sciencedirect sciencedirect 6flhqfh'luhfw sciencedirect · 2017. 1. 25. · damaged materials...

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Structural Integrity Procedia 00 (2016) 000–000 www.elsevier.com/locate/procedia

2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016.

XV Portuguese Conference on Fracture, PCF 2016, 10-12 February 2016, Paço de Arcos, Portugal

Thermo-mechanical modeling of a high pressure turbine blade of an airplane gas turbine engine

P. Brandãoa, V. Infanteb, A.M. Deusc* aDepartment of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa,

Portugal bIDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa,

Portugal cCeFEMA, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, 1049-001 Lisboa,

Portugal

Abstract

During their operation, modern aircraft engine components are subjected to increasingly demanding operating conditions, especially the high pressure turbine (HPT) blades. Such conditions cause these parts to undergo different types of time-dependent degradation, one of which is creep. A model using the finite element method (FEM) was developed, in order to be able to predict the creep behaviour of HPT blades. Flight data records (FDR) for a specific aircraft, provided by a commercial aviation company, were used to obtain thermal and mechanical data for three different flight cycles. In order to create the 3D model needed for the FEM analysis, a HPT blade scrap was scanned, and its chemical composition and material properties were obtained. The data that was gathered was fed into the FEM model and different simulations were run, first with a simplified 3D rectangular block shape, in order to better establish the model, and then with the real 3D mesh obtained from the blade scrap. The overall expected behaviour in terms of displacement was observed, in particular at the trailing edge of the blade. Therefore such a model can be useful in the goal of predicting turbine blade life, given a set of FDR data. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of PCF 2016.

Keywords: High Pressure Turbine Blade; Creep; Finite Element Method; 3D Model; Simulation.

* Corresponding author. Tel.: +351 218419991.

E-mail address: [email protected]

Procedia Structural Integrity 2 (2016) 793–800

Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer review under responsibility of the Scientific Committee of ECF21.10.1016/j.prostr.2016.06.102

10.1016/j.prostr.2016.06.102


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2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21.

21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy

Asymptotic Stress Field in the Vicinity of the Mixed-Mode Crack in Damaged Materials Under Creep Conditions

Stepanova Larisaa*, Yakovleva Ekaterinaa

aDepartment of Mathematical Modelling in Mechanics, Samara State University,Akad. Pavlov, 1, 443011 Russia

Abstract

The creep crack problems in damaged materials under mixed mode loading under creep-damage coupled formulation are considered. The class of the self-similar solutions to the plane creep crack problems in a damaged medium under mixed-mode loading is given. With the similarity variable and the self-similar representation of the solution for a power-law creeping material and the Kachanov- Rabotnov power-law damage evolution equation the near crack-tip stresses, creep strain rates and continuity distributions for plane stress conditions are obtained. The similarity solutions are based on the hypothesis of the existence of the completely damaged zone near the crack tip. It is shown that the asymptotical analysis of the near crack-tip fields gives rise to the nonlinear eigenvalue problems. The technique permitting to find all the eigenvalues numerically is proposed and numerical solutions of the nonlinear eigenvalue problems arising from the mixed-mode crack problems in a power-law medium under plane stress conditions are obtained. Using the approach developed the eigenvalues different from the eigenvalues corresponding to the Hutchinson-Rice-Rosengren (HRR) problem are found. Having obtained the eigenspectra and eigensolutions the geometry of the completely damaged zone in the vicinity of the crack tip is found for all values of the mixity parameter. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21.

Keywords: Damage (integrity) parameter; similarity variable; self - similar solution; mixed-mode loading; creep-damage coupling; nonlinear eigenvalue problem; eigenvalue spectrum

1. Introduction

Knowledge of stress, strain and displacement fields in the vicinity of the crack tip under mixed-mode loading conditions is important for the justification of fracture mechanics criteria and has attracted considerable attention

* Stepanova Larisa. Tel.: +7 -846 -334-54-41; fax: +7-846-334-54-41.



ScienceDirect


2452-3216 © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21.

21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy

Asymptotic Stress Field in the Vicinity of the Mixed-Mode Crack in Damaged Materials Under Creep Conditions

Stepanova Larisaa*, Yakovleva Ekaterinaa

aDepartment of Mathematical Modelling in Mechanics, Samara State University,Akad. Pavlov, 1, 443011 Russia

Abstract

The creep crack problems in damaged materials under mixed mode loading under creep-damage coupled formulation are considered. The class of the self-similar solutions to the plane creep crack problems in a damaged medium under mixed-mode loading is given. With the similarity variable and the self-similar representation of the solution for a power-law creeping material and the Kachanov- Rabotnov power-law damage evolution equation the near crack-tip stresses, creep strain rates and continuity distributions for plane stress conditions are obtained. The similarity solutions are based on the hypothesis of the existence of the completely damaged zone near the crack tip. It is shown that the asymptotical analysis of the near crack-tip fields gives rise to the nonlinear eigenvalue problems. The technique permitting to find all the eigenvalues numerically is proposed and numerical solutions of the nonlinear eigenvalue problems arising from the mixed-mode crack problems in a power-law medium under plane stress conditions are obtained. Using the approach developed the eigenvalues different from the eigenvalues corresponding to the Hutchinson-Rice-Rosengren (HRR) problem are found. Having obtained the eigenspectra and eigensolutions the geometry of the completely damaged zone in the vicinity of the crack tip is found for all values of the mixity parameter. © 2016 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Scientific Committee of ECF21.

Keywords: Damage (integrity) parameter; similarity variable; self - similar solution; mixed-mode loading; creep-damage coupling; nonlinear eigenvalue problem; eigenvalue spectrum

1. Introduction

Knowledge of stress, strain and displacement fields in the vicinity of the crack tip under mixed-mode loading conditions is important for the justification of fracture mechanics criteria and has attracted considerable attention

* Stepanova Larisa. Tel.: +7 -846 -334-54-41; fax: +7-846-334-54-41.


Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of the Scientific Committee of ECF21.

http://creativecommons.org/licenses/by-nc-nd/4.0/

http://creativecommons.org/licenses/by-nc-nd/4.0/

http://crossmark.crossref.org/dialog/?doi=10.1016/j.prostr.2016.06.102&domain=pdf

794 Stepanova Larisa et al. / Procedia Structural Integrity 2 (2016) 793–8002 Author name / Structural Integrity Procedia 00 (2016) 000–000

nowadays (Altenbach and Sadowski (2015), Barenblatt (2014), Bui (2006), Murakami (2012), Wei (2014), Kuna (2013), Soyarslan et al. (2016), Stepanova and Adylina (2014), Voyiadis and Kattan (2012), Stepanova and Yakovleva (2014), Tumanov et al (2015)). Damage field around a crack tip essentially affects the surrounding stress field, and hence governs the crack extension behavior in the material. This effect of the damage field is an important problem either in the discussion of stability and convergence in crack extension analysis. So far mainly crack problems for the pure opening mode I at symmetrical loading have been thoroughly treated (Murakami (2012), Wei (2014)). The corresponding fracture criteria have been obtained on the assumption that the crack continues to extend along its original line (two-dimensional case) or plane (three-dimensional case) in a straightforward manner on the ligament. Nowadays the analysis of mixed-mode loading of cracked structures in nonlinear materials is of particular interest. In engineering practice, there are plenty of examples and reasons leading to mixed-mode loading of cracked structures when mode I is superimposed by mode II and/or III, the symmetry (or antisymmetry) is violated and the situation is related to mixed-mode loading (Kuna (2013)). The type of loading on a structure (tension, shear, bending, torsion) can also change during service. For a crack this results in an alteration of opening mode I, II and III which is why the study of mixed-mode loads is of particular importance (Kuna (2013), Stepanova and Adylina (2014)). In linear fracture mechanics the principle of superposition allows to obtain solutions for mixed mode I/II crack problems whereas in nonlinear fracture mechanics many questions are still open (Richard et al. (2014), Stepanova and Adylina (2014), Stepanova and Yakovleva (2014), Torabi and Abedinasab (2015), Chousal and Moura (2013)). Analysis of the near crack-tip fields in power-law hardening (or power-law creeping) damaged materials under mixed-mode loading results in new nonlinear eigenvalue problems in which the whole spectrum of the eigenvalues and orders of stress singularity have to be determined (Stepanova and Igonin (2014), Stepanova (2008), Stepanova (2009)). The objective of this study is to analyze the crack-tip fields in a damaged material under mixed-mode loading conditions and to consider the meso-mechanical effect of damage on the stress-strain state near the crack tip.

2. Mathematical statement of the mixed-mode crack problem and basic equations

A static mixed mode crack problem under plane stress conditions is considered. The equilibrium equations and compatibility condition in the polar coordinate system can, respectively, be written as

( ) ( ), , , ,

, , ,

0, 2 0,

2 , , .rr r r rr r r r

r r rr rr r rr

r r

r r r rθ θ θθ θθ θ θ θ

θ θ θθ θθ

σ σ σ σ σ σ σ

ε ε ε ε

+ + − = + + =

= − + (1)

The constitutive equations are described by the power law

( ) ( )13 / 2 / /nij e ijB sε σ ψ ψ−=& , (2)

where ijs are the deviatoric stress tensor components; ,B n are material constants; ψ is an integrity (continuity)

parameter; ijε& are the strain components which for the plane stress conditions take the form:

( ) ( ) ( ) ( ) ( )1 1 12 / 2 , 2 / 2 , 3 / 2 .n n n n n nrr e rr e rr r e rB B Bθθ θθ θθ θ θε σ σ σ ψ ε σ σ σ ψ ε σ σ ψ− − −= − = − =& & & (3)

The Mises equivalent stress is expressed by 2 2 23e rr rr rθθ θθ θσ σ σ σ σ σ= + − + .

The constitutive model (2) is the phenomenological model of Kachanov and Rabotnov widely employed in creep damage theory and in damage analysis of high temperature structures (Murakami (2012), Riedel (1987)). The material parameters pertinent to Eqs. 2 for copper, the aluminium alloy, ferritic steels obtained from creep curves are given by Riedel (1987). By noting that the creep damage is brought about by the development of microscopic voids in creep process, L.M. Kachanov represented the damage state by a scalar integrity variable ( )0 1ψ ψ≤ ≤ where

1ψ = and 0ψ = signify the initial undamaged state and the final completely damaged state (or final fractured state), respectively ((Murakami (2012), Voyiadis and Kattan (2012), Voyiadis (2015), Stepanova and Igonin (2014)). L.M. Kachanov described the damage development by means of an evolution equation

Author name / Structural Integrity Procedia 00 (2016) 000–000 3

( )/ ,meAψ σ ψ= −& (4)

where ψ& denotes the time derivative, while A and m are material constants. The solution of Eqs. 1 – 4 should satisfy the traditional traction free boundary conditions on the crack surfaces

( ) ( ), 0, , 0.rr rθθ θσ θ π σ θ π= ± = = ± =

The mixed-mode loading can be characterized in terms of the mixity parameter pM which is defined as

( ) ( ) ( )0

2 / arctan lim , 0 / , 0 .pr

rM r rθθ θπ σ θ σ θ

→= = =

The mixity parameter pM equals 0 for pure mode II; 1 for pure mode I, and 0 1pM< < for different mixities of modes I and II. Thus, for combine-mode fracture the mixity parameter pM completely specifies the near-crack-tip fields for a given value of the creep exponent. By postulating the Airy stress function ( ),rχ θ expressed in the polar

coordinate system, the stress components state are expressed as: ,rrθθσ χ= , 2, / , /rr r r rθθσ χ χ= + ,

( ), / ,r rrθ θσ χ= − . As for the asymptotic stress field at the crack tip 0r → , one can postulate the Airy stress function and the continuity parameter as

( ) ( )1 1

0 0, ( ), , 1 ( )j j

j jj j

r r f r r gλ γχ θ θ ψ θ θ∞ ∞

+ +

= == = −∑ ∑ . (5)

First consider the leading terms of the asymptotic expansions (5): 1( , ) ( )r r fλχ θ θ+= , 1ψ = , where λ is indeterminate exponent and ( )f θ is an indeterminate function of the polar angle, respectively. In view of the asymptotic presentation for the Airy stress potential (5) the asymptotic stress field at the crack tip is derived as follows 1( , ) ( )ij ijr rλσ θ σ θ−= % , where 1λ − denotes the exponent representing the singularity of the stress field, and will be called the stress singularity exponent hereafter. According to Eq. 2 the asymptotic strain field as 0r → takes the form ( 1)( , ) ( ).n

ij ijr Br λε θ ε θ−= % The compatibility condition in Eq. 1 results in the nonlinear forth-order

ordinary differential equation (ODE) for the function ( )f θ :

( )[ ]{ } [ ] { }[ ] [ ] [ ]{ [ ]

( ) [ ]

22 2 2 4 2

22

2 2 2 2

1 ( 1)(2 ) 2 / 2 2 6 ( 1) 1 ( 1) ' ( 1)( 3)

( 1)(2 ) 2 ( 1) ( 1)( 2) 2 ( 1) ' ( 1) ( 1)

( 1) ' ( 1) / 2 ( 1) ' ( 1)

IVe e e e

e

f f n f f f n n f hf f f n n h

f f n f f f f f f f f

f ff ff f f

λ λ λ λ

λ λ λ λ λ λ λ

λ λ λ λ λ λ

′′ ′′− + − + + + − + − + + − − ×

′′ ′′ ′′′ ′′ ′′× + − + + − + + + + + + + + + +

′′ ′′ ′′′+ + + − + − + + + ( )( )

[ ] ( )} [ ][ ] [ ] [ ]

4

2 2 2

4 4

1 2

( 1) ( 1) 3 2( 1) ( 1)(2 ) 2 / 2

( 1) ( 1)(2 ) 2 ( 1) 1 ( 1) ( 1)(2 1) 0,

e

e

e e

f f f

f f f f f f n f h f f

nf f f n nf f f

λ λ λ

λ λ λ λ λ λ


′ ′′+ + − −

′′ ′′ ′′ ′ ′′′ ′ ′′′− + + + + + + − + − + −

′′ ′′− − + − + + − + − + − − =

(6)

where the following notations are adopted

[ ] [ ]( ) [ ][ ] [ ] [ ]

2 2 2 2 2 2

2 2 2

( 1) ( 1) ( 1) 1 3 , ( 1)

( 1) ( 1) ( 1) ( 1) / 2 ( 1) ( 1) / 2 3 .ef f f f f f f f h f f

f f ff f f f f f f f f

λ λ λ λ λ λ λ λ

λ λ λ λ λ λ λ λ λ λ

′′ ′′ ′ ′′= + + + + − + + + + = + + ×

′ ′′′ ′ ′ ′′′ ′′ ′ ′ ′′× + + + + − + + + − + + + +

The boundary conditions imposed on the function ( )f θ follow from the traction free boundary conditions on the crack surfaces:

( ) 0, ( ) 0f fθ π θ π′= ± = = ± = . (7)

3. Numerical solution of the nonlinear eigenvalue problem. Computational scheme. Eigenvalues and eigenfunctions

Thus the eigenfunction expansion method results in the nonlinear eigenvalue problem: it is necessary to find eigenvalues λ leading to nontrivial solutions of Eq. 6 satisfying the boundary conditions (Eq. 7). Therefore, the

Stepanova Larisa et al. / Procedia Structural Integrity 2 (2016) 793–800 7952 Author name / Structural Integrity Procedia 00 (2016) 000–000

nowadays (Altenbach and Sadowski (2015), Barenblatt (2014), Bui (2006), Murakami (2012), Wei (2014), Kuna (2013), Soyarslan et al. (2016), Stepanova and Adylina (2014), Voyiadis and Kattan (2012), Stepanova and Yakovleva (2014), Tumanov et al (2015)). Damage field around a crack tip essentially affects the surrounding stress field, and hence governs the crack extension behavior in the material. This effect of the damage field is an important problem either in the discussion of stability and convergence in crack extension analysis. So far mainly crack problems for the pure opening mode I at symmetrical loading have been thoroughly treated (Murakami (2012), Wei (2014)). The corresponding fracture criteria have been obtained on the assumption that the crack continues to extend along its original line (two-dimensional case) or plane (three-dimensional case) in a straightforward manner on the ligament. Nowadays the analysis of mixed-mode loading of cracked structures in nonlinear materials is of particular interest. In engineering practice, there are plenty of examples and reasons leading to mixed-mode loading of cracked structures when mode I is superimposed by mode II and/or III, the symmetry (or antisymmetry) is violated and the situation is related to mixed-mode loading (Kuna (2013)). The type of loading on a structure (tension, shear, bending, torsion) can also change during service. For a crack this results in an alteration of opening mode I, II and III which is why the study of mixed-mode loads is of particular importance (Kuna (2013), Stepanova and Adylina (2014)). In linear fracture mechanics the principle of superposition allows to obtain solutions for mixed mode I/II crack problems whereas in nonlinear fracture mechanics many questions are still open (Richard et al. (2014), Stepanova and Adylina (2014), Stepanova and Yakovleva (2014), Torabi and Abedinasab (2015), Chousal and Moura (2013)). Analysis of the near crack-tip fields in power-law hardening (or power-law creeping) damaged materials under mixed-mode loading results in new nonlinear eigenvalue problems in which the whole spectrum of the eigenvalues and orders of stress singularity have to be determined (Stepanova and Igonin (2014), Stepanova (2008), Stepanova (2009)). The objective of this study is to analyze the crack-tip fields in a damaged material under mixed-mode loading conditions and to consider the meso-mechanical effect of damage on the stress-strain state near the crack tip.

