Fracture Avoidance with Proper Use of Material

1
Fracture Avoidance with Proper Use of Material

• Pyramid of Egypt Schematic Roman Bridge Design
• The primary construction material prior to 19th
  were timber, brick and mortar
• Arch shape producing compressive stress ? stone
  have high compressive strength

Riley page 5 Anderson fig. 1-4, page 9 Gordon
fig. 14, page 188
2
Fracture Avoidance with Proper Use of Material
(cont)

• Roof spans and windows were arched to maintain
  compressive loading

Gordon plate 1 (after page 224) Anderson fig.
1-5
3
Fracture Avoidance with Proper Use of Material
(cont)

• Mass production of iron and steel (relatively
  ductile construction materials) ? feasible to
  build structures carrying tensile stresses

The Telfords Menai suspension bridge (1819)
The seven suspension bridge (wrought iron
suspension chains) (steel cable)
Gordon plate 11 plate 12
4
Stress Concentration, Fracture and Griffith
Theory

• Stress distribution around a hole in an infine
  plate was derived by G. Kirsch in 1898 using the
  theory of elasticity
• The maximum stress is three times the uniform
  stress
• Kt 3

Damage Tolerance Assessment Handbook fig. 2-1,
page 2-2
5
Stress Concentration, Fracture and Griffith
Theory (cont)

• C. E. Inglis (1913) investigated in a plate with
  an elliptical hole
• He derived or
• Modeling a crack with a ellipse means ? ? 0 ? Kt
  ? ? ? infinite stress
• Kt could not be used for crack problems

Damage Tolerance Assessment Handbook fig. 2-2
6
Stress Concentration, Fracture and Griffith
Theory (cont)

• A. A. Griffith (1920) used an energy balance
  analysis to explain the large reduction on the
  strength of glass
• 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

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Stress Concentration, Fracture and Griffith Theory
Damage Tolerance Assessment Handbook fig. 2-3 a
b
8
Stress Concentration, Fracture and Griffith
Theory (cont)
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Stress Concentration, Fracture and Griffith
Theory (cont)
Damage Tolerance Assessment handbook fig. 2-4 a
b
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Stress Concentration, Fracture and Griffith
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
11
Stress Concentration, Fracture and Griffith
Theory (cont)

• Energy balance

Energy stored in the body before crack extension
? (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
12
Stress Concentration, Fracture and Griffith
Theory (cont)

• Analyze a simplified geometry with a hole D 2a
• ?y ? everywhere outside the hole

Damage Tolerance Assessment Handbook fig. 2-5
13
Stress Concentration, Fracture and Griffith
Theory (cont)

• Strain energy density
• Total energy
• After crack extension of ?a (assume ? is
  constant)
• Elastic energy released
• Per unit of new crack area

Damage Tolerance Assessment Handbook fig. 2-5
14
Stress Concentration, Fracture and Griffith
Theory (cont)

• Energy released is used to break atomic bonds ?
  surface energy
• Surface energy (?e) is a material property
• Energy balance ?? crack growth if
• Griffith analysis based on Inglis solution yield
• and

15
Stress Concentration, Fracture and Griffith
Theory (cont)
Linear Elastic Fracture Mechanics (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

assumption r ltlt a LEFM valid if plasticity
remains small compared to the over all dimensions
of crack and cracked bodies
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Stress Concentration, Fracture and Griffith
Theory (cont)

• The term is given the symbol K (stress intensity
  factor)
• for an infinite plate
• The relation of K to G is
• for plane stress condition
• The use of G and KI leads to fracture criterion
  i.e. Gc and Kic i.e. fracture occur if
• G Gc or KI KIc

17
Stress Concentration, Fracture and Griffith
Theory (cont)
Stress Intensity Factor
for infinite plate
for other geometry
? can be obtained from 1. handbook
solution 2. approximate method 3.
numerical method
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Stress Concentration, Fracture and Griffith
Theory (cont)
Bannantine, fig. 3-4, page 92
19
Stress Concentration, Fracture and Griffith
Theory (cont)
Bannantine fig. 3-4, page 93 94
20
Stress Concentration, Fracture and Griffith
Theory (cont)
Loading Modes
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Stress Concentration, Fracture and Griffith
Theory (cont)

• Loading Modes (cont)
• Loading stresses terms for mode II
• Stresses terms for mode III

22
Extension of LEFM to Metals

• Griffith energy theory and Irwins stress
  intensity factor could explain the fracture
  phenomena for brittle solid
• For metals, beside surface energy absorption, the
  plastic energy absorption (?p) has to be added
• For typical metal, ?p ? 1000 ?e, thus ?e can be
  neglected
• It was not easy to translate energy concept into
  engineering practice

23
Extension of LEFM to Metals (cont)

• K concept was seen as the basis of a practical
  approach
• However, K is an elastic solution while at the
  crack tip plastic zone developed
• If it is assumed that the plastic zone at the
  crack tip is much smaller than the crack
  dimension ? K is still valid

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Extension of LEFM to Metals (cont)
Monotonic Loading
Plastic zone size
for ? 0
If ?y is equal to yield strength
or
plane stress
Corrected due to stress redistribution
plane stress
plane strain
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Plane Strain Fracture Toughness Testing

• Plane Strain Fracture Toughness Testing
• Standard test method include ASTM E399 Standard
  Test Methods for Plane Strain Fracture Toughness
  of Metallic Materials.
• Stringent requirement for plane strain condition
  and linear behaviour of the specimen.
• Specimen type permitted CT, SENB, arc-shaped and
  disk shape.

