concepts of fracture mechanics

CONCEPTS OF FRACTURE MECHANICS

G. Pluvinage(1) and S. Jallouf(2)

(1)Laboratoire de Fiabilité de Mécanique, Université de Metz, France (2)Faculty of Mechanical Engineering, University of Aleppo, Syria

Abstract: Fracture behavior is bounded by the two extremes of failure, linear elastic and fully plastic. By utilizing this concept a universal failure curve can be used to interpolate between the two extremes of failure. This Curve gives equivalent results to failure analyses basis on the J-integral or COD. The failure curve forms the basis of failure assessment diagram which reduces the complexities of elastic-plastic failure analyses. In this paper the general philosophy and principals behind any fracture mechanics assessment are discussed, with particular emphasis on the need to choose a route and define the security point according to particular problem specifications.

1. INTRODUCTION

Fracture mechanics is being increasing applied to assess safety

engineering and the concerted effort by designers, metallurgists, production and maintenance engineering, and inspectors to ensure safe operations without catastrophic fracture failures.

Several circumstances may be cited in order to justify the application of fracture mechanics theory to engineering structures. The most important circumstance is probably that all engineering structures contain cracks or crack-like flaws at some scale of examination. This is becoming increasingly obvious as more sophisticated non-destructive examination (NDE ) equipment and methods come into service.

Clearly, the safety of a structure can be guaranteed only when the engineering, the metallurgist and the inspector have worked together to ensure that the structure never contains critical combinations of stress, resistance to crack extension and flaw size, respectively Fig.1[4,5].

2 G. Pluvinage(1) and S. Jallouf(2)

It follows that the application of fracture mecanics techanics provides a

logical basis for the decisions which the aforementioned personnel have to make.

Recently it nas been estimated that a fracture mechanics saved over $100 million in the construction of the Forties Field oil production platforms in the North sea. Similarly, in a recent instance in the electrical power generation industry, a reported $32 million was saved by using fracture mechanics to prove that a cracked rotor in a 500 MW alternator coud be run safely until a replacement was available. On the other hand, it is much more difficult to quantify the economic advantages of fracture mechanics in relation to such considerations as safety, and the consequences of structural failures [1].

2.1 Basic Principals of Fracture Mechanics

Lectures in the field of fracture mechanics are mainly concerned with

linear fracture mechanics (L.F.M). In addition some aspects of elastoplastic fracture mechanics (E.P.F.M) are often added like descriptions of the J Integral and crack opening displacement concepts(C.O.D.) and crack tip opening displacement ( C. T. O. D. ). And Failure Assessment Diagram (FAD)

STRESS( Engineer )

CRACK SIZE (Inspector)

RESISTANCE TO CRACK EXTENTION

(Metallurgist)

Fig. 1. Basic fracture mechanics parameters and most closely associated personnel

Concepts of fracture mechanics 3

- E.P.F.M is applicable for defects which are not very short or very long. A general description of the fracture problems needs in addition to the ultimate strength criterion (for very short defects) and the instability criterion (for very long defects).

- E P F M can be considered as a modification of the lower bound describing the nocivity of a defect. This lower bound is relative to L.F.M with respect to the most severe defect (crack) and the worst situation (brittle fracture) [3].

2.2 Elastic stress distribution at crack tip 2.2.1 Irwin's Stress Intensity Approach Irwin developed in 1948 a series of linear elastic crack stress field

solutions using the mathematical procedures of Westergaard. Irwin showed that the stress field at the tip of a crack is characterized by a singularity of stress which decreases in proportion to the inverse square root of the distance from the crack tip. Later Irwin showed that the stess field in the region dominated by the singularity of stress can be regarded as the sum of three invariant stress patterns taken in proportions which depend upon loads, dimensions and shape factors. These mode are described Fig.2 [4].

Mode I or opening mode, Mode II or shearing mode, Mode III or antishearing mode.

mode I mode II mode III

Figure 2


2.3 ELASTIC-PLASTIC FRACTURE MECHANICS (EPFM) In recent years several attempts have been made to extend fracture

mechanics into the elastic-plastic regime. These began with plasticity corrections to linear-elastic fracture mechanics with modest success. However, these corrections proved insufficient to handle analytical modeling of many practical cracking problems from large-scale crack tip plasticity into fully plastic regimes.

This point is illustrated in Fig.3 which shows measured fracture stresses for crack tensile specimens of titanium and aluminum alloys, and two steels. Here the fracture stresses are normalized by the ultimate tensile stress, σu , and the crack lengths by (Kc/ σu )2 . The fracture stresses predicted from LEFM are in good agreement with the experimentally measured values for large cracks but become increasingly as crack size decreases. This behaviour is expected since at failure the plastic zone size is proportional to (Kc/ σy )2 which is small compared with large cracks but not compared with short ones.

On occasions there will be practical circumstances where failure analyses are required in the elastic-plastic regime between LEFM and fully plastic behaviour.

03

04

0.5

0.6

0.7

0.8

0.9

1.0

00

0.2

0 0.6

08

1.0

12

1.4

PLASTIC COLLAPSE LINE

LEFM

NO

RM

ALI

SED

FR

AC

TUR

E

NORMALISED CRACK LENGTH

Fig. 3 Fracture stress as a function of crack size


In this region the two criteria approach is at its weakest, in terms of both failure load predictions and determination of critical defect sizes. To obtain a more refined simulation of failure behavioure it is necessary to consider failure parameters whose applicability is equally valid in the presence of either small scale or large scale yielding conditions. Only then can a single fracture formula relevant to the whole range of failure be obtained. The first attempt at developing elastic-plastic models is termed the crack opening Displacement (COD) method [4].

2.3.1 Crack opening displacement Dugdale model

Dugdale considers an effective crack which is longer than the

physical one [2]. The crack edges in front of the physical crack carry uniform stresses equal to the yield stress Re on a distance Rp. This region is not really a crack and it size is chosen in order that the stress singularity disappears. This mean hat the stress intensity factor due to the global stress σg, KA has to be compensated by the stress intensity KB to the wedge stresses Re. KA = - KB

Figure 4 The stress intensity factor due to wedge stresses Re on a length Rp is equal to:

σg R

Re

Re

R

2a

2c


dxxaxa

xaxa

aR

Ka

R

eA

p

∫⎭⎬⎫

⎩⎨⎧

+−

+−+

=π

∫−

=a

eAxa

dxaRK0

22 )(2

π

The solution of this integral is:

K A = 2 R e . aπ

a r c c o s xa

o

a + R p

p

peA Ra

aarcRa

RK+

+= cos2

π

This stress intensity factor is equal to:

K B = σ g π a + R p = σ g π c

2c is the equivalent crack length:

σ g π c .= 2 R e

πc a r c c o s a

c

This leads to : ac

= c o sπ σ g

2 R e

Neglecting the higher terms in the series development of the cosine:

ac

⇒ 1 -π

2σ g

2

8 R e2


2.3.2 Crack opening displacement The displacement of the crack surface is given in LEFM by :

δ = 2 v =4 σ g

Ea

2- x

2

By applying a plastic zone correction

δ = 2 v =4 σ g

Ea + r y

2- x

2

The crack tip opening displacement at the tip of the physical crack is found for x = a. Since ry<< a

δ =4 σ g

Ea . r y

2- x

2

Substitution of ry yields:

r y2 =

σ g2

. a

2 R e 2

δ = 4π

.K I

2

E . R e

Alternatively the equation of the crack tip opening displacement comes from the Dugdale model:

δ = 8π

R eE

. σ g a L n s e cπ σ g

2 R e

Series expansion of the Ln sec yields if σg is small compared to Re:

δ =σ g

2π a

R e . a= K

2

R e . E

G = Re σ


2.3.3 J Integral Another powerful technique for analyzing elastic-plastic failure is based

on the J-integral. This path-independent integral was derived for nonlinear elastic materials as an expression for the rate of change of potential energy per unit thickness with respect to an incremental extension of crack. Tow things should be noted at this point. First the similarity with the strain energy release rate obtained in linear elasticity theory, J is a generalization of this concept to non-linear deformation. Secondly that the derivation is strictly true only for linear and non-linear elastic materials where unloading occurs along the same path as the initial loading. After plastic deformation real materials unload almost linear elastically, along a different path to the loading one. The energy interpretation is therefore not relevant if appreciable plastic deformation is present. Nevertheless since J is equal to the spatial derivative of an energy, it does represent, and can be interpreted as, a force on the crack tip and its associated plasticity [2,9]. Consider a non-linear elastic material, the stress-strain curve of which can be represented by the Ramberg- Osgood equation

ε

ε 0

=σ

σ 0

+ ασ

σ 0

Ν

where σ0 is the reference stress, ε0 the reference strain and N the strain hardening exponent.

