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					      Biomaterials
        Week 4
       10/6/2008
   Properties of materials
Chap 6 Diffusion
Chap 7 Mechanical Properties

                               1
Chap 6 Diffusion




                   2
       Why study diffusion
Heat treatment to improve their properties.
During heat treatment: always involve
atomic diffusion
Heat treating temperature and time, and/or
cooling rate are predictable, using
mathematics of diffusion and appropriate
diffusion


                                          3
        Learning Objectives
1. Name and describe the two atomic
  mechanisms of diffusion.
2. Distinguish between steady-state and
  nonsteady-state diffusion.
3. (a) Write Fick‘s first and second laws in
  equation form, and define all parameters.
  (b) Note the kind of diffusion for which
  each of these equations is normally
  applied.

                                               4
         Learning Objectives
4. Write the solution to Fick‘s second law for
  diffusion into a semi-infinite solid when the
  concentration of diffusing species at the surface
  is held constant. Define all parameters in this
  equation.
5. Calculate the diffusion coefficient for some
  material at a specified temperature, given the
  appropriate diffusion constants.
6. Note one difference in diffusion mechanisms for
  metals and ionic solids.

                                                      5
     Diffusion: introduction
Diffusion: the phenomenon of material
transport by atomic motion
Include:
– atomic mechanisms by which diffusion occurs
– Mathematics of diffusion
– Influence of temperature and diffusing species
  on the rate of diffusion



                                               6
Diffusion couple- copper-nickel
                 Figure 6.1
              (a) A copper—nickel diffusion
                 couple before a high—
                 temperature heat treatment
              (b) Schematic representations
                 of Cu (colored circles) and
                 Ni (gray circles) atom
                 locations within the diffusion
                 couple.
              (c) Concentrations of copper
                 and nickel as a function of
                 position across the couple.

                                              7
Diffusion couple- copper-nickel
                Figure 6.2
                (a) A copper—nickel
                diffusion couple after a
                high-temperature heat
                treatment, showing the
                alloyed diffusion zone.
                (b) Schematic
                representations of Cu
                (colored circles) and Ni
                (gray circles) atom locations
                within the couple.
                (c) Concentrations of
                copper and nickel as a
                function of position across
                the couple.
                                            8
          Diffusion couple
The heating couple is heated for an extended
period at an elevated temperature and cooled to
room temperature
Results indicated that copper atoms have
migrated or diffused into nickel, and nickel has
diffused into copper
Interdiffusion (impurity diffusion): one metal
diffuse into another
Self diffusion: pure metal, all atoms exchanging
positions are of the same type

                                                   9
  6.2 Diffusion mechanisms
Atomic perspective view: diffusion is just
the stepwise migration of atoms from
lattice site to lattice site.
For atom to make such a move:
(1) there must be an empty adjacent site
(2) the atom must have sufficient energy to
  break bonds with its neighbor atoms and then
  cause some lattice distortion during the
  displacement.
                                             10
  6.2 Diffusion mechanisms
At specific temperature some small
fraction of the total number of atoms in
capable of diffusive motion, by virtue of the
magnitude of their vibrational energies.
The fraction increase with rising
temperature




                                            11
Two dominate for metallic diffusion:
 Vacancy diffusion
 Interstitial diffusion:




                                   12
         Vacancy diffusion
The interchange of an atom from a normal lattice
position to an adjacent vacant lattice site or
vacancy
Necessitate the presence of vacancies, and the
extent to which vacancy diffusion can occur is a
function of the number of the defects that are
present




                                               13
Vacancy diffusion (Fig 6.3 a)




                                14
        Vacancy diffusion
Since diffusing atoms and vacancies
exchange positions, the diffusion of atoms
in one direction corresponding to motion of
vacancies in the opposite direction
Both self-diffusion and interdiffusion occur
by this mechanism




                                           15
         Interstitial diffusion
Atoms that migrate from an interstitial position to
a neighboring one that is empty.
Hydrogen, carbon, nitrogen, and oxygen which
have atoms that are small enough to fit into the
interstitial positions.
In most metal alloy, interstitial diffusion occurs
much more rapidly than diffusion by vacancy
mode.
Interstitial atoms are smaller and thus more
mobile.

