DSpace at VNU: A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems

DSpace at VNU: A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture...

(NS-FEM) for upper bound solutions of fracture problems

G R Liu1,2, L Chen1,∗,†, T Nguyen-Thoi1,3, K Y Zeng4and G Y Zhang2

1Center for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering,

National University of Singapore , 9 Engineering Drive 1, Singapore 117576, Singapore

2Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, Singapore 117576, Singapore

3Faculty of Mathematics and Computer Science , University of Science, Vietnam National University–HCM,

Hanoi , Vietnam

4Department of Mechanical Engineering , National University of Singapore, Singapore 117576, Singapore

SUMMARY

It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained

by the displacement compatible finite element method (FEM) together with the singular crack tip elements

It is, however, much more difficult to obtain the upper bound solutions for these problems This paper aims

to formulate a novel singular node-based smoothed finite element method (NS-FEM) to obtain the upperbound solutions for fracture problems In the present singular NS-FEM, the calculation of the systemstiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs)associated with nodes, which leads to the line integrations using only the shape function values along theboundaries of the SDs A five-node singular crack tip element is used within the framework of NS-FEM

to construct singular shape functions via direct point interpolation with proper order of fractional basis.The mix-mode stress intensity factors are evaluated using the domain forms of the interaction integrals.The upper bound solutions of the present singular NS-FEM are demonstrated via benchmark examplesfor a wide range of material combinations and boundary conditions Copyrightq 2010 John Wiley &Sons, Ltd

Received 6 August 2009; Revised 5 January 2010; Accepted 15 January 2010

KEY WORDS: numerical methods; meshfree method; upper bound; crack; stress intensity factor;

J -integral; energy release rate; NS-FEM; singularity

1 INTRODUCTION

In fracture analyses, it is important to evaluate fracture parameters, such as the stress intensity

factors (SIFs) or the energy release rate ( J -integral), which is the measure of the intensity of the

∗Correspondence to: L Chen, Center for Advanced Computations in Engineering Science (ACES), Department of

Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore.

†E-mail: g0700888@nus.edu.sg

finite element method (FEM) models, which are widely used in solving complicated engineeringproblems The models that give an upper bound can be one of the following models: (1) the stressequilibrium FEM model[4]; (2) the recovery models using a statically admissible stress field fromdisplacement FEM solution[5, 6]; and (3) the hybrid equilibrium FEM models [7] These threemodels, however, are known to have the following two common disadvantages: (1) the formulationand numerical implementation are computationally complicated and expensive and (2) there existspurious modes in the hybrid models or the spurious modes often occur due to the simple fact thattractions cannot be equilibrated by the stress approximation field Owing to these drawbacks, thesethree models are not yet widely used in practical applications, and are still very much confined inthe area of academic research.

In the linear fracture mechanics, the stresses and strains near the crack tip are singular:  ij∼

1/√r , ε ij ∼1/√r , (where r is the radial distance from the crack tip)[1] To capture the singularity

in the vicinity of the crack tip, the numerical simulation of cracks can be carried out with severaldifferent numerical approaches, such as FEM[8–10] and meshless methods [11–15] When thedisplacement compatible FEM is used, the eight-node quarter-point element or the six-node quarter-point element (collapsed quadrilateral) is often adopted to model the inverse√

r stress singularity

[8–10] However, to ensure that the singular elements are compatible with other standard elements,quadratic elements are required for the whole domain Otherwise, transition elements[16, 17] areneeded to bridge between the crack tip elements and the standard elements

Recently, Belytschko and Moes developed so-called extended finite element method (XFEM)

to model arbitrary discontinuities in meshes [18, 19] This method allows the crack to be trarily aligned within the mesh, and thus crack propagation simulations can be carried out withoutremeshing[20] Moreover, this extension exploits the partition of unity property of finite elements,which allows local enrichment functions to be easily incorporated into a finite element approx-imation while still preserving the classical displacement variational setting [21] However, theenrichment is only partial in the elements at the edge of the enriched subdomain, and consequently,the partitions-of-unity in the original XFEM is lost in the ‘transition’ zones, and hence ‘blending’elements need to be used in these transition zones[22]

arbi-More recently, Liu et al has generalized the gradient (strain) smoothing technique[23–25] andapplied it in the FEM context to formulate a cell-based smoothed finite element method (SFEM orCS-FEM)[26–28] In the CS-FEM, cell-based strain smoothing technique is incorporated to thestandard FEM formulation to reduce the over stiffness of the compatible FEM model The CS-FEM

has been developed for general n-sided polygonal elements (nCS-FEM)[29] dynamic analyses[30], incompressible materials using selective integration [31, 32], plate and shell analyses [33–36],and further extended for the XFEM to solve fracture mechanics problems in 2D continuum andplates[37]

To further reduce the stiffness, a node-based smoothed finite element (NS-FEM)[38, 39] has

been formulated using the smoothing domains (SDs) associated with nodes Liu et al.[38–42] hasshown that the NS-FEM has the very important property of producing upper-bound solutions, whichoffers a very practical means to bound the solutions from both above and below for complicatedengineering problems, as long as a displacement FEM model can be built Such bounds are obtained

Exploiting this special property of line integration, we now can further develop the NS-FEM forfracture analyses by computing the system stiffness matrix directly from the special basis shapefunctions which create the√

r displacement field, and thus obtain a proper singular stress field in

the vicinity of the crack tip

In this paper, a singular NS-FEM is formulated to obtain the upper bound solutions for themix-mode cracks Four schemes of SDs around the crack tip have been proposed based on thetriangular elements to model the singularity In addition, a five-node singular element is used withinthe framework of NS-FEM to construct singular shape functions for the SDs connected to the cracktip In our singular NS-FEM, the displacement field is at least (complete) linearly consistent, andthe enrichment near the crack tip is on top of the complete linear field Therefore, the partition-of-unity property as well as the linear consistency property are both ensured throughout the entireproblem domain, ensuring the stability and convergence of the solution Using the singular NS-FEMtogether with the singular FEM, we can now have a systematical way to numerically obtain bothupper and lower bounds of fracture parameters to crack problems Intensive benchmark numericalexamples for a wide range of material combinations and boundary conditions will be presented todemonstrate the interesting properties of the proposed method

2 BASIC EQUATIONSConsider a 2D static elasticity problem governed by the equilibrium equation in the domain∈R2

u =0 on u (6)The natural boundary condition is given by:

j =0, ∀i = j Then, the approximation of displacement field

for a 2D static elasticity problem is given by:

uh (x)=

i ∈ne n

where nen is the set of nodes of the element containing x, di =[d xi d yi]T is the vector of nodal

displacements, respectively, in x-axis and y-axis, and N i is a matrix of shape functions

Ni (x)= N i (x) 0

0 N i (x)

(10)

in which N i (x) is the shape function for node i Using Equations (4) and (9), the compatible strain

of FEM approximation is given by:

0())2 denotes the Sobolev space of functions with square integrable derivatives inand with vanishing values onu

Here, K is the system stiffness matrix of FEM that is assembled using:

A fundamental issue in modeling fracture mechanics problems is to simulate the singularity of stress

field near the crack tip In order to capture the singularity expressed as r −1/2 accurately without

using so many elements around the crack tip, the singular element is generally incorporated intothe standard FEM The theory and application of the different kinds of singular elements are welldocumented in[8–10] The most popular singular element is the eight-node quarter-point element orthe six-node quarter-point element (collapsed quadrilateral) These quarter-point quadratic elementsshift the corresponding mid-nodes to the quarter-point position as shown in Figure 1 However,

to ensure that the singular elements are compatible with other standard elements, the quadraticelements are required even for domains far away from the crack tip, which significantly increasesthe computational cost Otherwise, transition elements[16, 17] are needed to bridge between thecrack tip elements and the standard elements

4 THE IDEA OF SINGULAR NS-FEMDetailed formulations of the NS-FEM have been proposed in the previous work [38] Here, wemainly focus on the construction of a singular field near the crack tip using a basic mesh forthree-node linear triangular elements

/ 4

l l

Figure 1 The schematic of the eight-node and six-node quarter-point singular elements

s k

Ω crack tip

Centroid of triangle

Figure 2 Construction of node-based strain smoothing domains

4.1 Brief on the NS-FEM

In the NS-FEM, the domain is discretized using elements, as in the FEM However, we do not

use the compatible strains but the strains ‘smoothed’ over a set of non-overlap no-gap SDs Ns

associated with nodes, such that =Nn

k for node k is created by connecting sequentially the

mid-edge-points and the centroids of the surrounding triangles of the node as shown in Figure 2

where Ask= s d is the area of the SD s

k and s

k is the boundary of the SDs

k.Substituting Equation (9) into Equation (17), the smoothed strain can be written in the followingmatrix form of nodal displacements

