constitutive equations casa seminar wednesday 19 april 2006 godwin kakuba
TRANSCRIPT
Constitutive Equations
CASA Seminar Wednesday 19 April 2006
Godwin Kakuba
Outline• Introduction
– Continuum mechanics
– Stress
– Motions and deformations
– Conservation laws
• Constitutive Equations
– Linear elasticity
– Viscous fluids
– Linear viscoelasticity
– Placticity
• Summary
Introduction• Continuum mechanics
MatterMolecules
Atoms
Macroscopic scale
Introduction• Kinematics
• Stress
• Conservation laws
• Motions and deformations
Constitutive EquationsContinuum mechanics
Eqns that apply equally to all materialsEqns that describe the mechanical
behaviour of particular materials
•Linear elasticity
•Viscous fluids
•Viscoelasticity
•Plasticity
•Constitutive equations
Constitutive equations: Linear elasticity
Uniaxial loading: one dimensional elasticity
Constitutive equations: Linear elasticity
Linear elastic solid
a quadratic function
is equal to the rate at which mechanical work is done by the surface and body forces
Constitutive equations: Linear elasticity
Denote by thus (a) states that has the form
Consider a change of coordinate system,
Then,
We can also write
Constitutive equations: Linear elasticity
Interchanging i and j
Thus
independent constants
Constitutive equations: Linear elasticity
Alsoindependent elastic constants.
Using property and the energy conservation equation:
But and so
Constitutive equations: Linear elasticity
But
Hence
For an isotropic material
Constitutive equations: Newtonian viscous fluids
For a fluid at rest,
If the fluid is isotropic,
Constitutive equations of the form
Constitutive equations: Newtonian viscous fluids
For an incompressible viscous fluid,
or
For an ideal fluid,
or
If the stress is a hydrostatic pressure,
Constitutive equations: Linear viscoelasticity
Creep curve
Stress relaxation curve
Constitutive equations: Linear viscoelasticity
We consider infinitesimal deformations
Assuming the superposition principle, then
The inverse relation is
are stress relaxation functions.
are creep functions.
Constitutive equations: Plasticity
O C
A
B
Stress-strain curve in uniaxial tension
OA - linear relation between and
- Initial yield stress
OC - residual strain
Constitutive equations: Plasticity
For three-dimensional theory of plasticity
a yield condition
stress-strain relations for elastic behaviour
or
Thus
Constitutive equations: Plasticity
Plastic stress-strain relations
where
Hence
Constitutive equations: Summary
Linear elastic solid:
Isotropic material:
Newtonian fluid:
Viscoelasticity:
Plasticity: