fracture 1

MCEN90029 Advanced Mechanics of Solids Lecture L16 - 1

Lecture L16 Fracture

MCEN90029

Advanced Mechanics of Solids

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Summary •  Over the next 7 lectures, we will investigate concepts of

linear elastic fracture mechanics •  Fracture mechanics involves the study of stress and strain,

theories of elasticity and plasticity, and relates to the microscopic crystallographic defects found in real materials. Fracture mechanics primarily seeks to predict microscopic mechanical failure


Fracture Mechanics (an introduction)

•  Fracture mechanics is the study of structural failure by crack formation –  Analytical methods to calculate crack driving force –  Experimental methods to characterise material crack

resistance •  Study of fracture mechanics leads to improved component

design


History •  Fracture is a problem that has been around since

the first man-made structures •  In 1978, the US economic cost of fracture was

$119 billion

TYPE I: Negligence during design, construction and operation

TYPE II: Application of a new material, and an unexpected (unpredictable) result

Why do structures fail?


Type I failure •  The MSV Kurdistan oil tanker, which sustained brittle fracture when

sailing in the North Atlantic (1979). •  Combination of warm oil in taker and cold water on outer hull

produced thermal stresses. •  Fracture and crack propagation initiated from poorly welded keel


Type II failure •  Pinch clamping of polyethylene gas pipes have been known to

weaken the pipes, cause rupture, and gas leaking •  A small flaw on inner surface propagated through wall •  Stresses in pressurised pipe were sufficient to cause time-dependent

crack growth (over 6 years)

Fracture surface of failed pipe Pinch clamp


Type I and II failure •  The Challenger Space Shuttle disaster, caused by failure of an O-ring

seal in one of the main boosters, did not respond to the extreme temperature drops

•  Relatively new technology, where service experience was limited (Type II)

•  Engineers suspected issue, delaying launch (Type I)


A historical perspective •  Designing to avoid fracture is not new.

–  Pyramids –  The Roman’s bridge design

Schematic Roman bridge design. The arch shape causes loads to be transmitted as compressive stresses


Early fracture research •  Leonardo da Vinci’s experiments (1500 AD): The strength of iron

wires varied inversely with wire length (number of flaws proportional to wire length?)

•  Alan Arnold Griffith (1920’s) performed stress and fatigue analysis on an elliptic hole –  Flaws were unstable when strain energy change overcame material

surface energy - theories valid for brittle materials (glass) •  Westergaard (1938) developed the “stress intensity factor” •  Irwin (1956) modified Griffiths approach to include plastic flow •  Wells (1956) showed that failure occurs when crack sizes reach

critical values •  Rice (1968) developed the J-integral for energy release rate

Fracture mechanics is now an established engineering discipline!


The fracture mechanics approach to design •  In traditional approaches (a), a material was assumed to be adequate

if its strength was greater than the expected applied stress •  The fracture mechanics approach (b) has 3 variables (flaw size and

fracture toughness replace strength). Fracture mechanics quantifies all combinations of these variables

(b)

(a) But, there are two additional approaches… 1. The energy criterion 2. The stress-intensity approach


The energy criterion •  “Fracture (crack extension) occurs when the energy available for

crack growth is sufficient to overcome the resistance of the material” •  Energy release rate G is defined as the rate of change in potential

energy with crack area •  At fracture, G = Gc, the critical energy release rate (similar to fracture

toughness)

For a crack of length 2a in an infinite plate subject to a remote tensile stress:

€

G =πσ 2aE

E: Young’s modulus σ: Remotely applied stress a: Half crack length At fracture:

€

Gc =πσ f

2acE

(1)


The stress-intensity approach Below, an element near the tip of a crack is shown in an elastic material

KI = stress intensity factor, assumes crack tip conditions are linear elastic

KIc = critical stress intensity at which failure occurs. Fracture when KI = KIc

€

KI =σ πa (2)

Using (1) and (2):

€

G =KI2

E


Time-dependent crack growth and damage control

•  Fracture mechanics important in life prediction •  Rate of cracking related to stress intensity factor and critical crack size

In metals:

€

dadN

= C ΔK( )mda/dN = crack growth per cycle ΔK = stress-intensity range C,m = material constants

•  Initial crack size inferred from examination

•  Critical crack size from applied stress and fracture toughness

•  Allowable size (divide critical size by factor of safety) Structural flaw growing with time


Effect of material properties on fracture •  For low toughness, brittle fracture governs failure mechanism •  For high toughness, LEFM not valid (failure governed by plastic flow) •  For intermediate toughness, transition between LEFM and ductile

overload

Branches of fracture mechanics

(LEFM) Linear time-independent materials

Non-linear time-independent materials

Time-dependent materials


Fracture at the atomic level •  Fracture occurs when

sufficient work/energy breaks atomic bonds

•  Equilibrium (steady state spacing) occurs when PE minimum

Bond energy:

€

Eb = P dxx0

∞

∫

x = atomic spacing x0 = equilibrium spacing P = applied force Pc = cohesive force amplitude λ = half sine wave length

€

P = Pc sinπxλ

$

% &

'

( )


Fracture at the atomic level •  For small displacements,

force/displacement is linear

€

P = Pcπxλ

$

% &

'

( )

Thus, bond stiffness (gradient) is

€

k = Pcπλ

$

% & '

( )

Multiply by equilibrium distance, x0, and divide by cross sectional area, A

€

k⋅ x0 ⋅1A

= Pcπλ

%

& ' (

) * ⋅ x0 ⋅

1A

E =σ cπλ

%

& ' (

) * x0

σ c =Eλπx0

≈Eπ

Assuming x0 = λ

(1)


Fracture at the atomic level •  Surface energy:

€

γ s =12

Pc0

λ

∫ sin πxλ

&

' (

)

* + dx

(area under force/displacement curve)

€

γ s =12

σ c0

λ

∫ sin πxλ

'

( )

*

+ , dx

γ s =σ cλπ (2)

Equating (1) and (2)

Surface energy (per unit area)

€

γ s =σ cλπ

=σ c ⋅1π⋅σ cπx0E

'

( ) *

+ ,

∴σ c =Eγ sx0

(3)


Fracture at the atomic level •  What can we discern from this proof?

The theoretical cohesive strength of a material is approximately E/π BUT, experimental fracture strengths are usually 2 or 3 times below these these values (e.g Leonardo da Vinci). Why? - flaws in the materials

•  For fracture, fracture stress at atomic level must exceed cohesive strength σc.

•  Flaws lower global strength by magnifying local stresses

Thus, stress intensity must be defined!


Lecture summary

•  Today we introduced the concept of fracture •  We investigated the atomic basis of fracture •  In the next lecture we will discuss the stress

concentration effect of flaws

fracture 1

Documents

brittle fracture

history fracture

us economic cost of

limited type

launch type

operation type

study of structural

crack propagation