實驗力學研究室
Fundamentals
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First Principles
Body Under External Loading
• First Law: A body will remain at rest or will continue its straight
line motion with constant velocity if there is no unbalanced force
acting on it.
• Second Law: the acceleration of a body will be proportional to
the resultant of all forces acting on it and in the direction of the
resultant.
• Third Law: Action and reaction forces between interacting bodies
will be equal in magnitude, collinear, and opposite in direction.
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G, the center of gravity of the body.
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dv 
F  ma  m  G
dt
G= mv constitutes the linear momentum vector of a body.

F G

M  H
H is the angular momentum vector of the body.
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Constraining the body to uniaxial motion,


Fx  m x  c x  kx
Constraining the body to planar motion,
 F  ma
M  I 
G
G
G
aG is the vectorial acceleration of the center of gravity (c.g.) of the
body. IG is the mass moment of inertia of the body about an axis
normal to the plane of motion through the c.g., and α is the body’s
angular acceleration.
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Stress and Strain
What Is Stress?
For static equilibrium, τxy=τyx, τyz=τzy, and τzx=τxz.
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Principal Stresses
In a given loaded structure, a particular element orientation exists for
which all the shear stress components are zero. The normals to the
faces of an element in this orientation are called principal directions
and the stresses along these normals are the principal stresses.when
one of the principal stresses is zero, the stress state is considered to
be biaxial or plane stress. These problems can be deconstructed into
planar approximations in which the loading and boundary conditions
are in that plane and identical on any parallel plane.
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Plane Stress Case:
Maximum/minimum Principal Stresses
 1, 2 
 x   xy
2
  x  y 
2
   xy
 
 2 
2
Maximum/minimum Shear Stresses (45° away from the orientation
of principal stress)
 x  y 
 1, 2   


2
2
   xy 2

 x  y
2
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Triaxial Case:
 3  I1 2  I 2  I 3  0
where
I1   x   y   z
I 2   x y   y z   z x   2 xy   2 yz   2 zx
I 3   x y z  2 xy yz zx   x 2 yz   y 2 zx   z 2 xy
I1, I2, and I3 are called stress invariants. I1, or the first invariant, is
the internal hydrostatic pressure.
The maximum shear stress is given in terms of the maximum and
minimum principal stresses as follows.
 max 
1  3
2
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Strain

l
l
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Principal Strain
The strains that occur in the direction of principal stresses are known
as principal strains.
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Fundamental Stress States
Stress in Flexure

 My
I
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Stress in shear
VQ

Ib
where V is the shear force, b is the width of the stress section, and Q
is the first moment of area of the section about a transverse axis with
origin on the neutral axis.
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Stress in Torsion
Tr

J
Here, r is the redius from the torsional axis and J is the section’s
polar area moment of inertia.
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Tl

GJ
where G is the material’s modulus of rigidity.
The approximate formula for a rectangular section beam of width (w)
and thickness (t).
 max
T 
t 
 2  3  1.8 
wt 
w
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Stress in Pressure
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pi ri  po ro  ri ro  po  pi  r 2
t 
2
2
ro  ri
2
2
2
2
pi ri  po ro  ri ro  po  pi  r 2
r 
2
2
ro  ri
2
2
2
2
where σt is also known as the hoop stress in the cylinder, σr is radial
stress.
The longitudinal stress due to pressure on the end caps is constant
throughout the cylinder and is given by the equation.
l 
pi ri  po ro
2
2
ro  ri
2
2
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If the pressure created due to an equal-length cylindrical press-fit
was known, above equations could be used for obtaining the stress
state on both the outside and inside cylinders.
 1  R  ri

  1  ro  R
p   2
 vo    2
 vi 
2
2
R E r R
E R r
2

o

o
2
2
2

i

i
1

where ri is the inside cylinder’s internal radius, ro is the outside
cylinder’s external radius, R is the transition radius, and Ei, Eo, vi,
and vo are the inside and outside material Young’s moduli and
Poisson’s ratios, respectively. δ is the radial interference.
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Stress in Contact (Hertzian stresses )
Contact circular area resulting from the forced contact of two
spheres will be



3F 1  v1 E1  1  v2
3
a
8
1 d1  1 d 2
2
2
E
2
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At the center of this area, a maximum pressure pmax will occur of the
following magnitude.
pmax
3F

2a 2
For two cylinders of equal length l and diameters d1 and d2, the
resulting contact surface is a rectangle of length l and width 2b,
where
b



