lecture 5.3 fracture

8/2/2019 Lecture 5.3 Fracture

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Fracture and Toughness


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Stress-strain Behavior of Polymers

Brittle

Plastic

Highly elastic

Do not like ceramics and metals, polymer materials exhibit Various stress-strain behaviors, ranging from very brittle to highly

deformableModulus of elasticity ~7MPa to 4GPa.Plastic elongation can be >100%

very sensitive characteristics to strain rate, temperature, chemical

environment


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The area under the stress-strain curve is a measure of the

toughness and has units of energy per unit volume.

The tensile toughness, therefore, is an indication of the

energy that a material can absorb before breaking.


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Brittle Polymer

-Fracture in a brittle manner, eg Polystyrene or other glassy thermoplastics- In PS, the fracture and crack propagation occurs in the form ofcraze

CRAZE , under electron microscope reveals: load-bearing fibrils about

20 nm in diameter span the gap between the surface of the polymer.

Craze

Fibril

Molecular entanglements are essential, without them, there would be little to

stabilize the loaded fibrils If the polymer chains are too short to form effective entanglements, the material

is fragile


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Plastics - ductile or brittle

Whether a polymer is ductile or brittle in any given circumstancedepend its resistance

to yield, and to crazing and subsequent crack propagation

In thermoset, such as epoxy resin, the cross-linked resins show little yielding

under any conditions because the molecular network is unable to deform

sufficiently


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Increase Fracture resistance:

rubber modificationadd between 5 to 20% to form a small particles, between 0.1 to 5m

- rubber particles have low moduli, and therefore act as stress concentrators.

Problem with crazing-certain level of craze formation renders the component unserviceable- visibility is reduce in PMMA helicopter cabins- porosity can be a problem in ABS pipe

Crazing is precursor to fracture in a large number of polymers


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Fracture Mechanics

Failure is inhibited by energy-absorbing processes around the crack tip.

- unable to relieve the stress concentration or other mechanisms of

crack blunting.

A crack will spread only if the total energy of the system is lowered thereby

NEED to examine the change in the total internal energy of the system as the

crack begins to spread


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Fracture Mechanics

Brittle Solids fracture because the applied stress is amplified by

minute cracks. This crack occur naturally as a result of fabrication, solidification,

fatigue damage etc. This cracks are termed Griffith cracks.

Consider a thin sheet of thickness B and infinite width containinga sharp crack of length 2a transverse to a tensile stress applied byFixed grips.

The total energy of the system:

1. The elastic strain

2. The work of crack formation

2aB


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Fracture Mechanics

Elastic strain energyThe elastic strain energy per unit volume is

E is Youngs modulus.

o volume V of specimen, the total strain energy is

- the net effect of the insertion of a crack is a lowering of the

total strain energy of the sheet by

E

Ba22

+E2

2

+ E

V

2

2


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Fracture Mechanics

Work of Crack Formation

The work done per unit area of crack surface is GcAssume that no heat is dissipated

The total work of crack formation:

crack area x Gc= 2aBGc

The change in energy by the crack is

U = - (1)

Note: the crack spreads

AND the negative sign implies that the elastic strain energy

decrease.

caBGE

Ba

2

22

+


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Fracture Mechanics

The dependence ofU on a is shown in graph.

U

Ua

- (2B/E) a2

(2BGc)a

am

At small a the term linear in a dominates: it is positive and representsthe increase in the work of crack formation as the crack spreads

At large a the term in a2 dominates: it is negative and represents the

diminution in total strain energy as the crack spreads.

ais a central crack


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Fracture Mechanics

The crack is on the point of growth under stress , thework of crack propagation balances the decrease in

elastic strain energy;

(2)

Therefore:

(3)

This point is represent by the maximum in the U vs a plot which isstable at

(4)

( )C

aBGda

d

E

Ba

da

d2

22

=

2

c

m

EGa =

ma

cEGa =

2


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Fracture Mechanics

Foraabove amdU/da is negative, as a increases U

decreases- the total energy of the system decreases andthe crack will spread catastrophically.

a below amdU/da is positive, crack will not spread

because Uincreases with crack size.

If the stress is increased from zero, the fracture stressF is

F = (EGc/a)1/2 (plane-stress) (5)

Is known as Griffith equation.


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Fracture Mechanics

Gcis known as the fracture energy.

A more useful parameter is the plane-stress intensityfactorKc, which is defined as (for a wide sheet)

Kc =F (a)1/2 (plane-stress) (6)

From eq. 5 and 6 :

Kc = (EGc)1/2 (7)

The use of Kc to determine whether or not a given thin sheet willfracture under a stress implies that the size of the largest crack inthe sheet is known to designer.

The the stress intensity factorK=F (a)1/2

The crack will not spread for K < Kc


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Fracture Mechanics

Kc = (EGc)1/2(plane-stress)

Kc =F (a)1/2 (plane-stress)

The the stress intensity factorK=F (a)1/2

The crack will not spread for K < Kc


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Example 1

A sharp, central crack of length 60 mm in a wide, thin sheet of a glassy

plastic commences to propagate at F = 3.26 MPa.(a) CalculateKc

;

(b) Calculate Gc given that E=3GPa;

(c) Will a crack of length 5 mm in a similar sheet fracture under = 20 MPa ?

Answer

(a) Kc =F (a)1/2 = 3.26 ( x x 60 x 10-3)1/2 = 1.00 MPa m1/2

(b) = (106)2/3 X 109 = 333J/m2

(c) K= (a)1/2

E

KG

c

c

2

=


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Thank you

lecture 5.3 fracture

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