Basics Elements on Linear Elastic Fracture Mechanics and Crack Growth Modeling Sylvie Pommier

Basics elements on linear elastic fracture mechanics and crack growth modeling Sylvie Pommier

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Sylvie Pommier. Basics elements on linear elastic fracture mechanics and crack growth modeling. Doctoral. France. 2017. cel-01636731

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Sylvie Pommier, LMT (ENS Paris-Saclay, CNRS, Université Paris-Saclay) Fail Safe Damage Tolerant Design

• Consider the eventuality of damage or of the presence of defects,

• predict if these defects or damage may lead to fracture,

• and, in the event of failure, predicts the consequences (size, velocity and trajectory

of the fragments) 2 Foundations of fracture mechanics : The Liberty Ships

• 2700 Liberty Ships were built between 1942 and the end of WWII • The production rate was of 70 ships / day • duration of construction: 5 days • 30% of ships built in 1941 have suffered catastrophic failures • 362 lost ships The fracture mechanics concepts were still unknown

Causes of fracture:

hiver 1941 hiver • Welded Structure rather than bolted,

– offering a substantial assembly time gain but with a continuous path offered for

ships cracks to propagate through the structure. • Low quality of the welds (presence of

cracks and internal stresses) Liberty Liberty • Low quality steel, ductile/brittle transition around 0°C Liberty Ships, WWII, 1941, Brittle fracture

4 LEFM - Linear elastic fracture mechanics

Georges Rankine Irwin “the godfather of fracture mechanics »

• Stress intensity factor K

• Introduction of the concept of fracture toughness KIC • Irwin’s plastic zone (monotonic and cyclic) • Energy release rate G and Gc

(G in reference to Griffith) Georges Rankine Irwin Rankine Georges Historical context

Previous authors

Griffith A. A. - 1920 –"The phenomenon of rupture and flow in solids", 1920, Philosophical Transactions of the Royal Society, Vol. A221 pp.163-98

Westergaard H. M. – 1939 - Bearing Pressures and Cracks, Journal of Applied Mechanics 6: 49-53.

Muskhelishvili N. – 1954 - Ali Kheiralla, A. Muskhelishvili, N.I. Some Basic problems of the mathematical theory of elasticity. Third revis. and augmented. Moscow, 1949, J.Appl. Mech.,21 (1954), No 4, 417- 418. n.b. Joseph Staline died in 1953 Fatigue crack growth: De Havilland Comet

3 accidents 26/10/1952, departing from Rome Ciampino March 1953, departing from Karachi Pakistan 10/01/1954, Crash on the Rome-London flight (with passengers)

Paris & Erdogan 1961 They correlated the cyclic fatigue crack growth rate da / dN with the stress intensity factor amplitude DK

Introduction of the Paris’ law for modeling fatigue crack growth Fatigue remains a topical issue

8 Mai 1842 - Meudon (France) 3 Juin 1998 - Eschede (Allemagne) Fracture of an axle by fatigue Fracture of a wheel by Fatigue

8 Development of rules for the EASA certification

Aloha April, 28th 1988,

Los Angeles, June, 2nd 2006,

9 Rotor Integrity Sub-Committee (RISC)

UAL 232, July 19, 1989 Sioux City, Iowa • DC10-10 crashed on landing • In-Flight separation of Stage 1 Fan Disk • Failed from cracks out of material anomaly - Hard Alpha produced during melting • Life Limit: 18,000 cycles. Failure: 15,503 cycles. • 111 fatalities • FAA Review Team Report (1991) recommended: - Changes in Ti melt practices, quality controls - Improved mfg and in-service inspections - Lifing Practices based on damage tolerance

 AIA Rotor Integrity Sub-Committee (RISC) : Elaboration of AC 33.14-1 DL 1288, July 6, 1996 , Pensacola, Florida

• MD-88 engine failure on take-off roll • Pilot aborted take-off • Stage 1 Fan Disk separated; impacted cabin • Failure from abusively machined bolthole • Life Limit: 20,000 cycles. Failure: 13,835 cycles. • 2 fatalities • NTSB Report recommended ... - Changes in inspection methods, shop practices - Fracture mechanics based damage tolerance

Elaboration of AC 33.70-2 Why ? • To prevent fatalities and disaster

Where ? • Public transportation (trains, aircraft, ships…)

• Energy production (nuclear power plant, oil extraction and transportation …) Damage • Any areas of risk to public health and environment tolerance

How ? • Critical components are designed to be damage tolerant / fail safe

• Rare events (defects and cracks) are assumed to be certain (deterministic approach) and are introduced on purpose for lab. tests and certification Fracture mechanics

One basic assumption : The structure contains a singularity (ususally a geometric discontinuity, for example: a crack)

Two main questions : What are the relevant variables to characterize the risk of fracture and to be used in fracture criteria ?

What are the suitable criteria to determine if the crack may propagate or remain arrested, the crack growth rate and the crack path ?

