Crack Tip Plasticity II

Fracture Mechanics

Crack Tip Plasticity IIPresented by

Calvin M. Stewart, PhD

MECH 5390-6390

Fall 2020

Outline

• State of Stress • Crack Tip Region

• Influence on Fracture Behavior

• Influence on Fracture Toughness

• Additional Remarks

Crack Tip Plasticity in the Process Zone

State of StressCrack Tip Region

Crack Tip Region

• In our previous lecture, it was shown that the state of stress, i.e. plane stress or plane strain, affects the plastic zone size and shape.

• Applying yield criterion furnished the shapes in the figure, where plane stress and plane strain are different!!!!

• Let us now go into detail on the state of stress in the crack tip region.

Crack Tip Region

• Consider a through-thickness crack in a plate. We know that there is at least a biaxial (plane stress) condition, for which the elastic stresses in the x and y directions are given by

• shows that for small values of r both σx and σy will exceed the material yield stress. Thus a biaxial plastic zone will form at the crack tip.

Crack Tip Region

• Assuming in the first instance that there is a uniform state of plane stress and that the plastic zone is circular as in Irwin’s analysis, then a section through the plate in the plane of the crack gives the situation

With no strain hardening the material within the plastic zone should be able to flow freely (large strain) and contract in the thickness direction: however, the adjacent (and surrounding) elastic material cannot contract to the same extent. This phenomenon, called plastic constraint, leads to tensile stresses in the thickness direction on the plastic zone boundary, i.e. a triaxial stress condition which when unrelieved by deformation would correspond to plane strain.

Crack Tip Region

• For a plate of intermediate thickness that is neither fully in plane stress nor predominantly in plane strain these approximate variations are considerable, as

Crack Tip Region

• For Plane Stress Situations, we can show that the stresses in the vicinity of the crack cause large strain in the z direction.

• For “Thick” samples, the material at the crack tip middle is constrained (εzz=0), so that (σzz≠0) producing a Triaxial Stress State or “Crack Tip Triaxaility”

Crack Tip Region

• As such, it follows that a state of plane stress only exists at the free surface

Crack Tip Region

• Simple calculation of the stress state distribution for a certain plate thickness is not possible. However, there are empirical rules for estimating whether the condition is predominantly plane stress or plane strain:

• 1) Full plane stress may be expected if the calculated size of the plane stress plastic zone, i.e. 2ry in Irwin’s analysis, is of the order of the plate thickness.

• 2) Predominantly plane strain may be expected when the calculated size of the plane stress plastic zone, 2ry (the approximate value at the plate surfaces), is no larger than one-tenth of the plate thickness.

Crack Tip Region

• Consider the Mode I Crack in a Ductile Material

( )

0

zz

xx yy

Plane

Plane

=

+

For θ=0

2

0

0

22

Ixx yy

xy

zz I

K

r

Plane

KPlane

r

= =

=

=

Crack Tip Region

• Assuming ν=0.33

• At Yielding, using Von Mises

1 2

3

0

2

2

2

I

I

K

r

Plane

KPlane

r

= =

=

( ) ( ) ( )2 2 2

1 2 2 3 3 1

1

2von = − + − + −

( )

1

2

22

I

von

I

KPlane

r

KPlane

r

= −

Crack Tip Region

• Again von Mises,

• Similarly for Tresca,

• The result is the same for both cases!

( ) ( ) ( )2 2 2

1 2 2 3 3 1

1

2von = − + − + −

( )

1

2

22

I

von

I

KPlane

r

KPlane

r

= −

( )

2

1 22

I

tresca

I

KPlane

r

KPlane

r

= −

1 3tresca = −

Crack Tip Region

• Let’s Solve for the Plastic Zone Size, remembering Irwin’s Solution

• Using the previous slide we find ry

• where C is the plastic constrain factor, which describes the state of plane stress or plane strain.

( )

1

1

1 2

Plane

CPlane

= −

2

12 Iy

ys

Kr

=

21

,

1

2

I

y

ys

C Kr

− =

Crack Tip Region

• Now Given v=0.33

• Repeat again for -π ≤ θ ≤+π to generate yield surface for ductile materials

,

,

1

13

9

y planestress y

y planestrain y

Plane C r r

Plane C r r

= =

= =

Crack Tip Region

Effect of Thickness

• At the Elastic Limit ν approaches 0.5, The plane strain solution is NOT realistic

• Irwin’s proposed as an alternative C for plane strain,

• For real materials, the crack tip radius is not zero in which case

• Coupled with ν=0.5 means

3C =

21

,

1

2

I

y

ys

C Kr

− =

,

1

3y planestrain yr r=

y x

y z x 1 2 3

( )

1 1

1 2 0C

= = =

−

21

,

10

2

I

y

ys

C Kr

− = →

Planes of Maximum Shear Stress

Location of the planes of maximum shear stress at the tip of a crack for

a) plane stress and b) plane strain conditions.

z

yx

The Maximum Shear Stress Plane is between XZ-Plane and YZ-Plane Inclined at 45°

Plane StrainPlane Stress

A similar approach can be used for Plane Stress to find the following…

State of StressInfluence on Fracture Behavior

Influence on Fracture Behavior

Influence on Fracture Behavior

• Crack extension begins macroscopically flat but is immediately accompanied by small ‘shear lips’ at the side surfaces.

