Fracture Fra cture Avoidance Avoidanc e w ith P roper Use of M ateria ateriall
Pyramid of Egypt • •
Schematic Roman Bridge Design
The pri The prima mary ry con constr struct uction ion mat materi erial al pri prior or to 19t were timber, brick and mortar Arch Ar ch sha shape pe pro produ duci cing ng com compr pres essi siv ve stre stress ss → stone have high compressive strength
Riley; page 5 Anderson; fig. 1-4, page 9 Gordon; fig. 14, page 188
Fracture Fra cture Avoidance Avoidanc e w ith P roper Use of M ateria ateriall ’
•
Roof Roof spans spans and windows windows were arched arched to maintain maintain compress compressive ive loading loading
Gordon; plate 1 (after page 224) Anderson; fig. 1-5
Fracture Fra cture Avoidance Avoidanc e w ith P roper Use of M ateria ateriall ’
•
Roof Roof spans spans and windows windows were arched arched to maintain maintain compress compressive ive loading loading
Gordon; plate 1 (after page 224) Anderson; fig. 1-5
Fracture Fra cture Avoidance Avoidanc e w ith P roper Use of M ateria ateriall n ’ •
Mass production production of iron and steel steel (relative (relatively ly ductile ductile construct construction ion materials materials)) → feasible fea sible to build structure structuress carr in tensile tensile stresses stresses
e Te or s Menai suspension suspension ri ge 1819 (wrought iron suspension chains) Gordon; plate 11 & plate 12
e seven suspension suspension ri ge (steel cable)
Stress Concentrat ion, Fracture and r i f f i h Th r •
Stress distribution around a hole in an . 1898 using the theory of elasticity
•
The maximum stress is three times the uniform stress
•
K t = 3
Damage Tolerance Tolerance Assessment Handbook; fig. 2-1, page 2-2
Stress Concent ration, Fracture n r i f f i h Th r n ’ •
C. E. Inglis Inglis (1913) (1913) invest investiga igated ted in a plate plate with with an
•
He derived t
K t
= 1 + 2 a / b or
=
•
Mode Modelin ling g a crac crack k with with a elli ellips pse e me mean anss ρ → 0 → K t → ∞ → infinite stress
•
K t could not be used for crack problems
Damage Tolerance Assessment Handbook; fig. 2-2
Stress Concent ration, Fracture n riffi h Th r n ’
•
A. A. Griffith (1920) used an energy balance analysis to explain the large
•
Griffith proposed that the large reduction is due to the presence of microcracks
•
Griffith derived a relation between crack size and breaking strength by considering the energy balance associated with a small extension of a crack
Stress Concentratio n, Fracture and G riffith Theory
Damage Tolerance Assessment Handbook; fig. 2-3 a & b
Stress Concent ration, Fracture n riffi h Th r n ’ 1
1
2
2
Work Work 1
(
=
= 1
2
(
)( AL)
(
)(V )
) = train ener
densit
Stress Concentrati on, Fracture and G riffith Theory (cont’)
Damage Tolerance Assessment handbook; fig. 2-4 a & b
Stress Concentrati on, Fracture and G riffith Theory (cont’)
•
Crack length increase → plate becomes less stiff (more flexible) → slope of P vs x decreases → applied load drop
•
Change in energy stored is the difference in the shaded area
•
Release of elastic energy is used to overcome the resistance to crack growth
•
Rate of strain energy release = rate of energy absorption to overcome resistance to crack growth
Damage Tolerance Assessment Handbook; fig. 2-4b
Stress Concentrati on, Fracture and G riffith Theory (cont’)
Energy stored in the body before crack extension = Σ (energy remaining in the body after crack extension + work done on the body during crack extension + energy dissipated in irreversible processes) Damage Tolerance Assessment Handbook; fig. 2-4b
Stress Concentrati on, Fracture and G riffith Theory (cont’)
•
Analyze a simplified geometry with a hole D = 2a
•
σy = σ everywhere outside the hole Damage Tolerance Assessment Handbook; fig. 2-5
Stress Concentrati on, Fracture and G riffith Theory (cont’)
•
Strain energy density =
σ
2
2E
•
Total energy
σ
=
2
x vol
2E σ
U 1 =
2
[WLt −
π a
2
]
t
2 E After crack extension of ∆a (assume σ is constant)
U2 =
σ 2E
[
(
WLt − π a + ∆a
)2 t ]
Elastic energy released U1 − U 2 ≅
πσ
a t ∆a E
Per unit of new crack area G =
U1 − U 2 2 t ∆a
2
≅
πσ a
2E Damage Tolerance Assessment Handbook; fig. 2-5
Stress Concent ration, Fracture n riffi h Th r n ’
• • •
Surface energy ( γe) is a material property nergy a ance
G
σ •
s use to rea atom c on s → sur ace energy
nergy re ease
˙>
crac grow
≥ 2 γ e a
=
4E γ e
π
Griffith analysis based on Inglis solution yield 2
σ
a
=
γ e π
G
and
=
E
Stress Concent ration, Fracture n riffi h Th r n ’ Linear Elastic Fracture M echanics (LEFM ) •
In 1957 Irwin reexamined the problem of stress distribution around a crack
•
He analyzed an infinite plate with a crack
•
Using the theory of elasticity the stresses are dominated by x=
y=
=
K 2
r
K 2
r
K
ij=
2
r
cos
cos
sin
a 2
r
2
2
[1 − sin [1 + sin
cos
f ij ( )
2
2
cos
sin
sin
3 2
2
]
]
3
assumption r << a LEFM valid if plasticity remains small compared to the over all dimensions of crack and cracked bodies
Stress Concent ration, Fracture n riffi h Th r n ’
a K I •
=
a
for an infinite plate
The relation of K to G is 2
G
=
I
E
•
. . if G = Gc
or
K I = K Ic
c
c
. .