13 Classes of material behaviour : relevant variables

Linear elastic behaviour: linear elastic fracture mechanics (K)

Nonlinear behavior: non-linear fracture mechanics Hypoelasticity : Hutchinson Rice & Rosengren, (J) Ideally plastic material : Irwin, Dugdale, Barrenblatt etc.

Time dependent material behaviours: viscoelasticity, viscoplasticity (C*)

Complex non linear material behaviours : Various local and non local approaches of failure, J. Besson, A. Pineau, G. Rousselier, A. Needleman, Tvergaard , S. Pommier etc.

14 Classes of fracture mechanisms : criteria

• Brittle fracture

• Ductile fracture

• Dynamic fracture

• Fatigue crack growth

• Creep crack growth

• Crack growth by corrosion, oxydation, ageing

• Coupling between damage mechanisms

15 Mechanisms acting at very different scales of time and space, an assumption of scales separation

• Atomic scale (surface oxydation, ageing, …) • Microstructural scale (grain boundary corrosion, creep, oxydation, persistent slip band in fatigue etc… ) • Plastic zone scale or damaged zone (material hardening or softening, continuum damage, ductile damage...) • Scale of the structure (wave propagation …)

Atomic cohesion energy Brittle fracture 10 J/m2 energy 10 000 J/m2

16 Classes of relevant assumptions : application of criteria

Long cracks (2D problem, planar crack with a straight crack)

Curved cracks, branched cracks, merging cracks (3D problem, non- planar cracks, curved crack fronts)

Short cracks (3D problem, influence of free surfaces, scale and gradients effects)

Other discontinuities and singularities: • Interfaces / free surfaces, • Contact front in partial slip conditions, • acute angle ending on a edge,

17 Griffith’ theory Threshold for unsteady crack growth (brittle or ductile)

Relevant variable : energy release rate G

Criteria : An unsteady crack growth occurs if the cohesion energy released by the structure because of the creation of new cracked surfaces reaches the energy required to create these new cracked surfaces G = Gc

Data : critical energy release rate Gc Griffith’ theory

We x t : work of external forces

DU elastic : variation of the elastic energy of the structure

DU surface: variation of the surface energy of the structure

DU  DU elastic  DU surface  Wext

 DU surface  2 da  Wext  DU elastic

G  2 where Criteria : DU W G   elastic ext da 19 Evolution by Bui, Erlacher & Son

dU  Wext  Q where TdS  Q  0 dU  dF TdS  SdT in isothermal conditions dT  0

TdS  Q  dF Wext   0

 dFvolume  dFsurface Wext   0  G  Gc da  0 where Free energy instead of internal energy DF G  surface  2 Isothermal conditions instead of adiabatic c da Second principle DF W  G   volume ext da 20 J Integral (Rice) DF W  G   volume ext da

Eschelby tensor : energy density

J integral , (Rice’s integral if q is coplanar)

q vector: the crack front motion 21 J contour integral If the crack faces are free surfaces (no friction, no y fluid pressure …),

If volume forces can be neglected (inertia, electric field...) x

Then the J integral is shown to be independent of the choice of the selected integration contour

휕푢 퐺 = 퐽= 휑 푑푦 − 휎푛. Γ 푓푟푒푒 푒푛푒푟푔푦 푑푒푛푠푖푡푦 휕푥

22 Applications

C. Stoisser, I. Boutemy and F. Hasnaoui

23 • The crack faces must be free surfaces (no friction, no fluid pressure)

• Gc is a material constant (single mechanism, surfacic mechanism only)

• What if non isothermal conditions are considered ? Limitations • Unsteady crack growth criteria, non applicable to steady crack propagation,

• The surfacic energy 2 may be negligible compared with the energy dissipated in plastic work or continuum damage / localization process Linear Elastic Fracture Mechanics (LEFM)

Characterize the state of the structure where useful (near the crack front where damage occurs) for a linear elastic behavior of the material Preliminary remarks: From the discontinuity to the singularity Stress concentration factor Kt of an elliptical hole, With a length 2a and a curvature radius r 2a

 loc a Kt   1 2 r  r

2a  a r  0  loc  2     loc r

Singularity 26 Remarks: existence of a singularity

Geometry locally-self-similar → self-similar solution → principle of simulitude

r  0   r,  f r g  2a       r*   r * r f (r )  q  f (r)    r : distance to the discontinuity Warning: implicit choice of scale

27 Order of this singularity For a crack : =-0.5 Linear elasticity:

r   Br  r0

 r   Cr r0

  r rmax      0 Eelast  A r r rd dr r 0     r 0

r rmax 21 r 22 Eelast  2A r dr max r 0  Eelast  2 A r 0 r0 2  2 22 rmax Eelast  2A r 0 2  2 2  2  0    1   1   0 2  2  0    1 28 Non linear material behaviour ?

n          o   n 1 elastic  n=4 o   r   Ar  r0

r   Br n r0 

  r rmax  n Eelast  C r r rd dr r 0     r 0

r rmax 1n1 Eelast  2C r dr r 0  r 0 1 n  2  0 1n2 rmax 2 Eelast  2C r 0         1 n 2 1 n 29 A. Modes

B. Airy stress functions

C. Westergaard’s solution

D. Irwin’s asymptotic LEFM development

E. Stress intensity factor KI, KII, KIII

F. Williams analysis T, Tz, G G. Fracture Toughness

H. Irwin’s plastic zones

30 Fracture modes

Planar symmetric Planar anti-symmetric Anti-planar

31 Fracture modes

Tubes (pipe line)

32 Fracture modes

Various fractures in compression

33 Fracture modes

Various fractures in torsion

34 A. Modes

B. Airy stress functions

C. Westergaard’s solution

D. Irwin’s asymptotic LEFM development

E. Stress intensity factor KI, KII, KIII

F. Williams analysis T, Tz, G G. Fracture Toughness

H. Irwin’s plastic zones

35 Case of mode I Analysis of Irwin based on Westergaard’s analysis and Williams expansions

Planar Symmetric

36 Balance equation

Div  f  ra v

2D problem, quasi-static, no volume force    xx  xy  xz  0 x y z    xy  yy  yz  0 x y z    xz  yz  zz  0 x y z 37 Linear isotropic elasticity : E, n

E    n  xx 1n 2 xx yy 1n n E     Tr  1  yy   yy n xx  E E 1n 2 E    xy 1n xy

38 Compatibility equations  2 3u ux xx x   2  2 xx x  y x y 2 3 u y   yy  u y  yy   y 2 x y 2 x u 2 3 3 1  ux y     u   xy y  ux  xy     2   2  y x  xy y 2 x x 2 y

 2  2  2 2 xy  yy  xx xy  2 x  2 y 39 Combination

Compatibility + Linear elasticity 2 2 2 E   xy   yy   xx 2    xx  2  xx nyy  xy 2 x 2 y 1n E 2 2    n       2 yy 1n 2 yy xx 2 xy  yy  xx 2 2 E xy  x  y    xy 1n xy

Balance equations +   xx  xy  0 x y = 3 Equations, 3 unknowns   xy  yy  0

x y 40 Airy function F(x,y) -1862-

Balance equation Compatibility      2  2  2 xx  xy  xy  yy  0 2 xy  yy  xx x y x y xy  2 x 2 y

Assuming

 2 F   xx y 2  2 F  4 F  4 F  4 F  yy   2   0 x2  4 x  2 x 2 y  4 y  2 F    xy xy 1 equation, 1 unknow F(x,y)

41 Z(z) , z complex,  4 F  4 F  4 F  2   0 F=F(x,y)  4 x  2 x 2 y  4 y

A point in the plane is defined by a complex number z = x + i y Z a function of z : Z(z)=F(x,y)

 4Z  4Z   4 x  4 z Z (z) always fulfill all the  4Z  4Z equations of the problem    2 x 2 y  4 z Z(z) must verify the symmetry  4Z  4Z and the boundary conditions   4 y  4 z

42 A. Modes

B. Airy stress functions

C. Westergaard’s solution

D. Irwin’s asymptotic LEFM development

E. Stress intensity factor KI, KII, KIII

F. Williams analysis T, Tz, G G. Fracture Toughness

H. Irwin’s plastic zones

43 Irwin’s or Westergaard’s analyses

y 6 boundary or symmetry conditions 2 singularities, 2a S S 0 boundary conditions along the crack faces x Exact solution Taylor’s development with respect to the distance to the crack front Separated variables Similitude principle S

2D problem, plane (x,y) : Szz=n(Sxx+Syy) Symmetric with respect to y=0 & x=0

Away from the crack (x & y >> a) : sxx= S syy= S & sxy= 0

Singularities in y=0 x=+a & y=0 x=-a

44 Boundary conditions & Symmetries

    xx   yy  S,  xy  0 &

 2 F   xx y 2  2 F S   F   y 2  x2  a x  a y  a yy x2 2 2 3 4  2 F    xy xy S F   y 2  x2  a symmetries 2 4

45 Construction of Z(z)

 S 2 2  S 2 F  y  x  a4 Z  z  a 2 2 4

Relation  Z   Z  F  Re Z  yRe    Re Z  yIm    y   z 

2   2Z   3Z   F   R    yI    xx  xx e  2  m  3  y 2  z   z   2 F   2Z   3Z      R    yI   yy 2 yy e  2  m  3  x  z   z  2  F  3Z   xy     yR   xy xy e  3   z  46 Solution

