A new framework based on continuum damage mechanics and XFEM for high cycle fatigue crack growth simulations

Accepted Manuscript A New Framework Based on Continuum Damage Mechanics and XFEM for High Cycle Fatigue Crack Growth Simulations V.B. Pandey, I.V. Singh, B.K. Mishra, S. Ahmad, A. Venugopal Rao, Vikas Kumar PII: DOI: Reference:

S0013-7944(18)30820-8 https://doi.org/10.1016/j.engfracmech.2018.11.021 EFM 6234

To appear in:

Engineering Fracture Mechanics

Received Date: Revised Date: Accepted Date:

16 August 2018 27 October 2018 8 November 2018

Please cite this article as: Pandey, V.B., Singh, I.V., Mishra, B.K., Ahmad, S., Venugopal Rao, A., Kumar, V., A New Framework Based on Continuum Damage Mechanics and XFEM for High Cycle Fatigue Crack Growth Simulations, Engineering Fracture Mechanics (2018), doi: https://doi.org/10.1016/j.engfracmech.2018.11.021

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A New Framework Based on Continuum Damage Mechanics and XFEM for High Cycle Fatigue Crack Growth Simulations V. B. Pandey#, I. V. Singh#, B. K. Mishra#, S. Ahmad$, A. Venugopal Rao$, Vikas Kumar$ #

Department of Mechanical and Industrial Engineering,

Indian Institute of Technology Roorkee, Uttarakhand, India Phone: +91-1332-285888, Fax: +91-1332-285665, E-mail: [email protected] $

Defence Metallurgical Research Laboratory, DRDO, Hyderabad, India

ABSTRACT In this paper, we have developed a continuum damage mechanics (CDM) based methodology for high cycle fatigue crack growth simulations. A fatigue damage law is proposed and implemented in the framework of extended finite element method (XFEM). A new criterion is proposed based on damage evolution to identify the appropriate definition of stress triaxiality for acquiring the constraint effect on the stress state correctly. Few mesh regularization schemes are also employed for reducing the mesh sensitivity in the results. Simulations are performed on fracture specimens of different materials subjected to constant amplitude fatigue loading. The fatigue life of a turbine disc is also predicted under constant amplitude loading. The results obtained from present methodology (CDM and XFEM) are found in good agreement with the published experimental results. These simulations highlight that the continuum damage mechanics is a simple and effective tool to perform crack growth simulations under high cycle fatigue conditions. Keywords: High cycle fatigue crack growth; Damage mechanics; Stress triaxiality; XFEM; Non-local method 1. INTRODUCTION Fatigue crack propagation is one of the main failure mechanisms, which is experienced by almost every engineering component. Hence, the investigation of crack growth is necessary to predict the remaining life of a cracked component before its catastrophic failure. For this purpose, various methodologies have been developed in the past. Recently, a continuum damage mechanics (CDM) based methodology has been found to be very powerful to model the fracture and subsequent estimation of component life. To predict the life using CDM under fatigue loading, various fatigue damage laws have been proposed. 1

Qian et al. (1996) combined damage mechanics with finite element method (FEM) to model the fatigue crack growth at high temperature. Peerlings et al. (2000) proposed a fatigue damage law and used it in gradient enhanced non-local approach to simulate the fatigue crack growth. They concluded that this methodology is an effective way to estimate the remaining fatigue life of a cracked component. Wahab et al. (2001), Ashcroft et al. (2010) and Shenoy et al. (2010) also proposed various fatigue damage laws to investigate the crack growth in adhesive joints. They proved that the damage mechanics based fatigue crack growth methodology is better than the fracture mechanics based methodology. Oller et al. (2005) also used this approach and developed a fatigue damage law to solve the crack growth problems under thermo-mechanical fatigue loadings. Lestriez et al. (2007), Lee et al. (2011), Van Do et al. (2015) and Lee et al. (2016) adopted this methodology to simulate the fatigue crack growth in various engineering applications. All these damage models are the functions of either stress or strain components. These stress or strain components in any problem are highly influenced by the constraint. Hence, to estimate the remaining life of a component, the stress/strain state must be evaluated correctly. To predict the effect of constraint on stress state, stress triaxiality function is used in the literature. Many researchers reported that the present definition of stress triaxiality is inadequate to express the effect of constraint ahead of the crack tip accurately and hence, several suggestions and modifications have been reported (Wang, 1995; Schafer et al., 2000; Bao and Wierzbicki, 2004; Mehmanparast et al., 2014; Yu et al., 2016). These modifications are discussed in detail, in section 4. So far, the CDM in conjunction with FEM is used to simulate the crack growth. Although the combination of CDM and FEM is quite capable to provide accurate solutions, however it is found computationally expensive. Therefore, various numerical methods have been developed in past two decades to solve complex fracture problems. Some prevalent methods are cracking particle method (Rabczuk and Belytschko, 2007; Rabczuk et al., 2010), extended finite element method (Shedbale et al., 2013; 2016; Patil et al., 2017), extended isogeometric analysis (Ghorashi et al., 2015; Singh et al., 2018) and peridynamics (Ren et al., 2016; 2017). Apart from these methods, several other approaches have also been found which uses damage variable to model the fracture (Areias et al., 2013; 2016; 2018; Areias and Rabczuk, 2017). Poh and Sun (2017) presented a localizing gradient damage model that reduces the spurious damage growth. With the help of this method, a relatively sharp crack can be modeled. Although, all the above-discussed

2

methods model crack growth very well, nevertheless the implementation of CDM with extended finite element method (XFEM) is relatively straightforward. Therefore, it has been adopted in the present work. The combination of CDM and XFEM has been used in the literature before for elastic, plastic and creep crack growth problems (Seabra et al., 2013; Bansal et al., 2017a; 2017b; Beese et al., 2017; Pandey et al., 2017). Xu and Yuan (2009) studied mixed mode fatigue crack growth using cohesive zone damage model and XFEM. Bhamre et al. (2014) applied CDM with extended space-time method (XTFEM) for fatigue crack initiation and propagation. Zhan et al. (2017) used CDM and XFEM for fatigue life estimation. However, CDM was only used for crack nucleation and XFEM along with fracture mechanics was used for crack growth modeling. So far, the CDM along with XFEM is not used for the fatigue crack propagation. Therefore, in the present work, the CDM is combined with XFEM for the fatigue crack propagation. The main purpose of this work is to improve the methodology developed by Peerlings et al. (2000), Wahab et al. (2001) and others researchers so that a simple, effective and reliable tool based on CDM and XFEM can be developed for estimating the fatigue life of the components. The main novelties of this work are as follows, 

A continuum damage mechanics based new fatigue damage law is proposed. The details of the proposed damage model are provided in section 3.



It is known that the crack growth rate varies in the components. Based on the crack growth rate, the fracture specimens are described as high or low constraint specimens. The identification of a particular problem as high or low constraint is essential to perform accurate crack growth analysis. In literature, this identification is mainly done based on the geometry. However, it has been observed that the loading conditions also plays an important role to describe a problem as high constraint or low constraint. Therefore, we have proposed a new damage based criterion that identifies the problem as a high constraint or low constraint problem. The proposed criterion incorporates the effect of both geometry and loading via two definitions of stress triaxiality (Schafer et al., 2000). A detailed discussion on the proposed criterion is provided in section 4.



XFEM is combined with CDM to solve the fatigue crack growth problems. It is known that damage mechanics shows a pathological mesh dependent results in

3

crack growth problems. Hence, to minimize the mesh dependency, three regularization methods are employed, whose details are provided in section 6. This paper is organized as, section 2 shows a cracked body in the mathematical form. Fatigue damage model and its detailed formulation is provided in section 3. Section 4 provides the reason to select the concept of two definitions of stress triaxiality at the crack tip. In section 5, XFEM formulation is described for cracked body. Mesh regularization schemes are detailed in section 6 to reduce the mesh sensitivity. The implementation of the combined CDM and XFEM approach is provided in section 7. A jump in cycle approach and crack propagation strategy are also discussed in this section. In section 8, significance of the combined methodology and performance of the proposed algorithm are demonstrated by simulating the crack growth in several components. The effect of various parameters is also presented in this section. Finally, the conclusions drawn from the present analysis are summarized in section 9. 2. PROBLEM FORMULATION A homogeneous continuum domain    is shown in Fig. 1 with tractions imposed on  t and displacements prescribed on u . The crack surface  c is assumed to be traction free. In the presence of body force b , equilibrium equations and boundary conditions are (Singh et al., 2017), .σ  b  0 in 

(1a)

σ.n  t on  t

(1b)

σ.n  0 on  c

(1c)

u  u on u

(1d)

where, σ is the Cauchy stress tensor, t represents the applied traction vector with unit normal n , u is the displacement field and u is the prescribed displacement field vector. For small displacements, the kinematic relation can be presented as,

ε  ε(u)  s u

(2)

where,  s is the symmetric gradient operator. The linear constitutive relation between stress and strain can be defined by Hook’s law, σ = Cε

(3) 4

where, C is the Hooke’s tensor.

