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Chapter 5

Failures Resulting from Static Loading

Lecture 12

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Introduction to Fracture Mechanics

Linear elastic fracture mechanics (LEFM) analyzes crack growth during service
Assumes cracks can exist before service begins, e.g. flaw, inclusion, or defect
Attempts to model and predict the growth of a crack
Stress concentration approach is inadequate when notch radius becomes extremely sharp, as in a crack, since stress concentration factor approaches infinity
Ductile materials often can neglect effect of crack growth, since local plastic deformation blunts sharp cracks
Relatively brittle materials, such as glass, hard steels, strong aluminum alloys, and steel below the ductile-to-brittle transition temperature, benefit from fracture mechanics analysis

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Quasi-Static Fracture

Though brittle fracture seems instantaneous, it actually takes time to feed the crack energy from the stress field to the crack for propagation.
A static crack may be stable and not propagate.
Some level of loading can render a crack unstable, causing it to propagate to fracture.

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Quasi-Static Fracture

Foundation work for fracture mechanics established by Griffith in 1921
Considered infinite plate with an elliptical flaw
Maximum stress occurs at (±a, 0)

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Fig. 5−22

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Quasi-Static Fracture

Crack growth occurs when energy release rate from applied loading is greater than rate of energy for crack growth
Unstable crack growth occurs when rate of change of energy release rate relative to crack length exceeds rate of change of crack growth rate of energy

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Crack Modes and the Stress Intensity Factor

Three distinct modes of crack propagation

Mode I: Opening crack mode, due to tensile stress field
Mode II: Sliding mode, due to in-plane shear
Mode III: Tearing mode, due to out-of-plane shear

Combination of modes possible
Opening crack mode is most common, and is focus of this text

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Fig. 5−23

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Mode I Crack Model

Stress field on dx dy element at crack tip

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Fig. 5−24

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Stress Intensity Factor

Common practice to define stress intensity factor

Incorporating K_I, stress field equations are

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Stress Intensity Modification Factor

Stress intensity factor K_I is a function of geometry, size, and shape of the crack, and type of loading
For various load and geometric configurations, a stress intensity modification factor β can be incorporated

Tables for β are available in the literature
Figures 5−25 to 5−30 present some common configurations

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Stress Intensity Modification Factor

Off-center crack in plate in longitudinal tension
Solid curves are for crack tip at A
Dashed curves are for tip at B

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Fig. 5−25

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Stress Intensity Modification Factor

Plate loaded in longitudinal tension with crack at edge
For solid curve there are no constraints to bending
Dashed curve obtained with bending constraints added

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Fig. 5−26

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Stress Intensity Modification Factor

Beams of rectangular cross section having an edge crack

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Fig. 5−27

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Stress Intensity Modification Factor

Plate in tension containing circular hole with two cracks

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Fig. 5−28

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Stress Intensity Modification Factor

Cylinder loaded in axial tension having a radial crack of depth a extending completely around the circumference

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Fig. 5−29

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Stress Intensity Modification Factor

Cylinder subjected to internal pressure p, having a radial crack in the longitudinal direction of depth a

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Fig. 5−30

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Fracture Toughness

Crack propagation initiates when the stress intensity factor reaches a critical value, the critical stress intensity factor K_Ic
K_Ic is a material property dependent on material, crack mode, processing of material, temperature, loading rate, and state of stress at crack site
Also know as fracture toughness of material
Fracture toughness for plane strain is normally lower than for plain stress
K_Ic is typically defined as mode I, plane strain fracture toughness

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Typical Values for K_Ic

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Fracture toughness KIc for engineering metals lies in the range

20 ≤ KIc ≤ 200 MPa .√m;

For engineering polymers and ceramics, 1 ≤ KIc ≤ 5 MPa .√m.

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Brittle Fracture Factor of Safety

Brittle fracture should be considered as a failure mode for

Low-temperature operation, where ductile-to-brittle transition temperature may be reached
Materials with high ratio of S_y/S_u, indicating little ability to absorb energy in plastic region

A factor of safety for brittle fracture

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Example 5-6

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65 mm

12 m

50 MPa

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Example 5-6

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Fig. 5−25

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Example 5-6

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Example 5-7

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2.7 mm

1.4 m

2.8 m

4 MN (force)

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Example 5-7

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Example 5-7

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Fig. 5−26

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Example 5-7

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Note : You can use n=1.3 and compare with the allowable stress or use 1.3 for finding t as shown

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Problems: 5.1, 5.12, 5.36. and 5.84 from textbook