SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 December 1997 (Version 1c) compiled by Michael G. Jenkins, University of Washington page 1 / 36
NOTE: These notes represent selected highlights of ME354 and are not intended to replace conscientious
study, attendance of lecture, reading of the textbook, completion of homework assignments, and performance
of laboratory work. These notes are corrected, modified, and upgraded periodically with date and latest version
number appearing in the header.
Mechanics of Materials  a branch of mechanics that develops relationships
between the external loads applied to a deformable body and the intensity of internal forces
acting within the body as well as the deformations of the body
External Forces  classified as two types: 1) surface forces produced by a) direct
contact between two bodies such as concentrated forces or distributed forces and/or
b) body forces which occur when no physical contact exists between two bodies (e.g.,
magnetic forces, gravitational forces, etc.).
Internal Forces  non external forces acting in a body to resist external loadings
Support Reactions  surface forces that develop at the support or points of support
between two bodies. Support reactions may include normal forces and couple moments.
Equations of Equilibrium  mathematical expression of vector relations showing that
for a body not to translate or move along a path then
F
0
M
0
F
x
0
F
y
0
F
z
0
M
x
0
M
y
0
M
z
0
Some nomenclature used in these notes
Roman characters
a  crack length; A area; A
f
 final area; A
o
 original area; c  distance from
neutral axis to farthest point from neutral axis or Griffith flaw size; C center of Mohr's circle;
E elastic modulus (a.k.a., Young's modulus); F  force or stress intensity factor coefficient;
FS  factor of safety; G  shear modulus (a.k.a. modulus of rigidity); I  moment of inertia;
J  polar moment of inertia; K  strength coefficient for strain hardening; K  stress intensity
factor, k  bulk modulus; L  length; L
f
 final length; L
o
 original length; M or M(x)  bending
moment; m  metre (SI unit of length) or Marin factor for fatigue; N  Newton (SI unit of force)
or fatigue cycles; N
f
 cycles to fatigue failure; n  strain hardening exponent or stress
exponent; P  applied load; P
cr
 critical buckling load; P
SD
 SherbyDorn parameter;
P
LM
 LarsonMiller parameter; p  pressure; Q  first moment of a partial area about the
neutral axis or activation energy; R  radius of Mohr's circle or radius of shaft/torsion
specimen or stress ratio; S
f
 fracture strength; S
uts
or S
u
 ultimate tensile strength;
r  radius of a cylinder or sphere; S
y
 offset yield strength; T  torque or temperature;
T
mp
 melting temperature; t  thickness of cross section or time; t
f
 time to failure;
U  stored energy; U
r
 modulus of resilience; U
t
 modulus of toughness; V or V(x)  shear
force; v or v(x)  displacement in the "y" direction; w(x)  distributed load; x or X  coordinate
direction or axis; y or Y  coordinate direction or axis; z or Z  coordinate direction or axis;
Greek characters
 change or increment;  normal strain or tensoral strain component;
 normal strain at
;  angle or angle of twist;  engineering shear strain;
 Poisson's ratio;  angular velocity;  variable for radius or radius of curvature;
 normal stress;
,
,
 greatest, intermediate, and least principal normal stresses;
' effective stress;
 proportional limit, elastic limit, or yield stress;  shear stress;
max
 maximum shear stress;
o
 yield shear strength;  angle;
p
 principal normal
stress angle;
s
 maximum shear stress angle
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 December 1997 (Version 1c) compiled by Michael G. Jenkins, University of Washington page 2 / 36
Stress
Stress: i) the ratio of incremental force to incremental area on which the force acts such
that:
lim
A 0
F
A
.
ii) the intensity of the internal force on a specific plane (area) passing through
a point.