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Plane Strain Fracture Toughness Testing (cont)
Fracture Mechanics Testing Specimen
Configurations
27
Plane Strain Fracture Toughness Testing (cont)
Clevis for Compact Tension Specimen
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Plane Strain Fracture Toughness Testing (cont)

• Use an extensometer (e.g. clip gage) to detect
  the beginning of crack extension from the fatigue
  crack.

29
Plane Strain Fracture Toughness Testing (cont)

• Calculation of KQ for compact tension specimen
• where
• This KQ has to be checked with previous
  requirements

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Plane Strain Fracture Toughness Testing (cont)
Damage Tolerance Assessment Handbook fig. 2-13
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Plane Strain Fracture Toughness Testing (cont)
ASTM Standards fig. 1, page 410
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Plane Strain Fracture Toughness Testing (cont)

• Fatigue Pre-cracking
• Perform to obtain natural crack
• Fatigue load must be chosen
• such that the time is not very long
• plastic zone at the crack tip is small

33
Plane Strain Fracture Toughness Testing
(cont)
Instrumentation for Displacement and Crack Length
Measurements
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Plane Strain Fracture Toughness Testing (cont)

• Crack front curvature

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Plane Strain Fracture Toughness Testing (cont)

• Measure a1, a2 and a3 ?
• Any two of a1, a2 and a3 must not differ more
  than 10 from
• For straight notch ? asurface differ not more
  than 15 from and (asurface)left does not
  differ more than 10 from (asurface)right

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Plane Strain Fracture Toughness Testing (cont)

• Load displacement curves to determine PQ
• Additional Criteria
• Pmax/PQ lt 1.1

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Plane Strain Fracture Toughness Testing (cont)
Damage Tolerance Assessment handbook table 2-1,
page 2-31
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Plane Strain Fracture Toughness Testing (cont)
Damage Tolerance Assessment Handbook table 2-1,
page 2-32
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Plane Strain Fracture Toughness Testing (cont)

• Thickness Effect
• Plane strain condition occur for thick components
• For static material properties plane strain
  condition does not have influence
• For fracture toughness thickness have a strong
  influence

Thickness effect on fracture strength
Damage Tolerance Assessment Handbook fig. 2-16
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Plane Strain Fracture Toughness Testing (cont)

• Thickness Effect (cont)
• Specimen thicker than 1/2 inch ? plane strain
• For thinner stock KQ increases reaching a peak at
  thickness about 1/8 inch
• The peak KQ can exceed five times Kic
• After reaching the peak KQ declines at thickness
  lower than 1/8 inch
• Thickness effect can be explained with energy
  balance

Thickness effect on fracture strength
Damage Tolerance Assessment Handbook fig. 2-16
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Plane Strain Fracture Toughness Testing (cont)

• Thickness Effect (cont)
• ?Z 0 at free surface ? plane stress on the
  surface ? large plastic zone
• In the inside elastic material restrains
  deformation in Z direction
• For thick specimen interior deformation is almost
  totally restraint (?Z ? 0) ? plane strain
  condition
• Going inward from the surface, plastic zone
  undergoes transition from larger size to smaller
  size

Three-dimensional plastic zones shape
Damage Tolerance Assessment Handbook fig. 2-17a
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Plane Strain Fracture Toughness Testing (cont)

• Thickness Effect (cont)
• For decreasing thickness, ratio of plastic volume
  to total thickness increase
• Consequently energy absorption rate also
  increases for thinner plates
• While elastic strain energy is independent of
  thickness
• Thus for thinner plates more applied stress is
  needed to extend the crack

Plastic volume versus thickness
Damage Tolerance Assessment Handbook fig. 2-17b
43
Plane Strain Fracture Toughness Testing (cont)

• Thickness Effect (cont)
• Plane stress condition results in fracture
  surface having 45o angle to z axis ? shear lips
• For valid Kic test (plane strain condition) ?
  little or no evidence of shear lips

Typical Fracture Surface
Damage Tolerance Assessment Handbook fig 2-18
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Plane Strain Fracture Toughness Testing

• Thickness Effect (cont)
• For even thinner plates KQ declines due to the
  increase of strain energy release rate
• Additional strain energy release comes lateral
  local buckling of the plate around the crack

Lateral compression above Lateral buckling
and tearing and below the crack
Damage Tolerance Assessment Handbook fig. 2-19
2-20
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Plane Strain Fracture Toughness 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
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KIc of Aircraft Materials
Typical Yield Strength and Plane Strain Fracture
Toughness Values for Several Al Alloys
ASM Vol. 19 table 5, page 776
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KIc of Some Materials (cont)
Al Alloys 2124 and 7475 vs. 2024 and 7075
Application of Fracture Mechanics fig. 6-9, page
180
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KIc of Some Materials (cont)
Effect of Purity on KIc
ASM Vol. 19 table 6, page 777
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KIc of Aircraft Materials (cont)
Typical Yield Strength and Fracture Toughness of
High-Strength Titanium Alloy
ASM Vol. 19 table 3, page 831
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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

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The Net Section on Failure Criterion

• Stress concentration in ductile materials causes
  yielding which smoothed out the stress as applied
  load increased
• Failure is assumed to occur when stress at the
  net section was distributed uniformly reaching ?u
• For a plate width w containing a center crack of
  length 2a, the critical stress is

Net section failure criterion
Damage Tolerance Assessment Handbook fig. 2-34
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Kc of Aircraft Materials
Plane Stress Fracture Toughness (Kc) for Several
Al Alloys
ASM Vol. 19 fig. 10, page 779
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Crack Opening Displacement (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

J-Integral

• 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

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END