1. Continuum Mechanics (MTH487)
Lecture 29
Instructor
Dr. Junaid Anjum

2. Recap … The first and second laws of Thermodynamics
The Energy Equation
The principle of virtual work
: first law of thermodynamics
: second law of thermodynamics

3. Aims and Objectives The principle of virtual work (continued)
Direct statement
Converse statement
Some Examples
Linear elasticity

4. 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

5. Principle of virtual work
Let T be a statically admissible stress field corresponding to a kinematically admissible velocity field

6. Principle of virtual work

7. Principle of virtual work
If: T satisfies
the boundary condition on
the equations of motion (linear momentum balance)
then
the virtual work equation
is satisfied for all virtual velocity fields.

10. Principle of virtual work
If: T satisfies the virtual work equation
for all virtual velocity fields
then the stress field must satisfy
the boundary condition on
the equations of motion (linear momentum balance)

13. Principle of virtual work
T satisfies the virtual work equation
for all virtual velocity fields
the stress field must satisfy
the boundary condition on
the equations of motion (linear momentum balance)
The principle of virtual work forms the basis for the finite element
method in the mechanics of solids.
The principle of virtual work equations is the alternative representation
of equations of motion in integral form which is much better suited for
computer solution.
The principle of virtual work equations give an alternative way of
solving for stress field which satisfies equations of motion without
differentiating the stress (computational integration is preferred over
computational differentiation)

14. Principal of Virtual work
Problem 1: The Stress distribution in the shown plate is given by
where and are the coordinates inside the plate. Find the equilibrium body forces vector applied to the plate. Find the traction forces on the boundary edges A, B, C and D of the plate. Verify the principle of virtual work assuming a virtual velocity field

18. Linear Elasticity stress is a function of the strain
nonlinear
inelastic
linear
load
unload
Stress
Strain
load
unload
Stress
Strain
load
unload
Stress
Strain
stress is a function of the strain
complete recovery to the natural state upon
removal of the applied forces
: constitutive equation for elastic behavior
A body is linearly elastic if the stress depends linearly on the infinitesimal strain
where C, called the elasticity tensor
C: 81 components

19. Aims and Objectives
The principle of virtual work
Linear elasticity on