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Midterm

MATLS 1M03 Study Guide - Midterm Guide: Shear Modulus, Shear Stress, Modulus Guitars


Department
Materials Science and Engineering
Course Code
MATLS 1M03
Professor
Joey Kish
Study Guide
Midterm

Page:
of 2
= 1.6 x 10-19 C.
145 psi = 1MPa = 10
6
N / m
2
1 GPa = 10
9
N / m
2
There are three principal ways in which a load may be applied: namely, tension,
compression and shear.
Stress is a measure of an applied mechanical load or force, normalized to take into
account cross-sectional area.
Tension Tests:
Engineering Stress = F / A
0
F = Force applied perpendicular to the specimen cross section (N)
A
0
= the original cross-sectional area before any load is applied (m
2
)
= Engineering Stress (1MPa = 10
6
N / m
2
)
Engineering Strain = (l
i
– l
0
) / l
0
= l / l
0
l
0
= original length before any load is applied (m) l
i
= instantaneous length
= Engineering strain (unitless or m/m) (can also be expressed as a percentage)
Compression Tests: Same as tension just negative
Shear and Torsional Tests: = F / A
0
F = Force or load imposed parallel to the upper and lower faces (N)
A
0
= Area of faces (m
2
) = Shear stress
Shear strain is defined as the tangent of the strain angle .
Torsion is a variation of pure shear, wherein a structural member is twisted,
torsional forces produce a rotational motion about the longitudinal axis of one end
of the member relative to the other end.
Elastic Deformation
Stress-Strain Relationship = E
E = Constant of Proportionality / Modulus of elasticity / Young’s Modulus (GPa or
psi) E typically ranges from 45GPa to 407GPa
If there is a constant modulus then it is called elastic deformation. The greater the
modulus the stiffer the material, or the smaller the elastic strain that results from the
application of a given stress. Elastic deformation is nonpermanent. E is
proportional to dF/dr
Values of the modulus of elasticity for ceramic materials is higher than for metals
than for polymers. Increasing temperature causes the modulus of elasticity to
diminish.
= G G = shear modulus, slope of linear elastic region
Anelasticity is known as the time that a material takes to return to its original length
after a stress has been applied to it.
Poisson’s ratio (v) is the ratio of lateral and axial strains. v = -
x
/
z
= -
y
/
z
E = 2G(1+v)
Plastic deformation occurs when strains are over about 0.005, where permanent,
non-recoverable deformation occurs.
Most structures are designated to ensure that only elastic deformation will result
when a stress is applied. It is therefore desirable to know the stress level at which
plastic deformation begins, or were the phenomenon of yielding occurs. For metals
that experience this gradual elastic-plastic transition, the point of yielding may be
determined as the initial departure from linearity of the stress-strain curve; this is
sometimes called the proportional limit.
The stress corresponding to the intersection of this line and the stress-strain curve as
it bends over in the plastic region is defined as the yield strength
y
(MPa or psi).
The elastic-plastic transition is very well defined and occurs abruptly in what is
termed a yield point phenomenon.
After yielding, the stress necessary to continue plastic deformation in metals
increases to a maximum, and then decreases to the eventual fracture. The tensile
strength TS (MPa) is the stress at the maximum on the engineering stress-strain
curve.
Ductility is another important mechanical property. It is a measure of the degree of
plastic deformation that has been sustained at fracture. A material that experiences
very little or no plastic deformation upon fracture is termed brittle (opposite is
ductile). Brittle materials are approximately considered to be those having a
fracture strain of less than 5%. Ductility may be expressed quantitatively as either
percent elongation or percent reduction in area. As with modulus of elasticity, the
magnitudes of both yield and tensile strengths decline with increasing temperature;
just the reverse holds for ductility – it usually increases with temperature.
%EL = (l
f
– l
0
) / l
0
%RA = (A
0
– A
f
) / A
0
Resilience is the capacity of a material to absorb energy when it is deformed
elastically and then, upon unloading, to have this energy recovered.
