CHAPTER 3 – STRESS INTENSITY FACTOR, K

STRESS INTENSITY FACTOR, K

Consider a coordinate system for describing the stresses in the vicinity of a crack is shown in above drawing with Mode I loading.

K, characterizes the magnitude (intensity) of the stresses in the vicinity of an ideally sharp crack tip in a linear-elastic and isotropic material.

• For any case of Mode I loading, the stresses near the crack tip depend on r and θ as follows:

3cos 1 sin sin2 3 22

3cos 1 sin sin2 2 22

3sin cos cos2 2 22

Ixx

Iyy

Ixy

Kr

Kr

Kr

θ θ θσπ

θ θ θσπ

θ θ θτπ

⎛ ⎞= −⎜ ⎟⎝ ⎠⎛ ⎞= +⎜ ⎟⎝ ⎠

=

or ( )θπ

σ ijI

ij fr

K2

=

STRESSES NEAR THE CRACK TIP

( )0 (plane stress)

(plane strain, 0)

0

z

z x y z

yz zx

σ

σ ν σ σ ε

τ τ

=

= + =

= =

and

STRESSES NEAR THE CRACK TIP

( )2

Iij ij

K fr

σ θπ

=

For any case of Mode I loading, the stresses near the crack tip depends on r and θ :

KI is a measure of the severity of the crack, its definition in a formal mathematical sense is:

( ), 0lim 2I yr

K rθ

σ π→

=

It is generally convenient to express KI as:

( ); /IK YS a Y F a Lπ= =

Y is geometry constant depending on crack size (a) and a size parameter of the body (L), S is remotely applied stress

( ); /I gK YS a Y F a Lπ= =

The cracking situations shown here are ideal cases, in real application, the cracking situations are far more complicated.

However, these real crack situations have more practical meanings.

SOME EXAMPLES OF CRACKING

ELLIPTICAL CRACKS• Natural cracks occurring in practice are often initiated at corners

and edges. They tend to grow inwards and assume a quarter-elliptical or semi-elliptical shape.

Elliptical crack in aircraft-engine crank shaft

Corner crack in high strength steel lug

Surface flaws in fracture test specimens

D.Broek, Elementary Engineering Fracture Mechanics, Page 89, Figure 3.10, Page 91, Figure 3.12

(a) An embedded circular crack under uniform tension normal to the crack plane.

(b) Half-circular surface crack

(c) Quarter-circular corner crack

(d) Half—circular surface crack in shaft.

Case St Sb YFor small a

Limits±10% on Y

(a) P/4bt - 2/π = 0.637 a/t; a/b <0.5

(b) P/2bt 3M/bt2 0.728 a/t<0.4; a/b<0.3

(c) P/bt 6M/bt2 0.722 a/t<0.35;a/b<0.2

(d) 4P/πd2 32M/πd3 0.728 a/d<0.2 (t); a/d<0.35 (b)

(tension, P)(bending , M)

t

b

K FS aS SS S

π==

=

( )1.65

; 1 1.464 / 0.1D Da aK F S Q a c

Q cπ ⎛ ⎞= ≈ + ≤⎜ ⎟

⎝ ⎠This equation gives KD at point D for a uniform tension normal to the crack plane.

Crack growing from Notches

This is a situation of practical interest, a crack may grow from a stress concentration point, such as a hole, notch, or fillet, such as the case shown above of a pair of cracks growing from a circular hole in a wide plate.

If the crack is short compared with the hole radius, the solution is the same as for a surface crack in an infinity body, except that the stress is now ktS, being amplified by the stress concentration factor, if it is circular hole, then kt=3, therefore:

1.12A tK k S lπ=where l is the crack length measured from the hole surface.

If the crack has grown far from the hole, the solution is the same as for a single long crack of tip-to-tip length 2a, in this case:

BK FS aπ=Hence, for long cracks, the width of the hole acts as part of crack.

The exact K first follows KA, then falls below it and approaches to KB. In most cases, a crack at an internal or surface notch can be roughly approximated by using KA for crack lengths upto that where KA=KB, and then using KB for all longer crack lengths. The crack length where KA=KB, is labeled as l=l’ in above figure.

• The Griffith concept is one of the basic equation of fracture mechanics and it was established in 1921.

• Griffith noted that when a crack is introduced to a stressed plate of elastic material, a balance must be struck between the decrease in potential energy (related to the release of stored elastic energy and work done by movement of the external loads) and the increase in surface energy resulting from the presence of the crack.

• Likewise, an existing crack would grow by some increment if the necessary additional surface energy were supplied by the system.

• This “surface energy” arises from the fact that there is a nonequilibrium configuration of nearest neighbor atoms at any surface in a solid.

