PowerPoint Presentation · Title: PowerPoint Presentation Author: mashayekhi Created Date: 10/8/2018 2:44:01 PM

دانشکده مکانیک -دانشگاه صنعتي اصفهان مکانیک شکست

دانشگاه صنعتي اصفهان دانشکده مکانیک

Crack Tip Plasticity

1

دانشکده مکانیک -دانشگاه صنعتي اصفهان مکانیک شکست 2

Applies when non-linear deformation is confined to a small region surrounding the crack tip

Linear elastic fracture mechanics (LEFM)

Elastic-Plastic fracture mechanics (EPFM) :

Effects of the plastic zone negligible, linear asymptotic mechanical field.

Generalization to materials with a non-negligible

plastic zone size: elastic-plastic materials crack

Plastic zone

Elastic Fracture Contained yielding Full yielding Diffuse dissipation

D

L

a

, ,L a D B

B

L D a L D a

LEFM, KIC or GIC

fracture criterion EPFM, JC fracture

criterion

Catastrophic failure,

large deformations

بررسي محدوده پالستیک جلوی ترک


The Irwin Approach

رفتار پالستیک ناحیه جلوی ترک

Mode I loading of a elastic-

perfectly plastic material:

02

Iyy

K

r

Plane stress assumed

(1)

crack x ry

Y

rp

(1)

(2)

(2)

To equilibrate the two stresses distributions (cross-hatched region)

rp ?

Elastic:

Plastic correction ,yy Y pr r

ry : Intersection between the elastic distribution and the horizontal line yy Y S

2

IY S

y

K

r

0 0 2

y yr r

IY S p yy

Kr dr dr

r

yy

2

1

Ip

Y S

Kr

21

2

Iy

Y S

Kr


The Irwin Approach

Redistribution of stress due to plastic deformation:

Plastic zone length (plane stress):

Irwin’s model = simplified model for the extent of the plastic zone:

- Focus only on the extent of the plastic zone along the crack axis, not on its shape.

21

2

Iy

Y S

Kr

- Equilibrium condition along the y-axis not respected.

yy

real crack x

Y

fictitious crack

21

6

Iy

Y S

Kr

2

effyy

K

X

X

In plane strain, increasing of YS. : Irwin suggested in place of YS 3 Y S



The Irwin Approach

eff ya a r The effective crack:

2

1

2

Iy

Y S

Kr

For plane stress:

2

1

6

Iy

Y S

Kr

For plane strain:

The effective stress intensity is obtained by inserting aeff into the K

expression for the geometry of interest:

( )eff eff effK Y a a

Since the effective crack size is taken into account in the geometry

correction factor, Y, an iterative solution is usually required to solve for Keff.



Convergence after a few iterations…

Initial:

IK Y a a 2

01

2I

Y S

Ka a

m

1 plane stress

3 plane strainm

Yes

No

1i i

Algorithm:

( )

I

ieff i iK K Y a a

Application: Through-crack in an infinite plate (plane stress):

= 2 MPa, YS = 50 MPa, a = 0.1 m KI = 1.1209982

Keff= 1.1214469 4 iterations

Do i = 1, imax :

( )1 1

I

ii iK Y a a

2( )

1

2I

i

iY S

Ka a

m

( ) ( 1)

I I

i iK K

?

The Irwin Approach



The Strip Yield Model

The strip yield model was first proposed by Dugdale and Barenblatt.

This model approximates the elastic–plastic behavior by superimposing

two elastic solutions: a through crack under remote tension and a through

crack with closure stresses at the tip. Thus, the strip yield model is a

classical application of the principle of superposition.



The Strip Yield Model

( ) ( )I pK a r Load s:

Load sYS: closure( ) 2 arccos

p

I Y S Y S

p

a r aK K

a r

singular term = 0

( ) ( )I I Y SK K

cos2

p Y S

a

a r

2

8

Ip

Y S

Kr


Y S


eff pa a r The effective crack:

sec2

eff

Y S

K a



مقایسه تاثیر ناحیه پالستیک

: ILEFM K a

Comparison of Plastic Zone Corrections


The von Mises equation:

شکل ناحیه پالستیکPlastic Zone Shape

The principal stresses:

1

2

3

12 22

1 ( )2 22

0 (plane stress)

2(plane strain)

