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Constitutive Equations in Mechanical Behavior

Learn about constitutive equations in continuum mechanics and their application to various materials like linear elasticity, viscous fluids, viscoelasticity, and plasticity.

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Constitutive Equations in Mechanical Behavior

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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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