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Constitutive Equations CASA Seminar Wednesday 19 April 2006 Godwin Kakuba

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

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Introduction Continuum mechanics Matter Molecules Atoms Macroscopic scale

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Introduction Kinematics Stress Conservation laws Motions and deformations

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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

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Constitutive equations: Linear elasticity Uniaxial loading: one dimensional elasticity

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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

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

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

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

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Constitutive equations: Linear elasticity But Hence For an isotropic material

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

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

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Constitutive equations: Linear viscoelasticity Creep curve Stress relaxation curve

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

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

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

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Constitutive equations: Plasticity Plastic stress-strain relations where Hence

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

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