Modeling of Nonlinear Crack Growth in Steel and Aluminum Alloys by the Element Free Galerkin Method

Available online at www.sciencedirect.com

ScienceDirect Materials Today: Proceedings 5 (2018) 18805–18814

www.materialstoday.com/proceedings

ICMPC_2018

Modeling of Nonlinear Crack Growth in Steel and Aluminum Alloys by the Element Free Galerkin Method Showkat Ahmad Kantha*, G. A. Harmainb, Azher Jameelc a

Research Scholar, Department of Mechanical Engineering, National Institute of Technology Srinagar, 190006, India b Professor, Department of Mechanical Engineering, National Institute of Technology Srinagar, 190006, India c Assistant Professor, Department of Mechanical Engineering, Shri Mata Vaishno Devi University Katra, 182320, India

Abstract In this study, the modelling and simulation of the nonlinear fatigue crack growth phenomenon in steel and aluminium alloys has been carried out by employing the element free Galerkin method. The primary variable is approximated by the moving least square shape functions. For the consideration of the effect of various discontinuities present in the domain, appropriate enrichment functions have been added to the standard EFGM approximation. In order to keep track of the various discontinuities, the standard level set method is used. The present study includes the material and geometric nonlinearities in the mathematical formulations. The geometric nonlinearity has been modelled by employing the updated Lagrangian approach. The isotropic hardening has been used with the Ramberg-Osgood model to obtain the elasto-plastic behaviour of the material. Finally, several numerical problems have been solved by the element free Galerkin method to investigate fatigue crack growth in steel and aluminium alloys. © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization. Keywords: Element free Galerkin method; crack growth; fatigue life; critical crack length

1. Introduction Cracks are present in almost all engineering components. In order to check the safety and reliability of these structural components under cyclic loading conditions, study of fatigue crack growth is of utmost importance. Even

* Corresponding author. Tel.: +91-9906888967; E-mail address: [email protected] 2214-7853 © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization.

18806

S. A. Kanth et al. / Materials Today: Proceedings 5 (2018) 18805–18814

though an ample amount of analytical work has been done on some simple and common configurations like cracks in infinite and finite domains, but the realistic and practical geometries under actual loading conditions invoke the use of numerical methods. In order to model and simulate fatigue crack growth in structural components, several numerical techniques such as the boundary element method [1], the finite element method [2], mesh free methods [3,4] and the extended finite element method (XFEM) [5] have been developed till date. Among various numerical methods the finite element method has been the most dominant numerical method for solving various fracture mechanics problems but it experiences various difficulties while modelling crack growth problems which involve remeshing and mesh adaption. The element free Galerkin method (EFGM) enables the modelling of crack growth problems by extrinsic enrichment, intrinsic enrichment and smoothing techniques [6]. EFGM models different discontinuities by enriching the conventional displacement based approximation with additional enrichment functions [7]. In EFGM the shape functions are constructed by the moving least square approximation scheme [8]. EFGM has found a wide application in the various areas like fracture mechanics [9], metal forming [10], vibration analysis [11], heat transfer [12], contact analysis [13] and extrusion processes [14] and to analyze three-dimensional fracture mechanics problems [15] and bi-material interface cracks [16]. For the imposition of boundary conditions a number of techniques like Lagrangian multiplier [17], modified variational principle [18], and coupling of FE– EFGM [19] have already been proposed. The present work aims at analyzing the elasto-plastic fatigue crack propagation in steel and aluminum alloys by employing the element free Galerkin method. Several numerical problems have been solved and for the computation of fatigue life the standard Paris’ law is used. Maximum principal stress criterion is used to predict the direction of crack growth [8]. The stress intensity factors are obtained by the domain based interaction integral approach [20]. The Ramberg-Osgood model has been used to represent the elasto-plastic behavior of the material. Isotropic hardening has been assumed in the present study. For the simulation of the given problems, an efficient and accurate code has been composed in MATLAB. A rectangular plate with an edge crack has been considered for analysis. 2. The Element Free Galerkin Method (EFGM) In EFGM, the entire domain is discretized into a set of nodes. Let us consider a domain ‘Ω’ enclosed by the boundary′ ′. The boundary ‘ can further be divided into displacement ( ) and traction ( ) boundaries. Due to crack surface an additional boundary ( ) is produced. The applied tractions are represented by t and displacement field is given by u. The expression for equilibrium equation can be written as Δ. + =0, where represents the stress tensor and b denotes the body force per unit volume. The above equilibrium equation can be further expressed as, Ω

