States of matter
Solid:
• Crystalline
• Amorphous
Fluid:
• Liquid
• Gas
• Plasma
internal interaction
zz
z
yz
xz
zx
yx
xx
x
zy
xy
yy
y
In a medium, a set of parameters
leading to the forces exerted on an
infinitesimal cube element within the
medium, is called the stress tensor.
 ij 
dFij
dA
where is the i-th scalar component of the force exerted on the
j-th wall of the cube and dA is the area of one wall.
The SI unit of stress is the pascal (Pa).
Note: Only six independent components.
deformation
z
dyz
dzz
dxz
The deformation is described
by a strain tensor
dzy
dyy
y
dxy
dxx
x
dyx
dzx
ij 
d x i  j
da
where d(xi)j is the displacement of the j-th corner in the i-th
direction, and is the size of the cube (initial).
Hook's law
Within certain limits, the differential change in stress, caused
by external forces exerted on the medium, is a linear function
of the differential strain.
ij   ijkl kl
kl
or
ˆ  ˆ  ˆ
The proportionality tensor is called a modulus.
tension
z
The external forces, applied along a single
line to two opposite sides of the rod, cause a
uniform stress
dF
dL
dF
dL


Y


 zz
zz  Y 
A
L
Coefficient Y is called Young's modulus.
L
We can often approximate a finite change in
the related quantities using the above
y differential relation
x
-dF
F  Y  L
A
L
compression (uniaxial pressure)
z
The external forces are applied along a
single line to two opposite sides
F
F
L
   zz  Y   zz   Y 
A
L
L
L
The nonzero component of compressive stress
is called uniaxial pressure (P)
y
x
-F
dF
P  zz 
dA
Shear stress
z
dy
Tangential external forces applied to
two opposite sides of the object cause
a shear stress
dF
h
d
-dF
x
y
dy
dF
  yz  S   yz  S 
 S  d
h
A
Coefficient S is called the shear modulus.
Comment 1. Fluids in rest do not create shear stress.
Comment 2. The occurrence of a velocity dependent stress in a
moving fluid is called viscosity.
Hydrostatic pressure
z dF
dF
dF
y
dF
x
dF
Under hydrostatic pressure, all
shearing components of the stress
are zero and all compressive
components of stress are equal.
 dP  xx  yy
Hook’s law:
dV
dP  B 
V
dF
 zz  
A
fluid at rest
F0
F0  W
 field, pressure in
PIn
(h )agravitational
A
F0
h
P0
fluids dependsh on the pressure created
by an external
in
P0 A   force
gAdhand
' the depth
h
the fluid
0

W
 P0   gdh '
A
Ph   P0  gh
for uniform density:
P(h)
h
 gdh '  gh
0
F(h)
0
Pascal's principle
A change in the pressure applied to an enclosed
(incompressible) fluid is transmitted undiminished to every
portion of the fluid.
F1
A2
 F1
F2 
Hydraulic Press:
A1
A1
A2
y
Archimedes' principle
dA2
dBy  P1  dA1  cos 1  P2  dA2  cos 2 
 P1  dA  P2  dA  g  h  dA  gdV
2
dA
1
dA1
A body submerged (partially or completely)
in a fluid is buoyed up with a force equal in
magnitude to the weight of the fluid
displaced by the body
Ideal fluid
• nonviscous - there is no internal friction;
• flows steadily - at any point, the velocity of
the fluid does not depend on time;
• incompressible - its density does not
depend on pressure;
• irrotational - does not produce vortices
When the rate of flow is small (laminar flow), many fluids
can be approximated by the ideal fluid.
Bernoulli's equation
v2
y2
A2
dx2
v1
y1
A1
dx1
v 22
v12
 gy 2  P2 
 gy1  P1
2
2
For in ideal fluid, the sum of the
pressure, the kinetic energy per unit
volume, and the potential energy
per unit volume has the same value
at all points along a streamline.
from the work-energy theorem:

 A 2dx 2  v 22
  A1dx1  v12


gA
dx

y
 P2A2dx 2

 gA 2dx 2  y 2  

1
1
1   P1A1dx1 
2
2


 
Thermal contact
Two systems are in thermal (diathermic) contact, if they can
exchange energy without performing macroscopic work.
This form of energy transfer (random work) is called heat.
Mechanisms of Heat Transfer
1. Thermal Conduction
law of thermal conduction:
dQ
T
 kA 
dt
x

 more precisely :

A
dx
  
dQ
 kA  T 
dt

Mechanisms of Heat Transfer
1. Convection
natural convection:
resulting from differences in density
forced convection:
the substance is forced to move by a
fan or a pump.
The rate of heat transfer is directly related
to the rate of flow of the substance.
dQ = cTdm
Mechanisms of Heat Transfer
1. Radiation
Energy is transmitted in the form of
electromagnetic radiation.
Stefan’s Law
dQ
 AeT4
dt
 = 6  10-8 W/m2K
A – area of the source surface
e – emissivity of the substance
T – temperature of the source
E
B
Zeroth law of thermodynamics
Thermal Equilibrium:
If the systems in diathermic contact do not exchange
energy (on the average), we say that they are in thermal
equilibrium.
If both systems, A and B, are in thermal equilibrium
with a third system, C, then A and B are in thermal
equilibrium with each other.
Temperature
We say that two systems in thermal equilibrium have the same
temperature. (Temperature is a macroscopic scalar quantity
uniquely assigned to the state of the system.)
T  273.16 K  lim P
P3 0 P3
Gas Thermometer
h
T3 = 273.16 K is the temperature at
which water remains in thermal
equilibrium in three phases (solid,
liquid, gas).
The Celsius scale and, in the US, the Fahrenheit scale are often used.
TC  T  273.15 ;
TF  9 TC  32
5
Thermal expansion
For all substances, changing the temperature of a body
while maintaining the same stress in the body causes a
change in the size of the body.
D
dD
l
dl
linear expansion:
dl = ldl
The proportionality coefficient (T) is called
the linear thermal expansion coefficient.
volume expansion:
dV =VdV
The proportionality coefficient (T) is called
the volume thermal expansion coefficient.