Normal Stress: the intensity of the internal force acting normal to an incremental area
such that:
lim
A 0
F
n
A
Note: +tensile stress = "pulling" stress
and compressive stress = "pushing" stress
Shear Stress: the intensity of the internal force acting tangent to an incremental area
such that:
lim
A 0
F
t
A
General State of Stress: all the internal stresses acting on an incremental element
z
x
y
zy
zx
yx
xy
xz
yz
y
z
x
Note: A + acts normal to a positive face in the positive coordinate direction
and a +acts tangent to a positive face in a positive coordinate direction
Note: Moment equilibrium shows that
xy
yx
;
xz
zx
;
yz
zy
Complete State of Stress: Six independent stress components
(3 normal stresses,
x
;
y
;
z
and
3 shear stresses,
xy
;
yz
;
xz
) which uniquely
describe the stress state for each particular orientation
Units of Stress: In general:
Force
Area
F
L
2
,
In SI units,
Pa
N
m
2
or MPa 10
6
N
m
2
N
mm
2
In US Customary units,
psi
lb
f
in
2
or ksi 10
3
lb
f
in
2
kip
in
2
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 December 1997 (Version 1c) compiled by Michael G. Jenkins, University of Washington page 3 / 36
Stress Transformation
For the plane stress condition (e.g., stress state at a surface where no load is supported on
the surface), stresses exist only in the plane of the surface (e.g.,
x
;
y
;
xy
)
The plane stress state at a point is uniquely represented by three components acting on a
element that has a specific orientation (e.g., x, y) at the point. The stress transformation
relation for any other orientation (e.g., x', y') is found by applying equilibrium equations
( F 0 and
M 0 )
F
n
A and F
t
A
X
y
x'
y'
A
Ay=A sin
Ax=A cos
Rotated coordinate
axes and areas for
x and y directions
X
y
x'
y'
y Ay
s
x Ax
t
xy Ay
s
x' A
t
x'y' A
q
q
q
q
t
xyAx
Rotated coordinate
axes and components of
stress/forces for
original coordinate axes
F
x'
0
gives
x'
x
cos
2
y
sin
2
2
xy
cossin or
x'
x
y
2
x
y
2
cos2
xy
sin 2
F
y'
0
gives
x'y'
(
x
y
)cos sin
xy
(cos
2
sin
2
) or
x'y'
x
y
2
sin2
xy
cos2
Similarly, for a cut in the y' direction,
y'
x
sin
2
y
cos
2
2
xy
cossin or
y'
x
y
2
x
y
2
cos2
xy
sin 2
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 December 1997 (Version 1c) compiled by Michael G. Jenkins, University of Washington page 4 / 36
Principal Normal Stress  maximum or minimum normal stresses acting in principal
directions on principal planes on which no shear stresses act.
Note that
1
2
3
For the plane stress case
1,2
x
y
2
x
y
2
2
xy
2
and tan 2
p
2
xy
x
y
and
max
x
y
2
2
xy
2
,
ave
=
x
y
2
and tan 2
s
x
y
2
xy
Mohr's Circles for Stress States  graphical representation of stress
Examples of Mohr's circles
1
2
max
for xy plane
Mohr's circle for stresses in xy plane
1
3
max
=
Mohr's circle for stresses in xyz planes
2
1
3
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 December 1997 (Version 1c) compiled by Michael G. Jenkins, University of Washington page 5 / 36
Graphical Description of State of
Stress
In this example all stresses acting in axial
directions are positive as shown in Fig. 1.
2D Mohr's Circle
xy
y
x
X
Y
Fig. 1 Positive stresses acting
on a physical element.
As shown in Figs. 2 and 3, plotting actual
sign of the shear stress with x normal stress
requires plotting of the opposite sign of the
shear stress with the y normal stress on the
Mohr's circle.
x
y
xy
yface
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 December 1997 (Version 1c) compiled by Michael G. Jenkins, University of Washington page 6 / 36
1
2
Y
X
Direction
of
The direction of physical angle, is from
the xy axes to the principal axes.
Fig. 4  Orientation of physical element
with only principal stresses
acting on it.
Principal
Axis
Line of XY
Stresses
Direction of
Fig. 5  Direction of from the line of xy
stresses to the principal stress
axis.
Note that the sense (direction) of the
physical angle, is the same as on the
Mohr's circle from the line of the xy stresses
to the axes of the principal stresses.
strain except interchange variables as
Same relations apply for Mohr's circle for
and
2
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 7 / 36
Strain
Strain: normalized deformations within a body exclusive of rigid body displacements
Normal Strain: elongation or contraction of a line segment per unit length such that
lim
B A along n
A'B'
AB
AB
L
f
L
o
L
o
and a volume change results.
Note: +tensile strain = elongation
and compressive strain = contraction
Shear Strain: the angle change between two line segments such that
(
2
) '
h
(for small angles )
and a shape change results.
Note: +occurs if
2
'
and occurs if
2
'
General State of Strain: all the internal strains acting on an incremental element
A
xy
yx
x
y
Engineering shear strain,
xy
xy
yx
Complete State of Strain: Six independent strain components
(3 normal strains,
x
;
y
;
z
and
3 engineering shear strains,
xy
;
yz
;
xz
) which uniquely
describe the strain state for each particular orientation
Units of Strain: In general:
Length
Length
L
L
,
In SI units,
m
m
for and
m
m
or radian for
In US Customary units,
in
in
for and
in
in
or radian for
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 8 / 36
Strain Transformation
For the plane strain condition (e.g., strain at a surface where no deformation occurs normal
to the surface), strains exist only in the plane of the surface (
x
;
y
;
xy
)
The plane strain state at a point is uniquely represented by three components acting on a
element that has a specific orientation (e.g., x, y) at the point. The strain transformation
relation for any other orientation (e.g., x', y') is found by summing displacements in the
appropriate directions keeping in mind that
L
o
and h
x
y
y'
x'
Q
Q*
dy
x
x
dy
dx
ds
dy
dx
Rotated coordinate
axes and displacements
for x and y directions
Q
Q*
x'
y
x
x
x
y
y
{
x'
x'
dx
dy
ds
dy
cos
sin
ds
ds
dx
dy
Displacements in
the x' direction
for strains/ displacements
in the x and y directions
displacements in x'direction for Q to Q *
gives
x'
x
cos
2
y
sin
2
xy
cossin or
x'
x
y
2
x
y
2
cos2
xy
2
sin2
rotation of dx' and dy'
gives
x'y'
2
(
x
y
)cos sin
xy
2
(cos
2
sin
2
) or
x'y'
2
x
y
2
sin 2
xy
2
cos2
Similarly,
displacements in y'direction for Q to Q *
gives
y'
x
sin
2
y
cos
2
xy
cossin or
y'
x
y
2
x
y
2
cos2
xy
2
sin2
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 9 / 36
Principal Normal Strain  maximum or minimum normal strains acting in principal
directions on principal planes on which no shear strains act.
