1. Constitutive Equations CASA Seminar Wednesday 19 April 2006 Godwin Kakuba

2. Outline Introduction – Continuum mechanics – Stress – Motions and deformations – Conservation laws Constitutive Equations – Linear elasticity – Viscous fluids – Linear viscoelasticity – Placticity Summary

3. Introduction Continuum mechanics Matter Molecules Atoms Macroscopic scale

4. Introduction Kinematics Stress Conservation laws Motions and deformations

5. Constitutive Equations Continuum mechanics Eqns that apply equally to all materials Eqns that describe the mechanical behaviour of particular materials Linear elasticity Viscous fluids Viscoelasticity Plasticity Constitutive equations

6. Constitutive equations: Linear elasticity Uniaxial loading: one dimensional elasticity

7. 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

8. Constitutive equations: Linear elasticity Denote bythus (a) states that has the form Consider a change of coordinate system, Then, We can also write

9. Constitutive equations: Linear elasticity Interchanging i and j Thus independent constants

10. Constitutive equations: Linear elasticity Also independent elastic constants. Using property and the energy conservation equation: But and so

11. Constitutive equations: Linear elasticity But Hence For an isotropic material

12. Constitutive equations: Newtonian viscous fluids For a fluid at rest, If the fluid is isotropic, Constitutive equations of the form

13. Constitutive equations: Newtonian viscous fluids For an incompressible viscous fluid, or For an ideal fluid, or If the stress is a hydrostatic pressure,

14. Constitutive equations: Linear viscoelasticity Creep curve Stress relaxation curve

15. Constitutive equations: Linear viscoelasticity We consider infinitesimal deformations Assuming the superposition principle, then The inverse relation is are stress relaxation functions. are creep functions.

16. Constitutive equations: Plasticity OC A B Stress-strain curve in uniaxial tension OA - linear relation between and - Initial yield stress OC - residual strain

17. Constitutive equations: Plasticity For three-dimensional theory of plasticity a yield condition stress-strain relations for elastic behaviour or Thus

18. Constitutive equations: Plasticity Plastic stress-strain relations where Hence

19. Constitutive equations: Summary Linear elastic solid: Isotropic material: Newtonian fluid: Viscoelasticity: Plasticity: