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Constitutive Equations

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.

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Constitutive Equations

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  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 • Motions and deformations • Conservation laws

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

  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 by thus (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 Constitutive equations of the form For a fluid at rest, If the fluid is isotropic,

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

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

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

  16. Constitutive equations:Plasticity Stress-strain curve in uniaxial tension B A O C 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:

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