Large deformation analysis in the context of 3D compressible nonlinear elasticity using the VDQ method

Abstract

In this article, a new solution approach is developed to numerically compute large deformations of 3D hyperelastic solids based on the compressible nonlinear elasticity. The governing equations are derived by the minimum total potential energy principle, and the Neo-Hookean model is used for the hyperelastic character of material. One of the key novelties of the work is its formulation in which the tensor form of equations is replaced by an efficient matrix–vector form that can be readily utilized in the coding process. Moreover, the variational differential quadrature technique is adopted to arrive at the discretized governing equations in a direct way. Simple implementation, fast convergence rate, and computational efficiency are the main advantages of present approach. Through some examples, the accuracy and effectiveness of the proposed numerical approach are revealed.

Appendix

The discretization operator of $\left[\kern-0.15em\left[ {\blacksquare } \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }}$ computes the elements of field of ${\blacksquare }$ at the defined computational points in the domain of ${\mathfrak{C}}_{0}$ and discretizes in block form. The computational points are defined based on the Chebyshev distribution, and are discretized in the $X_{1}$, $X_{2}$ and $X_{3}$ directions as follows:

$$\begin{aligned} {\mathbf{X}}_{1}^{{cheb}} = {\text{Chebyshev}} - {\text{distribution}}(\left[ {0,L_{1} } \right],n_{1} ), \hfill \\ {\mathbf{X}}_{2}^{{cheb}} = {\text{Chebyshev}} - {\text{distribution}}(\left[ {0,L_{2} } \right],n_{2} ), \hfill \\ {\mathbf{X}}_{3}^{{cheb}} = {\text{Chebyshev}} - {\text{distribution}}(\left[ {0,L_{3} } \right],n_{3} ), \hfill \\ \end{aligned}$$

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$$\begin{aligned} {\mathbf{X}}_{1} = \left[ {\begin{array}{*{20}c} {{\mathbf{X}}_{1;1} } & {{\mathbf{X}}_{1;2} } & \cdots & {{\mathbf{X}}_{1;n} } \\ \end{array} } \right]^{\text{T}} = \left[\kern-0.15em\left[ {X_{1} } \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \left( {1_{{n_{3} *1}} { \circledast }1_{{n_{2} *1}} { \circledast }{\mathbf{X}}_{1}^{{cheb}} } \right), \hfill \\ {\mathbf{X}}_{2} = \left[ {\begin{array}{*{20}c} {{\mathbf{X}}_{2;1} } & {{\mathbf{X}}_{2;2} } & \cdots & {{\mathbf{X}}_{2;n} } \\ \end{array} } \right]^{\text{T}} = \left[\kern-0.15em\left[ {X_{2} } \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \left( {1_{{n_{3} *1}} { \circledast }{\mathbf{X}}_{2}^{{cheb}} { \circledast }1_{{n_{1} *1}} } \right), \hfill \\ {\mathbf{X}}_{3} = \left[ {\begin{array}{*{20}c} {{\mathbf{X}}_{3;1} } & {{\mathbf{X}}_{3;2} } & \cdots & {{\mathbf{X}}_{3;n} } \\ \end{array} } \right]^{\text{T}} = \left[\kern-0.15em\left[ {X_{3} } \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \left( {{\mathbf{X}}_{3}^{{cheb}} { \circledast }1_{{n_{2} *1}} { \circledast }1_{{n_{1} *1}} } \right)\varvec{ }\varvec{.} \hfill \\ \end{aligned}$$

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The discretized form of an arbitrary scalar field $f$ is defined as:

$$\begin{aligned} \mathbb{f} = \left[\kern-0.15em\left[ f \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \left[ {\begin{array}{*{20}c} {f_{1} } & {f_{2} } & \ldots & {f_{{n_{1} }} } & {f_{{n_{1} + 1}} } & \ldots & {f_{{n_{1} + n_{2} }} } \\& {f_{{n_{1} + n_{2} + 1}} } & \ldots & {f_{{n_{1} n_{2} }} } & \ldots & {f_{{n_{1} n_{2} n_{3} }} } \\ \end{array} } \right]^{\text{T}} , \end{aligned}$$