                                                  16
Interstitial diffusion (Fig 6.3 b)




                                     17
  6.3 Steady-state diffusion:
Diffusion flux (J): Mass (or, equivalently,
the number of atoms) M diffusing through
and perpendicular to a unit cross-sectional
area of solid per unit of time
                J = M/At               (6.1a)
– A: area across which diffusion occurring
– and t: elapsed diffusion time.



                                             18
  6.3 Steady-state diffusion
               J = (1/A) (dM/dt)           (6.1b)
– J: kg/m2 -s or atoms/m2 –s
– Steady state diffusion: Diffusion flux does not
  change with time
– Concentration of diffusion species on both
  surface of the plate are held constant




                                                    19
6.3 Steady state diffusion




                             20
   6.3 Steady-state diffusion
When concentration C is plotted vs.
position (or distance) within the solid x, the
resulting curve is termed the concentration
profile:
The slope at particular point on this curve
is the concentration gradient




                                             21
  6.3 Steady-state diffusion
   Concentration gradient: dC /dx    (6.2a)
In the present treatment, the concentration
profile is assumed to be linear,




                                          22
   6.3 Steady state diffusion
Concentration gradient =
      △C/ △x = (CA – CB)/(xA -xB)     (6.2b)
It is sometimes convenient to express
concentration in terms of mass of diffusing
species per unit volume of solid (kg/m3)




                                           23
   6.3 Steady state diffusion
For steady state diffusion in a single direction is
relatively simple, the flux is proportional to the
concentration gradient through the expression
             J = -D (dC)/ (dx)                   (6.3)
– D: diffusion coefficient (m^2)/sec
– Negative sign: the expression indicate the direction of
  diffusion is shown the concentration gradient, from
  high to a low concentration
Fick‘s first law


                                                        24
   6.3 Steady state diffusion
Driving force used in the context of what
compels a reaction to occur
For diffusion reaction: several such forces
are possible; but when diffusion is
according to Eq. 6.3, the concentration
gradient is the driving force
       J = -D (dC)/ (dx)                (6.3)


                                            25
Diffusion flux computation




                             26
6.4 Nonsteady-state diffusion
Most practical diffusion situations are
nonsteady-state ones
The diffusion flux and the concentration
gradient at some particular point in a solid
vary with time, with a net accumulation or
depletion of the diffusing species resulting.




                                            27
6.4 Nonsteady-state diffusion
Concentration profiles at three different diffusion times.
Under conditions of nonsteady state:




                                                             28
6.4 Nonsteady-state diffusion
The partial differential equation:
           (∂C/∂t) = ∂/ ∂x(D* ∂C/∂x)    (6.4a)
Fick‘s second law
If diffusion coefficient is independent of
composition
          (∂C/∂t) = (D* ∂2C/∂x2)        (6.4b)



                                            29
6.4 Nonsteady-state diffusion
For semi-infinite solid in which surface concentration is
held constant
Frequently, the source of the diffusing species is a gas
phase, the partial pressure of which is maintained at a
constant value.
The following assumptions are made:
 – Before diffusion, any of the diffusing solute atoms in
   the solid are uniformly distributed with concentration
   of C0
 – the value of x at the surface is zero and increases
   with distance into solid
 – the time is taken to be zero the instant before the
   diffusion process begins

                                                            30
6.4 Nonsteady-state diffusion
              (∂C/∂t) = (D* ∂2C/∂x2)
B.Cs: for t=0, C = C0 at 0 ≤ x ≤ ∞
For t > 0, C = Cs (the constant surface concentration) at
                          x =0
                     C = C0 at x= ∞
Solution
(Cx-C0)/(Cs-C0) = 1- erf (x/2√Dt)
– Cx represent the concentration at depth x after time t
– erf (x/2√Dt) is Gaussian error function
– erf(x) = 2/√π∫0xe-y2dy

