Maxwellian  models  oversimplified  viscoelas3c  responses  and  thus  give  approximate  predic3ons.  

FALL  2014:    EMCH  315     1  

Creep ε

εo

t

Real

Maxwell Model

To overcome limitations of Maxwell model alternative arrangements of elements have been proposed: e.g. ______________________ Model with_____________________ arrangement of spring and dashpot.

Most materials exhibit nonlinear creep rates

models steady-state creep (i.e.  constant  creep  rate  )  

Voight-Kelvin parallel

Voight-­‐Kelvin  Material  Model:  beLer  represents  ___________________  _______________________  creep.  

Ini3ally  the  dashpot  must  carry  the  en3re  force  because  the  spring  can  carry  a  force  only  when  extended.  The  force  in  the  V-­‐K  model  will  be  equal  to  the  force  in  the  dashpot________the  force  in  the  spring:  hence  ____________  {_______________________}    Strains  are  no  longer______________  as  the  dashpot  will  __________  the  spring  to  have  the  same  deforma3on  thus  deforma3on  compa3bility:      _____________      

FALL  2014:    EMCH  315     2  

σ

plus σ = σs + σd statically indeterminant

independent constrain

ε = εs = εd

non-steady state transient

Equilibrium  and  compa3bility  arguments  can  be  rewriLen  to  convey  that  the  response  is  ____________________.  

force  equilibrium:  σ(t) = σs(t) + σd(t)  

compa3bility:      ε(t) = εs(t) = εd(t)

stress-­‐strain  rela3onship  for  the  spring:    

 stress  strain  rela3onship  for  the  dashpot:  

FALL  2014:    EMCH  315     3  

_______________

_______________

_______________

_______________

σs(t) = Eεs(t)

dεd/dt = (1/η)σd(t)

σs(t) = Eε(t)

dε/dt = (1/η)σd(t)

time-dependent

To  determine  the  governing  cons3tu3ve  equa3on,  subs3tute  the  equa3ons  for  the  spring  and  dashpot  into  the  equilibrium  equa3on.  The  governing  stress-­‐strain  differen3al  equa3on:            Stress  depends  not  only  on  the  strain,  but  also  the  strain  rate  Solu3on  to  the  first-­‐order  linear  differen3al  equa3on  (see  next  slide)      

FALL  2014:    EMCH  315     4  

σ =η dε(t )dt

+ Eε(t )

ε(t ) = σE1− e− t /τ⎡⎣ ⎤⎦ where τ =η E

FALL  2014:    EMCH  315     5  

An  instantaneous  strain  cannot  be  imposed  as  the  dashpot  must  deform  prior  to  the  spring  in  the  V-­‐K  model.  

FALL  2014:    EMCH  315     6  

ε

t

Imposed condition σ

σo

t

Non-linear creep response

at t = 0 at time t as t → ∞

An  instantaneous  strain  ___________  be  imposed  because  the  model  can  only  deform  with  3me  as  the  dashpot  must  deform  prior  to  the  spring  in  the  V-­‐K  model.  

FALL  2014:    EMCH  315     7  

ε

t

ε(t ) = σE1− e− t /τ⎡⎣ ⎤⎦

cannot

ε∞ = σ/E

ε = 0

FALL  2014:    EMCH  315     8  

ε

t

Now we remove the stress at some arbitrary time

ε1

models __________________ creep: ____________ down as time passes

strain ___________ toward ____________ due to spring in parallel arrangement __________ on the dashpot εo

εo

εp

εp

t1

Maxwell Model

Voight-Kelvin

transient slows

decreases zero

pulling

FALL  2014:    EMCH  315     9  

ε

t

σ = Eε +η dεdt

= 0ε1

⇒η dεdt

= −Eε

The time-dependent strain can be determined after we remove the stress at a new t=0, while ε = ε1.

can’t account for __________________ for a constant strain, dε/dt = 0 and εE = σ

ε = ε1e− t /τ

stress relaxation

FALL  2014:    EMCH  315     10  

Viscoelas3c  Materials Their  uses Asphalt Roads  (moves  with  t  and  T) Silly  puLy Toys Polymeric  foams Seat  Cushions,  maLress  topper   Glass Dinnerware,  labware Rubber  Bands Hold  things  together Metals  (T  >Tm/2) W  filament  in  light  bulbs,  turbine  blades

Plas3c/polymer  items Sun  glasses Rubber  bumpers  in  cars Absorbs  energy  from  impact Wood Sagging  with  3me  as  viscoelas3c  response Nylon  Guitar  String Rockin’  Out! Steel  plates  coated  with                                          viscoelas3c  polymer

Damping  of  vibra3onal  energy

Discs  in  human  spines Slip  disc,  rupture  of  discs Skin Holds  us  together New  born  baby’s  skull Protects  the  brain  and  other  important  parts

Examples  and  Applica3ons  of  Viscoelas3c  Materials    

•  Creep  and  Recovery  

Materials  which  behave  elas3cally  at  room  temperature  oien  aLain  significant  viscoelas3c  proper3es  when  heated.  Such  is  the  case  with  metal  turbine  blades  in  jet  engines,  which  reach  very  high  temperatures  and  need  to  withstand  very  high  tensile  stresses.    Conven3onal  metals  can  creep  significantly  at  high  temperatures  and  this  has  led  to  the  development  of  creep-­‐resistant  alloys;  turbine  blades  are  now  oien  made  of  so  called  superalloys  which  contain  some  or  all  of  nickel,  cobalt,  chromium,  aluminum,  3tanium,  tungsten  and  molybdenum.  

FALL  2014:    EMCH  315     11  

Examples  and  Applica3ons  of  Viscoelas3c  Materials  

•  Creep  and  Recovery  

Disks  in  the  human  spine  are  viscoelas3c.  Creep  under  normal  body  weight  and  get  shorter  with  3me.  Lying  down  allows  the  spinal  disks  to  recover,  and  thus  most  people  are  taller  in  the  morning  than  in  the  evening.    

Skin  3ssue  is  viscoelas3c.  Ex.  Pinching  the  skin  at  the  back  of  the  hand;  it  takes  3me  to  recover  back  to  its  original  flat  posi3on.    The  longer  the  skin  is  held  in  the  pinched  posi3on,  the  longer  it  takes  to  recover.    The  more  rapidly  it  is  pinched,  the  less  3me  it  takes  to  recover  –  it  behaves  “more  elas3cally”.    

FALL  2014:    EMCH  315     12  

Examples  and  Applica3ons  of  Viscoelas3c  Materials  

•  Creep  and  Recovery  

Wood  is  viscoelas3c.  Creep  under  the  weight  of  the  roof  and  garavity  can  take  many  decades  or  centuries  to  be  no3ceable.    

Polymer  foams  used  in  seat  cushions.  Seat  cushions  creep  to  allow  progressive  conforma3on  of  the  cushion  to  the  body  shape.    

FALL  2014:    EMCH  315     13  

Examples  and  Applica3ons  of  Viscoelas3c  Materials  

•  Energy  Absorp3on    

Viscoelas3c  materials  have  the  property  of  absorbing  vibra3onal  energy,  damping  the  vibra3ons.  

Used  in  tall  buildings  which  can  vibrate  when  dynamically  loaded  by  wind  or  earthquakes.    

Viscoelas3c  materials  are  excellent  impact  absorbers.    A  peak  impact  force  can  be  reduced  by  a  factor  of  two  if  an  impact  buffer  is  made  of  viscoelas3c,  rather  than  elas3c  materials.    Elastomers  (any  of  various  substances  resembling  rubber)  are  highly  viscoelas3c  and  make  good  impact  absorbers.  