Note that
1
2
3
For the plane strain case
1,2
x
y
2
x
y
2
2
xy
2
2
and tan 2
p
xy
x
y
and
max
2
x
y
2
2
xy
2
xy
2
,
ave
=
x
y
2
and tan2
s
x
y
xy
Mohr's Circles for Strain States  graphical representation of strain
Examples of Mohr's circles
1
2
max
/2 for xy plane
Mohr's circle for strains in xy plane
1
3
max
=
Mohr's circle for strains in xyz planes
2
1
3
Strain Gage Rosettes
Rosette orientations and equations relating xy coordinate strains to
the respective strain gages of the rosette
45¡
60¡
60¡
a
b
c
a
b
c
45¡ Rectangular
60¡ Delta
x
y
x
x
a
x
a
y
c
y
1
3
(2
b
2
c
a
)
xy
2
b
(
a
c
)
xy
2
3
(
b
c
)
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 10 / 36
Continuum Mechanics and Constitutive Relations
Equations which relate stress and strain (a.k.a., Generalized Hooke's Law)
C
x
E
(1)
x
E
(1)(12)
(
x
y
z
)
y
E
(1)
y
E
(1)(12)
(
x
y
z
)
z
E
(1 )
z
E
(1 )(12)
(
x
y
z
)
xy
G
xy
yz
G
yz
xz
G
xz
S
x
1
E
x
(
y
z
)
y
1
E
y
(
x
z
)
z
1
E
z
(
x
y
)
xy
1
G
xy
yz
1
G
yz
xz
1
G
xz
C
S
1
and S
C
1
Elastic relation (1D Hooke's Law) =E
Plastic relation (Strain Hardening) =K
n
Poisson's ratio, = 
transverse
longitudinal
Plane stress :
z
0,
z
0
1
(
x
y
)
Plane strain :
z
0,
z
0 (
x
y
)
Stressstrain relations
for plane stress (x y plane)
x
E
(1
2
)
x
y
y
E
(1
2
)
y
x
z
xz
yz
0
xy
G
xy
Elastic Modulus, E=
Poisson's ratio, = 
lateral
longitudinal
Shear Modulus, G=
E
2(1)
Bulk Modulus, k =
x
y
x
3
x
y
x
E
3(12)
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 11 / 36
PLASTIC DEFORMATION
Non recoverable deformation beyond the point of yielding where Hooke's law
(proportionality of stress and strain) no longer applies. Flow curve is the true stress vs. true
strain curve describing the plastic deformation.
Simple Power Law
Strain
Strain
Hardening
p
e
Elastic: E (
o
)
Plastic: H
n
(
o
)
Approximate flow curves
T
RigidPerfectly Plastic
T
ElasticPerfectly Plastic
T
ElasticLinear Hardening
ElasticPower Hardening
Power
Linear
E
E
RambergOsgood Relationship
Total strain is sum of elastic and plastic
e
p
E
p
s H
p
n
E
H
1
n
Deformation Plasticity
eff
1
2
(
1
2
)
2
(
2
3
)
2
(
3
1
)
2
and
eff
2
3
(
1
2
)
2
(
2
3
)
2
(
3
1
)
2
Effective stresseffective strain curve is independent of the state of stress and is used to
estimate the stressstrain curves for other states of stress.
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 12 / 36
Failure Theories
Two types: Fracture and Yield Criteria. Generally used to predict the safe limits of a
material/component under combined stresses.
Factor of Safety,
FS
Material Strength
Component Stress
, Failure occurs if FS<1
Maximum Normal Stress Criterion
Fracture criterion generally used to predict failure of brittle materials.