The potential energy of a mechanical system is defined as

∫∫ −=ΠS

iiV

dSuTdVW*

**

xα

n

S

Fig. 5

Concepts of fracture mechanics 9 where W* is the strain energy density contains in a volume V*. This volume is enclosed in a surface S on which loads by unit surface Ti and displacements ui are acting.

When the crack is extended by an amount ∂a in the x direction, the

energy potential decreasing is :

∫∫ ∂∂∂

−∂∂

∂

∂Π∂

=Π−S

ii

V

ij

ij

adSxu

TadVx*

*.ε

εδ

According to the Green-Gauss theorem:

adSxu

TnWS

iix ∂⎟

⎠⎞

⎜⎝⎛

∂∂

−=Π− ∫ .*δ

∫= ijij dW εσ*

where nx is the normal in x direction.

aa∂

∂Π∂

=Π−δ

dSxu

TnWa S

iix∫ ⎟

⎠⎞

⎜⎝⎛

∂∂

−=∂Π∂

− ..*

Considering the path S and a normal n on one point of the

surface :

n = nx , ny = cos α, sin α

The surface element dS is equal to:


dS = dx2 +dy2 and nx dS = dy

∫Γ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−=

∂Π∂

− dSxu

TdWa y ..*

The Integral is called J Integral and is due to Rice:

∫Γ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−= dS

xu

TdWJ y ..*

In polar co-ordinates J can be expressed by:

θθπ

π

rdxuTCosWJ ∫

−

⎟⎠⎞

⎜⎝⎛

∂∂

−= *

The J integral has 3 properties J is related to the variation of potential energy, J is path independent, For non-linear behaviour J described the intensity of stress and strain at

crack tip. 2.4 Principe of failure assessment diagram

Fracture of components can be considered as belonging to one of

these three limit state : brittle fracture, elasto-plastic fracture and plastic collapse. Elasto-plastic fracture can be seen as an intermediate state between the two other and obtained by interpolation. The fracture assessment diagram is an attempt to apply this approach. The failure criterion is based on two parameters kr and Sr. The parameter kr is defined as the ration of equivalent elastic stress intensity factor to the fracture toughness [2]. kr =

KIe

K*

kr is equal to unity for brittle fracture and 0 for plastic collapse (K* is often taken as the fracture toughness KIc.

2.4.1 Assessment diagram drawing

The assessment diagram is plotted in co-ordinates kr and Sr . Two particular points of this diagram represent successively brittle fracture conditions (Sr = 0; kr = 1) and plastic collapse (Sr = 1; kr = O). The curve which defined the assessment diagram encloses between the co-ordinates a safe domain.


The loading conditions of a structure are represented by a point A of co-

ordinates krA ; S r

A . If this point is inside of his domain, this ensures the

structure’s integrity. If this point (B) is on the curve its co-ordinates are

krc ; S r

c and failure occurs Fig 5.

2.4.2 Properties of the assessment diagram By definition the co-ordinates of point A are proportional to the loading.

If the defect has a constant size, the representative point follows a straight line until it reaches the critical position. When the defect is evaluating, the loading path is more complex [2,4].

The safety factor is defined on figure as the ratio

Fs = OBOA .

1

0,8

0,6

0,4

0,2

10,80,60,40,20

kr

krA

krc

Src

SrA Sr

Fs = OBOA

Failure

Safe

Fig 6


2.4.3. R6 Failure Assessment Diagram To apply the R6 procedures, two parameters must be calculated for the

particular load, P, applied to the component containing a defect of size, a. These parameters are denoted Kr and Lr and are defined by:

),(

),(

1

1

yLr

cr

aPPL

KaPKK

σ=

=

where K1 is the elastic stress intensity factor, K1c is the fracture toughness, σy is the yield stress and PL is the value of P corresponding to plastic collapse of the component. Having evaluated Kr and Lr failure by fracture at the defect is avoided provided the assessment point (Lr , Kr ) lies within the region bounded by failure assessment curve as illustrated in Fig. 1. the pictorial representation of Fig. 7 is one of the advantages of R6 as it enables margins on load, for example, to be easily defined.