¯ek=

i ∈ns

where nsk is the set of nodes associated the SDs

k and ¯Bi (x k ) is termed as the smoothed strain

gradient matrix that is calculated by:

where Nseg is the number of segments of the boundarys

k , Ngau is the number of Gauss pointsused in each segment,w m,n is the corresponding weight of Gauss points, n h is the outward unit

normal corresponding to each segment on the SD boundary, and xm,n is the n-th Gaussian point

on the m-th segment of the boundarys

k.Because the NS-FEM is variationally consistent as proven (when the solutions are sought inthe(H1

0())2space) in [24], the assumed displacement uh and the smoothed strains ¯e satisfy the

smoothed Galerkin weak form:

crack tip

Figure 3 Scheme 1 of smoothing domains around the crack tip

4.2 Smoothing domains around the crack tip

It is well known that the FEM model using displacement-based compatible shape functions will

provide a stiffening effect to the exact model, and give the lower bound of the solution in strain

energy On the other hand, the strain smoothing operation used in an S-FEM will provide a

softening effect to the compatible FEM model and give the larger solution in strain energy than

that of the compatible FEM model Therefore, the battle between the softening and stiffeningeffects will determine the bound properties and accuracy of the proposed numerical method Liu

et al [38–42] have found that the softening effect depends on the number of elements associated with SD The more the elements that participate in a SD, the more the softening effect becomes.

The NS-FEM produces the upper-bound solutions, and is found in general to be overly soft[40],due to many elements participating in the node-based strain smoothing operation More importantly,

it is noticed that only one SDs

tip around the crack tip cannot adequately capture the singularity

of the stresses as shown in Figure 2 Therefore, the proper schemes of SDs around the crack tip

should be used to reduce the softening effect and capture the singularity In the present singular

NS-FEM, we propose four schemes for triangular-element-based SDs around the crack tip, asshown in Figure 6

• Scheme 1

In Scheme 1, only one layer of SD (from SD(1)to SD(7)) around the crack tip is used as shown

in Figure 3 Note that there are two kinds of SDs: inner and boundary ones, and each SD is createdbased on the edge connected directly to the crack tip

For the inner one, each SD is created by connecting sequentially the following points: (1) thecrack tip; (2) the centroid of one adjacent singular element of the edge; (3) the mid-edge-point;(4) the centroid of another adjacent singular element; and return to (1) the crack tip For example,the SD(2)filled with the blue shadow in Figure 3 is created by connecting sequentially #A, #C,

#D, #E, and #A

Figure 4 Scheme 2 of smoothing domains around the crack tip.

For the boundary one, each SD is created by connecting sequentially the following points: (1)the crack tip; (2) the mid-edge-point; (3) the centroid of adjacent singular element; and return

to (1) the crack tip For example, the SD(1)filled with the red shadow in Figure 3 is created by

connecting sequentially #A, #B, #C, and #A

• Scheme 2

Scheme 2 contains two layers of SDs at the crack tip as illustrated in Figure 4 Similar to theabove, each layer of SD is also based on the edge connected to the crack tip, and includes twokinds: inner and boundary SDs

(i) For the layer of SDs far away the crack tip (from SD(1)to SD(7):

Each inner SD is constructed by connecting (1) the centroid of one adjacent singular element

of the edge; (2) the mid-edge-point; (3) the centroid of another adjacent singular element; (4) the

1/8-centroid-point of the second adjacent singular element; (5) the edge-point; (6) the

1/8-centroid-point of the first adjacent singular element; and back to (1) the centroid of the first adjacentsingular element For example, the SD(2) filled with the blue shadow is created by connecting

sequentially #C, #D, #E, #J, #I, #H, and #C

Each boundary SD is created by connecting (1) the mid-edge-point; (2) the centroid of theadjacent singular element; (3) the 1/8-centroid-point of the adjacent singular element; (4) the

crack tip

Figure 5 Scheme 3 of smoothing domains around the crack tip

1/8-edge-point; and back to (1) the mid-edge-point For example, the SD (1) filled with the redshadow in Figure 4 is created by connecting sequentially #B, #C, #H, #G, and #B