FS
S
UTS
MAX(
1
,
2
,
3
)
Maximum Shear Stress (Tresca) Criterion
Yield criterion generally used to predict failure in materials which yield in shear (i.e. ductile
materials)
FS
(
o
= S
y
/2
0
/2)
MAX
1
2
2
,
2
3
2
,
1
3
2
Von Mises (Distortional Energy)
or Octahedral Shear Stress Criterion
Yield criterion generally used to predict failure in materials. which yield in shear (i.e. ductile
materials)
FS
(
o
S
y
)
'
'
1
2
(
1
2
)
2
(
2
3
)
2
(
3
1
)
2
'
1
2
(
x
y
)
2
(
y
z
)
2
(
z
x
)
2
6(
xy
2
yx
2
zx
2
)
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 13 / 36
Mechanical Testing
The results of materials tests (e.g. tensile, compressive, torsional shear, hardness, impact
energy, etc.) are used for a variety of purposes including to obtain values of material
properties for use in engineering design and for use in quality control to ensure materials
meet established requirements
Tensile Testing
P
Ao
Lo
=P/Ao
=(LiLo)/Lo
Mohr's Circle for Uniaxial
Tension
Elastic Modulus :
E
d
d
of the linear part of the stressstrain curve.
Yielding : Proportional limit,
p
; elastic limit; offset yield (S
ys
at 0.2% strain) where
o
is used
to generally designate the stress at yielding.
Ductility :
% elongation
L
f
L
o
L
o
x100
f
x100
or
%RA
A
o
A
f
A
o
x100
Necking is geometric instability at S
UTS
and
U
Strain hardening ratio =
S
UTS
o
where 1.4 is high and 1.2 is low.
Energy absorption (energy/volume):
Modulus of Resilience  Modulus of Toughness
= measure of the ability to  = measure of the ability to
store elastic energy  absorb energy without fracture
= area under the linear portion  = area under the entire
of the stressstrain curve  stressstrain curve
U
R
d
o
o
o
o
2
o
2
2E

U
T
d
o
f
(S
UTS
o
)
f
2
("flat"  curves)
or d
o
f
2S
UTS
f
3
(parabolic  curves)
Strainhardening:
T
K (
T
)
n
H(
T
)
n
H=K=strength coefficient and n = strain hardening exponent (0n1)
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 14 / 36
Representative stressstrain curves for tensile tests of brittle and ductile materials
Brittlle Material
Ductile Material
Strain
Strain
X
X
Sf
Su
Sy
E
Ur
Ut
E
Ur
Ut=Ur
Su=Sy=Sf
Table: Stressstrain definitions for tensile testing
PARAMETER FUNDAMENTAL
DEFINITION
PRIOR TO
NECKING
AFTER
NECKING
Engineering Stress
(
E
)
E
P
i
A
o
E
P
i
A
o
E
P
i
A
o
True Stress (
T
)
T
P
i
A
i
T
P
i
A
i
T
E
(1
E
)
T
P
i
A
neck
Engineering Strain
(
E
)
E
L
L
o
L
i
L
o
L
o
E
L
L
o
L
i
L
o
L
o
E
L
L
o
L
i
L
o
L
o
True Strain (
T
)
T
ln
L
i
L
o
T
ln
A
o
A
i
T
ln
L
i
L
o
T
ln
A
o
A
i
T
ln(1
E
)
T
ln
A
o
A
neck
Note: Subscripts: i=instantaneous, o=original; Superscripts: E=engineering, T=true
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 15 / 36
Hardness Testing
Resistance of material to penetration
Brinell
P=3000 kg
or 500 kg
D=10 mm
d
t
Steel or
tungsten
carbide
ball
BHN HB
P
Dt
2P
D D D
2
d
2
Vickers
P=1120 kg
d=L
136¡=Included
angle of faces
Diamond
pyramid
VHN HV
2P
L
2
sin
2
Rockwell
Requires Rockwell subscript to provide meaning to the Rockwell scale.
Examples of Rockwell Scales
Rockwell Hardness Indentor Load (kg)
A Diamond point 60
B 1.588 mm dia. ball 100
C Diamond point 150
D Diamond point 100
E 3.175 mm dia. ball 100
M 6.350 mm dia. ball 100
R 12.70 mm dia. ball 60
NotchImpact Testing
Resistance of material to sudden fracture in presence of notch
h1
h2
mass, m
IZOD
CHARPY
VNOTCH
IMPACT ENERGY=mg(h1h2)
TEMPERATURE
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 16 / 36
Torsion Testing
=TR/J
=R /L
2R
L
T
Torsional Shear Stress Torsional Shear Strain
TR
J
J
D
4
32
for solid shaft
J
(D
outer
4
D
inner
4
)
32
for tube
R
L
Shear Modulus :
G
=
E
2(1+ )
¥ For linear elastic behaviour, plane sections remain plane, so
R
L
and
TR
J
Modulus of Rupture (maximum shear stress) :
u
T
max
R
J
¥ For nonlinear behaviour, plane sections remain plane, so
R
L
but
TR
J
beyond linear region.
