Concept of Fracture Mechanics

Fracture Avoidance with Proper Use of Material

Pyramid of Egypt Schematic Roman Bridge Design

h• The primary construction material prior to 19th were timber, brick and mortar• Arch shape producing compressive stress → stone have high compressive

strength

Riley; page 5Anderson; fig. 1-4, page 9Gordon; fig. 14, page 188

Fracture Avoidance with Proper Use of Material (cont’)(cont )

• Roof spans and windows were arched to maintain compressive loading

Gordon; plate 1 (after page 224)Anderson; fig. 1-5

Fracture Avoidance with Proper Use of Material (cont’)(cont )

• Mass production of iron and steel (relatively ductile construction materials) →feasible to build structures carrying tensile stressesy g

h lf d’ b d ( ) h b dThe Telford’s Menai suspension bridge (1819) The seven suspension bridge

(wrought iron suspension chains) (steel cable)Gordon; plate 11 & plate 12

Stress Concentration, Fracture and Griffith TheoryGriffith Theory

• Stress distribution around a hole in aninfine plate was derived by G Kirsch ininfine plate was derived by G. Kirsch in1898 using the theory of elasticity

• The maximum stress is three times theuniform stress

• Kt = 3

Damage Tolerance Assessment Handbook; fig. 2-1, page 2-2

Stress Concentration, Fracture and Griffith Theory (cont’)and Griffith Theory (cont )

• C. E. Inglis (1913) investigated in a plate with anelliptical holeelliptical hole

• He derived orbaKt /21+=ρ/aK 21+=

• Modeling a crack with a ellipse means ρ → 0 →Kt → ∞ → infinite stress

ρ/aKt 21+=

• Kt could not be used for crack problems

Damage Tolerance Assessment Handbook; fig. 2-2


• A. A. Griffith (1920) used an energy balance analysis to explain the largereduction on the strength of glassreduction on the strength of glass

• Griffith proposed that the large reduction is due to the presence ofmicrocracks

• Griffith derived a relation between crack size and breaking strength byconsidering the energy balance associated with a small extension of a crack

Stress Concentration, Fracture and Griffith Theory

Damage Tolerance Assessment Handbook; fig. 2-3 a & b


))(( εσ LAPxWork 11== ))(( εσ LAPxWork22

==

1 ))(( ALWork σε21

=

))(( VWork σε21

=2

densityenergystrain=)(σε21 ygy)(2

Stress Concentration, Fracture and Griffith Theory (cont’)

Damage Tolerance Assessment handbook; fig. 2-4 a & b


• Crack length increase → plate becomes less stiff (more flexible) → slope of P vs xdecreases → applied load drop

• Change in energy stored is the difference in the shaded area

• Release of elastic energy is used to overcome the resistance to crack growth

• Rate of strain energy release = rate of energy absorption to overcome resistance tocrack growth

Damage Tolerance Assessment Handbook; fig. 2-4b


• Energy balance :

Energy stored in the body before crack extension = Σ (energy remaining in thebody after crack extension + work done on the body during crack extension +

• Energy balance :

energy dissipated in irreversible processes)



• Analyze a simplified geometry with a hole D = 2a

• σy = σ everywhere outside the hole



• Strain energy density =

E2σ2

• Total energy = volxE2

2σ

After crack extension of ∆a (assume σ is constant)

[ ]taWLtE

U 22

1 2πσ

−=

2

Elastic energy released

( )[ ]taaWLtE2

U 22

2 ∆+π−σ

=

2

Per unit of new crack areaE

ataUU2

21∆σπ

≅−

Damage Tolerance Assessment Handbook; fig. 2-5E2a

at2UUG

221 σπ≅

∆−

=


E l d i d t b k t i b d f• Energy released is used to break atomic bonds → surface energy

• Surface energy (γe) is a material property

E b l > k th if• Energy balance ˙> crack growth if

γ≥ e2G

πγ

=σ eE4a

• Griffith analysis based on Inglis solution yield

E2 a2σπandπ

γ=σ eE2a E

aG σπ=


• In 1957 Irwin reexamined the problem of stress distribution around a crackLinear Elastic Fracture Mechanics (LEFM)

p• He analyzed an infinite plate with a crack• Using the theory of elasticity the stresses are dominated by

]sinsin[cos23

21

22θθθ

πσ −=

rK

x

3θθθK]sinsin[cos

23

21

22θθθ

πσ +=

rK

y

23

22θθθτ coscossin

Kxy=

assumption r << a

2222πrxy

LEFM valid if plasticity remains small compared to the over all dimensions of crack and cracked bodies