GRIFFITH CRACK THEORY

THE GRIFFITH CONCEPT• Considering an infinite cracked plate of unit thickness with a

central transverse crack of length 2a. The plate is stressed to a stress σ and fixed at its ends as shown in the below. The load displacement diagram is given in the right.

• If the crack extends over a length dathe stiffness of the plate will drop (line OC), which means that some load will be relaxed since the ends of the plate are fixed. The elastic energy content will drop to a magnitude represented by area OCB. Crack propagation from a to a+da will result in an elastic energy release equal in magnitude to area OAC.

GRIFFITH CRACK THEORY

0 aU U U Uγ= − +Total potential

energy

Elastic energy of the

uncracked plate

Decrease in the elastic

energy caused by introducing the crack in the

plate

Increase in the elastic-surface energy caused

by the formation of the crack

surface

• For a large plate (thickness t) with a through thickness crack (length 2a) in the center and subjected to a tensile stress σ, Griffith estimated the surface energy term to be the product of the total crack surface area (2a*2*t), and the specific surface energy γs , which has units of energy/unit area.

• Griffith then used the stress analysis of Inglis for the case of an infinitely large plate containing an elliptical crack and computed the decrease in potential energy of the cracked plate to be (πσ2a2t)/E.

GRIFFITH CRACK THEORY• The change in potential energy of the plate associated with the introduction of a

crack may be written as:

satE

taUU γπσ 422

0 +−=−

• Determining the condition of equilibrium by differentiating the potential energy U with respect to the crack length and setting equal to zero

0242

=−=∂∂

Eatt

aU

sπσγ

• Therefore:E

as

22 πσγ = or

aE sπγσ 2

= for plane stress

and ( )212

νπγσ−

=a

E s for plane strain

THE GRIFFITH CONCEPT• The condition for crack growth is:

dadW

dadU

=

U is the elastic energy and W is the energy required for crack growth.

• For an elliptical flaw

Ea

dadU 22πσ

=

per unit plate thickness, where E is elastic modulus.

• Usually dU/da is replaced by:

EaG

2πσ=

which is called “elastic energy release rate” per crack tip, G is also called the crack driving force.

• The energy consumed in crack propagation is denoted by R=dW/da which is called the crack resistance.

• The energy condition states that G must be at least equal to R before crack propagation can occur.

THE GRIFFITH CONCEPT• If R is a constant this means that G must exceed a certain critical

value GIc. Hence crack growth occurs when :

aEGorG

Ea Ic

cIcc

πσπσ

==2

• Griffith derived his equation for glass, which is a very brittlematerial. Therefore he assumed that R consisted of surface energy only.

• In ductile materials, such as metals, plastic deformation occurs at the crack tip. Much work is required in producing a new plastic zone at the tip of the advancing crack.

• The plastic zone has to be produced upon crack growth the energyfor its formation can be considered as energy required for crackpropagation. This means that for metals R is mainly plastic energy, the surface energy is so small and can be neglected.

THE GRIFFITH CONCEPT• The energy criterion is a necessary criterion for crack extension.

It need not be a sufficient criterion.

• Even if sufficient energy for crack propagation can be provided,the crack will not propagate unless the material at the crack tip is ready to fail: the material should be at the end of its capacity to take load and to undergo further straining.

• Since and therefore this gives: aKI πσ=Ic

c GE

a=

2πσ

GE

K=

2

• For the case of plane strain:

( ) ( ) IcIc

II G

EKandG

EK

=−=−2

22

2 11 νν

THE GRIFFITH CONCEPT

• It is important to recognize that the Griffith relationship was derived for an elastic material containing a very sharp crack.

• The radius of the crack is assumed to be very sharp.

• As such, the Griffith relationship, as written, should be considered necessary but not sufficient for failure.

• The crack tip radius also would have to be atomically sharp to raise the local stress above the cohesive strength.

• The Griffith equation and its underlying premise are basically sound and represent a major contribution to the fracture literature.

MODIFIED GRIFFITH EQUATION• Griffith equation is valid only for ideally brittle solids. He obtained good

agreement between his theory and experimental fracture strength of glass, but the Griffith theory severely underestimates the fracture strength of metals.

• Irwin (1948) and Orowan (1948) independently modified the Griffith expression to account for materials that are capable of plastic flow. The revised expression is give by:

( )a

E psc π

γγσ

+=

2

where γp is the plastic work per unit area of surface created, and is typically much larger than γs.

• In an ideally brittle solid, a crack can be formed merely by breaking atomic bonds; γs reflects the total energy of broken bonds in a unit area. When a crack propagates through a metal, however, dislocation motion occurs in the vicinity of the crack tip, resulting in additional energy dissipation.