Stress Concent ration, Fracture n riffi h Th r n ’ Stress I ntensit Factor
= K
=
for
a
for
β can be obtained from :
infinite plate other geometry 1. handbook solution
3. numerical method
Stress Concent ration, Fracture n riffi h Th r n ’
Bannantine, fig. 3-4, page 92
Stress Concentrati on, Fracture and G riffith Theory (cont’)
Bannantine; fig. 3-4, page 93 & 94
Stress Concentrati on, Fracture and G riffith Theory (cont’)
Stress Concentrati on, Fracture and G riffith Theory (cont’) Loadin
M odes cont’
Loading stresses terms for mode II
K II
=−
xy
•
sin
2
=
K II
=
K II
sin
2
2
2 2
2
2
r
cos
−
Stresses terms for mode III xz
=−
yz
=−
III
2
r
III
2
r
sin cos
+ cos
2
2
2
2
cos
cos
3 2 3
2
2
3 2
Extension of LEFM to M etals
•
Griffith energy theory and Irwin’s stress intensity factor could explain the fracture phenomena for brittle solid
•
For metals, beside surface energy absorption, the plastic energy absorption (γp) has to be added
a , •
p
e,
=
e
p
e
It was not easy to translate energy concept into engineering practice
Extension of LEFM to M etals (cont’)
• • •
K concept was seen as the basis of a practical approach , If it is assumed that the plastic zone at the crack tip is much smaller than the crack dimension → K is still valid
Extension of LEFM to M etals (cont’) y
K
=
2
r
cos
2
[1 − sin
2
sin
2
]
θ = 0
for y
K
=
2
r
y
ys
r
*
p
K
=
2
=
1 2
or
r * p
⎛ K ⎞ ⎜ ⎟ ⎜ ys ⎟ ⎝
2
r
*
p
=
K 2 2
ys
2
plane stress
Corrected due to stress redistribution
r p r p
= =
1
1 3
⎛ K ⎞ ⎜ ⎟ ys
⎛ K ⎞ ⎜ ⎟ ⎜ ys ⎟ ⎝ ⎠
2
2
lane stress plane strain
P lane Strain Fracture Toughn ess Testing
P l an e S tr ai n Fr act ur e To ug hn es s Tes ti ng • Standard test method include ASTM E399: “Standard Test Methods for Plane Strain Fracture Toughness of M etallic M aterials ”. • Stringent requirement for plane strain condition and linear behaviour of the specimen. • Specimen type permitted: CT, SENB, arc-shaped and disk shape.
P lane Strain Fracture Toughn ess Testing (cont’) Fracture Mechanics Testing
Specimen Configurations
P lane Strain Fracture Toughn ess Testing (cont’) Cl ev is fo r Com pact Ten si on Sp eci m en
P lane Strain Fracture Toughn ess Testing (cont’) •
Use an extensometer (e.g. clip gage) to detect the beginning of crack extension from the fatigue crack.
P lane Strain Fracture Toughn ess Testing (cont’)
• Calculation of K Q for compact tension specimen
K Q
=
PQ 1 / 2
f (
a
)
where a f ( W )=
+ W a
.
+
.
a W
−
a W
.
(1 − ) a W
+
3 2
• This K Q has to be checked with previous requirements
.
a W
−
.
a W
P lane Strain Fracture Toughn ess Testing (cont’)
Damage Tolerance Assessment Handbook; fig. 2-13
P lane Strain Fracture Toughn ess Testing (cont’)
ASTM Standards; fig. 1, page 410
P lane Strain Fracture Toughn ess Testing (cont’) Fatigue Pre-crack ing •
Perform to obtain natural crack
•
Fatigue load must be chosen : 0
0
such that the time is not very long
P lane Strain Fracture Tough ness Testing (cont’) Instrum entation for Displacement and Crack Length M easurements
P lane Strain Fracture Toughn ess Testing (cont’) • Crack front curvature
P lane Strain Fracture Toughn ess Testing (cont’)
•
a
=
a1 + a 2 + a3
• Any two of a1, a2 and a3 must not differ more than 10% from • For strai ht notch → a differ not more than 15% from does not differ more than 10% from (a surface)right
a and a
P lane Strain Fracture Toughn ess Testing (cont’) • Load displacement curves to determine P Q Additional Criteria »
< 2
»
»
.
2.5
⎛ K ⎞ < a ⎜ ys ⎟ ⎝ ⎠ ⎜ ⎜
K Q
⎟ < B ⎟
P lane Strain Fracture Toughn ess Testing (c
P lane Strain Fracture Toughn ess Testing (cont’)
Damage Tolerance Assessment Handbook; table 2-1, page 2-32
P lane Strain Fracture Toughn ess Testing (cont’) c ness • • •
ec
Plane strain condition occur for thick components or s a c ma er a proper es p ane s ra n con
on oes no
For fracture toughness thickness have a strong influence
Thickness effect on fracture strength Damage Tolerance Assessment Handbook; fig. 2-16
ave n uence
P lane Strain Fracture Toughn ess Testing (cont’) Thickness Effect (cont’) •
Specimen thicker than 1/2 inch → plane strain
•
For
thinner
stock
KQ
increases
1/8 inch •
The peak K Q can exceed five times ic
Thickness effect on fracture strength
• •
After reaching the peak K Q declines at thickness lower than 1/8 inch c ness e ec can e exp a ne w
Damage Tolerance Assessment Handbook; fig. 2-16
energy a ance
P lane Strain Fracture Toughn ess Testing (cont’) Thickness Effect (cont’) •
σZ = 0 at free surface → plane stress
•
In the inside elastic material restrains deformation in Z direction
•
For thick deformation
specimen is almost
interior totally
condition Three-dimensional plastic zones shape
•
Going inward from the surface, plastic zone undergoes transition from larger size to smaller size
Damage Tolerance Assessment Handbook; fig. 2-17a
P lane Strain Fracture Toughn ess Testing (cont’) Thickness Effect (cont’)
Plastic volume versus thickness
Damage Tolerance Assessment Handbook; fig. 2-17b
•
For decreasing thickness, ratio of p as c vo ume o o a c ness increase
•
Conse uentl ener absor tion rate also increases for thinner plates
•
While
•
Thus for thinner plates more applied stress is needed to extend the crack
elastic
strain
energy
is
P lane Strain Fracture Toughn ess Testing (cont’) Thickness Effect (cont’)
Typical Fracture Surface Damage Tolerance Assessment Handbook; fig 2-18
•
Plane stress condition results in fracture surface havin 45o an le to z axis → shear lips
•
For valid K ic test (plane strain condition) little or no evidence of shear li s
P lane Strain Fracture Toughn ess Testing Temperature Effect •
Fracture toughness depends on temperature
•
However Al alloys are relatively insensitive over the range of aircraft service temperature condition
•
Many alloy steels exhibit a sharp transition in the service temperature range
Fracture toughness versus temperature Damage Tolerance Assessment Handbook; fig. 2-21
K I c of Aircraft M aterials Values for Several Al Alloys
ASM Vol. 19; table 5, page 776
K I c of Som e Materials (cont’) .
Application of Fracture Mechanics; fig. 6-9, page 180
K I c of Som e Materials (cont’) Effect of Purity on K Ic
ASM Vol. 19; table 6, page 777
K I c of Aircraft M aterials (cont’) Typical Yield Strength and Fracture Toughness of High-Strength Titanium Alloy
ASM Vol. 19; table 3, page 831
Failure in Large Scale Yielding •
Strength assessment for structures do not meet small scale yielding condition : 1. R-curve method 2. Net section failure 3. Crack tip opening displacement 4. J-integral 5. Energy density → mixed mode loading 6. Plastic collapse → for 3D cracks
The N et Section on Failure Criterion •
ress concen ra on n uc e ma er a s causes y e the stress as applied load increased
ng w c smoo e ou
•
Failure is assumed to occur when stress at the net section was distributed uniformly reaching σu
Net section failure criterion •
For a plate width w containing a center crack of length 2a, the critical stress is σc =
w − 2a w
σ f
Damage Tolerance Assessment Handbook; fig. 2-34
K c of Aircraft M aterials Plane Stress Fracture Toughness (K c) for Several Al Alloys
ASM Vol. 19; fig. 10, page 779
Crack Opening Displacem ent (COD) •
Applied load will cause a crack to open, the crack opening displacement can be used as a parameter
•
At a critical value of COD fracture occur
•
Developed for steels
-I n
r l
•
J-integral is an expression of plastic work (J) done when a body is loaded
•
J-integral can be calculated from elastic plastic calculation
•
At a critical value of J fracture occur