The shape of the failure assessment diagram is given in R6 and is based

on a number of factors. The first is avoidance of fracture under linear elastic conditions. This is achieved by the criterion:

Kr ≤ 1 …………(3) Which in view of eqn. (1) is

……….(1)

……….(2)

0 0.5 1.0 1.5 Lr

0.5

1.0 Kr

X ( Lr ,Kr )

Failure Avoided

Fig. 7 The R6 failure assessment diagram illustrating failure avoidance when an assessment point ( Lr,Kr) lies inside the diagram


K1 ≤ K1c …..…..(4) The second factor is avoidance of failure by plastic collapse. This is

achieved by limiting max

rr LL ≤ …………(5) where max

rL is defined in terms of the material flow stress, σ , and yield stress by

y

rLσσ

=max ………(6)

since plastic collapse load is directly proportional to yield stress, inequality (5) combined with the definitions of eqns. (2) and (6) restricts the applied load to:

),( σaPP L≤ ……..(7) The flow stress is used in this collapse limit to allow for hardening above

the yield stress. A typical value of flow stress. Which is often used, is the mean of the yield stress and the ultimate stress. However, R6 in not generally prescriptive about the choice of flow stress.

Equations (3) and (5) define limits on the failure assessment diagram. The difficult area to define is the intermediate elastic-plastic region. When detailed analyses methods are used, the parameter J is often evaluated and failure is avoided if :

J ( P, a ) ≤ J1c ……….(8) Here J1c a material property related to fracture toughness by

1

21

1 EKJ c

c = ………(9)

where E1 is Young's modulus, E, divided by (1-ν2 ) where ν is poisson's ratio, in plane strain; E1 = E in plane stress. Similarly, in the limit of elastic behaviour, the calculated value of J, Jel say, is related to the stress intensity factor, K1, by:

1

21

EKJel = …………….(10)

combining eqns. (1) and (8)-(10), the failure avoidance criterion of eqn. (8) can be written in the form:

Kr ≤ f(Lr) ……….(11) Where

2/1

3 )( ⎟⎠⎞

⎜⎝⎛=

JJLf el

r …………(12)


Here the subscript 3 has been introduced in eqn (12) as this definition of the failure assessment curve, )( rLf , is termed option 3 in R6. In evaluating eqn (12), both J and Jel are evaluated at the same load. The overall definition of the failure assessment diagram in terms of the limits of inequalities (3), (5) and (11) is shown in Fig. 8.

The option 3 curve of eqn (12) may be evaluated from finite-element

solutions for J or from experimental estimates of J, the FAD in this option is dependent on the material and the geometry.

The option 2 curve This option is dependent only on the stress/strain curve of the material

and is defined as:

2/12

2 2)(

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+=

ref

refr

ref

refr E

LELf

εσ

σε

…….(13)

where σref = Lr σy ………..(14) and εref is the true strain obtained from the uniaxial tensile stress/strain

curve at a true stress level σref . The first term in eqn. (13) describes both elastic and fully plastic behaviour. The second term is a plasticity correction term for small scale yielding which is phased out as the first term becomes much greater than unity.

0 0.5 1.0 1.5 Lr

0.5

1.0 Kr

Kr = f(Lr) )

Fig. 2 The R6 failure assessment diagram defined by three limiting criteria, avoidance of: elastic failure; plastic collapse; elastic-plastic fracture

Kr = 1 Lr = maxrL


The option 1 curve this option is given by :

[ ] ( )[ ]max

1

max621

0)(

65.exp7.03.014.01)(

rrr

rrrrr

LLforLfLLforLLLf

f=

≤−+−=…….(15)

This curve was chosen as an empirical fit to option 2 curves for a variety of materials, but biased towards the lower bound. With this curve, only the yield and flow stresses are needed to define the upper limit, max

rL , of eqn. (6) rather than detailed stress/strain data. Thus, the curve is independent of both material and geometry.

