Page 195 - T. Anderson-Fracture Mechanics - Fundamentals and Applns.-CRC (2005)

1656_C004.fm Page 175 Thursday, April 21, 2005 5:38 PM

Dynamic and Time-Dependent Fracture 175

Recent work by Nakamura et al. [18, 19] quantified inertia effects in laboratory specimens and
showed that these effects can be neglected in many cases. They observed that the behavior of a
dynamically loaded specimen can be characterized by a short-time response, dominated by discrete
waves, and a long-time response that is essentially quasistatic. At intermediate times, global inertia
effects are significant but local oscillations at the crack are small, because kinetic energy is absorbed
by the plastic zone. To distinguish short-time response from long-time response, Nakamura et al. defined
a transition time t when the kinetic energy and the deformation energy (the energy absorbed by the
τ
specimen) are equal. Inertia effects dominate prior to the transition time, but the deformation energy
dominates at times significantly greater than t . In the latter case, a J-dominated field should exist near
τ
the crack tip and quasistatic relationships can be used to infer J from global load and displacement.
Since it is not possible to measure kinetic and deformation energies separately during a fracture
mechanics experiment, Nakamura et al. developed a simple model to estimate the kinetic energy
and the transition time in a three-point bend specimen (Figure 4.2). This model was based on the
Bernoulli-Euler beam theory and assumed that the kinetic energy at early times was dominated by
the elastic response of the specimen. Incorporating the known relationship between the load-line
displacement and the strain energy in a three-point bend specimen leads to an approximate rela-
tionship for the ratio of kinetic to deformation energy:


E k = Λ Wt  2
˙
∆()
U  c  o t ∆()   (4.2)
where
E = kinetic energy
k
U = deformation energy
W = specimen width
∆ = load line displacement
∆ = displacement rate
˙
c = longitudinal wave speed (i.e., the speed of sound) in a one-dimensional bar
o
Λ = geometry factor, which for the bend specimen is given by
SBEC
Λ= (4.3)
W

where S is the span of the specimen.

FIGURE 4.2 Three-point bend specimen.