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Chapter 6

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					             Chapter 6:
         Mechanical Properties
ISSUES TO ADDRESS...
• Stress and strain: What are they and why are
   they used instead of load and deformation?
• Elastic behavior: When loads are small, how much
   deformation occurs? What materials deform least?
• Plastic behavior: At what point does permanent
   deformation occur? What materials are most
   resistant to permanent deformation?
• Toughness and ductility: What are they and how
   do we measure them?


                                                   Chapter 6 - 1
            Elastic Deformation
     1. Initial         2. Small load               3. Unload

                                bonds
                                stretch

                                                    return to
                                                    initial
                  d
                       F
                            F             Linear-
                                          elastic
Elastic means reversible!                           Non-Linear-
                                                    elastic
                                               d
                                                      Chapter 6 - 2
    Plastic Deformation (Metals)
     1. Initial   2. Small load                    3. Unload
                       bonds
                       stretch                            planes
                       & planes                           still
                       shear                              sheared


                                               dplastic
                  delastic + plastic

                                  F
                  F
Plastic means permanent!      linear                 linear
                              elastic                elastic
                                                               d
                                        dplastic
                                                       Chapter 6 - 3
               Engineering Stress
• Tensile stress, s:         • Shear stress, t:
                     Ft                  Ft          F

Area, A                     Area, A               Fs


                                       Fs
                     Ft
                                Fs              Ft
       Ft  lb f  N           t=         F
    s=    = 2 or                Ao
       Ao  in    m2
    original area
    before loading
                           Stress has units:
                            N/m2 or lbf/in2
                                              Chapter 6 - 4
         Common States of Stress
• Simple tension: cable
F                   F
A o = cross sectional
area (when unloaded)
    F
 s=    s                 s
    Ao
                                                  Ski lift (photo courtesy
• Torsion (a form of shear): drive shaft          P.M. Anderson)


            M           Fs   Ao
Ac
                                      Fs
                                  t =
                                      Ao
  M
       2R                         Note: t = M/AcR here.
                                                        Chapter 6 - 5
 OTHER COMMON STRESS STATES (1)
• Simple compression:



 Ao


                                           Canyon Bridge, Los Alamos, NM
                                           (photo courtesy P.M. Anderson)


                                      F
                                   s=
                                           Note: compressive
  Balanced Rock, Arches                    structure member
  National Park
  (photo courtesy P.M. Anderson)
                                      Ao   (s < 0 here).



                                                               Chapter 6 - 6
OTHER COMMON STRESS STATES (2)
• Bi-axial tension:           • Hydrostatic compression:




Pressurized tank                     Fish under water   (photo courtesy
(photo courtesy                                         P.M. Anderson)
P.M. Anderson)
                   sq > 0

                     sz > 0        sh< 0

                                                        Chapter 6 - 7
                Engineering Strain
• Tensile strain:                                           • Lateral strain:
                                                    d/2
                                                                       -dL
        e = d                                                      eL =
           Lo                                          Lo               wo
                                       wo

                           dL /2
• Shear strain:
         q
       x                          g = x/y = tan q

   y                90º - q
                                                               Strain is always
       90º                                                     dimensionless.
                Adapted from Fig. 6.1 (a) and (c), Callister 7e.          Chapter 6 - 8
                          Stress-Strain Testing
 • Typical tensile test                                                 • Typical tensile
    machine                                                                specimen



                                                                                          Adapted from
extensometer                                   specimen                                   Fig. 6.2,
                                                                                          Callister 7e.




                                                                                      gauge
                                                                                      length




 Adapted from Fig. 6.3, Callister 7e. (Fig. 6.3 is taken from H.W.
 Hayden, W.G. Moffatt, and J. Wulff, The Structure and Properties of
 Materials, Vol. III, Mechanical Behavior, p. 2, John Wiley and Sons,
 New York, 1965.)                                                                   Chapter 6 - 9
            Linear Elastic Properties
• Modulus of Elasticity, E:
 (also known as Young's modulus)

• Hooke's Law:
      s=Ee             s                        F
                                   E

                                       e
                         Linear-
                         elastic               F
                                           simple
                                           tension
                                           test

                                             Chapter 6 - 10
                  Poisson's ratio, n
                                     eL
• Poisson's ratio, n:

            eL
        n=- e
                                                     e

 metals: n ~ 0.33                     -n
 ceramics: n ~ 0.25
 polymers: n ~ 0.40

 Units:                 –n > 0.50 density increases
 E: [GPa] or [psi]
                        –n < 0.50 density decreases
 n: dimensionless                  (voids form)

                                                  Chapter 6 - 11
        Mechanical Properties
• Slope of stress strain plot (which is
  proportional to the elastic modulus) depends
  on bond strength of metal




                            Adapted from Fig. 6.7,
                            Callister 7e.


