lecture 9 fracture of solids - arizona state university 440_516 mechanical...liberty ship ss...
TRANSCRIPT
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Liberty ship SS Schenectady was a tanker built during WWII. It underwent brittle hull fracture in ice cold water before leaving the shipyard.
Tensile fracture of a brass rod showingcharacteristic cup-cone.
Fracture of Solids
Ductile fracture surface of brass showing so-called “void coalescence”.
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Fracture of SolidsTheoretical tensile strength of an elastic solid
max max
max
max
max
2sin @ / 4;
2for small / ; ; where /
; for / 22 2/
xx
xx eE e x a
xE E a
x a aE
ps s l s sl
pl s sl
l ls lp p
s p
æ ö= = =ç ÷è ø
æ ö= = =ç ÷è ø
æ öæ ö æ ö= = »ç ÷ç ÷ ç ÷è øè ø è ø
=
a
s
s
U(r)
r
F(r)
r
l/4
F(r) = − ddrU (r)
λ
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Fracture of Solids
Work of fracture
2g a
s
s
gg
σ
e = (x / ao )
E
σ max
Utotal = 2γ = σ max sin2π xλ0
λ /2
∫ dx =λσ max
π
λ = 2πγσ max
σ max =Eγao
⎛
⎝⎜⎞
⎠⎟
1/2
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Theoretical strength of a defect-free solid
σ max =Eγao
⎛
⎝⎜⎞
⎠⎟
1/2
Work of fracture for a defect-free solid
Take E = 10 G Pa = 1010 Pa, g = 2 J m-2, ao = 3 x 10-10 m.
σ max =1010 × 23×10−10
⎛⎝⎜
⎞⎠⎟
1/2
≈ 8×109Pa = 8 GPa
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Stress Concentrations
¥s
¥s2 4 2
2 4 2
2 4
2 4
4 2
4 2
1 1 3 4 cos22
1 1 3 cos22
3 21 sin 22
rr
r
a a ar r r
a ar r
a ar r
q
ss q
ss q
ss q
¥
¥
¥
é ùæ ö æ ö= - + + -ê úç ÷ ç ÷
è ø è øë ûé ùæ ö æ ö
= + - +ê úç ÷ ç ÷è ø è øë ûæ ö
= - - +ç ÷è ø
Airy stress function2 4 2
2 cos24 4 2r a a
rj s q
æ ö= - - +ç ÷
è ø
qqs
qqqs
rrs
a
max
@ ; = /2 and 3 /3
2
0r rr
r a
q
q p p
s ss s s=
=
= =
=
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Stress Concentrations
For the circular flaw 3tK =
Def: Stress Concentration factor, Kt
maxt
nom
Kss
=
Elliptical flaw
2a2b
( )max 1 2 /a bs s= +
2 4 2
2 4 2
2 4
2 4
4 2
4 2
1 1 3 4 cos22
1 1 3 cos22
3 21 sin 22
rr
r
a a ar r r
a ar r
a ar r
q
ss q
ss q
ss q
¥
¥
¥
é ùæ ö æ ö= - + + -ê úç ÷ ç ÷
è ø è øë ûé ùæ ö æ ö
= + - +ê úç ÷ ç ÷è ø è øë ûæ ö
= - - +ç ÷è ø
@r = a; θ = π / 2 and 3π / 2σθθ = 3σ nom
Kt =3σ nom
σ nom
= 3
Kt = (1+ 2a / b)The radius of curvature at a is,
2 /b ar =
( )1/ 2max 1 2 /as s ré ù= +ë ûso,
for
a >> b
σ max = 2σ a / ρ( )1/2
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Stress field of an Elliptical flaw
x
y
ab
( )( )( )2
0
sinh 2 cosh 2 cosh 2 2
cosh 2 1
yy
yy o
along ybbs s
s a a as a
= =
+ -=
-ao
ao is the ellipse defined by the flaw.
bo
The Cartesian coordinates, x, y areconnected to the elliptical coordinatesa, b by
( )coshx iy a ia b+ = +
2a
( ) ( )( )
( ) ( )
2
2 22 2
/1 / 11 / 1 /1 / 1 /
yy b ar a
r a b ar a b a
ss
é ù+ ê ú= +
ê úé ù+ - -é ù+ - - ê úë ûë ûë û
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Stress field of an Elliptical flaw
r is the distance along y = 0. The relative values of r/a and b/a compared to1 determine the behavior of and define regions of interest.
In the region r/a < 1 the leading term (dominant) including the bluntnesscontribution of b/a is,
( ) ( )( )
( ) ( )
2
22
/1 1/ 2 // 2 /
yy b a
b a r ab a r a
ss
é ù» +ê ú
+ê ú+ ë ûr < a
As b/aÞ 0 (a sharp “crack”) the stress field decays as (a/r)1/2.
