Thermal stress intensity factor expressions for functionally graded cylinders with internal circumferential cracks using the weight function method

Accepted Manuscript Thermal stress intensity factor expressions for functionally graded cylinders with internal circumferential cracks using the weight function method Iman Eshraghi, Nasser Soltani PII: DOI: Reference:

S0167-8442(15)30068-9 http://dx.doi.org/10.1016/j.tafmec.2015.09.003 TAFMEC 1621

To appear in:

Theoretical and Applied Fracture Mechanics

Please cite this article as: I. Eshraghi, N. Soltani, Thermal stress intensity factor expressions for functionally graded cylinders with internal circumferential cracks using the weight function method, Theoretical and Applied Fracture Mechanics (2015), doi: http://dx.doi.org/10.1016/j.tafmec.2015.09.003

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Thermal stress intensity factor expressions for functionally graded cylinders with internal circumferential cracks using the weight function method Iman Eshraghi1, Nasser Soltani Intelligence Based Experimental Mechanics Center, Department of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

Abstract In this paper, the weight function method is used to derive mathematical expressions in terms of the Gauss hypergeometric function for the mode-I thermal stress intensity factor of functionally graded cylinders with internal circumferential cracks. To determine the weight function coefficients, a unique function is fitted to reference stress intensity factors obtained from finite element analysis. Effects of the internal convection cooling coefficient and the material power law index are investigated, as well. It is shown that the thermal stress intensity factors predicted by the developed mathematical expression are in good agreement with those directly obtained from finite element analysis. Results of this study may be used in material selection, design optimization, safety assessment against fracture, and fatigue life evaluation of functionally graded cylinders. Keywords: Circumferential crack; Functionally graded material; Hollow cylinder; Thermal stress intensity factor; Weight function.

1

Corresponding Author, Email: [email protected] (Iman Eshraghi), Tel:+989121433335, Address: North Karegar Ave., Jalal Ale Ahmad Blvd., Tehran, Tehran, Iran

1

Nomenclature

( , ; ; ) ℎ

( = 1,2,3) ,

( , ) , ,

( = 1,2,3) ( = 1, … , 4) ( = 1, … , 4) ℬ

( = 1, … ,4) ( ) χ

crack depth area of domain integral evaluation fitting parameters Biot number, Bi = Rih∞/ki elastic modulus modified plane stress/strain elastic modulus Gauss hypergeometric function convection coefficient J-integral value thermal conductivity mode-I stress intensity factor normalized mode-I stress intensity factor reference stress intensity factor weight function coefficients a smooth function, which varies from unity to zero radial and longitudinal directions in the cylindrical coordinate radius of any material point inside the cylinder weight function upper and lower surfaces of the crack in the integration domain thickness of the cylinder temperature distribution in the cylinder radial and longitudinal components of the displacement field strain energy density normalized reference stress intensity factors geometric parameters in the stress intensity factor expression coefficient of thermal expansion crack depth to cylinder wall thickness ratio β = a/t parameters dependent on λ Kronecker delta constants dependent on thermal and mechanical boundary conditions power law index for functionally graded material Poisson’s ratio reference uniform stress applied to the crack faces axial stress component generic material property

Subscripts

( ) ( ) ( ) ( )

inner surface of the cylinder outer surface of the cylinder crack tip location fluid inside the cylinder

2

1.

Introduction

In many engineering applications, such as the power generation and chemical industries, structural components with cylindrical geometry subject to severe temperature gradients are often used. As a result, thermal stresses comparable in magnitude to the applied mechanical stresses exist in these structures. With the presence of small cracks, thermo-mechanical loads can result in the crack growth and final leakage or failure of the structure. Therefore, structural integrity assessment of these components is necessary when a flaw or crack exists. Circumferential cracks are often formed at the joint sections of pressure vessels and pipes. For example, cracks are often detected at a girth weld of a pipe due to the incomplete penetration of the weld material into the interface. Therefore, circumferential cracks in pipes and pressure vessels with homogeneous material properties have been the subject of research for many years (see, e.g., [1–5]). Under extremely high temperature working conditions, functionally graded materials (FGMs) are capable of minimizing the effect of thermal stresses by a smooth change in their thermal and mechanical material properties along one (or more) direction(s). To date, researchers have investigated the behavior of functionally graded (FG) cylinders subjected to thermal and mechanical loads (see, e.g., [6–9]) and in some application cases thermal shocks have been identified as the source of the crack initiation in FG structures [10]. These cracks can grow under further thermal and/or mechanical loads and cause premature failure of the structure. Therefore, integrity assessment of FGMs with cracks subject to thermal and/or mechanical loads is necessary. Analytical and numerical approaches have been used to obtain stress intensity factor (SIF) and thermal SIF in FGMs. Martínez-Pañeda, and Gallego [11] performed numerical and experimental studies to understand the quasi-static crack initiation and crack path growth in FG planar structures. Afsar and Anisuzzaman [12] analyzed pressurized thick-walled FG cylinders with two diametrically-opposed edge cracks. They considered the effect of residual thermal strains in their analysis to obtain SIFs. Recently, Guo and Noda [13] and Zhang et al. [14] studied the problem of thermal shock in FG structures with general thermo-mechanical material properties. They used the perturbation method along with the interaction energy integral method to analyze this problem. The weight function method has been used to obtain SIF of homogeneous cylindrical structures with cracks. With the weight functions given for a crack configuration, it is possible to obtain SIFs for arbitrary loadings applied to the crack face [15,16]. This method 3