: ( ) Ω=

:

Ω

Ω+

Ω

:

Γ

(1)

The modified displacement based EFGM approximation for the modelling of various crack growth problems can be expressed as, (χ) = ∑

Ψ (χ)

+∑

Ψ (χ)[H(χ) − H( ) ]

+∑

Ψ (χ) ∑

[ (χ) −

( )]

(2)

where n denotes the set of nodes in the domain, depicts the standard nodal degree of freedom, shows the set of split nodes, shows the enriched degrees of freedom associated with the split nodes, shows the set of tip nodes and represents the enriched degrees of freedom associated with tip nodes. Ψ denotes the standard EFGM shape function. The Heaviside jump function creates a discontinuous displacement field and its value varies from +1 on one side and -1 on the other side of the crack surface. For the enrichment of the tip nodes, crack-tip enrichment functions (χ) are employed. The crack tip enrichment functions for isotropic linear elastic materials are defined in local crack tip coordinates (r and ) as given in [21]. ( ) = [√ cos

, √ sin , √ cos sin , √ sin cos ]

(3)

S. A. Kanth et al. / Materials Today: Proceedings 5 (2018) 18805–18814

18807

The second enrichment(√ sin ), function provides the required discontinuous displacement field near the crack tip. While the accuracy of the results is improved by the other three enrichment functions [21]. The discrete system denotes a of equation as [ ]{ } = { } is obtained by substituting Eq. (2) in the equilibrium equation, where } . and vector of nodal unknowns, defined as { } = { are the elemental stiffness matrix and nodal force vector, defined as [

; {

]=

}={

}

(4)

The sub-matrices in the above equation are given by, = = =

(B )

Ω Ω

Ω , where r,s = u,a,b

Ψ ( )(χ) − H( )

Ω

Ψ (

)( ) −

( )

Ω+ Ω+

(5)

Ψ ( )(χ) − H( ) Γ Ψ (

)( ) −

(6)

( ) Γ ; (

The B-matrix relating the strains with displacements is defined as { } = [ ]{

= 1, 2, 3, 4)

(7)

}, where { } represents the strain

tensor. The B-matrix is defined as, =

Ψ 0 Ψ

0 Ψ Ψ

(8)

(Ψ ( ( ) − ( ))) 0 = (Ψ ( ( ) − ( ))) (Ψ ( ( ) − ( ))) 0 = (Ψ ( ( ) − ( )))

0 (Ψ ( ( ) − ( ))) (Ψ ( ( ) − ( ))) 0 (Ψ ( ( ) − ( ))) (Ψ ( ( ) − ( )))

(9)

(10)

3. Elasto-Plastic Analysis Von-Mises yield criterion with isotropic strain hardening [22, 23] is used to determine the yielding of the material. The behavior of the material will be partly elastic and partly elastic after a certain amount of initial yielding. Thus, we can write the total increment in the strain as the sum of the plastic strain component( ) and elastic strain component( ) , =(

) + (

)

(11)

The plastic strain increment ( ) produced is proportional to the stress gradient of the plastic potential function (Q). The plastic potential function and the yield function ( ) are equal according to the associated theory of plasticity. Hence, the increment in plastic strain ( ) can be written as, (

) =

(12)

18808

S. A. Kanth et al. / Materials Today: Proceedings 5 (2018) 18805–18814

Where

denotes the plastic multiplier. Since at the stress point under consideration, the vector

is normal to the

yield surface therefore Eq. (12) is termed as the normality condition, Thus, Eq. (11) can be written as, =

+

+

(13)

Where denotes shear modulus, is hydrostatic stress, is Poisson’s ratio, is Kronecker delta and is deviatoric stress component. Using the concept of work hardening and strain hardening hypothesis, the Von-Mises yield function can be written as, ( , )= ( )− ( )

(14)

Where is referred as the Cauchy stress vector and manipulations, the stress–strain relations are obtained as,

denotes the hardening parameter. After mathematical

dσ = D dε Where

(15)

denotes the elasto-plastic constitutive matrix.