GENERAL SOLUTION FOR STRESS FIELD

• In the vicinity of the crack tip the higher order terms can be neglected and the stress field at the crack tip are obtained as:

( )π

θσ2

, 11

1 KCfr

Cijij ==

• The general analysis shows that the stress fields surrounding mode I crack tips are always of the same form. It only remains to find K1for a particular configuration.

MODE II CRACK SOLUTIONS• Mode II crack problem

( ) 0,2

3sin2

sin12

cos2

23cos

2cos

2sin

2

23cos

2cos2

2sin

2

==+=

⎥⎦⎤

⎢⎣⎡ −=

=

⎥⎦⎤

⎢⎣⎡ +

−=

yzxzyyxxzz

IIxy

IIyy

IIxx

rK

rK

rK

ττσσυσ

θθθπ

τ

θθθπ

σ

θθθπ

σ

• For an infinite cracked plate with uniform in-plane shear stress τ at infinity:

aKII πτ=

MODE III CRACK SOLUTIONS• For Mode III crack problem:

02

cos2

,2

sin2

====

=−

=

xyzzyyxx

IIIyz

IIIxz r

Kr

K

τσσσ

θπ

τθπ

τ

SUPERPOSITION PRINCIPLE• Since the stress field equations are the same for all mode I cases,

the stress intensity factor for a combination of load systems p, q, r can be obtained simply by superposition:

....+++= IrIqIpI KKKK

• Similar cases for mode II and mode III.

• In a combination of different modes loading, the superposition is not allowed.

• The superposition principle can sometimes be used to derive stress intensity factors.

( ) IIIIIItotal KKKK ++≠

EXAMPLES OF SUPERPOSITION

• Considering a crack with internal pressure

• Case a shows a plate without a crack under uniaxial tension, since no crack, then the KIa=0.

• A cut of length 2a is made in the center of the plate. This is allowed if the stresses previously transmitted by the cut material are applied as external stresses to the slit edges (case b, with KIb=0).

• Case b is a superposition of a plate with a central crack under uniaxial tension σand a plate with a crack having distributed forces σ at its edges.

aKK

KKK

IdIe

IbIeId

πσ−=−=

==+ 0

EXAMPLES OF SUPERPOSITION

• Considering the case of a crack emanating from a loaded rivet hole. The hole is small with respect to the crack.

• This case can be obtained from a superposition of three other cases.

IeIdIbIa KKKK −+=

• Since it is obvious that IeIa KK =

• The stress intensity: ( )a

WaKKK IdIbIa πσπσ

221

21 +=+=

EFFECTS OF SAMPLE THICKNESS

Effect of thickness on fracture toughness of an alloy steel heat treated to the high strength

EFFECTS OF SAMPLE THICKNESS• Experimentally-measured fracture toughness, KQ, decreases

with increasing specimen thickness, t, as illustrate in previous slide.

• This is because the behavior is affected by the plastic zone at the crack tip in a manner that depends on thickness.

• Once the thickness obeys the following relationship, involving the yield strength, σ0, no further decrease is expected.

2

0

2.5 QKt

σ⎛ ⎞

≥ ⎜ ⎟⎝ ⎠

EFFECTS OF YIELD STRENGTH

Fracture toughness vs yield strength for AlSl4340 steel quenched and tempered to various strength levels.

EFFECT OF TEMPERATURE

Fracture toughness and yield strength vs temperature for a nuclear pressure vessel steel.

Fracture toughness vstemperature for a silicon nitride ceramic

EFFECT OF TEMPERATURE

Fracture toughness vs. temperature for several steels used for turbine-generator rotors

EFFECT OF TEMPERATURE• Fracture toughness generally increases with temperature.• In metals with BCC structure, there is a distinct temperature-transition

behavior.• The physical mechanisms of fracture are different in below and above

the temperature-transition.• Below the temperature-transition, the fracture mechanism is identified

as cleavage, and above it, as dimpled rupture.

Cleavage fracture surface Dimpled rupture surface

EFFECT OF LOADING RATE

A higher rate of loading usually lowers the fracture toughness, having an effect similar to decreasing the temperature. The effect can be thought of as causing temperature shift in the KIC behavior.

EFFECT OF MICROSTRUCTURE• Small variations in chemical composition or processing of a given material can significantly affect fracture toughness. This is because they usually have effects on a microscopic level that facilitate fracture.

• Fracture toughness is generally more sensitive than other mechanical properties to anisotropy and planes of weakness introduced by processing.

• Nonmetallic inclusions and voids may also become elongated and/or flattened so that they also cause the fracture properties to vary with direction.

Designation for crack plane and growth direction in rectangular sections.

Corresponding effects on fracture toughness in plates of three aluminum alloys.

MIXED-MODE FRACTURE

• If a crack is not normal to the applied stress, or if there is a complex state of stress, a combination of fracture Models I, II and III may exist.

• This example shows a combined Modes I and II.