Instead
1
2R
3
(/L)
dT
d(/L)
3T
Modulus of Rupture (maximum shear stress) when dT/d(/L) = 0 so
u
3T
max
2R
3
Table: Comparison of stresses and strains for tension and torsion tests
Tension Test Torsion Test
1
max
;
3
2
0
1
3
;
2
0
max
1
2
max
2
max
2
1
2
max
max
1
;
2
3
1
2
max
1
3
;
2
0
max
3
1
2
max
1
3
2
1
effective stress
eff
1
2
(
1
2
)
2
(
2
3
)
2
(
3
1
)
2
effective strain
eff
2
3
(
1
2
)
2
(
2
3
)
2
(
3
1
)
2
1
s 3
1
e
1
e
2
3
1
3
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 17 / 36
Compression Testing
P
Ao
Lo
=P/Ao
=(LiLo)/Lo
P
1
No necking and maximum load may not occur since pancaking allows load to keep
increasing. For many metals and polymers, the compressive stress and strain relations are
similar to those in tension (including elastic constants, ductility, and yield). For other
materials, such as ceramics, glasses, and composites (often at elevated temperatures),
compression behavior may be quite different than tensile behavior.
In an ideal column (no eccentricity) the axial load, P, can be increased until failure occurs
wither by fracture, yielding or buckling. Buckling is a geometric instablity related only to the
elastic modulus (stiffness) of the material and not the strength.
P
cr
2
EI
(KL)
2
or
cr
2
E
(KL/r)
2
where (L/r) is the slenderness ratio
and (KL/r) is the effective slenderness ratio
Sometimes, L
e
=KL is the effective length.
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 18 / 36
Creep and Time Dependent Deformation
Time dependent deformation under constant load or stress at temperatures greater than 30
and 60% of the melting point (i.e. homologous temperatures, T/T
mp
>0.30.6)
TIME, t
STRAIN
III
II
steadystate
I
dt
d
d
dt
.
TIME, t
.
min
.
I
II
III
CREEP
STRAIN
RATE,
ú
min
A
n
exp( Q/RT)
min
.
log
log
n
Stress exponent, n, from isothermal tests:
ú
min
B
n
so that
log
ú
min
logB n log
or
n
log ú
min,1
log ú
min,2
log
1
log
2
Activation energy, Q, from isostress tests:
min
.
1/T
ln
(Q/R)
ú
min
Cexp(Q/RT )
so that
ln
ú
min
lnC (Q/R) (1/T )
or
Q
R(ln ú
min,1
ln ú
min,2
)
1
T
1
1
T
2
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 19 / 36
Long term predictions from short term results
 valid only if the creep/creep rupture
mechanism does not change over time. Ruleofthumb: shorttime test lives should be at
least 10% of the required longterm design life. Creep rupture occurs by the coalescence of
the diffusional damage (creep cavitation by inter or intragranular diffusion and oxidation)
which is manifested during secondary (steadystate creep).
Stressrupture
Empirical relation
A t
f
N
Important where creep deformation is tolerated but rupture is to be avoided.
Failure time, t
f
Stress
N
MonkmanGrant
Empirical relation
ú
min
t
f
C or
ú
min
Ct
f
m
where m 1 if the relation is applicable.
Important where total creep deformation
(i.e.
ú
min
t
f
)
is of primary concern.
min
.
t
m
f
SherbyDorn
Assumes that Qf( or T) and suggests that the creep strains for a given stress form a
unique curve if plotted versus the temperature compensated time,
t exp(Q/RT)
.
A common physical mechanism is assumed to define the timetemperature paramter such
that the SherbyDorn parameter
P
SD
log logt
f

log (e)
R
Q
1
T
P
SD
LarsonMiller
Assumes that Q=f() and suggests that the creep strains for a given stress form a unique
curve if plotted versus the temperature compensated time,
f
t
f
exp(Q/RT)
.
A common physical mechanism is assumed to define the timetemperature parameter such
that the LarsonMiller parameter
P
LM
log (e)
R
Q T(logt
f
+C)
P
LM
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 20 / 36
Material Damping
Energy dissipation during cyclic loading  internal friction which is material, frequency,
temperature dependent.
u=internal damping energy
=
d
t
t
a
a
Dynamic Modulus :
E
*
a
a
Phase Angle :
Loss Coefficient :
Q
1
tan
u
2U
e
Storage Modulus:
'
a
E
*
cos (where
'
at
a
)
Elastic Energy:
U
e
1
2
'
a
at
a
maximum extension
Fracture
Fracture is the separation (or fragmentation) of a solid body into two or more parts under the
action of stress (crack initiation and crack propagation) Presence of cracks may weaken the
material such that fracture occurs at stresses much less than the yield or ultimate strengths.
Fracture mechanics is the methodology used to aid in selecting materials and designing
components to minimize the possibility of fracture from cracks.
ALLOWABLE
STRESS,
CRACK LENGTH, a
Low K
High K
Ic
Ic
ALLOWABLE
STRESS,
CRACK LENGTH, a
a
t
= transition crack length
between yield and fracture
Cracks lower the material's tolerance (allowable stress) to fracture.