)(θππσσ ijij fra

2=


• The term is given the symbol K (stress intensity factor)• The term is given the symbol K (stress intensity factor)

for an infinite plateaπσ

aK I πσ=

• The relation of K to G is

for plane stress conditionK 2 for plane stress condition

• The use of G and KI leads to fracture criterion i e G and Ki i e fracture occur

EKG I=

• The use of G and KI leads to fracture criterion i.e. Gc and Kic i.e. fracture occurif

G = Gc or KI = KIc


Stress Intensity FactorStress Intensity Factor

f faK πσ= for infinite plate

for other geometry

aK πσ=

aK πβσ= g y

β can be obtained from : 1. handbook solution

2. approximate method2. approximate method

3. numerical method


Bannantine, fig. 3-4, page 92


Bannantine; fig. 3-4, page 93 & 94


Loading ModesLoading Modes


Loading Modes (cont’)g ( )Loading stresses terms for mode II

]coscos[sin 32 θθθσ +−=K II

x ][2222πrx

3θθθσ coscossinK IIy = 2222πry

]sinsin[cos 31 θθθτ −=K II

• Stresses terms for mode III

]sinsin[cos22

122π

τrxy

θK22θ

πτ sin

rK III

xz −=

θK22θ

πτ cos

rK III

yz −=

Extension of LEFM to Metals

• Griffith energy theory and Irwin’s stress intensity factor could explain thefracture phenomena for brittle solid

• For metals, beside surface energy absorption, the plastic energy absorption(γp) has to be added

γγ )(E +2

• For typical metal γ ≅ 1000 γ thus γ can be neglected

πγγ

σ)( peE

a+

=2

• For typical metal, γp ≅ 1000 γe, thus γe can be neglected

• It was not easy to translate energy concept into engineering practice

Extension of LEFM to Metals (cont’)

• K concept was seen as the basis of a practical approach

• However K is an elastic solution while at the crack tip plastic zone developed• However, K is an elastic solution while at the crack tip plastic zone developed

• If it is assumed that the plastic zone at the crack tip is much smaller than thecrack dimension → K is still valid

Extension of LEFM to Metals (cont’)Plastic zone size Monotonic LoadingPlastic zone size Monotonic Loading

for θ = 0

]sinsin[cos22

122

θθθπ

σ −=r

Ky

for θ 0

If σ is equal to yield strengthr

Ky π

σ2

=

If σy is equal to yield strength

orp

ysr

K*π

σ2

=p

Kr 2

2

2σ

π =*

plane stress

ysσ2

21

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ys

pK

rσπ

*

plane stress

Corrected due to stress redistribution2

21 ⎟

⎠

⎞⎜⎜⎝

⎛=p

Kr

σπ p a e st ess

plane strain

2 ⎠⎜⎝ ysσπ 2

31

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ysp

Kr

σπ

Plane Strain Fracture Toughness Testing

Plane Strain Fracture Toughness Testing

f• Standard test method include ASTM E399: “Standard Test Methods forPlane Strain Fracture Toughness of Metallic Materials”.

• Stringent requirement for plane strain condition and linear behaviour of thespecimen.

• Specimen type permitted: CT, SENB, arc-shaped and disk shape.

Plane Strain Fracture Toughness Testing (cont’)

Fracture Mechanics Testing

Specimen Configurations


Clevis for Compact Tension Specimen


• Use an extensometer (e.g. clip gage) to detect the beginning of crackextension from the fatigue crack.


• Calculation of KQ for compact tension specimen

)(/ Waf

BWP

K QQ 21=

whereWBW

( ) ( ) ( ) ( ) ( )( 657214321364488602 432( ) ( ) ( ) ( ) ( )(( )231

657214321364488602 432

Wa

Wa

Wa

Wa

Wa

Wa

Waf

−

−+−++=

.....)(

• This KQ has to be checked with previous requirements




ASTM Standards; fig. 1, page 410


Fatigue Pre-cracking• Perform to obtain natural crack• Fatigue load must be chosen :

0 such that the time is not very longplastic zone at the crack tip is small0 plastic zone at the crack tip is small


Instrumentation for Displacement and Crack Length Measurements


• Crack front curvature


• Measure a1, a2 and a3 → 3321 aaaa ++

=Measure a1, a2 and a3 →

• Any two of a1, a2 and a3 must not differ more than 10% from

• For straight notch → asurface differ not more than 15% from and (asurface)left

3

aa

g surface ( surface)leftdoes not differ more than 10% from (asurface)right

a


• Load displacement curves to determine PQ

Additional Criteria

» Pmax/PQ < 1.1Pmax/PQ < 1.1

KQ <⎞

⎜⎛

2

52» ays

Q

⎞⎛

<⎟⎠

⎜⎜⎝

2

52σ

.

