Nonlinear bending analysis of hyperelastic Mindlin plates: a numerical approach
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O R I G I NA L PA P E R
R. Ansari · R. Hassani · M. Faraji Oskouie · H. Rouhi
Nonlinear bending analysis of hyperelastic Mindlin plates: a numerical approach
Received: 17 July 2019 / Revised: 2 April 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
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. Latin symbols Matrix notation ei A p∗q Ai j AP∗Q;r ∗s AMN AMN;i j
Scalar base vectors p ∗ q matrix i, j element of matrix A P ∗ Q-block matrix where each block is an r ∗ s matrix M,N block element of block-matrix A i, j element of M,N block element of block-matrix A
Operators AT = trans (A) ,
Transpose of A
R. Ansari (B) · R. Hassani · M. Faraji Oskouie Faculty of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran E-mail: [email protected] H. Rouhi Department of Engineering Science, Faculty of Technology and Engineering, East of Guilan, University of Guilan, 44891-63157 Rudsar, Vajargah, Iran R. Ansari Center of Excellence for Mathematical Modelling, Optimization and Combinational Computing (MMOCC), University of Guilan, Rasht, Iran
R. Ansari et al.
A = diag (A) · ˆ = Aq A A = Ad ˘ = Ab A = d (A) δ (A) ∇X · A = Div (A) ∇x (A) = Grad (A) A,B = ∂A ∂B = ∂B A A,B ⊗ ◦ ⎧ ⎨ A ⊗ B =Ai j Bkl ei ⊗ e j ⊗ ek ⊗ el A⊗B =Aik B jl ei ⊗ e j ⊗ ek ⊗ el , ⎩ A⊗B =A B e ⊗ e ⊗ e ⊗ e il jk i j k l
Diagonal of vector A Dot product (simple inner product) Reduction in tensor A to a q-order tensor Discretization of matrix A on d-domain Each-block diagonal of block-matrix A Discretization of (= Greek-letters) matrix on d-domain Increment of A Variation of A Divergence with respect to material position vector Gradient with respect to material position vector Partial derivative of A with respect to B Di
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