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 21 / 36
Griffith Theory of Brittle Fracture
A crack will propagate when the decrease in elastic strain energy is at least equal to the
energy required to create the new fracture surfaces
2c
W
t
Ue
Us
U
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 22 / 36
In general
MODE I
OPENING
MODE
MODE II
SLIDING
MODE
MODE III
TEARING
MODE
K F a
Y a
a
where F, Y, and are geometry correction factors
Subscripts on K refer to fracture mode:
K
I
=Mode I, opening mode
K
II
= Mode II, sliding mode
K
III
=Mode III, tearing mode
Note:
=
K
2
E'
where E' = E (plane stress)
and E'= E/(1
2
) (plane strain)
Plane strain fracture toughness
K
Ic
is the critical stress intensity factor in plane strain conditions at stress intensity factors
below which brittle fracture will not occur. The plane strain fracture toughness, K
Ic
, is a
material property and is independent of geometry (e.g. specimen thickness).
Fracture toughness in design
Fracture occurs when
K
Ic
K
I
F a
where F is the geometry correction factor for the particular crack geometery.
Designer can choose a material with required K
Ic
,
OR design for the stress, ,to prevent fracture,
OR choose a critical crack length, a, which is detectable (or tolerable).
Cyclic Fatigue
Fatigue is failure due to cyclic (dynamic) loading including timedependent failure due to
mechanical and/or thermal fatigue. Fatigue analysis may be stressbased, strain based, or
fracture mechanics based.
Stressbased analysis
t
min
max
m
a
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 23 / 36
max
Maximum stress
min
Minimum stress
m
Mean stress =
max
min
2
Stress range =
max
min
a
Stress amplitude =
2
(
max
m
) (
m
min
)
Note: tension and compression . Completely reversed R 1,
m
0.
R Stress ratio
min
max
A = Amplitude ratio
a
m
1R
1R
SN Curves
Stress (S)fatigue (N
f
) life curve where gross stress, S, may be presented as
,
a
,
max
,or
m
. High cycle N
f
>10
5
(sometimes 10
2
10
4
) with gross stress elastic. Low
cycle N
f
<10
2
10
4
with gross elastic plus plastic strain.
10
10
S
log N
f
6
8
e
Ferrous and Tibased alloys
Nonferrrous materials
(e.g Al or Cu alloys) ( @ 10 cycles)
= fatigue limit or endurance limit ( @10 cycles)
e
6
8
e
Fatigue factors
Recall stress concentration factor:
k
t
LOCAL
REMOTE
S
log N
f
e
e
NOTCHED
UNNOTCHED
Fatigue strength reduction factor:
k
f
e
UN NOTCHED
e
NOTCHED
Notch sensitivity factor,
q
k
f
1
k
t
1
where q=0 for no
notch sensitivity, q=1 for full sensitivity.
qÒ as notch radius, , Ò and qÒ as S
UTS
Ò
Generally, k
f
<< k
t
for ductile materials and sharp notches but k
f
k
t
for brittle materials and
blunt notches. This is due to i) steeper d/dx for sharp notch so average stress in fatigue
process zone is greater for the blunt notch, ii) volume effect of fatigue which is tied to
average stress over larger volume for blunt notch, iii) crack cannot propagate far from a
sharp notch because steep stress gradient lowers K
I
quickly. In design, avoid some types of
notches, rough surfaces, and certain types of loading. Compressive residual stresses at
surfaces (from shot peening, surface rolling, etc.) can increase fatigue lives.
Endurance limit,
e
is also lowered by factors such as surface finish (m
a
), type of loading
(m
t
), size of specimen (m
d
), miscellaneous effects (m
o
) such that:
e
'
m
a
m
t
m
d
m
o
e
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 24 / 36
Note that
e
can be estimated from the ultimate tensile strength of the material such that:
e
m
e
S
UTS
where m
e
=0.40.6 for ferrous materials.
For design purposes:
Effect of mean stress for constant amplitude completely reversed stress.
a
m
e
S
S
ys
uts
Goodman
Soderberg
Goodman:
a
e
1
m
S
UTS
Soderberg, use S
YS
instead of S
UTS
.
If factor of safety and/or fatigue factor are used:
For brittle materials, apply k
f
to
e
, k
f
k
t
to S
UTS
, and FS to S
UTS
and
e
.
a
e
FS¥k
f
1
m
(S
UTS
/(k
f
k
t
)FS)
For ductile materials, apply k
f
to
e
and FS to S
UTS
and
e
.
a
e
FS¥k
f
1
m
(S
UTS
/FS)
Effect of variable amplitude
about a constant mean stress.
t
a3
a2
a1
N
N
N
1
2
3
N
f
N
N
N
1
2
3
a1
a2
a3
a
f
f
f
PalmgrenMiner Rule (Miner's Rule)
N
1
N
f1
N
2
N
f2
N
3
N
f3
N
j
N
fj
1
Fatigue crack growth
The fatigue process consists of 1) crack initiation, 2) slip band crack growth (stage I crack
propagation) 3) crack growth on planes of high tensile stress (stage II crack propagation)
and 4) ultimate failure.