22

***

I

I

I

KCos Sin

r

KCos Sin

r

KCos

r

1/22 2 2

1 2 1 3 2 3

1( ) ( ) ( ) *( )

2 e

2

2

1 2, ( *)2 2

*

xx yy xx yy

xy

Substituting the Mode I stress fields into Eq. (**):


Plastic Zone Shape

2

2

2

2 2

1 3( ) 1 for plane stress

4 2

1 3( ) (1 2 ) (1 ) for plane strain

4 2

Iy

Y S

Iy

Y S

Kr Cos Sin

Kr Cos Sin

شکل ناحیه پالستیک


Crack tip plastic zone shapes estimated from the elastic solutions and the von Mises yield

criterion: (a) Mode I, (define the approximate boundary between elastic and plastic behavior).

Plastic Zone Shape

2

1

4

p

I

Y

r

K

Plane strain

Plane stress (= 0.0)

increasing 0.1, 0.2, 0.3, 0.4, 0.5

شکل ناحیه پالستیک


Plane strain or plane stress?

In general, the conditions ahead of a crack tip are neither plane stress nor

plane strain. There are limiting cases where a two dimensional

assumptions are valid, or at least provides a good approximation.

The nature of the plastic zone that is formed ahead of a crack tip plays a

very important role in the determination of the type of failure that occurs.

Since the plastic region is larger in PSS than in PSN, plane stress failure

will, in general, be ductile, while, on the other hand, plane strain fracture

will be brittle, even in a material that is generally ductile. This

phenomenon explains the peculiar thickness effect, observed in all

fracture experiments, that thin samples exhibit a higher value of fracture

toughness than thicker samples made of the same material and operating

at the same temperature. From this it can be surmised that the plane stress

fracture toughness is related to both metallurgical parameters and

specimen geometry while the plane strain fracture toughness depends

more on metallurgical factors than on the others.

ای و کرنش صفحه ای اعتبار حالت تنش صفحه


Due to presence of crack tip,

stress in a direction to normal to

crack plane yy will be large

near the crack tip. This stress

would in turn tries to contract in

x and z direction. But the

material surrounding it will

constraint it, inducing stresses in

x and z direction, there by a

triaxial state of stress exists near

the crack tip. This leads to plane

strain condition at interior. At

the plate surface zz is zero and

zz is maximum. This leads to

plane stress condition at exterior.

بروز تنش سه محورهPlane strain or plane stress?


The state of stress is also dependent on size of plate thickness.

If the plastic zone size is small compared to the plate thickness, plane

strain condition exists. If the plastic zone size is larger than the plate

thickness, plane stress condition prevails. As the loading is increased,

plastic zone size also increases leading to plane stress conditions.

Plane strain or plane stress?

ای و کرنش صفحه ای اعتبار حالت تنش صفحه


تاثیر ضخامت بر ضریب شدت تنش بحراني

Effect of plate thickness on fracture toughness


Limits of LEFM

As per ASTM standard LEFM is applicable for components of size

2

2

2

2.5

or B 2.5

or b 2.5

I

ys

I

ys

I

ys

Ka

Kt

KW

As per ASTM standard fracture toughness testing can be done on

Specimens of size 2

, , 2.5

Ic

ys

Ka B W

تاثیر ضخامت بر ضریب شدت تنش بحراني


1) 1D approximation (3) L corresponding to 0pr

Remarks:

22

1 2

2

I

Y S

KL

Thus, in plane strain:

2) Significant difference in the size and shape of mode I plastic zones.

For a cracked specimen with finite thickness B, effects of the boundaries:

in plane stress:

21

2

I

Y S

KL

- Essentially plane strain in the in the central region.

Triaxial state of stress near the crack tip:

- Pure plane stress state only at the free surface.

B

Evolution of the plastic zone shape through the thickness:

جمع بعدی تاثیر ناحیه پالستیک جلوی ترک


4) Solutions for rp not strictly correct, because they are based on a purely elastic:

3) Similar approach to obtain mode II and III plastic zones:

Alternatively, Irwin plasticity correction using an effective crack length …

Stress equilibrium not respected.

جمع بعدی تاثیر ناحیه پالستیک جلوی ترک

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PowerPoint Presentation · Title: PowerPoint Presentation Author: mashayekhi Created Date: 10/8/2018 2:44:01 PM