2\xi ^{2}+\xi ^{3}} \right) \\ s_3 =l\left( {\eta -\xi \eta } \right) &{} s_4 =l\left( {\zeta -\zeta \xi } \right) \\ s_5 =3\xi ^{2}-2\xi ^{3} &{} s_6 =l\left( {-\xi ^{2}+\xi ^{3}} \right) \\ s_7 =l\xi \eta &{} s_8 =l\xi \zeta \hbox { } \\ \end{array}} \right\} \end{aligned}$$

(A.2)

where $\xi =x/l$, $\eta = y/l$, and $\zeta = z/l$ are the dimensionless parameters of the element and l is the element length.

Fully parameterized ANCF thick plate element

The fully parameterized ANCF thick plate element, proposed in [37], is a 4-node element with 48 degrees of freedom. The nodal coordinates $\mathbf{e}^{jk}$ at the node k of the finite element j are defined as

$$\begin{aligned} \mathbf{e}^{jk}=\left[ {\mathbf{r}^{jk^{\mathrm{T}}} \;\mathbf{r}_x^{jk^{\mathrm{T}}} \;\mathbf{r}_y^{jk^{\mathrm{T}}} \;\mathbf{r}_z^{jk^{\mathrm{T}}} } \right] ^{\mathrm{T}}\;\;\;\;\;\;k=1,\ldots ,4, \end{aligned}$$

(A.3)

where $\mathbf{r}^{jk}$ is the absolute position vector at node k of the element j and $\mathbf{r}_x^{jk}$, $\mathbf{r}_y^{jk}$, and $\mathbf{r}_z^{jk}$ are the position vector gradients obtained by differentiation with respect to the spatial coordinates x, y, and z, respectively. The shape function matrix of this element can be written as ${\varvec{S}} =\big [s_1 \mathbf{I}\hbox { }s_2 \mathbf{I}\hbox { }s_3 \mathbf{I}\hbox { }s_4 \mathbf{I}\hbox { }s_5 \mathbf{I}\hbox { }s_6 \mathbf{I}\hbox { }s_7 \mathbf{I}\hbox { }s_8 \mathbf{I}\hbox { }s_9 \mathbf{I}\hbox { }s_{10} \mathbf{I}\hbox { }s_{11} \mathbf{I}\hbox { }s_{12} \mathbf{I}\hbox { }s_{13} \mathbf{I}\hbox { }s_{14} \mathbf{I}\hbox { }s_{15} \mathbf{I}\hbox { }s_{16} \mathbf{I}\big ]$, where

$$\begin{aligned} \left. {\begin{array}{l@{\quad }l} s_1 =-\left( {\xi -1} \right) \left( {\eta -1} \right) \left( {2\eta ^{2}-\eta +2\xi ^{2}-\xi -1} \right) \hbox { } &{} s_2 =-a\xi \left( {\xi -1} \right) ^{2}\left( {\eta -1} \right) \\ s_3 =-b\eta \left( {\eta -1} \right) ^{2}\left( {\xi -1} \right) \hbox { } &{} s_4 =t\zeta \left( {\xi -1} \right) \left( {\eta -1} \right) \hbox { } \\ s_5 =\xi \left( {2\eta ^{2}-\eta -3\xi +2\xi ^{2}} \right) \left( {\eta -1} \right) \hbox { } &{} s_6 =-a\xi ^{2}\left( {\xi -1} \right) \left( {\eta -1} \right) \hbox { } \\ s_7 =b\xi \eta \left( {\eta -1} \right) ^{2}\hbox { } &{} s_8 =-t\xi \zeta \left( {\eta -1} \right) \\ s_9 =-\xi \eta \left( {1-3\xi -3\eta +2\eta ^{2}+2\xi ^{2}} \right) \hbox { } &{} s_{10} =a\xi ^{2}\eta \left( {\xi -1} \right) \hbox { } \\ s_{11} =b\xi \eta ^{2}\left( {\eta -1} \right) \hbox { } &{} s_{12} =t\xi \eta \zeta \\ s_{13} =\eta \left( {\xi -1} \right) \left( {2\xi ^{2}-\xi -3\eta +2\eta ^{2}} \right) \hbox { } &{} s_{14} =a\xi \eta \left( {\xi -1} \right) ^{2} \\ s_{15} =-b\eta ^{2}\left( {\xi -1} \right) \left( {\eta -1} \right) \hbox { } &{} s_{16} =-t\eta \zeta \left( {\xi -1} \right) \hbox { } \\ \end{array}} \right\} \end{aligned}$$

