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Nonlinear bending analysis of hyperelastic Mindlin plates: a numerical approach

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Abstract

In this paper, a numerical solution strategy is proposed for studying the large deformations of rectangular plates made of hyperelastic materials in the compressible and nearly incompressible regimes. The plate is considered to be Mindlin-type, and material nonlinearities are captured based on the Neo-Hookean model. Based on the Euler–Lagrange description, the governing equations are derived using the minimum total potential energy principle. The tensor form of equations is replaced by a novel matrix–vector format for the computational aims. In the solution strategy, based on the variational differential quadrature technique, a new numerical approach is proposed by which the discretized governing equations are directly obtained through introducing differential and integral matrix operators. Fast convergence rate, computational efficiency and simple implementation are advantages of this approach. The results are first validated with available data in the literature. Selected numerical results are then presented to investigate the nonlinear bending behavior of hyperelastic plates under various types of boundary conditions in the compressible and nearly incompressible regimes. The results reveal that the developed approach has a good performance to address the large deformation problem of hyperelastic plates in both regimes.

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Abbreviations

\({\mathbf {e}}_{i}\) :

Scalar base vectors

\({\mathbf {A}}_{p*q}\) :

\(p*q\) matrix

\(A_{ij}\) :

i, j element of matrix \({\mathbf {A}}\)

\(\mathbb {A}_{{{\varvec{P}}}*{\varvec{Q}};r*s}\) :

\({\varvec{P}}*{\varvec{Q}}\)-block matrix where each block is an \(r*s\) matrix

\({\mathbf {A}}_{{\varvec{MN}}}\) :

\({{\varvec{M,N}}}\) block element of block-matrix \(\mathbb {A}\)

\({\mathbf {A}}_{{\varvec{MN;}}ij}\) :

i, j element of \({{\varvec{M,N}}}\) block element of block-matrix \(\mathbb {A}\)

\({\mathbf {A}}^{\mathrm {T}}={\mathrm {trans}} \left( {\mathbf {A}} \right) {\mathrm {,}}\) :

Transpose of \({\mathbf {A}}\)

\(\langle \mathbf{A}\rangle ={\mathrm {diag}}\left( {\mathbf {A}} \right) \) :

Diagonal of vector \({\mathbf {A}}\)

\(\cdot \) :

Dot product (simple inner product)

\(\hat{\mathbf{A}}=\left\lceil {\mathbf {A}} \right\rceil _{q}\) :

Reduction in tensor \({\mathbf {A}}\) to a q-order tensor

\({\mathbb {A}}=\llbracket {\mathbf {A}}\rrbracket _{d}\) :

Discretization of matrix \({\mathbf {A}}\) on d-domain

\(\breve{{\mathbb {A}}}=\langle {\mathbb {A}}\rangle _{\hbox {\char 098}}\) :

Each-block diagonal of block-matrix \(\mathbb {A}\)

\(\overline{\overline{\blacksquare }}=\llbracket \blacksquare \rrbracket _{d}\) :

Discretization of \(\blacksquare (=\hbox {Greek-letters})\) matrix on d-domain

\(\Delta \left( {\mathbf {A}} \right) \) :

Increment of \({\mathbf {A}}\)

\(\delta \left( {\mathbf {A}} \right) \) :

Variation of \({\mathbf {A}}\)

\(\nabla _{{\mathbf {X}}}\cdot \mathbf {A}=\mathrm {Div}\left( {\mathbf {A}} \right) \) :

Divergence with respect to material position vector

\({\nabla }_{{\mathbf {x}}}\left( {\mathbf {A}} \right) ={\mathrm {Grad}} \left( {\mathbf {A}} \right) \) :

Gradient with respect to material position vector

\({\mathbf {A}}_{,{\mathbf {B}}}=\frac{\partial {\mathbf {A}}}{\partial {\mathbf {B}}}=\partial _{{\mathbf {B}}}{\mathbf {A}}\) :

Partial derivative of \({\mathbf {A}}\) with respect to \({\mathbf {B}}\)

\(\mathbb {A}_{,\mathbb {B}}\) :

Discretized form of \({\mathbf {A}}_{,{\mathbf {B}}}(=\llbracket {\mathbf {A}}_{,{\mathbf {B}}} \rrbracket _{d})\)

\(\otimes \) :

Tensorial product

\({\circledast }\) :