2 F 1  v1 E1  1  v2
l
1 d1  1 d 2
2
2
E
2
The maximum pressure occurs along the long center line of the
rectangle.
2F
pmax 
bl
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Stress in thermal Expansion
 x   y   z   T 
The constant of proportionality α is know as the material’s
coefficient of thermal expansion. For a straight beam constrained at
both ends, the resulting compressive stress at a distance from the
ends is given by the next equation.
   T E
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Stress Concentration Factors
The stress concentrations will also appear close to unplanned
irregularities in the part, such as cracks and pits.
 max
Kt 
o
 max
K ts 
o
σo and τo are the nominal stresses found in the part without the
feature.
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Material Properties
Types of Materials
• Isotropic. Properties are the same in any direction or at any cross
section.
• Anisotropic. Properties differ in two or more directions.
• Orthotropic. Specific type of anisotropic in which planes of
extreme values are orthogonal(i.e.,perpendicular to one another).
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Common Material Properties (modulus of elasticity
(E), modulus of rigidity (G), and Poisson’s ratio (v))
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Point P, known as the proportional limit. Point E, the elastic limit.
Point Y is the yield point of the material, corresponding to its yield
strength (Sy). Point U indicates the maximum stress that can be
achieved by the material. This corresponds to its ultimate or tensile
strength. Fracture point (F), which marks the fracture strength (SF)
of the material.
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Ductile versus Brittle Material Behavior
If permanent set (plastic deformation) is obtainable, the material is
said to exhibit ductility. For ductile materials , the ultimate tensile
and compressive strengths have approximately the same absolute
value. Brittle materials on the other hand are stronger in
compression than in tension.
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Brittle materials exhibit the behavior described below.
• A graph of stress versus strain is a smooth, elastic curve until
failure which manifests as fracture. Materials behaving in this
manner do not have a “yield strength.”
• Compressive strength is usually many times greater than tensile
strength.
• Modulus of rupture is approximately the same as tensile strength.
• Rapid crack propagation along cleavage planes occurs with no
noticeable plastic deformation.
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Rules of thumb used to determine if brittle or ductile behavior
should be expected are summarized below.
• If the percent elongation is at or below 5%, assume brittle
behavior.
• If the published ultimate compressive strength is greater than the
ultimate tensile strength, assume brittle behavior.
• If no yield strength is published, suspect brittle behavior.
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Failure Modes
Typical Failure Modes
• Fracture. Fracture is said to occur when new cracks appear or
existing cracks are extended. A brittle fracture is one that exhibits
little or no permanent (plastic) deformation.
• Yielding. A body which experiences stresses in excess of the
yield strength is said to have failed only when this yielding
compromises the integrity or function of the part. Yielding near
stress concentrations is not considered a failure if it produces
localized strains which merely redistribute the stress, whereupon
yielding ceases.
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• Insufficient stiffness. Parts must be stiff enough to hold
tolerances and support required loads. Moving parts may have
undesirable resonant frequencies if they are too flexible.
• Buckling. The sudden loss of stability or stiffness under applied
load. Stress levels need not be high for buckling to occur.
• Fatigue. Parts that are subject to variable loading will lose
strength with time and may fail after a certain number of cycles.
• Creep. Bodies under load gradually deform over time. The
apparent modulus property is derived form empirical creep data
for various materials and may be used to compensate for the
effects of creep.
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Classic Failure theories
Ductile Failure theory
• Maximum normal stress theory. Failure occurs whenever σ1 or
σ3 equals the failure strength of the material in tension or
compression, respectively.
• Maximum shear stress theory ( Tresca criterion ). Yielding
begins when the maximum shear stress becomes equal to onehalf the yield strength. Failure in tension of ductile materials
occurs on one of the 45°maximum shear planes. Annealed
ductile materials tend to fail according to this theory.
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• Distortion energy ( Von Mises-Hencky ) theory (Suitable for
entire stress state). Probably the most widely used, this theory
predicts that failure by yielding will occur whenever the von
Mises, or effective stress (σ’ ), equals the yield strength of the
materials.
 vm
  1   2 2   2   3 2   1   3 2 


2


1
2
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Brittle Failure Theory
• Maximum normal stress. Similar to that defined for ductile
materials. Failure occurs when the ultimate strength, not yield, is
reached.
• Coulomb-Mohr theory. Fracture occurs when the maximum and
minimum principal stresses combine for a condition which
satisfies the following:
1
S ut

