8 Damage Mechanics

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116 Introduction to Thermodynamics of Mechanical Fatigue

where is the fraction of life required for crack initiation. Having known the value of for different stress levels, the bi-linear model provides good agreement with experimentalresults (Manson and Halford 1981). In fact, at low stress levels most of the life is expended

for crack initiation, while at high stresses major fraction of life is spent for crack propagation(Stephens et al. 2000). This way, the bi-linear damage model provides a satisfactory predic-tion for low-to-high or high-to-low loading conditions. Figure 7.1schematically shows thebi-linear damage model compared with linear Miners rule.

Nonlinear damage models have also been developed to improve the Miners rule andaccount for load sequence. For example, Marco and Starkey (1954) proposed a nonlineardamage model in the form of

D n

N

i

i

i

=

(7.3)

where iis the power exponent and varies as the stress level changes. Figure 7.1shows aschematic of a nonlinear damage model along with the linear and bi-linear models. Notethat efforts made to improve the Miners rule have been only partially successful andno model can claim to work perfectly in a complex variableamplitude loading. In fact,Stephens et al. (2000), stated, Consequently, the PalmgrenMiner linear damage rule isstill dominantly used in fatigue analysis or design in spite of its many shortcomings. Acomprehensive review of available damage models is given by Fatemi and Yang (1998).

7.1.1 ENTROPY-BASEDDAMAGEVARIABLE

The damage models reviewed in the previous section rely on counting the number ofcycles, ni, or more precisely the fraction of life, ni/Ni. This provides an extremely easytechnique for assessing the cumulative fatigue damage in practice since counting the

n/N

1.00.80.60.40.20.0

D

0.0

0.2

0.4

0.6

0.8

1.0

Line

armod

el

Bi-line

armod

el

Nonlin

earm

odel

FIGURE 7.1 Damage variable versus life fraction.

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117Damage Mechanics

cycles can be readily performed by using a counter. However, note that simply countingthe cycles cannot account for parameters that significantly influence the damage accu-mulation such as alteration of the state of stress (axial, biaxial or multi-axial), variablefrequency, and changes in environmental conditions (temperature, humidity, etc.). To

overcome these shortcomings, energy-based damage models provide promising resultsby bringing into play the role of, for example, the hysteresis energy dissipation, w. Asimple form of energy-based damage model can be defined by tallying up the energy dis-sipation per cycle as (Kliman 1984):

=

D w

Wf (7.4)

where Wf is the energy at fracture. Utilizing the energy-based damage model, one can

accumulate the hysteresis energy dissipated per cycle in each loading sequence. This way,the effect of different stress states, frequency, and so forth, is naturally taken into account.

In the sense of degradation and damage accumulation, the concept of tallying entropyis more fundamental than the energy dissipation. Recently, attempts have been made tolink the damage variable,D, to entropy accumulation (Amiri, Naderi, and Khonsari 2011;Naderi and Khonsari 2010a, 2010b). Both entropy flow and entropy generation are provento be good candidates for evaluation of damage.

Amiri, Naderi, and Khonsari (2011) reported a series of bending fatigue tests to evalu-ate damage parameters in which the entropy exchange with the surroundings, deS, was

employed. As discussed inChapter 2, the rate of entropy exchange with the surroundingscan be evaluated from Equation (2.25). The accumulated entropy flow was computed byintegrating Equation (2.25) over cycles (time) as

= =

S

d S

dtdt

hA T T

Tdt

( )e

e

t t

0

0

0

(7.5)

Note that if the integration is performed from the beginning, t = 0, to the failure, t = tf, then

the entropy evaluated from Equation (7.5)represents the total entropy flow at failure, Sf.Figure 7.2shows the normalized number of cycles as a function of normalized entropy

exchange for different displacement amplitudes of bending fatigue test. Results areadapted from Amiri, Naderi, and Khonsari (2011). The material used for the specimen isAluminum 6061-T6 and the frequency of tests is 10 Hz. The normalized number of cyclesis defined as the ratio of the number of cycles to the number of cycles to failure, N/Nf,and the normalized entropy exchange is defined as the ratio of accumulation of entropyexchange to the entropy exchange at the failure, Se/Sf. Results show a linear relationshipbetween normalized number of cycles and normalized accumulation of entropy exchange,

expressed as

=

S

S

N

N

e

f f

(7.6)

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Taking advantage of the above linear relationship, a new damage equation is definedbased on the entropy exchange as follows (Amiri et al. 2011):

( )

=

D

S

S

S

1

lnln 1

f

e

f (7.7)

Figure 7.3illustrates the damage evolution for Stainless Steel 304 at displacement ampli-tude of = 48.26 mm. Shown in this figure also are the results of the work of Duyi andZhenlin (2001) for comparison. Duyi and Zhenlin employed the exhaustion of static frac-ture toughness to arrive at the following damage model:

= D

EU

NN

N

1

ln( )ln 1a T

f f

2

0

(7.8)

where UT0is the static toughness of undamaged material,Eis the elastic modulus, and aisthe stress amplitude.Figure 7.3shows the nonlinear increase of the damage variable. Closeto final failure, the damage variable increases drastically, representing the critical damage,

Dc. This concept can be utilized as an indication of imminent fracture as discussed next.Figure 7.4shows the evolution of the damage parameter for Aluminum 6061-T6 under-

going a bending fatigue test at three different displacement amplitudes, . Amiri, Naderi,and Khonsari (2011) defined the critical condition in fatigue as a sudden increase in damageparameter which, in turn, is a consequence of a sharp increase in entropy exchange. Thiscondition is shown in Figure 7.4by a dashed line. Based on the entropy approach, theydetermined that the critical damage parameter for Aluminum 6061-T6 and Stainless Steel304 are approximately 0.3 and 0.2, respectively. Critical damage is a material property andis independent of the loading amplitude.

Normalized Entropy Flow, Se/Sf

1.00.80.60.40.20.0

Norma

lize

dNum

bero

fCycles

,N/Nf

0.0

0.2

0.4

0.6

0.8

1.0

= 38.1 mm

= 45.72

= 41.91

= 44.45

= 48.26

= 49.53

FIGURE 7.2 Normalized number of cycles against normalized entropy exchange. (Reproducedfrom Amiri, M., Naderi, M., and Khonsari, M.M.,Int. J. Damage Mech.,20, 89112, 2011.)



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119Damage Mechanics

Similarly, Naderi and Khonsari (2010a, 2010b) used the concept of entropy generationdiS, instead, to derive a damage evolution model. Recall Equation (5.57) for entropy genera-tion during the course of fatigue. The accumulated entropy generation can be evaluatedby integrating Equation (5.57) over cycles (time) as

= =

Jdt

T

T

Tdt

:q

t

p

t

0

2

0

(7.9)

1000050000

0.2

0.4

0.6

0.8

1.0

Number of Cycles

DamageVariab

le,

D

Entropic approach (Amiri et al., 2011)

Eq. (8.8), Duyi and Zhenlin (2001)

FIGURE 7.3 Evolution ofDfor Stainless Steel 304 at displacement amplitude of = 48.26 mm.(Reproduced from Amiri, M., Naderi, M., and Khonsari, M.M.,Int. J. Damage Mech.,20, 89112,2011.)

Number of Cycles

103102 104

DamageVaria

ble,

D

0.2

0.0

0.4

0.6

0.8

1.0

= 49.53 mm

= 44.45

= 38.1

Critical Damage

FIGURE 7.4 Critical damagefor Aluminum 6061-T6. (Reproduced from Amiri, M., Naderi, M.,and Khonsari, M.M.,Int. J. Damage Mech., 20, 89112, 2011.)



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Note that if the integration is performed from the beginning, t = 0, to the failure, t = tf,entropy generation evaluated from Equation (7.9)represents the total entropy generation atfailure, f. Naderi and Khonsari used this definition to arrive at the following equation forthe damage variable:

( )=

D

D

ln 1ln 1

c

c f f (7.10)

whereDcdenotes the critical value of damage and crepresents the entropy accumulationup to the critical condition.Figure 7.5shows the evolution of the damage variable for twodifferent loading sequences for Aluminum 6061-T6. Results are adapted from Naderi andKhonsari (2010a). Each sequence includes three loading stages. Plot (a) corresponds to thesequence from high to intermediate to low loads, while plot (b) is in reversed order. This

figure clearly illustrates the capability of entropy generation in addressing the effect of loadsequence on the damage variable. In fact, in test (a) where the load level increases, the dam-age induced in the sample is more pronounced than test (b). This is due to the fact that byapplying higher load first, followed by lower load, greater damages in the form of macro-cracks is induced in the sample, which results in a higher value ofD. This is in accordancewith the concept of CDM, which is described next.