3. CONTINUUM DAMAGE MODEL When a component is subjected to external forces, internal damage is accumulated. The resistance of the component to the applied load decreases, which may be quantified by a damage variable

 

in continuum damage mechanics. Generally, the evolution of

damage variable  is defined as a function of stress or strain components as,

  f  ,  

(4)

The evolution of this damage variable primarily depends on the external conditions. The present study focuses on the estimation of remaining life of a component having a crack, subjected to high cycle fatigue loading. So a fatigue damage model is presented in the next sub-section. 3.1 Fatigue Damage Model To predict the life of components under high cycle fatigue loading, various damage laws have been proposed by several researchers. Models presented by Chaboche and Lense (1988), Paas et al. (1993), Xiao et al. (1998), Lemaitre et al. (1999), Peerlings et al. (2000), Wahab et al. (2001) are most popular in the literature. These damage laws are tabulated in Table 1. Most of these models were used to evaluate fatigue life up to crack initiation. Among these, the models provided by Chaboche and Lense (1988) and Lemaitre et al. (1999) are most prominent in literature. However, they are not much suitable due to large number of parameters. It can be seen from Table 1 that the stress triaxiality function was neglected in most of the damage evolution laws (Paas et al., (1993); Xiao et al., (1998); Peerlings et al., (2000)). Although stress triaxiality remains constant for proportional loading, still it should be involved in the damage evolution equations (Lemaitre and Desmorat, 2005). It is known that in a growing crack, stress triaxiality changes with geometry and loading conditions. Hence, the stress triaxiality should be considered in the damage evolution equation. It is also observed from Table 1 that damage laws were presented either in power-law form or exponential form. Murakami and Liu (1995) argued that exponential function reduces mesh dependency and provides a better fit to the experiment results. Therefore, a new damage law is proposed which is the modified form of Wahab et al. (2001) model. The multiaxial form of the proposed damage law is given as, 5



 d M 1  e  dN 



 max,eq   min,eq 



e Rv0.5 

(5)

where, M ,  ,  are material constants,  is the damage variable,  max,eq ,  min,eq are the maximum and minimum equivalent stresses respectively and Rv is the stress triaxiality function which is defined as, Rv 

2 1     3 1  2  T 2 3

(6) where, T 

 H ,max is the stress triaxiality,  is Poisson’s ratio,  H ,max is the Hydrostatic  eq ,max

stress corresponding to the maximum stress. Table 1: Existing fatigue damage laws Total material Stress triaxiality parameters function

Damage law Chaboche and Lesne (1988)

 d  AII  1      1  1     .    dN   M 0 1  3b2 H ,mean 1     AII 

1 3  Sij ,max  Sij ,min    Sij ,max  Sij ,min  2  2 

 1 a

AII  AII  u   e,max

0.5



5

No

3

No

2

No

5

Yes

3

No

AII   l 0 1  3b1 H ,mean 

Paas et al. (1993)

   d  C  dN  1

 1

Xiao et al. (1998) d B (1  R)2 q  eq2 q,max  dN q (1   )2 q

Lemaitre et al. (1999) 2 s 1 2 s 1   d 2 Rv  max  k f    f (1  k )    dN C (1  k )(2s  1)  2 ES (1  k ) 2 (1   ) 2  s  

Peerlings et al. (2000)

   d  Ce dN  1

 1

6

Wahab et al. (2001)  m

  eq ,max   eq ,min  d  A  dN (1   )  

3

Rv / 2

Yes

3.1.1 Determination of the material constants For uniaxial condition, the fatigue damage law written in Eq. (5) becomes



 d M 1  e  dN 

   



e

(7)

Arranging the above equation and applying the limits



 d N M 1  e  e    0 o



   



 M 1  e 1 1         e 0 







dN

   

 1 1  M 1 e  1    e  

(8)



   

N



(9)

(10)

N

The component will reach the end of its life when damage variable approaches unity. So substituting   1 and N  N f in the above equation, the final life of the component will be Nf 

1 M   

(11)



Taking log of both sides  1  log  N f     log     log   M 

(12) The value of M and  can be obtained by comparing with the S-N curve. To determine  , rearranging the Eq. (10) and substituting the value of N f from Eq. (11) in Eq. (10) e 

1



1  1  e 

 NN

f

(13)

7

Taking logs of both sides,



 N  log 1  1  e   Nf   1





(14) The value of  is obtained by comparing the results from  vs N curve. In the present work, crack growth simulations are performed on components made of steel and aluminum alloy. The experimental damage evolution curve is taken from Hua and Socie, (1984) for steel and from Djebli et al., (2013) for aluminum alloy. The damage evolution obtained from Eq. (14) is calculated for both steel and aluminum. The value of

  6.1 is considered for steel and   3.2 is taken for aluminum to mimic the results. Figure 2 shows that the present model captures the damage evolution quite effectively. In the Fig. 3, the proposed model is compared with Wahab et al. (2001) model. Figure 3 shows that the present model provides good match with the experimental data and produces better results as compared to Wahab et al. (2001) model. There are two reasons for this observation. The first one is the exponential nature of damage evolution equation rather than power-law. The second one is that the proposed model uses a parameter  , which is determined from  vs N curve while Wahab et al. (2001) model does not use  vs N curve for evaluating any parameter. The details of the Wahab et al. (2001) model are given in Appendix. 3.1.2 Effect of stress ratio on material constants Three material constants are used in order to define the damage evolution law for a material subjected to high cycle fatigue. Out of these three constants, two constants (M and

 ) are obtained from S-N curve. It is well known that S-N curve depends on stress ratio (R). Therefore, the values of M and  will also change with the stress ratio. S-N curves for commonly used materials subjected to different stress ratios can be found in fatigue handbooks. Sometimes, the S-N curve at different values of R is not available for a material. In that case, ASME elliptical or modified Goodman criteria can be used to obtain the values of M and  . Third constant  is obtained from  vs N curve as explained earlier. The value of  is assumed constant with respect to the stress ratio in the present work. 4. STRESS STATE AT CRACK TIP

8

To predict the life of a cracked component, most commonly used fracture specimens are compact tension (CT), C-shaped tension (CST), single edge notch bending (SENB), single edge notch tension (SENT), double edge notch tension (DENT) and middle crack tension (MT). Experiments on these specimens show that the crack growth rate varies in these specimens. The reason is that the constraint levels are different in these specimens. Based on the crack growth rate behavior, these specimens are generally referred as high constraint (CT), intermediate constraint (CST, SENB) and low constraint (SENT, DENT, MT) specimens. The crack growth rate is maximum in high constraint specimens and minimum in low constraint specimens (Mehmanparast et al., 2014). It is further noticed that apart from the shape of the specimen, size of specimen, its thickness, magnitude of applied load, loading condition and crack length affect the state of stress at the crack tip. These conditions decide whether the crack front is under plane stress or plane strain. The assumption of plane stress/plane strain significantly affects the constraint at crack tip. Hence, the division of components based on geometry alone is not sufficient, the effect of loading conditions should also be incorporated. To capture the effect of constraint on stress state, stress triaxiality is used in the literature, which is defined as T 

H where  H is the hydrostatic component and  eq is the equivalent von eq

Mises stress. The effect of stress triaxiality is examined in the cracked and notched specimens. It is reported for cracked and notched specimens that the aforementioned definition of stress triaxiality is inadequate to predict the experimental behavior effectively. Therefore, the following attempts were made in the literature to capture the effect of constraint on stress state near the crack tip: a)

Use of alternative definitions of stress triaxiality. i.

T

1 (Liu and Murakami, 1998; Schafer et al., 2000; Tamura et al., 2009)  eq

where,  1 is the maximum principal stress. ii.

T

1   r      A   eq 

where,  r and   are the radial and circumferential stress components respectively. The value of A is taken 2 for plane stress and 3 for plane strain (Sun et al., 1991). 9

iii.

T  A Bp where, A and B are constants which depend on geometry alone. p represents the effective plastic strain rate (Bonora et al., 2005). 2

iv.

  f T   1  k  H  where, k is a material constant (Wang, 1995).   eq 

b) Reduce the critical value of damage parameter with respect to geometry (Yu et al., 2016). c) The specimens are divided into two regions i.e. high constraint and low constraint. Hence, two sets of material parameters are used for each region (Mehmanparast et al., 2014; Quintero and Mehmanparast, 2016). d) Use of different damage dissipation potential functions for different regions of triaxiality (Bao and Wierzbicki, 2004; 2005). Stress triaxiality defined in a(i) was introduced to capture the effect of constraint on state of stress for the brittle materials. However, the researchers have found it quite suitable for ductile materials also. Definitions used in a(ii, iii, iv) have been applied in the plastic region only. It is noted that a(i, ii, iv) do not depend on geometry of the problem whereas a(iii) captures the effect of geometry through geometry dependent constants A and B. The effect of loading was not considered in this equation. Moreover, the reduction in critical value of damage parameter with respect to geometry as suggested in (b), makes the solution geometry dependent, which is then applicable only for known shapes. Hence, generalization of this concept becomes difficult for problems with complex geometry. A similar problem arises with the division of material parameters based on the shape of specimen viz. high or low constraint. Furthermore, introducing different damage potentials for different regions of triaxiality is used only for the notched specimens so far. This section emphasizes that a crack growth problem is generally classified into high or low constraint problem based on its geometry. Since, loading conditions have a great influence on the constraints of the problem hence the stress triaxiality definition should not be decided only on the basis of geometry. Therefore, in order to predict the damage evaluation rate correctly, two definitions of stress triaxiality (Schafer et al., 2000) are employed in the present work. Out of these two definitions, one is used for high constraint problem T   H  eq  and other one is used for low constraint problem T  1  eq  . The

10

selection of triaxiality definition (i.e. high or low constraint) depends on the damage growth rate in a particular problem. The numerically obtained damage evolution rate or crack growth rate should be consistent with the experimental observations. The definition of the stress triaxiality function is chosen accordingly. To implement this methodology, a damage-based criterion is proposed that can distinguish whether the problem is a high or low constraint problem. The detailed implementation of this criterion is discussed in the algorithm under section 7.

5. EXTENDED FINITE ELEMENT METHOD Extended finite element method (XFEM) captures the discontinuity by adding enrichment functions using partition of unity in the standard finite element. In this study, extrinsic enrichment approach is used for displacement approximation. Four noded bilinear elements are used for performing the simulations. The displacement approximation at any point is given as (Patil et al., 2017) n

p

i 1

j 1

u(x)   Ni (x)ui   N j (x)  (x)  (x j )  d j std FEM approx

enrichment

(15) where, N (x) represents the standard shape functions of the finite element method, u i is the vector of standard degree of freedom,  (x) is the enrichment function, d j are the additional degree of freedoms corresponding to enriched part. To represent the crack in XFEM, two kinds of enrichment functions are used: one for the completely cracked elements (i.e. split elements) and one for the partially cracked elements (i.e. tip elements). The enrichment functions are selected based on the local behavior of problem. For an element completely intersected by the crack, Heaviside signed function is defined as,





 1 if x  x .n  0 H  x   else 1

(16) where, x is a Gauss point, x is the nearest point to x on the discontinuity and n is the unit normal drawn at x 11

For modeling the crack tip behavior, branch functions obtained from the analytical solution using linear elastic fracture mechanics theory, are used which are given as,

       (x)  1 ,  2 , 3 ,  4   r  cos ,sin , cos sin  ,sin sin   2 2 2 2   (17) where, polar coordinates (r , ) are measured from the crack tip. Therefore, the displacement approximation presented in Eq. (15) becomes, n

u(x)   Ni (x)ui  i 1

p



j split 1

N j (x)  H (x)  H (x j )  a j 

4

 N (x)   (x)   (x )b k

k

k

(18)

k tip 1

where, H (x) is the Heaviside function, a j are the additional degree of freedoms associated with Heaviside jump function,  (x) represents the branch function and b k are the additional degree of freedoms associated with branch function. 5.1 Discrete Equations The strong form of boundary value problem mentioned in Eq. (1) is presented in its weak form as,

 σ :  d    b. u d    t . u d 

(19)

t

By substituting Eq. (18) in the above equation, the discrete form is obtained a,

 B

T

σ d   NT b d   NT t d 

t

(20) Above equation can also be written as,

K u  f 

(21)

where, K is the global stiffness matrix, u represents the displacement vector and f is the force vector respectively.

 

e For a particular element, the elemental stiffness matrix K and elemental force vector

 f  is given as, h

 Kijuu  K ije   Kijau  Kijbu 

Kijua Kijaa Kijba

Kijub   Kijab  Kijbb 

(22a)

12



f h  fiu fia fib1 fib 2 fib3 fib 4



T

(22b) The submatrices and vectors that appear in the Eq. (22) are K ijpq 

  Bi

p



 CB T

q i

d ,

where,

p,

q

=

u,

a,

b

e

(23)

 Ni b d    Ni t d 

fiu 

e

i

(24) fia 

 Ni  H (x)  H (xi )  b d    Ni  H (x)  H (xi )  t d 

e

i

(25) fib 

 Ni   (x)  (xi )  b d    Ni

e

  (x)  (xi )  t d 

where,   1, 2,3, 4

i

(26) where, Bui , Bia , Bbi , Bbi  are the strain displacement matrices. 6. MESH REGULARIZATION Localization is a quite common problem in continuum damage mechanics, which occurs in the crack propagation. This localization leads to the mesh dependent results in crack growth problems. There are two major reasons by which mesh dependence arises (Murakami and Liu, 1995), (i) Stress singularity near the crack tip. (ii) Strain localization due to material softening. In the present work, crack is modeled using XFEM. It is clear from Eq. (5) that the damage evaluated at the Gauss points is a function of the state of stress at that Gauss point. As the mesh becomes finer, the position of Gauss point moves towards the crack tip, and the value of stress increases dramatically due to stress singularity considered in Eq. (17). Hence, the value of damage variable also increases on these Gauss points, which leads to mesh dependent results. Three regularization schemes (i.e. stress limitation method, nonlocal integral formulation and reduction in the critical value of damage parameter) proposed by Murakami and Liu (1995) are used in the present work to reduce the mesh dependency in the results. 13

6.1 Stress Limitation Method As pointed out in the last section, the value of stress increases dramatically as Gauss point moves towards the crack tip. For fine meshes, the stress state reaches to very high value at the Gauss points which are near the crack tip. Therefore, to reduce the mesh sensitivity in results, the state of stress has to be restricted to a realistic value. For this purpose, a stress limitation criterion is applied, which states that the value of stress measures like  1 ,  H ,

 eq cannot exceed the ultimate strength ( u ) of that material. If anyone of stress measures (  1 ,  H ,  eq ) at any Gauss point approaches the ultimate strength, the corresponding stress measure is taken as  u at that Gauss point. 6.2 Non-local Integral Formulation Bazant and Pijaudier-Cabot (1988) developed a weighted averaging scheme to diminish the localization. In a numerical sense, this formulation is based on the assumption that the crack growth criterion can not depend only on the failure of a single Gauss point. For a crack to grow, some region near the crack tip must be damaged. This assumption is in line with the experimental observations. For implementing this formulation, a circular area in front of the crack tip is considered. The center of this circle lies at lc distant from the crack tip as shown in Fig. 4. This circle is drawn in such a manner that it touches the crack tip. To obtain the weighted average value, a new damage parameter Dcp is established at the center of this circle. The value of Dcp is the weighted average of damage variable  of those Gauss points which lies inside the circle. Dcp is defined as

Dcp 

 ii i

 i i

 x gp  xc  exp and   x gp , xc   1.5 lc2  2  lc3  1

2

   

(27)

where, x gp are coordinates of the Gauss points and x c are the coordinates of the center of the circle. Similar to damage variable  , Dcp also lies between 0  Dcp  1. If the value of

Dcp exceeds its critical value ( DcpC ) , crack growth occurs. lc is a length scale parameter. It is demonstrated in literature that the value of lc can be approximately taken as the mesh size. Once the value of lc is selected for a mesh, this value becomes constant and provides

14

mesh independent results for relatively fine meshes (Murakami and Liu, 1995; Seabra, 2013; Duddu and Waisman, 2013; Pandey et al., 2017). 6.3 Reduction in the Critical Value of Damage Parameter In continuum damage mechanics, a Gauss point is considered fail when the value of the damage variable attains some critical value. Theoretically, the critical value of the damage variable is unity. However, in experiments, the critical value is found in the range of 0.2 to 0.5 (Lemaitre and Desmorat, 2005). It is shown in the literature that the mesh sensitivity can be reduced by decreasing the critical value of damage variables i.e.  C and DcpC . The parameters discussed in this section (i.e. lc ,  C and DcpC ) are the numerical parameters. To obtain the values of these parameters, simulated results are compared with the experiments. Thereafter, these values are kept constant for other geometries and other loading conditions. Researchers have shown that mesh insensitivity can be obtained by incorporating these regularization schemes (Wang and Waisman, 2016; Pandey et al., C 2017). The value of   0.99 is considered for all simulations and the values of other two

numerical parameters are given in the respective crack growth problem. 7. NUMERICAL IMPLEMENTATION In the present study, XFEM and continuum damage mechanics are combined together to estimate the fatigue life. The advantage of this combination is that a sharp crack can be easily modeled by employing XFEM using a relatively coarse mesh. This combined methodology also makes the crack growth analysis simple. To apply this procedure, a FE displacement approximation is enriched using XFEM. After applying boundary conditions, strain and stress components are calculated at each Gauss point. Since crack has been already defined by XFEM, so damage variable is used as a post-processer for the life assessment. The damage variable is not included in the constitutive relation, which helps in avoiding ill-conditioning of the stiffness matrix. The obtained values of stress or strain components are substituted in the damage evolution equation to calculate the fatigue life. As the damage reaches its critical value near an existing crack tip, the crack length is extended. Both plane stress and plane strain problems are solved in the present work. Forward Euler time integration scheme is employed for damage evolution for maintaining the simplicity of the algorithm. For reducing the mesh sensitivity, mesh regularization schemes are employed. The procedure for modeling the fatigue crack growth is explained 15

in the following algorithm. A schematic representation of the algorithm is also given in the form of a flowchart in Fig. 5.

Algorithm: XFEM implementation of the fatigue damage model. (i) For a given mesh and crack position, initialize damage variable   0 at each Gauss point. (ii) Perform elastic analysis using XFEM

Kumax  Fmax and Kumin  Fmin where, Fmin  R  Fmax . R is the stress ratio. Obtain σ max and σ min (iii) To apply jump in cycle, a small value of  N is assumed. Let  N  10 (iv) Checking for stress triaxiality definition Initialize N  0 while N  1000 for n  1 to nGauss point Evaluate 1,max ,  H ,max ,  eq,max ,  eq,min Compute the damage rate  from Eq. (5) Calculate incremental damage variable and update as, N

     N .   N

N N

         N

N

end for Compute Dcp from Eq. (27) Update cycle: N  N  N end while C if Dcp  0.01Dcp

T

 H ,max  eq ,max

T

 1,max  eq ,max

else

16

end if (v) Reinitialize N  0 (vi) Reinitialize damage variable   0 (vii) Fatigue analysis while a  a final Follow steps (ii) and (iii). C while Dcp  Dcp

for n  1 to nGauss point Evaluate 1,max ,  H ,max ,  eq,max ,  eq,min Compute the damage rate  from Eq. (5) Calculate incremental damage variable and update as, N

     N .  

N N

N

         N

N

end for Compute Dcp from Eq. (27) to apply non-local integral formulation. Update cycle: N  N  N end while Update crack length: a  a  a Compute the crack direction end while

7.1 Jump in Cycle The evolution of damage variable is given in Eq. (5). It can be seen from this equation that if the damage is evaluated for each cycle, the simulation becomes computationally more expensive. To reduce the effort in simulation, jump in cycle concept is quite popular. In this approach, it is assumed that damage remains constant for a block of loading cycles. In the present study, forward Euler scheme is applied to calculate the increment in damage variable for a given number of cycles. Though the forward Euler is not unconditionally stable. However, Zhang et al. (2012) demonstrated that forward Euler method can provide stable results for small increments of  N . Since the application of forward Euler is quite

17

straightforward hence, it is implemented in the present work to integrate the damage equation. 7.2 Location of a New Crack Tip To identify the crack direction for a new crack segment, few points ahead of the crack tip are selected along the circumference of a semi-circle as shown in Fig. 6. Then by computing the triaxiality on these points, the direction of new crack can be achieved. To compute the triaxiality on these new circumferential points, first the location of a particular circumferential point is identified in the element. Next, the value of stress triaxiality is evaluated at the circumferential point by performing the interpolation between the Gauss points of that element and circumferential point. Using this procedure, stress triaxiality is evaluated for all the points on the circumference. The point which has the maximum value of triaxiality provides the direction of crack growth. To improve the accuracy, slightly large and small radius semi-circles are employed for this purpose and then an averaged direction is taken for crack propagation. 7.3 Crack Propagation Strategy In the XFEM, as the crack advances, the standard finite element becomes an enriched element (tip or split element). Therefore, the type of element changes as the crack grows. The shape functions of enriched elements are not polynomial and hence standard Gauss quadrature is not applicable for such elements. However, by employing subtriangularization scheme, the Gauss quadrature can be applied in an enriched element. While implementing sub-triangularization with crack growth, the number of Gauss points and their position changes in an element. It can be seen from Eq. (5) that the damage is calculated at the Gauss points. With the continuous change in number and position of these Gauss points, an accurate evaluation of damage variable on these new Gauss points is necessary. Therefore, to transfer the data from old Gauss points to new ones, a data transfer scheme is adopted in this work. To implement the data transfer, all the Gauss points in the elements are grouped in old as well as in new configuration. Then by employing element wise interpolation in both the configurations, the proper data transfer is established. 8. NUMERICAL RESULTS AND DISCUSSION This section provides the implementation of the XFEM in conjunction with damage mechanics to evaluate the crack growth under constant amplitude fatigue loading. Standard fracture specimens (compact tension, four point bend, single edge notched tension and

18

middle crack tension) are selected for the fatigue crack growth studies. Specimen dimensions (in mm) and their geometric specifications are provided in Table 2 and Fig. 7 respectively. The specimens are made of steel and aluminum alloys. The tensile properties of these materials are listed in Table 3. As discussed in section 3, S-N curve and  vs N curve are required to extract the fatigue data for simulation. The availability of  vs N curve is not as common as S-N curve for different materials. Hence, the value of  is taken as 6.1 for steel and 3.2 for aluminum. These values are kept constant irrespective of the grade of the material. The effect of different values of  is also shown on the steel as well as aluminum. Material constants for the fatigue damage law are listed in Table 4. Table 2: Specimen geometry and dimensions Specimen

Material

L

H

B

ainitial

a final

60

10

10

35

9.1

-

-

CT

T10CuNiCr180 steel (Negru et al., 2013)

62.5

CT

Al A356-T6 alloy (Stephens, 1988)

88.9 69.6

SENB SENT MT

Al 2024-T351 alloy (Benachour et al., 2010) Al 7005 alloy (Ma et al., 2006)

25

10

10

-

-

90

108

10

45

60

25

2.95

2

13

Al 6063-T6 alloy (Kumar and Garg, 1985) 87.5

Table 3: Tensile properties for various materials E (GPa)



σu (MPa)

205.68

0.30

891.44

Al A356-T6 alloy (Stephens, 1988)

70

0.33

289

Al 2024-T351 alloy (Benachour et al., 2010)

74

0.33

477

Al 7005 alloy (Ma et al., 2006)

72

0.33

350

70.632

0.33

224.85

Material T10CuNiCr180 steel (Negru et al., 2013)

Al 6063-T6 alloy (Kumar and Garg, 1985)

Table 4: Material constants for fatigue damage law of different materials Material T10CuNiCr180 steel Al A356-T6 alloy

Stress ratio (R)

M





0.1

3.3884×10-15

3.68

6.1

0.1

2.2387×10-19

5.70

3.2

11.6

3.2

-30

0.5

1×10

19

Al 2024-T351 alloy

0.1

1.2023×10-30

9.75

3.2

Al 7005 alloy

0.5

5.0119×10-54

21.82

3.2

Al 6063-T6 alloy

0.1

3.9811×10-41

15.68

3.2

8.1 Fatigue Crack Growth in CT Specimen Two different studies are presented in this sub-section. In the first study, a CT specimen made of steel is used for performing the mesh sensitivity and parametric analysis. In the second study, a CT specimen made of aluminum is chosen to show the effect of different stress ratios on the crack growth evolution. 8.1.1 T10CuNiCr180 steel CT specimen Negru et al. (2013) performed the experimental crack growth on a CT specimen made of T10CuNiCr180 steel subjected to 10 kN. The crack growth study was conducted for the stress ratio of R  0.1 . The S-N curve was also provided by Negru et al. (2013). Material constants for the fatigue damage law obtained from this S-N curve are listed in Table 4. Dimensions of the specimen and material constants are detailed in Tables 2-4. The boundary conditions for CT specimen are shown in Fig. 7. A 79×79 mesh is taken to C discretize the domain. The value of numerical parameters i.e. lc and Dcp are taken as 0.58

mm and 0.15 respectively to mimic the experimental results. The crack growth results presented in Fig. 8 show that the XFEM results agree well with the experimental data. The stress distribution in CT specimen is shown in Fig. 9 for increase in crack length. Apart from the validation, mesh sensitivity analysis and effect of ‘  ’ on the crack growth is also performed. (a) Mesh sensitivity analysis A mesh sensitivity analysis is performed to establish the importance of mesh regularization scheme. For this, the same study as discussed above is performed on various meshes. The value of lc (0.58 mm) is selected for 65×65 mesh and then it remains constant for fine meshes. The results of crack length vs number of cycle is plotted in Fig. 10 for four different meshes. It can be observed from these results that the use of regularization scheme reduces the mesh sensitivity effectively. To study the influence of arbitrary mesh on fatigue crack growth, the CT specimen is discretized with an unstructured mesh as shown in Fig. 11. The results obtained for unstructured as well as structured mesh are plotted with experimental data in Fig. 12.

20

Figure 12 shows that fatigue life is found nearly same with small deviation between unstructured and structured meshes. (b) Effect of ‘  ’ on the crack growth for steel The consequences of the material parameter (  ) are demonstrated by performing simulations on a CT specimen subjected to 10 kN at R  0.1 . The other material and numerical parameters are kept same as discussed previously. Four values of ‘  ’ are selected and the effect of ‘  ’ is shown on both damage evolution curve and crack growth curve. From the results presented in Fig. 13, it is observed that for the steel, the variation in ‘  ’ does not affect the crack growth results much. 8.1.2 A356-T6 aluminum alloy CT specimen This example is used for showing the capability of the present methodology to model the crack growth for different stress ratios. This study also shows that the numerical parameters obtained from one simulation can be further used for the simulation of other studies on same material. To study the effect of stress ratios, Stephens (1988) performed the experimental crack growth on CT specimen (A356-T6 aluminum alloy) at R  0.1 and R  0.5 . For R  0.1 and R  0.5 , the maximum applied load is taken as 4144.44 N and 3230 N respectively. The initial and final crack lengths are 23 mm and 38 mm respectively for R  0.1 whereas these values are 30 mm and 45 mm respectively for R  0.5 (Stephens, 1988). Tajiri et al. (2015) provided the S-N curve for A356-T6 aluminum alloy and the material constants for the fatigue damage law obtained from this S-N curve are provided in Table 4. Dimensions of the specimen along with material constants are provided in Tables 2-4. The mesh size is taken as 87×69 for this problem. C The values of lc  0.6 mm and Dcp  0.4 are obtained by comparing the simulated

results with experimental results for R  0.1 . Then these numerical parameters are used to predict the crack growth for R  0.5 . The crack growth results obtained by XFEM simulations along with the available experiment results are presented in Fig. 14. The results obtained from simulations are found in excellent agreement with the experimental ones. This study concludes that the present methodology can be used to obtain the results for different stress ratios. 8.2 Fatigue Crack Growth in SENB Specimen

21

This is another example which shows that the numerical parameters obtained from one simulation can further be applied to the simulate other studies made of same material. A four-point bend specimen made of 2024-T351 aluminum alloy is selected. The SENB specimen is subjected to 1285 N fatigue load at R  0.1 . The initial and final crack length is taken as 3.21 mm and 7.42 mm respectively. The material constants for fatigue damage law listed in Table 4, are obtained from the S-N curve of 2024-T351 aluminum alloy, provided in a handbook (MIL-HDBK-5J, 2003). Dimensions of the specimen and material constants are given in Tables 2-4. The distance between the points of force application is 21 mm. 249 elements are taken in length direction and 49 elements are used C in vertical direction to discretize the SENB specimen. The value of lc and Dcp are taken

as 0.10 mm and 0.107 respectively. The values of numerical parameters are selected such that they can reproduce the experimental results. To ensure the applicability of the numerical parameters, another simulation is performed on same component subjected to 1145 N. The initial and final crack lengths for this study are taken as 3.35 mm and 7.42 mm respectively. The other parameters are same as in the previous study. The experimental results obtained by Benachour et al. (2010) are compared with the XFEM results in Fig. 15. Figure 15 shows that the crack growth results obtained by XFEM simulation are in accordance with the experiment results. The effect of ‘  ’ on the crack growth for aluminum and the consequences of jump in cycle on fatigue crack growth are also studied for this problem. (a) Effect of ‘  ’ on the crack growth for aluminum alloy The effect of the material parameter (  ) is demonstrated by performing the simulations on a four-point bend specimen subjected to 1285 N at R  0.1 . The other material and numerical parameters are kept same as discussed above. Five values of ‘  ’ are selected and the effect of ‘  ’ is shown in Fig. 16 for both damage evolution and crack growth curve. It is observed that crack growth results vary with the variation in ‘  ’ for aluminum alloy. For steel, the damage evolution and crack growth results do not vary much as shown in Fig. 13. The reason can be found from damage evolution equation as shown in Eq. (16). For   0 , this equation provides a straight line. As the value of ‘  ’ increases, the curve starts to shift downwards as can be seen in Fig. 13(a) and Fig. 16(a). Since the nature of equation in exponential. Hence, for the small values of ‘  ’, the larger variation in curves

22

can be expected as compared to higher values of ‘  ’ as can be seen in Fig. 17. This explanation is also true for the crack growth curves (Fig. 13(b) and Fig. 16(b)). (b) Effect of jump in cycle The effect of jump in cycle (  N ) is demonstrated by performing simulations on a fourpoint bend specimen subjected to 1285 N at R  0.1 . The other material and numerical parameters are kept same as discussed above. The results obtained for four values of  N are presented in Fig. 18. It is observed that for small values of  N , the variation in crack growth results is negligible. 8.3 Fatigue Crack Growth in SENT Specimen A SENT specimen made of 7005 aluminum alloy and subjected to 29.63 MPa at R  0.5 is selected for fatigue crack growth simulation. For this study, Ma et al. (2006) provided the experimental fatigue crack growth results. The S-N curve for 7005 aluminum alloy is obtained from Shih and Chung (2003). Material constants for the fatigue damage law is obtained from this S-N curve. These material constants are given in Table 4. Dimensions of the specimen and material constants are given in Tables 2-4. A 45×53 mesh size is used to discretize the SENT specimen.

lc and DcpC are taken as 1.0 mm and 0.22 respectively to reproduce the experimental results. The crack growth results obtained by XFEM show a good match with experiment as shown in Fig. 19. Stress distribution in SENT specimen with growing crack is also shown in Fig. 20.

(a) Effect of increment in crack length To consider the effect of crack length increment on fatigue crack growth, four values of

a are applied on the SENT specimen. For this case, only the incremental crack length is varied and all the other parameters are kept same as discussed above. The fatigue life obtained for four values of a are plotted in Fig. 21. Figure shows that the slight variation can be seen for small crack growth increments, which further reduces with the increase in crack growth increment size. 8.4 Fatigue Crack Growth in MT Specimen Kumar and Garg (1985) performed the experimental crack growth study on a MT specimen (6063-T6 aluminum alloy) subjected to 122.6 MPa at R  0.1 . Nanninga (2008)

23

provided the S-N curve for 6063-T6 aluminum alloy and material constants for the fatigue damage law obtained from this S-N curve is reported in Table 4. Dimensions of the specimen along with material constants are given in Tables 2-4. The mesh size is taken as C 39×139 for this problem. The values of lc and Dcp are obtained as 1.0 mm and 0.575

respectively. A comparison of crack growth results obtained by XFEM with the experimental results is shown in Fig. 22. Figure 22 shows that the life of a cracked MT specimen can be effectively computed by the present approach. 8.5 Significance of Proposed Damage based Criterion This study is conducted to show the importance of proposed damage based criterion that distinguish the problem as high or low constraint problem. For this, two studies (CT specimen as discussed in section 8.1.2 and SENT specimen as discussed in section 8.3) are performed by considering both definitions one by one. These two specimens are specifically chosen because traditionally, CT is considered as a high constraint specimen and SENT is considered as a low constraint specimen. The obtained results are presented in Fig. 23(a) for CT and in Fig. 23(b) for SENT. It can be seen from the simulated results that for CT specimen, the experimental crack growth curve could be reproduced when the simulated results are obtained by considering it as low constraint while SENT has to be considered as high constraint problem to mimic the experiment results. The reason of this behavior is the amplitude of applied load that affects the state of stress near the crack tip. Therefore, this study concludes that the specimen should be considered as high or low constraint based on the damage evolution rate near the crack tip. Hence, the proposed damage based criterion is an obligatory step which should be incorporated in the studies to predict the experiment behavior correctly. 8.6 Mixed Mode Fatigue Crack Growth To further demonstrate the capability of proposed methodology, a mixed mode problem (CT specimen with hole) is solved. The CT specimen is subjected to 10 kN force at R  0.1 . The dimensions of the CT specimen are shown in Fig. 24. The material is taken as

T10CuNiCr180 steel. Hence, the material properties, constants for fatigue damage law and numerical parameters remain same as discussed in section 8.1.1. The fatigue life of CT specimen with hole predicted by the present methodology is presented in Fig. 25. In Fig. 26, various stress plots for growing crack are also provided to show the diverted crack path in the presence of hole. 24

8.7 Fatigue Crack Growth under Pure Mode-II Loading A SENT specimen subjected to 42 MPa shear load at R  0.1 is studied for solving a pure mode-II problem. The dimensions of the SENT specimen are provided in Fig. 27. The material is considered as 7005 aluminum alloy. Hence, all material constants and parameters are taken same as in section 8.3. A 50×50 mesh is taken to discretize the SENT specimen. The fatigue life and crack path obtained by combined methodology are presented in Fig. 28 and Fig. 29 respectively. The examples solved in section 8.6 and 8.7 establishes the capability of proposed framework to solve the fatigue crack growth under different types of loading environment. 8.8 Fatigue Crack Growth in Turbine Disc The proposed framework is finally used to estimate the remaining fatigue life of a cracked turbine disc. A schematic illustration of a turbine disc is shown in Fig. 30 (a) (Kumar et al., 2018). An initial crack of 0.9 mm at 51° is assumed at each hole on the disc so that a quarter model can be used to simulate the fatigue crack growth in the turbine disc. The quarter model along with boundary conditions is shown in Fig. 30 (b). A 212 MPa distributed load at R  0.1 is applied on the outer periphery of turbine disc. The material for disc is taken as T10CuNiCr180 steel and hence, the material properties and constants for fatigue damage law are taken from Table 3 and Table 4. For T10CuNiCr180 steel, the numerical parameter are considered same as in section 8.1.1. The remaining fatigue life of cracked turbine disc obtained by combined methodology is presented in Fig. 31. This study confirms that the proposed framework can also be used for predicting the crack growth in real life problems. 9. CONCLUSIONS AND FUTURE SCOPE In this work, a new methodology is proposed based on continuum damage mechanics and XFEM for fatigue crack growth simulations. A new fatigue damage law is proposed to estimate the fatigue crack growth life. A criterion based on the damage evolution is proposed to select the definition of stress triaxiality. The damage law is implemented in the XFEM framework to efficiently simulate the crack growth. To make the results mesh independent, mesh regularization approaches are adopted. Forward Euler time marching scheme is used for the integration of damage variable to keep the algorithm simple. The proposed methodology is validated by comparing the simulated results with the previously published experimental data. Four standard fracture specimens (i.e. compact tension, four

25

point bend, single edge notched tension and middle crack tension specimen made of steel and aluminum alloys) are used to simulate the fatigue crack growth. The effect of stress ratio and other load values is investigated on the results. It is demonstrated that the numerical parameters obtained from a particular simulation can be further used for other loading conditions. Several parametric studies are conducted to find the consequences of the parameters on the fatigue life. This work shows that fatigue crack growth life can be easily determined by present methodology. In future, this methodology can be used to determine fatigue life under variable amplitude loading, fretting fatigue, and overload conditions. Apart from various loading scenarios, uncertainty and sensitivity analysis (Vu-Bac et al., 2016; Hamdia et al., 2017; 2018) can be done for proposed damage model to observe the effect of various input parameters on fatigue crack growth. The proposed framework can also be extended for 3D so that the damage based fatigue crack growth approach can be used for solving the real life problems.

