Mechanical Behaviour of Materials

Dr.-Ing. 郭 瑞 昭

Chapter 14 Fracture mechanics

Facture Mechanics

Fracture toughness

Applied stress

Flaw size

aIcK

σ

A material fracture depends on temperature, the stress state and with time, and the environment.

Theoretical facture strength

Stress required to separate two atomic layers

Under normal stress a material is to cleave, when the fracture surface is perpendicular to the applied stress.

The atoms are separated along the direction of the applied stress.

Theoretical facture strength: stress intensity approach

Eadx

ddd

dadx

xdth

==

=

⎟⎠

⎞⎜⎝

⎛=

0

0

/.3

.2

2sin.1

σεσ

ε

πσσ

⎟⎠

⎞⎜⎝

⎛=dx

ddxd

a thπ

σπσ 2

cos2

0

Ead

th02π

σ =

0

2

→

⎟⎠

⎞⎜⎝

⎛=x

thddxd

σπσ

102EE

th ≈≈π

σ

Theoretical facture strength: Energy criterion

γπ

σ 22sin2/

0=⎟

⎠

⎞⎜⎝

⎛∫d

th dxdx

0aE

thγ

σ =

( )0

2

aE

thγ

σ =

⎟⎠

⎞⎜⎝

⎛= xdthπ

σσ2sin

Theoretical Cleavage Strength

Crack-initiated fracture: stress concentration

The fundamental requisite for the propagation of a crack is that the stress at the tip of the crack must exceed the theoretical cohesive strength of the material.

The stress concentration factor (SCF) is as the ratio of the maximum stress to the applied stress.

a

Kσσmax=

Stress concentration factor (stress approach)

Inglis: The stress rises dramatically near the hole and has a maximum value at the edge of the hole. The maximum value is given:

⎟⎠

⎞⎜⎝

⎛ +σ=σba

a 21max

ρσ≅⎟⎟

⎠

⎞⎜⎜⎝

⎛

ρ+σ=σ

aaaa 221max

a<<ρab2

=ρ

having a radius of curvature

ρσ=σ

aa2max

External applied stressstress concentration factor

ρaK 2=

a

Kσσmax=stress concentration factor

Stress concentration factor (stress approach)

⎟⎠

⎞⎜⎝

⎛ +σ=σba

a 21max

Facture modes

Shear mode

Irwin’s fracture analysis: stress intensity factor

⎥⎦

⎤⎢⎣

⎡ −=23

sin2

sin12

cos2

θθθ

πσ

rKI

x

⎥⎦

⎤⎢⎣

⎡ +=23

sin2

sin12

cos2

θθθ

πσ

rKI

y

23

cos2

sin2

cos2

θθθ

πτ

rKI

xy =

0=zσ

( )yxz σσυσ +=

0== zxyz ττ

plane stress

plane strain

Irwin proposed the stress state around an Infinitely sharp crack in a semi-infinite elastic solid

Stress intensity factor

aKI πσ=

Stress intensity factor in a semi-infinite body is given:

Stress intensity factor for finite body is given:

afKI πσ=f depends on the specimen geometry and is >1 for small crack

Fracture occurs when KI reaches a critical value, KIc, which is a material property.

( ) 2/1afKIc

fπ

=σ 1=f cIc EGK =

Comparison between K and KIC

KI (the stress intensity factor): provides a complete description of the state of stress, strain and displacement over some region of the body, is dependent of the crack length and the geometry of the body.

K (the stress concentration factor): determines the magnitude of the stress at a single point, is independent of the crack length.

⎟⎠

⎞⎜⎝

⎛⋅⋅=wafaKI πσ

a

Kσσmax=

Griffith theory (energy criterion)

γ⋅⋅=Δ 4surf taU

( )Etata

EU

222

2

elast 2

2σπ

πσ

=⎟⎟⎠

⎞⎜⎜⎝

⎛=Δ

EtaatU

22

total

4σπ

γ −=Δ

0 2422

total =σπ

−γ=Δ

Etat

daUd

2a

aE

c πγ

σ2

=

Fracture stress

thickness

Griffith theory-conti.

Crack in (a) thin (t1) and (b) thick (t2) plates. Note the plane-stress state in (a) and the plane-strain state in (b).

aE

f πγ

σ2

= aE

f πνγ

σ)1(

22−

=

Plane stress Plane strain

Fracture toughness

Orowan theory: energy including plastic energy

aEG

c

π=σ

pEG +γ= 2cIncluding the plastic work in generating the fracture surface

aEGK cIcc πσ== Fracture toughness

aE

f πγ

σ2

=

aKc

fπ

σ =

Shear fracture: Plane stress

000

=

≠

≠

z

y

x

σ

σ

σ

σpr

Remarks: 1.The toughness of very thin sheets is quite low, that is, the type of low-energy fracture. 2. Plane stress toughness is not a material property. 3. Biaxial stress state

Slant-type fracture

Flat fracture: Plane strain

σpr 0

00

=

≠

≠

z

y

x

ε

ε

ε

( )yxz

y

x

σσνσ

σ

σ

+=

≠

≠

00

Remarks: 1. The triaxial stress state of plane strain reduces the plastic zone size in comparison to the plane stress zone size. 2. The triaxial stress state is pronounced at the boundary between the plastic and elastic zones.

Plastic zone size

π

⎟⎠⎞

⎜⎝⎛

=2

2Ic

pYK

rPlane stress

Plane strain

π

⎟⎠⎞

⎜⎝⎛

=6

2Ic

pYK

r

σpr

εpr

Effect of thickness on Kc

The thickness of the specimen should be much greater than the radius of the plastic zone for plane stress:

prt <

KIc

prt ⋅≤ 2

2

5.2 ⎟⎠

⎞⎜⎝

⎛≥YKt Ic

pr

Fracture toughness testing

The following are the fracture toughness parameters commonly obtained from testing

• K (stress intensity factor) can be considered as a stress-based estimate of fracture toughness. K depends on geometry (the flaw depth, together with a geometric function, which is given in test standards for each test specimen geometry).

• CTOD (crack-tip opening displacement) can be considered as a strain-based estimate of fracture toughness. However, it can be separated into elastic and plastic components. The elastic part of CTOD is derived from the stress intensity factor, K. The plastic component is derived from the crack mouth opening displacement (measured using a clip gauge).

• J (J-integral) is an energy-based estimate of fracture toughness. It can be separated into elastic and plastic components. As with CTOD, the elastic component is based on K, while the plastic component is derived from the plastic area under the force-displacement curve.

Determination of fracture toughness

Measurement of fracture toughness

Fracture toughness: Charpy test

An increased loading rate has an similar effect to decreased temperature

Fracture toughness vs. strength

tough

brittle