7.2 CONTINUUM DAMAGE MECHANICS (CDM)

Continuum damage mechanics is a relatively new branch of solid mechanics, which deals withanalysis and characterization of a materials defect at micro- to mesoscale. The early studies dateback to the works of Kachanov (1958) and major contributions were made later by Krajcinovic(1984), Lemaitre (1985), Kachanov (1986), and Chaboche (1988). Since then, CDM has beengrowing rapidly. Lemaitre (2002) presented the main landmarks of the development of the

Number of Cycles

0 7000600050004000300020001000

DamageVaria

ble,

D

0.0

0.2

0.4

0.6

0.8

1.0

H-I

I-L

L-I I-H

H-I : High to IntermediateI-L : Intermediate to LowL-I : Low to Intermediate

I-H : Intermediate to High(a)

(b)

FIGURE 7.5 Evolution of Dfor variable-amplitude loading. (Reproduced from Naderi, M. andKhonsari, M.M.,J. Mater. Sci. Eng., A527, 61336139, 2010a.)



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121Damage Mechanics

science of CDM from 1958 to 2000. In fact, the development of CDM-based damage modelsis rooted in irreversible thermodynamics analysis by taking into account the evolution of statevariables, state potential, and dissipation potential (Lemaitre and Desmorat 2005). In whatfollows, we illustrate the application of the CDM model proposed by Bhattacharya (1997) and

Bhattacharya and Ellingwood (1998, 1999) to predict crack initiation in fretting-fatigue androlling-fatigue. The advantages of this approach are that this model is obtained from the lawsof thermodynamics and that it uses the bulk material properties to predict the crack initiation.This model avoids the use of empirical relations that exist in the literature on fretting- androlling-fatigue. Before delving into formulations, let us introduce basic definitions essential inCDM modeling.

7.2.1 DAMAGEVARIABLE, D(n)

In the CDM context, the damage variable,D

(n

), is generally described by a scalar, secondorder, and fourth order tensor on the elemental areaA0, with normal vector n. It manifestsitself in the gradual loss of effective cross-sectional area(seeFigure 7.6)and is defined as

nD

A A

A( )

0

0

=

(7.11)

For a pristine material with effective cross-sectional area = A0, Equation (7.11)yieldsD(n) = 0. For the isotropic damage, wherein the damage variable is independent of thenormal vector,D(n) reduces to a scalar denoted by D. To simplify the formulations, let

us assume that the damage variable is scalar. Similar to the definition for effective cross-sectional area, the notion of effective stress can be defined based onD

=

D1 (7.12)

Bhattacharya (1997) used the concept of effective stress along with the RambergOsgoodlaw to derive the relationship between stress and strain ,

= + E K

M

(7.13)

Defects

Elemental area, A0

Normal vector, n

Specimen

A0

FIGURE 7.6 Definition of damage variable on elemental areaA0.



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whereEand Kare elastic modulus and strain hardening modulus of pristine material, respec-tively, andMrepresents the hardening exponent. Note that the first term on the right-handside of Equation (7.13)represents the elastic strain, e, and the second term plastic strain, p.According toEquation (7.13)the following relationship between eand pcan be derived

=

K

Ee p

M

1

(7.14)

From Equation (7.12)throughEquation (7.14), the effective stress can be written as

= K D(1 ) p

M

1

(7.15)

Similar to the Equation (5.17), Bhattacharya (1997) defined the Helmholtz free energy as afunction of the damage variable,= (D), inwhichDis, in turn, a function of strain,D =

D(). He then arrived at the following equation for effective stress:

=

=

d

d D

dD

d (7.16)

SubstitutingEquation (7.15) into Equation (7.16)and rearranging the results, we obtain

=

dD

d

K D

D

(1 )

/

pM1

(7.17)

For the case of damage in uniaxial monotonic loading, Bhattacharya (1997) derived thefollowing equation for /D:

=

+

+ +

D

K

E

K

M2 1 1

3

4pM M

pM M

f

2 2

0

21

1

0

1 1

(7.18)

where 0 is the threshold plastic strain and f is the fracture strength. Substitution ofEquation (7.18) into Equation (7.17) yields

=

+

+

+

+ +

dD

D

K d

K

E

K

M

1

2 1 1/

3

4

pM

pM M

pM M

f

1

2 2

0

21

1

0

1 1

(7.19)

The closed-form solution of the differential equation inEquation (7.19)can be obtainedfor uniaxial cyclic loading withDito be the damage variable at the ith number of the cycle.Without going through the mathematical derivations, let us present the final form of the

equation for calculation of damage variable (Bhattacharya 1997):

D

D F S

D S

1 1 ;

;i

i i e

i e

1 max

1 max

( )=

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