6. Creep and fatigue failure‡：このマークが付してある著作物は、第三者が有する著作物ですので、同著作物の再使用、同著作物の二次的著作物の創作等については、著作権者より直接使用許諾を得る必要があります。

6.1 Creep as a phenomena

CreepStrain increases under a constant stress
Tertiary creep

Elapsed time

Secondary or
steady creep
Primary creep

Axial stress
Axial strain

Creep strain

Elapsed time

TimeTime

Instantaneous strain

Why creep?
Failure of rock mass around tunnel
Failure of rock slopewith trigger
pore pressure, thermal stress, freeze thaw cycle, weathering, etc.

without trigger
creep

Constant strain rate

Axial stress
CreepCircumferential strain Axial strain

Creep stress and time to failure

Creep limit?
Creep stresslog (life expectancy)

Q: Time to failure of a rock is 3.6 × 103 s and 3.6 ×
104 s under 100 MPa and 80 MPa of creep stress,
respectively. Estimate time to failure under 85 MPaof creep stress.
A： 20230 s

Strain rate in secondary creep

ε& = Aσ nε&: strain rateAσ: constant: creep stressn：constant (20～50)

6.2 Mechanical properties of bodies

6.2.1 Elasticity, viscosity and plasticity

Elasticity
Elastic body deforms when subjected to force
Deformation vanishes when force is removed
Example of elastic bodies
Springs, rubber

Linear elastic
Nonlinear elastic

Force

Deformation

TimeDeformation

TimeForceLinear elastic body
Spring isused as anelement bodyε=σE

Linear elastic
Nonlinear elastic

Force

Deformation

Time

Deformation

TimeForceViscosity
Deformation continues under constant force
Deformation does not vanish when force is removed
Oil, candy etc.

Force

Deformation
rate

Deformation

Time

Newtonviscosity

TimeForceNewtonian bodyσ = ηε&σηε&

Strain rateτμγ&

Shear stress
Viscosity
Rate of shear

Normal stressNormal viscosityτ = μγ&An element body for Newtonian body
Dashpot is used

Newtonviscosity

Force

Deformation
rate

Deformation

TimeTime

Force

Plasticity
Yield and deformation occurs at a stressDeformation does not vanish when stres is removed
Clay etc.

A body which does not deform until yield point
Slider is used for an element body

A body which shows elastic deformation until yieldpoint
Perfectly plastic body

Perfectly
plastic body

Force
Yield point

Yield point
Force

Yield
strength

Deformation

TimeTimeSlider

Deformation

Slider?

Force
Yield point

Yield strengthm

Deformation

Yield strength = Maximum value of frictional force = μmg
where μ is the friction coefficient6.2.2 Mechanical modelMaxwell body

One of the models which shows viscoelasticity
Stress relaxation occurs when stress is kept
constant
Strain increases under constant stress

Sticky candy etc.
StressStrainTimeTime

dε sdσ=Edt
dt
Ttal strain ε isε = εs + εd

σ = Eε s

Thereforedε dε s dε d=+dt
dt
dt
1 dσ σ=+E dt η（Maxwell equation）

σ = ηε&d

Constant strain
Assume that strain is kept constant ind ε 1 dσ σ=+dt E dt ηThen,
1 dσ σ+0=E dt ησ1 dσ=−ηE dt
1 dσ
1= − dt
ηE σtln σ= − + C1
ηE⎛ Et⎞⎛ Et ⎞+ C1 ⎟⎟ = exp(C1 ) exp⎜⎜ − ⎟⎟⎝ η⎠⎝ η ⎠Assume that σ = σ 0 at t = 0,σ = exp⎜⎜ −⎛ Et ⎞⎟⎟η⎝⎠σ = σ 0 exp⎜⎜ −σ0StressTime

Stress relaxation

Constant stress
Assume constant stress indε 1 dσ σ=+dt E dt ηThen,
Straindε σ=dt ηε=Since ε =σEε=σt + C1ηat t = 0,σE+σtηTime

Viscous deformation

Voigt body (Kelvin body)
One of the models for visco-elasticbody
Strain increases with decreasing strain rate
under constant stress
Strain decreases with decreasing strain rate
when stress is removed
Stress removal

Nylon etc.
StrainTime

σ s = EεTotal stress σ isσ = σs +σdTherefore,dεdt（Voigt or Kelvin equation）

σ = Eε + η

σ d = ηε&In the case where stress is kept constant
between t = 0 and t ＝ t１
σ = Eε + η⎛t⎞
− Eε = exp C 2 exp⎜⎜ − E ⎟⎟ − ση⎠⎝dεdtdε= − Eε + σdtdε1= dt− Eε + σ ηln( − Eε + σ ) t−= + C1Eηtln( − Eε + σ ) = − E + C 2η⎛t⎞ σε = C 3 exp⎜⎜ − E ⎟⎟ +η⎠ E⎝Assuming that ε = 0 when t = 0,⎛t⎞ σexp⎜⎜ − E ⎟⎟ +Eη⎠ E⎝⎛σ⎧t ⎞⎫ε = ⎨1 − exp⎜⎜ − E ⎟⎟ ⎬E⎩η ⎠⎭⎝ε =−η⎛t⎞− Eε + σ = exp C 2 exp⎜⎜ − E ⎟⎟η⎠⎝retarded elasticityσStress removal

StrainTimeThen, unloaded
Substituting σ = 0, t = t1 and strain at t = t1⎛t⎞ σε = C 3 exp⎜⎜ − E ⎟⎟ +η⎠ E⎝Hence,σ⎧t1 ⎞ ⎫⎛⎨1 − exp⎜⎜ − E ⎟⎟ ⎬Eη ⎠⎭⎛t⎞⎝ε= ⎩exp⎜⎜ − E ⎟⎟η⎠t ⎞⎛⎝exp⎜⎜ − E 1 ⎟⎟η⎠⎝t ⎞⎫t − t1 ⎞⎛⎛σ⎧⎟⎟ε = ⎨1 − exp⎜⎜ − E 1 ⎟⎟ ⎬ exp⎜⎜ − EE⎩ηη⎠⎝⎠⎭⎝

into

ε=t1 ⎞ ⎫⎛1exp
E⎟⎟ ⎬⎜−−⎨⎜ηE⎩⎠⎭⎝σ⎧we gett1 ⎞ ⎫t ⎞⎛⎛1exp
E⎟⎟ ⎬ = C 3 exp⎜⎜ − E 1 ⎟⎟⎜−−⎨⎜η ⎠⎭η⎠E⎩⎝⎝σ⎧
Therforet1 ⎞ ⎫⎛
1expE⎟⎟ ⎬−−⎜⎨⎜ηE⎩⎠⎭⎝C3 =t ⎞⎛
exp⎜⎜ − E 1 ⎟⎟η⎠⎝retarded elasticityσ⎧Stress removal

StrainTime

St. Venant body
Elasto-plastic body
Elastic deformation until
yield point
Plastic deformation after
yield point
Tooth paste, mild steel etc.

Yield point

StressStrain

Bingham body
A model for Visco-elasto-plasticbody
Elastic deformation until yield point
Viscous deformation after yield point

StressStrain

TimeTime

Mechanical model for creep deformation ofrock
Burgers model forprimary and steadycreep
Tertiary creep can
not be easily
represented.
Strain

S1

S2

D2

D2

D1
S1

S2TimeD1

6.3 Example of creep test

Creep behavior in circumferential strain

Damage of rock

The same strength can be attained when the specimen is reloaded
before minimum circumferential strain rate is attained.

Peak stress is less than that in constant strain rate test when the specimen
is reloaded after minimum circumferential strain rate is attained.

6.4 Mechanism of creep

Main causes of creep
Change in distance between mineral particles
Migration of pore water, shear displacement, detachment etc.

Deformation of mineral particles
Pressure solution, diffusion, dislocation etc.

Subcritical crack growth
Stress corrosion etc.

Migration of pore water
Finite velocity in migration of pore water due to pressure
gradient by specimen deformation.
Hydraulic conductivity
Viscosity
Temperature

The primary creep can be explained by homogenization ofpore pressure.
Effect of temperature can be explained.

Pressure solution
Quartz etc. can be dissolved in pore water by high stressconcentration at contact points.
The solution migrates by grain boundary diffusion etc.
Quartz etc. are precipitated at low stress-region.
Primary creep can be explained by the increase of contacted area.
Effects of temperature, pH etc. can be explained?

Diffusion creep

Homogenization of density
Voids: from a grain boundary under tension to that undercompression
Atoms: from a grain boundary under compression to that undertension

Let L be the grain size. The density gradient isσ,

LIn the diffusion inside the grain, displacement rate is
proportional to density gradient, Cσ,

Lstrain rate is, dividing the disdplacement rate by the grain size,

ε& = CσL2
In the grain boundary diffusion, cross sectional area is WL
WL
where W is the thickness of grain boundary. The area is 2 times
Lthe grain boundary diffusion case. Therefore,CσWε& = 3L

Strain rate in diffusion creep
Inversely proportional to square or cube of the grain size
Proportional to creep stress

Diffusion creep can be observed for metals and ceramics
under low creep stresses.
Existence of diffusion creep in minerals and rocks has notbeen proven.