• As the crack extends (which it does very quickly at instability) the shear lips widen to cover the entire fracture surface, which then becomes fully slanted either as single or double shear.

• Single shear is associated with plane stress.

• > 90% Flat + Double shear is associated with plane strain.

Influence on Fracture Behavior

Plane Stress Transitional Plane Strain

Influence on Fracture Behavior

• Planes of maximum shear stress, play an important role. Experimental studies indicate that under (a) plane strain conditions a ‘hinge’ type deformation is followed by flat fracture, whereas under (b) plane stress slant fracture occurs by shear after a hinge type initiation.

Influence on Fracture BehaviorEffect of Temperature

Effect of Thickness

304SS subjected to Liquid Metal EmbrittlementExhibits Transitional Thickness Fracture Surface

Effect of Environment

State of StressInfluence on Fracture Toughness

Influence on Fracture Toughness

• The critical stress intensity, Kc, depends on specimen thickness.

• Beyond a certain thickness, when the material is predominantly in plane strain and under maximum constraint, the value of Kc tends to a limiting constant value.

• This value is called the plane strain fracture toughness, KIc, and may be considered a material property.

Influence on Fracture Toughness

Plane Strain Fracture Toughness

Influence on Fracture Toughness

• In experiments, to ensure the plane strain fracture toughness is obtained. The following rule of thumb has been adopted.

• where B is the thickness, a is the crack length, and (W-a) is the ligament length.

( )

2

, , 2.5 Ic

ys

Ka B W a

−

Proportional to plastic zone size

ASTM Plane Strain Criterion(W-a) a

B

Additional Remarks

Additional Remarks

• The plastic zone can be experimental measured using techniques such as electron microscopy. This techniques are mostly used to study the development of plastic zones during fatigue crack growth in order to obtain more insight into the mechanics of crack extension.

• Finite element analysis is a great tool for estimating the plastic zone size and shape in components subject to complex loading conditions and crack geometries which cannot be estimated analytically.

Additional Remarks

• The Plane Strain Fracture Toughness depends on Temperature, Load rate, and NOT on Thickness.

• -Irwin et al. developed model for experimental data [1967]

• This empirical theory developed for test samples has difficulties in modeling real components.

( )1/2

2

1 1.4

1

crit Ic Ic

IcIc

ys

K K

K

B

= +

=

,where B is thickness!

Specimen Components

Summary➢ The state of stress in the crack tip region is greatly influenced by the thickness of the

component (i.e. the state of plane stress versus plane strain).

➢ A Triaxial state of stress arises under the plane strain condition.

➢ Empirical rules existing to determine if a component is under plane stress or plane strain. These rules also tell us about the planes of maximum shear stress within the component.

➢ The State of stress contributes to the formation of the fracture surface with plane stress samples exhibit single shear and plane strain sample exhibiting > 90% flat + double shear lips.

➢ The Critical Stress Intensity, Kc, depends on specimen thickness.

➢ The Plane Strain Fracture Toughness, KIc is the limiting value of Kc where it becomes incentive to thickness.

Homework 6

1. A thin steel plate (σys=40 ksi) that is 6 inches wide contains a central crack 2.0 inches in length.• (a) if the plane-stress fracture toughness of the steel is 55 ksi √in what is the

maximum stress that can be supported by the plate (including effects of local yielding on the crack tips)?

• (b) What is the longest crack the plate can support without failure at an applied stress of 20 ksi?

Homework 6

Homework 6

References

• Janssen, M., Zuidema, J., and Wanhill, R., 2005, Fracture Mechanics, 2nd Edition, Spon Press

• Anderson, T. L., 2005, Fracture Mechanics: Fundamentals and Applications, CRC Press.

• Sanford, R.J., Principles of Fracture Mechanics, Prentice Hall

• Hertzberg, R. W., Vinci, R. P., and Hertzberg, J. L., Deformation and Fracture Mechanics of Engineering Materials, 5th Edition, Wiley.

• https://www.fracturemechanics.org/

https://www.fracturemechanics.org/

CONTACT INFORMATION

Calvin M. Stewart

Associate Professor

Department of Mechanical Engineering

The University of Texas at El Paso

500 W. University Ave, Suite A126, El Paso, TX 79968-0521

Ph: 915-747-6179

cmstewart@utep.edu

me.utep.edu/cmstewart/

mailto:cmstewart@utep.eduhttp://me.utep.edu/cmstewart/