# Large-deformation plasticity analysis using the edge-based ... An edge-based smoothed finite element

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APCOM & ISCM 11-14th December, 2013, Singapore

1

Large-deformation plasticity analysis using the edge-based smoothed finite element method

*J. Liu¹, Z.Q. Zhang2 1Department of Mechanical Engineering, National University of Singapore, 117576, Singapore.

2Institute of High Performance Computing, Singapore. *Corresponding author: nickliujun@gmail.com

Abstract

An edge-based smoothed finite element method (ES-FEM) using 3-node triangular element was recently proposed to improve the accuracy and convergence rate of the standard finite element method (FEM) for 2D elastic solid mechanics problems. In this research, ES-FEM is extended to the large-deformation plasticity analysis, and a selective edge-based / node-based smoothed finite element (selective ES/NS-FEM) method using 3-node triangular element is adopted to address the volumetric locking problem. Validity of ES-FEM for large-deformation plasticity problem is proved by benchmarks, and numerical examples demonstrate that, the proposed ES-FEM and selective ES/NS-FEM method possess (1) superior accuracy and convergence properties for strain energy solutions comparing to the standard FEM using 3-node triangular element (FEM-T3), (2) better computational efficiency than FEM-T3 and similar computational efficiency as FEM using 4-node quadrilateral element and 6-node quadratic triangular element, (3) a selective ES/NS-FEM method can successfully simulating severe element distortion problem, and address volumetric locking problem in large-deformation plasticity analysis. Keywords: Large-deformation plasticity, Finite element method (FEM), Edge-based smoothed finite element method (ES-FEM), Volumetric locking, Gradient smoothing.

Introduction

Numerical simulation of finite-deformed material has attracted numerous research efforts. The finite element method (FEM) has been developed during last decades to deal with material and geometric nonlinear problem. Usually, lower order elements, especially 3-node linear triangular element (T3) for 2-D problem and 4-node linear tetrahedron element (T4) for 3-D problem, are attractive in practical engineering problems because of their intrinsic simplicity, easy preprocessing, and lower requirement on solution regularity. However, the use of T3 or T4 element is highly limited in the large deformation plasticity analysis, because of disadvantages such as convergence problem, element distortion and volumetric locking. Therefore, developing techniques to optimize the linear triangular element is a significant job, and lower order elements with superior accuracy and convergence properties are powerful tools for the simulations of contact-impact, crack propagations, material fracturing progressing, large scale multi-physics etc. In order to overcome the limitations of FEM, various technologies have been proposed. The strain smoothing technique was used for stabilizing the nodal integrated meshfree method (Chen JS, Wu CT et al. 2001) and then applied in the natural element method (Yoo JW, Moran B et al. 2004). Liu et al. has generalized a gradient (strain) smoothing technique (Liu GR 2008) and applied it in general meshfree settings to accommodate discontinuous shape functions (Liu GR 2009). Applying the same technique to the finite element method, an edge-based smoothed finite element method (ES-FEM)(Liu GR, Nguyen-Thoi T et al. 2009) has been formulated based on the gradient (strain) smoothing technique for static, free and forced vibration analyses in 2D plane strain and plane stress problems. The system stiffness matrix of this method is computed using the gradient smoothing technique over the smoothing domains associated with the edges of the element, which endows its superior convergence properties, computational accuracy and efficiency, spatial and temporal stability.

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In this paper, the edge-based smoothed finite element (ES-FEM) is extended to be applied for large strain plasticity analysis, and a selective ES/NS-FEM approach is used to address the volumetric locking problem. Compared to standard FEM, this smoothed technique could be able to use fewer elements to get more precise and stable results, and better convergence property; these properties can decrease the computational cost significantly.

2. Edge-based smoothed finite element method for finite strain plasticity

2.1 Basic ES-FEM theory and formulation

In ES-FEM, the domain discretization is still based on T3 element in standard FEM, but the integration required in the virtual principle is performed based on the smoothing domains associated with the edges instead of on the triangular element in standard FEM. In this method, the closed

problem domain Ω is divided into sN smoothing domains with 1s sN

k k=Ω = Ω and 0 s s i jΩ ∩Ω =

when i j≠ , where sN is the number of smoothing domains equal to the total number of element edges located in the entire problem domain. For triangular elements, the smoothing domain skΩ associated with the element edge k can be created by connecting two endpoints of the edge to centroids of adjacent elements as shown in Fig.1.