FALL  2014:    EMCH  315     14  

Examples  and  Applica3ons  of  Viscoelas3c  Materials  

•  Energy  Absorp3on    Viscoelas3c  materials  are  used  in  automobile  bumpers,  on  computer  drives  to  protect  from  mechanical  shock,  in  helmets  (the  foam  padding  inside),  in  wrestling  mats,  etc.    Viscoelas3c  materials  are  also  used  in  shoe  insoles  to  reduce  impact  transmiLed  to  a  person's  skeleton.    The  car3lage  at  the  ends  of  the  femur  and  3bia,  in  the  knee  joint,  is  a  natural  shock  absorber.  In  an  osteoarthri3c  knee,  the  car3lage  has  degraded  -­‐  some3mes  the  bones  grind  against  each  other  causing  great  pain.  Synthe3c  viscoelas3c  materials  can  be  injected  directly  into  an  osteoarthri3c  knee,  enveloping  car3lage-­‐deficient  joints  and  ac3ng  as  a  lubricant  and  shock  absorber  

FALL  2014:    EMCH  315     15  

FALL  2014:    EMCH  315     16  

The behavior of many viscoelastic materials lies between the behavior of a spring or a dashpot and is described by a combination of the Hookean (spring) and Newtonian (dashpot) elements. A couple of possibilities:

σ

σ

time __________ response

time __________ response

The  models  described  so  far  provide  a  qualita3ve  illustra3on  of  the  viscoelas3c  behavior  of  polymers.  

•  The  Maxwell  element  is  the  most  suited  to  represent  fluid  polymers:  the  permanent  flow  predominates  on  the  longer  term,  while  the  short-­‐term  response  is  elas3c.  

•  The  Voight-­‐Kelvin  element,  with  an  added  spring  and,  if  necessary,  a  dashpot,  is  beLer  suited  to  describe  the  nature  of  a  solid  polymer.  

FALL  2014:    EMCH  315     17  

Both  models,  the  Maxwell  element  and  the  Voight-­‐Kelvin  element,  are  limited  in  their  representa3on  of  the  actual  viscoelas3c  behavior.  

Maxwell  •  Capable  of  modeling  both  

stress  relaxa3on  and  creep.  •  Predicts  that  the  stress  relaxes  

to  zero:  In  reality,  the  stress  levels  off  at  some  finite  value.  

•  Predicts  that  creep  strain  accumulates  linearly  with  3me;  in  reality,  strain  can  accumulate  non-­‐linearly.    

Voight-­‐Kelvin  •  Predicts  the  more  realis3c  

case  of  transient  (non-­‐linear)  creep.  

•  Predicts  recovery.  •  Cannot  impose  instantaneous  

strain,  thus,  cannot  model  stress  relaxa3on.  

FALL  2014:    EMCH  315     18  

– elas3c  (ability  to  stretch  and  return  to  its  original  length)  

– plas3c  (permanent  deforma3on)  – viscoelas3c  (elas3c  and  3me-­‐dependent  response)  – creep  (length  changes  over  3me  under  constant  load)  

FALL  2014:    EMCH  315     19  

General  descrip3ons  of  mechanical  responses  are  

E  MCH  315    Mechanical  Response  of  Engineering  Materials  

Lecture  27    Creep  I Chap.  10  

FALL  2014:    EMCH  315     Lecture  27:  Slide  20  

FALL  2014:    EMCH  315     21  

FALL  2014:    EMCH  315     22  

Creep Experiment

Defini3on:    _______________  that  increases  as  a  func3on  of  _____________  and  _____________  under  ________________.  For  _____________  solids,  service  _____________________  exceed  ___________  of  Tm.  