At infinity At infinity S       S,    0 Z   z 2  a xx yy xy 2 4 Solution:  2Z   3Z    R    yI   xx e  2  m  3   z   z   2Z   3Z    R    yI   yy e  2  m  3   z   z   3Z    yR   xy e  3   z  Valid for any 2D problem, with symmetries along the planes y=0 & x=0, and biaxial BCs

47 A. Modes

B. Airy stress functions

C. Westergaard’s solution

D. Irwin’s asymptotic LEFM development

E. Stress intensity factor KI, KII, KIII

F. Williams analysis T, Tz, G G. Fracture Toughness

H. Irwin’s plastic zones

48 Exact solution for a crack Singularities in y=0 x=+a   2Z   3Z  & y=0 x=-a   R    yI   xx e  2  m  3   z   z   S 2 Z  z  a4   2Z   3Z  2   R    yI   + + yy e  2  m  3  1 1   z   z  Z  Sz  3Z  z  a z  a   yR   z xy e  3   z 

 Z 1  Sz 2  a2  2 z

Exact solution 49 Asymptotic solution - Irwin-

y Local coordinates (r,), r → 0 Exact Solution i z  a  re  r Z 1   Sz 2  a2  2 x z

2 2   Z Sz  Z Sa S a i  2 2 1 2  1  e z z 2  a2  2 z 2arei  2 2r

3 2 3 2 3  Z  Sa i  Z  Sa 1 S a 2 3  3    e z 2 2 2 3 3 z  a  z 2arei  2 r 2r

50 Asymptotic solution - Irwin-

2   Z S a i  2Z   3Z   e 2   R    yI   2 xx e  2  m  3  z  z   z  2r  2Z   3Z    R    yI   yy e  2  m  3  z z 3 3     i  Z 1 S a 2 3   Z  3   e   yR   z r xy e  3  2r  z  Westergaard’s stress function : y S a    3   xx  cos 1 sin sin  2 r 2  2 2  r  S a    3   yy  cos 1 sin sin  x 2 r 2  2 2  S a    3   xy  cos sin cos  2 r  2 2 2  51 Error associated to this Taylor development along =0

푆푦푦 푎 + 푟 퐾퐼 푎 + 푟 Exact solution 휎푦푦 푟, 휃 = 0 = = 푟 2푎 + 푟 휋푎푟 2푎 + 푟

Asymptotic solution 2 3 푟 퐾퐼 3 푟 5 푟 5 휎푦푦 푟, 휃 = 0 = 1 + + + 푂 푟2 푒푟푟표푟~ 2휋푟 4 푎 32 푎 4 푎

Error 1 term 0.1 Erreur = 1%

0.01 2 terms 푟 1 term = ퟎ. ퟎퟏퟑ 푎 0.001 푟 2 terms = ퟎ. ퟐퟗ 푎 10 4 푟 3 terms = ퟎ. ퟔퟗ 0.1 0.2 0.3 0.4 0.5 푎 r/a 52 A. Modes

B. Airy stress functions

C. Westergaard’s solution

D. Irwin’s asymptotic LEFM development

E. Stress intensity factor KI, KII, KIII

F. Williams analysis T, Tz, G G. Fracture Toughness

H. Irwin’s plastic zones

53 Mode I, non equi-biaxial conditions

T  S  S xx yy Equibiaxial Biaxial (Superposition) 

 

K I  S yy a 54 Stress intensity factors

K I    3  Similitude principle  xx  cos 1 sin sin  T (geometry locally planar, with a 2 r 2  2 2 

straigth crack front, self-similar, K I    3   yy  cos 1 sin sin  singularity) 2 r 2  2 2  K   3   I cos sin cos Same KI & T → Same local field xy 2 r 2 2 2

KI &T Spatial distribution, given Crack geometry and once for all, in the crack boundary conditions front region  gij() f(r)=r

55 von Mises stress field

D Tr r,  3 D D  r,    r,  1  eq r,    r, : r,  3 2

Plane stress, Mode I, T=0 Plane strain, Mode I, T=0

56 von Mises stress field

Mechanisms controlled by shear . Plasticity, . Visco-plasticity . Fatigue

T  Sxx  S yy

T / K = -10 m-1/2 T / K = -5 m-1/2 T / K = 0 m-1/2 T / K = 5 m-1/2 T / K = 10 m-1/2

Plane strain, Mode I

57 Hydrostatic pressure Fluid diffusion (Navier Stokes), Tr r,  Diffusion creep (Nabarro-Herring) Chemical diffusion

Plane stress, Mode I, T=0 Plane strain, Mode I, T=0

58 Hydrostatic pressure

Fluid diffusion (Navier Stokes), Tr r,  Diffusion creep (Nabarro-Herring) Chemical diffusion

T  Sxx  S yy

T / K = -10 m-1/2 T / K = -5 m-1/2 T / K = 0 m-1/2 T / K = 5 m-1/2 T / K = 10 m-1/2

Plane strain, Mode I 59 Other T components, in Mode I

General triaxial Equibiaxial Superposition Superposition loading plane strain non equibiaxial non plane strain conditions conditions

