Crack Interactions Study Under Thermal Load Using EFGM and XFEM

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ISSN (Print): 2319-3182, Volume-1, Issue-1, 2012

42

Crack Interactions Study Under Thermal Load Using EFGM and XFEM

Himanshu Pathak & Akhilendra Singh

Department of Mechanical Engineering, Indian Institute of Technology Patna, Patna, India E-mail : [email protected]

Abstract - Multiple voids exists in all engineering component which interact with each other and affects the stress intensity factor. During their service life engineering component exposed to thermal loading. The thermal loading creates the singular stress field at the crack tip, which causes the sudden failure of component and loss of human life. Thus, the study of crack interactions under thermal loading is of great importance. In the present work, crack interaction study has been performed under mixed mode thermal loading using element free Galerkin method (EFGM) and extended finite element method (XFEM). In EFGM and XFEM, the domain of interest is discretized by scattered nodes and standard FEM element respectively without physically having any crack in the domain, and the presence of a crack in the domain is ensured by extrinsic enrichment only. In extrinsic enrichment technique, additional functions were added with the standard displacement based approximation within partition of unity (PU) framework. These additional functions were taken from the theoretical background of the problem. The model problems including edge crack and edge crack interactions under thermal loading solved by both EFGM and XFEM.

Keywords - EFGM; XFEM; SIF; thermal crack interactions

I. INTRODUCTION

Modeling of crack problem is an essential element in the life prediction of engineering component. Due to importance of this problem researchers are always trying to model arbitrary crack growth in the wide variety of situations. In addition, most of the engineering components such as thermal barrier of aerospace structure, internal combustion engine, thermal power plant combustion chamber etc. are subjected to thermal load during their service life. The presence of void and pore within the engineering component makes the problem complicated due to their combined effect of thermal and mechanical effect. In the last decades,

different numerical approaches have been used for crack modeling Finite Element Method and Boundary Element Method. The development of Quarter Point Element makes a significant advancement in the use of Finite Element Method for LEFM problem. However Finite element method (FEM) is not well-suited for certain classes of problems such as crack propagation &

moving discontinuities, moving phase boundaries, phase transformation, large deformations, modeling of multi- scale phenomena. In Boundary Element Method, remeshing issue makes the method inconvenient for crack grow simulation problem. To overcome these problems, a number of computational methods have been developed in last decade. Hence a new meshfree technique Element Free Galerkin Method EFGM [1]

comes into picture to handle discontinuous domain problem, which constructs approximations from a set of nodal data and associated weight functions with compact support on the domain. Another efficient computational method extended finite element method XFEM developed by the Belytschko et al [2] where the method incorporates both the discontinuous Heaviside function and near tip asymptotic field through a partition of unity method. In both methods, standard displacement based approximation is enriched by additional functions using partition of unity, which results in additional degree of freedom.

So far, very little effort has been made to study the effect of crack interactions using meshfree methods [3- 4] and extended finite element method. Moreover, static and moving crack problems have not been analyzed under thermal load. Therefore, in the present work, XFEM and EFGM have been extended to obtain the numerical solution of crack interactions problem under thermal loading with different crack domain configuration. The extrinsic enrichment technique within partition of unity (PU) framework has been

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43 implemented in EFGM and XFEM to study the effect of crack interactions.

II. NUMERICAL METHODS:EFGM AND XFEM The EFGM requires moving least square MLS approximants functions [1] to approximate an unknown function. Using MLS approximation, the unknown field variable u is approximated by MLS approximation

x

u^h over the domain :

x a x p x x

x _j ^T

m

j j

h p a

u

1

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The mesh free shape function Φ_I(x) is defined as

I m T

j

jI j

I p

Φ (x) ( )(A ( )B( )) p A ¹B

1

1 x x

x (2)

The quadratic spline weight function is used in the present analysis; it can be written as function of normalized radius r as

1 0

1 0 3 8 6 ) 1 ( ) (

4 3 2

r r r r r r

w

w x x_I (3)

Extended Finite Element Method (XFEM) is the modified form of Partition of Unity Finite Element Method (PUFEM) and Generalized Finite Element Method (GFEM). The formulation of XFEM is similar to finite element method. It uses Langrage interpolation function to interpolate the field variables as well as geometry. The field variable u is approximated by Lagrange interpolation u(x)[2], which is given as:

j n

j ju u

1

(x) a (x) p

(x) ^T (4)

III. NUMERICAL FORMULATION

Consider a body bounded by domain contains the strong discontinuity (see Fig. 1). The equilibrium and boundary conditions for linear elastostatic problem may be described as

0

. b (5a)

where, is Cauchy stress tensor, bis body force vector and . is divergence operator. The boundary conditions for the problem domain can be defined as:

t n

. on Γ_t (5b)

u

u on Γ_u

The weak form of governing equation can be written as:

0 ,

: _s

t

d d

d u.b u.t W u

ε _u (6)

The last term W_u u, _s comes in weak form due to langrage multiplier used to enforce essential boundary condition in EFGM; in case of XFEM formulation this term will vanish.

