A VDQ‐based multifield approach to the 2D compressible nonlinear elasticity

Abstract

Summary A numerical multifield methodology is developed to address the large deformation problems of hyperelastic solids based on the 2D nonlinear elasticity in the compressible and nearly incompressible regimes. The governing equations are derived using the Hu‐Washizu principle, considering displacement, displacement gradient, and the first Piola‐Kirchhoff stress tensor as independent unknowns. In the formulation, the tensor form of equations is replaced by a novel matrix‐vector format for computational purposes. In the solution strategy, based on the variational differential quadrature (VDQ) technique and a transformation procedure, a new numerical approach is proposed by which the discretized governing equations are directly obtained through introducing derivative and integral matrix operators. The present method can be regarded as a viable alternative to mixed finite element methods because it is locking free and does not involve complexities related to considering several DOFs for each element in the finite element exterior calculus. Simple implementation is another advantage of this VDQ‐based approach. Some well‐known examples are solved to demonstrate the reliability and effectiveness of the approach. The results reveal that it has good performance in the large deformation problems of hyperelastic solids in compressible and nearly incompressible regimes.

Studying nonlinear vibrations of composite conical panels with arbitrary-shaped cutout reinforced with graphene platelets based on higher-order shear deformation theory

“…Also, the discretized form of E l and E nl are given byin which bold-scriptD2dX1 and bold-scriptD2dX2, respectively, show 2D differential operators in X1 and X2 directions (Hassani et al, 2019a, 2019b).…”

“…In addition, based on the related integral operator, boldC1eq, boldC2eq, boldC3eq, and boldC4eq are computed aswhere trueS2dX1X2 is the integral operator with respect to X1 and X2 in the 2D configuration domain, and trueS1dX3 denotes the integral operator with respect to X3 (in the thickness direction) (Hassani et al, 2019, 2019b).…”

“…In addition, the each-block-diagonal form of P 1 and P 2 can be presented as Also, the discretized form of E l and E nl are given by Note: FSDT: first-order shear deformation theory; HSDT: higher-order shear deformation theory. in which D 2d X 1 and D 2d X 2 , respectively, show 2D differential operators in X 1 and X 2 directions (Hassani et al, 2019a(Hassani et al, , 2019b.…”

“…Nevertheless, the limitation of VDQ is its incapability to consider polygon and concave domains. Later, the VDQ method was enabled for considering arbitrary-shaped polygon domains by the mapping method in Ansari et al (2019), Hassani et al (2019aHassani et al ( , 2019b and Torabi et al (2019). Recently, Ansari et al (2020aAnsari et al ( , 2020b combined the ideas of VDQ and finite element method (FEM) to propose a numerical approach capable of solving the nonlinear bending and thermal postbuckling problems of plates with arbitrary shape including hole.…”