                                                     Chapter 6 - 12
           Other Elastic Properties
                             t                     M
• Elastic Shear
   modulus, G:
                                  G                     simple
                                      g                 torsion
      t=Gg                                              test


                                               M
• Elastic Bulk               P                     P
   modulus, K:
           V                         V P                  P
    P = -K                            Vo
           Vo                K                            pressure
                                                          test: Init.
                                                          vol =Vo.
• Special relations for isotropic materials:              Vol chg.
                                                           = V
          E                 E
    G=                K=
       2(1 + n)          3(1 - 2n)
                                                       Chapter 6 - 13
         Young’s Moduli: Comparison
                               Graphite
                Metals                           Composites
                               Ceramics Polymers
                Alloys                             /fibers
                               Semicond
         1200
         1000                   Diamond
          800
          600
                                Si carbide
          400   Tungsten        Al oxide                  Carbon fibers only
                Molybdenum      Si nitride
E(GPa)    200
                Steel, Ni
                Tantalum         <111>
                                                          CFRE(|| fibers)*
                Platinum        Si crystal
                Cu alloys        <100>                    Aramid fibers only
          100   Zinc, Ti
           80   Silver, Gold
                                Glass -soda               AFRE(|| fibers)*     Based on data in Table B2,
                Aluminum                                  Glass fibers only
           60
           40
                Magnesium,
                Tin                                       GFRE(|| fibers)*     Callister 7e.
                                Concrete                                       Composite data based on
109 Pa     20                                             GFRE*
                                                          CFRE*
                                                                               reinforced epoxy with 60 vol%
                                                                               of aligned
                                Graphite                  GFRE( fibers)*
           10                                                                  carbon (CFRE),
            8                                             CFRE( fibers) *
            6                                             AFRE( fibers) *
                                                                               aramid (AFRE), or
                                              Polyester                        glass (GFRE)
            4                                   PET
                                                 PS                            fibers.
                                               PC         Epoxy only
            2
                                               PP
            1                                  HDPE
          0.8
          0.6                                             Wood(     grain)
                                               PTFE
          0.4

          0.2                                  LDPE                                             Chapter 6 - 14
     Useful Linear Elastic Relationships
• Simple tension:                 • Simple torsion:
                                             2ML o
 d = FL o d = -n Fw o                 a=
           L                                    4
     EA o        EA o                        pr o G
         F                                   M = moment
               d/2                           a = angle of twist
Ao
                Lo                            Lo
          wo

                                       2ro
  dL /2
• Material, geometric, and loading parameters all
   contribute to deflection.
• Larger elastic moduli minimize elastic deflection.
                                                      Chapter 6 - 15
         Plastic (Permanent) Deformation
                                (at lower temperatures, i.e. T < Tmelt/3)

• Simple tension test:
                                     Elastic+Plastic
  engineering stress, s              at larger stress



                  Elastic
                  initially
                                      permanent (plastic)
                                      after load is removed


                          ep          engineering strain, e

                               plastic strain    Adapted from Fig. 6.10 (a),
                                                 Callister 7e.


                                                                        Chapter 6 - 16
                 Yield Strength, sy
• Stress at which noticeable plastic deformation has
   occurred.
                             when ep = 0.002
        tensile stress, s
                                                   sy = yield strength
   sy

                                                   Note: for 2 inch sample
                                                         e = 0.002 = z/z
                                                          z = 0.004 in

                 engineering strain, e
   ep = 0.002        Adapted from Fig. 6.10 (a),
                     Callister 7e.
                                                                 Chapter 6 - 17
                                  Yield Strength : Comparison
                                                      Graphite/
                                    Metals/                                                                                                        Composites/
                                                      Ceramics/                                                                       Polymers
                                    Alloys                                                                                                           fibers
                                                      Semicond
                           2000
                                   Steel (4140) qt

                           1000
Yield strength, sy (MPa)



                                   Ti (5Al-2.5Sn) a




                                                                                                                                                                         in ceramic matrix and epoxy matrix composites, since
                            700    W (pure)




                                                                            since in tension, fracture usually occurs before yield.




                                                                                                                                                                             in tension, fracture usually occurs before yield.
                            600    Cu (71500) cw
                            500    Mo (pure)
                            400    Steel (4140) a
                                   Steel (1020) cd
                            300
                                                                                                                                                                                                                                  Room T values
                                                        Hard to measure ,




                                                                                                                                                      Hard to measure,
                                   Al (6061) ag
                            200    Steel (1020) hr                   ¨
                                   Ti (pure) a
                                   Ta (pure)
                                   Cu (71500) hr                                                                                                                                                                                 Based on data in Table B4,
                                                                                                                                                                                                                                 Callister 7e.
                            100                                                                                                                                                                                                  a = annealed
                                                                                                                                             dry
                             70                                                                                                       PC
                                                                                                                                                                                                                                 hr = hot rolled
                             60    Al (6061) a                                                                                         Nylon 6,6                                                                                 ag = aged
                             50                                                                                                       PET
                                                                                                                                                                                                                                 cd = cold drawn
                             40                                                                                                       PVC humid
                                                                                                                                                                                                                                 cw = cold worked
                                                                                                                                      PP
                             30                                                                                                       HDPE                                                                                       qt = quenched & tempered

                             20

                                                                                                                                      LDPE
                                   Tin (pure)                                                                                                                                                                                                  Chapter 6 - 18
                             10
                    Tensile Strength, TS
• Maximum stress on engineering stress-strain curve.
                                                       Adapted from Fig. 6.11,
                                                       Callister 7e.
          TS
                                                       F = fracture or
             sy
                                                            ultimate
      engineering




                                                            strength
         stress




                         Typical response of a metal
                                                       Neck – acts
                                                       as stress
                                                       concentrator
                               strain
                      engineering strain
• Metals: occurs when noticeable necking starts.
• Polymers: occurs when polymer backbone chains are
  aligned and about to break.
                                                            Chapter 6 - 19
                               Tensile Strength : Comparison
                                                      Graphite/
                                     Metals/                                   Composites/
                                                      Ceramics/    Polymers
                                     Alloys                                      fibers
                                                      Semicond
                             5000                                                   C fibers
                                                                                Aramid fib
                             3000                                               E-glass fib
Tensile strength, TS (MPa)


                             2000   Steel (4140) qt
                                                                                AFRE(|| fiber)
                             1000   W (pure)         Diamond                    GFRE(|| fiber)
                                    Ti (5Al-2.5Sn)aa
                                    Steel (4140)                                CFRE(|| fiber)
                                    Cu (71500) cw     Si nitride
                                    Cu (71500) hr    Al oxide
                                    Steel (1020)
                             300              ag
                                    Al (6061) a
                                    Ti (pure)
                             200    Ta (pure)                                                     Room Temp. values
                                    Al (6061) a
                             100                     Si crystal                 wood(|| fiber)    Based on data in Table B4,
                                                       <100>       Nylon 6,6
                                                      Glass-soda   PC PET                         Callister 7e.
                              40                                   PVC          GFRE( fiber)      a = annealed
                                                      Concrete     PP
                              30                                                CFRE( fiber)      hr = hot rolled
                                                                                AFRE( fiber)
                                                                      HDPE                        ag = aged
                              20                      Graphite
                                                                   LDPE                           cd = cold drawn
                                                                                                  cw = cold worked
                              10                                                                  qt = quenched & tempered
                                                                                                  AFRE, GFRE, & CFRE =
                                                                                                  aramid, glass, & carbon
                                                                                                  fiber-reinforced epoxy
                                                                                wood (   fiber)
                                                                                                  composites, with 60 vol%
                                                                                                  fibers.
                               1                                                                                Chapter 6 - 20
                              Ductility
                                                    Lf - Lo
• Plastic tensile strain at failure:          %EL =         x 100
                                                      Lo
                          smaller %EL
     Engineering
     tensile
     stress, s               larger %EL                   Ao
                                                    Lo          Af          Lf
Adapted from Fig. 6.13,
Callister 7e.



