Fracture Behavior

In January 1943 the one-day old Liberty Ship, SS Schenectady, had just completed successful sea trials and returned to harbor in calm cool weather when . . . "Without warning and with a report which was heard for at least a mile, the deck and sides of the vessel fractured just aft of the bridge superstructure. The fracture extended almost instantaneously to the turn of the bilge port and starboard. The deck side shell, longitudinal bulkhead and bottom girders fractured. Only the bottom plating held. The vessel jack- knifed and the center portion rose so that no water entered. The bow and stern settled into the silt of the river bottom."

Approximately 2700 ships were built from 1942 to the end of WWII, using prefabricated all welded construction. At the start of the program 30% of Liberty Ships suffered catastrophic fracture.

What factors attributed to the failures?

• _____________________________________________________________________

• _____________________________________________________________________

• ______________________________________________________________________.

Section

6

52

Fracture Mechanics

How to design something to withstand:

• ____________________________

• ____________________________

• ____________________________

Griffith’s energy relation

English aeronautical engineer, A.A.Griffith, recognized that theoretical predictions of the stresses at the tip of a crack approach infinity (thus any material with a crack would fail). He developed a ______________________________ approach to predict failure where he assumed that the growth of a crack requires creation of surface energy (which is the same as the surface tension of the material, γ, times the total crack surface area created).

A necessary condition for crack growth is that the negative change in potential energy is greater than or equal to the change in the surface energy generated with the creation of a new crack:

dAdUdUdUd γ2431 ≥−−=Π−

53

For a center notched panel, with crack length of 2a, Griffith Equation is

Ea2

2 πσγ = 2/1)2(

⎟⎠⎞

⎜⎝⎛=

aE

πγσ

This only works well for __________________________materials.

For other materials (i.e. plastics, metals, composites) other energy absorbing components are added to the equation, such as plastic work Wpl and the strain energy release rate, G, is defined to include requirements for all kinds of energy, including surface energy.

2/12/1)2(⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=

aGE

aEWpl

ππγ

σ

So for this case (center notched panel), E

aG2πσ

=

The strain energy release rate, G, can be understood as the_________________________________ __________________________________________________________ . The units of G are __________________________ .

For failure, Gc, the critical strain energy release rate (also fracture energy) is defined as:

Ea

G fc

2πσ= where fσ is the applied fracture stress.

Gc can be interpreted as ____________________________.

The fracture energy, Gc, is a _____________________________, while G is a _________________________ which is a function of applied loads and geometry.

Modes of Loading

There are three ways of applying force to enable a crack to propagate:

__________ __________ ____________

54

Stress Intensity Factor

Another approach to fracture mechanics is the Stress Intensity Factor (KI, KII, and KIII) which, for the center notched panel is:

aKI πσ=

Combining this with the equation for G above, we see that K and G are related as:

EGK = cc EGK = this is actually only the case for plane stress (thin samples)

For ______________________ (thick samples compared to the plastic zone size) K and G are related as:

21 ν−=

EGK where E and ν are the Young’s modulus and the Poisson’s ratio respectively.

The stress intensity factor, K, is a __________________________ and is a measure of the stress singularity at the crack tip. It depends on ___________________________. The critical stress intensity factor, KC, also called the _____________________, is a ________________________. When ICI KK ≥ then ____________________________.

In general, 2/1aYK CC σ= where Y is a geometrical factor that corrects for the sample shape and crack length.

Note, crack tip creates a 1/sqrt(r) singularity. The Yield stress prevents the stress from approaching infinity. How is the plastic zone size, ry, determined?

55

Mode I

Mode II

Mode III

56

Stress Intensity Factor for various geometries

Infinite Plate with a Center Through Crack under Tension

Infinite Plate with a Hole and Symmetric Double Through Cracks under Tension

Semi-infinite Plate with an Edge Through Crack under Tension

Infinite Stripe with a Center Through Crack under Tension

Infinite Stripe with an Edge Through Cracks under Tension

Infinite Stripe with Symmetric Double Through Cracks under Tension

57

Stress intensity factor is to ______________________ as stress is to ____________________.