has been widely used to develop closed form SIF expressions or tabulated SIF results for various crack geometries which are available in the codes of standard for the structural integrity assessment of cracked structures with homogeneous material properties [17]. Analytical approaches for the analysis of fracture problems in FGMs [18] are limited to specific geometries of the cracked body, crack configuration, and loading conditions. Therefore, to obtain SIFs for complex geometries and load cases, numerical approaches such as the boundary domain element method [19], standard FE method with the energy integral methods [20,21] or the extended finite element method [22] have been used. Energy-based FE methods can accurately determine SIFs for cracked FG structures; however, extensive FE analysis is required for every individual crack geometry and its loading condition. By combining the weight function method and the energy-based FE method it is possible to reduce the number of FE analyses required for a given crack geometry under different loading conditions in FG structures [23,24]. There are few works in the literature which have applied the weight function method to FG cracked structures. Bahr et al. [25] applied the weight function method to determine SIFs in FG structures with residual stresses. Shi et al. [26] proposed basic weight function equations for two-dimensional FG cracked structures using Betti’s reciprocal theorem. Seifi [27] calculated residual stresses for autofrettaged FG cylinders and obtained SIFs at the deepest and surface points of a semi-elliptical axial crack using the weight function approach. Recently, the authors have applied the weight function method to FG cylinders with internal circumferential cracks and have shown the accuracy of the method in predicting the SIFs [28]. A modified form of the J-integral for axisymmetric FG cylinders under applied mechanical loads was proposed in [28] and was used to obtain reference SIFs from the FE analysis results. In this work, the weight functions proposed by Glinka and Shen [29] are used to calculate thermal SIFs of circumferential cracks in FG cylinders. The FG cylinder is under internal and external pressure loads as well as internal convection cooling. Thermo-mechanical material properties of the FG cylinder vary through the wall thickness according to a power law equation. The steady state temperature distribution and the resulting thermal stresses in the cylinder are discussed. Using the weight function expression and the axial component of the thermal stress, analytical expressions in terms of the Gauss hypergeometric function are derived for the mode-I thermal SIF of the FG cracked cylinder. The weight function coefficients are determined using reference SIF results obtained from the FE analyses for 4

three reference crack surface loads. The modified form of the J-integral in domain form for axisymmetric FG structures, presented in [28], is further extended to include the effect of thermal loading and is implemented in the post-processing step of the FE analysis to calculate the reference SIFs. Furthermore, a unique fitting function is introduced to interpolate the reference SIFs for all three reference load cases and for arbitrary values of the crack depth to cylinder thickness ratio and the material power law index not covered in the FE modeling matrix. Coefficients of the proposed curve fitting equation are determined using the obtained reference SIFs from the FE modeling. Comparison of the thermal SIF results predicted by the weight function method with those of the direct FE analysis results is presented. Effects of the Biot number, the crack depth to cylinder thickness ratio, and the FG power law index on the thermal SIF results are discussed, as well. 2.

Thermal stresses for FG cylinders subject to internal cooling

An infinitely long thick-walled cylinder with inner radius Ri, outer radius Ro, wall thickness t = Ro ˗Ri, and a complete internal circumferential crack of depth a is shown in Figure 1-(a). The cylinder is subjected to internal cooling with a convection heat transfer coefficient of h∞ and internal fluid temperature T∞. The cylinder outer surface temperature is kept constant at To and internal pressure pi and external pressure po are applied to the inner and the outer surfaces of the cylinder, respectively. Since the cylinder is assumed to be very long, plane strain conditions exist in the cylinder. The cylinder is made of a FG material with thermal and mechanical properties that vary through the cylinder wall thickness according to the following equation

χ=χ

(1)

where χ is a generic material property representing the cylinder thermal conductivity k, the cylinder elastic modulus E, and the cylinder thermal expansion coefficient α. The Poisson’s ratio is assumed to be constant everywhere in the cylinder, i.e. ν = 0.3. In Eq. (1), χi is the material property value at the cylinder inner radius, r = Ri , and λ is the FGM power law index representing the variation of material properties along the cylinder wall thickness. For a homogeneous cylinder with λ = 0, all material properties are constant and identical to the cylinder inner surface material properties, i.e. χ = χi.

5

For the FG cylinder discussed above with thermal conductivity given by Eq. (1), the steady state temperature distribution is given by [30]

= −

ln +

+

; for ≠ 0

(2)

; for = 0

and the resulting axial component of the stress in the un-cracked cylinder for the plane strain condition is [7,30]

where

=

/

⎧ ⎪ (1 + )(1 − 2 ) = ⎨ ⎪ ( + ⎩(1 + )(1 − 2 )

; for ≠ 0

(3)

ln ) ; for = 0

. The constant terms A1, A2 , ηj, and γj in Eqs. (2) and (3) can be determined

from the thermal and mechanical boundary conditions applied to the inner and the outer surfaces of the cylinder which are given in the Appendix. 3.

Determination of SIFs using the weight function approach

With a simple integral equation, the weight function method calculates the SIF of a cracked body using the stress distribution along the crack face location in the un-cracked body. Because the weight function is dependent on the crack geometry, the key in this method is to determine the proper mathematical form of the weight function. It is worth mentioning that the weight function is dependent on the crack geometry and it is independent of the applied loads. If the weight function for a pressurized cylinder with internal circumferential crack is given by s(r,a) and the normal stress distribution along the crack face in the un-cracked body due to thermo-mechanical loads is given by σzz(r), the mode-I SIF for the internally pressurized cylinder can be calculated from the following integral equation

=

(

( )+

) ( , )



where a is the crack depth. Note that in Eq. (4) the internal pressure pi is added to the axial stress component σzz(r) because the crack is internal to the cylinder body and the internal

6

(4)

pressure acts on the crack faces in addition to the axial stress component resulting from thermo-mechanical loads [31,32]. For the cylinder with circumferential internal crack shown in Figure 1-(a), the mathematical form of the weight function is given by [29,33]

( , )=

2

√2

1−

−

⁄

+

+

1−

⁄

−

+

1−

−



(5)

In Eq. (5) three unknown coefficients Mj (j=1, 2, 3) exist. To determine these coefficients, it is sufficient to use three reference load cases (reference stresses) with known reference SIFs, Kr. This can be done by substituting Eq. (5) and the reference SIFs, Kr, into Eq. (4). Similar to the work of Jones and Rothwell [33] who suggested use of the uniform, linear, and quadratic surface pressure loads as the reference load cases, the following three reference load cases can be used here for the FG cylinder shown in Figure 1 ( )=

Reference load Case (1) – uniform distribution Reference Load Case (2) – linear distribution

( )=

Reference Load Case (3) – quadratic distribution

( )=

(6) −

(7)

−

(8)

It should be noted that the reference load cases given by Eqs. (6) to (8) are symmetric with respect to the crack line. Dimensionless reference SIFs for the above three reference load cases, also known as the boundary correction factors, are defined as

=

⁄ √

(j=1, 2,

3), where σ0 is the uniform applied stress. With the help of the boundary correction factors Yj, the coefficients Mj in Eq. (5) can be determined as follow [33]

= −

48 + √2 6 5

= 21 − √2 = −

105 4

64 + √2 24 5

− 39 −

+ 42

315 2

+

− 132

315 2

+ 126

7

(9)

(10)

(11)

Substituting the axial component of the stress given in Eq. (3) and the weight function given in Eq. (5) into Eq. (4) and integrating Eq. (4), the thermal SIF expression for the FG cylinder (λ ≠ 0) is obtained as follows

where

=

2

2+

+

( ⁄Λ)

+

1 2

+

2 ( ⁄Λ) 3

+

−

+3

= 2√Λ tanh 1

+2

√Λ

( ⁄Λ)

1 ln Λ Λ

+ =Λ + +

In Eq. (13) tanh ( ) = ln

1

+1 +

ln

√Λ +

ln

√Λ − 1

− 1 ;

1 − √Λ tanh

Λ

;

+2

1 + √Λ tanh 1

1

3 5 ,− ; ;Λ 2 2

tanh

+

1

Ζ ; ≠ 0

(1 + )(1 − 2 )

1 3 ,− ; ;Λ + 2 2

= 2 ( ⁄Λ)

+

2 3

√Λ

;

−

+ 1

Λ

(14)

= −1 Λ

+ √Λ

(15)

= −2

; | | < 1, Λ = ⁄(

+ ), and

( + 1) ⋯ ( + − 1) ( + 1) ⋯ ( + ( + 1) ⋯ ( + − 1)

8

(13)

≠ −1, −2

Gauss hypergeometric function which is defined as follows [34]

( , ; ; ) =1+

(12)

( , ; ; ) is the − 1)

!



(16)

The series in Eq. (16) converges if c is a positive integer and |x| < 1. Similarly, the following thermal SIF expression can be obtained for the homogeneous cylinder with λ = 0

=

2

+

where

(1 + )(1 − 2 )

=

2+

= 2 (ln −

−

+

16 4 + 9 3 3 1 + 4 2

− 2) + 4

−

−

+

ln

Λ

2 ln 3

+

−

2 3

(

) ; = 0

+

+

1 2

(18)

⁄Λ tanh

4

ln

3√

Λ

(17)

−1

( ⁄Λ)

⁄

√Λ tanh

√Λ

(19)

1 ( ⁄Λ) ln( ⁄Λ) + (1 + Λ) ln 2 2Λ

Similar thermal SIF expressions for the homogeneous cylinder (λ = 0) with internal circumferential crack were derived previously in [3,32]. As mentioned earlier, the coefficients Mj in Eqs. (12) to (19) can be determined from the reference SIFs. The reference SIFs can be directly obtained from the FE analysis of the cracked FG cylinder which is discussed in the following section. 4.

Determination of SIFs using finite element analysis

As discussed in section 3, the three reference SIFs, corresponding to the uniform, linear, and quadratic pressure loads given in Eqs. (6) to (8), can be used to determine the weight function coefficients Mj . In this section, the reference SIFs are obtained from the FE analysis of the cracked FG cylinder. In this regard, nodal displacements obtained from the FE analysis of the cracked FG cylinder are used to calculate the reference SIFs using a modified form of the Jintegral. In addition, thermal SIFs for the FG cylinder subject to thermo-mechanical loads are obtained from the FE analysis to evaluate the accuracy of the predicted thermal SIFs from the developed weight function method. A curve fitting function that can be used as a substitute for the FE analysis to calculate the reference SIFs is presented and discussed, as well.

9

4.1.

J-integral for FG cylinders with circumferential cracks

For a FG cylinder with axisymmetric loading conditions the modified domain form of the Jintegral is given by [28]

=

1

ℬ

−

ℬ

ℬ

+

−

−



(20)

+

where u is the displacement field, σ is the stress tensor, Rtip is the radius of the crack front with respect to the axis of the cylinder, and A is the domain of integration ahead of the crack tip. In Eq. (20), the function q varies smoothly from unity on the crack front to zero on the boundary of the domain A, r and z denote the radial and the axial directions, respectively, subscripts

and ℬ range over r and z, SU and SL denote the top and the bottom crack faces

located in the integration domain, respectively, and W is the strain energy density corresponding to the applied thermo-mechanical loads. The term

in Eq. (20) is the

explicit derivative of the strain energy density with respect to the r coordinate which is defined by

=

+

+

(21)

where E and α are the elastic modulus and the thermal expansion of the FG material, respectively, which are dependent on the spatial coordinate r, and T is the temperature distribution through the cylinder thickness. The reader is referred to reference [28] for more details regarding the modified J-integral for axisymmetric FG geometries and to references [35,36] for details on the numerical evaluation of the J-integral using FE results. Once the J-integral value is obtained, the mode-I SIF can be calculated as follows

=



10

(22)

where

=

and

=

conditions, respectively, and

/(1 −

and

) for the plane stress and the plane strain

are the elastic modulus and the Poisson’s ratio

values at the crack tip location, respectively. 4.2.

Finite element modeling

The infinitely long thick-walled cylinder, shown in Figure 1-(a), with inner radius Ri, outer radius Ro, wall thickness t, and a complete internal circumferential crack of depth a was considered in all of the FE modeling cases. ABAQUS Standard commercial software [37] was used for the FE model generation. ABAQUS user subroutines UMATHT and UMAT were used to implement the cylinder thermal and mechanical properties gradation, given by Eq. (1), respectively, and to assign the corresponding generic property values to each integration point of each element. Furthermore, the process of FE model crack generation and analysis for different crack depth to the cylinder wall thickness ratios, different FG gradation parameters, and different Biot number values as the thermal boundary condition was automated by developing an ABAQUS Python script. ABAQUS quadratic axisymmetric elements DCAX8 for the steady state heat transfer analysis and quadratic axisymmetric elements CAX8 for the static analysis with full integration option were used in all of the FE analyses. As shown in Figure 1-(b), the reflective symmetry property of the longitudinal section of the cylinder made it possible to model half of the cylinder section in all of the FE models. The half-length of the cylinder was considered to be L/2 = 100×Ro in all FE simulations so that condition of an infinitely long cylinder is satisfied. The far end of the cylinder was fixed in the axial direction and symmetric boundary conditions were applied to the un-cracked face of the cylinder as shown in Figure 1-(b). Mesh convergence studies suggested that the element size in the first five contour regions around the crack tip should not exceed 1 percent of the crack depth. A typical FE mesh of the cylinder longitudinal section with details of the mesh near the crack tip is shown in Figure 2 where the cylinder elastic modulus at the outer surface is four times its elastic modulus at the inner surface along the thickness direction. In the FE modeling of the FG cylinder, the FGM power law index λ varied from -3 to 3 in increments of 1. For each λ value, the crack depth to cylinder thickness ratio, β = a/t, varied from 0.1 to 0.9 with increment size of 0.05. To determine the weight function coefficients Mj the three reference load cases given in Eqs. (6) to (8) were applied in the FE models and the corresponding reference SIFs were obtained by evaluating the modified J-integral discussed 11

in section 4.1. Table 1 lists a summary of the FE modeling matrix where 357 FE runs were performed to determine the weight function coefficients. Additional FE analyses were performed for the thermo-mechanical load cases that will be described later in section 5.3. These additional FE runs were only used to compare the SIFs predicted by the present weight function method given in Eqs. (12) and (17) with the direct FE analysis SIF results. 4.3.

Extraction of SIFs from the FE analysis of the FG cylinders

The J-integral expression discussed in section 4.1 was used in the post processing step of the FE analysis to obtain the corresponding SIFs for the cracked FG cylinder. The J-integral was numerically evaluated for the six domains enclosed by the six closest contours around the crack tip in all of the FE models. The first two domains closest to the crack tip were disregarded and the average of the last four domains was used as the energy release rate value. Plane strain condition was assumed in Eq. (22) to derive SIFs from the numerical values of the J-integral. It should be noted that the FE modeling matrix given in Table 1 covers finite number of the crack depth to cylinder thickness ratios and finite number of the FG power law indexes. Once the reference SIFs are obtained from the FE analyses, a curve fitting function can be employed to calculate the boundary correction factors Yj given in Eqs. (9) to (11). This curve fitting function can be used to calculate the boundary correction factors Yj for any arbitrary value of the crack depth to cylinder thickness ratio β and any arbitrary value of the FG power law index λ without the need for additional FE analysis. The proposed curve fitting function is as follows

=

+ exp =

(

− 1)( 2

1.0 + + 2)

+

;

(23)

+8

With the help of a nonlinear least squares scheme, coefficients Ajl (j=1, 2, 3; l=0, 1, …, 16) of the fitting function in Eq. (23) can be determined based on the extracted FE results for the FE modeling matrix listed in Table 1. Numerical values of Ajl in Eq. (23) are listed in Table 2 for FG hollow cylinders with Ro = 2×Ri. Note that Eq. (23) when used in combination with Eqs. (9) to (19) gives the thermo-mechanical SIFs of FG and homogeneous cylinders with internal circumferential cracks without the need for any FE simulation and additional post-processing. 12

This is a very simple and fast approach for the calculation of SIFs of FG cylinders for their structural integrity assessment. Also the method can be extended to include other types of loading conditions once the axial stress distribution on the crack face location in the uncracked geometry is determined either explicitly or numerically. 5.

Results and discussions

In this section SIFs for the homogeneous and FG cylinders are calculated using the present weight function method and the obtained results are compared to the available SIF results reported in the literature. Next, thermal SIF results for the thermally stressed pressurized FG cylinders are obtained using the weight function approach presented in this work. 5.1.

SIF results for the homogenous cylinder (λ = 0)

As shown in Figure 3, normalized thermo-mechanical SIFs, defined as KN = KI / pi(πa)0.5, for the homogeneous hollow cylinders with internal crack are obtained using the present weight function method and the results are compared to those reported in [3]. Internal and external pressure of pi = 10 MPa and po = 0.1 MPa, are applied to the cylinder, respectively, and the cylinder inner surface is subjected to convection cooling T∞ = –100 ˚C. Results are reported for four different combinations of the cylinder outer radius to inner radius ratios and Biot number values. Good agreement is observed between the present weight function technique results with those reported in [3]. Furthermore, to investigate the accuracy of the presented fitting function, normalized SIF results are calculated using the present weight function method for a pressurized homogeneous cylinder with pi = 10 MPa, po = 0.1 MPa, and Ro = 2×Ri are listed in Table 3. No internal cooling is considered in this example. It should be noted that in Table 3, the “WF by FE” results are obtained from the weight functions given in Eq. (5) with their coefficients Mj, given in Eqs. (9) to (11), determined based on the boundary correction factors Yj obtained from the FE analysis whereas the “WF and Eq. (23)” SIF results are obtained from the same weight functions with their coefficients Mj determined using the boundary correction factors Yj predicted by the curve fitting technique given by Eq. (23). For comparison, SIF results directly obtained from the FE analysis of the cylinder are also reported in Table 3. In addition, SIF results for the same cylinder with same loading conditions reported in references [32,38] are listed in Table 3 for comparison. From this table, a very good agreement is observed between the presented weight function results and those reported in references [32,38]. 13

Table 4 lists the normalized reference SIF results,

=

⁄ √

, = 1, 2, 3, for the

homogeneous cylinder with different crack depth to cylinder thickness ratios, β. Numerical results from the literature are also reported in this table for comparison. These results correspond to the three reference load cases, given in Eqs. (6) to (8) applied to the crack face as pressure loads while other pressure and thermal loads are set to zero. Good agreement is observed between the results of the present work and the SIF results of the references listed in this table. It should be noted that from the Table 4 results one can conclude that the “WF by FE” approach is self-consistent. This means that the weight function approach reproduces the FE results based on which its coefficients Mj were initially determined. 5.2.

SIF results for FG cylinders

In this section two examples are considered to assess the accuracy of the proposed weight function approach for the calculation of SIFs for FG cylinders. It should be noted that the expression given in Eq. (12) is not applicable to these two verification examples because different material gradation profiles, other than Eq. (1), and different crack face stress distribution are considered in these two examples. We apply the presented weight function approach and calculate the weight function coefficients and the related SIFs using the reference SIFs and the stress distribution for the un-cracked geometry directly obtained from the FE analysis of these two examples. For the first verification example, the SIF results reported by Gu et al. [39] for a solid FG cylinder with a central circular crack under remote tension is considered. Two gradation profiles for the cylinder elastic modulus, i.e. a linear and an exponential variation along the radial direction, is considered as follow [39] Linear: ( ) = (

−

) +

Exponential: ( ) =

(24) (25)

Table 5 shows normalized J integral results, defined as JN = JE0/σ2a, obtained using the present weight function method for the FG cylinder with crack depth of radius a=25 mm, cylinder radius b = 50 mm, cylinder length L = 250 mm, and E0 = 200 GPa and Eb = 4000 GPa. Results reported by Gu et al. [39] are listed in this table, as well. Note that SIFs 14

obtained by the weight function approach can be converted to J-integral values using Eq. (22). J-integral results directly obtained from the FE analysis of the cylinder are also listed in Table 5. Good agreement is observed between the results of the current study and those reported by Gu et al. [39]. For the second example, the FG hollow cylinder with an internal circumferential crack under remote tensile load described in reference [40] is considered here. The cylinder elastic modulus variation is assumed to be as follows [40] ( )=

+(

−

)

−

(26)

where subscripts c and m denote the ceramic and metal constituent materials, respectively, and n is the gradient exponent of FG material. Normalized SIFs reported in [40] are listed in Table 6. The present weight function analysis results are also listed in this table. It is worth mentioning that in [40] the SIFs are derived using the displacement correlation technique whereas in this work the J-integral in domain form is used to obtain reference SIFs and then weight function technique is used to calculate SIF values corresponding to remote tension load. As can be seen from Table 6, good agreement is observed between the results of the two works. The above verification examples show the accuracy of the proposed weight function technique for the calculation of SIFs for both the homogenous and the FG cylinders with axisymmetric cracks. Furthermore, it is shown that the proposed weight function technique is applicable to material gradation profiles other than the one considered in the present work given in Eq. (1). 5.3.

Calculation of thermal SIFs for FG cylinders

In this section, thermo-mechanical SIF results for FG hollow cylinders are presented. It should be noted that in all of the FE models of this section the cylinder outer radius is set to twice of the inner radius, Ro = 2×Ri. Furthermore, the cylinder thermal conductivity, elastic modulus, and thermal expansion coefficient at the inner radius are set to ki = 20 W/m˚C, Ei = 70 GPa, and αi = 23×10–6 1/˚C, respectively. Table 7 summarizes the normalized reference SIFs, also known as the boundary correction factors, corresponding to the three reference load cases given in Eqs. (6) to (8). These results are obtained from the FE analysis of the FG hollow cylinder for different power law indices λ and different crack depth to cylinder 15

thickness ratios β. To illustrate the accuracy of the present weight function technique and the proposed curve-fitting equation, i.e. Eq. (23), reference SIFs results obtained from these methods are also reported for a selected set of the loading cases and crack depth to thickness ratios. The relative difference between the predicted and the direct FE normalized SIF is defined as −

Relative Difference (percent) =





× 100

(27)

The maximum relative difference between the predicted SIFs by the weight function and the proposed fitting scheme given in Eq. (23) and those directly obtained from the FE analysis is less than 2 percent for all of the cases listed in Table 7. This shows that the proposed curve fitting function given in Eq. (23) accurately predicts the boundary correction factors and therefore the SIFs. Also, results obtained from the “WF by FE” approach are in good agreement with the direct FE analysis results indicating the self-consistency of the presented weight function approach. The reference SIFs in Table 7 were then used to calculate thermal SIFs for the FG cylinder subjected to internal pressure pi = 5 MPa, external pressure po = 0.2 MPa, and internal cooling boundary condition with T∞ = ‒50 ˚C, Tref = 0 ˚C, and To = Tref,. By changing the Biot number value, defined as Bi = Rih∞/ki, it is possible to study the effect of the internal cooling on the SIF results. It is worth mentioning that when the cylinder inner surface temperature is Ti = T∞, the Biot number is Bi = ∞ and when there is no internal cooling applied to the cylinder, the Biot number is Bi = 0. Figure 4 shows the normalized SIF results KN for Bi = 0 and for different FGM power law indices λ obtained from the direct FE analysis and predicted by the weight function approach combined with the proposed curve fitting technique. In this figure, the direct FE analysis results are shown in symbols and the predicted results from the weight function method combined with the curve fitting Eq. (23) are shown by lines. Similarly, Figure 5, Figure 6, and Figure 7 show the normalized SIF results for the Biot numbers Bi = 0.1, Bi = 10, and Bi = 100, respectively. The maximum value of the relative difference, in percent, between the predicted SIFs from the weight function method and the direct FE analysis SIF results is also reported in the figures legend. It should be mentioned that the relative difference values reported for the “FE” in the figures legends indicate the magnitude of maximum relative 16

difference (in percent) between the direct FE SIF results and the SIF results calculated from the weight function method whose coefficients Mj were determined using the boundary correction factors Yj directly obtained from the FE analysis. The maximum relative difference for all cases presented in Figure 4 to Figure 7 is less than 0.5% when the SIFs are calculated using the weight function approach and the boundary correction factors directly obtained from the FE analysis. Also, the relative difference values reported after the “WF” in the figures legends are the maximum relative difference (in percent) between the direct FE analysis SIF results and the predicted SIF results by the weight function method where the boundary correction factors Yj were determined using the curve fitting function given in Eq. (23). The maximum relative difference for all cases presented in Figure 4 to Figure 7 is less than 2% when the curve fitting technique is used. Figure 4 suggests that in the absence of thermal loads, SIFs of shallow cracks are more affected by the variation of the material power law index, λ, than the SIFs of deep cracks. Also, FG cylinders with lower λ values have larger SIFs for shallow cracks while this trend reverses for FG cylinders with deep cracks. It should be noted that SIFs for FG cylinders with 0.75 < β < 0.8 are not affected too much by the variation of the power law index λ when the cylinder is subjected to internal and external pressure loads only. For λ ≤ 0, increasing the crack depth to cylinder thickness ratio β causes drop in the SIFs, initially. This trend reverses for mid values of β. For the case of λ > 0, SIFs monotonically increase with increasing β. Thermal SIFs for FG cylinders subjected to thermo-mechanical load with Bi = 0.1 are shown in Figure 5 for different λ values. From Figure 5 it can be seen that for shallow cracks, FG cylinders with smaller λ values have higher SIF values. This trend reverses for the deeper cracks. Figure 6 shows the SIF results for Bi = 1 and different λ values. It can be seen from this figure that increasing the Biot number from 0.1 to 1 results in considerably higher SIF values. Contrary to the trend seen before in Figure 5 (for Bi = 0.1), for this case with Bi = 1 deep cracks are more affected by the variation of the power law index λ compared to shallow cracks. Therefore, one may conclude that for low Biot numbers shallow cracks are more sensitive to the variation of the FGM power law index and for high Biot numbers deep cracks are more affected by the variation of the power law index λ. Finally, a large Biot number value, i.e. Bi = 100, was considered and the corresponding thermal SIF results are shown in Figure 7. For FG cylinders with negative λ values, thermal SIFs are relatively small and the thermal SIF curves shown in Figure 7 are close to each 17

other. On the other hand, thermal SIFs are considerably higher for FG cylinders with positive λ values and deep cracks. Note that for Bi = 100, thermal SIFs increase with the FGM power law index at fixed β value. From Figure 5 to Figure 7, it can be observed that the relative difference between the predicted results and the direct FE analysis results is not very much affected by the variations of the Biot number. To investigate more the effect of the Biot number on thermal SIFs, normalized SIFs were obtained for three fixed values of the cylinder power law index λ, i.e. λ = –3, λ = 0, and λ = 2, and the Biot number for each λ value varied from 0 to ∞. Figure 8, Figure 9, and Figure 10, show the thermal SIF results with various Biot numbers for λ = –3, λ = 0, and λ = 2, respectively. As expected, in all of the three cases for a given β value, thermal SIF increases when the Biot number increases from 0 to ∞. As shown in Figure 8, for the case of the FG cylinder with λ = –3, increasing β from 0.1 to 0.6 causes reduction in thermal SIFs for all of the Biot number values. Further increase in β (beyond 0.75) results in negligible increase in thermal SIFs for this case. Also, shallow cracks with β = 0.1 have the highest thermal SIFs when Bi ≥ 0.1 for this case. For the case of the homogeneous cylinder (λ = 0), thermal SIFs are almost close to each other for deep and shallow cracks as shown in Figure 9. For the case of the FG cylinder with λ = 2, thermal SIFs monotonically increase with increasing β irrespective of the Biot number value as shown in Figure 10. For all of the three cases shown in Figure 8 to Figure 10, the maximum relative difference between the predicted results by the weight function and the direct FE analysis results is less than 1%. 6.

Conclusions

Using the weight function approach, mathematical expressions were developed based on the Gauss hypergeometric function for thermal SIFs of FG cylinders with internal circumferential cracks subjected to thermo-mechanical loads. The weight function coefficients were determined using the reference SIFs obtained from the FE analysis of the FG cylinder for three reference load cases applied to the crack face. A modified form of the J-integral in domain form for axisymmetric nonhomogeneous structures was employed in the postprocessing step of the FE analysis of the FG cylinder to extract the thermal SIFs. Because the FE modeling for reference SIFs was done for finite number of the material power law indexes 18

and finite number of the crack depth to cylinder thickness ratios, a unified curve fitting scheme was further introduced to calculate the reference SIFs for arbitrary values of the material gradation index and the crack depth to cylinder thickness ratio. The proposed curve fitting technique removes the need for additional FE analysis required for the determination of the weight function coefficients. Effects of the crack depth, the FGM power law index, and the Biot number on the thermal SIFs were investigated in detail. The following conclusions can be drawn from the results presented in this work. 1- In the absence of thermal loads, FG cylinders with smaller FGM power law index have higher SIFs for shallow and medium crack depths. For deep cracks, SIFs are higher for FG cylinders with larger FGM power law index values. 2- With the presence of internal thermal cooling, FG cylinders with shallow cracks and small FGM power law index values experience higher thermal SIFs when the Biot number is small. 3- For high Biot number values, FG cylinders with larger FGM index experience higher thermal SIFs for all of the crack depth to cylinder thickness ratios. 4- For a given crack depth to cylinder thickness ratio, increasing the Biot number always results in higher thermal SIFs. 5- Thermal SIFs of FG cylinders with shallow cracks and negative FGM power law index are more sensitive to the variation of the Biot number. Also, for FG cylinders with positive FGM power law index and deep cracks, thermal SIFs are more sensitive to the variation of the Biot number. The followings may be considered for the extension of the present work. 1- The present weight function method can be applied to mixed-mode crack problems by adding additional terms to the weight function expressions to account for the shearing mode as discussed in [41]. The method presented in this work can be extended to the mixed-mode fracture of FG structures under thermo-mechanical loads. However, the mixed-mode SIFs cannot be calculated using the J-integral technique because the J-integral is a single scalar value. Instead, an interaction energy integral technique, as discussed in [42,43], may be employed to calculate the mode I and the mode II reference SIFs required for the determination of the weight function coefficients.

