Constitutive Equations
CASA Seminar Wednesday 19 April 2006
Godwin Kakuba
Outline
• Introduction
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Continuum mechanics
Stress
Motions and deformations
Conservation laws
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Linear elasticity
Viscous fluids
Linear viscoelasticity
Placticity
• Constitutive Equations
• Summary
Introduction
• Continuum mechanics
Matter
Molecules
Atoms
Macroscopic scale
Introduction
• Kinematics
• Stress
• Motions and deformations
• Conservation laws
Constitutive Equations
Continuum mechanics
Eqns that apply equally to all materials
•Constitutive equations
•Linear elasticity
•Viscous fluids
•Viscoelasticity
•Plasticity
Eqns that describe the mechanical
behaviour of particular materials
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
Consider a change of coordinate system,
Then,
We can also write
has the form
Constitutive equations: Linear elasticity
Interchanging i and j
Thus
independent constants
Constitutive equations: Linear elasticity
Also
independent 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
Constitutive equations of the form
For a fluid at rest,
If the fluid is isotropic,
Constitutive equations:
Newtonian viscous fluids
If the stress is a hydrostatic pressure,
For an incompressible viscous fluid,
or
For an ideal fluid,
or
Constitutive equations:
Creep curve
Stress relaxation curve
Linear viscoelasticity
Constitutive equations:
Linear viscoelasticity
We consider infinitesimal deformations
Assuming the superposition principle, then
are stress relaxation functions.
The inverse relation is
are creep functions.
Constitutive equations:
Plasticity
Stress-strain curve in uniaxial tension
B
A
O
C
OA - linear relation between
- Initial yield stress
OC - residual strain
and
Constitutive equations:
Plasticity
For three-dimensional theory of plasticity
a yield condition
stress-strain relations for elastic behaviour
or
Thus
Constitutive equations:
Plastic stress-strain relations
where
Hence
Plasticity
Constitutive equations:
Linear elastic solid:
Isotropic material:
Newtonian fluid:
Viscoelasticity:
Plasticity:
Summary