FALL  2014:    EMCH  315     23  

Permanent strain time temperature constant stress

temperatures (absolute) 30 - 40%

Response

tr

ε0

ε

t

σ

t

Imposed condition

TIME

crystalline

Three  stages  of  creep  behavior:    Stage I

FALL  2014:    EMCH  315     24  

t1

ε1

I II III

ε2

t2 tr

ε0

ε

t

Primary Stage -- a.k.a. ______________

slope of ε vs t plot is the strain rate: Slope initially high then slope __________ until it becomes constant Decreasing transient creep strain rate due to _____________ processes:

at the ____________- thermally activated softening (or annealing) dominates

_____________ strain hardening

facilitates increasing creep resistance

decreases

opposing beginning

eventually

Three  stages  of  creep  behavior:    Stage II

FALL  2014:    EMCH  315     25  

t1

ε1

I II III

ε2

t2 tr

ε0

ε

t

Secondary Stage -- a.k.a. ______________

II.

continued strain-hardening and ongoing annealing processes reach _______________ equilibrium 90% of life

initiates when ε becomes _______________ after Stage Iconstant

steady - state

Three  stages  of  creep  behavior:    Stage III

FALL  2014:    EMCH  315     26  

t1

ε1

I II III

ε2

t2 tr

ε0

X rupture

ε

t

___________________ III. initiates at end of stage II when increases in an _________ manner deformation becomes _________ by the formation of a ________ as in a tension test thus represents onset of structural _________ leading to fracture/rupture terminates at time-to-rupture (tr)

ε unstable

localized neck

instability

Two  types  of  design  situa3ons  occur  in  engineering  prac3ce.      

•  _____________________–  limits  on  total  amount  of  strain  tolerated  by  structural  element  (e.g.  minimal  clearances)  –  ____________________  

•  _________________________–  life3me  of  the  component  prior  to  failure  is  cri3cal  (creep  strains  can  be  tolerated)  –  _____________________  

We  want  to  determine    

FALL  2014:    EMCH  315     27  

(requires predictive equations for ________________.

(requires predictive equations for ______________.)

ε = f (material properties, σ, T, t).

steady-state creep

creep rupture

Strain-­‐limited  design  and  predic3ve  equa3ons  for  steady-­‐state  creep.  

FALL  2014:    EMCH  315     28  

ε tot (t) =The total strain is calculated as

I II

ε

t

ε ssε0

3 contributions to the total strain. (i)  instantaneous ____:

strain generated at loading

(ii)  transient _______ : time dependent contribution β and η depend on stress, temperature, and material

(iii)  steady-state _____ contribution when strain rate is constant

The  transient  term  can  be  considered  negligible  rela3ve  to  the  steady-­‐state  term.  

FALL  2014:    EMCH  315     29  

I II III

ε

t

ε ss

calculation for the steady-state or linear creep rate

ε1,t1ε2,t2

ε tot (t) = εo +

ε ss =

Steady-­‐state  creep  rates                      depend  on  stress  and  temperature.  

FALL  2014:    EMCH  315     30  

What we want to know is the functional dependence of ____________________________________so that we can use this for design.

Raising either σ or T ________

Different constant stress levels at the same temperature.

The same constant stress levels for different temperatures.

ε ss

FALL  2014:    EMCH  315     31  

An equation of the form:

Where: A = coefficient, _________________________. ΔH = activation energy k = Boltzman’s constant (k = 1.38x10-23 J/K or

k = 6.79x10-23 in-lb/R) T = __________________________ ____________ R = 460 + °F

Ae -ΔH/kT εss =

describes steady state creep.

K = 273 + °C

(potential energy necessary to ____________

1/T

ln ε

ss

constant σ Ta

−1

Tb−1

Tc−1

material- and stress- dependent

temperature (absolute degrees)

drive creep

FALL  2014:    EMCH  315     32  

Consider two sets of data were collected each at different levels of constant stress.

1/T

ln ε

ss

σ1 σ2

same slope different slope

1/T

ln ε

ss

σ1 σ2

-ΔH k

-ΔH k ΔH (σ ) = ΔH

Activation energy can be stress dependent so _______________

ε ss = Ae kT

New  material  parameter                                                  .  

