# Assessment of thermal fatigue crack growth in the high cycle

## Transcript Of Assessment of thermal fatigue crack growth in the high cycle

Assessment of thermal fatigue crack growth in the high cycle domain under sinusoidal thermal loading An application – Civaux 1 case

V. Radu E. Paffumi N. Taylor K.-F. Nilsson

EUR 23223 EN - 2007

The Institute for Energy provides scientific and technical support for the conception, development, implementation and monitoring of community policies related to energy. Special emphasis is given to the security of energy supply and to sustainable and safe energy production.

European Commission Joint Research Centre Institute for Energy

Contact information Address: N. Taylor E-mail: [email protected] Tel.: +31-224-565202 Fax: +31-224-565641 http://ie.jrc.ec.europa.eu http://www.jrc.ec.europa.eu

Legal Notice Neither the European Commission nor any person acting on behalf of the Commission is responsible for the use which might be made of this publication.

A great deal of additional information on the European Union is available on the Internet. It can be accessed through the Europa server http://europa.eu/

JRC 41641

EU 23223 EN ISSN 1018-5593 ISBN : 978-92-79-08218-4 DOI : 10.2790/4943

Luxembourg: Office for Official Publications of the European Communities © European Communities, 2007 Reproduction is authorised provided the source is acknowledged

Printed in The Netherlands

Assessment of thermal fatigue crack growth in the high cycle domain under sinusoidal thermal loading An application – Civaux 1 case

V. Radu E. Paffumi N. Taylor K.-F. Nilsson

3

CONTENTS

Abstract ................................................................................................................................................. 4 Nomenclature .................................................................................................................................... 5

1. Introduction ....................................................................................................................................... 7 2. Thermal stresses developed under sinusoidal thermal loading in pipes.......................................... 11 3. Thermal fatigue crack growth approach.......................................................................................... 15 4. Fatigue life associated with the critical frequencies for thermal stress ranges (Civaux 1 case) ..... 18

4.1. Description of the Civaux 1 case.............................................................................................. 18 4.2 The stress intensity factors for internal surface cracks in pipe for a highly nonlinear stress distribution ...................................................................................................................................... 20 4.3 Application on the Civaux 1 case.............................................................................................. 22

4.3.1 Critical frequencies for maximum stress ranges ................................................................ 23 4.3.2 Stress intensity factor solution for long axial crack under hoop thermal stress................. 30 4.3.3 Stress intensity factor solution for fully circumferential crack under axial thermal stress ............................................................................................................................................ 37 4.3.4 Fatigue life assessment for crack growth ........................................................................... 44 5. Summary and Conclusions.............................................................................................................. 46 References ........................................................................................................................................... 47 Appendix 1: Thermal stress components for a pipe subject to sinusoidal thermal loading……….....49 Appendix 2: The first hundred roots of the transcendental equation (Civaux pipe geometry)………52 Appendix 3: Specific critical frequencies associated with thermal stress components for a pipe

subject to sinusoidal thermal loading…………………...……………………………...53 Appendix 4: Benchmarking the stress intensity factor (KIaxial) for a long axial crack in a pipe under

internal pressure.......................................................................................................…...58 Appendix 5: Derivation of KI for a long axial crack under hoop thermal stress……………………60 Appendix 6: Derivation of KI for fully circumferential crack under axial thermal stress……….….62

4

Abstract

The assessment of fatigue crack growth due to cyclic thermal loads arising from turbulent mixing presents significant challenges, principally due to the difficulty of establishing the actual loading spectrum. So-called sinusoidal methods represents a simplified approach in which the entire spectrum is replaced by a sine-wave variation of the temperature at the inner pipe surface. The amplitude can be conservatively estimated from the nominal temperature difference between the two flows which are mixing; however a critical frequency value must be determined numerically so as to achieve a minimum predicted life. The need for multiple calculations in this process has lead to the development of analytical solutions for thermal stresses in a pipe subject to sinusoidal thermal loading, described in a companion report. Based on these stress distributions solutions, the present report presents a methodology for assessment of thermal fatigue crack growth life. The critical sine wave frequency is calculated for both axial and hoop stress components as the value that produces the maximum tensile stress component at the inner surface. Using these through-wall stress distributions, the corresponding stress intensity factors for a long axial crack and a fully circumferential crack are calculated for a range of crack depths using handbook K solutions. By substituting these in a Paris law and integrating, a conservative estimate of thermal fatigue crack growth life is obtained. The application of the method is described for the pipe geometry and loadings conditions reported for the Civaux 1 case. Additionally, finite element analyses were used to check the thermal stress profiles and the stress intensity factors derived from the analytical model. The resulting predictions of crack growth life are comparable with those reported in the literature from more detailed analyses and are lower bound, as would be expected given the conservative assumptions made in the model.

5

Nomenclature

a l ri , ro

- crack depth - wall-thickness of the pipe - inner and outer radii of the pipe

θ

- temperature change from the reference temperature

To

- reference temperature

r

- radial distance

k

- thermal diffusivity

λ

- thermal conductivity

ρ

- density

c

- specific heat coefficient

F(t)

- function of time representing the thermal boundary condition applied

on the inner surface of the cylinder

Jυ(z) , Yυ(z) - Bessel functions of first and second kind of order υ

θ0

- amplitude of temperature wave

ω

- wave frequency in rad/s

f

- wave frequency in Hz

t

- time variable

sn

- positive roots of the transcendental equation (kernel of finite Hankel

transform )

ε r

- radial strain

ε θ

- hoop strain

ε z

- axial strain

σ r

- radial stress

σ θ

- hoop stress

σ z

- axial stress

x

- radial local coordinate originating at the internal surface of the

component

σ 0 -uniform coefficient for polynomial stress distribution

σ1 -linear coefficient for polynomial stress distribution

6

σ 2 -quadratic coefficient for polynomial stress distribution

σ 3 -third order coefficient for polynomial stress distribution

σ 4 -fourth order coefficient for polynomial stress distribution

E

- Young’s modulus

α

- coefficient of the linear thermal expansion

ν

- Poisson’s ratio

u

- radial displacement

da

- increment of crack growth for a given cycle

dN

C

- fatigue crack growth law parameter

n

- fatigue crack growth law exponent

∆Kmax=Kmax- Kmin - maximum stress intensity factor range

Kmax

- maximum stress intensity factor

Kmin

- minimum stress intensity factor

∆Kth

- the threshold stress intensity factor range

∆K = ∆K

eff (1− R)m

- effective stress intensity range

R = K min K max

- stress intensity factor ratio

m

- parameter in the ∆Keff expression

σ VM

- effective stress range intensity (Von Mises equivalent stress)

∆S range

- effective equivalent stress range intensity

G0, G1, G2, G3, G4 - the influence coefficients of hoop stress distribution G’0, G’1, G’2, G’3, G’4 - the influence coefficients of axial stress distribution KIaxial - the Mode I stress intensity factor for an infinite longitudinal surface crack KIcirc - the Mode I stress intensity factor for a fully circumferential surface crack

7

1. Introduction

Quantifying the thermal fatigue damage and subsequent crack growth which may arise due to thermal stresses from turbulent mixing or vortices in light water reactor (LWR) piping systems remains a demanding task and much effort continues to be devoted to experimental, FEA and analytical studies [1, 2, 3, 4].

The problem of thermal fatigue in mixing areas arises in pipes where a turbulent mixing or vortices produce rapid fluid temperature fluctuations with random frequencies. The results in temperature fluctuations can be local or global and induce random variations of local temperature gradients in structural walls of pipe, which lead to cyclic thermal stresses [5, 6]. These cyclic thermal stresses are caused by oscillations of fluid temperature and the strain variations result in fatigue damage, cracking and crack growth.

The response of structures to thermal loads depend on the heat transfer process. In certain components the pipe wall does not respond to high frequency fluctuation of fluid temperature because of heat transfer loss, and low frequency components of fluctuation may not cause large thermal stresses because of thermal homogenization [7,8]. Numerical simulations of the type of thermal stripping1 and high-cycle thermal fatigue that can occur at tee junctions of LWR piping systems have shown that the critical oscillation frequency of surface temperature is the range 0.1-1 Hz [ 5, 6, 9, 10, 11].

In a previous work [12] an analytical set of solutions was developed for thermal stresses in a hollow cylinder subject to sinusoidal thermal loading based on the Hankel transform, properties of Bessel’s functions and the thermoelasticity governing equations. The solution of the time-dependence of temperature in a hollow cylinder allows the derivation of analytical solutions for the associated thermal stresses and their profiles through the wall-thickness.

1 Thermal striping is defined as effect of a rapid random oscillation of the surface temperature inducing a corresponding fluctuation of surface stresses and strains in adjacent metal.

8

In the present paper, the Civaux 1 case [1] was used to assess the application of these analytical thermal stress solutions in crack growth life assessment in the high cycle thermal fatigue domain. The time-dependent analytical solution for thermal stresses in pipe components were used to analyze critical frequencies for axial and hoop stresses as well as for von Mises equivalent stress intensities. Each critical frequency has been derived based on the maximum range of thermal stresses. The maximum stress intensity factor range ∆KImax is considered for two types of crack: a long axial crack and fully circumferential crack on inner surface of the pipe. The fatigue crack growth approach is based on the stress intensity factor solutions expressed in terms of a fourth order polynomial stress distribution through thickness. The crack growth analyses use a Paris-law type equation. Finally, the predictions are compared with the results of other analyses of the Civaux case reported in the literature. Figure 1 shows a flow-chart which describes the steps performed for analysis of thermal fatigue crack growth due to sinusoidal thermal loading.

