Page 45 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

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2 Linear Elastic Fracture

Mechanics

The concepts of fracture mechanics that were derived prior to 1960 are applicable only to materials
that obey Hooke’s law. Although corrections for small-scale plasticity were proposed as early as
1948, these analyses are restricted to structures whose global behavior is linear elastic.
Since 1960, fracture mechanics theories have been developed to account for various types of
nonlinear material behavior (i.e., plasticity and viscoplasticity) as well as dynamic effects. All of
these more recent results, however, are extensions of linear elastic fracture mechanics (LEFM).
Thus a solid background in the fundamentals of LEFM is essential to an understanding of more
advanced concepts in fracture mechanics.
This chapter describes both the energy and stress intensity approaches to linear fracture mechan-
ics. The early work of Inglis and Griffith is summarized, followed by an introduction to the energy
release rate and stress intensity parameters. The appendix at the end of this chapter includes
mathematical derivations of several important results in LEFM.
Subsequent chapters also address linear elastic fracture mechanics. Chapter 7 and Chapter 8
discuss laboratory testing of linear elastic materials, Chapter 9 addresses application of LEFM to
structures, Chapter 10 and chapter 11 apply LEFM to fatigue crack propagation and environmental
cracking, respectively. Chapter 12 outlines numerical methods for computing stress intensity factor
and energy release rate.

2.1 AN ATOMIC VIEW OF FRACTURE

A material fractures when sufficient stress and work are applied at the atomic level to break the
bonds that hold atoms together. The bond strength is supplied by the attractive forces between atoms.
Figure 2.1 shows schematic plots of the potential energy and force vs. the separation distance
between atoms. The equilibrium spacing occurs where the potential energy is at a minimum. A
tensile force is required to increase the separation distance from the equilibrium value; this force
must exceed the cohesive force to sever the bond completely. The bond energy is given by

E = ∫ x o ∞ Pdx (2.1)
b

where x is the equilibrium spacing and P is the applied force.
o
It is possible to estimate the cohesive strength at the atomic level by idealizing the interatomic
force-displacement relationship as one half of the period of a sine wave:
 π x 
P P = c sin  λ  (2.2a)

where the distance λ is defined in Figure 2.1. For the sake of simplicity, the origin is defined at x .
o
For small displacements, the force-displacement relationship is linear:
 π x 
P P = c  λ  (2.2b)

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