• This is complex because the crack may change directions so that it does not grow in its original plane, and also because the two fracture modes so not act independently, but rather interact.

Combined mode fracture in two aluminum alloys

EXAMPLE 3.1A theoretical calculation based on the cohesion force suggest that the tensile strength of glass should be 10 GPa. However, a tensile strength of only 1.5% of this value is found experimentally. Griffith supposed that this low value was due to the presence of cracks in the glass. Calculate the size 2a of a crack normal to the tensile direction in a plate. (Young’s modulus E=70 GPa, and surface tension γs=0.5J/m).

Example 3.1 Solution:The experimental tensile stress is σ=10e9*0.015=150e6=150 MPa.

Based on the Griffith energy relation: σf=(2Eγs/πa)1/2, so that the half length of the crack is a=(2Eγs/πσf

2)=(2*70e9*0.5/3.14*(150e6)2)=9.9e-7 m, so the size of a crack normal to the tensile direction should be

2a=1.98e-6m = 1.98 mm

EXAMPLE 3.2Rocket motor casings may be fabricated from either of two steels: (a) low alloy steel: yield strength 1.2 GPa, toughness 70 MPam1/2; (b) maraging steel: yield strength 1.8 GPa, toughness 50 MPam1/2; The relevant specification requests a safety factor of n=1.5. Calculate the minimum defect size which will lead to brittle fracture in service for each material.

A safety factor n=1.5 means that the maximum applied stress should be σa= yield-strengh/1.5.

Assume it is a large plate with central defect, therefore, at failure, we can use KIc=σa√πac, this gives: ac=(KIc/σa)2/π=(nKIc/σf)2/π. So

(a) 2ac=2/π*(1.5*70e6/1.2e9)2=4.9 mm

(b) 2ac=2/π∗(1.5*50e6/1.8e9)2=1.1 mm

Which materials should be used?

Example 3.2 Solution:

EXAMPLE 3.3A plate of maraging steel has a tensile strength of 1900 MPa. Calculate the reduction in strength caused by a crack in this plate with a length 2a=3 mm oriented normal to the tensile direction. (Elastic modulus E=200 GPa, surface tension γs=2 J/m2, plastic energy per unit crack surface area γp=2x104J/m2.

A1: Assuming plane stress condition, if we use the Griffith theory for brittle materials, then we have σa=(2Eγs/πa)1/2=(2*200e9*2/π*(3e-3/2))=13.03 MPa. Only 0.68% of original tensile strength, this obviously underestimates the residual strength.

A2: If we use the modified Griffith theory for metals, we have: σa=(2E(γs+γp)/πa)1/2=(2*200e9*(2+2e4)/π∗(3e-3/2))=1303 MPa. This gives the reduction of the strength by (1900-1303)/1900=31.42%. This is more reasonable estimation and showing why the Griffith theory needs to be modified if applied to metals.

Example 3.3 Solution:

EXAMPLE 3.4σ

σ

W

a

Material Yield strength (N/mm2)

Tensile strength (N/mm2)

Fracture Toughness (N/mm3/2)

Steel 4340 1470 1820 1500

Maragingsteel

1730 1850 2900

Al 7075-T6

500 560 1040

Consider a plate with an edge crack (see figure). The plate thickness is such a plane strain condition is present (Given: W=1000 mm, other material properties are given in following table).

Answer following questions: (1) Does fracture occur at stress σ=2/3σys and a crack length a=1 mm? (2) What is the critical defect size at a stress σ=2/3σys? (3) What is the maximum stress for a crack length a=1 mm without permanent consequences?

Example 3.4 SolutionPlane strain stress intensity factor for edge crack is KI=Cσ√πa, with C=1.12.

(a) If a=1 mm, and σ=2/3σys, then for the three materials:

Steel 4340: KI=1945.5 N/mm3/2 > KIc=1500 N/mm3/2, fracture occurs;

Maraging steel: KI=2289.5 N/mm3/2 < KIc=2289.5 N/mm3/2, no fracture;

Al 7075-T6: KI=661.7 N/mm3/2 < KIc=1040 N/mm3/2, no fracture.

(b) If σ=2/3σys, the critical defect size can be determined by ac=(KIc/1.12σ)2/π, so:

Steel 4340: ac=0.595 mm; Maraging steel: ac=1.604 mm;

Al 7075-T6: ac=2.470 mm.

(c) If a=1 mm, without permanent consequences, the critical stress can be calculated as σa=KIc/(1.12√πa), so:

Steel 4340: σa=755.6 N/mm2; Maraging steel: σa=1460.8 N/mm2;

Al 7075-T6: σa=523.9 N/mm2 > σys, yield will occur, so that the maximum stress can not higher than yield strength, i.e., σa=500 N/mm2.