A better understanding of the physics of entropy
generation at multiple length scales (micro--to
macroscale) during fatigue damage may provide
more insight into the stochastic nature of fatigue
failure.An accurate evaluation of entropy generation at multiple length scales
might therefore shed more light on the physics of
fatigue crack initiation.Since entropy generation is an additive function, the total entropy generation can be presented
as the sum of entropy changes associated with the
material and process at corresponding scale levels:
dStot m = + dS icro dSmeso + dSmacro (2)
where Stot = total entropy generation of the system
under study; Smicro = entropy generation at microscale; Smeso = entropy generation at mesoscale; and
Smacro = entropy at macroscale.As a consequence of
the ongoing cyclic slip process, the Persistent Slip
Bands (PSBs) evolve and interact with high-angle
Grain Boundaries (GBs), the result of which leads
to dislocation pile-ups, static extrusions in the
form of ledges/steps at the GB, stress concentration, and ultimately crack initiation.We present entropy generation of dislocation
activities to find answers for some fundamental questions: whether fatigue damage is a submartingale or strictly increasing function with
time; whether entropy, instead of its counterpart,
energy function, provides additional insights into
our understanding of fatigue damage evolution;
and whether entropy can be used in a fundamental
way to predict the evolution of damage enabling
fatigue life predictions.Regarding
fatigue, damage may be defined as a reduction in the
Young's modulus, as a cumulative number of cycles
ratio, as a reduction of load-carrying capacity, as
crack length, or as released strain energy (Amiri &
Modarres 2014.) In this paper, we consider fatigue
damage as slip irreversibilities that exist in a material
and accumulate during fatigue loading.3 Entropy as damage metric
3.1 Various forms of degradation
The Degradation-Entropy Generation (DEG)
theorem developed by Bryant et al. (2008) states
that the rate of degradation is related to the irreversible entropy generations by the underlying dissipative physical processes that degrade materials.That is, a structural transformation takes
place in the dislocation ensemble upon crack initiation such that randomly-distributed-dislocation
density reduces due to self-organization and the
formation of the patterned planar-dislocationwall structure or microcracks.This criterion known as
Lyapunov's theory of stability may provide a crack
initiation criterion based on the entropy of the
system by defining a control parameter x which
is changing from equilibrium state A to equilibrium state B. Control parameter x is defined in the
context of statistical thermodynamics by Jarzynski (2011).However, the formation of the patterned planar-dislocation-wall structures at microscale and submicron may be considered as the
configuration entropy that decreases and results in
increased orderliness of the system, i.e. dSmicro0.Ostoja-Starzewski & Malyarenko
(2014) use the Doob-Meyer decomposition which
says that under mild technical conditions any submartingale is the sum of a martingale (M) and an
increasing process (G).This concept has been stablised within the context
of statistical mechanics which indicated that that
the deterministic nature of non-decreasing dissipation function is the average manifestation of a
statistically fluctuating dissipation on very small
scales.As mentioned above, in calculation of entropy, it is of paramount importance to
consider behavior of entropy evolution at multiple
length scales, because entropy generation at small
scales is a stochastic phenomenon with possible
deviation from the second law of thermodynamics.Amiri & Modarres (2014) have
invoked this hypothesis and have shown that the
rate of damage in various processes such as fatigue,
wear, corrosion, radiation damage and creep can be
evaluated based on the rate of entropy generation.Stewart et al. (2006) employed
non-equilibrium thermodynamics theory to predict
the initiation of dynamic recrystallisation (DRX)
for nickel and steels.3.3 Fatigue crack initiation
As mentioned earlier, it is widely accepted that
strain localization due to increasing dislocation
density during cyclic forward and reverse loading is a precursor to fatigue crack initiation.Equation (2) suggests that even if bulk stress and
strain are in the elastic range, the vicinity of inherent micro defects deformation can be locally plastic.From definition
of Gibbs free energy and its relation to entropy of
the system G = H-TS, where H is enthalpy, T is
temperature and S is entropy, we will derive the
4104
necessary entropic condition for crack initiation
based on instability criterion of the system:
1
2
0 2 ?However, the decrease of entropy and increase of
orderliness at microscale is compensated by the
entropy increase at the mesoscale due to the formation and propagation of cracks, i.e. dSmeso0.Ni
 is the
fatigue nucleation life associated with virtual crack
length a. Mathematically, to maximize the Gibbs
free energy, one can write:
d G
da Ni
?The in situ neutron diffraction study by Huang
et al. (2010) states that dislocation self-organization
arises possibly during the formation of a microcrack.2.1 Entropy balance at multiple length scales
Let us confine our discussion to length scale that
spans from micro to macro.In an open system capable of exchanging heat
and matter with its surroundings the change of the
total entropy, dS consists of sum of two parts:
dS = d S ( ) e i +d S ( ) d S ( )i ; 0 (1)
where d(e)
S is the entropy exchange with the surroundings (or reversible entropy change) and d(i)
S is the entropy generation within the system (or
irreversible entropy change).This suggests that even if bulk material undergoes a reversible process, at lower scales it might
experience irreversibilities that are not observed at
continuum level.This theorem states that there is one-to-one correspondence between the rate of entropy generation and the rate of degradation relating to each
other through a scaling factor called degradation
coefficient.As
outlined in the previous section, the entropy generation is a submartingale meaning that at small
scales the entropy generation can become negative
that gives rise to behavior not seen under the restriction of the conventional second law of thermodynamics.As discussed by
Mughrabi (2009), the mechanisms of cyclic microplasticity, based on the glide of dislocations, are
responsible for the fatigue phenomena.In fact,
cyclic slip irreversibilities in a microstructural sense
occur not only at the surface but also in the bulk
at the dislocation scale which contribute to surface
fatigue damage.Employing non-equilibrium thermodynamics, we may develop the necessary formulations for evaluations of entropy generation during
fatigue damage.The rate of entropy generation due to dislocations activities can be expressed
as (Huang et al. 2009):
dS Gb d
T = +
+ ?As stated earlier, if
a nonequilibrium stationary system loses its stability, the excess entropy should satisfy the inequality
in (7).= constants.Therefore, d??????1
2
?2 ??2).2).? ) = ??