4. Large Deformation Analysis During elasto-plastic large deformation analysis, usually two types of nonlinearity occur: (1) geometric nonlinearity due to large deformations and (2) material nonlinearity due to plasticity. In order to model the geometric nonlinearity two approaches have been developed these are the total Lagrangian and updated Lagrangian approach. In the present paper updated Lagrangian approach is employed for the modelling of geometric nonlinearity. In this approach current configuration is used as a reference configuration. The configuration in which the quantity occurs is denoted by the left superscript on a quantity, and the configuration with respect to which the indicates that the quantity occurring in quantity is measured is represented by a left subscript. Thus configuration is the quantity measured in configuration . Thus for the elasto-plasticity with large deformation equilibrium equation can be written as [24], ∇. +

=0

Ω

(16)

The essential boundary conditions are defined as, =

Γ

(17)

Where denotes the Cauchy stress tensor and is the body force per unit volume. By the incorporation of the essential boundary conditions in the updated Lagrangian approach, Eq. (16) can be expressed in the variational form as, Ψ( ) = )

Ω

Ω −

Ω

Γ =0

Ω −

Γ −

Γ −

( −

(18)

Where are the updated Green–Lagrange strain tensor and are updated second Piola–Kirchhoff stress tensor. denotes the vector of Lagrange multiplier, introduced in order to enforce the essential boundary conditions. In terms of nodal values, the Lagrange multiplier can be expressed as, ( )=

( )

( )=

( )

(19)

S. A. Kanth et al. / Materials Today: Proceedings 5 (2018) 18805–18814

=

Now using

+

,

= Φ

18809

and Eq. (19), Eq. (18) can be expressed as,

Ψ( ) = Ω+ (Φ

−

)

Ω −

Ω−

Γ−

Γ =

Where(

) +(

) =

can be further divided into linear and nonlinear

)

(21)

+

Ψ( ) =

(

)

=

Ω+

Ω

Γ−

Γ −

(20)

The incremental updated Green–Lagrange strain tensor terms as, = (

Φ

+ (

Ω

. Now using Eq. (21), Eq. (20) can be written as )

Ω +

Γ −

Φ

Φ

(

Ω

Γ

)

−

Ω − Γ

=0

Ω−

Ω

(22)

Now, Eq. (22) can be further written as, Ω

(

)

Γ

−

Φ =

Where,

(

Ω+ Γ

(

Ω

)

)

(

Ω

)

Ω −

+

−

Γ −

Φ

=0

Ω and

(23) =

Ω −

Ω

Γ.

The first and second terms of Eq. (23) can be written alternatively as, (

Ω

)

(

)

Ω=

Ω

{ (

) }

(

)

Ω =

Ω =

Ω

(24) (

Ω

= Where

)

Ω=

Ω

{ (

Ω and

Ω

) } =

Ω =

Ω =

Ω

Ω

Ω

denotes the material tangent stiffness matrix and

(25) (26)

denotes the geometric stiffness matrix.

is the

matrix of Cauchy stress components and B, G represent the Cartesian shape function derivatives matrix. We have

B =

,

0

0

,

,

,

,

;G =

0

0 ,

,

0

0

,

;

=

.

(27)

18810

S. A. Kanth et al. / Materials Today: Proceedings 5 (2018) 18805–18814

5. Fatigue Crack Growth Analysis By utilizing the domain based interaction integral approach [25, 26] in mixed mode crack growth problems the stress intensity factors (SIFs) have been computed. For any two equilibrium states of a cracked specimen, the domain based interaction integral can be defined as, ( , )

( )

( )

=

+

( )

( )

−

( , )

(28)

where ‘q’ denotes the smooth weight function. The value of ‘q’ is ‘0’ along the contour and ‘1’ at the crack tip. ( , )

represents the mutual strain energy and is defined as, ( , )

( ) ( )

=

+

( ) ( )

=

( ) ( )

( ) ( )

=

(29)

In the above equation, the actual state is represented by state ‘1’ and state ‘2’ represents the auxiliary state respectively. The stresses and strains are represented as stress intensity factors ( , )

=

and ( ) ( )

∗

where for plane stress,

∗

=

and respectively. The relation between the mixed mode

and the domain based interaction integral can be written as, +

( ) ( )

and for plane strain,

(30) ∗

=

. By the judicious selection of the auxiliary states i.e.

state 1 and 2, eq. (30) can be used for the evaluation of the individual mixed mode stress intensity factors. By ( )

choosing 0,

( )

= 1,

( )

= 0, we can obtain the mode-I stress intensity factor and similarly we choose

= 1 for mode-II stress intensity factor. Thus, we get

( )

=

( ,I) ∗

( )

and

=

( ,II) ∗

( )

=

.

6. Numerical results and Discussions 6.1. Fatigue Crack Growth in Aluminum Alloys The fatigue crack growth analysis of a square plate, shown in Fig. 1(a), with dimensions 100 mm × 200 mm, containing an inclined edge crack of length mm, inclined at an angle , under tension is presented in this section. The material properties chosen for analysis are given in the Table 1. Fig. 1(b) shows the EFGM representation of the given domain. Fig. 2. Shows the fatigue life diagram for different aluminum alloys. Fatigue life diagram for aluminum alloy (Al 2024-T3) at various crack angles is presented in Fig. 3. Similarly Fig. 4 shows the fatigue life diagrams for aluminum alloy (Al 6061-T6) at various crack angles and Fig.5. presents the fatigue life diagrams for aluminum alloy (Al 7075-T6) at various crack angles. 100 90

b

80

a

70 60 50

L

40 30 20 10 0

L

0

20

40

60

80

100

Fig. 1. (a) Edge crack in a square plate; (b) EFGM domain representation for an edge cracked square plate.

S. A. Kanth et al. / Materials Today: Proceedings 5 (2018) 18805–18814

18811

Table 1: Materials Properties of various steels and aluminium alloys

(

)

(

)

(

)

(

√ )

200

0.30

415

725

77

0.20

5.74 × 10

2.25

193

0.30

215

505

124

0.70

2.178 × 10

3.25

200

0.30

345

550

110

0.20

7.46 × 10

3.00

−

73.1

0.33

345

485

26

0.18

5.85 × 10

3.59

−

68.9

0.33

276

310

29

0.17

1.04 × 10

3.17

−

71.7

0.33

505

572

25

0.11

7.64 × 10

3.70

34

Al 2024-T3 Al 6061-T6 Al 7075-T6 Fatigue Life

32 30

Crack length (mm)

28 26 24 22 20 18 16 14 -0.5

0

0.5

1

1.5 2 No of cycles

2.5

3

3.5

4 4

x 10

Fig. 2. Fatigue life diagrams for different aluminium alloys. 32

00 150

30

250

Crack length (mm)

28

350 Fatigue Life

26 24 22 20 18 16 14 -0.5

0

0.5

1

1.5 2 2.5 No of cycles

3

3.5

4

4.5 4

x 10

Fig. 3. Fatigue life diagrams for aluminium alloy (Al 2024-T3) at various crack angles.

18812

S. A. Kanth et al. / Materials Today: Proceedings 5 (2018) 18805–18814

34

00 150

32

250

30

350 Fatigue Life

Crack length (mm)

28 26 24 22 20 18 16 14 -0.5

0

0.5

1

1.5 2 No of cycles

2.5

3

3.5

4 4

x 10

Fig. 4. Fatigue life diagrams for aluminium alloy (Al 6061-T6) at various crack angles. 32

00 150

30

250

Crack length (mm)

28

350 Fatigue Life

26 24 22 20 18 16 14 -2000

0

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 No of cycles

Fig. 5. Fatigue life diagrams for aluminium alloy (Al 7075-T6) at various crack angles.

6.2. Fatigue Crack Growth in Steel Alloys The fatigue crack growth analysis of the same square plate shown in Fig.1 (a), made of steel, is presented in this section. The material properties chosen for analysis are given in the Table 1. Fig. 1(b) shows the EFGM representation of the given domain. Fatigue life diagram for steel alloy (AISI 440A) at various crack angles is presented in Fig. 6. Similarly Fig. 7 shows the fatigue life diagrams for steel alloy (AISI 304) at various crack angles and Fig. 8 presents the fatigue life diagrams for steel alloy (AISI 446) at various crack angles respectively. 35

00 150 250

Crack length (mm)

30

350 Fatigue Life

25

20

15

-0.5

0

0.5

1

1.5 No of cycles

2

2.5

3

3.5 4

x 10

Fig. 6. Fatigue life diagrams for stainless steel (AISI 440A) at various crack angles.

S. A. Kanth et al. / Materials Today: Proceedings 5 (2018) 18805–18814

35

18813

00 150 250

Crack length (mm)

30

350 Fatigue Life

25

20

15

-1000

0

1000

2000 3000 No of cycles

4000

5000

6000

Fig. 7. Fatigue life diagrams for stainless steel (AISI 304) at various crack angles. 45

00 150

40

250 350 Fatigue Life

Crack length (mm)

35

30

25

20

15 0

2

4

6 No of cycles

8

10

12 4

x 10

Fig. 8. Fatigue life diagrams for stainless steel (AISI 446) at various crack angles.

7. Conclusions The present study models the nonlinear fatigue crack growth phenomenon in steel and aluminum alloys by employing the element free Galerkin method. The primary variable is approximated by the moving least square shape functions. For the consideration of the effect of various discontinuities present in the domain, appropriate enrichment functions have been added to the standard EFGM approximation. The geometric nonlinearity has been modelled by employing the updated Lagrangian approach. The isotropic hardening has been used with the RambergOsgood model to obtain the elasto-plastic behavior of the material. Several numerical problems have been solved by the element free Galerkin method to investigate fatigue crack growth in steel and aluminum alloys. The results clearly indicate that the proposed technique can be efficiently and accurately used to model fatigue crack growth in engineering specimens. Because of the meshless approach, the element free Galerkin method eliminates the drawbacks associated with the finite element meshes. References [1] A. Portela, M. Aliabadi, D. Rooke, Int. J. Numer. Methods Eng. 33 (1991) 1269–87. [2] S. Cheung, A. R. Luxmoore, Eng. Fract. Mech. 77 (2003) 1153–69. [3] M. Pant, I. V. Singh, B. K. Mishra, Int. J. Mech. Sci. 52 (2010) 1745–55.

18814

S. A. Kanth et al. / Materials Today: Proceedings 5 (2018) 18805–18814

[4] M. Duflot, H. N. Dang, J. Comput. Appl. Math 168 (2004) 155–64. [5] T. Belytschko, T. Black, Int. J. Numer. Methods Eng. 45 (1999) 601–20. [6] N. Moes, J. Dolbow, T. Belytschko, Int. J. Numer. Methods Eng. 46 (1999) 131–50. [7] S. Dag, Eng. Fract. Mech. 73 (2006) 2802–28. [8] P. Lancaster, K. Salkauskas, Math Comput. 37 (1981) 141–58. [9] Y. Xu, S. Saigal, Comput. Methods. Appl. Mech. Eng. 154 (1998) 331–43. [10] G. Yanjin, W. Xin, G. Zhao, L. Ping, Int. J. Adv. Manuf. Technol. 42 (2009) 83–92. [11] G. Liu, TT. Nguyen, K. Lam, J. Sound Vib. 320 (2009) 1100–30. [12] A. Jameel, G.A. Harmain, Streng. Mater. 48 (2016) 294–307. [13] G. Y. Li, T. Belytschko, Eng. Comput. 18 (2001) 62–78. [14] W. Xin, Z. Guoqun, W. Weidong, G. Yanjin, L. Ping, Chin. J. Mech. Eng. 20 (2007) 26–31. [15] R. Brighenti, Eng. Fract. Mech. 72 (2005) 2808–20. [16] H. Pathak, A. Singh, I. V. Singh, Int. J. Mech. Mater. Des. 8 (2012) 9–36. [17] T. Belytschko, L. Gu, YY. Lu, Modell Simul. Mater. Sci. Eng. 2 (1994) 519–34. [18] YY. Lu, T. Belytschko, L. Gu, Comput. Methods Appl. Mech. Eng. 113 (1994) 397–414. [19] Y. Krongauz, T. Belytschko, Comput. Methods Appl. Mech. Eng. 131 (1996) 133–45. [20] S. Bordas, J. G. Conley, B. Moran, J. Gray, E. Nichols, Eng. Comput. 23 (2007) 25–37. [21] A. Yazid, N. Abdelkader, H. Abdelmadjid, Appl. Math Modell 33 (2009) 4269–82. [22] E. Hinton, D.R.J. Owen, Pineridge Press Limited, 1980. [23] S. Kumar, I.V. Singh, B.K. Mishra, Int. J. Mech. Mater. 10 (2014) 165-177. [24] J.N. Reddy, Oxford University Press, 2009. [25] EE. Gdoutos, Kluwer Academic Publications, 2nd ed. (2005). [26] A. Jameel, G.A. Harmain, Int. J. Fatigue. 81 (2015) 105–116.