Fatigue cracks initiate at free surfaces (external or internal) and initially consist of slip band
extrusions and intrusions. Fatigue striations (beach marks) on fracture surfaces represent
successive crack extensions normal to tensile stresses when 1 mark1N but marksN
f
.
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 26 / 36
Shafts in Torsion
=T /J
= /L
2R=2c=D
L
T
2D Mohr's Circle for Pure Torsion
Torsional Shear Stress Torsional Shear Strain
T
J
where J= polar moment of intertia =
2
dA
dA
max
Tc (or c
o
)
J
L
max
c
L
Shear Modulus :
G
d
d
=
E
2(1+)
For linear elastic behaviour, plane sections remain plane, so
L
and
T
J
Special cases
J
D
4
32
=
c
4
2
for solid shaft
J
(D
outer
4
D
inner
4
)
32
=
(c
o
4
c
i
4
)
2
for tube
Power transmission
P
T
P =power (S.I. units, P= W= N¥m/s, US Customary, P=HP = 550 ft¥lb/s
T = torque
=
d
dt
=angular velocity, rad/s ( =RPM
2
60
)
Angle of twist
T(x)dx
J(x)G
0
L
(in general)
TL
JG
(at x =L for constant T, J, G)
TL
JG
(for multiple segments for different T, J, G)
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 27 / 36
Pressure Vessels
pr
t
pr
2t
p
Thin wall cylindrical
pressure vessel
p
pr
2t
Thin wall spherical
pressure vessel
Thin wall refers to a vessel with inner radius to wall thickness ratio, r/t, of greater than 10.
For cylindrical vessel with internal gage pressure only,
At outer wall,
1
pr
t
(hoop);
2
pr
2t
(longitudinal);
3
0 (radial)
,
At inner wall,
1
pr
t
(hoop);
2
pr
2t
(longitudinal);
3
p (radial)
For spherical vessel with internal gage pressure only,
At outer wall,
1
pr
2t
(hoop);
2
pr
2t
(longitudinal);
3
0 (radial)
,
At inner wall,
1
pr
2t
(hoop);
2
pr
2t
(longitudinal);
3
p (radial)
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 28 / 36
Beams
x
y
+w(x)
+M
+V
+V
+M
Beam Sign Convention
R
M
v=0
dx
dv
=0
Support Condition Force Reaction Boundary Condition
R
v=0
dx
dv
0
R
v=0
dx
dv
0
Fixed
Roller
Pinned
M=0
M=0
R=0
M=0
v0
dx
dv
0
Free
FBD, Shear Diagram and Moment Diagram
FBD:
F
0,
M 0
Shear Diagram (V):
dV
dx
w(x)
Moment Diagram (M):
dM
dx
V
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 29 / 36
Bending strain and stress
Neutral Axis
=Centroid
+M
y
x
My
y
I
Radius of Curvature
+M
Normal Stress and Strain
y
y
c
max
where
max
c
= 
My
I
and
max
=
Mc
I
y distance from neutral axis
radius of curvature of neutral axis
c = distance from neutral axis to point furthest
from neutral axis
M= bending moment
I = moment of inertia of cross section= y
2
dA
dA
Shear Stress
VQ
It
V shear force
Q ydA'
A'
y'
A'where A' portion of cross section
I = moment of inertia of entire cross section
t = thickness of cross section at point of interest
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 30 / 36
Compare normal and shear stress distributions
Neutral Axis
=Centroid
+M
y
x
My
I
+M
+V
+V
VQ
It
h
b
c
o
c
o
c=
c
i
Rectangular Cross Section
I =
bh
3
12
max
6M
bh
2
max
3V
2A
3V
2(bh)
Circular Cross Section
I=
c
4
4
max
2M
c
3
max
4V
3A
4V
3(c
2
)
Tubular Cross Section
I =
(c
o
4
c
i
4
)
4
max
2Mc
o
(c
o
4
c
i
4
)
max
2V
A
2V
(c
o
2
c
i
2
)
Beam Deflections
Moment Curvature
1
M
EI
Equations for Elastic Curve
EI
d
4
v
dx
4
=w(x)
EI
d
3
v
dx
3
=V(x)
EI
d
2
v
dx
2
= M(x)
Need to integrate equations for elastic curve for find v(x) and dv(x)/dx in terms of M(x), V(x),
w(x), and constants of integration. The specific solution for the elastic curve is then found by
applying the boundary conditions. Note that v=dv/dx=0 for fixed support, v=0 but dv/dx0 for
simple support, and v=max or min when dv/dx=0 at maximum moment (i.e. inflection point).
