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

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

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)

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

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

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

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

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

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

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

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

Principle of virtual work

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

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