A compatible triangular element including vertex rotations for plane elasticity analysis

Allman, Dan

doi:10.1016/0045-7949(84)90197-4

Cited by 438 publications

(223 citation statements)

References 6 publications

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“…Researchers have attempted to introduce rotational degrees of freedom to classical finite elements by considering special form of displacement functions along each element edge to improve their performance in bending problems [34,35]. This strategy leads often to zero energy deformation modes and to singular element stiffness matrix even if the rigid body motions are constrained.…”

Section: Introductionmentioning

confidence: 99%

Discontinuous Cell Method (DCM) for the simulation of cohesive fracture and fragmentation of continuous media

Cusatis

Rezakhani

Schauffert³

2017

Engineering Fracture Mechanics

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In this paper, the Discontinuous Cell Method (DCM) is formulated with the objective of simulating cohesive fracture propagation and fragmentation in homogeneous solids without issues relevant to excessive mesh deformation typical of available Finite Element formulations. DCM discretizes solids by using the Delaunay triangulation and its associated Voronoi tessellation giving rise to a system of discrete cells interacting through shared facets. For each Voronoi cell, the displacement field is approximated on the basis of rigid body kinematics, which is used to compute a strain vector at the centroid of the Voronoi facets. Such strain vector is demonstrated to be the projection of the strain tensor at that location. At the same point stress tractions are computed through vectorial constitutive equations derived on the basis of classical continuum tensorial theories. Results of analysis of a cantilever beam are used to perform convergence studies and comparison with classical finite element formulations in the elastic regime. Furthermore, cohesive fracture and fragmentation of homogeneous solids are studied under quasi-static and dynamic loading conditions. The mesh dependency problem, typically encountered upon adopting softening constitutive equations, is tackled through the crack band approach. This study demonstrates the capabilities of DCM by solving multiple benchmark problems relevant to cohesive crack propagation. The simulations show that DCM can handle effectively a wide range of problems from the simulation of a single propagating fracture to crack branching and fragmentation.

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Section: Introductionmentioning

confidence: 99%

Discontinuous Cell Method (DCM) for the simulation of cohesive fracture and fragmentation of continuous media

Cusatis

Rezakhani

Schauffert³

2017

Engineering Fracture Mechanics

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“…Note that the nodal rotations r (k) are introduced as additional degrees of freedom which are to be interpreted as relative rotations superimposed on the displacement field and the average rotation generated by the nodal displacements u (k) and v (k) . This notion of "hierarchical rotations" has been discussed in [1,2] and elaborated on in detail in [9]. Hence the nodal rotations are formally defined as…”

Section: Methods Of Analysismentioning

confidence: 99%

“…For the interpolation of the displacements due to the nodal rotations, the original Allman's triangle uses a quadratic function [1]. Later, a cubic interpolation has been proposed [2].…”

Section: Methods Of Analysismentioning

confidence: 99%

“…One well-known example for this behavior is shear locking of the constant stress triangle (CST) element [3]. In order to circumvent these problems, several strategies have been proposed, such as the assumed strain concept (e.g., [4,10,14]) or the introduction of additional degrees of freedom, e.g., drilling DOF's [1,7,12,15,18]. A very general overview is given, e.g., by [9].…”

Section: Introductionmentioning

confidence: 99%

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A family of triangular and tetrahedral elements with rotational degrees of freedom

Bucher

2017

Acta Mech

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A purely kinematic approach to the formulation of plane stress/strain triangular elements with three nodes as well as tetrahedral elements with four nodes including rotational degrees of freedom is presented. The class of elements is shown to perform comparably well in several well-established test cases. Essentially, the displacement interpolation functions are cubic allowing for more flexibility in the displacement-based formulation. The in-plane triangle can be augmented by a plate bending element to form an efficient triangular shell element.

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“…To this aim it is convenient to include in the formulation, even as an enrichment of the shape functions, the so called drilling rotations. It is worth mentioning that the Allman's like models [4] appear to be particularly simple and the choice of assuming a quadratic interpolation function for the side transversal displacement is effective but, as counterpart, the side interpolation becomes insensitive to the nodal rotation average and the resultant element can lead to defective rank. Moreover by using Allman's like interpolation the choice of referring to relative nodal rotations(i.e.…”

Section: Introductionmentioning

confidence: 99%

Composite Fem Models for Limit and Shakedown Analysis

Leonetti¹,

Garcea²,

Nguyen‐Xuan³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. The paper improved S-FEMs formulations with an enriched displacement field, making use of modified Allman's shape functions. This mixed interpolation is the natural context in performing lower bound strategy for shakedown, limit analysis and elastoplastic analysis. The model takes advantages from the simplicity and few addressed requirements for good performances in nonlinear analysis. The simple assumption made for the stress field regards the convenience of using self-equilibrated stress interpolations in Cartesian coordinates. In the proposed composite elements the stress is discontinuous on the element and across their sides and the mesh of the elements is coincident with the discretization of the geometry. This stress interpolation is able to address the discontinuities in the plastic strain and, in such a way, to define in their description a finer mesh with respect the basic grid.

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A compatible triangular element including vertex rotations for plane elasticity analysis

Cited by 438 publications

References 6 publications

Discontinuous Cell Method (DCM) for the simulation of cohesive fracture and fragmentation of continuous media

Discontinuous Cell Method (DCM) for the simulation of cohesive fracture and fragmentation of continuous media

A family of triangular and tetrahedral elements with rotational degrees of freedom

Composite Fem Models for Limit and Shakedown Analysis

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