                                                           31
6.4 Nonsteady-state diffusion




                                32
6.4 Nonsteady-state diffusion
The concentration parameters appear in Eq 6.5
are noted in Fig 6.6, a concentration profile
taken at a specific time




                                                33
6.4 Nonsteady-state diffusion
Equ. 6.5 thus demonstrates the
relationship among concentration, position,
and time.
– That Cx being a function of the dimensionless
  parameter x/√Dt,
– maybe determined at any time and position if
  parameters C0, Cs and D are known



                                              34
6.4 Nonsteady-state diffusion
Suppose that it is desired to achieve some
specific concentration of solute, C1 in an
alloy;
The left hand side of Equ 6.5
(Cx-C0)/(Cs-C0) = 1- erf (x/2√Dt)
now becomes
(C1 – C0 )/ ( Cs – C0 ) = constant


                                         35
6.4 Nonsteady-state diffusion
This being the case, the right hand side of
this same expression is also a constant,
       And       x/2√Dt = constant
          Or x2 / Dt = constant




                                              36
6.4 Nonsteady-state diffusion




                                37
38
39
40
6.5 Factors influence diffusion:
Diffusing species
Temperature




                                   41
          Diffusing species
The magnitude of the diffusion coefficient D is
indicative of the rate at which atom diffuse
Coefficient, both self and interdiffusion, for
several metallic systems are listed in Table 6.2
The diffusion species and host material
influence the diffusion coefficient
This comparison also provides a contrast
between rates of diffusion via vacancy and
interstitial modes
Self diffusion occurs by a vacancy mechanism,
whereas carbon diffusion in iron is interstitial

                                                   42
Diffusion data




                 43
                Temperature
The diffusion rate increase 6 order of magnitude from
500 to 900C
The temperature dependence of diffusion coefficients is
related to temperature according to
                D = D 0 exp ( - Qd / RT)
where
D0 = a temperature-independent preexponential (m2/s)
Qd = the activation energy for diffusion (J/mol, cal/mol. or
eV/atom)
R = the gas constant. 8.31 J/moI-K, 1.987 cal/mol-K, or
8.62 X 10-5 eV/atom-K
T = absolute temperature (K)


                                                           44
               Temperature
The activation energy may be thought of as that energy
required to produce the diffusive motion of one mole of
atoms.
A large activation energy results in a relatively small
diffusion coefficient.
Table 6.2 also contains a listing of D0 and Qd values for
several diffusion systems.
Taking natural logarithms of Equation 6.8 yields




                                                            45
               Temperature
Since D0,Qd and R are all constants. Equation 6.9b takes
on the form of an equation of a straight line:
                          y = b + mx
where y and x are analogous, respectively, to the
variables log D and l/T.
Thus, if log D is plotted versus the reciprocal of the
absolute temperature. a straight line should result,
having slope and intercept of -Qd/2.3R and log D0,
respectively.
This is, in fact, the manner in which the values of Qd, and
D0 are determined experimentally.
From such a plot (or several alloy systems (Figure 6.7), it
may be noted that linear relationships exist for all cases
shown.
                                                         46
Diffusion coefficient vs. reciprocal
           temperature




                                       47
Diffusion coefficient determination




                                      48
Design example




                 49
50
Design example




                 51
Design example




                 52
   6.6 Other diffusion paths
Atomic migration may also occur along
dislocations, grain boundaries, and external
surfaces.
These are sometimes called ‗short-circuit‖
diffusion path inasmuch as rates are much faster
than for bulk diffusion.
However, in most situations short— circuit
contributions to the overall diffusion flux are
insignificant because the cross-sectional areas
of these paths are extremely small.