In the region, r/a >1 ( )( )
2
2
1 /12 /
yy b a
r a
ss
+= +
+
and the stress field scales as (a/r)2
σ yy ≈σ (a / 2r)1/2 r < a: near field
r > a: far field
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Sharp Cracks
2a
r
q
Central crack of length 2a in an infinite plate under uniform tension. Theleading terms for r <<a,
( ) ( ) ( ){ } ( )( ) ( ) ( ){ } ( )( ) ( ) ( ) ( ) ( )
1/ 2 1/ 2
1/ 2 1/ 2
1/ 2 1/ 2
/ 2 cos / 2 1 sin / 2 sin 3 / 2
/ 2 cos / 2 1 sin / 2 sin 3 / 2
/ 2 sin / 2 cos / 2 cos 3 / 2
xx
yy
xy
a r O r
a r O r
a r O r
s s q q q
s s q q q
s s q q q
é ù= - +ë û
é ù= + +ë û
é ù= +ë û
Def: Stress Intensity Factor, K
K as p=
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( )( ) ( ){ }
( )( ) ( ){ }
( )( ) ( ) ( )
1/ 2
1/ 2
1/ 2
cos / 2 1 sin / 2 sin 3 / 22
cos / 2 1 sin / 2 sin 3 / 22
sin / 2 cos / 2 cos 3 / 22
xx
yy
xy
Kr
Kr
Kr
s q q qp
s q q qp
s q q qp
é ù= -ë û
é ù= +ë û
é ù= ë û
Sharp Cracks
The stress intensity factor, K, defines the strength of the crack in much the same way as the Burgers vector defines the strength of a dislocation.
These equations are often written in short-hand as:
σ ij =K2πr
fij (θ )
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Crack Loading Modes
Mode I:Opening
Mode II:In-plane shear
Mode III:Out-of plane or longitudinal shear
IKIIK IIIK
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Units of K
K Y as p= Y = numerical factor depending on geometry and loading
MPa m
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Sharp Cracks
Mode I stress field
Cartesian coordinates
( )( ) ( ){ }
( )( ) ( ){ }
( )( ) ( ) ( )
1/ 2
1/ 2
1/ 2
cos / 2 1 sin / 2 sin 3 / 22
cos / 2 1 sin / 2 sin 3 / 22
sin / 2 cos / 2 cos 3 / 22
xx
yy
xy
K
r
K
r
K
r
s q q qp
s q q qp
s q q qp
= -é ùë û
= +é ùë û
= é ùë û
Cylindrical polar coordinates
( ) ( )( )( ) ( )
2
3
2
cos / 2 1 sin / 2
cos / 22
sin / 2 cos / 2
rrI
r
Krqq
q
q qss q
ps q q
ì é ù+ë ûü ïï ï=ý íï ïþ ïî
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Mode II stress field
Sharp Cracks
Cartesian coordinates
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )
sin / 2 2 cos / 2 cos 3 / 2
sin / 2 cos / 2 cos 3 / 22
cos / 2 1 sin / 2 sin 3 / 2
ì- +é ùü ë ûïï ï=ý íï ï
-é ùþ ï ë ûî
xxII
yy
xy
Kr
q q qss q q q
ps q q q
Mode III stress field
Cartesian coordinates
( )( )sin / 2
cos / 22xz III
yz
Kr
qss qp
-ìü ï=ý íïþ î
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Griffith theory of fracture
Mechanical work during crack extension
2a
dada
2(a+da)
iT
1
2 3T
u1u 2u
Apply tractions loading the crack system
1 21 11/ 2elU Tu® =
1T
During crack extension the work done is
( )2 31 2 1LoadU T u u® = -
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Griffith theory of fracture
Mechanical work during crack extension
The elastic energy at location 3 is just
1 31 21/ 2elU Tu® =
The change in elastic energy is just
( )1 3 1 21 2 11/ 2el elU U T u u® ®- = -
Then
2Load elU U= D
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Griffith theory of fracture
2a
dada
2(a+da)
iTConsider a crack of length 2ain a 2D plate of infinite extentunder external boundary tractions.
The total energy, UT of thesystem is composed of 3 terms.
T el surf loadU U U U= + -
Here U load is the work doneby the applied loads Ti on thesystem.
We have shown that
2load elW U U= = D
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Griffith theory of fracture
For a thin plate of unit thickness under load the excess elastic energy in the system owing to the presence of the crack is,
2 2 /elU a Eps=
The surface energy of the crack system is
4surfU ag=
The total energy may now be written as
( )2 2 / 4
T el surf
T
U U W U
U E aap gs
= -
+-
+
=
U
aa*
Excess elastic energy of a cracked solid
Uelasticno crack = 1
2σεV ;σ = Eε
Uelasticno crack
unit thickness= σ 2
2EA
Uelasticcrack
unit thickness= σ 2
2EA+ πσ 2a2
Eexcess elastic energy!"#
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Griffith theory of fracture
We can find the position of the energy for fracture, corresponding to the maximum in UT,
( )
2
1/ 2
1/ 2
2
2 / 4 0
2
21
T
G
G
d U a Eda
Ea
Ea
p s g
gs sp
gsp n
= - + =
é ù= = ê úë û
é ùê ú=
-ê úë û
For the case of plane strain
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Griffith theory of fracture
Def: Crack extension force
( )eldG U Wda
= - -
G is the (negative) change of potential energy per unit crack extension per unitwidth.
( )2
2eld aG W Uda E
ps gº - = =
Energy/area = Force/length
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Griffith theory of fracture
( )
1/ 2
2
21GGEKa gs pn
é ùê ú= =
-ê úë û
What is a “typical value” of K at which fracture of a brittle solid is predicted
Plane Strain
We can rearrange this to read
( )
1/ 2
2
21GE
ags
p n
é ùê ú=
-ê úë û
E ~ 1011 Pa, g ~ 1Jm-2
Many materials exhibit critical K values 10 -100 times larger then this. Why?
KG = 0.45 MPa m
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This is the order of magnitude value we expect for the critical stress intensity for fracture of a metal based on the Griffith theory.
Is there a mathematical limit to the value of r in the equation for the crack-tip stress field?
σ ij =K2πr
fij (θ )
Many materials exhibit critical K values 10 -100 times larger then this. Why?
Is there a physical limit to the value of r in the equation for the crack-tip stress field?
KG = 0.45 MPa m
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0.45 MPa mGK
Many materials exhibit critical K values 10 -100 times larger then this. Why?
This is the order of magnitude value we expect for the critical stress intensity for fracture of a metal based on the Griffith theory.
Is there a mathematical limit to the value of r in the equation for the crack-tip stress field?
σ ij =K2πr
fij (θ )
Is there a physical limit to the value of r in the equation for the crack-tip stress field?
Many materials exhibit critical K values 10 -100 times larger then this. Why?
YES! an interatomic spacing
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Crack tip plasticity
σ ij =K2πr
fij (θ )
Based on the crack tip stress field, will there (always?) be a region nearthe crack tip where the yield stress of the metal is exceeded.
Let’s ignore the term, and set equal to the yield strength.
fij (θ )σ ij
Estimate of the plastic zone size
fij (θ ) = 1 σ ij =σ yield =σ y
rplastic = rp =K 2
2πσ y2
A typical yield strength of a high strength steel is 300 MPa.
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Crack tip plasticityEstimate of the plastic zone size
fij (θ ) = 1 σ ij =σ yield =σ y
rplastic = rp =K 2
2πσ y2
A typical yield strength of a high strength steel is 300 MPa.
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Crack tip plasticity
σ ∼ r−1/2σ y
rplastic = rp =K 2
2πσ y2
x
y
rplastic
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Crack tip plasticityEstimate of the plastic zone size
fij (θ ) = 1 σ ij =σ yield =σ y
rplastic = rp =K 2
2πσ y2
The smallest possible size of the plastic zone corresponds to about a lattice spacing. Let’s examine what the value of would have to be, to correspond to a lattice spacing at the typical value of the critical Griffith value of the stress intensity factor, K = 0.45 MPa.
σ y
rp ≈ ao = 3×10−10 m
σ y =K2πrp
= K
2πao
4.5×105Pa2π × 3×10−10
≅ 1×1010 Pa
How does this compare to the theoretical fracture stress of a solid?
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Crack tip plasticityEstimate of the plastic zone size
fij (θ ) = 1 σ ij =σ yield =σ y
rplastic = rp =K 2
2πσ y2
The smallest possible size of the plastic zone corresponds to about a lattice spacing. Let’s examine what the value of would have to be, to correspond to a lattice spacing at the typical value of the critical Griffith value of the stress intensity factor, K = 0.45 MPa.
σ y
Theoretical strength of an elastic solid
ao : Lattice spacing
Smallest possible rp is about 1 nm.
rp ≈ ao = 3×10−10 m
σ y =K2πrp
= K
2πao
4.5×105Pa2π × 3×10−10
≅ 1×1010 Pa
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Crack tip plasticity on a microscopic scale
Dislocations define the plastic zone size
Crack tip dislocation interactions and dislocation–dislocation interactions via the Peach-Kohler equation.
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Crack tip plasticity on the atomic scale
First principles based calculations: Density Functional Theory and Molecular Dynamic Simulations.
On the atomic scale. The crack tip serves as a source for dislocations. As the tip emits dislocations it become less sharp or “blunted.”
In general we can see that there are multiple length scales associated with crack tip stress/strain fields and elastic-plastic fracture.