60 Full solutions KI, KII, KIII, T, Tz & G

Mode I Mode II Mode III

K    3   K III  I  K II    3    cos 1 sin sin T  xz  sin  G xx    xx  sin 2  cos cos  2 r 2  2 2  2 r 2  2 2  2 r 2

K    3  K III  I K II   3   cos  yy  cos 1 sin sin    sin cos cos yz 2 r 2  2 2  yy 2 r 2 2 2 2 r 2

K I   3 K   3 4K III r    cos sin cos II   u  sin xy  xy  cos 1 sin sin  z 2 r 2 2 2 2 r 2  2 2  2 2 2 K r  I K II r  ux  cos   cos  u  sin 2   cos  2 2 2 x 2 2 2 K r  I K II r  u y  sin   cos  u  cos 2   cos  2 2 2 y 2 2 2

Déformation plane Contrainte plane

 zz n  xx  yy Tz (3n )     (3 4n ) 1n 

61 von Mises stress field

3 D D D Tr r,   r,   r, : r,  r,    r,  1 eq       3 2

Mode I Mode II

62 Summary

- Exact solutions for the 3 modes, determined for one specific geometry - Taylor development, 1st order → asymptotic solution generalized to any other cracks - First order - Solution expressed with separate variables f (r) g () and f (r) self-similar - Solution : f (r) a power function, r, with  = - 1/2 - Higher Orders - A unique stress intensity factor for all terms - The exponent of (r/a) increasing with the order of the Taylor’s development - Boundary conditions - Singularity along the crack front, symmetries, planar crack and straight front - no prescribed BCs along the crack faces, - Boundary conditions defined at infinity

6 independent components of the stress tensor at infinity → 6 degrees of freedoms in MLER: KI, KII, KIII and T, Tz, and G A. Modes

B. Airy stress functions

C. Westergaard’s solution

D. Irwin’s asymptotic LEFM development

E. Stress intensity factor KI, KII, KIII

F. Williams analysis T, Tz, G G. Fracture Toughness

H. Irwin’s plastic zones

64 Williams expansion

 4 F  4 F  4 F A self-similar solution in 4 the form is sought directly 4  2 2 2  4   F  0  x  x y  y as follows : y Fr,   r 2 g 

r  x

2 2 1   F  1  F 2  g  2 F  r      2 r  g  r  r r  r  r 2  2  2 2 2 4 2  g  2  g   g  4 F    2 2r 2 g  2r 2   2 r 2  r 2  2  2  4 2 4  2 2  g   g  4 F  r 2    2 2 g     2  2          2 4     

65 Williams expansion

y A self-similar solution in the form is sought directly as follows : r  2 x Fr,   r g 

2 4  2 2  g   g  4 F  r 2    2 2 g     2  2    0        2 4     

Dans ce cas g() doit vérifier

4 2 d g  2 d g  2  2    2   2   2 g   0 d 4 d 2

66 Williams expansion

r  x

4 2 d g  2 d g  2  2    2   2   2 g   0 d 4 d 2

ip The solution is sought as follows : g   Ae

 p4  2    22  p2  2   22  0  p2  2 p2    22  p   p    2

67 Williams expansion

r  x

Fr,   Rer 2 Aei  Bei  Cei2  Dei2 

Boundary conditions are defined along the crack faces which are defined as free surface (fluid pressure & friction between faces are excluded)

 2 F  r,     r,     0  r 2   F   r r,       r,     0 r  r 

68 Williams expansion

r  x

Fr,   Rer 2 Aei  Bei  Cei2  Dei2 

Re Aei  Bei  Cei2  Dei2  0  r,     0   Re Aei  Bei  Cei2  Dei2  0 Re Aei  Bei    2Cei2    2Dei2  0  r,     0  r Re Aei  Bei    2Cei2    2Dei2  0

69 Williams expansion

r  x

A sery of eligible solutions is 2 1  n obtained :  n    n   n even g   B cos 1   D cos 1   2    2    n    n   n odd g   Asin 1   C sin 1  n  2    2   1 Fr,   r 2 g  La solution en contrainte s’exprime alors à partir des dérivées d’ordre 2 de F, toutes les valeurs de n sont possibles, tous les modes apparaissent

70 Williams versus Westergaard

- The boundary conditions are free surface conditions along the crack faces (apply on 3 components of the stress tensor), no boundary condition at infinity → absence of T, Tz, and G

- Super Singular terms → missing BCs

- The first singular term of the Williams expansion is identical to the first term of the Taylor expansion of the exact solution of Westergaard

- The stress intensity factors of the higher order terms are not forced to be the same as the one of the first term, - advantage, leaves some flexibility to ensure the compatibility of the solution with a distant, non-uniform field - drawbacks, it replaces the absence of boundary conditions at infinity by condition of free surface on the crack, and it lacks 3 BCs, it is obliged to add constraints T, Tz, and G arbitraitement A. Modes

