Page 194 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 174 Thursday, April 21, 2005 5:38 PM

174 Fracture Mechanics: Fundamentals and Applications

There are two major classes of dynamic fracture problems: (1) fracture initiation as a result of
rapid loading, and (2) rapid propagation of a crack. In the latter case, the crack propagation may initiate
either by quasistatic or rapid application of a load; the crack may arrest after some amount of unstable
propagation. Dynamic initiation, propagation, and crack arrest are discussed later in this chapter.

4.1.1 RAPID LOADING OF A STATIONARY CRACK
Rapid loading of a structure can come from a number of sources, but most often occurs as the
result of impact with a second object (e.g., a ship colliding with an offshore platform or a missile
striking its target). Impact loading is often applied in laboratory tests when a high strain rate is
desired. The Charpy test [14], where a pendulum dropped from a fixed height fractures a notched
specimen, is probably the most common dynamic mechanical test. Dynamic loading of a fracture
mechanics specimen can be achieved through impact loading [15, 16], a controlled explosion near
the specimen [17], or servo-hydraulic testing machines that are specially designed to impart high
displacement rates. Chapter 7 describes some of the practical aspects of high rate fracture testing.
Figure 4.1 schematically illustrates a typical load-time response for dynamic loading. The load
tends to increase with time, but oscillates at a particular frequency that depends on specimen geometry
and material properties. Note that the loading rate is finite, i.e., a finite time is required to reach a
particular load. The amplitude of the oscillations decreases with time, as kinetic energy is dissipated
by the specimen. Thus, inertia effects are most significant at short times, and are minimal after
sufficiently long times, where the behavior is essentially quasistatic.
Determining a fracture characterizing parameter, such as the stress-intensity factor or the J
integral, for rapid loading can be very difficult. Consider the case where the plastic zone is confined
to a small region surrounding the crack tip. The near-tip stress fields for high rate Mode I loading
are given by Equation 4.1.
Kt()
σ = I (4.1)
ij
2 πr
where (t) denotes a function of time. The angular functions f are identical to the quasistatic case
ij
and are given in Table 2.1. The stress-intensity factor, which characterizes the amplitude of the
elastic singularity, varies erratically in the early stages of loading. Reflecting stress waves that pass
through the specimen constructively and destructively interfere with one another, resulting in a
highly complex time-dependent stress distribution. The instantaneous K depends on the magnitude
I
of the discrete stress waves that pass through the crack-tip region at that particular moment in time.
When the discrete waves are significant, it is not possible to infer K from the remote loads.
I

FIGURE 4.1 Schematic load-time response of a rap-
idly loaded structure.