2. Mathematical statement of the mixed-mode crack problem and basic equations

A static mixed mode crack problem under plane stress conditions is considered. The equilibrium equations and compatibility condition in the polar coordinate system can, respectively, be written as

( ) ( ), , , ,

, , ,

0, 2 0,

2 , , .rr r r rr r r r

r r rr rr r rr

r r

r r r rθ θ θθ θθ θ θ θ

θ θ θθ θθ

σ σ σ σ σ σ σ

ε ε ε ε

+ + − = + + =

= − + (1)

The constitutive equations are described by the power law

( ) ( )13 / 2 / /nij e ijB sε σ ψ ψ−=& , (2)

where ijs are the deviatoric stress tensor components; ,B n are material constants; ψ is an integrity (continuity)

parameter; ijε& are the strain components which for the plane stress conditions take the form:

( ) ( ) ( ) ( ) ( )1 1 12 / 2 , 2 / 2 , 3 / 2 .n n n n n nrr e rr e rr r e rB B Bθθ θθ θθ θ θε σ σ σ ψ ε σ σ σ ψ ε σ σ ψ− − −= − = − =& & & (3)

The Mises equivalent stress is expressed by 2 2 23e rr rr rθθ θθ θσ σ σ σ σ σ= + − + .

The constitutive model (2) is the phenomenological model of Kachanov and Rabotnov widely employed in creep damage theory and in damage analysis of high temperature structures (Murakami (2012), Riedel (1987)). The material parameters pertinent to Eqs. 2 for copper, the aluminium alloy, ferritic steels obtained from creep curves are given by Riedel (1987). By noting that the creep damage is brought about by the development of microscopic voids in creep process, L.M. Kachanov represented the damage state by a scalar integrity variable ( )0 1ψ ψ≤ ≤ where

1ψ = and 0ψ = signify the initial undamaged state and the final completely damaged state (or final fractured state), respectively ((Murakami (2012), Voyiadis and Kattan (2012), Voyiadis (2015), Stepanova and Igonin (2014)). L.M. Kachanov described the damage development by means of an evolution equation


( )/ ,meAψ σ ψ= −& (4)

where ψ& denotes the time derivative, while A and m are material constants. The solution of Eqs. 1 – 4 should satisfy the traditional traction free boundary conditions on the crack surfaces

( ) ( ), 0, , 0.rr rθθ θσ θ π σ θ π= ± = = ± =

The mixed-mode loading can be characterized in terms of the mixity parameter pM which is defined as

( ) ( ) ( )0

2 / arctan lim , 0 / , 0 .pr

rM r rθθ θπ σ θ σ θ

→= = =

The mixity parameter pM equals 0 for pure mode II; 1 for pure mode I, and 0 1pM< < for different mixities of modes I and II. Thus, for combine-mode fracture the mixity parameter pM completely specifies the near-crack-tip fields for a given value of the creep exponent. By postulating the Airy stress function ( ),rχ θ expressed in the polar

coordinate system, the stress components state are expressed as: ,rrθθσ χ= , 2, / , /rr r r rθθσ χ χ= + ,

( ), / ,r rrθ θσ χ= − . As for the asymptotic stress field at the crack tip 0r → , one can postulate the Airy stress function and the continuity parameter as

( ) ( )1 1

0 0, ( ), , 1 ( )j j

j jj j

r r f r r gλ γχ θ θ ψ θ θ∞ ∞

+ +

= == = −∑ ∑ . (5)

First consider the leading terms of the asymptotic expansions (5): 1( , ) ( )r r fλχ θ θ+= , 1ψ = , where λ is indeterminate exponent and ( )f θ is an indeterminate function of the polar angle, respectively. In view of the asymptotic presentation for the Airy stress potential (5) the asymptotic stress field at the crack tip is derived as follows 1( , ) ( )ij ijr rλσ θ σ θ−= % , where 1λ − denotes the exponent representing the singularity of the stress field, and will be called the stress singularity exponent hereafter. According to Eq. 2 the asymptotic strain field as 0r → takes the form ( 1)( , ) ( ).n

ij ijr Br λε θ ε θ−= % The compatibility condition in Eq. 1 results in the nonlinear forth-order

ordinary differential equation (ODE) for the function ( )f θ :

( )[ ]{ } [ ] { }[ ] [ ] [ ]{ [ ]

( ) [ ]

22 2 2 4 2

22

2 2 2 2

1 ( 1)(2 ) 2 / 2 2 6 ( 1) 1 ( 1) ' ( 1)( 3)

( 1)(2 ) 2 ( 1) ( 1)( 2) 2 ( 1) ' ( 1) ( 1)

( 1) ' ( 1) / 2 ( 1) ' ( 1)

IVe e e e

e

f f n f f f n n f hf f f n n h

f f n f f f f f f f f

f ff ff f f

λ λ λ λ


λ λ λ λ λ λ

′′ ′′− + − + + + − + − + + − − ×

′′ ′′ ′′′ ′′ ′′× + − + + − + + + + + + + + + +

′′ ′′ ′′′+ + + − + − + + + ( )( )

[ ] ( )} [ ][ ] [ ] [ ]

4

2 2 2

4 4

1 2

( 1) ( 1) 3 2( 1) ( 1)(2 ) 2 / 2

( 1) ( 1)(2 ) 2 ( 1) 1 ( 1) ( 1)(2 1) 0,

e

e

e e

f f f

f f f f f f n f h f f

nf f f n nf f f

λ λ λ

λ λ λ λ λ λ


′ ′′+ + − −

′′ ′′ ′′ ′ ′′′ ′ ′′′− + + + + + + − + − + −

′′ ′′− − + − + + − + − + − − =

(6)

where the following notations are adopted

[ ] [ ]( ) [ ][ ] [ ] [ ]

2 2 2 2 2 2

2 2 2

( 1) ( 1) ( 1) 1 3 , ( 1)

( 1) ( 1) ( 1) ( 1) / 2 ( 1) ( 1) / 2 3 .ef f f f f f f f h f f

f f ff f f f f f f f f

λ λ λ λ λ λ λ λ

λ λ λ λ λ λ λ λ λ λ

′′ ′′ ′ ′′= + + + + − + + + + = + + ×

′ ′′′ ′ ′ ′′′ ′′ ′ ′ ′′× + + + + − + + + − + + + +

The boundary conditions imposed on the function ( )f θ follow from the traction free boundary conditions on the crack surfaces:

