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```									      Biomaterials
Week 4
10/5/2009
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.
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
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
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

20
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

21
In the present treatment, the concentration
profile is assumed to be linear,

22
△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
For steady state diffusion in a single direction is
relatively simple, the flux is proportional to the
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
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
J = -D (dC)/ (dx)                (6.3)

25
Diffusion flux computation

26
Most practical diffusion situations are
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
Concentration profiles at three different diffusion times.

28
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
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.
– 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
(∂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

32
The concentration parameters appear in Eq 6.5
are noted in Fig 6.6, a concentration profile
taken at a specific time

33
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
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
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

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
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
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
environment
 Testing of materials: ASTM (American
Society for Testing Materials)

67
7.2 Concepts of Stress and Strain
    Principal ways
be applied:
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
 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
 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
 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

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

 Influence of porosity on strength is rather
dramatic: 10 vol% porosity decrease 50%
strength
 σfs= σ0exp(-nP)

113
Porosity on the flexural strength

114
115
Test I time change
 Date: 11/7/2007
 Time: 9:10 AM, 80 minutes
 Format: Close book, Calculator needed.
Dictionary allowed
1. Multiple choice
2. Definition of general terms
3. Calculation: formula and table will be given if
needed

116
117
Bending test

118
Mechanical behavior: Polymer
 Mechanical characteristics are sensitive to:
 Some modification of the testing
techniques, specimen configurations are
necessary.
 Especially for highly elastic materials, such
as rubber

119
3 typical stress strain curves
 A: Brittle polymer
 B: Plastic polymers
 C: Totally elastic: rubber like elasticity,
elastomers

120
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

121
Polymer vs. metal
   E:
    7M – 4G
(polymer)
    100s G
(metal
   Tensile
strength:
    100Mpa
(Polymer)
    4100 MPa
(Metal)

122
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

123
PMMA (poly methyl methacrylate)

124
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

125
Viscoelastic deformation Viscoelastic
deformation

126
Viscoelastic deformation Viscoelastic
deformation

Relaxation Modulus   Creep Modulus   127
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
128
Viscoelastic deformation
   C: Viscoelastic: instantaneous elastic strain
once load, fellow 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
129
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
130
Viscoelastic relaxation modulus
 The decrease of Et (r)
with time
 Lower Et (r) level with
increasing
temperature

131
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

132
Relaxation modulus:
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.

133
Relaxation

134
Relaxation modulus:

135
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)

136
Creep

137
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

138
Rockwell hardness test
 Most common, simple
 Several different scales from combination of
 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

139
Hardness test technique

140
Rockwell hardness scale

141
Superficial Rockwell hardness scale

142
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

143
Other hardness tests
 Brinell hardness test
 Knoop and Vickers micro hardness test

144
Hardness conversion-   ASTM E140

145
Correlation between hardness and TS

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

146
Design/safety factors
 σw = σy/N
 N: factor of safety, 1.2 to 4

147
Biomaterials HW #4
Due: Next week (10/24)
1. Q. 6.8
2. Q.6.11
3. Of the materials listed in Table 7.3.
1. Which is the least strong? Why?
2. Which is the least stiff? Why
3. Which will experience the smallest percent reduction
in area? And why?
4. Which one is the hardest? And why?

4. 7.28 (use SI unit)
5. 7.46
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