19

2- In this study, the FGM power law index was assumed to be identical for both the mechanical and the thermal properties of the FG cylinder. However, the approach presented here is general and can be used to employ separate power law indices for the thermal and for the mechanical properties. In addition, the present work is capable of including various forms of material gradation profiles different from the one given in Eq. (1) as well as accounting for other types of thermal and mechanical boundary conditions. Also, other cracked geometries in non-homogeneous materials, such as cracked FG plates, can be analyzed using the weight function approach presented in this work. 3- In certain scales, material interfaces are more or less present in FGMs as discussed in [44]; therefore, when dealing with fracture of FGMs at these scales, the existence of interface in the material should be taken into consideration. Methods based on the interaction energy integrals [44] combined with continuum-based approaches such as the extended FEM (XFEM) [45] may be used to study the fracture problem of materials with strong or weak interfaces when the integral domain contains arbitrary interfaces. The weight function method is still applicable to these problems provided the stress distribution in the un-cracked body is known and the reference SIFs are correctly calculated for the determination of the weight function coefficients. For the purpose of determining the reference SIFs, the methods presented in references [44, 45] can be used. 4- Results of the present work can be used for the selection of material profile, design optimization, and safety assessment of FG cylinders subject to thermo-mechanical loads in order to prevent early fracture. Acknowledgements The authors would like to thank Dr. Mohammad Amin Eshraghi for his helpful comments by reading the initial draft of the manuscript. Appendix The constant terms in Eqs. (2) and (3) discussed in section 2 for FG cylinders, λ ≠ 0, are given as follow.

=

1−

−

+

20

(A. 1)

=

+

(A. 2)

= (1 +

)

= (1 +

=

= (2 + )

−

=2

+

+ − 1 ; ,

=

=

(A. 3)

)

(A. 4)

(1 + )

−

(1 + )

(A. 6)

+ − 1 ;

=− ± − +1 2 4 1− 2 (1 + )

( + 1)(2 + 1) + =−

(1 + )

( + 1) + = =

−

1−

1+ − −

(A. 5)

= 2 ; /

=

(A. 7)

(A. 8)

(A. 9)

− 1 (1 − )

(A. 10)

− 1 (1 − )

(A. 11)

− −

(A. 12)

where

=−

= [(1 − )

(1 + )(1 − 2 ) −

−

+ ]

(1 − ) + 1

+ (1 + )

= , ; = 1,2

21



− (1 + )

(A. 13)

−

(A. 14)

=

It should be noted that in the above equations

/

, Bi = Rih∞/ki is the Biot number, and

Tref is the uniform stress free initial temperature of the cylinder. For the case of the homogenous cylinder, λ = 0, the constants are given as follow.

=

=2

+

=

=2 =

=

−

= 1;

(1 + )(1 − 2 ) −[

−

ln

1+ −

(A. 15)

+ ln

(A. 16) −

1+

(A. 17) (A. 18)

(1 + ) 2(1 − )

where

=−

1

−

(A. 19)

− −

=

(A. 20)

2 −1 ,

+ (1 + )

− (1 + ) = ,

22

] ln

−

(A. 21)

− (1 − )

(A. 22)

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26

Figure Captions Figure 1. Schematic of a circumferential crack in a pressurized long hollow cylinder subject to internal cooling (a), and the corresponding boundary conditions and loadings applied in the FE model (b) Figure 2. A typical FE mesh of the FG cylinder with variation of the elastic modulus in the cylinder section (top) and near the crack tip region (bottom) Figure 3. Comparison of normalized thermo-mechanical SIFs for a homogeneous cylinder; lines: present analysis, symbols: Ref. [3] Figure 4. Normalized SIF for the FG cylinder with different material power law indexes and Bi = 0 (no internal cooling) Figure 5. Normalized SIF for the FG cylinder with different material power law indexes and Bi = 0.1 Figure 6. Normalized SIF for the FG cylinder with different material power law indexes and Bi = 1 Figure 7. Normalized SIF for the FG cylinder with different material power law indexes and Bi = 100 Figure 8. Normalized thermal SIF for the FG cylinder with λ = –3 subjected to thermomechanical loads with various Biot numbers Figure 9. Normalized thermal SIF for the homogeneous cylinder λ = 0 subjected to thermomechanical loads with various Biot numbers Figure 10. Normalized thermal SIF for the FG cylinder with λ = 2 subjected to thermomechanical loads with various Biot numbers

27

Table 1. FE modeling matrix for weight function determination

Parameter

Variation range

# of variations

Crack geometry (β = a/t)

0.1 to 0.9 with increment 0.05

17

FG power law index (λ)

-3 to 3 with increment 1

7

Total number of FE runs

357 Reference load cases (6), (7), (8)

Uniform, Linear, Quadratic

3

(for the determination of the weight function coefficients)

28

Table 2. Curve fitting function coefficients for the three boundary correction factors.

Fitting Coefficient Aj0 Aj1 Aj2 Aj3 Aj4 Aj5 Aj6 Aj7 Aj8 Aj9 Aj10 Aj11 Aj12 Aj13 Aj14 Aj15 Aj16

j=1 0.6506 -0.7087 4.578 5.948 -32.56 75.48 -82.88 34.6 -0.02175 -2.906 3.238×10-4 0.05498 3.187 -1.088×10-6 -9.214×10-5 -0.03231 -1.242

j=2 -0.8814 -0.1022 0.2016 6.781 -20.21 55.9 -70.26 30.88 -0.01335 -2.828 6.181×10-5 0.03026 3.024 -7.324×10-8 2.572×10-4 -0.01491 -1.102

29

j=3 -0.01658 -4.109 -0.9491 59.48 -181.3 283.5 -227.9 73.65 -0.01076 -2.36 3.992×10-5 0.02237 1.996 1.731×10-6 2.921×10-4 -9.392×10-3 -0.4949

Table 3. Comparison of the normalized SIFs for the pressurized homogeneous cylinder

β = a/t

Present (FE)

Present

Present

(WF by FE)

(WF and Eq. (23))

[32]

[38]

