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This and the next 2 images are from:
Rabczuk T, Areias PMA, Belytschko T (Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208-311, U.S.A),
“A meshfree thin shell method for nonlinear dynamic fracture”, Int J Numer Methods Eng 72: 524–548, 2007
ABSTRACT: A meshfree method for thin shells with finite strains and arbitrary evolving cracks is described. The C1 displacement continuity requirement is met by the approximation, so no special treatments for fulfilling the Kirchhoff condition are necessary. Membrane locking is eliminated by the use of a cubic or quartic polynomial basis. The shell is tested for several elastic and elasto-plastic examples and shows good results. The shell is subsequently extended to modelling cracks. Since no discretization of the director field is needed, the incorporation of discontinuities is easy to implement and straight forward.
INTRODUCTION:
This paper describes a meshfree method for treating the dynamic fracture of shells. It includes both geometric and material nonlinearities and also includes a meshfree fluid model, so that complex fluid-structure interaction problems are feasible. Here we illustrate this capability with the fracture of a fluid-filled cylinder that is impacted by a penetrating projectile. The shell formulation is based on the Kirchhoff-Love (KL) theory.
A meshfree thin shell based on the imposition of the KL constraints was first proposed by Krysl and Belytschko [1]. However, the shell was developed for small deformations, small strains and elasticity. Three dimensional modelling of shear deformable shells and degenerated shells in a meshfree context was studied by Noguchi et al. [2], Li et al. [3] and Kim et al. [4]. Usually, a low order polynomial basis was used, e.g. in [3], a trilinear polynomial basis was applied and the method was applied to several non-linear problems. However, for thin structures, three dimensional modelling of shell structures is computationally expensive. Garcia et al. [5] developed meshfree methods for plates and beams; the higher continuity of meshfree shape functions was exploited for Mindlin-Reissner plates in combination with a p-enrichment. Wang and Chen [6] proposed a meshfree method for Mindlin-Reissner plates. Locking is treated using second order polynomials for the approximation of the translational and rotational motion in combination with a curvature smoothing stabilization. Kanok-Nukulchai et al. [7] addressed shear locking for plates and beams in the element-free Galerkin method.
We develop a meshfree thin shell that combines classical shell theory with a continuum based shell. The kinematic assumptions of classical KL shell theory is adopted. We make use of the generality provided by the continuum description, so that constitutive models developed for continua are easily applicable to shells. The formulation is valid for finite strains.
We include in the shell the capability to model cracks, which are modeled either by cracked particles as in Rabczuk and Belytschko [8] or by a local partition of unity [9, 10, 11]. Due to the higher order continuity of meshfree methods, which enables the use of Kirchhoff- Love shell theories in pristine form, the incorporation of discontinuities is very simple and straight forward. The director field is not discretized, which simplifies the incorporation of discontinuities. In our meshfree model, the concepts for modelling cracks in continua can be adopted directly to shells.
The paper is arranged as follows: First, the kinematics of the shell is described. Then, the meshfree method, the element-free Galerkin (EFG) method, is reviewed and the concept how to incorporate continuum constitutive models is described. The extensions to modelling cracks is described in section 5. Finally, we test the meshfree shell for different elastic linear and nonlinear problems, plastic problems and cracking problems.
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