HRR Integral

Recall with is the integration constant.

1

1

0 20

,n

ijn

EJn

I r

1

020

,

n

n

ijn

EJn

E I r

nI

Recall the material equation, Now the singularity, unlike varies as a function of n and state of stress.

0 0 0

n

IK

HRR Integral

Recall

with is the integration constant, as shown in the Figure below.

1

1

0 20

,n

ijn

EJn

I r

1

020

,

n

n

ijn

EJn

E I r

nI

Recall the material equation,

Now the singularity, unlike , varies as a function of n and state of stress.0 0 0

n

IK

HRR Integral, cont.

Angular variation of dimensionless stress for n=3 and n=13.

HRR Integral, cont.

Note the singularity is of the strenth . For the specific case of n=1 (linearly elastic), we have singularity.

Note also that the HRR singularity still assumes that the strain is infinitesimal, i.e., , and not the finite strain . Near the tip where the strain is finite, (typically when ), one needs to use the strain measure . 

1

11 n

r

1r

1, ,2ij i j j iu u 1

, , , ,2ij i j j i k i k jE u u u u 0.1ij E

Some consequences of HRR singularity

In elastic-plastic materials, the singular field is given by

(with n=1 it is LEFM)

stress is still infinite at . the crack tip were to be blunt then since it is now a free surface. This is not the case in HRR field. HRR is based on small strain theory and is not thus applicable in a region very close to the crack tip.

1

1

1

1

1

2

n

ij

n

ij

Jk

r

Jk

r

0r 0 at 0xx r

HRR Integral, cont.

Large-strain crack tip finite element results of McMeeking and Parks.Blunting causes the stresses to deviate from the HRR solution close to the crack tip.

HRR Integral, cont.

Some additional notes on CTOD and J-integral CTOD is based on Irwin’s formulation

Or based on strip yield (Dugdale’s model)

2

12

2

1

2 2

4 42

Ip

YS

py I

Iy

YS YS

Kr

rKu K

K Gu

E

8ln sec

2ys

ys

ys

a

E

G

CTOD is generally determined by three point bend test.

HRR Integral, cont.

0

( )

with

and ij

xi

ij ij i ij j

uJ wdy t ds

x

w dV t n

J-Integral

• Derived for non-linear elastic material and hence is not valid directly for elasto-plastic material during unloading part.

• Assumes deformation theory, and hence needs proportional loading.

• Assumes small strain formulation (no large strain, no large rotation admissible).

• J is path independent when there is no traction or displacement in the crack faces.

• J cannot be evaluated at the tip where there is singularity since the theory assumes that inside the domain the region should be analytic.

HRR Integral, cont.

Large Strain Zone

HRR singularity still predicts infinite stresses near the crack tip. But when the crack blunts, the singularity reduces. In fact at for a blunt crack. The following is a comparison when you consider the finite strain and crack blunting. In the figure, FEM results are used as the basis for comparison.

0 at 0xx r

Large-strain crack tip finite element results of McMeeking and Parks.Blunting causes the stresses to deviate from the HRR solution close to the crack tip.

The peak occurs at and

decreases as . This corresponds to approximately twice the width of CTOD. Hence within this region, HRR singularity is not valid.

0x

J

1x

HRR Integral, cont.

Evaluation of J

Early works of Landes and Begley were based on the definition

1 UJ

B a

Obtain curve for different crack sizes .

Compute from the previous results.

Plot (see Fig.3-13 below).

The method is not useful there are too many specimens and tests.

P 1 2 3 4a a a a

dU

da

vs. J

HRR Integral, cont.

HRR Integral, cont.

Evaluation of J using DENT (Double edge notched tension)

Let the deep notch where H represents a function. Now,

12

0

12

0

plel12

0

2pl1 1

2/0

Note that in this case,

dA=2 da =-2 db

J= P

P

P

P

P

P P

P

P

dPa

dPb

dPb b

KdP

E b

( ) ( )PP bb H A

/

/

21/

0

( ) ( ).

From (A) ( )

1

and finally (integrating by parts)

1 2

2

P

P P P Pb b b

P

P Pb

b

P PP

P b

P P

H Hb

HP

Pb b P

KJ P d P

E b

HRR Integral, cont.

HRR Integral, cont.

It can be shown that

In general for any configuration

where is a dimensionless constant.

2

/0

2 P

IP

KJ M d

E b

2

/I P PK U

JE Bb

P

Evaluation of J using Edge Cracked Plate in bending

HRR Integral, cont.

Relationship between J and CTOD In LEFM . However in the case of small scale yielding (SSY), where m is a dimensionless constant that depends on the state of stress and the material properties.

J G YSJ m G

Contour along the boundary of the strip yield zone ahead of a crack tip.

HRR Integral, cont.

Note that in the figure is a contour along the strip yield zone. If the cohesive zone extent is considerably larger than the cohesive zone width, i.e., then and the traction on the crack face . Then,

. For a fixed value of ,

0;dy

1, 0leads to y x z yyn n n t

ii

uJ wdy t ds

x

Define a new coordinate system with = -x

( )

( )

yy yy

y yu u

If the cohesive zone length is small compared to other dimensions, then

Since

0

( )2 ( ) y

yy

uJ d

22 ( ); y

y

uu d d

0

( )yyJ d

For the strip yield model, since

The above relationship is valid for plane stress with elastic-perfectly plastic materials.

