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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. dSmicro < 0.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. dSmeso > 0.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).? ) = ??

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.
2.1 Entropy balance at multiple length scales
Let us confine our discussion to length scale that
spans from micro to macro. For the sake of clarity of discussion, submicron scales (e.g., nano and
atomic scales) are not discussed in this work.
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 form of entropy
balance is commonly used in continuum mechanics
with the requirement from the second law of thermodynamics as d(i)
S > 0 for an irreversible process.
Here, we focus our attention to the irreversible
part, (i)
S and continue our discussion around this
term. For the sake of simplicity let us drop the subscript (i) in the entropy generation term and refer
to it by S hereafter.
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.
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. 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 indicated that for an
irreversible process, M ≠ 0 and G > 0 with randomly
occurring negative entropy generation. Figure 1
shows a schematic of the entropy generation for an
irreversible process as a submartingale.
It is shown that at continuum level the entropy
increases monotonically to agree with the second
law of thermodynamics, however at microscale it
is possible that entropy generation violated the
Figure 1. Evolution of entropy generation as a
submartingale.
second law as defined in continuum mechanics.
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. The relevance of the above discussion to
the study of damage processes and in particular
fatigue damage is discussed next.
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.
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. Therefore, if each and every irreversible process is identified in a degrading system,
the rate of degradation can be determined via the
rate of entropy generation. This theorem is however developed within the context of continuum
mechanics. An immediate conclusion from this
theorem is that:
Entropy generation ≡ Damage
This means that entropy generation can be used
as damage metric. 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.
4103
However, an interesting question that arises is
whether this hypothesis is applicable when dealing
with damage processes at small spatial scales. 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. Therefore, it is of interest to investigate
the evolutionary behavior of entropy generation at
very small scales and its correspondence to damage
evolution. For this purpose we merely discuss one
damage process, i.e., fatigue damage.
3.2 Fatigue damage
Fatigue damage is accompanied by transformation which is governed by the laws of thermodynamics. The irreversible nature of fatigue damage
may be attributed to dislocations. 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.
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.
Plastic deformation is a typically irreversible process which can be described by non-equilibrium
thermodynamics. Stewart et al. (2006) employed
non-equilibrium thermodynamics theory to predict
the initiation of dynamic recrystallisation (DRX)
for nickel and steels. They found that the critical
conditions for DRX are reaching certain critical
values of stored energy and entropy generation rate
simultaneously. Employing non-equilibrium thermodynamics, we may develop the necessary formulations for evaluations of entropy generation during
fatigue damage. 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.
During a strain interval dε, there are dρ dislocations generated at the subgrain boundaries, and the
same amount of dislocations is annihilated at such
boundaries in the meantime. Therefore, dρ can be
expressed in terms of a generation process and an
annihilation process. The rate of entropy generation due to dislocations activities can be expressed
as (Huang et al. 2009):
dS Gb d
T = +

• 
  
   
  
   1
  2
  β 2 α ρ (3)
  where G = the shear modulus; T = temperature;
  b = magnitude of the Burgers vector; and α and
  β = constants. Therefore, evaluation of entropy
  generation in fatigue process comes down to evaluation of dislocation accumulation.
  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. To elaborate on these fundamental questions, let us first discuss the physics
  of fatigue damage accumulation and fatigue crack
  initiation.
  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.
  Considering the stored energy variations is of
  prime interest since they correspond to internal
  energy variations induced by stain localization.
  Generally speaking, these variations and those
  of thermoelastic sources are directly linked to
  the thermodynamic potential chosen to describe
  the material state. Considering Gibbs free energy
  change (∆G) from a state of dislocation pile up
  to that of virtual crack nucleation of size a, one
  can write the energy balance in the following form
  (Tanaka & Mura 1981):
  ∆G W A = − e i − + NWs s A
  2 γ (4)
  where We and Ws
  respectively represent the elastic energy release and cyclic internal stored energy.
  Parameter A is the area of virtual crack (πa2
  ) and
  γs
  is the surface energy of the material. 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
  ∆ = 0 (5)
  from which the number of cycles to fatigue crack
  initiation Ni
  is determined (details of formulations
  are given in Tanaka & Mura 1986). 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 δ S < (6)
  Note that δ2
  S denotes the second variation of
  S. As the information within the system becomes
  too complex, the system is no longer able to remain
  organized and becomes unstable. This instability
  within the system stimulates a bifurcation event
  that leads to new states of order because both
  descendant branches have less entropy than the
  ancestral branch (Fig. 2). 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). States A and B are referred to as initial
  damage free and crack initiation states, respectively.
  Evolution of the non-equilibrium fatigue process
  between states A and B is controlled by the evolution of control parameter x. In this investigation,
  the control parameter x is defined as the pile up of
  dislocations within a grain. 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. The thermodynamic state of the crack initiation is associated
  with the minimization of Gibbs free energy (or in
  our investigation), beginning of instability of the
  sub-system (Fig. 2). It can be shown that:
  1
  2
  d 2
  dt
  S X J
  k
  k k (δ δ ) = ∑ δ (7)
  where δXk and δJk are deviations of thermodynamic forces and fluxes from stationary state. This
  equation is called the excess entropy generation.
  If the excess entropy production is non-negative,
  the given state of the system is stable; otherwise,
  the system loses its stability. Through the application of the excess entropy production a thermodynamic system in general, and a fatigue system in
  particular, can lose stability and initiate a crack.
  Using entropy generation in Eq. (3) with instability
  condition given in Eq. (7), a necessary condition
  for crack initiation based on the control parameter
  x can be developed. In this sense, the entropy is a
  measure of stability of the system and more fundamental than energy.
  We also articulate that entropy generation in
  fatigue damage is a submartingale meaning that
  at very small temporal and spatial scales it is possible that entropy generation becomes negative.
  This statement requires further attention. It is to
  be noted that the definition of damage varies at
  different geometric scales as the observable field
  variables describing the damage change. 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. Therefore,
  the entropy generation specifically pertains to slip
  accumulations and movement of dislocations.
  The in situ neutron diffraction study by Huang
  et al. (2010) states that dislocation self-organization
  arises possibly during the formation of a microcrack. 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. As stated earlier, if
  a nonequilibrium stationary system loses its stability, the excess entropy should satisfy the inequality
  in (7). Here, the nonequilibrium stationary system
  refers to the dislocations reaching a stable position.
  Therefore, inequality in (7) provides a necessary
  condition for a fatigue system to form patterned
  planar-dislocation-wall structure or microcracks.
  If, however, self-organization occurs the entropy
  generation is minimal in a stationary state. This
  decrease in entropy generation constitutes an
  apparent paradox, because it seems to violate the
  second law of thermodynamics. However, as discussed earlier, in analysis of a nonequilibrium system, entropy generation on multiple length scales
  should be considered.
  For example, in high cycle fatigue where the
  Figure 2. Representation of entropy bifurcation. stress level is very low and bulk material deforms
  4105
  elastically, no change of entropy is expected, i.e.
  dSmacro = 0. 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. dSmicro < 0.
  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. dSmeso > 0.
  Therefore the total entropy generation should be
  written in the form of Eq. (2). An accurate evaluation of entropy generation at multiple length scales
  might therefore shed more light on the physics of
  fatigue crack initiation.