(ii) For the layer of SDs connected directly to the crack tip (from SD(8) to SD(4) ):

Each SD is the left part of the region of the SD in Scheme 1 subtracting the region of thecorresponding SD in the layer far away from the crack tip defined in Scheme 2 For instance, theinner SD(8) filled with the black shadow is constructed by connecting sequentially #A, #H, #I,

#J, and #A, and the boundary SD(7) filled with the green shadow is constructed by connecting

sequentially #A, #G, #H, and #A in Figure 4

• Scheme 3

As given in Figure 5, one layer of SD (from SD(1) to SD(6) ) based on the singular element is

used around the crack tip in Scheme 3 The SD is just one-third of the region of the singularelement connected to the crack tip For example, the SD(2)filled with the blue shadow in Figure 5

is constructed by the node set containing #A, #D, #E, and #F

• Scheme 4

In Scheme 4, two layers of SDs are constructed near the crack tip based on the singular elements

as illustrated in Figure 6

(i) For the layer of SDs far away from the crack tip (from SD(1)to SD(6) ), each one is created by

connecting (1) the mid-edge-point of one edge; (2) the centroid; (3) the mid-edge-point of anotheredge; (4) the 1/8-edge-point of the second edge; (5) the 1/8-centroid-point; (6) the 1/8-edge-point

of the first edge; and back to (1) the mid-edge-point of first edge For instance, the SD(2)with the

blue shadow is created by connecting sequentially #D, #E, #F, #K, #J, #I, and #D as shown inFigure 6

(ii) For the layer of SDs connected directly to the crack tip (from SD(7)to SD(2 )), each one is

the left part of the one-third region of the singular element connected to the crack tip Therefore,the SD(8)with the black shadow in Figure 6 is constructed by the node set containing #A, #I, #J,

and #K

A B

Figure 6 Scheme 4 of smoothing domains around the crack tip

Note that for Schemes 1 and 2, each SD is created based on the edge connected to the cracktip, and the number of elements associated with one SD is 2; whereas for the Schemes 3 and 4,each SD is created based on the elements, and the number of elements associated with one SD is

just 1 As found by Liu et al.[38–42], the more the elements that are associated, the more the

softening effect becomes Therefore, the softening effect of Schemes 3 and 4 is smaller than that

of Schemes 1 and 2, which means that Schemes 3 and 4 reduce the softening effect They thus

provide a more excellent accuracy compared to Schemes 1 and 2 In addition, due to the stresssingularity near the crack tip, it is conceivable that the two-layer SDs can simulate the change ofstress more accurately As a result, Scheme 4 produces the best accuracy among the four proposedschemes in this numerical method

4.3 Singular shape function

As regards the shape function, since the stress singularity at the crack tip of a crack is of inversesquare root type, the polynomial basis shape functions cannot represent the stress and strain field

at the crack tip In this work, a five-node singular element containing the crack tip is used withinthe framework of NS-FEM to construct special basis (singular) shape functions With this singular

3 2

3

Figure 7 Node arrangement near the crack tip

shape function for interpolation, we can create the√

r displacement field and thus obtain a 1 /√r

singular stress field in the vicinity of the crack tip As shown in Figure 7, one node is added toeach edge of the triangular elements connected to the crack tip The location of the added node is

at the one-quarter length of the edge from the crack tip Based on this setting, a field function u (x)

at any point of interest on an edge of a singular element can be constructed by the three nodesalong the edge:

u (x)= 3

i=1p i (x)a i =a0+a1r +a2

√

where r is the radial coordinate originated at the crack tip and a i is the interpolation coefficient for

p i (x) corresponding to the given point x The coefficients a i in Equation (25) can be determined

by enforcing Equation (25) to exactly pass through nodal values of the three nodes along the edge:

4

5

x G

y G

Figure 8 The schematic of the five-node element for Scheme 4 of smoothing domains

where l is the length of the element edge and  i (i=1, 2, 3) are the shape functions for thesethree nodes on the edge

Figure 8 shows a five-node element connected to the crack tip using Scheme 4 of the smoothingdomains It can be seen that the five-node element is constructed by two kinds of SDs: the crack tip

SD and the normal SD To perform the point interpolation within the crack tip SD, it is assumed

that the field function u varies in the same way as given in Equation (25) in the radial direction.

