On numerically accurate finite element solutions in the fully plastic range

Nagtegaal, J. C.; Parks, David M.; Rice, James R.

doi:10.1016/0045-7825(74)90032-2

Cited by 902 publications

(345 citation statements)

References 9 publications

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“…(11,14) are commonly evaluated by Gauss integration which requires the use of background integration cells. It is normal to use rectangular and triangle cells and high order quadratures.…”

Section: Stabilized Conforming Nodal Integrationmentioning

confidence: 99%

“…Current research is focussing on developing limit analysis tools which are sufficiently efficient and robust to be of use to engineers working in practice. However, when FEM is applied some of the well-known characteristics of mesh-based methods can lead to problems: the solutions are often highly sensitive to the geometry of the original mesh, particularly in the region of stress or displacement/velocity singularities; furthermore, volumetric locking may occur in plane strain and 3D problems [11]. Although adaptive schemes with the h-version [12][13][14][15][16] or p-version FEM [17,18] have been used to try to overcome such disadvantages, and show immense promise, the schemes quickly become complex and a large number of elements are generally required to obtain accurate solutions.…”

Section: Introductionmentioning

confidence: 99%

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Limit analysis of plates using the EFG method and second‐order cone programming

Gilbert

Askes

2009

Numerical Meth Engineering

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Published paperLe, Canh V., Gilbert, Matthew and Askes, Harm (2009) SUMMARYThe meshless Element-Free Galerkin (EFG) method is extended to allow computation of the limit load of plates. A kinematic formulation which involves approximating the displacement/velocity field using the moving least squares technique is developed. Only one displacement variable is required for each EFG node, ensuring that the total number of variables in the resulting optimization problem is kept to a minimum, with far fewer variables being required compared with finite element formulations. A stabilized conforming nodal integration scheme is extended to plastic plate bending problems. The evaluation of integrals at nodal points using curvature smoothing stabilization both keeps the size of the optimization problem small and also results in stable and accurate solutions. Difficulties imposing essential boundary conditions are overcome by enforcing directly displacements at the nodes. The formulation can be expressed as the problem of minimizing a sum of Euclidean norms subject to a set of equality constraints. This non-smooth minimization problem can be transformed into a form suitable for solution using Second-Order Cone Programming (SOCP). The procedure is applied to several benchmark problems and is found in practice to generate good upper bound solutions for benchmark problems.

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“…(11,14) are commonly evaluated by Gauss integration which requires the use of background integration cells. It is normal to use rectangular and triangle cells and high order quadratures.…”

Section: Stabilized Conforming Nodal Integrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Limit analysis of plates using the EFG method and second‐order cone programming

Gilbert

Askes

2009

Numerical Meth Engineering

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“…Figure 8 shows the discretized specimens with their respective initial specific volume field v := 1+e. The rectangular domains were discretized using 2000 trilinear hexahedral elements equipped with the B-bar method for nonlinear kinematics using the so-called mean dilation technique [47,48] (see Reference [49] for a survey on the B-bar method).…”

Section: Simulations With Cubical Specimensmentioning

confidence: 99%

Capturing strain localization in dense sands with random density

Andrade

Borja

2006

Numerical Meth Engineering

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SUMMARYThis paper presents a three-invariant constitutive framework suitable for the numerical analyses of localization instabilities in granular materials exhibiting unstructured random density. A recently proposed elastoplastic model for sands based on critical state plasticity is enhanced with the third stress invariant to capture the difference in the compressive and extensional yield strengths commonly observed in geomaterials undergoing plastic deformation. The new three-invariant constitutive model, similar to its two-invariant predecessor, is capable of accounting for meso-scale inhomogeneities as well as material and geometric nonlinearities. Details regarding the numerical implementation of the model into a fully nonlinear finite element framework are presented and a closed-form expression for the consistent tangent operator, whose spectral form is used in the strain localization analyses, is derived. An algorithm based on the spectral form of the so-called acoustic tensor is proposed to search for the necessary conditions for deformation bands to develop. The aforementioned framework is utilized in a series of boundary-value problems on dense sand specimens whose density fields are modelled as exponentially distributed unstructured random fields to account for the effect of inhomogeneities at the meso-scale and the intrinsic uncertainty associated with them.

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“…In fact, for such problems first-order shape functions (bior tri-linear interpolations) used to approximate the displacement field over a finite element exhibit convergence issues, B Gerhard A. Holzapfel holzapfel@tugraz.at 1 see, e.g., Hughes [19], Zienkiewicz and Taylor [40] and Wriggers [38]. One way to avoid this problem is to use a mixed variational formulation that hinges on the Hu-Washizu principle, where the master field appears together with additional subsidiary conditions, as first introduced by Nagtegaal et al [21] and discussed in Brezzi and Fortin [4] for small strains. It was later extended to finite strain problems by Simo et al [32] and is also well documented in the literature, by, e.g., Miehe [20] and Wriggers [38].…”

Section: Introductionmentioning

confidence: 99%

On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials

2018

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Quasi-incompressible behavior is a desired feature in several constitutive models within the finite elasticity of solids, such as rubber-like materials and some fiber-reinforced soft biological tissues. The Q1P0 finite element formulation, derived from the three-field Hu-Washizu variational principle, has hitherto been exploited along with the augmented Lagrangian method to enforce incompressibility. This formulation typically uses the unimodular deformation gradient. However, contributions by Sansour (Eur J Mech A Solids 27: [28][29][30][31][32][33][34][35][36][37][38][39] 2007) and Helfenstein et al. (Int J Solids Struct 47:2056-2061 conspicuously demonstrate an alternative concept for analyzing fiber reinforced solids, namely the use of the (unsplit) deformation gradient for the anisotropic contribution, and these authors elaborate on their proposals with analytical evidence. The present study handles the alternative concept from a purely numerical point of view, and addresses systematic comparisons with respect to the classical treatment of the Q1P0 element and its coalescence with the augmented Lagrangian method by means of representative numerical examples. The results corroborate the new concept, show its numerical efficiency and reveal a direct physical interpretation of the fiber stretches.

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On numerically accurate finite element solutions in the fully plastic range

Cited by 902 publications

References 9 publications

Limit analysis of plates using the EFG method and second‐order cone programming

Limit analysis of plates using the EFG method and second‐order cone programming

Capturing strain localization in dense sands with random density

On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials

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