Fig.1 Triangular elements and the smoothing domains (shaded areas) associated with edges in ES-FEM(Liu GR,

Nguyen-Thoi T et al. 2009)

In edge-based smoothing domains, the smoothed gradient of displacement field iu can be obtained by

( ) ( ) s k

i i ku u d Ω

∇ = ∇ Φ Ω∫ x x (1) where ( )iu∇ x is the gradient of the displacement field iu , and ∇ is defined as a smoothed gradient operator. skΩ is the smoothing domain associated with the edge k. ( )kΦ x is a given smoothing function that satisfies at least unity property

( ) 1 s k

k d Ω

Φ Ω =∫ x (2) In ES-FEM, a simple local constant smoothing function can be used in the calculation (Liu GR, Nguyen-Thoi T et al. 2009)

1/ , ( )

0,

s s k k

k s k

A x x

∈Ω Φ =

∉Ω x (3)

3

where skA is the area of the smoothing domain s kΩ , and is calculated by

1

1 3

sd

s k

n s e k i

i A d A

=Ω

= Ω = ∑∫ (4)

where sdn is the number of elements around the edge k and s kA is the area of the

thj element around the edge k. Fig.1 shows that 1sdn = when edge k is a boundary edge, and 2sdn = when edge k is an inside edge. Using divergence theorem on Eq. (1), it can be obtained that

1 ( ) s k

i is k

u u d A Γ

∇ = Γ∫ x n (5) where n is the outward normal vector of the smoothing domain boundary skΓ . In the ES-FEM-T3, the displacement field is interpolated by the linear FEM shape function, and can be written in the following form

L

i I Ii I G

u N u ∈

= ∑ (6) where IN is the shape function of node I at reference configuration, Iiu is the displacement component of node I. LG is the set of the so-called supporting nodes of the smoothing domain

s kΩ .

Therefore, the smoothed gradient of displacement filed can be formulated by substituting Eq. (6) into (5)

1 1 s s k k

L L L

i I Ii I Ii Ii Iis s I G I G I Gk k

u N u d N d u b u A AΓ Γ∈ ∈ ∈

∇ = Γ = Γ =

∑ ∑ ∑∫ ∫n n (7)

where Iib is the smoothed derivatives of shape function as 1

s k

Ii I is k

b N n d A Γ

= Γ∫ (8) Naturally, the smoothed strain ε in the domain ( )kΩ associated with edge k can now be obtained using Iib

0 0 Ix

Iy I I I

Iy Ix

b b

b b

= =

ε u B u (9)

2.2 Formulating the large deformation plasticity model

In continuum mechanics, the deformation gradient has the form of = ∇ +F u I (10)

where u is the displacement field tensor, ∇ is the gradient operator, and I is the identity matrix. The smoothed deformation gradient associated with edge k based on the smoothed domain can be defined as:

( ) ( ) ( )( ) ( )

1 1( ) ( ) ( ) k k kk k kX X d X d dA AΩ Ω Ω

= Φ Ω = Ω = ∇ Ω+∫ ∫ ∫F F F u I (11) Applying divergence theorem to Eq. (11) in the current configuration yields

1 s k

s k

d A Γ

= ⊗ Γ + = +∫F u n I e I (12)

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where e represents the smoothed gradient field of displacement, i.e. iu∇ given in Eq. (5). n is the outward normal vector. Also, the smoothed rate of deformation tensor could be calculated as

1−=L FF (13) The smoothed deformation rate and smoothed continuum spin can be additively decomposed as

e p= +D D D (14) e p e= + =W W W W (15)

The smoothed Jaumann stress rate could be obtained by

( ) 2e etrκ µ ∇

= +T D I D (16)

where κ and µ are conventional Lame elastic constants. The 2J associated flow theory has the form of

p Fλ ∂= ∂

D T

(17)

where λ is the plasticity multiplier. The consistent tangent matrix can be solved by (Zienkiewicz OC and Taylor RL 2000)

0 2 3 22 1

3 ep T T

tr tr iH

µ λ µ µ λκ µ µ

∆ ∆ = + − − − +

C mm I nn T T

(18)

The meaning of parameters mentioned in Eq. (18) can be found in reference (Zienkiewicz OC and Taylor RL

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