APPENDIX A.1 Wahab et al. (2001) damage model To evaluate the life of component subjected to fatigue load, Wahab et al. (2001) presented a damage model. The multiaxial form of this model is given as,  m

d   max,eq   min,eq   A  dN 1   

Rv /2

(A1) where, A,  , m are material constants and Rv is the stress triaxiality function which is defined as,

  2 Rv  1     3 1  2   H ,max  3   eq ,max 

2

(A2) A.1.1 Determination of the material constants The above-written fatigue damage law can be given under uniaxial condition as,

26

 m

d     A  dN  1  

(A3) Arranging the above equation and applying the limits 

 1   

 m

N

d   A   

0

 m

dN

o

(A4)





1   m 1 1      m 1



 0

 A   

 m

N

(A5) 





1   m 1  m  1  A    N 1      m 1

(A6) The component will reach its final life when damage variable reaches unity. So substituting   1 and N  N f in the above equation, the final life of the component will be Nf 

1 A   m  1  

 m

(A7) Taking logs of both sides

  1 log  N f      m  log     log   A   m  1    (A8) Therefore, the value of A and  can be obtained by comparing with S-N curve. To determine the relationship between damage variable and number of cycles, rearranging Eq. (A6)

1   

  m1

 1  A   m  1  

 m

N

(A9) Substituting Eq. (A7) in Eq. (A9) 1

 N   m1   1  1    Nf 

(A10) 27

Eq. (A10) represents the damage evolution with normalized fatigue life. ACKNOWLEDGMENT This work is financially supported by Defence Metallurgical Research Laboratory (DMRL), Defence Research and Development Organisation (DRDO), Hyderabad, through grant no. DGNSM/04/4019/DMR305/CARS/XFEM dated November 24, 2014. REFERENCES Areias, P. and Rabczuk, T. (2017): Steiner-point free edge cutting of tetrahedral meshes with applications in fracture, Finite Elements in Analysis and Design, Vol. 132, pp. 27-41. Areias, P., Msekh, M.A. and Rabczuk, T. (2016): Damage and fracture algorithm using the screened Poisson equation and local remeshing, Engineering Fracture Mechanics, Vol. 158, pp. 116-143. Areias, P., Rabczuk, T. and Dias-da-Costa, D. (2013): Element-wise fracture algorithm based on rotation of edges, Engineering Fracture Mechanics, Vol. 110, pp. 113-137. Areias, P., Reinoso, J., Camanho, P.P., de Sá, J.C. and Rabczuk, T. (2018): Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation, Engineering Fracture Mechanics, Vol. 189, pp. 339-360. Ashcroft, I.A., Shenoy, V., Critchlow, G.W. and Crocombe, A.D. (2010): A comparison of the prediction of fatigue damage and crack growth in adhesively bonded joints using fracture mechanics and damage mechanics progressive damage methods, The Journal of Adhesion, Vol. 86(12), pp. 1203-1230. Bansal, M., Singh, I.V., Mishra, B.K., Sharma, K. and Khan, I.A. (2017a): A two-scale stochastic framework for predicting failure strength probability of heterogeneous materials, Composite Structures, Vol. 179, pp. 294-325. Bansal, M., Singh, I.V., Mishra, B.K., Sharma, K. and Khan, I.A. (2017b): A stochastic XFEM model for the tensile strength prediction of heterogeneous graphite based on microstructural observations, Journal of Nuclear Materials, Vol. 487, pp. 143-157. Bao, Y. and Wierzbicki, T. (2004): On fracture locus in the equivalent strain and stress triaxiality space, International Journal of Mechanical Sciences, Vol. 46(1), pp. 81-98. Bao, Y. and Wierzbicki, T. (2005): On the cut-off value of negative triaxiality for fracture, Engineering Fracture Mechanics, Vol. 72(7), pp. 1049-1069. Bazant, Z.P. and Pijaudier-Cabot, G. (1988): Nonlocal continuum damage, localization instability and convergence, Journal of Applied Mechanics, Vol. 55(2), pp. 287-293. Beese, S., Loehnert, S. and Wriggers, P. (2017): 3D ductile crack propagation within a polycrystalline microstructure using XFEM, Computational Mechanics, pp. 1-18. Benachour, M., Hadjoui, A., Benguediab, M. and Benachour, N. (2010): Effect of the amplitude loading on fatigue crack growth, Procedia Engineering, Vol. 2(1), pp. 121-127.

28

Bhamare, S., Eason, T., Spottswood, S., Mannava, S. R., Vasudevan, V. K. and Qian, D. (2014): A multi-temporal scale approach to high cycle fatigue simulation, Computational Mechanics, Vol. 53(2), pp. 387-400. Bonora, N., Gentile, D., Pirondi, A. and Newaz, G. (2005): Ductile damage evolution under triaxial state of stress: theory and experiments, International Journal of Plasticity, Vol. 21(5), pp. 981-1007. Chaboche, J.L. and Lesne, P.M. (1988): A non‐linear continuous fatigue damage model, Fatigue & Fracture of Engineering Materials & Structures, Vol. 11(1), pp. 1-17. Djebli, A., Aid, A., Bendouba, M., Amrouche, A., Benguediab, M. and Benseddiq, N. (2013): A non-linear energy model of fatigue damage accumulation and its verification for Al-2024 aluminum alloy, International Journal of Non-Linear Mechanics, Vol. 51, pp. 145-151. Duddu, R. and Waisman, H. (2013): A nonlocal continuum damage mechanics approach to simulation of creep fracture in ice sheets, Computational Mechanics, Vol. 51(6), pp. 961-974. Ghorashi, S.S., Valizadeh, N., Mohammadi, S. and Rabczuk, T. (2015): T-spline based XIGA for fracture analysis of orthotropic media, Computers & Structures, Vol. 147, pp. 138-146. Hamdia, K.M., Ghasemi, H., Zhuang, X., Alajlan, N. and Rabczuk, T. (2018): Sensitivity and uncertainty analysis for flexoelectric nanostructures, Computer Methods in Applied Mechanics and Engineering, Vol. 337, pp. 95-109. Hamdia, K.M., Silani, M., Zhuang, X., He, P. and Rabczuk, T. (2017): Stochastic analysis of the fracture toughness

of

polymeric

nanoparticle

composites

using polynomial

chaos

expansions, International Journal of Fracture, Vol. 206(2), pp. 215-227. Hua, C.T. and Socie, D.F. (1984): Fatigue damage in 1045 steel under constant amplitude biaxial loading, Fatigue & Fracture of Engineering Materials & Structures, Vol. 7(3), pp. 165-179. Kumar, M., Ahmad, S., Singh, I. V., Rao, A. V., Kumar, J. and Kumar, V. (2018): Experimental and numerical studies to estimate fatigue crack growth behavior of Ni-based super alloy, Theoretical and Applied Fracture Mechanics, Vol. 96, pp. 604-616. Kumar, R. and Garg, S.B.L. (1985): Influence of applied stress ratio on fatigue crack growth in 6063-T6 aluminium alloy, International Journal of Pressure Vessels and Piping, Vol. 20(1), pp. 65-76. Lee, C.H., Chang, K.H. and Van Do, V.N. (2016): Modeling the high cycle fatigue behavior of Tjoint fillet welds considering weld-induced residual stresses based on continuum damage mechanics, Engineering Structures, Vol. 125, pp. 205-216. Lee, C.S., Kim, M.H., Lee, J.M. and Mahendran, M. (2011): Computational study on the fatigue behavior of welded structures, International Journal of Damage Mechanics, Vol. 20(3), pp. 423-463. Lemaitre, J. and Desmorat, R. (2005): Engineering damage mechanics: ductile, creep, fatigue and brittle failures, Springer Science & Business Media, New York, USA. Lemaitre, J., Sermage, J.P. and Desmorat, R. (1999): A two scale damage concept applied to fatigue, International Journal of Fracture, Vol. 97(1), pp. 67-81.

29

Lestriez, P., Bogard, F., Shan, J.L. and Guo, Y.Q. (2007): Damage evolution on mechanical parts under cyclic loading, In AIP Conference Proceedings, Vol. 908(1), pp. 1389-1394. Liu, Y. and Murakami, S. (1998): Damage localization of conventional creep damage models and proposition of a new model for creep damage analysis, JSME International Journal Series A, Vol. 41(1), pp. 57-65. Ma, S., Zhang, X.B., Recho, N. and Li, J. (2006): The mixed-mode investigation of the fatigue crack in CTS metallic specimen, International Journal of Fatigue, Vol. 28(12), pp. 1780-1790. Mehmanparast, A., Davies, C.M., Webster, G.A. and Nikbin, K.M. (2014): Creep crack growth rate predictions in 316H steel using stress dependent creep ductility, Materials at High Temperatures, Vol. 31(1), pp. 84-94. MIL-HDBK-5J (2003): Metallic Materials and Elements for Aerospace Vehicle Structures, U.S. Department of Defense, United States of America. Murakami,

S. and Liu,

Y.

