Hybrid equilibrium element with high‐order stress fields for accurate elastic dynamic analysis

Abstract

In the present article the two-dimensional hybrid equilibrium element formulation is initially developed, with quadratic, cubic, and quartic stress fields, for static analysis of compressible and quasi-incompressible elastic solids in the variational framework of the minimum complementary energy principle. Thereafter, the high-order hybrid equilibrium formulation is developed for dynamic analysis of elastic solids in the variational framework of the Toupin principle, which is the complementary form of the Hamilton principle. The Newmark time integration scheme is introduced for discretization of the stress fields in the time domain and dynamic analysis of both the compressible solid and quasi-incompressible ones. The hybrid equilibrium element formulation provides very accurate solutions with a high-order stress field and the results of the static and dynamic analyses are compared with the solution of the classic displacement-based quadratic formulation, showing the convergence of the two formulations to the exact solution and the very satisfying performance of the proposed formulation, especially for analysis of quasi-incompressible elastic solids.

A coupled plasticity-damage cohesive-frictional interface for low-cycle fatigue analysis

“…[36]. The interface damage law has been recently developed in an hybrid equilibrium based formulation in [37,38] and in a multi-physics framework in [39,40,41].…”

Equilibrium finite element and error estimation for frictional contact problem based on linear complementarity problem formulation

“…However, it is difficult to deal with multiply‐connected domains and in linear elasticity problems, the stress functions should be $$$ {C}^1 $$$‐continuous, turning into another difficult interpolation problem. In order to circumvent these difficulties, the hybrid equilibrium element is subsequently proposed 20‐28 . It constructs the equilibrated stress field by polynomial basis functions in each element, and introduces auxiliary weighting functions, also known as elementary boundary displacements, to guarantee the interelement traction continuity on each internal element boundary 29 .…”

“…In order to circumvent these difficulties, the hybrid equilibrium element is subsequently proposed. [20][21][22][23][24][25][26][27][28] It constructs the equilibrated stress field by polynomial basis functions in each element, and introduces auxiliary weighting functions, also known as elementary boundary displacements, to guarantee the interelement traction continuity on each internal element boundary. 29 As is noteworthy, the hybrid equilibrium element is plagued by spurious kinematic modes (SKMs).…”

“…To eliminate the SKMs, the macro-element/super-element technique 20,29 and the restraining approach 25 have been developed. Up till now, the hybrid equilibrium element method has been successfully applied to many problems, including elastostatics, 28,29 plates, 22,26 delamination/crack propagation, 23 elastodynamics, 24 and folded plates/shells. 25 Recently, an alternative traction-based equilibrium element 30,31 is proposed for the EFEM.…”

Inter-Element Crack Propagation with High-Order Stress Equilibrium Element

“…The equilibriumbased formulation for elastic-dynamic analysis is developed in Ref. 26 . The proposed formulation provides a solution of the elastic problem with a quadratic stress fields that strongly verifies the homogeneous equilibrium equations and inter-element equilibrium equations 27,28,29,30,31,32,33,34 .…”