0.1

1.276

1.276

1.283

1.279

1.267

0.3

1.259

1.259

1.264

1.255

1.263

0.5

1.341

1.341

1.336

1.320

1.336

0.7

1.578

1.578

1.577

1.516

1.584

30

Table 4. Comparison of reference SIFs (boundary correction factors) for the homogeneous cylinder

β = a/t

0.1

0.3

0.5

0.7

Present

Present

Present

[3]

[38]

[33]

(FE)

(WF by FE)

(WF and Eq. (23))

Y1

1.070

1.070

1.076

1.068

1.063

1.068

Y2

0.066

0.066

0.067

-

0.067

0.065

Y3

0.005

0.005

0.005

-

-

0.005

Y1

1.057

1.057

1.061

1.056

1.06

1.056

Y2

0.198

0.198

0.199

-

0.198

0.197

Y3

0.046

0.046

0.046

-

-

0.046

Y1

1.125

1.125

1.121

1.125

1.122

1.125

Y2

0.347

0.347

0.347

-

0.347

0.346

Y3

0.134

0.134

0.134

-

-

0.133

Y1

1.324

1.324

1.323

1.324

1.329

1.324

Y2

0.554

0.554

0.554

-

0.555

0.553

Y3

0.292

0.292

0.292

-

-

0.291

31

Table 5. Comparison of normalized J-integral results for a FG solid cylinder with central circular crack

Elastic module variation

JN [39]

Present (FE)

Present (WF by FE)

Linear (Eq. (24))

0.0464

0.0462

0.0462

Exponential (Eq. (25))

0.0358

0.0357

0.0357

32

Table 6. Comparison of normalized SIFs for FG cylinders with internal circumferential crack under remote tension

Ri/Ro

Em/Ec

=

a/t n = 0.25 [40]

Present

[40]

(WF by FE) 0.7

0.8

2.0

1.2

/ √

n=1

Present

n=2 [40]

(WF by FE)

Present (WF by FE)

0.1

0.8950

0.8958

0.7434

0.7432

0.7929

0.7926

0.7

1.6042

1.6037

1.5074

1.5095

1.4209

1.4228

0.2

1.1489

1.1398

1.1018

1.0943

1.1142

1.1089

0.6

1.6526

1.6569

1.6111

1.6163

1.5890

1.5942

33

Table 7. Normalized reference SIFs for the functionally graded cylinder

β

0.1

Load case

method

(1)

-3

-2

-1

1

2

3

FE

1.146

1.118

1.093

1.051

1.034

1.020

WF by FE

1.146

1.118

1.093

1.051

1.034

1.020

WF and Eq. (23)

1.153

1.127

1.102

1.053

1.030

1.009

FE

0.069

0.068

0.067

0.066

0.065

0.064

0.005

0.005

0.005

0.005

0.005

0.005

1.140

1.108

1.079

1.028

1.008

0.990

(2)

0.174

0.171

0.167

0.162

0.160

0.158

(3)

0.033

0.033

0.032

0.032

0.031

0.031

1.149

1.123

1.100

1.062

1.045

1.030

(2)

0.282

0.277

0.273

0.266

0.263

0.261

(3)

0.086

0.085

0.084

0.083

0.082

0.082

(2) (3) 0.25

0.4

0.5

0.6

0.75

0.9

λ

(1)

(1)

FE

FE

(1)

FE

1.174

1.154

1.138

1.115

1.106

1.097

(2)

FE

0.359

0.354

0.350

0.345

0.343

0.341

WF by FE

0.359

0.354

0.350

0.345

0.343

0.341

WF and Eq. (23)

0.360

0.355

0.351

0.344

0.342

0.340

(3)

FE

0.138

0.136

0.135

0.133

0.133

0.132

(1)

FE

1.224

1.210

1.203

1.202

1.205

1.209

(2)

0.447

0.443

0.440

0.438

0.439

0.439

(3)

0.448

0.444

0.440

0.437

0.437

0.439

1.395

1.393

1.400

1.444

1.479

1.519

(2)

0.626

0.624

0.624

0.636

0.645

0.657

(3)

0.353

0.351

0.351

0.355

0.359

0.364

1.995

2.010

2.037

2.136

2.212

2.308

1.038

1.043

1.052

1.087

1.114

1.149

FE

0.680

0.682

0.686

0.704

0.718

0.735

WF by FE

0.680

0.682

0.686

0.704

0.718

0.735

WF and Eq. (23)

0.680

0.681

0.686

0.704

0.718

0.736

(1)

(1)

FE

FE

(2) (3)

34

(a)

(b)

Figure 1. Schematic of a circumferential crack in a pressurized long hollow cylinder subject to internal cooling (a), and the corresponding boundary conditions and loadings applied in the FE model (b)

Figure 2. A typical FE mesh of the FG cylinder with variation of the elastic modulus in the cylinder section (top) and near the crack tip region (bottom)

Figure 3. Comparison of normalized thermo-mechanical SIFs for a homogeneous cylinder; lines: present analysis, symbols: Ref. [3]

Figure 4. Normalized SIF for the FG cylinder with different material power law indexes and Bi = 0 (no internal cooling)

Figure 5. Normalized SIF for the FG cylinder with different material power law indexes and Bi = 0.1

Figure 6. Normalized SIF for the FG cylinder with different material power law indexes and Bi = 1

Figure 7. Normalized SIF for the FG cylinder with different material power law indexes and Bi = 100

Figure 8. Normalized thermal SIF for the FG cylinder with λ = −3 subjected to thermomechanical loads with various Biot numbers

Figure 9. Normalized thermal SIF for the homogeneous cylinder λ = 0 subjected to thermomechanical loads with various Biot numbers

Figure 10. Normalized thermal SIF for the FG cylinder with λ = 2 subjected to thermomechanical loads with various Biot numbers

Highlights ·

Mathematical expressions are derived for the thermal SIF of cracked FG cylinders.

·

The weight function approach is used for the thermal fracture analysis.

·

A fitting function is introduced to calculate the reference SIFs.

·

Effects of the internal cooling and the FGM index on the thermal SIFs are studied.

·

Predicted SIFs by the weight function agree well with direct FE SIF results.

35