In order to model the two-dimensional body with a crack and material discontinuity, extrinsic enriched displacement approximation [2] can be written as:





 











t r

s j n

j n

j j n

j j j

n

j j

h u H a b c

u (x) (x) (x) (x) (x)

4

1 1

(7)

where, u_jis the field variable for the continuous part of EFGM solution, a_jis the additional degree of freedom (DOF) associated with Heaviside function

x) (

H for split domain of influence, b_jis the additional DOF associated with Level set enrichment function

x)

( used to simulate material discontinuity and c_j additional DOF stands for four asymptotic enrichment functions for crack tip. In the approximation equation, nis the set of all nodes within the domain of influence, nsare the set of all nodes whose support is cut by the crack face within the domain of influence, n_rare those sets of nodes whose support is cut by the weak discontinuity and n_tare sets of nodes associated with crack tip domain of influence. By introducing the enriched displacement approximation in weak form of governing equation and using of arbitrariness of nodal variation, a following set of discrete equation is obtained as:

q f λ u 0 G

G

K (EFGM) (8)

K u f (XFEM)

For the case of plane stress in an isotropic plate with coefficient of thermal expansion subjected to a temperature difference T , the thermal strain matrix [6]

given by

0 T T

T (9)

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44 A. Enrichment Functions

The Heaviside function [5] is utilized for modeling of strong discontinuity present within the domain by both EFGM and XFEM. In EFGM, those nodes whose domain of influence is cut by crack face are modeled by Heaviside function where as in XFEM the elements which are completely cut (split element) by crack face are modeled by Heaviside function. The definition of the Heaviside function for the purpose of computation may be defined as:

otherwise 1

0 . (

1 if n

H x x*)

x (10)

Crack tip enrichment functions [5] are used to model the crack tip singularity and this allows effective representation of the crack tip field in computation. The enrichment function consists of Westergaard’s solution to capture the singularity on the crack tip.

sin 2 sin , sin 2 cos , 2 cos , 2 sin

(x) r r r r (11)

B. Interaction Integral for Numerical SIF

Now a day, the domain based interaction energy integral method is widely used for evaluating the individual stress intensity factors (K_I,K_II) in mixed mode loading conditions. For crack problem under thermal loading, J-integral remains globally path independent if material homogeneity exists in the direction parallel to crack surface. The domain form of interaction integral [6] can be written as

dA q u u

J _j

A

i aux ij aux i ij j aux ik

ik 1 ,1 ,1) ,

( (12)

Where, q is scalar weight function and its value is one at crack tip and zero at the contour. ^aux , ^aux are the auxiliary Cauchy stress tensor and auxiliary engineering strain tensor. The Interaction integral related with mixed mode stress intensity factor [6] by the following relation:

aux II II aux I

* I

2 K K K K

E

J (13)

In the above equation K_Iâux and K_IIâux are auxiliary field stress intensity factor, E* is equivalent Young’s modulus. By introducing, K_Iâux =1, K_IIâux =0 and vice- versa, mixed mode SIF (KI and KII) can be calculated.

IV. NUMERICAL RESULTS AND DISCUSSION A rectangular plate with an edge crack of

200mm

100mm under 1MPa traction force Fig.1. has been solved by EFGM and XFEM for different sets of nodes. The obtained SIF normalized by exact solution of the problem and plotted against the different nodal points. Fig.2. shows the convergence of both methods.

Fig.1 : Edge crack domain

Fig. 2 : SIFs variation on different sets of node The uniformly distributed 1800 nodes have been taken for all simulation work. The material property used in the simulation is given in Table 1.

TABLE I. MATERIAL PROPERTY Sl No.

Youngs’

Modulus E (MPa)

Poisson Ratio υ

Coefficient of thermal expansion

α (/^oC)

01. 200×10³ 0.3 11.7×10^-6

In case of EFGM, Lagrange multiplier is used to enforce essential boundary condition in addition scaling parameter taken as 1.2. Few edge crack problems have been solved to study the effect of crack interaction under sudden temperature change T 100^oC. The major crack length is taken as a 40mm in all cases. The

0 500 1000 1500 2000 2500 3000 3500 4000

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

Nodes

Normalized SIF ^{Exact KI}

Exact KII XFEM KI XFEM KII EFGM KI EFGM KII

θ

T

L/ 2 L

W a

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45 single edge crack under sudden temperature change domain shown in Fig. 3.

Fig. 3 : Edge crack under thermal load

Fig. 4: SIFs variation on different crack orientation From the SIF plot Fig 4., mode-I SIF decreases with crack orientation and mode-II SIF increases. The obtained SIF are in best agreement of Intrinsic Enriched Element Free Galerkin Method [4]. The multiple edge cracks on opposite face domain shown in Fig. 5. The size of minor crack varied from 10mm to 40mm with increment of 5mm .

Fig. 5: Edge crack interaction under thermal load

Fig. 6 : SIFs variation at major crack tip In problem domain Fig.5, mode-I SIF at major crack tip decreases with minor crack length increment whereas SIF at minor crack tip increases as shown in Fig.6. and Fig. 7.

Fig. 7: SIFs variation at minor crack tip The numerical SIF has been calculated at minor edge crack tip. From the SIF plot Fig. 7, the mode-I SIF increases with minor crack increment. The numerically obtained SIF by EFGM and XFEM are in good agreement with each other. The reason of change in SIF is the change in stress field at major and minor crack tip due to minor crack tip increment. The stress and strain field shown in Fig. 8. The singularity in both stress and strain field capture at crack tip.

Fig. 8(a): Stress contour σxx

-10 0 10 20 30 40 50 60 70 80

Crack Orientation (degree)

SIF MPa-sqrt(m)

Ref (KI) Ref (KII) XFEM (KI) XFEM (KII) EFGM (KI) EFGM (KII)

5 10 15 20 25 30 35 40 45

68 69 70 71 72 73 74 75 76 77 78

Crack increment (mm)

SIF MPa-sqrt(m)

XFEM (KI) EFGM (KI)

5 10 15 20 25 30 35 40 45

25 30 35 40 45 50 55 60 65 70 75 80

Crack increment (mm)

SIF MPa-sqrt(m)

XFEM (KI) EFGM (KI)

-350 -300 -250 -200 -150 -100 -50 0

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46 Fig. 8(b): Strain contour εxx

Fig. 8(c): Stress contour σxy

Fig. 8(d): Strain contour ε_xy

Fig. 8(e): Stress contour σ_yy

Fig. 8(f) : Strain contour εyy

V. CONCLUSION

In this work, crack interaction problems subjected to thermal load are simulated and analyzed by EFGM and XFEM under mixed mode loading condition. The extrinsic partition of unity enrichment technique has been successfully used to model thermally loaded crack domain problem. During the problem formulation, Heaviside enrichment and asymptotic crack tip enrichment function extrinsically incorporated within partition of unity framework. From the SIF analysis, it can be concluded that result obtained by XFEM and EFGM are in good agreement to each other. Moreover, with the increase in number of nodes, the both method start converging for all nodal points in the domain. Due to accuracy and simplicity in implementation both methods are suitable to model complicated crack interaction geometry.

VI. REFERENCES

[1] T. Belytschko, Y. Y. Lu and L. Gu, Element-free Galerkin methods, Int J Numer Meth Engng, vol. 37, 1994, pp. 229-256.

[2] N. Moes, J. Dolbow and T. Belytschko, A finite element method for crack growth without remeshing, Int J Numer Meth Engng, vol. 46, 1999, pp.135–150.

[3] B. Muravin, The application of element free Galerkin method in the investigation of crack interactions, Ph.D.

Thesis, Tel-Aviv University, Israel 2003.

[4] M. Pant, I. V. Singh and B. K. Mishra, A numerical study of crack interactions under thermo-mechanical load using EFGM, Journal of Mechanical Science and Technology, vol. 25, 2011, pp. 403-413.

[5] S. Mohammadi, Extended finite element method for fracture analysis of structures, (Blackwell Publishing 2008).

[6] H. Pathak, A. Singh, I. V. Singh, Numerical simulation of bi-material interfacial cracks using EFGM and XFEM, Internation Journal of Mechanics and Materials in Design, vol. 8, 2012, pp. 9-36.

-16 -14 -12 -10 -8 -6 -4 -2 x 10^-4

-200 -150 -100 -50 0 50 100 150 200

-3 -2 -1 0 1 2 3 x 10^-3

-300 -200 -100 0 100 200 300

-1 -0.5 0 0.5 1 1.5 x 10^-3