( ) 0, ( ) 0f fθ π θ π′= ± = = ± = . (7)

3. Numerical solution of the nonlinear eigenvalue problem. Computational scheme. Eigenvalues and eigenfunctions

Thus the eigenfunction expansion method results in the nonlinear eigenvalue problem: it is necessary to find eigenvalues λ leading to nontrivial solutions of Eq. 6 satisfying the boundary conditions (Eq. 7). Therefore, the


order of the stress singularity is the eigenvalue and the angular variations of the field quantities correspond to the eigenfunctions. When we consider mode I loading or mode II loading conditions symmetry or antisymmetry requirements of the problem with respect to the crack plane at 0θ = are utilized. Due to the symmetry (or antisymmetry) the solution is sought for one of the half-planes. In analyzing the crack problem under mixed-mode loading conditions the symmetry or antisymmetry arguments can not be used and it is necessary to seek for the solution in the whole plane π θ π− ≤ ≤ . To find the numerical solution one has to take into account the value of the mixity parameter. For this purpose in the framework of the proposed technique Eq. 6 is numerically solved on the interval [ ]0,π and the two-point boundary value problem is reduced to the initial problem with the initial conditions reflecting the value of the mixity parameter

( 0) 1,f θ = = ( ) ( )( 0) 1 / / 2 ,pf tg Mθ λ π′ = = + ( ) 0,f θ π= = ( ) 0.f θ π′ = = The first initial condition is the normalization condition. The second condition follows from the value of the

mixity parameter specified. At the next step the numerical solution of Eq. 6 is found on the interval [ ],0π− with the following boundary conditions

( ) 0,f θ π= − = ( ) 0,f θ π′ = − = ( 0) 1,f θ = = ( ) ( )( 0) 1 / / 2 .pf tg Mθ λ π′ = = + The analogous approach has been realized by Stepanova (2008) where the near mixed-mode crack-tip stress field

under plane strain conditions was analyzed. It is assumed that the eigenvalue of the problem considered equals the eigenvalue of the classical HRR problem /( 1)n nλ = + . However, it turns out that when we construct the numerical solution for the mixed-mode crack problem the radial stress component ( , )rr rσ θ has discontinuity at 0θ = whereas for the cases of pure mode I and pure mode II loadings when 1pM = and 0pM = are valid the radial stress component is continuous at 0θ = . Numerical analysis carried out previously for mixed-mode crack problem under plane strain conditions leads to the continuous angular distributions of the radial stress component

( , 0)rr rσ θ = (Stepanova and Yakovleva (2014)). Thus one can compute the whole set of eigenvalues for plane stress conditions from the continuity requirements of the radial stress components on the line extending the crack. In accordance with the procedure proposed the spectrum of the eigenvalues λ is numerically obtained. Results of computations are shown in Tables 1-3 where the new eigenvalues λ computed and the values of the functions

( 0)f θ′′ = , ( 0)f θ′′′ = , ( )f θ π′′ = − and ( )f θ π′′′ = − numerically obtained for the different values of the mixity

parameter pM and the creep exponent n are given. The angular distributions of the stress components for different values of creep exponent n and for all values of the mixity parameter pM are shown in Fig. 1.

Table 1. Eigenvalues for different values of mixity parameter for plane stress conditions 2n = . pM λ ( 0)f θ′′ = ( 0)f θ′′′ = ( )f θ π′′ = − ( )f θ π′′′ = −

0.9 -0.3003200 -0.25428500 -0.52319280 0.36781000 0.41793000

0.8 -0.2860900 0.30988600 -0.65543910 -0.14222000 1.23657500

0.7 -0.2678900 -0.40297913 -0.80444475 -0.37921000 0.54939150

0.6 -0.2609300 -0.46493199 1.03847110 -0.54340000 0.46094200

0.5 -0.2523320 -0.52217930 -1.40075019 -0.72780000 0.42459230

0.4 -0.2436980 -0.57136233 -1.98711539 -0.97155000 0.40989380

0.3 -0.2370190 -0.61089207 -2.95625279 -1.35116000 0.41294900

0.2 -0.2324790 -0.64000914 -4.76598544 -2.08610169 0.44422935

0.1 -0.2298723 -0.65774480 -9.82544937 -4.26300089 0.57184713


-3 -2 1-1 0 2 3 q-0.4

-0.2

0.2

0.4

0.6

0.8

srr

0.40.30.20.1

=0.05M p ---------------

0.950.90.80.70.60.5 ---

---------------

n=6a)

-3 -2 1-1 0 2 3 q

-0.8

0.10

-0.04

0.06

0

sq

0.8

-0.06

-0.02

0.02

0.04

0.40.30.20.1

=0.05M p ---------------

0.950.90.80.70.60.5 ---

---------------

n=6

rb)

-3 -2 1-1 0 2 3 q

0

0.04

0.02

-0.02

-0.04

-0.06

-0.08

-0.10

sqq

0.40.30.20.1

=0.05M p ---------------

0.950.90.80.70.60.5 ---

---------------

n=6

c)

-3 -2 1-1 0 2 3 q

0

-0.4

-0.2

0.2

0.4

0.6

0.8

srr

0.30.20.1

=0.05M p ------------

0.950.90.80.70.60.5 ---

---------------

n=8

0.4 ---

0 7 ---

0.2

d)

-3 -2 1-1 0 2 3

-0.10

0.10

-0.05

0.05

0

n=8

q

sr

0.40.30.20.1

=0.05M p ---------------

0.950.90.80.70.60.5 ---

---------------

0.4

qe)

-3 -2 1-1 0 2 3

0

0.04

0.02

-0.02

-0.04

-0.06

-0.08

-0.10

-0.12

-0.14

0.40.30.20.1

=0.05M pn=8

---------------

0.950.90.80.70.60.5 ---

---------------

sqq

q

0 959

0 66 ---

g)