V. Radu E. Paffumi N. Taylor K.-F. Nilsson

EUR 23223 EN - 2007

The Institute for Energy provides scientific and technical support for the conception, development, implementation and monitoring of community policies related to energy. Special emphasis is given to the security of energy supply and to sustainable and safe energy production.

European Commission Joint Research Centre Institute for Energy

Contact information Address: N. Taylor E-mail: [email protected] Tel.: +31-224-565202 Fax: +31-224-565641 http://ie.jrc.ec.europa.eu http://www.jrc.ec.europa.eu

Legal Notice Neither the European Commission nor any person acting on behalf of the Commission is responsible for the use which might be made of this publication.

A great deal of additional information on the European Union is available on the Internet. It can be accessed through the Europa server http://europa.eu/

JRC 41641

EU 23223 EN ISSN 1018-5593 ISBN : 978-92-79-08218-4 DOI : 10.2790/4943

Luxembourg: Office for Official Publications of the European Communities © European Communities, 2007 Reproduction is authorised provided the source is acknowledged

Printed in The Netherlands

Assessment of thermal fatigue crack growth in the high cycle domain under sinusoidal thermal loading An application – Civaux 1 case

V. Radu E. Paffumi N. Taylor K.-F. Nilsson

3

CONTENTS

Abstract ................................................................................................................................................. 4 Nomenclature .................................................................................................................................... 5

1. Introduction ....................................................................................................................................... 7 2. Thermal stresses developed under sinusoidal thermal loading in pipes.......................................... 11 3. Thermal fatigue crack growth approach.......................................................................................... 15 4. Fatigue life associated with the critical frequencies for thermal stress ranges (Civaux 1 case) ..... 18

4.1. Description of the Civaux 1 case.............................................................................................. 18 4.2 The stress intensity factors for internal surface cracks in pipe for a highly nonlinear stress distribution ...................................................................................................................................... 20 4.3 Application on the Civaux 1 case.............................................................................................. 22

4.3.1 Critical frequencies for maximum stress ranges ................................................................ 23 4.3.2 Stress intensity factor solution for long axial crack under hoop thermal stress................. 30 4.3.3 Stress intensity factor solution for fully circumferential crack under axial thermal stress ............................................................................................................................................ 37 4.3.4 Fatigue life assessment for crack growth ........................................................................... 44 5. Summary and Conclusions.............................................................................................................. 46 References ........................................................................................................................................... 47 Appendix 1: Thermal stress components for a pipe subject to sinusoidal thermal loading……….....49 Appendix 2: The first hundred roots of the transcendental equation (Civaux pipe geometry)………52 Appendix 3: Specific critical frequencies associated with thermal stress components for a pipe

subject to sinusoidal thermal loading…………………...……………………………...53 Appendix 4: Benchmarking the stress intensity factor (KIaxial) for a long axial crack in a pipe under

internal pressure.......................................................................................................…...58 Appendix 5: Derivation of KI for a long axial crack under hoop thermal stress……………………60 Appendix 6: Derivation of KI for fully circumferential crack under axial thermal stress……….….62

4

Abstract

The assessment of fatigue crack growth due to cyclic thermal loads arising from turbulent mixing presents significant challenges, principally due to the difficulty of establishing the actual loading spectrum. So-called sinusoidal methods represents a simplified approach in which the entire spectrum is replaced by a sine-wave variation of the temperature at the inner pipe surface. The amplitude can be conservatively estimated from the nominal temperature difference between the two flows which are mixing; however a critical frequency value must be determined numerically so as to achieve a minimum predicted life. The need for multiple calculations in this process has lead to the development of analytical solutions for thermal stresses in a pipe subject to sinusoidal thermal loading, described in a companion report. Based on these stress distributions solutions, the present report presents a methodology for assessment of thermal fatigue crack growth life. The critical sine wave frequency is calculated for both axial and hoop stress components as the value that produces the maximum tensile stress component at the inner surface. Using these through-wall stress distributions, the corresponding stress intensity factors for a long axial crack and a fully circumferential crack are calculated for a range of crack depths using handbook K solutions. By substituting these in a Paris law and integrating, a conservative estimate of thermal fatigue crack growth life is obtained. The application of the method is described for the pipe geometry and loadings conditions reported for the Civaux 1 case. Additionally, finite element analyses were used to check the thermal stress profiles and the stress intensity factors derived from the analytical model. The resulting predictions of crack growth life are comparable with those reported in the literature from more detailed analyses and are lower bound, as would be expected given the conservative assumptions made in the model.

