maximum entropy and zero entropy generation and is represented by point E in the
figure. If the system is maintained in a nonequilibrium state by supplying energy to
it, two processes are usually considered. Near equilibrium, the system will evolve
from its initial state A to a state B of minimum rate of entropy generation. If the
system is far from equilibrium, it may move from its initial state C to a state D, but at
present, no description of this state or the process in terms of entropy production is
available. The only accepted governing principle is energy conservation.
Whaley (1983) proposed a model for fatigue crack nucleation using the irreversible thermodynamics to quantify the damage caused by plastic strain. Whaley (1983)
model is based on the hypothesis that entropy gain which results from dynamic
irreversible plastic strain is a material constant. This was later proven experimentally
by Naderi et al. (2010) and, Imani and Modarres (2015). Whales (1983) postulated
that structural damage by fatigue could be quantified by a single material parameter,
the critical entropy threshold of fracture. Whaley (1983) postulated that the critical
entropy threshold of fracture is therefore related to the irreversible part of the fracture
energy. The critical entropy threshold of static fracture is a random variable and the
variability can be quantified by a confidence interval. According Whaley (1983), the
confidence interval for the entropy threshold just comes from the variance of the
plastic strain. It is noteworthy that Whaley (1983) assumed that only plastic strain
generates irreversible entropy, of course that is not accurate.
Naderi et al. (2010) were able to prove experimentally that total cumulative
entropy generation is constant at the time of failure and is independent of sample
geometry, load, and loading frequency. They named this critical entropy value
Fatigue Fracture Entropy (FFE). Figure 4.2 shows experimental fatigue fracture
Fig. 4.2 Fatigue fracture entropy versus the number of cycles to failure for different bending
fatigue tests of Al 6061-T6 with different specimen thickness, frequencies, and displacement
amplitudes. Fatigue fracture entropy remains at roughly 4 MJ m
À3 K
À1
, regardless of thickness,
load, and frequency. Displacement amplitude varied from 25 mm to 50 mm. Filled circle, thickness
¼ 6.35 mm, f ¼ 10 Hz; filled diamond, thickness ¼ 3.00 mm, f ¼ 10 Hz; filled star, thickness ¼ 4.82
mm, f ¼ 10 Hz; unfilled circle, thickness ¼ 6.35 mm, f ¼ 6.5 Hz; unfilled triangle, thickness ¼ 4.82
mm, f ¼12.5 Hz; unfilled star, thickness ¼ 6.35 mm, f ¼ 6.5 Hz; unfilled diamond, thickness ¼ 6.35
mm, f ¼ 12.5 Hz. After Naderi et al. (2010)
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4 Unified Mechanics Theory
figure. If the system is maintained in a nonequilibrium state by supplying energy to
it, two processes are usually considered. Near equilibrium, the system will evolve
from its initial state A to a state B of minimum rate of entropy generation. If the
system is far from equilibrium, it may move from its initial state C to a state D, but at
present, no description of this state or the process in terms of entropy production is
available. The only accepted governing principle is energy conservation.
Whaley (1983) proposed a model for fatigue crack nucleation using the irreversible thermodynamics to quantify the damage caused by plastic strain. Whaley (1983)
model is based on the hypothesis that entropy gain which results from dynamic
irreversible plastic strain is a material constant. This was later proven experimentally
by Naderi et al. (2010) and, Imani and Modarres (2015). Whales (1983) postulated
that structural damage by fatigue could be quantified by a single material parameter,
the critical entropy threshold of fracture. Whaley (1983) postulated that the critical
entropy threshold of fracture is therefore related to the irreversible part of the fracture
energy. The critical entropy threshold of static fracture is a random variable and the
variability can be quantified by a confidence interval. According Whaley (1983), the
confidence interval for the entropy threshold just comes from the variance of the
plastic strain. It is noteworthy that Whaley (1983) assumed that only plastic strain
generates irreversible entropy, of course that is not accurate.
Naderi et al. (2010) were able to prove experimentally that total cumulative
entropy generation is constant at the time of failure and is independent of sample
geometry, load, and loading frequency. They named this critical entropy value
Fatigue Fracture Entropy (FFE). Figure 4.2 shows experimental fatigue fracture
Fig. 4.2 Fatigue fracture entropy versus the number of cycles to failure for different bending
fatigue tests of Al 6061-T6 with different specimen thickness, frequencies, and displacement
amplitudes. Fatigue fracture entropy remains at roughly 4 MJ m
À3 K
À1
, regardless of thickness,
load, and frequency. Displacement amplitude varied from 25 mm to 50 mm. Filled circle, thickness
¼ 6.35 mm, f ¼ 10 Hz; filled diamond, thickness ¼ 3.00 mm, f ¼ 10 Hz; filled star, thickness ¼ 4.82
mm, f ¼ 10 Hz; unfilled circle, thickness ¼ 6.35 mm, f ¼ 6.5 Hz; unfilled triangle, thickness ¼ 4.82
mm, f ¼12.5 Hz; unfilled star, thickness ¼ 6.35 mm, f ¼ 6.5 Hz; unfilled diamond, thickness ¼ 6.35
mm, f ¼ 12.5 Hz. After Naderi et al. (2010)
128
4 Unified Mechanics Theory