(A.4)

In this equation, $\xi =x/a$, $\eta =y/b$, and $\zeta =z/t$ are the dimensionless parameters of the element and a, b, and t are the element length, width, and thickness respectively.

ANCF thin plate element

The ANCF thin plate element, proposed in [38], is a 4-node element with 36 degrees of freedom. The nodal coordinates $\mathbf{e}^{jk}$ at node k of the finite element j are as follows:

$$\begin{aligned} \mathbf{e}^{jk}=\left[ {\mathbf{r}^{jk^{\mathrm{T}}} \;\mathbf{r}_x^{jk^{\mathrm{T}}} \;\mathbf{r}_y^{jk^{\mathrm{T}}} } \right] ^{\mathrm{T}}\;\;\;\;\;\;k=1,\ldots ,4, \end{aligned}$$

(A.5)

where $\mathbf{r}^{jk}$ is the absolute position vector at node k of the element j and $\mathbf{r}_x^{jk}$ and $\mathbf{r}_y^{jk}$are the position vector gradients obtained by differentiation with respect to the spatial coordinates x and y, respectively. The shape function matrix of this element can be written as $\mathbf S =\left[ {s_1 \mathbf{I}\hbox { }s_2 \mathbf{I}\hbox { }s_3 \mathbf{I}\hbox { }s_4 \mathbf{I}\hbox { }s_5 \mathbf{I}\hbox { }s_6 \mathbf{I}\hbox { }s_7 \mathbf{I}\hbox { }s_8 \mathbf{I}\hbox { }s_9 \mathbf{I}\hbox { }s_{10} \mathbf{I}\hbox { }s_{11} \mathbf{I}\hbox { }s_{12} I} \right]$, where

$$\begin{aligned} \left. {\begin{array}{l@{\quad }l} s_1 =-\left( {\xi -1} \right) \left( {\eta -1} \right) \left( {2\eta ^{2}-\eta +2\xi ^{2}-\xi -1} \right) \hbox { } &{} s_2 =-a\xi \left( {\xi -1} \right) ^{2}\left( {\eta -1} \right) \\ s_3 =-b\eta \left( {\eta -1} \right) ^{2}\left( {\xi -1} \right) \hbox { } &{} s_4 =\xi \left( {2\eta ^{2}-\eta -3\xi +2\xi ^{2}} \right) \left( {\eta -1} \right) \hbox { } \\ s_5 =-a\xi ^{2}\left( {\xi -1} \right) \left( {\eta -1} \right) \hbox { } &{} s_6 =b\xi \eta \left( {\eta -1} \right) ^{2}\hbox { } \\ s_7 =-\xi \eta \left( {1-3\xi -3\eta +2\eta ^{2}+2\xi ^{2}} \right) &{} s_8 =a\xi ^{2}\eta \left( {\xi -1} \right) \\ s_9 =b\xi \eta ^{2}\left( {\eta -1} \right) \hbox { }&{} s_{10} =\eta \left( {\xi -1} \right) \left( {2\xi ^{2}-\xi -3\eta +2\eta ^{2}} \right) \hbox { } \\ s_{11} =a\xi \eta \left( {\xi -1} \right) \hbox { }&{} s_{12} =-b\eta ^{2}\left( {\xi -1} \right) ^{2}\left( {\eta -1} \right) \hbox { } \\ \end{array}} \right\} \end{aligned}$$

(A.6)

In this equation, $\xi =x/a$ and $\eta =y/b$ are the dimensionless parameters of the element, and a and b are the element length and width, respectively.