(1995): Mesh-dependence in local approach to creep

fracture. International Journal of Damage Mechanics, Vol. 4(3), pp. 230-250. Nanninga, N.E. (2008): High cycle fatigue of AA6082 and AA6063 aluminum extrusions, Doctoral dissertation, Michigan Technological University, United States of America. Negru, R., Marsavina, L., Muntean, S. and Pasca, N. (2013): Fatigue behaviour of stainless steel used for turbine runners, In Advanced Engineering Forum, Vol. 8, pp. 413-420 Oller, S., Salomón, O. and Oñate, E. (2005): A continuum mechanics model for mechanical fatigue analysis, Computational Materials Science, Vol. 32(2), pp. 175-195. Paas, M.H.J.W., Schreurs, P.J.G. and Brekelmans, W.A.M. (1993): A continuum approach to brittle and fatigue damage: theory and numerical procedures, International Journal of Solids and Structures, Vol. 30(4), pp. 579-599 Pandey, V.B., Singh, I.V., Mishra, B.K., Ahmad, S., Rao, A.V. and Kumar, V. (2017): Creep crack simulations

using

continuum

damage

mechanics

and

extended

finite

element

method, International Journal of Damage Mechanics, DOI. 1056789517737593. Patil, R.U., Mishra, B.K. and Singh, I.V. (2017): A new multiscale XFEM for the elastic properties evaluation of heterogeneous materials, International Journal of Mechanical Sciences, Vol. 122, pp. 277-287. Peerlings, R.H.J., Brekelmans, W.A.M., De Borst, R. and Geers, M.G.D. (2000): Gradientenhanced damage modelling of high-cycle fatigue, International Journal for Numerical Methods in Engineering, Vol. 49(12), pp. 1547-1569. Poh, L.H. and Sun, G. (2017): Localizing gradient damage model with decreasing interactions, International Journal for Numerical Methods in Engineering, Vol. 110(6), pp. 503-522. Qian, Z., Takezono, S. and Tao, K. (1996): A nonlocal damage mechanics approach to high temperature fatigue crack growth, Engineering Fracture Mechanics, Vol. 53(4), pp. 535-543. Quintero, H. and Mehmanparast, A. (2016): Prediction of creep crack initiation behaviour in 316H stainless steel using stress dependent creep ductility, International Journal of Solids and Structures, Vol. 97, pp. 101-115.

30

Rabczuk, T. and Belytschko, T. (2007): A three-dimensional large deformation meshfree method for arbitrary evolving cracks, Computer Methods in Applied Mechanics and Engineering, Vol. 196(29), pp. 2777–2799. Rabczuk, T., Zi, G., Bordas, S. and Nguyen-Xuan, H. (2010): A simple and robust threedimensional cracking-particle method without enrichment, Computer Methods in Applied Mechanics and Engineering, Vol. 199(37-40), pp. 2437-2455. Ren, H., Zhuang, X. and Rabczuk, T. (2017): Dual-horizon peridynamics: a stable solution to varying horizons, Computer Methods in Applied Mechanics and Engineering, Vol. 318, 7pp. 62-782. Ren, H., Zhuang, X., Cai, Y. and Rabczuk, T. (2016): Dual‐horizon peridynamics, International Journal for Numerical Methods in Engineering, Vol. 108(12), pp. 1451-1476. Schafer, B.W., Ojdrovic, R.P. and Zarghamee, M.S. (2000): Triaxiality and fracture of steel moment connections, Journal of Structural Engineering, Vol. 126(10), pp. 1131-1139. Seabra, M.R., Šuštarič, P., Cesar de Sa, J.M. and Rodič, T. (2013): Damage driven crack initiation and propagation in ductile metals using XFEM, Computational Mechanics, Vol. 52, pp. 161179. Shedbale, A.S., Sharma, A.K., Singh, I.V. and Mishra, B.K. (2016): Modeling and Simulation of Metal Forming Processes by XFEM, Applied Mechanics and Materials, Vol. 829, pp. 41-45. Shedbale, A.S., Singh, I.V. and Mishra, B.K. (2013): Nonlinear simulation of an embedded crack in the presence of holes and inclusions by XFEM, Procedia Engineering, Vol. 64, pp. 642-651. Shenoy, V., Ashcroft, I.A., Critchlow, G.W. and Crocombe, A.D. (2010): Fracture mechanics and damage mechanics based fatigue lifetime prediction of adhesively bonded joints subjected to variable amplitude fatigue, Engineering Fracture Mechanics, Vol. 77(7), pp. 1073-1090. Shih, T.S. and Chung, Q.Y. (2003): Fatigue of as-extruded 7005 aluminum alloy, Materials Science and Engineering: A, Vol. 348(1), pp. 333-344. Singh, S.K., Singh, I.V., Bhardwaj, G. and Mishra, B.K. (2018): A Bézier extraction based XIGA approach for three-dimensional crack simulations, Advances in Engineering Software, Vol. 125, pp. 55-93 Singh, S.K., Singh, I.V., Mishra, B.K., Bhardwaj, G. and Bui, T.Q. (2017): A simple, efficient and accurate Bézier extraction based T-spline XIGA for crack simulations, Theoretical and Applied Fracture Mechanics, Vol. 88, pp. 74-96. Stephens, R.I. (1988): Fatigue and fracture toughness of A356-T6 cast aluminum alloy, Society of Automotive Engineers, 400 Commonwealth Drive, Warrendale, PA 15096, Vol. SP-760. Sun, J., Deng, Z.J. and Tu, M.J. (1991): Effect of stress triaxiality levels in crack tip regions on the characteristics of void growth and fracture criteria, Engineering Fracture Mechanics, Vol. 39(6), pp. 1051-1060. Tajiri, A., Uematsu, Y., Kakiuchi, T. and Suzuki, Y. (2015): Fatigue crack paths and properties in A356-T6 aluminum alloy microstructurally modified by friction stir processing under different conditions, Frattura ed Integrita Strutturale, Vol. 34, pp. 347-354.

31

Tamura, H., Sasaki, E., Yamada, H., Katsuchi, H. and Chanpheng, T. (2009): Involvements of stress triaxiality in the brittle fracture during earthquakes in steel bridge bents, International Journal of Steel Structures, Vol. 9(3), pp. 241-252. Van Do, V.N., Lee, C.H. and Chang, K.H. (2015): High cycle fatigue analysis in presence of residual stresses by using a continuum damage mechanics model, International Journal of Fatigue, Vol. 70, pp. 51-62. Vu-Bac, N., Lahmer, T., Zhuang, X., Nguyen-Thoi, T. and Rabczuk, T. (2016): A software framework

for

probabilistic

sensitivity

analysis

for

computationally

expensive

models, Advances in Engineering Software, Vol. 100, pp. 19-31. Wahab, M.A., Ashcroft, I.A., Crocombe, A.D. and Shaw, S.J. (2001): Prediction of fatigue thresholds

in

adhesively

bonded

joints

using

damage

mechanics

and

fracture

mechanics, Journal of Adhesion Science and Technology, Vol. 15(7), pp. 763-781. Wang, T.J. (1995): An engineering approach to remove the specimen geometry constraint dependence of elastic-plastic fracture toughness, Engineering Fracture Mechanics, Vol. 51(5), pp. 701-706. Wang, Y. and Waisman, H. (2016): From diffuse damage to sharp cohesive cracks: A coupled XFEM framework for failure analysis of quasi-brittle materials, Computer Methods in Applied Mechanics and Engineering, Vol. 299, pp. 57-89. Xiao, Y.C., Li, S. and Gao, Z. (1998): A continuum damage mechanics model for high cycle fatigue, International Journal of Fatigue, Vol. 20(7), pp. 503-508. Xu, Y. and Yuan, H. (2009): On damage accumulations in the cyclic cohesive zone model for XFEM analysis of mixed-mode fatigue crack growth, Computational Materials Science, vol. 46(3), pp. 579-585. Yu, F., Ben Jar, P.Y., Hendry, M.T. (2016): Stress triaxiality effect on damage evolution of highstrength steels, 24th International Congress of Theoretical and Applied Mechanics (XXIV ICTAM), 21-26 August, Montreal, Canada. Zhan, Z., Hu, W., Li, B., Zhang, Y., Meng, Q. and Guan, Z. (2017): Continuum damage mechanics combined with the extended finite element method for the total life prediction of a metallic component, International Journal of Mechanical Sciences, Vol. 124, pp. 48-58. Zhang, T., McHugh, P.E. and Leen, S.B. (2012): Finite element implementation of multiaxial continuum damage mechanics for plain and fretting fatigue, International Journal of Fatigue, Vol. 44, pp. 260-272.