                          Engineering tensile strain, e


• Another ductility measure:                      Ao - Af
                                          %RA =           x 100
                                                    Ao

                                                               Chapter 6 - 21
                              Toughness
• Energy to break a unit volume of material
• Approximate by the area under the stress-strain
   curve.
Engineering               small toughness (ceramics)
tensile                                      large toughness (metals)
stress, s
Adapted from Fig. 6.13,                                  very small toughness
Callister 7e.                                            (unreinforced polymers)



                          Engineering tensile strain,     e

      Brittle fracture: elastic energy
      Ductile fracture: elastic + plastic energy
                                                                  Chapter 6 - 22
                              Resilience, Ur
• Ability of a material to store energy
   – Energy stored best in elastic region

                                               ey
                                      Ur =         sde
                                               0
                                   If we assume a linear
                                   stress-strain curve this
                                   simplifies to

                                           1
                                       Ur @ sy e y
                                           2
    Adapted from Fig. 6.15,
    Callister 7e.
                                                          Chapter 6 - 23
               Elastic Strain Recovery




Adapted from Fig. 6.17,
Callister 7e.

                                         Chapter 6 - 24
                                     Hardness
• Resistance to permanently indenting the surface.
• Large hardness means:
  --resistance to plastic deformation or cracking in
     compression.
  --better wear properties.
                        apply known force               measure size
   e.g.,                                                of indent after
   10 mm sphere                                         removing load

                                                              Smaller indents
                        D                          d          mean larger
                                                              hardness.

 most       brasses         easy to machine              cutting    nitrided
 plastics   Al alloys       steels          file hard     tools     steels     diamond


                              increasing hardness
                                                                                Chapter 6 - 25
       Hardness: Measurement
• Rockwell
  – No major sample damage
  – Each scale runs to 130 but only useful in range
    20-100.
  – Minor load 10 kg
  – Major load 60 (A), 100 (B) & 150 (C) kg
     • A = diamond, B = 1/16 in. ball, C = diamond


• HB = Brinell Hardness
  – TS (psia) = 500 x HB
  – TS (MPa) = 3.45 x HB

                                                     Chapter 6 - 26
            Hardness: Measurement
Table 6.5




                                    Chapter 6 - 27
          True Stress & Strain
Note: S.A. changes when sample stretched

• True stress   sT = F Ai           sT = s1 + e
• True Strain   eT = ln i  o    eT = ln1 + e




                                         Adapted from Fig. 6.16,
                                         Callister 7e.




                                              Chapter 6 - 28
                           Hardening
• An increase in sy due to plastic deformation.
         s
                                         large hardening
       sy
          1
       sy                                small hardening
          0




                                               e
• Curve fit to the stress-strain response:
                                      hardening exponent:
                 sT = K eT  n       n = 0.15 (some steels)
                                      to n = 0.5 (some coppers)
     “true” stress (F/A)          “true” strain: ln(L/Lo)
                                                            Chapter 6 - 29
  Variability in Material Properties
• Elastic modulus is material property
• Critical properties depend largely on sample flaws
  (defects, etc.). Large sample to sample variability.
• Statistics
                                           n
                                         xn
   – Mean                            x=
                                         n
                                                      1
                                    n           2
                                      x i - x  
                                                      2

   – Standard Deviation           s=
                                     n -1 
                                                 
         where n is the number of data points
                                                          Chapter 6 - 30
           Design or Safety Factors
• Design uncertainties mean we do not push the limit.
• Factor of safety, N              Often N is
                            sy         between
               sworking =              1.2 and 4
                             N
• Example: Calculate a diameter, d, to ensure that yield does
   not occur in the 1045 carbon steel rod below. Use a
   factor of safety of 5.
                                                            d
                        sy
          sworking =                1045 plain
                                       carbon steel:
                        N           sy = 310 MPa                     Lo
  220 ,000 N        5               TS = 565 MPa
    
   p d2 / 4
                                           F = 220,000N
                d = 0.067 m = 6.7 cm
                                                         Chapter 6 - 31
                     Summary
• Stress and strain: These are size-independent
   measures of load and displacement, respectively.
• Elastic behavior: This reversible behavior often
   shows a linear relation between stress and strain.
   To minimize deformation, select a material with a
   large elastic modulus (E or G).
• Plastic behavior: This permanent deformation
   behavior occurs when the tensile (or compressive)
   uniaxial stress reaches sy.
• Toughness: The energy needed to break a unit
   volume of material.
• Ductility: The plastic strain at failure.


                                                   Chapter 6 - 32
           ANNOUNCEMENTS
Reading:


Core Problems:


Self-help Problems:




                           Chapter 6 - 33

				
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posted:11/30/2011
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