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States of matter. Solid:. Fluid:. Liquid Gas Plasma. Crystalline Amorphous. z.  zz.  yz.  xz.  zy.  zx.  yy.  yx.  xy. y.  xx. x. internal interaction.

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States of matter
States of matter

Solid:

Fluid:

  • Liquid

  • Gas

  • Plasma

  • Crystalline

  • Amorphous


Internal interaction

z

zz

yz

xz

zy

zx

yy

yx

xy

y

xx

x

internal interaction

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.

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

dzy

dyy

y

dxy

dxx

dzx

dyx

x

deformation

The deformation is described by a strain tensor

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.

or

Hook's law

The proportionality tensor is called a modulus.


Tension

z caused by external forces exerted on the medium, is a linear function of the differential strain.

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

dF

dL

L

Coefficient Y is called Young's modulus.

y

-dF

x

tension

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


Compression uniaxial pressure

z caused by external forces exerted on the medium, is a linear function of the differential strain.

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

F

L

L

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

y

-F

x

compression (uniaxial pressure)


Shear stress

z caused by external forces exerted on the medium, is a linear function of the differential strain.

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

dy

h

d

dF

y

-dF

x

Shear stress

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 caused by external forces exerted on the medium, is a linear function of the differential strain.

dF

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

dF

dF

dF

dF

x

y

Hydrostatic pressure

Hook’s law:


Fluid at rest

F caused by external forces exerted on the medium, is a linear function of the differential strain.0

F0

h

W

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

F(h)

fluid at rest

P0

for uniform density:

P(h)


Pascal s principle

F caused by external forces exerted on the medium, is a linear function of the differential strain.1

Pascal's principle

A change in the pressure applied to an enclosed (incompressible) fluid is transmitted undiminished to every portion of the fluid.

F2

Hydraulic Press:

A1

A2


Archimedes principle

y caused by external forces exerted on the medium, is a linear function of the differential strain.

2

1

Archimedes' principle

dA2

dA

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
Ideal fluid caused by external forces exerted on the medium, is a linear function of the differential strain.

  • 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

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.

Bernoulli's equation

v2

y2

A2

dx2

v1

y1

A1

dx1

from the work-energy theorem:


Thermal contact
Thermal contact energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline

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
Mechanisms of Heat Transfer energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline

1. Thermal Conduction

law of thermal conduction:

A

dx


Mechanisms of Heat Transfer energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline

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


B energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline

E

Mechanisms of Heat Transfer

1. Radiation

Energy is transmitted in the form of electromagnetic radiation.

Stefan’s Law

 = 6  10-8 W/m2K

A – area of the source surface

e – emissivity of the substance

T – temperature of the source


Zeroth law of thermodynamics
Zeroth law of thermodynamics energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline

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

h energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline

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.)

Gas Thermometer

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.

;


Thermal expansion

dl energy per unit volume, and the potential energy per unit volume has the same value at all points along a streamline

l

D

dD

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.

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.


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