Stress σ=P/A

Stress Intensity Factor (KI)

___________ σy

___________ (KIC)

58

Experimental Fracture Mechanics Example double cantilever beam DCB testing to evaluate interlaminar fracture toughness

Lbao

h

∗=ba

PG23δ

• DCB specimen was first introduced to evaluate adhesives

• Energy release rate is determined by:

59

Data Reduction

U

P

δ

U = 12 Pδ = 1

2δ 2

C

δ = C(a)P

Fixed Grips

BG = −∂Π∂a

|δ = −∂U∂a

|δ = 12

δC

⎛ ⎝

⎞ ⎠

2 dCda

= 12 P( )2 dC

daEq. (1)

a

x

d2uy

dx2 =MEI

=P(a − x)

EI

uy =PEI

ax2

2−

x3

6⎛ ⎝ ⎜ ⎞

⎠

duy

dx= 0uy = 0 at x=0

uy(a) =4Pa3

EBh3I =

112

Bh3since

δ = 2 • uy =8PEB

ah

⎛ ⎝

⎞ ⎠

3

Eq. (2)

M P

60

E =8PδB

ah

⎛ ⎝

⎞ ⎠

3

E =8PδB

ah

⎛ ⎝

⎞ ⎠

3

G =12a2

EB2h3δC

⎛ ⎝

⎞ ⎠

2

=3δ 2 Eh3

16a4G =12a2

EB2h3δC

⎛ ⎝

⎞ ⎠

2

=3δ 2 Eh3

16a4

C =δP

=8

EBah

⎛ ⎝

⎞ ⎠

3

Eq. (2) δ = ...

C =δP

=8

EBah

⎛ ⎝

⎞ ⎠

3

Eq. (2) δ = ...

G =12a2

EB2h3 P( )2

Eq. (1) G(C(a))= ...

G =12a2

EB2h3 P( )2

Eq. (1) G(C(a))= ...

G =3δP2aB

G =3δP2aB

Cube root of specimen compliance plotted versus crack length

G =3δP2ba∗

a* = a + Δ

Correction for both shear deformation and root rotation by adding a length Δ to a, where Δ can be determined experimentally by plotting C1/3

61

Typical load-displacement curve for virgin and healed reference specimens (8H satin weave)

0

10

20

30

40

50

60

0 10 20 30 40 50

δ (mm)

DCPD & catalystinjected

Virgin

Healed

A

C

B:

E

D

• A: Crack propagation commences ahead of precrack

• B: Loading of virgin is completed and catalyzed DCPD is injected into delamination

• C: Crack propagation commences for the healed specimen

• D: The crack has propagated through the entire healed region

• E: Further loading creates a new “virgin” crack ahead of previously healed region.

0

500

1000

1500

50 55 60 65 70 75 80 85 90

a (mm)

Specimen No.

12345678AVG.

62

Mode II delamination testing End-Notched Flexure (ENF) Test

Mixed-Mode Bending (MMB) Test

63

Other Experimental Methods for Fracture Behavior • Tensile Measurements

• Impact Testing

o Izod, Charpy, Drop Tower

• Fatigue Testing

o Governed by Paris law

minmax KKK

KAdnda m

−=Δ

Δ=

where A and m are material dependent parameters.

In all cases, the properties measured are highly dependent on ______________________.

64

There are a number of mechanisms by which polymers and composites fail. One of the common modes (especially in rubber toughened systems) is _________________________. As seen below, fibril strands span the craze (ahead of the crack) absorbing energy.

x( t )

v0

W

H

SpecimenSupport

Flag

InstrumentedTup

Velocitygate

Example Problem:

In the compliance calibration of an edge cracked fracture toughness testpiece of a tough composite material, it was observed that a load of 100 kN produced a displacement between the loading pins of 0.3000 mm when the crack length was 24.5 mm and 0.3025 mm when the crack length was 25.5 mm. The fracture load of an identical testpiece, containing a crack of length 25.0 mm is 158 kN. Calculate the critical strain energy release rate, GIC, and the plane-strain fracture toughness, KIC, of the material. All test pieces were 25 mm thick. The following elastic constants are given for the material: ν=0.3 and E=70 GPa.

Example Problem:

The mode I energy release rate, GI, for a crack (blister) growing from internal pressure, P, is given as

aa

tEPGI

2

32

3)1( 43

22 where E is the effective Young’s modulus, is the effective

Poisson’s ratio, a is the crack length, and t is the laminate thickness. Determine whether crack growth will be stable or unstable for a monotonically increasing pressure loading.

t 

P 

a 

Substrate Laminate