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whose derivatives are calculated as follows based on the numerical differentiation of GDQ [20] :

$$\mathbb{f}_{{,{\mathbf{X}}_{1} }}^{\left( r \right)} = \left[\kern-0.15em\left[ {f_{{,X_{1} }}^{\left( r \right)} } \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{X_{1} }}^{\left( r \right)} \left[\kern-0.15em\left[ f \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{X_{1} }}^{\left( r \right)} \mathbb{f}, \quad \mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{X_{1} }}^{\left( r \right)} = \left( {{\mathbf{I}}_{{n_{3} }} { \circledast }{\mathbf{I}}_{{n_{2} }} { \circledast }\mathop {\mathbf{\mathcal{D}}}\limits^{1d} \,_{{X_{1} }}^{\left( r \right)} } \right),$$

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$$\mathbb{f}_{{,{\mathbf{X}}_{2} }}^{\left( r \right)} = \left[\kern-0.15em\left[ {f_{{,X_{2} }}^{\left( r \right)} } \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{X_{2} }}^{\left( r \right)} \left[\kern-0.15em\left[ f \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{X_{2} }}^{\left( r \right)} \mathbb{f}, \quad \mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{X_{2} }}^{\left( r \right)} = \left( {{\mathbf{I}}_{{n_{3} }} { \circledast }\mathop {\mathbf{\mathcal{D}}}\limits^{1d} \,_{{X_{2} }}^{\left( r \right)} { \circledast }{\mathbf{I}}_{{n_{1} }} } \right),$$

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$$\mathbb{f}_{{,{\mathbf{X}}_{3} }}^{\left( r \right)} = \left[\kern-0.15em\left[ {f_{{,X_{3} }}^{\left( r \right)} } \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{X_{3} }}^{\left( r \right)} \left[\kern-0.15em\left[ f \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{X_{3} }}^{\left( r \right)} \mathbb{f}, \quad \mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{X_{3} }}^{\left( r \right)} = \left( {\mathop {\mathbf{\mathcal{D}}}\limits^{1d} \,_{{X_{3} }}^{\left( r \right)} { \circledast }{\mathbf{I}}_{{n_{2} }} { \circledast }{\mathbf{I}}_{{n_{1} }} } \right),$$

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$$\mathbb{f}_{{,{\mathbb{X}}}}^{\left( r \right)} = \left[\kern-0.15em\left[ {f_{{,{\mathbf{X}}}}^{\left( r \right)} } \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{\mathbf{X}}}^{\left( r \right)} \left[\kern-0.15em\left[ f \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{\mathbf{X}}}^{\left( r \right)} \mathbb{f}, \quad \mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{\mathbf{X}}}^{\left( r \right)} = \left[ {\begin{array}{*{20}c} {\mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{X_{1} }}^{\left( r \right)} } \\ {\mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{X_{2} }}^{\left( r \right)} } \\ {\mathop {\mathbf{\mathcal{D}}}\limits^{3d} \,_{{X_{3} }}^{\left( r \right)} } \\ \end{array} } \right],$$

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in which

$$\mathop {\mathbf{\mathcal{D}}}\limits^{1d} \,_{{X_{k} }}^{\left( r \right)} = \left[ {D_{k\,\,ij}^{\left( r \right)} } \right],\varvec{ }$$