Dislocation creep

Edge dislocation

制限資料
飯田修一（1965）、技術者のための固体物性、第2版、丸善、p. 228図11-2

制限資料
飯田修一（1965）、技術者のための固体物性、第2版、丸善、p. 228図11-3

Screw dislocation

制限資料
飯田修一（1965）、技術者のための固体物性、第2版、丸善、p. 229図11-4

Strain rate by dislocation is proportional to density and velocityof dislocation.
ε& = C ' ρ v
Density and valocity of dislocation is proportional tosquare of stress and stress, respectively.

ε& = C ' ' σ 3Strain rate by dislocation creep is proportional to cube ofstress.
Creep strain rate of minerals and rocks under hightemperature sometimes proportional to the cube of stress.
Existence of dislocation has not proven

Stress corrosion
Corrosion of elongated and weakened atomic bonds bysurround substances.
Substances which cause corrosion
H2O for SiO2
CH3CN for MgF2

O
SiOOO

Si+H2OOOSiOOO

HHOSiOCrack velocity and stress intensity factor
Region 1
Stress corrosion rate increases
with stress intensity
factor→crack velocity increases

Region 2

Region 3Region 1Region 2
Crack velocity is almostconstant since diffusion
velocity of the corrosionmaterial determines thecorrosion rate.
This region would not beobserved clearly forinhomogeneous materials.

Region 3
Failure due to mechanicalelongation of atomic distance.

log V
Fracture toughnesslog KIUpper limit of crack velocity
0.38 times P-wave velocity
due to the limit of energy
supply rate.
The upper limit of velocity
of fracture propagation infault earthquake is velocity
of Reyleigh wave (0.88～0.96 times S-wavevelocity)
Difference in new fracture
and propagation of slip
along already fractured fault.

Region 2

Region 3Region 1

log V
Fracture toughnesslog KI

Activation energy
Activation energy

Potential

Heat of reaction

Reactant

Activated complex

ProductReaction coordinate

Activation energy has to be exceeded for a reaction
The higher the energy, the higher the probability to exceed the activationenergyIncrease in reaction rate - increase of crack velocity

Number of reaction molecules and the density of corrosion material have apositive correlation in a certain density range

How to increase energy?
Raising temperature(enthalpy)
Elongation(potential)

Atomic distance

Potential

Aomic force⎛ − E act + αK I ⎞V = V0 a ( H 2 O) exp⎜
⎟RT⎝⎠

α=2V

*π rc

Crack velocity inRegion 1V0 (m/s) is a constant depending material and conditionsa(H2O) is the activity of water that is nearly equal to the relative
humidity
Eact (J/mol) is the activation energy per 1 mol
KI is the stress intensity factor for Mode I (Pa⋅m0.5)
R is the gas constant (8.3144 J K-1 mol-1)
T is the absolute temperature (K)V* is the volume of activation
rc is the radius of curvature at crack tipValue in the parenthesis is below zero.
V0 is the value of V when the value in the parenthesis is zero, namely,
infinite high temperature or αKI = Eact
Crack velocity decreases with decrease od temparature from infinite
high temperature or decrease of stress intensity from Eact/α

Q: Crack velocity of a rock under a small stress is 1 mm/s
and 10 mm/s at 30°C and 200°C, respectively. Estimate
crack velocity at 0°C.
A: 0.497 mm/s

Explanation of primary creep based on stresscorrosion

σ1KI =σ1FπlαThe longer the crackgrows,
The smaller the stressintensity factor becomes
Crack velocity decreases
Strain rate decreases≅FK I = σ πaσ

When the crack grows
under tensile stress, stressintensity increases.
Primary creep shouldn't
appear but was observed.

2aExplanation of tertiary creep based on stresscorrosion
When crack density exceeds
a certain limit, stressintensity factor increases
due to interferences between
the cracks
Crack velocity increases.
Strain rate increases.

6.5 Fatigue failureFatigue failure?

Fatigue failures ofmetallic parts is causes ofmany accidents of
airplanes, nuclear power
plants.

StressFailure due to cyclicloadingTime

S-N curve
SN:stress amplitude:number of cycles
Fatigue limit
Slog NCharacteristics of fatigue crack growthda/dN :increment of
crack length in aloading cycle

ΔKth:lower

limit ofstress intensity factor
amplitude

Kmax = KIC
log da/dN

ΔKth

Crack never grow understress intensity factor
below the limit.
log ΔK

Mechanism of fatigue

Loading process

Unloading process

Characteristics of fatigue crack制限資料

制限資料

Striation pattern

PLAMOS web-site内「http://www.plamos.com/html/support/s_10_03_04.html」に掲載してある電顕写真Fatigue crack grows in plastic regions which were createdby the crack itself.
There are many oxidized or corroded materials at crack
surfaces.
They prevent crack growth under very small stressintensity amplitude

Fatigue crack of rock
Time to failure
becomes
maximum at acertain stressamplitude.

1
Creep

σmin / σmax

0log (life expectancy)