B. Airy stress functions

C. Westergaard’s solution

D. Irwin’s asymptotic LEFM development

E. Stress intensity factor KI, KII, KIII

F. Williams analysis T, Tz, G G. Fracture Toughness

H. Irwin’s plastic zones

72 J contour integral The J integral is shown to be independent of the choice of the selected integration contour y 휕푢 퐺 = 퐽= 휑푑푦 − 휎푛. Γ 휕푥

The integration contour G can be chosen inside the domain of x validity of the Westergaard’s stress functions to get G in linear elastic conditions 1 − 휈2 1 + 휈 Energy release rate 퐺 = 퐾2 + 퐾2 + 퐾2 퐸 퐼 퐼퐼 퐸 퐼퐼퐼

1 − 휈2 퐺 = 퐾2 Fracture toughness 푐 퐸 퐼푐 73 A. Modes

B. Airy stress functions

C. Westergaard’s solution

D. Irwin’s asymptotic LEFM development

E. Stress intensity factor KI, KII, KIII

F. Williams analysis T, Tz, G G. Fracture Toughness

H. Irwin’s plastic zones

74 Mode I, LEFM, T=0

Syy

Syy Syy

Syy

75 LEFM stress field (Mode I)

Von Mises equivalent deviatoric stress

76 Irwin’s plastic zones size, step 1: rY

Along the crack plane, =0 K K  r,  0   r,  0  I ,  r,  0  2n I ,  r,  0  0 xx yy 2 r zz 2 r xy

K 2 K 1 2n  p r,  0  I 1n   r,  0  I H 2 r 3 eq 2 r

Yield criterion :  eq rY ,  0   Y

2 2 1 2n  K I rY  2 2  Y

77 Irwin’s plastic zones size, step 2: balance

Hypothesis: when plastic deformation occurs, the stress tensor remains proportionnal to the LEFM one

yy(r,=0)

Y

Elastic field

rY rp r 78 Limitations

Crack tip blunting modifies the proportionnality ratio between the components of the stress and strain tensors

FE results, Mesh size 10 micrometers, Re=350 MPa, Rm=700 MPa, along the crack plane 79 Irwin’s plastic zones size, step 2: balance

yy(r,=0)

2 2 1 2n  K I rpm  2rY  2   Y

Y

Elastic field

rY rp r

rr r K max pm  r K max I dr  Y dr  I dr  2 r  1 2n   2 r  r  r0 r0 rrpm Y

80 Irwin’s plastic zone versus FE computations

Ideally elastic-plastic material Y=600 MPa, E=200 GPa, n=0.3 plane strain, along the plane =0

81 Irwin’s plastic zone versus FE computations

Ideally elastic-plastic material Y=600 MPa, E=200 GPa, n=0.3 plane strain, along the plane =0

82 Irwin’s plastic zone versus FE computations

Ideally elastic-plastic material Y=600 MPa, E=200 GPa, n=0.3 plane strain, along the plane =0

Mode I, Monotonic and cyclic plastic zones

)

MPa Stress ( Stress

Plastic strain (%) 84 Mode I, Monotonic and cyclic plastic zones

Monotonic plastic zone

2 2 1 2n  K I max rmpz  2   Y

Cyclic plastic zone

2 2 1 2n  DK I rcpz  2  4 Y

85 T-Stress effect

T  S  S xx yy 

 

K I  S yy a

86 T-Stress effect

87 T-Stress effect

K I  S yy a

T  S yy

1/2 Irwin’s plastic zone, Y=400 MPa, KI=15MPa.m

88 Ductile fracture

Measurement of the crack tip opening angle at the onset of fracture

89 Example of the effect of a T-Stress for long cracks

90 Example of the effect of a T-Stress for long cracks

0.48 % Carbon Steel [Hamam,2007] 91 Fatigue, and crack growth modeling Measurements

Crack length increasing

COD

J.Petit

Potential drop COD Direct optical measurements  Digital image correllation

93 a

da/dN = f(a)

a N Load cycle N

Fmax

R=Fmin/Fmax Fop DF DFeff Fmin

94 Paris’ law max K I  K IC

A - threshold regime B – Paris’ regime C - unstable fracture

DKeff  DKth

Subcritical crack growth if DK is over the non propagation threshold DKeff MPa m

95 Fatigue – Threshold regime

[Neumann,1969] 96 Fatigue – Threshold regime

Titanium alloy TA6V [Le Biavant, N18 nickel based superalloy at room 2000]. The fatigue crack grows temperature, [Pommier,1992]. The along slip planes. crack grows at the intersection between slip planes