                                               53
 6.7 DIFFUSION IN IONIC AND
   POLYMERIC MATERIALS
Ionic Materials
For ionic compounds, the phenomenon of diffusion is more
complicated than for metals inasmuch as it is necessary to consider
the diffusive motion of two types of ions that have opposite charges.
Diffusion in these materials usually occurs by a vacancy mechanism
(Figure 6.3a).




                                                                   54
 6.7 DIFFUSION IN IONIC AND
   POLYMERIC MATERIALS
And, as we noted in Section 5.3. in order to maintain charge
neutrality in an ionic material, the following may be said about
vacancies: (1) ion vacancies occur in pairs [as with Schottky defects
(Figure 5.3)]. (2) they form in nonstoichiometric compounds (Figure
5.4), and (3) they-are created by substitutional impurity ions having
different charge states from the host ions (Example Problem 5.2).




                                                                    55
           Ionic Materials
In any event, associated with the diffusive motion of
electrical charge.
And in order to maintain localized charge neutrality in the
vicinity of this moving ion, it is necessary that another
species having an equal and opposite charge
accompany the ion‘s diffusive motion.
Possible charged species include another vacancy, an
impurity atom, or an electronic carrier [i.e., a free
electron or hole (Section 12.6)].
It follows that the rate of diffusion of these electrically
charged couples is limited by the diffusion rate of the
slowest moving species.

                                                         56
         Ionic Materials
When an external electric held is applied across
au ionic solid, the electrically charged ions
migrate (i.e.. diffuse) in response to forces that
are brought to bear on them.
And, as we discuss in Section 12.16. this ionic
motion gives rise to an electric current.
Furthermore, the electrical conductivity is a
function of the diffusion coefficient (Equation
12.23).
Consequently, much of the diffusion data for
ionic solids come from electrical conductivity
measurements.
                                                 57
         Polymeric Materials
For polymeric materials, we are more interested in the
diffusive motion of small foreign molecules (e.g., 02, H20,
CO2,. CH4) between the molecular chains than in tile
diffusive motion of atoms within the chain structures.
A polymer‘s permeability and absorption characteristics
relate to the degree to which foreign substances diffuse
into the material.
Penetration of these foreign substances can lead to
swelling and/or chemical reactions with the polymer
molecules, and often to a depreciation of the material‘s
mechanical and physical properties (Section 16.11).



                                                          58
         Polymeric Materials
Rates of diffusion are greater through amorphous
regions than through crystalline regions: the structure of
amorphous material is more ―open.‖
This diffusion mechanism may be considered to be
analogous to interstitial diffusion in metals— that is, in
polymers, diffusive movement from one open amorphous
region to an adjacent open one.

Foreign molecule size also affects the diffusion rate:
smaller molecules diffuse faster than larger ones.
Furthermore, diffusion is more rapid for foreign
molecules that are chemically inert than for those that
react with the polymer.
                                                          59
         Polymeric Materials
For some applications low diffusion rates through
polymeric materials are desirable, as with food and
beverage packaging and with automobile tires and inner
tubes.

Polymer membranes are often used as filters to
selectively separate one chemical species from another
(or others) (e.g.. the desalinization of water).

In such instances it is normally the case that the diffusion
rate of the substance to be filtered is significantly greater
than that for the other substance (s).

                                                           60
     Summary: Diffusion
Diffusion mechanism: step-wise atom
motion, self diffusion, interdifffusion,
vacancy, interstitial
Steady state diffusion
Non-steady state diffusion
Factors influence diffusion



                                           61
Chap 7 Mechanical
   Properties




                    62
Why study mechanical properties
 It is incumbent on engineers to understand how
  the various mechanical properties are measured
  and what these properties represent;
 They may be called upon to design
  structures/components using predetermined
  materials such that unacceptable levels of
  deformation and/or failure will not occur.
 We demonstrate this procedure with respect to
  the design of a tensile-testing apparatus in
  Design Example 7.1.

                                                   63
          Learning Objectives
1. Define engineering stress and engineering strain.
2. State Hooke‘s law, and note the conditions
  under which it is valid.
3. Define Poisson‘s ratio.
4. Given an engineering stress - strain diagram,
  determine (a) the modulus of elasticity, (b) the
  yield strength (0.002 strain offset), and (c) the
  tensile strength, and (d) estimate the percent
  elongation.

                                                  64
           Learning Objectives
5. For the tensile deformation of a ductile
  cylindrical specimen, describe changes in
  specimen profile to the point of fracture.
6. Compute ductility in terms of both percent
  elongation and percent reduction of area for a
  material that is loaded in tension to fracture.
7. Compute the flexural strengths of ceramic rod
  specimens that have been bent to fracture in
  three-point loading.
8. Make schematic plots of the three characteristic
  stress—strain behaviors observed for polymeric
  materials.
                                                      65
         Learning Objectives
9. Name the two most common hardness-
  testing techniques; note two differences
  between them.
10. (a) Name and briefly describe the two
  different microhardness testing techniques,
  and (b) cite situations for which these
  techniques are generally used.
11. Compute the working stress for a ductile
  material.
                                           66
    7.1 Mechanical properties- Introduction

 Subject to force or load
 Consideration: applied load, duration,
  environment
 Testing of materials: ASTM (American
  Society for Testing Materials)




                                              67
7.2 Concepts of Stress and Strain
    Principal ways
     which load may
     be applied:
1.   Tensile loading
2.   Compression
3.   Shearing
4.   Torsion



                                68
          Standard Tension Test
   Normally: circular, rectangular also used
   Narrow uniform center region
   Standard diameter: 12.8 mm,
   length: 4 times diameter, 60 mm
   Gauge length: 50mm




                                                69
     Engineering stress and strain
   Engineering stress: σ= F/A0

   Engineering Strain: ε= (li-l0)/l0= ∆l/ l0




                                                70
71
Geometric consideration of the stress
               state
   Normal stress:
    σ‘= σcos2θ
    = σ(1+cos 2 θ)/2

   Shear stress:
    τ‘= σsin θ cosθ
    = σ(sin 2 θ/2)


                                    72
73
74
          Stress-strain behavior
   σ=Eε
    E: modulus of
    elasticity




                                   75
          Stress-strain behavior
   Nonlinear behavior:
    Tangent or secant
    modulus




                                    76
                          Fig 7.6
         Stress strain behavior
 Atomic scale: small changes in the interatomic
  spacing and stretching of interatomic bonds.
 Modulus of elasticity: measure of the resistance
  to separation of adjacent atoms/ions/molecules
 Modulus is proportional to the slope of the
  interatomic force-separation curve at equilibrium
  spacing:
 E proportional to (dF/dr)r0



                                                  77
Force vs. interatomic separation




                                   78
79
            Stress strain behavior
 As would be expected, the imposition of compressive,
  shear, or torsional stresses also evokes elastic behavior.
 The stress—strain characteristics at low stress levels are
  virtually the same for both tensile and compressive
  situations, to include the magnitude of the modulus of
  elasticity.
 Shear stress and strain are proportional to each other
  through the expression


   where G is the shear modulus, the slope of the linear
    elastic region of the shear stress—strain curve, Table 7.1
    also gives the shear moduli for a number of the common
    metals.
                                                             80
            7.4 Anelasticity
 Anelasticity: elastic deformation continue
  after load applied, and upon load release
  some finite time is required for complete
  recovery.
 Time dependent: loading rate dependent
 Viscoelastic behavior




                                               81
                   Anelasticity
 Up to this point. it has been assumed that elastic
  deformation is time independent, that is, that an applied
  stress produces an instantaneous elastic strain that
  remains constant over the period of time the stress is
  maintained.
 It has also been assumed that upon release of the load
  the strain is totally recovered, that is. that the strain
  immediately returns to zero.
 In most engineering materials, however. there will also
  exist a time-dependent elastic strain component. That is,
  elastic deformation will continue after the stress
  application, and upon load release some finite time is
  required for complete recovery.
                                                          82
                  Anelasticity
 This time-dependent elastic behavior is known
  as anelasticity and it is due to time-dependent
  microscopic and atomistic processes that are
  attendant to the deformation.
 For metals the anelastic component is normally
  small and is often neglected.
 However, for some polymeric materials its
  magnitude is significant; in this case it is termed
  viscoelastic behavior, which is the discussion
  topic of Section 7.15.

                                                    83
     Elastic behavior of materials
 Tensile stress in z direction
 Constriction in the lateral
  direction
 Poisson ratio:
  ν=-εx/ εz= =-εy/ εz
 Theoretically: isotropic
  material ν=0.25, max= 0.5
 Many metal and alloy: 0.25 to
  0.35


                                     84
               Poisson ratio
 For isotropic materials:
                  E=2G(1+ν)
 In most metal: G is about 0.4E
 Elastically anisotropic: E varies with
  crystallographic direction,




                                           85
Mechanical Behavior:
      Metals




                       86
     Mechanical Behavior: Metals
 For most metallic materials, elastic deformation persists
  only to strains of about 0.005.
 As the material is deformed beyond this point, the stress
  is no longer proportional to strain (Hooke‘s law, Equation
  7.5. ceases to he valid), and permanent, nonrecoverable.
  or plastic deformation occurs.
 Figure 7.10a plots schematically the tensile stress—
  strain behavior into the plastic region for a typical metal.
 The transition from elastic to plastic is a gradual one for
  most metals: some curvature results at the onset of
  plastic deformation. which increases more rapidly with
  rising stress.

                                                            87
88
    Mechanical Behavior: Metals
 From an atomic perspective, plastic deformation
  corresponds to the breaking of bonds with
  original atom neighbors and then reforming
  bonds with new neighbors as large numbers of
  atoms or molecules move relative to one
  another; upon removal of the stress they do not
  return to their original positions.
 This permanent deformation for metals is
  accomplished by means of a process called slip,
  which involves the motion of dislocations as
  discussed in Section 8.3.
                                                89
    Mechanical properties of Metal
 Yielding: proportional limit (Fig 7.10 a & b)
 Tensile strength: the stress at the maximum on
  the engineering stress-strain curve. (Fig 7.11)
 Ductility: percent elongation or percent
  reduction in area
 Ductility:
       Deform plastically before fracture
       Degree of allowable deformation during fabrication
   Brittle materials: fracture strain less than 5%
                                                             90
92
     Temperature on mechanical
            properties
 Table 7-2: room temperature values of
  yield strength, tensile strength, and
  ductility
 Sensitive to prior deformation, presence of
  impurities, and/or heat treatment
 Modulus is not sensitive to them
 Yield and tensile strength decline with
  temp.
 Ductility increase with temp.
                                            93
Table 7.2 Room-Temperature Mechanical
         Properties (in Tension)




                                    94
Table 7.2 Room-Temperature Mechanical
         Properties (in Tension)




                                    95
Table 7.2 Room-Temperature Mechanical
         Properties (in Tension)




                                    96
Temperature effects on stress strain curves
   Engineering stress-strain behavior for iron at 3
    temperatures: Tensile strength, Yield strength ↘;
    ductility ↗ with tempearture↗




                                                        97
      Resilience and toughness
 Resilience: capacity of a material to
  absorb energy when it is deformed
  elastically.
 Modulus of resilience: Ur = ∫0εσdε
 Linear elastic region: Ur = 1/2 σyεy
 Ur = 1/2 σyεy=1/2 σy(σy/E)= σy2/2E
 Resilience materials: high yield strengths
  and low moduli, such as spring
                                               98
Resilience




             99
Toughness: materials absorb energy
          up to fracture




                                 100
        True stress and strain
 Pass point M, material become weaker
  (Fig 7.11)
 In fact, increase in strength, however,
  reduction in cross sectional area
 Resulting reduction in the load-bearing
  capacity
 True stress: σT : load divided by the
  instantaneous cross-sectional area Ai
                                            101
102
          True stress and strain
                       σT = F/Ai
                       εT = ln(li/l0)


