1. Continuum Mechanics (MTH487)
Lecture 28
Instructor
Dr. Junaid Anjum

2. Aims and Objectives The first and second laws of Thermodynamics
Some definitions (entropy, specific heat, internal
energy, kinetic energy etc )
Law of Conservation of Energy
The Energy Equation
Some Examples
The principle of virtual work

3. Law of Conservation of Energy
The material time derivative of the kinetic plus internal energies is equal to the sum of the rate of work of the surface and body forces, plus all other energies that enter or leave the body per unit time.
: kinetic energy
: mechanical power (work done by body and surface forces)

4. Law of Conservation of Energy
stress power
: stress work
: internal energy
: specific internal energy

5. Law of Conservation of Energy
For a thermomechanical continuum
: heat supply
Rate at which thermal energy is added
: heat flux
Fourier’s law of heat conduction
: temperature gradient
: thermal conductivity
: first law of thermodynamics

6. Law of Conservation of Energy
: thermal energy balance
: energy equation

7. Second Law of Thermodynamics
: specific entropy per unit mass
: net external entropy supplied
The net entropy production within V must be non-negative
: second law of thermodynamics

8. Stress power
Problem 1: For a certain continuum at rest, the stress is given by
where is a constant. Use the continuity equation to show that for this case the stress power may be expressed as

10. Energy equation
Problem 2: Show that one way to express the rate of change of kinetic energy of the material currently occupying the volume V is by the equation

12. Energy equation
Problem 3: Consider a continuum for which the stress is and which obeys the heat conduction law Show that for this medium the energy equation takes the form

14. Energy equation
Problem 4: If mechanical energy only is considered, the energy balance can be derived from the equations of motion. Verify that one form of the result is

16. Principle of virtual work

17. Principle of virtual work
Consider a body B whose boundary is given by S. Let the body be subjected to surface forces and body force b.
displacement boundary conditions are prescribed on the part of the
boundary,
traction boundary conditions are applied on the portion of the
boundary
A displacement field is kinematically admissible if
satisfies the displacement boundary condition on
is continuously differentiable
A statically admissible stress field
satisfies the equations of motion
traction boundary condition on
A kinematically admissible velocity field
satisfies on

18. Aims and Objectives The first and second laws of Thermodynamics
The Energy Equation
The principle of virtual work
: first law of thermodynamics
: second law of thermodynamics