97 Fatigue – Threshold regime – fracture surface

“pseudo-cleavage” facets at the initiation site

98 Fatigue – Threshold regime – fracture surface

INCO 718

99 Paris’ law max K I  K IC

A - threshold regime B – Paris’ regime C - unstable fracture

DKeff  DKth

Subcritical crack growth if DK is over the non propagation threshold DKeff MPa m

100 Paris’ regime : crack growth by the striation process

[Laird,1967], [Pelloux, 1965]

316L

TA6V

OFHC INCO 718 INCO

101 102 103 104 105 106 107 108 109 110 111 Crack growth is governed by crack tip plasticity

112 Consequences

• the quantities of LEFM (KI, KII, KIII) control the behavior of the K-dominance area

• which controls the behavior of the plastic zone

• which controls crack growth by pure fatigue

113 • Introduction

• History effects in mode I • Observations • Long distance effects • Short distance effects • Modelling Outline • History effects in mixed mode • Observations • Crack growth rate • Crack path • Simulation • Modelling 114 Long distance effect (overload)

Constant amplitude fatigue idem + 1 OL (factor 1.5)

idem + 1 OL (factor 1.8)

(mm)

length Crack

Number of cycles CCT, 0.48% carbon steel, [Hamam et al. 2005] 115 Long distance effect (residual stresses)

Kopening

116 • Introduction

Short distance effect (repeated overloads)

(mm)

aOL –

length idem after 1 OL (factor 2) idem after 10 OL (factor 2)

Constant amplitude fatigue Crack

Number of cycles

CT, 316L austenitic stainless steel, [Pommier et al] 118 Short distance effect (block loadings)

1 99

100 9900

119 • If the plastic zone is well constrained inside the K- dominance area

• It is subjected to strain controlled conditions by the elastic bulk,

• Mean stress relaxation

• Material cyclic hardening • Introduction

• A very small plastic zone produces very large effects on the fatigue crack growth rate and direction

• Finite element method : elastic plastic material, very fine mesh required, 3D cracks, huge number of cycles to be modelled, tricky post-treatment

• Fastidious and time consuming

122 A simplified approach is needed: the elastic-plastic behaviour of the plastic zone is condensed a non-local elastic-plastic model tailored for cracks

elastic plastic FE + POD Linear elastic FE analyses for 3D cracks Method

d  f  ,... dt Constitutive model LOCAL

Scale transition Generation of evolutions of r (CTOD) versus KI

Expérimental input n°2

dr Tensile Push  g dK I , K I ... pull test dt

Expérimental input n°1 da dr   dt dt Crack growth model, Fatigue crack growth including history effects, experiment da/dt : rate of production of cracked area per unit length of the crack front

da da dr  DCTOD    dN dt dt

Adjust the coefficient a using one constant amplitude fatigue crack growth experiment

125 Single overload : long range retardation

126 Block loading : short range retardation

127 Stress ratio (mean stress) effect (R>0)

128 Stress ratio (mean stress) effect (R<0)

129 Random loading simulations

number of blocks

130 • Introduction

푑푎 = 퐶∆퐾푚 푑푁 푒푞 1 푛 푛 푛 푛 훥퐾푒푞 = ∆퐾퐼 + 훽∆퐾퐼퐼 + 훾∆퐾퐼퐼퐼

Same values of Kmax, Kmin, DK for each mode Fatigue crack growth experiments

Crack growth rate

Crack path

132 Load paths in mixed mode I+II

133 Load paths in mixed mode I+II+III

134 ∞ 푓 (2푎) 푓 (2푎) 0 퐾퐼 퐼 퐼 퐹푋 ∞ = 푓 (2푎) −푓 (2푎) 0 퐾퐼퐼 퐼퐼 퐼퐼 퐹푌 ∞ 0 0 푓퐼퐼퐼(2푎) 퐾퐼퐼퐼 퐹푍

135 Experimental protocol

6 actuators hydraulic testing machine - ASTREE

136 Fatigue crack growth in mixed mode I+II+III

137 Crack path – mode I+II+III

138 Mode III contribution

139 Mode III contribution

140 Mode III contribution

141 FE model and boundary conditions

Periodic BC along the two faces normal to the crack front

Prescribed displacements based on LEFM stress intensity factors ∞ 푰 ∞ 푰푰 ∞ 푰푰푰 푲푰 풖풃풄_풏풐풎, 푲푰푰풖풃풄_풏풐풎 ,푲푰푰푰풖풃풄_풏풐풎

Elastic plastic material constitutive behaviour (kinematic and isotropic hardening identified experiments)

142 Crack : locally self similar geometry → locally self similar solution 풇 휶풓 =풌 휶 풇 풓

풓 − Small scale yielding 풇 풓 ퟎ 풑 풓→∞ 풇풊 풓 = 풇풊 ퟎ 풆

V퐞퐥퐨퐜퐢퐭퐲 퐟퐢퐞퐥퐝 ∶ 풇 풓 = ퟎ 퐟퐢퐧퐢퐭퐞

143 Cumulated equivalent plastic strain

144 radial distribution 풄 풄 푷푶푫ퟐ → 풖풊 (푷) ≈ 퐟 풓 품풊 (휽)