If no volume change
                       Aili = A0l0
                       
         True and engineering stress and strain
                  σT = σ(1+ ε)
                   εT = ln (1+ ε)
              Only valid before necking

                                                   103
            True stress and strain
   Corrected stress within the neck is lower than the stress
    from applied load and real neck cross-sectional area




                   Corrected stress: take into account the
                   complex stress state without neck region



                                                              104
        True stress and strain
 True Stress-strain curve from onset of
  plastic deformation to beginning of
  necking can be approximated by
 σT= K εTn
 Table 7-4




                                           105
106
    Mechanical behavior- Ceramics
 Flexural strength
 Stress strain behavior not ascertained by
  tensile test:
     Difficult to prepare specimens having
      requiring geometry
     Difficult to grip brittle materials without
      fracture them
     0.1% strain, perfect align to avoid bending is
      difficult
                                                       107
Bending test




               108
       Bending (Flexure) test
 Transverse bending
 Rod specimens bent in 3-point or 4-point
 Top part under compression
 Bottom part under tension
 Tension strength is about 1/10 of
  compression strength
 The stresses at fracture using this flexure
  test: Flexural strength, modulus of rupture,
  fracture strength, or bend strength
                                            109
           Flexural strength
 Stress: Mc/I
 For rectangular cross section:
                σfs=3FfL/2bd2

 Circular cross section
                 σfs=3FfL/πR3
 The tensile stresses of ceramics in Table
  7.2 are flexural stress
                                              110
Porosity on the mechanical properties of ceramics

 Ceramics fabrication: from powder
 During ensuring heat treatment, much of
  this porosity will be eliminated.
 In case of some residual porosity:
  deleterious influence on elastic properties
  and modulus
 The modulus is function of volume fraction
  porosity
 E= E0 (1-1.9P + 0.9 P2)
                                                111
Influence of porosity on the modulus




                                   112
Porosity on the mechanical properties
             of ceramics
   Porosity decrease the flexural strength:
     Reduce  the cross sectional area
     Stress concentrators (about 2)

 Influence of porosity on strength is rather
  dramatic: 10 vol% porosity decrease 50%
  strength
 σfs= σoexp(-nP)


                                               113
Porosity on the flexural strength




                                    114
115
Bending test




               116
    Mechanical behavior: Polymer
 Mechanical characteristics are sensitive to:
  loading rate, temperature, environment
 Some modification of the testing
  techniques, specimen configurations are
  necessary.
 Especially for highly elastic materials, such
  as rubber


                                              117
     3 typical stress strain curves
A: Brittle polymer
B: Plastic polymers
C: Totally elastic: rubber like elasticity, elastomers




                                                         118
              Plastic polymer
 Yield point taken as a
  maximum on the
  curve, beyond the
  termination of the
  linear-elastic region
 Tensile strength: at
  fracture
 Tensile strength may
  be greater or less
  then yield strength

                                119
                 Polymer vs. metal
   E:
        7M – 4G
         (polymer)
        100s G
         (metal
   Tensile
    strength:
        100Mpa
         (Polymer)
        4100 MPa
         (Metal)

                                     120
    Temperature effect at polymer
 Polymers are more sensitive to temperature.
 Increase temperature produce:
       Decrease elastic modulus
       Reduce tensile strength
       Enhance ductility
   Decreasing rate of deformation has the same
    influence on the stress-strain behavior as
    increasing temperature, softer and ductile


                                                  121
PMMA (poly methyl methacrylate)




                              122
      Viscoelastic deformation
 An amorphous polymer: like glass at low
  temperature; elastic, conformity to
  Hooke‘s law, σ= E ε
 Rubbery solid: intermediate temperature;
  combine elastic and viscous, viscoelaticity
 Viscous liquid: higher temperature;
  viscous or liquidlike behavior


