CHPP Fracture
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Transcript of 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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