풓 −풑 풇풊 풓 = 풇풊 ퟎ 풆

145 POD based post treatment

푒 Solution of an elastic FE analyses with 푢푖 (푃) ∞ 1/2 푲풊 =1MPa.m for each mode

풗푬푭_풊 푷, 풕 . 풖풆(푷) 풆 풆 푷흐푫 풊 흂풊 푷, 풕 = 푲풊 풕 풖풊 (푷) 푲풊 풕 = 풆 풆 푷흐푫 풖풊 (푷). 풖풊 (푷)

풓é풔풊풅풖_풊 푬푭_풊 풆 풗 푷, 풕 = 풗 푷, 풕 − 흂풊 푷, 풕

146 POD based post treatment

풓é풔풊풅풖_풊 푬푭_풊 풆 풗 푷, 풕 = 풗 푷, 풕 − 흂풊 푷, 풕

풓é풔풊풅풖_풊 풄 푷푶푫ퟏ → 풗 푷, 풕 ≈ 흆 풊 풕 . 풖풊 (푷)

푐 푐 푃푂퐷2 → 푢푖 (푃) ≈ f 푟 푔푖 (휃) 1 푔푐 휃 = 휋 = −푔푐 휃 = −휋 = 퐼푦 퐼푦 2

lim 푓 푟 =1 푟→0

147 POD based post treatment

ퟑ 풆 풄 풗 푷, 풕 = 푲풊 풕 . 풖풊 (푷) + 흆 풊 풕 . 풖풊 (푷) 풆 풄 풊=ퟏ 흂풊 푷,풕 흂풊 푷,풕

푲 풕 풊 Intensity factors, non-local variables 흆 풊 풕

풖풆(푷) 풊 Field basis / weigthing functions tailored for 풄 cracks in elastic plastic materials 풖풊 (푷)

148 FE Simulations and results

149 150 Crack propagation law

풂 풏∗ = 휶 풕 ⋀흆

In mode I, this law derives from the CTOD equation

In mode I+II+III, it derives from the Li’s model

151 FE Simulations and results

152 153 Intensity factor evolutions

154 Mode III contribution ?

A Mode III load step increases the amplitude of Mode I and of Mode II plastic flow

155 156 Approach FE model 풗 푷, 풕

Material constitutive law, local and tensorial

휀 = 푓 휎 , 푒푡푐.

Crack tip region 휌 = 휌 , 휌 constitutive law, non-local 퐼 퐼퐼 and vectorial ∞ ∞ ∞ ∞ 휌 = 푔 퐾 , 푒푡푐. 퐾 = 퐾 퐼 , 퐾 퐼퐼

- Elastic domain (internal variables) - Normal plastic flow rule - Evolution equations 157 Elastic domain : generalized Von Mises Criterion

∞ 푋 2 ∞ 푋 2 퐾퐼 − 퐾퐼 퐾퐼퐼 − 퐾퐼퐼 푓푌 = 푌 + 푌 − 1 퐾퐼 퐾퐼퐼

퐺퐼 퐺퐼퐼 푓푌 = 푌 + 푌 − 1 퐺퐼 퐺퐼퐼

2 푠푖푔푛 퐾∞ − 퐾푋 퐾∞ − 퐾푋 퐺 = 푖 푖 푖 푖 푖 퐸∗

158 Model

Yield criterion

ퟐ ퟐ ퟐ 푲∞ − 푲푿 푲∞ − 푲푿 푲∞ − 푲푿 풇 = 푰 푰 + 푰푰 푰푰 + 푰푰푰 푰푰푰 − ퟏ 풀 ퟐ 풀 ퟐ 풀 ퟐ 푲푰 푲푰푰 푲푰푰푰

푮푰 푮푰푰 푮푰푰푰 풇 푮푰, 푮푰푰, 푮푰푰푰 = 풀 + 풀 + 풀 − ퟏ 푮푰 푮푰푰 푮푰푰푰

Flow rule 풔풊품풏풆 푮 풊 흆 풊 = 흀 풀 푮풊

Evolution equation 횪 푲푴−ퟏ 푲푿 푿 푿풆풒 푲 = 푪 흆 − 푴−ퟏ 풅 흆 풅 풘풉풆풓풆 풅 = 푿 ퟏ + 횪 푲푿풆풒 푲풆풒

159 Conclusions

• Fatigue crack growth experiments in Mixed mode I+II+III non proportionnal loading conditions

• Result : A load path effect is observed on fatigue crack growth and on the crack path

• Adding a mode III step to mixed mode I+II fatigue cycles increases the fatigue crack growth rate

• Elastic-plastic FE analyses show that accounting for plasticity allows predicting the load path effect and the effect of mode III Plasticity

• A simplified model has been developped to replace non-linear FE analyses

160 161