»B

K

ys

Q <⎟⎟⎠

⎞⎜⎜⎝

⎛2

52σ

.ys ⎠⎝


Damage Tolerance Assessment handbook; table 2-1, page 2-31


Damage Tolerance Assessment Handbook; table 2-1, page 2-32

Plane Strain Fracture Toughness Testing (cont’)Thi k Eff tThickness Effect

• Plane strain condition occur for thick components

F t ti t i l ti l t i diti d t h i fl• For static material properties plane strain condition does not have influence

• For fracture toughness thickness have a strong influence


Thickness effect on fracture strength


Thickness Effect (cont’)

• Specimen thicker than 1/2 inch →plane strain

• For thinner stock KQ increasesreaching a peak at thickness aboutreaching a peak at thickness about1/8 inch

• The peak KQ can exceed five timesKKic

Thickness effect on fracture strength

• After reaching the peak KQ declines at thickness lower than 1/8 inch

Thickness effect can be e plained ith ene g balance• Thickness effect can be explained with energy balance


Plane Strain Fracture Toughness Testing (cont’)Thickness Effect (cont’)

• σZ = 0 at free surface → plane stresson the surface → large plastic zoneon the surface → large plastic zone

• In the inside elastic materialrestrains deformation in Z direction

• For thick specimen interiordeformation is almost totallyrestraint (σZ ≈ 0) → plane strainrestraint (σZ ≈ 0) → plane straincondition

Three-dimensional plastic zones shape

• Going inward from the surface, plastic zone undergoes transition from largersize to smaller size

Damage Tolerance Assessment Handbook; fig. 2-17a


Thickness Effect (cont’)

• For decreasing thickness, ratio ofplastic volume to total thicknessplastic volume to total thicknessincrease

• Consequently energy absorption rateq y gy palso increases for thinner plates

• While elastic strain energy isindependent of thicknessindependent of thickness

• Thus for thinner plates more appliedstress is needed to extend the crack

Plastic volume versus thickness


Plane Strain Fracture Toughness Testing (cont’)Thickness Effect (cont’)

• Plane stress condition results in fracturesurface having 45o angle to z axis →g gshear lips

• For valid Kic test (plane strain condition)→ little or no evidence of shear lips→ little or no evidence of shear lips

Damage Tolerance Assessment Handbook; fig 2-18

Typical Fracture Surface

Plane Strain Fracture Toughness TestingTemperature Effect

• Fracture toughness depends on temperature

• However Al alloys are relatively insensitive over the range of aircraft servicetemperature condition

• Many alloy steels exhibit a sharp transition in the service temperature range


Fracture toughness versus temperature

KIc of Aircraft MaterialsTypical Yield Strength and Plane Strain Fracture ToughnessTypical Yield Strength and Plane Strain Fracture ToughnessValues for Several Al Alloys

ASM Vol. 19; table 5, page 776

KIc of Some Materials (cont’)Al Alloys 2124 and 7475 vs 2024 and 7075Al Alloys 2124 and 7475 vs. 2024 and 7075

Application of Fracture Mechanics; fig. 6-9, page 180

KIc of Some Materials (cont’)

Effect of Purity on KIc


KIc of Aircraft Materials (cont’)

Typical Yield Strength and Fracture Toughness of High-Strength Titanium Alloy


Failure in Large Scale Yielding

• Strength assessment for structures do not meet small scale yielding condition:

1. R-curve method

2. Net section failure

3. Crack tip opening displacement

4. J-integral

5. Energy density → mixed mode loading

6. Plastic collapse → for 3D cracks

The Net Section on Failure Criterion

St t ti i d til t i l i ldi hi h th d t• Stress concentration in ductile materials causes yielding which smoothed outthe stress as applied load increased

• Failure is assumed to occur when stress at the net section was distributeduniformly reaching σu

Net section failure criterion

• For a plate width w containing a center crack of length 2a, the critical stress is

Damage Tolerance Assessment Handbook; fig. 2-34fc wa2wσ

−=σ

Kc of Aircraft Materials

Plane Stress Fracture Toughness (Kc) for Several Al Alloys

ASM Vol. 19; fig. 10, page 779

Crack Opening Displacement (COD)

• Applied load will cause a crack to open, the crack opening displacement canbe used as a parameter

• At a critical value of COD fracture occur

• Developed for steels

J-IntegralJ Integral

• J-integral is an expression of plastic work (J) done when a body is loaded

• J-integral can be calculated from elastic plastic calculation

• At a critical value of J fracture occur

END

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Concept of Fracture Mechanics

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