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: