XFEM Modeling of Fracture Mechanics in ...

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ScienceDirect Procedia Materials Science 3 (2014) 1257 – 1262

20th European Conference on Fracture (ECF20)

XFEM modeling of fracture mechanics in transversely isotropic FGMs via interaction integral method E. Golia,* , M.T. Kazemib a,b

Department of Civil Engineering, Sharif University of Technology, Tehran, Iran

Abstract Functionally graded materials are advanced composites with smoothing variation in the mechanical and thermal properties. In recent years, FGMs have been used in various fields of engineering especially in the high-tech applications. Due to importance of applications of FGMs, their behavior in the cracked or defected conditions must be evaluated. In order to reach this aim, extended finite element as a robust tool for modeling of discontinuity is used in this research. Besides, interaction integral method is employed to evaluation the stress intensity factors in the case of transversely isotropic FGMs. As will be shown in this study, the couple system of interaction integral method and XFEM is represented a useful framework for evaluating fracture parameters in an anisotropic non-homogenous medium such as transversely isotropic FGMs. As the last but not the least case, computational efforts largely is reduced by using the extended finite element method in this study. © 2014Published The Authors. Published by Elsevier Ltd.CC BY-NC-ND license. © 2014 by Elsevier Ltd. Open access under Selection andpeer-review peer-review under responsibility ofNorwegian the Norwegian University of Science and Technology Department of Selection and under responsibility of the University of Science and Technology (NTNU), (NTNU), Department Structural of StructuralEngineering. Engineering Keywords: : Fracture mechanics; Functionally graded materials; Transversely isotropic medium; Interaction integral method; XFEM

1. Introduction Generally, homogenous materials aren't compatible with modern technology needs, so new and advanced materials have been used frequently. Functionally graded materials are a new class of composites in which the volume fraction of constituent materials vary smoothly, giving a non-uniform microstructure with continuously graded macro properties (Kim and Paulino 2003). Due to special applications of FGMs, they have been studied in

* Corresponding author. Tel.:+98-912-826-0040. E-mail address:[email protected]

2211-8128 © 2014 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering doi:10.1016/j.mspro.2014.06.204

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E. Goli and M.T. Kazemi / Procedia Materials Science 3 (2014) 1257 – 1262

the field of Fracture mechanics by researchers. Recently, the interaction integral method on cracked isotropic/orthotropic FGMs has been used in many of research for mechanical (Kim and Paulino 2003, 2002, Kubair et al. 2005, Dolbow and Gosz 2002) and thermal (Kawasaki and Watanabe 2002, Noda and Jin 1993, Guo et al. 2012, Kim and KC 2008) loading conditions. Also, interaction integral method and XFEM, as a powerful numerical tool on the discontinuous fields, was used for orthotropic FGMs under mechanical and thermal loadings (Hosseini et al. 2013). In this paper interaction integral method and constitutive relations of such a non-homogenous transversely isotropic medium have been implemented to XFEM for evaluating stress intensity factors in a transversely isotropic FGM and the main purpose of this research is extending available researches to transversely isotropic FGMs. In the first section, constitutive relation of transversely isotropic material is presented. Then, J-integral theory and extraction stress intensity factors have been discussed in sections 3 and 4, respectively. Section 5 represent the brief introduction of XFEM. Numerical example is represented in section 6 that shows numerical results and output figures. The final parts of this paper are conclusion and references. 2. Constitutive relation Equation (1) is represented in order to introduction of Hooke's law in the compliance form for a transversely isotropic material.

H xx ½ °H ° ° yy ° °°H zz °° ® ¾ °H yz ° °H zx ° ° ° °¯H xy °¿

ª 1 « E « p « Xp « « Ep « X « pz « Ep « « 0 « « « 0 « « « « 0 ¬

Xp Ep

1 Ep

X pz

X zp Ez

X zp Ez

0

0

0

0

Ep

1 Ez

0

0

0

0

1 2G zp

0

0

0

0

1 2G zp

0

0

0

0

º 0 » » » 0 » » V xx ½ » °V ° yy ° 0 »° » °°V zz °° » ®V ¾ yz ° 0 »° » °V zx ° ° »° °V xy ¿° 0 »¯ » » 1 Xp » E p »¼

(1)

where H , E , X , G and V denote strain component, Young's modulus, Poisson ratio, shear modulus and stress component, respectively. In addition, p is used for material properties that are respected to plane of symmetry and z for symmetric axis that is normal to mentioned plane. Similar to orthotropic medium, there is the condition for the transversely isotropic domain as

X pz

X zp

Ep

Ez

(2)

It should be remember that, for a transversely isotropic functionally graded material, from each point to other point, material properties can change. 3. J-integral theory The common EDI form of J- integral is presented in equation (3), (Kim and Paulino 2003) .

J

³ V u ij

A

.

i ,1

^

`

1 w G1 j q , j dA ³ V ij u i ,1 j H ij ,1 C ijpq ,1H ij H pq q dA 2 A

(3)

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where u , w , G , A , C and q show displacement field, strain energy, Kronecker delta symbol, surface of domain integral, fourth order stiffness tensor and a weighted function, respectively. This weighted function should be smooth from outer edge of surface A to the inner edge of it with the value 0 on the outer bound and 1 on the inner. Surface A and other requirements are shown in Fig. 1.

Fig. 1. Surface

A

and function

q

(Goli and Kazemi 2013)

For applying J-integral on non-homogenous medium such as FGMs, one imaginary, non-physical field should be introduce as an auxiliary field. This auxiliary field (in incompatible form) satisfies the equilibrium and constitutive relations but violates the compatibility equation (Kim and Paulino 2003).

V

0, V

aux ij , j

aux ij

C ijkl H

aux kl

, H

aux ij

z

1 wu iaux

(

2 wx j

wu aux j wx i

)

(4)

Now by substituting terms of actual and auxiliary in the equation (3), we have (kim and paulino 2003)

J act J aux M

J where 2012).

(5)

M is known as interaction integral and is introduced in equation (6) for the incompatible form (Guo et al.

M

³ ^V

act ij

A

`

^

`

aux u iaux V ijaux u iact,1 V ikaux H ikact G1 j q , j dA ³ V ijact ª s ijmn tip S ijmn º V mn q dA ,1 ,1 A

¬

¼

(6)

4. Extraction stress intensity factors from M-integral Fracture energy and stress intensity factor are two important index in the concept of fracture mechanics. Equation (7) connects these two index to each other (Kim and Paulino 2003).

G where

c11K I2 c12 K I K II c 22 K II2

(7)

K I and K II denote stress intensity factor for mode I and II of fracture, respectively. Also other coefficient

are introduced as

c11

(

c12

(

c 22

(

a22

tip

2

a22

tip

2

a11

)Im (( P1tip P2tip ) / ( P1tip P2tip )) a11

(8)

tip

)Im ((1) / ( P1tip P2tip )) (

2

) Im ( P1tip P2tip )

(9)

tip

2

) Im ( P1tip P2tip )

(10)

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where Pi is the root of characteristic equation of anisotropic medium. On the other hand, M integral relates to stress intensity factors of actual and auxiliary fields with,

M local

1 , K IIaux

0 for one mode and K Iaux

2c11K I K Iaux c12 K I K IIaux K Iaux K II 2c 22 K II K IIaux

With considering

K Iaux

(11)

0 , K IIaux

1 for other mode, two simple

equations are gained for evaluating stress intensity factors in the actual field (Kim and Paulino 2003). I M local ® II ¯ M local

2c11K I c12 K II

(12)

c12 K I 2c 22 K II

The auxiliary fields of an orthotropic medium for incompatible form, are used in this study for transversely isotropic problem. These auxiliary fields are given in other references, e.g Kim and Paulino 2002, 2003. 5. Extended finite element method XFEM as a numerical tool is used in the this research. The framework of XFEM has many advantages such as no need to remeshing, high accuracy and reproducing stress singularity in the crack tip. Displacement field is discreted by XFEM framework as NOD

u

¦N x u i

i 1

i

§

¦ N x ¨ ¦ F x b t

t N

tip

©j

j

1NTF

tj

· ¸ ¦ N i x H ] as ¹ sN

(13)

H

First term of equation (13) is the same term of ordinary FEM. Second term is added to common term for reproducing singular values in stress fields and last term helps to the method for simulating discontinuity of displacement field along the crack faces (Belytschko and Black 1999). a and b are added degrees of freedom to common DOFs. Also H is Heaviside function and F is a set of four specific tip enrichment functions as

F

^

T

T

T

T

r sin , r cos , r sin sin T , r cos sin T 2 2 2 2

`

(14)

6. Numerical example In this section In this section a transversely isotropic functionally graded material is considered to illustrate application of interaction integral in this kind of advanced materials. Problem is solved solved in the plane strain condition. Configuration, boundary conditions and dimensions are shown in Figure (2).

Fig. 2. A cracked transversely isotropic FGM

Some assumption, material properties and their functions of variation are represented as,

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W L

X top p

1

h

5

3 W

8

,

X top pz

, Etop p

0.31, X bot p

E p y = Etop p

E

Xp y = X

X

top p

Etop z

204GPa, Ebot p

0.22, X top pz

§©Wy

90.43GPa, Ebot z

116.36GPa

0.14 E

· , E y = Etop z ¸ z ¹

bot p

Etop ¨ p

bot p

§y · X ¨ ¸ , X pz y = X top pz ©W ¹

J

top p

E

bot z

X

bot pz

§©Wy

Etop ¨ z

· ¸ ¹

E

§y · X ¨ ¸ ©W ¹

(15)

J

top pz

where top and bot are related to top and bottom edges of problem (Dag et al. 2010). Variation of material properties along y-direction is depicted in Figure (3-a). In addition, Figure (3-b) shows the mesh that is used for this example. Domain is descritized with 2262 nodes and 2156 quadratic elements.

Fig. 3. (a) Variation of material properties along y-direction; (b) XFEM mesh

By implementing mentioned theory and relations in the XFEM framework, stress intensity factors are gained as represented in Table (1). Results are normalized with Eq. (16) and presented for various ratios of a /W .

K In

KI

(16)

P Sa

Table (1). The normalized SIFs for a transversely isotropic FGM

E ,J

E

E

2, J =1.5

3, J =2

a /W

K In

0.25

0.458

0.3

0.430

0.35

0.346

0.25

0.503

0.3

0.452

0.35

0.351

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Finally, Fig. 4 shows stress fields and exaggerated deformation shape for the case a /W

J

1.5

0.25 , E

2 and .

Fig. 4. (a)

Vx

(b)

Vy

(c)

V xy

(d) exaggerated deformation shape

7. Conclusion In this paper, the incompatible form of the interaction integral method has been implemented for a transversely isotropic functionally graded material subjected to mechanical loading. Also extended finite element method as a nonpareil tool in the computational Fracture mechanics area, has been hired to work with interaction integral method. Finally, the stress intensity factors for the different crack lengths and various material properties variations have been gained by this couple system of interaction integral and XFEM that it shows efficiency of presented framework. References Belytschko, T., Black, T., 1999. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 45, 601–620. Dag, S., Arman, E.E., Yildirim, B., 2010. Computation of thermal fracture parameters for orthotropic functionally graded materials using Jkintegral. International Journal of Solids and Structures 47, 3480–3488. Dolbow, J., Gosz, M., 2002. on the computation of mix-mode stress intensity factors in functionally graded materials. International Journal of Solids and Structures 39, 2557–2574. Goli, E., Kazemi, M.T., 2013. A study on effect of the material properies function on mechanical performance of uncracked/ cracked FGMs. International Conference on Civil Engineering Architecture & Urban Sustainable Development, Tabriz , Iran, 356. Guo, L., Guo, F., Yu, H., Zhang, L., 2012. An interaction energy integral method for nonhomogeneous materials with interfaces under thermal loading. International Journal of Solids and Structures 49, 355–365. Hosseini, S.S., Bayesteh, H., Mohammadi, S., 2013. ThermomechanicalXFEMcrackpropagationanalysisoffunctionallygraded materials. Materials Science and Engineering A 561, 285–302. Kawasaki, A., Watanabe, R., 2002. Thermal fracture behavior of metal/ceramic functionally graded materials. Engineering Fracture Mechanics 69, 1713–1728. Kim, J.H., KC, A., 2008. A generalized interaction integral method for the evaluation of the T-stress in orthotropic functionally graded materials under thermal loading. Journal of Applied Mechanics 75, 051112. Kim, J.H., Paulino, G.H., 2003. The interaction integral for fracture of orthotropic functionally graded materials: evaluation of stress intensity factors. International Journal of Solids and Structures 40, 3967–4001. Kim, J.H., Paulino, G.H., 2002. Finite element evaluation of mixed mode stress intensity factors in functionally graded materials. International Journal for Numerical Methods in Engineering 53, 1903–1935. Kubair, D.V., Geubelle, P.H., Lambros, J., 2005. Asymptotic analysis of a mode III stationary crack in a ductile functionally graded material. Journal of Applied Mechanics 72, 461–467. Noda, N., Jin, Z.H., 1993. Steady thermal stresses in an infinite nonhomogenous elastic solid containing a crack. Journal of Thermal Stresses 16, 181–196.