States of matter

Solid:• Liquid• Gas• Plasma

Fluid:

• Crystalline• Amorphous

internal interaction

x

y

z

xy

yy

zy

xx

yx

zx

xz

yz

zzIn a medium, a set of parameters leading to the forces exerted on an infinitesimal cube element within the medium, is called the stress tensor.

ijijdF

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

dzx

dyx

dxx

x

dzy dyy

dxy

y

dxzdzz

dyz

z

The deformation is described by a strain tensor

ad

xd jiij

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

The proportionality tensor is called a modulus.

Within certain limits, the differential change in stress, caused by external forces exerted on the medium, is a linear function of the differential strain.

kl

klijklij ˆˆˆor

tension

L

dL

-dF

dF

x

y

zThe external forces, applied along a single line to two opposite sides of the rod, cause a uniform stress

zzzz Y

Coefficient Y is called Young's modulus.

We can often approximate a finite change in the related quantities using the above differential relation

FA

Y LL

A

dF

L

dLY

compression (uniaxial pressure)

L

L

x

y

z

-F

FThe external forces are applied along a single line to two opposite sides

L

LYY

A

F zzzz

The nonzero component of compressive stress is called uniaxial pressure (P)

dA

dFP zz

Shear stress

h

dy

x

y

z

-dF

dF Tangential external forces applied to two opposite sides of the object cause a shear stress

yzyz SA

dFh

dyS dS

Coefficient S is called the shear modulus.

d

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

dF

dF

dF

dF

dF

x

y

z

Under hydrostatic pressure, all shearing components of the stress are zero and all compressive components of stress are equal.

zzyyxx dPA

dF

Hook’s law:

V

dVBdP

fluid at restF0

h

F0

F(h)

P0

P(h)

)h(P A

WF0

A

'gAdhAPh

00

h

00 'gdhP

gh'gdhh

0

for uniform density:

In a gravitational field, pressure in fluids depends on the pressure created by an external force and the depth in the fluid

ghPhP 0 W

Pascal's principleA change in the pressure applied to an enclosed (incompressible) fluid is transmitted undiminished to every portion of the fluid.

F1

A1

F2

A2

11

2 FA

A

Hydraulic Press:

Archimedes' principle

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

ydB

dA1

dA2

1

2

dA

y

222111 cosdAPcosdAP

dAPdAP 21 dAhg gdV

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

A1

A2

v1

v2

dx2

dx1

y1

y2

from the work-energy theorem:

222

2222 ydxgA

2

vdxA

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.

11

21

22

22 Pgy

2

vPgy

2

v

111

2111 ydxgA

2

vdxA 111 dxAP 222 dxAP

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:

x

TkA

dt

dQ

Tk

dt

dQ A :precisely more

A

dx

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 = cTdm

Mechanisms of Heat Transfer

1. Radiation

Energy is transmitted in the form of electromagnetic radiation.

EBStefan’s Law

4AeTdt

dQ

= 6 10-8 W/m2K

e – emissivity of the substance

A – area of the source surface

T – temperature of the source

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.

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.)

Temperature

T K PPP

273163 0 3

. lim

h

T3 = 273.16 K is the temperature at which water remains in thermal equilibrium in three phases (solid, liquid, gas).

Gas Thermometer

The Celsius scale and, in the US, the Fahrenheit scale are often used.

T TC 27315. T TF C 95

32;

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.

lD

dl linear expansion:

dl = ldlThe 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.

dD