Fig. 1.Angular distributions of the stress components for different values of the mixity parameter and for different values of the creep exponent: (a) angular distribution of the radial stress rrσ for 6n = ; (b) angular distribution of the shear stress rθσ for 6n = ; (c) angular distribution of the

tangential stress θθσ for 6n = ; (d) angular distribution of the radial stress rrσ for 8n = ; (e) angular distribution of the shear stress rθσ for

8n = ; (g) angular distribution of the tangential stress θθσ for 8n = .


order of the stress singularity is the eigenvalue and the angular variations of the field quantities correspond to the eigenfunctions. When we consider mode I loading or mode II loading conditions symmetry or antisymmetry requirements of the problem with respect to the crack plane at 0θ = are utilized. Due to the symmetry (or antisymmetry) the solution is sought for one of the half-planes. In analyzing the crack problem under mixed-mode loading conditions the symmetry or antisymmetry arguments can not be used and it is necessary to seek for the solution in the whole plane π θ π− ≤ ≤ . To find the numerical solution one has to take into account the value of the mixity parameter. For this purpose in the framework of the proposed technique Eq. 6 is numerically solved on the interval [ ]0,π and the two-point boundary value problem is reduced to the initial problem with the initial conditions reflecting the value of the mixity parameter

( 0) 1,f θ = = ( ) ( )( 0) 1 / / 2 ,pf tg Mθ λ π′ = = + ( ) 0,f θ π= = ( ) 0.f θ π′ = = The first initial condition is the normalization condition. The second condition follows from the value of the

mixity parameter specified. At the next step the numerical solution of Eq. 6 is found on the interval [ ],0π− with the following boundary conditions

( ) 0,f θ π= − = ( ) 0,f θ π′ = − = ( 0) 1,f θ = = ( ) ( )( 0) 1 / / 2 .pf tg Mθ λ π′ = = + The analogous approach has been realized by Stepanova (2008) where the near mixed-mode crack-tip stress field

under plane strain conditions was analyzed. It is assumed that the eigenvalue of the problem considered equals the eigenvalue of the classical HRR problem /( 1)n nλ = + . However, it turns out that when we construct the numerical solution for the mixed-mode crack problem the radial stress component ( , )rr rσ θ has discontinuity at 0θ = whereas for the cases of pure mode I and pure mode II loadings when 1pM = and 0pM = are valid the radial stress component is continuous at 0θ = . Numerical analysis carried out previously for mixed-mode crack problem under plane strain conditions leads to the continuous angular distributions of the radial stress component

( , 0)rr rσ θ = (Stepanova and Yakovleva (2014)). Thus one can compute the whole set of eigenvalues for plane stress conditions from the continuity requirements of the radial stress components on the line extending the crack. In accordance with the procedure proposed the spectrum of the eigenvalues λ is numerically obtained. Results of computations are shown in Tables 1-3 where the new eigenvalues λ computed and the values of the functions

( 0)f θ′′ = , ( 0)f θ′′′ = , ( )f θ π′′ = − and ( )f θ π′′′ = − numerically obtained for the different values of the mixity

parameter pM and the creep exponent n are given. The angular distributions of the stress components for different values of creep exponent n and for all values of the mixity parameter pM are shown in Fig. 1.


0.9 -0.3003200 -0.25428500 -0.52319280 0.36781000 0.41793000

0.8 -0.2860900 0.30988600 -0.65543910 -0.14222000 1.23657500

0.7 -0.2678900 -0.40297913 -0.80444475 -0.37921000 0.54939150

0.6 -0.2609300 -0.46493199 1.03847110 -0.54340000 0.46094200

0.5 -0.2523320 -0.52217930 -1.40075019 -0.72780000 0.42459230

0.4 -0.2436980 -0.57136233 -1.98711539 -0.97155000 0.40989380

0.3 -0.2370190 -0.61089207 -2.95625279 -1.35116000 0.41294900

0.2 -0.2324790 -0.64000914 -4.76598544 -2.08610169 0.44422935

0.1 -0.2298723 -0.65774480 -9.82544937 -4.26300089 0.57184713


-3 -2 1-1 0 2 3 q-0.4

-0.2

0.2

0.4

0.6

0.8

srr

0.40.30.20.1

=0.05M p ---------------

0.950.90.80.70.60.5 ---

---------------

n=6a)

-3 -2 1-1 0 2 3 q

-0.8

0.10

-0.04

0.06

0

sq

0.8

-0.06

-0.02

0.02

0.04

0.40.30.20.1

=0.05M p ---------------

0.950.90.80.70.60.5 ---

---------------

n=6

rb)

-3 -2 1-1 0 2 3 q

0

0.04

0.02

-0.02

-0.04

-0.06

-0.08

-0.10

sqq

0.40.30.20.1

=0.05M p ---------------

0.950.90.80.70.60.5 ---

---------------

n=6

c)

-3 -2 1-1 0 2 3 q

0

-0.4

-0.2

0.2

0.4

0.6

0.8

srr

0.30.20.1

=0.05M p ------------

0.950.90.80.70.60.5 ---

---------------

n=8

0.4 ---

0 7 ---

0.2

d)

-3 -2 1-1 0 2 3

-0.10

0.10

-0.05

0.05

0

n=8

q

sr

0.40.30.20.1

=0.05M p ---------------

0.950.90.80.70.60.5 ---

---------------

0.4

qe)

-3 -2 1-1 0 2 3

0

0.04

0.02

-0.02

-0.04

-0.06

-0.08

-0.10

-0.12

-0.14

0.40.30.20.1

=0.05M pn=8

---------------

0.950.90.80.70.60.5 ---

---------------

sqq

q

0 959

0 66 ---

g)

Fig. 1.Angular distributions of the stress components for different values of the mixity parameter and for different values of the creep exponent: (a) angular distribution of the radial stress rrσ for 6n = ; (b) angular distribution of the shear stress rθσ for 6n = ; (c) angular distribution of the

tangential stress θθσ for 6n = ; (d) angular distribution of the radial stress rrσ for 8n = ; (e) angular distribution of the shear stress rθσ for

8n = ; (g) angular distribution of the tangential stress θθσ for 8n = .



