A mixed finite element model for plane strain elastic—plastic analysis Part II. Application to the 4-node bilinear element

Curnis, Antonio; Corradi, Leone

doi:10.1016/s0045-7825(96)01099-7

Cited by 15 publications

(19 citation statements)

References 10 publications

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“…As indicated, the implementation of the mixed finite element is based on suitable independent approximations of both kinematic and static fields . The stress field at each generic element i contains the in‐plane stresses

σ^{i T} (x) \in ℜ^{3} = [σ_{x}, σ_{y}, σ_{italicxy}]

, which can be described as functions of the generalized stresses Q i ∈ ℜ 5 by

σ^{i} (bold-italicx) MathClass-rel= {MathClass-ops}^{i} (bold-italicx) {bold-italicQ}^{i} MathClass-punc,

where

{MathClass-ops}^{i} (bold-italicx) MathClass-rel\in ℜ^{3 MathClass-bin\times 5} MathClass-rel= \frac{1}{Ω} ([], \begin{matrix} falsenonefalsearray \\ arraycenter I_{3} & arraycenter s_{1} MathClass-open(x MathClass-close) \end{matrix}) MathClass-punc,

x \in ℜ^{2} = (ξ, η)

contains natural coordinates ranging from − 1 to 1, I 3 ∈

ℜ^{3 \times 3}

is a 3 × 3 identity matrix, s 1 ( x ) ∈ ℜ 3 × 2 is a stress matrix, and Ω is the element area.…”

Section: The Elastoplastic Problem and Its Finite Element Modelmentioning

confidence: 99%

“…By substituting Equation into Equation , we can write the in‐plane plastic strains π i ( x ) solely as functions of the generalized plastic strains

{bold-italicp}^{i normalT} MathClass-rel= ([], \begin{matrix} falsenonefalsearray \\ arraycenter p_{0}^{T} & arraycenter p_{1}^{T} \end{matrix})

π^{i} (bold-italicx) MathClass-rel= {bold-italich}_{3}^{i} (bold-italicx) {bold-italicp}^{i} MathClass-punc,

where

{bold-italich}_{3}^{i} (bold-italicx) MathClass-rel\in ℜ^{3 MathClass-bin\times 5} MathClass-rel= ([], \begin{matrix} falsenonefalsearray \\ arraycenter I_{3} & arraycenter h_{1} MathClass-open(x MathClass-close) + h_{2} MathClass-open(x MathClass-close) R \end{matrix})

is a condensed strain matrix. The complete descriptions of key quantities (such as s 1 ( x ), h 1 ( x ), h 2 ( x ), and R ) are given in .…”

Section: The Elastoplastic Problem and Its Finite Element Modelmentioning

confidence: 99%

“…For the von Mises material, the stepwise holonomic runs with Δ t = 0.05 (solid line in Figure ) predicted accurately (without any locking behavior) the maximum load of α ≈ 0.996, about 0.4 % less than the reference result of α = 1 (dotted line) . The first yield was also found at α = 0.620.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Key features of our approach are as follows: The evolutive elastoplastic analysis of the nonholonomic (path‐dependent) problem for the suitably discretized structure is formulated in a conventional stepwise holonomic (path‐independent) fashion . The particular finite element used is the mixed element developed by Capsoni and Corradi for plane strain von Mises problems . This element advantageously provides a locking‐free capability as well as coarse‐mesh accuracy.…”

Section: Introductionmentioning

confidence: 99%

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A direct complementarity approach for the elastoplastic analysis of plane stress and plane strain structures

Tangaramvong

Tin‐Loi

Song

2012

Numerical Meth Engineering

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SUMMARY This paper presents a direct complementarity approach for carrying out the elastoplastic analysis of plane stress and plane strain structures. Founded on a traditional finite‐step formulation, our approach, however, avoids the typically cumbersome implementation of iterative predictor–corrector procedures associated with the ubiquitous return mapping algorithm. Instead, at each predefined step, the governing formulation—cast in its most natural mathematical programming format known as a mixed complementarity problem—is directly solved by using a complementarity solver run from within a mathematical modeling system. We have chosen the industry‐standard General Algebraic Modeling System/PATH mixed complementarity problem solver that is called from within the General Algebraic Modeling System environment. We consider both von Mises and Tresca materials, with perfect or hardening (kinematic and isotropic) behaviors. Our numerical tests, five (benchmark) examples of which are presented in this paper, have been carried out using models constructed from the mixed finite element of Capsoni and Corradi (Comput. Methods Appl. Mech. Eng. 1997; 141:67–93), which beneficially offers a locking‐free behavior and coarse‐mesh accuracy. The results indicate, in addition to an isochoric locking‐free behavior, good accuracy and the ability to circumvent the difficult singularity problem associated with the corners of Tresca yield surfaces. Copyright © 2012 John Wiley & Sons, Ltd.

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σ^{i T} (x) \in ℜ^{3} = [σ_{x}, σ_{y}, σ_{italicxy}]

, which can be described as functions of the generalized stresses Q i ∈ ℜ 5 by

σ^{i} (bold-italicx) MathClass-rel= {MathClass-ops}^{i} (bold-italicx) {bold-italicQ}^{i} MathClass-punc,

where

{MathClass-ops}^{i} (bold-italicx) MathClass-rel\in ℜ^{3 MathClass-bin\times 5} MathClass-rel= \frac{1}{Ω} ([], \begin{matrix} falsenonefalsearray \\ arraycenter I_{3} & arraycenter s_{1} MathClass-open(x MathClass-close) \end{matrix}) MathClass-punc,

x \in ℜ^{2} = (ξ, η)

contains natural coordinates ranging from − 1 to 1, I 3 ∈

ℜ^{3 \times 3}

is a 3 × 3 identity matrix, s 1 ( x ) ∈ ℜ 3 × 2 is a stress matrix, and Ω is the element area.…”

Section: The Elastoplastic Problem and Its Finite Element Modelmentioning

confidence: 99%

“…By substituting Equation into Equation , we can write the in‐plane plastic strains π i ( x ) solely as functions of the generalized plastic strains

{bold-italicp}^{i normalT} MathClass-rel= ([], \begin{matrix} falsenonefalsearray \\ arraycenter p_{0}^{T} & arraycenter p_{1}^{T} \end{matrix})

π^{i} (bold-italicx) MathClass-rel= {bold-italich}_{3}^{i} (bold-italicx) {bold-italicp}^{i} MathClass-punc,

where

{bold-italich}_{3}^{i} (bold-italicx) MathClass-rel\in ℜ^{3 MathClass-bin\times 5} MathClass-rel= ([], \begin{matrix} falsenonefalsearray \\ arraycenter I_{3} & arraycenter h_{1} MathClass-open(x MathClass-close) + h_{2} MathClass-open(x MathClass-close) R \end{matrix})

is a condensed strain matrix. The complete descriptions of key quantities (such as s 1 ( x ), h 1 ( x ), h 2 ( x ), and R ) are given in .…”

Section: The Elastoplastic Problem and Its Finite Element Modelmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A direct complementarity approach for the elastoplastic analysis of plane stress and plane strain structures

Tangaramvong

Tin‐Loi

Song

2012

Numerical Meth Engineering

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“…Equations (28)- (30) resulting from the present mixed formulation can be cast in the standard form [32]   …”

Section: Discrete Governing Equationsmentioning

confidence: 99%

Finite elements with embedded displacement discontinuity: a generalized variable formulation

Bolzon

Corigliano

2000

Int. J. Numer. Meth. Engng.

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A variationally consistent generalized variable formulation for enhanced strain finite elements

Perego

2000

Commun. Numer. Meth. Engng.

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The so‐called enhanced strain finite elements are based on the enrichment of the standard compatible strain field by the introduction of additional, non‐compatible strains. This class of elements can be derived starting from a partial Hu–Washizu variational principle. However, since in the original enhanced strain formulation the stress field is eliminated from the formulation, a separate least‐squares procedure had to be implemented for a variational derivation of the stress field. A three‐field generalized variable approach incorporating strain enhancement is proposed in the present paper within the context of linear elastic structural problems. It is shown how the original two‐field enhanced strain method can be naturally recovered by suitably choosing the strain model. For this case a straightforward, but still variationally consistent stress recovery is proposed. Copyright © 2000 John Wiley & Sons, Ltd.

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A mixed finite element model for plane strain elastic—plastic analysis Part II. Application to the 4-node bilinear element

Cited by 15 publications

References 10 publications

A direct complementarity approach for the elastoplastic analysis of plane stress and plane strain structures

A direct complementarity approach for the elastoplastic analysis of plane stress and plane strain structures

Finite elements with embedded displacement discontinuity: a generalized variable formulation

A variationally consistent generalized variable formulation for enhanced strain finite elements

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