Kronecker product

\(\circ \) :

Hadamard product

\(\left\{ {\begin{array}{l} {\mathbf {A}}\otimes {\mathbf {B=}}A_{ij}B_{kl}{\mathbf {e}}_{i}\otimes {\mathbf {e}}_{j}\otimes {\mathbf {e}}_{k}\otimes {\mathbf {e}}_{l} \\ {\mathbf {A}}\overline{\otimes }{\mathbf {B=}}A_{ik}B_{jl}{\mathbf {e}}_{i}\otimes {\mathbf {e}}_{j}\otimes {\mathbf {e}}_{k}\otimes {\mathbf {e}}_{l} \\ {\mathbf {A}}\underline{\otimes }{\mathbf {B=}}A_{il}B_{jk}{\mathbf {e}}_{i}\otimes {\mathbf {e}}_{j}\otimes {\mathbf {e}}_{k}\otimes {\mathbf {e}}_{l} \\ \end{array}} \right. ,\) :

Modified dyadic products of two second-order tensors

\(w_{0}\) :

Total energy density with respect to material unit volume

\(\varPi \) :

Total energy

\(\varPi ^\mathrm{ext}\) :

External energy term

\(\varPi ^\mathrm{int}\) :

Internal energy term

\(\varPi ^\mathrm{lag}\) :

Lagrange energy term

\({\mathbf {x,X}}\) :

Spatial/material position vector of a material point

\(d{\mathbf {x}},d{\mathbf {X}}\) :

Spatial/material line element vector

\(\forall _{t},A_{t},S_{t}\) :

Spatial volume/area/line

\(\forall _{0},A_{0},S_{0}\) :

Material volume/area/line

\(dV_{t},dV_{0}\) :

Spatial/material volume element

\(\mathfrak {B}_{0}\) :

Material configuration

\(\partial \mathfrak {B}_{0}^{\blacksquare }\) :

Material boundary configuration on \(\blacksquare \)

\({\mathbf {H}}\) :

Displacement gradient

\({\mathbf {F}}\) :

Deformation gradient

J :

Determinant of deformation gradient

\({\mathbf {P}}\) :

First Piola-Kirchhoff stress tensor

\({\mathbf {b}}_{{0}}\) :

Spatial body force vector per material unit volume

\(\nu \) :

Poisson’s ratio

E :

Young’s modulus

\(\mu \) :

Shear modulus

\(\kappa \) :

Bulk modulus

\(n_{1},n_{2},n_{3}\) :

Grid numbers in \(X_{1}\), \(X_{2}\) and \(X_{3}\) directions

n :

Total grid size \((=n_{1}\times n_{2}\times n_{3})\)

\({\mathbf {A}}^{{\circ \mathrm{n}}}\) :

\(=\underbrace{\mathbf{A}\circ \mathbf{A}\circ \ldots \circ \mathbf{A}}\)

\(\mathbf {1}_{m*n}\) :

\({=\left[ {\begin{array}{*{20}c} 1 &{} \cdots &{} 1\\ \vdots &{} \ddots &{} \vdots \\ 1 &{} \cdots &{} 1\\ \end{array} } \right] }_{m*n}\)

\(\mathbf {0}_{m*n}\) :

\({=\left[ {\begin{array}{*{20}c} 0 &{} \cdots &{} 0\\ \vdots &{} \ddots &{} \vdots \\ 0 &{} \cdots &{} 0\\ \end{array} } \right] }_{m*n}\)

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

    R. Ansari, R. Hassani & M. Faraji Oskouie

  2. Department of Engineering Science, Faculty of Technology and Engineering, East of Guilan, University of Guilan, 44891-63157, Rudsar, Vajargah, Iran

    H. Rouhi

  3. Center of Excellence for Mathematical Modelling, Optimization and Combinational Computing (MMOCC), University of Guilan, Rasht, Iran

    R. Ansari

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  1. R. Ansari
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  2. R. Hassani
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Correspondence to R. Ansari.

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Ansari, R., Hassani, R., Faraji Oskouie, M. et al. Nonlinear bending analysis of hyperelastic Mindlin plates: a numerical approach. Acta Mech 232, 741–760 (2021). https://doi.org/10.1007/s00707-020-02756-x

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  • DOI: https://doi.org/10.1007/s00707-020-02756-x

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