0.9 -0.25560 0.0629300 -0.84040608 0.91827825 -0.44735992

0.8 -0.24450 0.0618000 -0.99744294 0.72118794 -0.34292779

0.7 -0.23350 0.0629600 -1.18109399 0.47043046 -0.19390281

0.6 -0.22020 0.0270200 -1.38664305 -0.2687000 0.33821000

0.5 -0.21079 0.0250800 -1.68869370 -0.4582000 0.27883500

0.4 -0.20527 0.0794000 -2.16329580 -0.7040500 0.36147500

0.3 -0.20303 0.1962682 -2.96732416 -1.1093000 0.53612990

0.2 -0.20573 0.4170900 -4.60578260 -1.9332800 0.91471369

0.1 -0.21570 0.8971300 -9.66506720 -4.5057200 2.13143060


0.9 -0.2300000 0.12630 -0.95970043 1.00961648 -0.63151712

0.8 -0.2180000 0.13638 -1.12682114 0.81269900 -0.50297035

0.7 -0.2088000 0.15539 -1.32590338 0.58885377 -0.36077682

0.6 0.19930000 0.15484 -1.55328928 -0.16700000 0.69325000

0.5 -0.1907800 0.15920 -1.85600631 -0.37120000 0.23952300

0.4 -0.1887000 0.24360 -2.34861936 -0.61155000 0.37189580

0.3 -0.1933200 0.40705 -3.19053122 -1.01899000 0.61801287

0.2 -0.2061950 0.70100 -4.94411518 -1.86793000 1.14461341

0.1 -0.2304187 1.34989 -10.72665207 -4.59496000 2.87542559

The method proposed has been applied to nonlinear eigenvalue problems arising from the problem of the determining the near crack-tip fields in the damaged materials. In continuum damage mechanics (Ochsner (2016), Altenbach and Sadowski (2015), Murakami (2012), Kuna (2013), Voyiadis (2015), Voyiadis and Kattan (2012), Zhang and Cai (2010)), the damage state at an arbitrary point in the material is represented by a properly defined integrity variable ( )θψ ,r . The integrity parameter reaches its critical value at fracture. According to this notion, a crack in a fracture process can be modeled with the concept of a completely damaged zone in the vicinity of the crack tip. Namely a crack can be represented by a region where the integrity state has attained to its critical state

crψψ = , i.e., by the completely damaged zone (CDZ). Then the development of the crack and its preceding damage can be elucidated by analyzing the local states of stress, strain and damage. The CDZ may be interpreted as the zone of critical decrease in the effective area due to damage development. Inside the completely damaged zone the damage involved reaches its critical value (for instance, the damage parameter reaches unity) and a complete fracture failure occurs. In view of material damage stresses are relaxed to vanishing (Voyiadis (2015), Stepanova and Igonin (2014), Stepanova and Adulina (2014), Stepanova and Yakovleva (2014)). Therefore, one can assume that the stress components in the CDZ equal zero. Outside the zone damage alters the stress distribution substantially compared to the corresponding non-damaging material. Well outside the CDZ the continuity parameter is equal to 1. Asymptotic remote boundary conditions are the asymptotic approaching the HRR solution. Dimensional analysis of the system formulated shows that the damage mechanics equations must have similarity solutions of the form

( ) 1/ ˆ( , , ) ( , ),mij ijr t At Rσ θ σ θ−= ),(ˆ),,( θψθψ Rtr = ,

where ( ) ( 1) / /n mnR r At BI C− + ∗= is the similarity variable. It should be noted that the remote boundary conditions

can be formulated in a more general form ( ) ),(~,, nrCtr ijs

ij θσθσ =∞→ , where the stress singularity exponent s is


unknown and has to be determined as a part of solution, C% is the amplitude of the stress field at infinity defined by the specimen configuration and loading conditions. For the power-law creep constitutive relations, the power-law damage evolution equation and the more general remote boundary conditions the self-similar variable

( ) CAtrR sm ~)/(1= can be introduced. After introducing the self-similar variable the equilibrium equations, the constitutive equations, the compatibility condition retain their forms, whereas the damage evolution equation becomes ( )ˆ ˆˆ, / m

R eR smψ σ ψ= − (the superscript ˆ is further omitted). The asymptotic solution outside the completely

damaged zone ( )R →∞ is sought in the form 1

0 0( , ) ( ), ( , ) 1 ( ),j j

j jj jR R f R R gλ γχ θ θ ψ θ θ∞ ∞+

= == = −∑ ∑

where , , ( ), ( )j j j jf gλ γ θ θ should be found as a part of the solution. The asymptotic representation of the stress components outside the CDZ has the form

1 ( )0

( , ) ( )k kij ijk

R Rλσ θ σ θ∞ −=

=∑ .

The kinetic law of damage evolution permits to obtain: ( ) ( )010 1 λλλγ −+−= kmk . The creep strain rates in the

vicinity of the crack tip can be represented as ( )( )0 1 ( )0

( , ) ( )n km kij ijk

R R λε θ ε θ∞ − +

== ∑& & . The compatibility equation results

in the following set of ODEs ( ) ( ) ( )( )( ) ( ) ( ) ( )

, 02 1 1 , 1k k k kk R RR k RR k k ks s s s s n kmθ θθ θθε ε ε ε λ+ = − − + = − +& & & (8)

Having obtained the numerical solutions of Eqs. 8 and the solutions of the nonlinear eigenvalue problems earlier considered one can find the configuration of the completely damaged zone modeled in the neighborhood of the crack tip through the equation

,0)(1),(0

=−= ∑ =θθψ γ

j

k

jgRR j 1, 2,...k =

One can compare the boundaries of the CDZ given by the two-term and three-term asymptotic expansions of the integrity (continuity) parameter. It is turned out that if the asymptotic remote boundary condition is postulated as the condition of the asymptotic approaching the HRR-field then the shapes of the CDZ given by the two-term asymptotic expansion and three-term asymptotic expansions differ essentially from each other. The new stress asymptotic behavior results in the contours of the CDZ which converge to the limit contour. The new far field stress asymptotic can be interpreted as the intermediate stress asymptotics valid for times and distances at which effects of the initial and boundary conditions on the stress and damage distributions are lost. The geometry of the completely damage zone for different values of the mixity parameter is shown in Fig. 2 where 1,2k = is designed the boundary of the CDZ built by the use of the 1k + - term asymptotic expansion of continuity.

0-1 1 X1-2

0

-0.4

0.4

X2

0.8

-0.8

n=6Mp=0.3

crack surfaces

1 2

a)

0-0.5-1-1.5 0.5 1 X1-2

0

-0.2

0.2

-0.4

0.4

0.6

X2

0.8

-0.6

n=6Mp=0.5

1 2

crack surfaces

b)

0-0.5-1-1.5 0.5 1

0

-0.2

0.2

-0.4

0.4

0.6

X2

X1

n=6Mp=0.7

1 2

crack surfaces

c)

Fig. 2. Geometry of the completely damaged zone for different values of the mixity parameters: (a) for the mixity parameter 0.3pM = ; (b) for the mixity parameter 0.5pM = ; (c) for the mixity parameter 0.7pM = . The red line shows the boundary of the CDZ obtained by the two-term asymptotic expansion of the integrity


parameter whereas the blue line shows the boundary of the CDZ obtained by the three-term asymptotic expansion of the integrity parameter. From Fig. 2 it can be seen that the boundary of the CDZ determined by the use of the 1k + -term asymptotic expansion of continuity is very close to the boundary built by the k -term asymptotic expansion of the continuity parameter whereas the HRR stress field results in the boundary of the CDZ given by the two-term expansion which differs substantially from the boundary of the CDZ given by the three-term asymptotic expansion of the integrity parameter by the form and dimensions.

4. Summary

Asymptotic crack-tip fields in damaged materials are developed for a stationary plane stress crack under mixed mode loading conditions. The asymptotic solutions are obtained by the use of the similarity variable and the similarity presentation of the solution. On the basis of the self-similar representation of the solution the near crack-tip stress, creep strain rate and continuity distributions are given. It is shown that meso-mechanical effect of damage accumulation near the crack tip results in new intermediate stress field asymptotic behavior and requires the solution of nonlinear eigenvalue problems. To attain eigensolutions a numerical scheme is worked out and the results obtained provide the additional eigenvalues of the HRR problem. By the use of the method proposed the whole set of eigenvalues for the mode crack in a power law material under mixed mode loading can be determined. The self-similar solutions are based on the idea of the existence of the completely damaged zone near the crack tip. The higher order terms of the asymptotic expansions of stresses, creep strain rates and continuity parameter allowing to obtain the contours of the completely damaged zone in the vicinity of the crack tip are derived and investigated.

References

Altenbach, H., Sadowski, T., 2015. Failure and Damage Analyses of Advanced Materials, Springer, Berlin. Barenblatt, G.I., 2014. Flow, Deformation and Fracture: Lectures on Fluid Mechanics and the Mechanics of Deformable Solids for

Mathematicians and Physicists, Cambridge University Press, Cambridge. Bui, H.D., 2006. Fracture Mechanics. Inverse Problems and Solutions, Springer, Dordrecht. Chousal, J.A.G., de Moura, M.F.S.F. 2013. Mixed mode I+II continuum damage model applied to fracture characterization of bonded joints. Int.

J. of Adhesion and Adhesives 41, 92-97. Kuna, M., 2013. Finite Elements in Fracture Mechanics. Theory-Numerics-Applications. Springer, Dordrecht. Murakami, S., 2012. Continuum Damage Mechanics. A Continuum Mechanics Approach to the Analysis of Damage and Fracture. Springer,

Dordrecht. Ochsner, A., 2016. Continuum Damage and Fracture Mechanics. Springer Science + Business Media, Singapore. Richard, H.A., Schramm, B., Schrimeisen, N.-H., 2014. Cracks on Mixed Mode loading – Theories, experiments, simulations. International

Journal of Fatigue 62, 93 – 103. Riedel, H., 1987. Fracture at High Temperature, Springer – Verlag, Berlin. Soyarslan, C, Richter, H., Bargmann, S., 2016. Variants of Lemaitre damage model and their use in formability prediction of metallic materials.

Mechanics of Materials 92, 58-79. Stepanova, L.V., Adylina, E.M. 2014. Stress-strain state in the vicinity of a crack under mixed loading. Journal of Applied Mechanics and

Technical Physics 55(5), 885-895. Stepanova, L.V., Igonin, S.A., 2014. Perturbation method for solving the nonlinear eigenvalue problem arising from fatigue crack growth

problem in a damaged medium. Applied Mathematical Modelling 38(14), 3436-3455. Stepanova, L.V., 2008. Eigenspectra and orders of stress singularity at a mode I crack tip for a power-law medium. Comptes Rendus. Mecanique

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1332-1347. Stepanova, L.V., Yakovleva, E.M., 2014. Mixed-mode loading of the cracked plate under plane stress conditions. PNRPU Mechanics Bulletin 3,

129-162. Torabi, A.R., Abedinasab, S.M., 2015. Brittle fracture in key-hole notches under mixed mode loading. Experimental study and theoretical

predictions. Engineering Fracture Mechanics 134, 35 – 53. Tumanov, A.V., Shlyannikov, V.N., Chandra Kishen, J.M., (2015). An automatic algorithm for mixed mode crack growth rate based on drop

potential method. International Journal of Fatigue 81, 227 – 237. Voyiadis, G.Z., Kattan, P.I., 2012. Damage Mechanics with Finite Elements: Practical Applications with Computer Tools. Springer, Berlin. Voyiadis, G.Z., 2015. Handbook of Damage Mechanics, Springer – Verlag, New York. Wei, R.P., 2014. Fracture Mechanics. Integration of Mechanics, Materials Science and Chemistry, Cambridge University Press, Cambridge. Zhang, W., Cai, Y., 2010. Continuum Damage Mechanics and Numerical Applications. Springer Science & Business Media, Heidelberg.

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