3
S uc
1
where Sut and Suc represent the ultimate tensile and compressive
strengths, and both σ3 and Suc are always negative, or in compression.
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• Modified Mohr theory. Fracture occurs as defined in the
Coulomb-Mohr theory except in the fourth quadrant condition
where σ1 is in tension and σ2 is in compression.
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Other Failure Theories
Buckling
Pcr 
 2 EI
Le
2
where E is the modulus of elasticity of the column’s materials, I is
the smallest or least moment of inertia of its cross-sectional area, and
Le is its effective length. The last term, Le=KL, depends on the actual
length L of the column and an effective length factor K, which is
assigned according to the constraint conditions of the column ends.
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r as the smallest radius of gyration of the column’s cross-sectional
area (A).
I
r
A
A corresponding critical stress(σcr) may be calculated as seen in the
next equation.
 2E
 cr 
Le / r 2
If a slenderness ratio (Le/r) is defined as
Le
 2E

r
Sy
the column is considered Euler, and a critical load must be calculated
and recorded.
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For nonEuler columns,
 cr 
 2 Et
Le / r 2
a tangent modulus variable (Et) has taken the place of the elastic
modulus (function of location), below the yield point, Et=E as
expected.
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Fatigue
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Endurance or fatigue limit (Se) is defined as the maximum cyclic
stress which a part can sustain for an “ infinite” number for cycles.
The endurance limit of the actual rotating beam speciman is
designated as Se'. The correlation between Se and Se' is
S e  k a kb k c k d k e S e
'
Here, ka is a surface factor, kb is a size factor, kc is a load factor, kd is
a temperature factor, and ke is an all encompassing, other
miscellaneous effects factor.
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Because nenferrous metals and alloys lack an endurance limit ( the
strength of material never stablizes but keep decreasing with time). A
fatigue strength (Sf ’) is usually reported for 50(107) cycles of
reversed stress. This strength is often as low as 1/4 Sut for some
aluminum alloys.
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To obtain the fatigue strength at N cycles for a part experiencing
alternating or completely reversed stress, you can curve-fit the S-N
curve using the following equation:
S f  aN b
where a and b are provided by
2

0.9 Sut 
a
Se
0.9 Sut
1
b   log
3
Se
Note that Se' may be substituted for Se in above equation to predict
Sf '.
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If the completely reversed stress has an amplitude (σa), the
corresponding number of cycles of life is calculated via the next
equation.
1b

 
N  a 
 a 
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When the mean stress (σm) is at a level other than zero, the cyclic
loading is classified as fluctuating stress case. One of the most
accepted equations that provides a solution to this scenario is the
modified Goodman relation:
a
m
1


S e Sut n
where Sut is the ultimate tensile strength of the material and n is the
safety factor used in the design.
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When a ductile material is subjected to a fatigue-type loading, there
are basic structural changes that occur. In chronological order, the
changes are summarized below.
1. Crack initiation. A crack begins to form within the material.
2. Localized crack growth. Local extrusions and intrusions occur at
the surface of the part because plastic deformations are not
completely reversible.
3. Crack growth on planes of high tensile stress. The crack
proceeds across the section at those points of greatest tensile
stress.
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4. Ultimate ductile failure. When the crack reduces the effective
cross section to a size that cannot sustain the applied loads, the
sample ruptures by ductile failure.
All the modifying factors that affect the endurance life of a part, are
summarized below.
• Stress concentrators. General part features as described in the
“Stress and Strain” section, which cause high local stresses and
thus decrease fatigue life.
• Surface roughness. Smooth surfaces are more crack resistant
because roughness creates stress concentrators.
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• Surface conditioning. Hardening processes tend to increase
fatigue strength while plating and corrosion protection tend to
diminish fatigue strength.
• Environment. A corrosive environment greatly reduces fatigue
strength. A combination of corrosive attack and cyclic stresses is
called corrosion fatigue.
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Creep
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The stress state for a viscoelastic material can be expressed as
d
  E  
dt
The four stages of creep failure, as shown in figure, are described
below.
• Instantaneous elongation. Normal deformation under applied
load.
• Primary creep. Material strain hardens under load to decrease
creep rate.
• Secondary creep. Material elongates at a steady rate, called
minimum creep rate.
• Tertiary creep. Due to necking and formation of voids, elongation
proceeds at an increasing rate until fracture.
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The secondary phase is of significant interest to engineers because it
dominates the actual creep process from a time standpoint. Creep
strength is defined as the stress which produces a minimum creep
rate of 10-5 % per hour.
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