In the tangential direction, however, it is assumed to vary linearly For points 6 and 7 which arethe midpoints of lines 2–3 and 4–5, the field functions can be evaluated simply as follows:

u6=1

u7=1

present singular NS-FEM which uses both the crack tip SD and the normal SD, a proper number ofGauss points hence need to be used on the boundary segments of the crack tip SD, which depend

on the order of the assumed displacement field (or shape function) along these boundary segments.Specifically as shown in Figure 8, for the crack tip SD filled with the blue shadow surrounded bythe set of boundary segments of AB-BC-CD-DE-EF-FA, there are two kinds of segments: (1) one

is nearly along the tangential direction (AB, BC, DE, and EF) and (2) the other is along the radialdirection (CD and FA) The displacement in the tangential direction is assumed linearly, henceone Gauss point is enough for the segments (AB, BC, DE, and EF) along this direction Whereas,the displacement in the vicinity of the crack tip possesses the√

r behavior in the radial direction,

and thus five Gauss points are used for the segments (CD and FA) to ensure accuracy

We now construct specifically the shape function for Gauss points on two kinds of boundarysegments

(i) For the Gauss point G x on a segment nearly along the tangential direction (for example AB,this point is also on the line 1−−o as shown in Figure 8), the field function is interpolated as:

It is also noted that since the derivatives of the singular shape term (1/√r ) are not required

to calculate the system stiffness matrix, the formulation of the present five-node element is muchsimpler Moreover, it does not need to use the quadratic elements or transition elements[16, 17].Therefore, it can be very easily incorporated into the standard NS-FEM

5 DOMAIN INTERACTION INTEGRAL METHODS

In the linear elasticity, the general form of J -contour integral, which is identical to the energy release rate of potential energy G, for a two-dimensional crack can be written as[45]:

2 ij ε ij  xj − ij *u i

*x



n jd, i = x or y, j = x or y (38)

where a is the crack length and

is the potential energy of the model

The SIFs are computed using the domain forms of the interaction integrals [18, 19, 46] Forthe general mixed-mode cracks in an isotropic material, the relationship between the value of the

J -integral and the SIFs can be given by:

1 =1, K1(2) =0 and evaluating I = I1, we can compute K1and proceed in an

analo-gous manner to evaluate K2

The contour integral in Equation (41) is not the best form suited for numerical calculations

We, therefore, recast the integral into an equivalent domain form by multiplying the integrand by

a sufficiently smooth weighting function q which takes a value of unity on an open set containing the crack tip and vanishes on an outer prescribed contour C0as shown in Figure 9(a) Assumingthat the crack faces are traction free, the interaction integral may be written as:

I=

C

 (1) ik ε (2) ik  xj − (1) ij *u

(2) i

(2) ij

*u (1) i

*x

where the contour C =+C++C−+C0and m is the unit outward normal to the contour C Now

using the divergence theorem and passing to the limit as the contour is shrunk to the crack tip,gives the following equation for the interaction integral in domain form:

where we have used the relations m j =−n j on and m j =n j on C+, C−, and C0

For the numerical evaluation of the above integral, as shown in Figure 9(b), the domaind is

then set to be the collection of all the elements that have a node within a radius of r d =r k h e and

this element set is denoted as Nd he is the characteristic length of an element touched by thecrack tip and the quantity is calculated as the square root of the element area

The weighting function q that appears in the domain form of the interaction integral is set as follows: if a node n i that is contained in the element e ∈ Nd lies outside d, then q i=0; if node

n i lies in d, then q i =1 As the gradient of q appears in Equation (44), the elements set Nd

in

with all the nodes insidedas shown in Figure 9(b) contribute nothing to the interaction integral,

and non-zero contribution to the integral is obtained only for elements set Neffd with an edge thatintersects the boundary*d Therefore, Equation (44) can be given by:

I=−N

d eff

m=1

 d eff,m

 (1) ik ε ik (2)  xj − (1) ij *u

(2) i

The weighting function q that appears in the domain form... on C+, C−, and C0

For the numerical evaluation of the above integral, as shown in Figure 9(b), the domaind is

then... following equation for the interaction integral in domain form:

where we have used the relations m j =−n j on and m j =n j on