$$\begin{aligned} & D_{k\,\,\,ij}^{\left( r \right)} = \left\{ {\begin{array}{*{20}l} {\frac{{P_{i} }}{{\left( {{\mathbf{X}}_{{{\mathbf{K}};i}} - {\mathbf{X}}_{{{\mathbf{K}};j}} } \right)P_{j} }}} \hfill & {i \ne j \; {\text{and}} \; r = 1} \hfill \\ {r\left( {D_{k\,\,ij}^{\left( 1 \right)} D_{k\,\,ii}^{{\left( {r - 1} \right)}} - \frac{{D_{k\,\,\,ij}^{{\left( {r - 1} \right)}} }}{{\left( {{\mathbf{X}}_{{{\mathbf{K}};i}} - {\mathbf{X}}_{{{\mathbf{K}};j}} } \right)}}} \right)} \hfill & {i \ne j \; {\text{and }} \; r = 2,3, \ldots ,n_{k} - 1} \hfill \\ { - \mathop \sum \limits_{j = 1; j \ne i}^{{n_{k} }} D_{k\,\,ij}^{\left( r \right)} } \hfill & {i = j \; {\text{and }} \; r = 1,2,3, \ldots ,n_{k} - 1} \hfill \\ \end{array} } \right. \\ & P_{i} = \mathop \prod \limits_{j = 1; j \ne i}^{{n_{k} }} \left( {{\mathbf{X}}_{{{\mathbf{K}};i}} - {\mathbf{X}}_{{{\mathbf{K}};j}} } \right). \\ \end{aligned}$$

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Also, the numerical integration in the VDQ method is performed based on the Taylor series as follows [20]:

$$\begin{aligned} \mathop \int \limits_{{{\mathfrak{C}}_{0} }} f {\text{d}}X_{1} {\text{d}}X_{2} {\text{d}}X_{3} = \mathop {\mathbf{\mathcal{S}}}\limits^{3d}_{{X_{1} X_{2} X_{3} ;{\mathfrak{C}}_{0} }} \left[\kern-0.15em\left[ f \right]\kern-0.15em\right]_{{{\mathfrak{C}}_{0} }} = \mathop {\mathbf{\mathcal{S}}}\limits^{3d}_{{X_{1} X_{2} X_{3} ;{\mathfrak{C}}_{0} }} \mathbb{f}, \hfill \\ \mathop {\mathbf{\mathcal{S}}}\limits^{3d}_{{X_{1} X_{2} X_{3} ;{\mathfrak{C}}_{0} }} = \left( {\mathop {\mathbf{\mathcal{S}}}\limits^{1d}_{{X_{3} }} { \circledast }\mathop {\mathbf{\mathcal{S}}}\limits^{1d}_{{X_{2} }} { \circledast }\mathop {\mathbf{\mathcal{S}}}\limits^{1d}_{{X_{1} }} } \right), \hfill \\ \end{aligned}$$

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where $\mathop {\mathbf{\mathcal{S}}}\limits^{1d}_{{X_{1} }}$, $\mathop {\mathbf{\mathcal{S}}}\limits^{1d}_{{X_{2} }}$, and $\mathop {\mathbf{\mathcal{S}}}\limits^{1d}_{{X_{3} }}$ are calculated as:

$$\begin{aligned} & \mathop {\mathbf{\mathcal{S}}}\limits^{1d}_{{X_{K} }} = \mathop \sum \limits_{r = 0}^{\infty } {\mathbf{M}}^{\left( r \right)} \mathop {\mathbf{\mathcal{D}}}\limits^{1d} \,_{{X_{K} }}^{\left( r \right)} , \\ & {\mathbf{M}}^{\left( r \right)} = \left[ {\begin{array}{*{20}c} {\frac{{\left( {{\mathbf{X}}_{{{\mathbf{K}};2}} - {\mathbf{X}}_{{{\mathbf{K}};1}} } \right)^{r + 1} }}{{2^{r + 1} \left( {r + 1} \right)!}}} & \ldots & {\frac{{\left( {{\mathbf{X}}_{{{\mathbf{K}};i + 1}} - {\mathbf{X}}_{{{\mathbf{K}};i}} } \right)^{r + 1} - \left( {{\mathbf{X}}_{{{\mathbf{K}};i - 1}} - {\mathbf{X}}_{{{\mathbf{K}};i}} } \right)^{r + 1} }}{{2^{r + 1} \left( {r + 1} \right)!}}} & \ldots & {\frac{{\left( {{\mathbf{X}}_{{{\mathbf{K}};n_{k} - 1}} - {\mathbf{X}}_{{{\mathbf{K}};n_{k} }} } \right)^{r + 1} }}{{2^{r + 1} \left( {r + 1} \right)!}}} \\ \end{array} } \right]. \\ \end{aligned}$$

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