chpp fracture

8/6/2019 CHPP Fracture

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Fracture Mechanics ofPolymers

Rowan W. Truss

The University of Queensland


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Polymer Fracture Mechanics

fracture: creation of new surfaceswithin a solid

compare yield & deformation:maintains continuity

Fracture mechanics usually dealswith brittle fracture: little plasticdeformation before fracture


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Energy Approach:Basic concept

n Creation of a new surface requiresenergy (R): surface energy + local

deformation/rearrangements etc.n Energy supplied by: stored elastic

energy + work done by external

forcesn Other energy loss terms: kinetic

energy, bulk plastic deformation, etc


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Energy balance

P = load

= displacementU = internal energy

J or G = strain energy release rate

Ek = kinetic energy

Ep = plastic deformation energy

P d = dU + J dA + dEp +dEk

At fracture J = R


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Linear Elastic FractureMechanics

n Elastic energy associated withplastic deformation small

n Linear obeys Hookes Lawn Stress Strain

n Quasi-static kinetic energy term

small


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Tearing of Rubbers

Energy balance concepts of Griffith(brittle glasses) extended to tearing of

rubbers by Rivlin & Thomas -1952at crack growth

= - dE/dA

= energy required to produce unit areaof crack, A

dE/dA = energy release rate per unitarea of crack


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Tearing of Rubbers

Ideally rubbers are non-linearelastic

little energy dissipation remotefrom the crack tip

Note: analysis is not dependent onlinear elastic behaviour


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W is strain energy density atstrain , VC volume of region C

example: pure shear sample

Region A:

contains crack - unstressed

EA = 0

(strain energy density)Region B: around crack tip -

complex stress field,EB = ?

Region C: in pure shearEC = VC W

Region D: near surface,stress state complex

ED

= ?

A B C D


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pure shear sample

when crack extends An Size of Regions B and D remain the same,

n

Region A expands at expense of Region Ci.e. material with strain energy density, W,

is converted material with zero strain

E = -WVC = -W t l0 aand

-dE/dA = = 1/2 W l0


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Development of LinearElastic Fracture Mechanics

n all materials are imperfect

i.e. they contain flaws or small

cracksn these cracks can grow to cause

brittle fracture

n cracks propagate only whenspecific energy conditions met


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LEFM energy balance

n linear elastic solid,containing crack of

length, an loaded to P, with a

load point deflection

n

work done by load is1/2P

n stored as elasticstrain energy, U


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LEFM energy balance

n Crack grows by da, requiresenergy Rn

New surface energyn Localised plastic deformation

n

Energy comes fromn Work done by external loads

n Release of strain energy


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LEFM energy balance

define the strain energy release rate,G

G = Pd/da dU/da

which reduces to

G = P2dC/da

where C = compliance = /P

1c (Irwin- Kies)


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Compliance methods

Obtain G1c from the fracture load and thechange in compliance with crack length

Rate of change of compliance with cracklength

n Measured experimentally

n Calculated from elasticity theory

n Finite elements calculations


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alternative approach:Stress analysis

stresses at point near tip of a crack

x= {K /r} fx ()

y= {K /r} fy ()

z= {K /r} fz ()


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Stress intensity factor, K1

n Stress field around crack tip describedby term, K (Stress intensity factor)

n 1 refers to mode 1 opening

n Crack grows when stress field reachessome critical dimension,

ie at critical K, K1c (fracture toughness)

K1c = Y (a)1/ 2


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K1c

and G1c

n K1c and G1c are related through themodulus, E

K1c2 = E* G

where E = E , plane stress

E = E/(1-2) , plane strain


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Plastic Zone

n Crack causes stress concentration

n High stresses yielding & plasticdeformation at crack tip

n near centre of thick section - high constraint-zone is small

n at surface no constraint - large zone

n

Measured K depends on size of plastic zonen Minimum value of K obtained when

specimen so thick that effect of large zoneat the edges negligible

(so called plane strain conditions)


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Zone shape

n Assuming vonMises criterion foryield, plastic zoneis rounded lobe atcrack tip

n most polymers

form extendedzone coplanar withthe crack

n

CRAZE


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Craze Microstructure

n voids and polymerfibrils across the

surfaces (40-60%void)

n can still support load- not true crack

n final fracture occursby tearing mid-rib ofcraze


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Fracture surface - SEM

n fracture surfaceshows remnantsof high local

plasticityn local plasticity

absorbs energy

n toughening

mechanism


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plane strain

Plastic zone size, Rp

= 1/2(K1c

/y

)2 plane stress

= 1/6 (K1c/y)2 plane strain

= /8 (K1c/y)2 line zone

To ensure plane strain need Rp to be severaltimes smaller than specimen dimensions

a, W-a, B > 2.5(K1c/y)2


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Strategies to obtain planestrain K

1c

n Increase the sample dimensions (notalways possible)

n decrease the temperature (y increaseswith decreasing T faster than K1c)

n increase the pressure (as for decreasing T)

n

apply brittle surface layer or estimate theenergy from plane stress region


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J testing

n Samples loaded to different displacements togive different amounts of crack growth, a

n specimens unloaded, broken open, measure a

nJ computed from area under load-displacement curve

J = 2 (U-Ui)/B(W-a)

where U - energy at given displacement,

Ui indentation energy,

B, W - specimen thickness and width resp.


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J testing

n Plot J v. an construct blunting line

J = COD = 2 y ay is yield stress

n intersection of blunting line and J- a

line taken as Jc


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Essential work of fracture

Deep notched samples

assumes failure energy partitioned

into two:n essential work of fracture, we , thin

process zone co-planar with notch,scales as ligament area

n plastic energy in yielded zone,scales as ligament area squared


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Essential work of fracture

i.e.

U = we B(W-2a) + wp (W-2a)2

where is a geometricconstant

plot of U/ B(W-2a) v. (W-2a) shouldgive straight line with intercept of weand slope of wp


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chpp fracture

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