In particular the three options in R6 for the failure assessment curve have been evaluated: option 3 which depends on both material stress/strain behaviour and on geometry; option 2 which depends only on material stress/strain response; and option 1 which is independent of both material and geometry.

R6 Revision 4 The major new release of R6, revision 4, is due in 2000. Revision 4 will

adopt the same basic structure as the SITAP procedure, but will be divided into five chapters [7,8]:

Chapter I Basic procedures Chapter II Inputs to basic procedures Chapter III Alternative approaches Chapter IV Compendia Chapter V Validation

2.5 A16 failure assessment diagram In the method proposed by the Code RCC-MR annex A16, the reference

stress is an estimation of the limit net stress. This reference stress divided by the yield stress Re defined the Sr parameter [2]:

S r =σref

Re

The critical value of the kr parameter is given by :

k rc

=J e

c

J Ic= 1

A


with

A = φ +εr efεel

φ = 0,5. Sr2.εelεr ef

Φ is a plastic zone correction. The reference strain is defined on a universal and non - dimensional stress-strain curve given by the following relationship:

εr efεel

= fσr ef

Re

εel =σr ef

E

for 0 ≤

εr ef

εel

≤ 1

εr ef

εel

= 0,9845+1,3.Sr- 2 ,427.Sr2

+2,971Sr3

forεr ef

εel

≥ 1

εr ef

εel

= 641,46- 1967.Sr +2188. Sr2

- 1044. Sr3+184,37. Sr

4

The failure assessment diagram is presented in figure IT exhibits a ”tail” for high value of Sr.


Figure 52

3. Conclusion Elastoplastic Fracture Mechanic leads to conditions for fracture sensitive to geometry and loading conditions. More than 30 fracture criteria exists including the well-known J Integral and C.O.D.And different equations for the failure assessment diagram are proposed in the literature. They are listed in table 1.

REFERENCES

1. David Broek, The Practical Use of fracture Mechanics, Kluwer Academic Publishers,

Northerlands, 1989. 2. Guy Bluvinage, Exercices de Mecanique elastoplastique de la Rupture, Cepadues-

editions, CEPAD 1995, France 3. Guy Bluvinage, La Rupture du Bois et de ses Composites, CEPAD 1992, France 4. G. G. CHELL, Developments in Fracture Mechanics-1, Applied Science Publishers LTD

1979 5. G. C. Sih. E. Sommer, W. DAHI., Application of Fracture Mechanics to Materials and

structures, Martinus Nijhoff Publishers 1984 6. R. A. Ainsworth, Failure assessment diagrams for use in R6 assessments for austenitic

components, Int. J. Pres. Ves. & Piping 65 (1996) 303-309 7. P.J. Budden, The R6 procedure: recent developments and comparison with alternative

approaches, Int. J. Pres. Ves. & Piping 77 (2000) 895-903 8. R. A. Ainsworth, Failure assessment diagrams for high temperature defect assessment,

Int. J. Pres. Ves. & Piping 62 (1999) 95-109 9. FLAW GROWTH & Fracture, Proceedings of the tenth National symposium on Fracture

Mechanics' ASTM, American Society for testing and Materials 1977


Table 1

Irwin

KC*

=σg

cπa

1-

σg

c

σ0

2

kr = 1 -S r

2

2

Dugdale

KD*

= 8 Re2. aπ

. Ln 1

cosπ.σg

C

2 Re

kr =

18

π2

. Ln 1

cosπ.S r

2

Newmann KN

*= σN

c. πa

k r = 1- mN S R

R6

Kc*

= σg

c. πaeff

kr = 1- 0,14 S r2

.

0,3 + 0,7exp -0,65S r6

Tangent

stress KC

*= σg

cπa . Fσ

aW

σg

c

σo

2

+ ασg

c

σo

n + 1kr =

S r

S r2

+ αS rn+ 1

EL -PL HBK KJ

*= EJel a0 . P

PL

2

+EJpl a,n PPL

n + 1kr =

S r

HeS r2

+ HnS rn+ 1

NUREG -O744 KC

* =σg

cπa . Fσ

aW

1-Fσ2

βσg

c

σo

2

k r = 1 -Fσ

2

β S r2

concepts of fracture mechanics

Documents

failure assessment diagram

fracture mechanics

plastic regime

wedge stresses

rp

crack size

engineering structures

mode iii