Constitutive Equations CASA Seminar Wednesday 19 April 2006 Godwin Kakuba Outline • Introduction – – – – Continuum mechanics Stress Motions and deformations Conservation laws – – – – 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