U
r
= d = ½
y
y
=
2y
/ 2E (J/m
3
)
Toughness is a mechanical term that is used in several contexts; loosely speaking, it
is a measure of the ability of a material to absorb energy up to fracture energy up to
fracture.
Sometimes it is more meaningful to use a true stress-true strain scheme. True stress
T
is defined as
T
= F / A
i
True stain
T
= ln (l
i
/ l
0
)
If A
i
l
i
= A
0
l
0
then
T
= (1 + ) and
T
= ln (1 + )
T
= K
Tn
n and K are constants. n is called the strain-hardening exponent.
Another mechanical property that may be important to consider is hardness, which
is a measure of a material’s resistance to localized plastic deformation (eg dent or
scratch)
= Resistivity (
.
m) V = voltage (V) I = Current (Amps) R = Resistance (V/A)
= electrical conductivity (
.
m)
-1
p = the number of holes per m
3
l = distance between two points at which the voltage is measured A = cross-
sectional area J = current density = electric field intensity
E
F
= Fermi energy E
g
= Band Gap V
d
= drift velocity
e
= constant of proportionality, electron mobility (m
2
/ V
.
s)
n = electron concentration or free electrons / m
3
(m
-3
)
C = Heat capacity T = Temperature Q = Energy A = temperature-independent
constant (
.
m) l = length
l
= linear coefficient of thermal expansion (W / m
.
k)
E = modulus of elasticity q = heat flux k = thermal conductivity
CV = constant volume CP = constant pressure CP > CV
Ohm’s Law V = IR
= V*A / I*l = 1 / J =  = V / l
Metals are good conductors, typically having conductivities on the order of 107 (
.
m)
-1
. At the other extreme are materials with very low conductivities, ranging
between 10
-10
and 10
-20
(
.
m)
-1
; these are electrical insulators. Materials with
intermediate conductivities, generally from 10
-6
to 10
4
(
.
m)
-1
, are termed
semiconductors.
Within most solid materials a current arises from the flow of electrons, which is
termed electric conduction. In addition, for ionic materials a net motion of charged
ions is possible that produces a current; such is termed ionic conduction.
The magnitude of the electrical conductivity is strongly dependent on the number of
electrons available to participate in the conduction process.
As atoms come within close proximity of one another, electrons are acted upon, or
perturbed, by the electrons and nuclei of adjacent atoms. This influence is such that
each distinct atomic state may split into a series of closely spaced electron states in
the solid, to form what is termed an electron energy band.
The energy corresponding to the highest filled state at 0 K is called the Fermi
Energy, E
f
.
Two final band structures are similar; one band (the valance band) that is
completely filled with electrons is separated from an empty conduction band, and
an energy band gap lies between them. For very pure materials, electrons may not
have energies within this gap. The difference between the two structures lies in the
magnitude of the energy gap; for materials that are insulators, the band gap is
relatively wide, whereas for semiconductors it is narrow. The Fermi energy for
these two band structures lies within the band gap – near its center.
Only electrons with energies greater than the Fermi energy may be acted on and
accelerated in the presence of an electric field. These are the electrons that
participate in the conduction process, which are termed free electrons. Another
charged electronic entity called a hole is found in semiconductors and insulators.
Holes have less energy than E
f
and also participate in electric conduction.
For insulators and semiconductors, empty states adjacent to the top of the filled
valance band are not available. To become free, therefore, electrons must be
promoted across the energy band gap and into empty states at the bottom of the
conduction band.
Increasing the temperature of either a semiconductor or an insulator results in an
increase in the thermal energy that is available for electron excitation.
When an electric field is applied, a force is brought to bear on the free electrons; as
a consequence, they all experience an acceleration in a direction opposite to that of
the field, by virtue of their negative charge.