FALL  2014:    EMCH  315     33  

φ =εoσ o

m

ε ssεo

= σσ o

⎛⎝⎜

⎞⎠⎟

m

e−ΔH (σ )

kT ε ss = φσ me

−ΔHkT ⇒

We consider cases when the instantaneous load can be considered quasi-static and the strain rate is __________:

ε ss =

σσ o

⎛⎝⎜

⎞⎠⎟

m

e−ΔH (σ )

kT

____________ creep rate equation.

Experimentally  determined  rela3onships  for  the  stress  dependence  of  the  

steady-­‐state  creep  rate  are  of  the  form  ________  

Normalized (or non-dimensionalized) form:

ϕ, m, and H are material parameters.

FALL  2014:    EMCH  315     34  

Shown below is the steady-state strain rate creep response for Al, determined at 230oC. What is the equation relating the steady-state strain rate to stress? What are the values of the two parameters for Al?

Recalling  that                                                    we  can  rewrite  the  Dorn-­‐Miller  creep  rate  equa3on  as        and  set    

Rewrite a simplified form of the Dorn-Miller equation as

In-class problem (solution method for HW6 #1)

ε ss ∝σ m

ε ss =

1σ o

⎛⎝⎜

⎞⎠⎟

m

e−ΔH (σ )

kT σ m

B = 1σ o

⎛⎝⎜

⎞⎠⎟

m

e−ΔH (σ )

kT

NOTE: At a single temperature there is no way to say anything about temperature dependence of the strain rate. Using this form we can simply solve for two

parameters B and m

FALL  2014:    EMCH  315     35  

Take the logarithm of both sides “m” is the slope and can be calculated as sub

log εss = logB + m logσ

Solve for two parameters B and m

m =

log10 ε ss1 − log10 ε ss2( )log10σ 1 − log10σ 2( ) =

m typically ranges from 4.0 ≤ m ≤6.0

(σ 1, ε ss1)

B =ε ssσ m =

FALL  2014:    EMCH  315     36  

ln ε ss = ln

σσ o

⎛⎝⎜

⎞⎠⎟

m

+ − ΔH (σ )k

⎛⎝⎜

⎞⎠⎟1T

Taking the natural logarithm of ε ss :

In-class problem: Determine the complete Dorn-Miller steady state creep rate equation.

ε ss =

σσ o

⎛⎝⎜

⎞⎠⎟

m

e−ΔH (σ )

kT

can determine ΔH(σ ) from the slope of ln ε ss vs 1/T

By analyzing experimental data, we can ascertain whether the activation energy ΔH(σ) is indeed stress dependent.

we must determine _______________.

FALL  2014:    EMCH  315     37  

slope

δ (ln ε ss )δ (1 /T ) σ

= − ΔH (σ )k

Compare steady-state creep data for two different constant stress levels.

δ (ln ε ss )δ (1 /T ) σ =4 ksi

= δ (ln ε ss )δ (1 /T ) σ =6ksi

FALL  2014:    EMCH  315     38  

since  slopes  are  iden3cal  ⇒____________and activation energy is _______________

slope is defined by δ (ln ε ss )

δ (1 /T ) σ

=

for one value of σ, substitute Boltzman’s constant k (J/K or in-lb/R) in order to determine ΔH

Note units for k Convert T to R (T in °F) or Convert T to K (T in °C)

HW6 # 3: Given ΔH(σ) Plot

By  analyzing  data  for  different  stress  levels,  we  can  determine  the                                                                                                            .  

•  Compare  the  same  steady-­‐state  creep  data  for  

FALL  2014:    EMCH  315     39  

Taking the logarithm of both sides to the base 10log10 ε ss = log10 B +m log10σ

remaining terms; .

different constant temperatures.

Use simplified D-M eqnε ss = Bσ

m

known data points will establish to determine σo

We  can  determine  the                                                  Dorn-­‐Miller  steady  state  creep  rate  equa3on                                                                                                                                                              

With  m,  subs3tute  previously  determined                and  use                data  points  to  determine                .  

FALL  2014:    EMCH  315     40  

�

ε ss =

σσ o

⎛⎝⎜

⎞⎠⎟

m

e−ΔHkT

ΔH known σo

complete

εss ; T ; σ �

(see Example Prob. 10.2 for solution).