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 31 / 36
Statically Indeterminate
AxiallyLoaded Members
L
L
AC
CB
C
A
B
F
F
P
B
A
F
0 so F
A
F
B
P 0
But F
A
and F
B
are unknown
so
Use loaddisplacement relation and compatibility
at the common point C
F
A
L
AC
AE
F
B
L
CB
AE
0
TorsionallyLoaded Members
L
L
AC
CB
C
A
B
T
T
A
T
B
M
0 so T
A
T
B
T 0
But T
A
and T
B
are unknown
so
Use torquetwist relation and compatibility
at the common point C
T
A
L
AC
JG
T
B
L
CB
JG
0
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 32 / 36
Beams
L
L
AC
CB
C
A
B
P
M
0 and
F 0
But there are additional supports not needed
for stable equilibrium which are redundants
and determine the degree of indeterminacy
so
First determine redundant reactions, then
use compatibility conditions to determine
redundants and apply these to beam to solve
for the remaining reactions using equilibrium
If use method of integration, integrate the
differential equation,
d
2
v
dx
2
M
EI
twice to
find the internal moment in terms of x (i.e., M(x)).
The redundants and constants of integration are
found from the boundary conditions.
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 33 / 36
Engineering Materials
Classes and various aspects of engineering materials.
Size scales and disciplines involved in the study of engineering materials.
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 34 / 36
Crystals, structures, defects and dislocations, theoretical strength
Four common crystal structures: (primitive) cubic, bodycentered cubic, facecentered cubic,
and hexagonal close packed.
a) amorphous b) crystalline
Examples of a) amorphous (without form) and b) crystalline structures
Types of point defects Types of line defects (dislocations)
[a) edge dislocations and b) screw dislocations]
Maximum Cohesive Strength
max
E
E
10
Upper
Bound
max
E
s
a
0
Lower
Bound
Maximum Shear Stress at Slip
max
Gb
2a
o
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 35 / 36
Strengthening Mechanisms
Grain Boundary Strengthening
Mechanism: GB is region of disturbed lattice with steep strain gradients
High angle = high fracture energy plus diffusion sites
Low angle = edge dislocations climb
T
eq
is equicohesive temp where GB is weaker than grain and d is the grain diameter.
Result: At R.T. As d
UTS
AND as d
UTS
such that
o
i
kd
1/2
(HallPetch Eq. where
o
is yield stress,
i
is friction
stress and k is the"locking" parameter
At H.T. If T>T
eq
as d
UTS
BUT if T<T
eq
as d
UTS
Yield Point Phenomenon
Strain
Upper Yield
Lower Yield
Strain
Hardening
Lders Bands
are bands of
yielded material
Mechanism: Lders bands of yielded and unyielded
material with C and N atoms forming
atmospheres (interstitials) to pin dislocations
and forcing new dislocations to form.
Result: Upper yield point followed by lower yield point
before strain hardening.
Strain Aging
Mechanism: C and N atoms form atmospheres (interstitials) to pin dislocations and forcing
new dislocations to form BUT diffusion of interstitials can repin dislocations.
Result: Upper yield point and lower yield point return even if material is strain hardening.
Strain
Strain
At R.T., No strain age and no YP
Aged at T or after days at R.T., YP returns
YP returns
for load/unload
load/unload
shows no YP
Solid Solution Strengthening
Mechanism: Atomiclevel interstitial and substitutional solute atoms provide resistance to
dislocation motion as dislocations bend around regions of high energy.
Result: Level of stress strain curve increases and yield strength increases.
Two Phase Aggregates
Mechanism: Microstructurallevel solid solution (dispersed structure) or particulate
additions (aggregated structures). Super saturation of particles in a matrix
where hard particles block slip in a ductile matrix and localized strain
concentration raise yield strength due to plastic constraint.
Result: Yield strength increases, hardness increases
Bounds on properties: Isostrain:
m
=
p
=
c
so
c
=V
p
p
+V
m
m
Isostress:
m
=
p
=
c
so
c
=V
p
p
+V
m
m
SOME SALIENT ASPECTS OF ME354
MECHANICS OF MATERIALS LABORATORY
30 Decmber 1997 (Version c) compiled by Michael G. Jenkins, University of Washington page 36 / 36
Strengthening Mechanisms (cont'd.)
Fiber Strengthening
Mechanism: Discrete fibers carry load and directional properties "toughen" composite.
Discrete matrix transmits load to fibers and protects fibers.
Result: High strength to weight ratio, directional properties
Bounds on properties: Isostrain:
m
=
p
=
c
so
c
=V
p
p
+V
m
m
Isostress:
m
=
p
=
c
so
c
=V
p
p
+V
m
m
Martensite Strengthening
Mechanism: Fine structure and high dislocation density provide effective barriers to slip
with C atoms strongly bound to dislocations and restrict dislocation motion.
Result: Hardness and strength increase
Amount of Cold Work
%EL
%RA
Suts
Sys
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