Several parameters are used to describe the extent of this scattering; these include
the drift velocity and the mobility of an electron. The drift velocity v
d
represents the
average electron velocity in the direction of the force imposed by the applied field.
v
d
=
e
= n||
e
Matthiessen’s rule
total
=
t
+
i
+
d
t = thermal i = impurity d = deformation
=
0
+ aT
i
= A*c
i
(1 – c
i
)
i
=
V
+
V
Intrinsic semiconductors are those in which the electrical behaviour is based on the
electronic structure inherent to the pure metal. When the electrical characteristics
are dictated by impurity atoms, the semiconductor is said to be extrinsic.
= n|e|
e
+ p|e|
h
Chapter 19 – Thermal Properties
Heat Capacity is a property that is indicative of a material’s ability to absorb heat
from the external surroundings; it represents the amount of energy required to
produce a unit temperature rise.
C = dQ / dT
Specific Heat (often denoted by lowercase c) is sometimes used; this represents the
heat capacity per unit mass and has various units.
Only certain energy values are allowed (the energy is said to be quantized), and a
single quantum of vibrational energy is called a phonon.
C
v
= AT
3
A is a constant C
v
= constant volume Temperature is
in degrees Calvin
l
f
– l
0
/ l
0
=
l
* (T
f
– T
0
)
V / V
0
=
v
T
Brittle materials (ceramics) may experience fracture as a consequence of non-
uniform dimensional changes in what is termed thermal shock, as discussed later in
the chapter.
Thermal conduction is the phenomenon by which heat is transported from high-to-
low temperature regions of a substance. The property that characterizes the ability
of a material to transfer heat is the Thermal Conductivity.
q = -k (dT/dx) J = -D (dC/dx)
q = heat flux (W/m
2
) k = constant (W/m
2
) dT/dx = temperature gradient
A thermal conductivity is associated with each of these mechanisms, and the total
conductivity is… k = k
l
+ k
e
k
l
= lattice vibration k
e
= electron thermal conductivities
Wiedemann-Franz law: L = k / T
= electrical conductivity T = temperature L = 2.44x10-8
.
W/(K)
2
Thermal Stress are stresses induced in a body as a result of changes in temperature.
= E
l
(T
0
-T
f
) = E
l
T
Thermal Shock Resistance
TSR =
f
k / E
l
Tutorials
Whenever the energy of an electric field in a metal is greater than the Fermi Energy
(E
f
), electrons will jump to the conduction band.
Competing Effects:
As the temperature of a material increases, more charge carriers are promoted or
created in the material. This is good for conductivity. However, as the temperature
goes up those charge carriers scatter more – colliding with each other. This is bad
for conductivity.
Semiconductors that can promote charge carriers by absorbing light energy are
called photoconductors.
E = hc /
E = V / E
T
(h
ph
/
ph
+ h
T
/
T
+ h
p
/
p
)
The end result is that the hot side of the conductor becomes slightly positively
charged, and the cooler side becomes slightly negatively charged.
Seebeck Effect – two different metals are joined and kept at the same temperature.
If the temperatures are different, there will be a heat current in each metal.
Which of the following tests are most useful for quality control purposes, the
requirements being that they are both quick and cheap? Circle the letter of each
answer that applies. Marks will be deducted for incorrect answers.
a) tensile test b) Charpy impact test c) Fracture toughness test d) Hardness test
Some of the following terms describes a material property, i.e. they are parameters
whose value is characteristic of the material only (and not dependent of the size and
shape of the component), while others are not. Circle the letter of each parameter
which is a material parameter. Circle all those that apply. Marks will be deducted
for incorrect answers. a) resolved shear stress b) hardness c) elastic stiffness d)
fracture toughness
Solute strengthening depends on the difference in size between solute and host,
independent of which one is larger.
Materials deformation is usually caused by dislocation motion.
The strategy of using dislocations to impede the dislocation motion is named cold
work hardening or strain hardening
When referring to the fracture of materials, fatigue means materials fail because
they are subject to cyclic loading.
A Poisson’s ratio of 0.5 means that deformation occurs at a constant volume.
Foams typically have a Poisson’s ratio close to zero. Poisson’s ratio is a material
parameter.
Yield Stress (Y) Ultimate Tensile Stress (U) Elongation (E)