Design Contents |

Preliminary Matters

Design Methodology

..brain storming

..evaluation matrix

..QFD

Statistical Considerations

..variability in materials

..variability in dimensions

..variability in loading

..preferred sizes

Tolerances

Design Factor

Introduction to Failure

Failure Theories

Application of von Mises

..criterion in 2 D

Stress Concentration

..and notch sensitivity

Failure Under Combined Loading

..combined bending and torsion

Failure Under Cyclic Loading

..fatigue

..fracture mechanics

Instability - Buckling

Concentrically Loaded Strut

..slender columns

..Euler formula

..effective length

..short and intermediate columns

Eccentrically Loaded Strut

.. theory

Shock Loading

..deflection

..stress

FAILURE UNDER CYCLIC LOADING - Fatigue, Fracture and Crack Growth Rates |

**3 Fracture Mechanics -
3.1 Introduction**

In the 1960s when higher strength steels started to be introduced, it was noted that components often failed suddenly at loads that were well below those that would have caused yielding. Investigation these failures led to the development of 'fracture mechanics'.

Changes in component geometry such as those caused by grooves, give rise to stress concentration effects. As long as the geometry is known, these effects can be computed and taken into account during design. However many components contain defects, often cracks, where the end of the crack is very sharp, its radius is not known and can not be measured. For situations like these, the stress concentration can not be determined and an alternative approach is needed. The approach is to use 'fracture mechanics'.

The way in which a cracked component is loaded can be idealised in one of three ways: . For the vast majority of engineering applications it is 'mode I', the 'opening mode', which is of interest.

The stress near a crack tip can be characterised by a single parameter, the stress intensity factor,
where Q is a factor depending of specimen and crack geometry ('geometry correction factor'), a is the crack depth (or half the crack length).
Fracture occurs when the value of K reaches K_{c} the fracture toughness of the material.

K_{c} can be found from laboratory tests and applied to real structures of different geometries. K_{Ic} (the subscript 'I'
meaning mode I) is a function of the material thickness and reaches a minimum when the material is thick enough to provide full restraint
(plane strain). Extensive values can be found in the literature (see below).

Although strictly the stress intensity applies only to linear elastic fracture mechanics (LEFM), some crack tip plasticity is common in structural materials. Provided this plastic zone is only small <a/15 or < B/15 (in configurations shown on right: b = B) then the stress intensity is not significantly affected away from the crack tip and LEFM and K can be used.

Structural materials absorb significant energy in the plastic zone near the crack tip which makes them tough, however brittle materials have no, or a very small crack tip plastic zone and have low toughness.

Crack growth rates in components subjected to **cyclic loading** have been investigated in a number of structural materials and all
structural metals have the same 'sigmoidal' shaped crack growth curves when the crack growth is plotted against the stress intensity factor range
(K_{max} - K_{min}) on log axes.

**3.2 How Fracture Mechanics is used in Design and Maintenance Operations**

Fracture mechanics may be used to assess if a defect present in a structure will cause failure during the load application or for cyclic
loading, how many load cycles will be required to grow the crack to the critical length that causes failure.

**3.3 Geometry Correction Factors** - for configurations shown left and below.

These are for mode I loading and normally are only valid when b>>a. For a short single edge through crack of depth a, Q also = 1.12

Data, often obtained from finite element analysis, have then been plotted as curves and equations based on these curves have been
derived and are used in computer software to calculate residual lives of cracked components. Such equations
are used in the applet for the crack growth calculation of the elliptical surface crack, see later section: 'Computation'.

More accurate values for the shape factor 'Q', than shown here are used for the calculation in the applet,
see later sections.

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David J Grieve, 20th February 2014.

**Contact the Author:**

Please contact me for comments and / or corrections or to purchase the book, at: davejgrieve@aol.com