 

HRR Integral, cont.

ysJ

( )yy ys

More rigorous analysis based on HRR Integral Shih showed that when HRR integral can be assumed to be valid, then

where is a dimensionless constant. For non-hardening materials, . Thus in summary, there is a unique relationship between J and CTOD. Both the quantities are valid for elastic-plastic materials under certain assumptions.

0

nd J

nd 1nd

HRR Integral, cont.

3.4 Crack Growth Resistant Curves

Schematic J resistance curve for a ductile material.

HRR Integral, cont.

Both J and CTOD increase as the crack begins to grow and display a rising R-curve. R represents the resistance of the material to crack growth. Initially there is a stable crack growth and then it becomes unstable.

Note that and critical CTOD indicate the initiation of crack growth. The initiation cannot be precisely defined in practice. For generic materials, R-curve may provide a better indication of the toughness of the material rather than just . If we define a tearing modulus, then

ICJ

ICJ

RT

0

2R

R

E JT

a

Stability and R-curve for elastic materials

 We know from the Griffith theory, in perfectly brittle elastic materials crack extension starts when where is the surface energy per side of the crack face. Since the crack size a extends after initiation, the strain energy release rate (or crack driving force) G increases leading to an unstable crack growth in ideally elastic brittle material. However if the material is not ideal elastic, then the resistance to crack growth R increases as shown in the following figure.

2 fG w fw

HRR Integral, cont.

Schematic driving force R-curve diagrams

The left figure shows a material with a flat R-curve, while the right shows that of rising R-curve. In the flat type, while the crack is stable at stress level , it becomes unstable at a stress of . On the rising curve, the crack continues to grow from with stress levels . Thus the conditions for stable crack growth is

12

0 to ca a

1 4 to

and dG dR

G Rda da

HRR Integral, cont.

Unstable crack growth occurs when

Some notes on R-curve for elastic materials:

For a flat R-curve, one can define a critical value of .

For a rising R-curve, it is difficult to estimate the precise value of G at which crack starts to grow. This provides no information on the growth process or the R-curve.

dG dR

da da

critical cG G

HRR Integral, cont.

Reasons for R-curve

Resistance to fracture may arise from the energy needed to create new surfaces, additional inelastic irrecoverable dissipative energy needed.

For a flat R-curve, one can define a critical value of

If and is a constant, then R-curve is flat (only in ideal brittle material).

For ductile fracture in metals, plastic zone size increases with crack growth and hence R-curve rises. For infinitely large specimen, plastic zone size eventually reaches a steady state; R-curve then reaches a steady state.

Falling R-curve is possible if strain-rate effect reduces the size of plastic zone with crack growth.

State of stress alters the shape of R-curve. Plane stress has steeper R-curve compared to plane strain. R-curve may be affected by free boundary effects. Displacement control produces more stable than load control.

2 IR W

2R

HRR Integral, cont.

Schematic J resistance curve for a ductile material.

The driving force is expressed in terms of applied tearing modulus

Condition for stable crack growth

20

( )app

E JT

a

and R app RJ J T T

HRR Integral, cont.

Unstable crack growth when

app RT T

2 2 3

2 2

2 3

2 3 31

20

2 and

3

2 J 2| |

P

2 2| |

Phence

2 4 2{1 ( )}

3

p a

p a

app m

P a PaJ G

BEI EI

J P a a

a BEI BEI

Pa a

a EI BEI

P a a aT

BI EI EI

HRR Integral, cont.

J-controlled fracture

J-controlled fracture established crack tip conditions for J as well as CTOD.

HRR Integral, cont.

L---characteristic length scale; uncrack ligment in SSY, K and J characterize crack tip. r is where singularity. For monotonic, quasistatic loading J-dominated zone occurs in the plastic zone. Well inside the plastic zone (J-controlled), elastic singularity is not valid. However inside the plastic zone, HRR solution is valid and the stresses . Now finite strain occurs within 2 from tip. Here HRR field is not valid.

1r

11nr

In SSY, K characterizes crack tip, but singularity is not valid upto the tip.

Here J is valid for elastic-plasticity, but the

1r

HRR Integral, cont.

In elastic-plastic conditions, J is still approximately valid. Here K-field is not valid. As plastic zoone increases in size, K-dominated zone disappears but J-dominated zone persists. Here K has no meaning; j-integral and CTOD are valid parameters. Large scale yielding

Size of finite strain zone increases. J is not unique. Single parameter fracture mechanics does not exist. Critical J exhibit a size and geometry dependence.

HRR Integral, cont.

Note that in elastic unloading, non-linear elastic modeling (J-integral) is not valid. For J-controlled fracture, non proportional plasticity must be embedded within J-controlled region.

J-controlled crack growth

HRR Integral, cont.

SSY controlled crack growth