32

t

Ω

u

u

t

c

Fig. 1: Schematic representation of homogeneous continuum cracked body

1.0

Damage variable ()

0.8

Steel 1045 (Exp., Hua and Socie, 1984) Al 2024 (Exp., Djebli et al., 2013) Present damage model (Steel 1045) Present damage model (Al 2024)

0.6

0.4 



0.2

0.0 0.0

0.2

0.4

0.6

0.8

Normalized number of cycle (N/Nf)

Fig. 2: Damage evolution curve for steel and aluminum alloy

33

1.0

1.0

Damage variable ()

0.8

Steel 1045 (Exp., Hua and Socie, 1984) Present damage model Wahab et al. (2001) model

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Normalized number of cycles (N/Nf)

(a) 1.0

Al 2024 (Exp., Djebli et al., 2013) Present damage model Wahab et al. (2001) model

Damage variable ()

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Normalised number of cycles (N/Nf)

(b) Fig. 3: Comparison of damage evolution curves for present damage model and Wahab et al. (2001) damage model with experimental data (a) Steel and (b) Aluminum alloy

34

lc

Dcp Gauss point

Crack Tip

Fig. 4: Representation of Gauss points (red colored star) for the calculation of D cp

35

Initialize mesh, crack,   0 Elastic solution Variables σij ,max , σij ,min are known. Assume N . Initialize N  0 Evaluate 1,max , H ,max , eq,max , eq,min Compute

N

   ,

N N

  , Dcp

Update cycle N  N  N

No

Check if N  1000 Yes

Yes

T

No

Check if Dcp  0.01DcpC

H  eq

T

1  eq

Reset N  0 ,   0 Elastic solution Variables σij ,max , σij ,min are known. Evaluate 1,max , H ,max , eq,max , eq,min Compute

N

   ,

N N

  , Dcp

Update cycle N  N  N

No

Check if Dcp  DcpC Yes Update crack length If a  a final

No

Yes End Fig. 5: Flowchart of high cycle fatigue crack growth solution procedure

36

Crack

lc

Split element

Tip element

Gauss points Circumferential points

Fig. 6: Selection of a point to find crack growth direction

37

P

H a

P

P W

B

L

(a) P/2

P/2 X

H a

2L

(b)

H a

2L 2a

P

0.5H

L

2H

(c)

(d)

Fig. 7: Schematic illustrations of test specimens: (a) CT (b) SENB (c) SENT (d) MT

38

35

Exp. (Negru et al., 2013) XFEM (curve fit)

Crack length (mm)

30

25

20

15

10 0

50,000

100,000

150,000

200,000

250,000

Number of cycles

Fig. 8: A comparison of experimental and XFEM fatigue crack growth curves for CT specimen

39

(a)

(b)

(c)

(d)

Fig. 9: Stress distribution



yy ,max

 in CT specimen for growing crack at (a) a = 10.25 mm (b) a =

17.50 mm (c) a = 23.50 mm (d) a = 31.50 mm

40

65x65 mesh 69x69 mesh 75x75 mesh 79x79 mesh

35

Crack length (mm)

30

25

20

15

10 0

50,000

100,000

150,000

200,000

250,000

300,000

Number of cycles

Fig. 10: Mesh sensitivity analysis to show the effect of mesh regularization schemes

Fig. 11: Discretization of CT specimen through unstructured mesh

41

35 Exp. (Negru et al., 2013) Unstructured mesh Structured mesh

Crack length (mm)

30

25

20

15

10 0

50,000

100,000

150,000

200,000

250,000

Number of cycles

Fig. 12: Effect of unstructured and structured mesh on fatigue crack growth

42

1.0

Damage variable ()

0.8

   

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Normalized number of cycles (N/Nf)

(a)    

35

Crack length (mm)

30

25

20

15

10 0

50,000

100,000

150,000

200,000

250,000

300,000

Number of cycles

(b) Fig. 13: Effect of ‘  ’ on fatigue life in a steel specimen (a) damage evolution (b) fatigue crack growth

43

40 Exp. (Stephens, 1988) XFEM (curve fit)

Crack length (mm)

36

32

28

24

0

200,000

400,000

600,000

Number of cycles

(a) Exp. (Stephens, 1988) XFEM (curve fit)

44

Crack length (mm)

40

36

32

28 0

200,000

400,000

600,000

800,000

Number of cycles

(b) Fig. 14: A comparison of experimental and XFEM fatigue crack growth curves for CT specimen at (a) R  0.1 and (b) R  0.5

44

9 1285 N (Exp., Benachour et al., 2010) 1145 N (Exp., Benachour et al., 2010) 1285 N XFEM (curve fit) 1145 N XFEM (curve fit)

8

Crack length (mm)

7

6

5

4

3 0

100,000

200,000

300,000

400,000

500,000

Number of cycles

Fig. 15: A comparison of experimental and XFEM fatigue crack growth curves for four point bend specimen

45

1.0     

Damage variable ()

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Normalized number of cycles (N/Nf)

(a)

8

    

Crack length (mm)

7

6

5

4

3 0

50,000

100,000

150,000

200,000

Number of cycles

(b) Fig. 16: Effect of ‘  ’ on fatigue life in an aluminum alloy specimen (a) damage evolution (b) fatigue crack growth

46

1.0

0.8

-e

-

0.6

0.4

0.2

0.0 0

2

4

6

8

10



Fig. 17:  vs 1  e curve.

8 dN = 05 cycles dN = 10 cycles dN = 15 cycles dN = 20 cycles

Crack length (mm)

7

6

5

4

3 0

50,000

100,000

150,000

200,000

Number of cycles

Fig. 18: Effect of jump in cycle on fatigue crack growth

47

60 Exp. (Ma et al., 2006) XFEM (curve fit)

Crack length (mm)

56

52

48

44 0

40,000

80,000

120,000

160,000

Number of cycles

Fig. 19: A comparison of experimental and XFEM fatigue crack growth curves for SENT specimen

48

(a)

(b)

(c)

(d)

Fig. 20: Stress distribution



yy ,max

 in SENT specimen for different crack length at (a) a = 45.25

mm (b) a = 48.00 mm (c) a = 50.00 mm (d) a = 54.00 mm

49

60 a = 0.10 mm a = 0.15 mm a = 0.20 mm a = 0.23 mm

Crack length (mm)

56

52

48

44 0

40,000

80,000

120,000

160,000

Number of cycles

Fig. 21: Effect of crack growth increment on fatigue crack growth

14 Exp. (Kumar & Garg, 1985) XFEM (curve fit)

12

Crack length (mm)

10

8

6

4

2 0

5,000

10,000

15,000

20,000

25,000

Number of cycles

Fig. 22: A comparison of experimental and XFEM fatigue crack growth curves for MT specimen

50

40 Exp. (Stephens, 1988)

Crack length (mm)

36

32

28

24

0

500,000

1,000,000

1,500,000

2,000,000

120,000

160,000

Number of cycles

(a) 60 Exp. (Ma et al., 2006)

Crack length (mm)

56

52

48

44 0

40,000

80,000 Number of cycles

(b) Fig. 23: Effect of stress triaxiality definition on fatigue crack growth for (a) CT specimen (b) SENT specimen

51

P

ϕ7

8.1

8.3

40 19

P

23 9.2 29.5 40

Fig. 24: Schematic illustration of CT specimen with hole along with dimensions (in mm) and boundary conditions

Crack length (mm)

32

28

24

20

0

10,000

20,000

30,000

40,000

Number of cycles

Fig. 25: Predicted fatigue crack growth curve for CT specimen under mixed mode loading

52

(a)

(b)

(c)

(d)

Fig. 26: Stress distribution



yy ,max

 in CT specimen for different crack length under mixed mode

loading at (a) a = 19.00 mm (b) a = 22.50 mm (c) a = 26.25 mm (d) a = 30.25 mm

53

50 a = 25

25

50

Fig. 27: Schematic illustration of SENT specimen subjected to pure mode II load along with boundary conditions

40

Crack length (mm)

36

32

28

24 0

7,500

15,000

22,500

30,000

Number of cycles

Fig. 28: Predicted fatigue crack growth curve for SENT specimen under mode II failure

54

(a)

(b)

(c)

(d)

Fig. 29: Stress distribution



yy ,max

 in SENT specimen for different crack length subjected to

pure mode II loading at (a) a = 25.00 mm (b) a = 32.00 mm (c) a = 42.80 mm (d) a = 52.00 mm

55

(a)

Φ5 PCD 150

25

66

15

19

(b) Fig. 30: (a) Schematic representation of an actual turbine disc (b) quarter model along with loading and boundary conditions (Kumar et al., 2018)

56

8

Crack length (mm)

6

4

2

0 0

50,000

100,000

150,000

200,000

250,000

Number of cycles

Fig. 31: Predicted fatigue crack growth in turbine disc

57

300,000

Research Highlights 1. A continuum damage mechanics (CDM) and XFEM based methodology is developed for high cycle fatigue crack growth simulations. 2. A new damage model is proposed for the evaluation of fatigue crack growth life. 3. A new criterion is proposed based on the damage evolution to identify the appropriate definition of stress triaxiality. 4. A non-local CDM approach is implemented to reduce the mesh sensitivity. 5. The present methodology is found quite successful for fatigue crack growth simulations.

58