                                                123
Viscoelastic deformation




                           124
      Viscoelastic deformation




Relaxation Modulus   Creep Modulus   125
      Viscoelastic deformation
 B: elastic deformation: instantaneous,
  total deformation occurs the instant the
  stress is applied or release; upon release,
  the deformation is totally recovered
 D: viscous deformation: deformation or
  strain is not instantaneous; response to an
  applied stress, deformation delay or
  depends on time. Also the deformation is
  not reversible
                                            126
         Viscoelastic deformation
   C: Viscoelastic: instantaneous elastic strain
    once load, followed by a viscous time-
    dependent strain, a form of anelasticity

   Silicone polymer (silly putty): roll into a
    ball and drop onto a surface, it bounce
    elastically; it can also be pulled like a
    rubber
                                                  127
    Viscoelastic relaxation modulus
 Stress relaxation: a specimen is initially
  strained rapidly in tension to a low stain
  level, the stress necessary to maintain the
  strain level is measured as function of
  time, which temperature is hold constant
 relaxation modulus Er(t) = σ(t) /ε0
 The magnitude of the relaxation modulus
  is a function of temperature
                                            128
    Viscoelastic relaxation modulus
 The decrease of Et (r)
  with time
 Lower Et (r) level with
  increasing
  temperature




                                      129
                         Viscoelastic
   Amorphous polystyrene:
       Glassy region: rigid and
        brittle, independent of
        temp.
       Leathery or glass transition:
        deformation time
        dependent and not total
        recoverable
       Rubbery; elastic and
        viscous present
       Rubbery flow and viscous
        flow


                                        130
Relaxation




             131
            Relaxation modulus:
 crystalline, lightly cross-link, and amorphous
Polystyrene materials having several molecular
  configurations
 (C): amorphous
 (B): lightly cross linked atactic polystyrene, the
  rubbery region form a plateau that extends to
  the temperature at which the polymer
  decomposes, no melting
 (A): totally crystalline isotactic PS: glass temp.
  less obvious (small volume of amorphous),
  relaxation modulus is relatively high.

                                                       132
         Relaxation modulus:
crystalline, lightly cross-link, and amorphous




                                                 133
           Viscoelastic creep
 Viscoelastic creep: susceptible to time-
  dependent deformation when stress level
  is maintained constant
 Stress is applied instantaneously and is
  maintained at control level while strain is
  measured as function of time
 Creep compliance: Ec(t) = σ0/ε(t)


                                                134
Creep




        135
                       Hardness
 Measure of a material‘s resistance to localized
  plastic deformation (small dent or scratch)
 Natural minerals: Mohs scale
 Reasons for hardness test:
       Simple and inexpensive
       Relatively nondestructive
       Other mechanical properties may be estimated from
        hardness data, like tensile strength



                                                            136
        Rockwell hardness test
 Most common, simple
 Several different scales from combination of
  various indenters and loads
 Indenters include all metal alloy and conical
  diamond (Brale)
 Hardness number is determined by difference in
  depth of penetration resulting from the
  application of initial minor load followed by a
  larger major load

                                                137
Hardness test technique




                          138
Rockwell hardness scale




                          139
Superficial Rockwell hardness scale




                                      140
      Rockwell hardness scale
 When specifying Rockwell and superficial
  hardness: both hardness number and
  scale symbol must be indicated
 80 HRB: Rockwell hardness of 80Kg on B
  scale
 60 HR30W: superficial hardness of 60 on
  the 30W scale


                                             141
        Other hardness tests
 Brinell hardness test
 Knoop and Vickers micro hardness test




                                          142
Hardness conversion-   ASTM E140




                                   143
    Correlation between hardness and TS

 For Steel:
 TS(MPa) = 3.45 X HB
       HB: Brinell hardness
   TS(psi)=500 X HB




                                          144
          Design/safety factors
 σw = σy/N
 N: factor of safety, 1.2 to 4




                                  145
146

				
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