5

Nomenclature

a l ri , ro

- crack depth - wall-thickness of the pipe - inner and outer radii of the pipe

θ

- temperature change from the reference temperature

To

- reference temperature

r

- radial distance

k

- thermal diffusivity

λ

- thermal conductivity

ρ

- density

c

- specific heat coefficient

F(t)

- function of time representing the thermal boundary condition applied

on the inner surface of the cylinder

Jυ(z) , Yυ(z) - Bessel functions of first and second kind of order υ

θ0

- amplitude of temperature wave

ω

- wave frequency in rad/s

f

- wave frequency in Hz

t

- time variable

sn

- positive roots of the transcendental equation (kernel of finite Hankel

transform )

ε r

- radial strain

ε θ

- hoop strain

ε z

- axial strain

σ r

- radial stress

σ θ

- hoop stress

σ z

- axial stress

x

- radial local coordinate originating at the internal surface of the

component

σ 0 -uniform coefficient for polynomial stress distribution

σ1 -linear coefficient for polynomial stress distribution

6

σ 2 -quadratic coefficient for polynomial stress distribution

σ 3 -third order coefficient for polynomial stress distribution

σ 4 -fourth order coefficient for polynomial stress distribution

E

- Young’s modulus

α

- coefficient of the linear thermal expansion

ν

- Poisson’s ratio

u

- radial displacement

da

- increment of crack growth for a given cycle

dN

C

- fatigue crack growth law parameter

n

- fatigue crack growth law exponent

∆Kmax=Kmax- Kmin - maximum stress intensity factor range

Kmax

- maximum stress intensity factor

Kmin

- minimum stress intensity factor

∆Kth

- the threshold stress intensity factor range

∆K = ∆K

eff (1− R)m

- effective stress intensity range

R = K min K max

- stress intensity factor ratio

m

- parameter in the ∆Keff expression

σ VM

- effective stress range intensity (Von Mises equivalent stress)

∆S range

- effective equivalent stress range intensity

G0, G1, G2, G3, G4 - the influence coefficients of hoop stress distribution G’0, G’1, G’2, G’3, G’4 - the influence coefficients of axial stress distribution KIaxial - the Mode I stress intensity factor for an infinite longitudinal surface crack KIcirc - the Mode I stress intensity factor for a fully circumferential surface crack

7

1. Introduction

Quantifying the thermal fatigue damage and subsequent crack growth which may arise due to thermal stresses from turbulent mixing or vortices in light water reactor (LWR) piping systems remains a demanding task and much effort continues to be devoted to experimental, FEA and analytical studies [1, 2, 3, 4].

The problem of thermal fatigue in mixing areas arises in pipes where a turbulent mixing or vortices produce rapid fluid temperature fluctuations with random frequencies. The results in temperature fluctuations can be local or global and induce random variations of local temperature gradients in structural walls of pipe, which lead to cyclic thermal stresses [5, 6]. These cyclic thermal stresses are caused by oscillations of fluid temperature and the strain variations result in fatigue damage, cracking and crack growth.

The response of structures to thermal loads depend on the heat transfer process. In certain components the pipe wall does not respond to high frequency fluctuation of fluid temperature because of heat transfer loss, and low frequency components of fluctuation may not cause large thermal stresses because of thermal homogenization [7,8]. Numerical simulations of the type of thermal stripping1 and high-cycle thermal fatigue that can occur at tee junctions of LWR piping systems have shown that the critical oscillation frequency of surface temperature is the range 0.1-1 Hz [ 5, 6, 9, 10, 11].

In a previous work [12] an analytical set of solutions was developed for thermal stresses in a hollow cylinder subject to sinusoidal thermal loading based on the Hankel transform, properties of Bessel’s functions and the thermoelasticity governing equations. The solution of the time-dependence of temperature in a hollow cylinder allows the derivation of analytical solutions for the associated thermal stresses and their profiles through the wall-thickness.

1 Thermal striping is defined as effect of a rapid random oscillation of the surface temperature inducing a corresponding fluctuation of surface stresses and strains in adjacent metal.

8

In the present paper, the Civaux 1 case [1] was used to assess the application of these analytical thermal stress solutions in crack growth life assessment in the high cycle thermal fatigue domain. The time-dependent analytical solution for thermal stresses in pipe components were used to analyze critical frequencies for axial and hoop stresses as well as for von Mises equivalent stress intensities. Each critical frequency has been derived based on the maximum range of thermal stresses. The maximum stress intensity factor range ∆KImax is considered for two types of crack: a long axial crack and fully circumferential crack on inner surface of the pipe. The fatigue crack growth approach is based on the stress intensity factor solutions expressed in terms of a fourth order polynomial stress distribution through thickness. The crack growth analyses use a Paris-law type equation. Finally, the predictions are compared with the results of other analyses of the Civaux case reported in the literature. Figure 1 shows a flow-chart which describes the steps performed for analysis of thermal fatigue crack growth due to sinusoidal thermal loading.