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Continuum Mechanics (MTH487)
Lecture 30 Instructor Dr. Junaid Anjum
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Recap … The principle of virtual work Linear elasticity on
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Aims and Objectives Linear elasticity (continued) Hooke’s law
Hooke’s law for isotropic media Elastic constants (Young’s modulus, Poisson’s ratio, bulk modulus, shear modulus) Physical interpretation of elastic constants problems
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Linear Elasticity stress is a function of the strain
nonlinear inelastic linear load unload Stress Strain load unload Stress Strain load unload Stress Strain stress is a function of the strain complete recovery to the natural state upon removal of the applied forces : constitutive equation for elastic behavior A body is linearly elastic if the stress depends linearly on the infinitesimal strain where C, called the elasticity tensor C: 81 components
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Linear Elasticity 36 components
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Linear Elasticity Strain as a function of stress
where the fourth-order tensor A is known as the compliance tensor; it is the inverse of C and therefore not a tensor !!!
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Linear Elasticity energy equation reduces to adiabatic, isothermal
: strain energy (purely mechanical)
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Linear Elasticity : strain energy density
stress in terms of strain energy linearized form …
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Linear Elasticity 21 components
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Linear Elasticity Hooke’s law for isotropic media
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Linear Elasticity Problem 1. Show that for an isotropic linear elastic solid the principal axes of the stress and strain tensors coincide, and develop an expression for the relationship among their principal values.
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Linear Elasticity Hooke’s law for isotropic media: Elastic constants
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Linear Elasticity Hooke’s law for isotropic media: Elastic constants
: Young’s modulus : Poisson’s ratio
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Linear Elasticity Physical interpretations …
Hooke’s law for isotropic media: Elastic constants : shear modulus : bulk modulus Physical interpretations … Uniaxial state of stress
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Linear Elasticity Physical interpretations …
Hooke’s law for isotropic media: Elastic constants Physical interpretations … Simple Shear Uniform triaxial tensions (or hydrostatic compression) relates pressure to the change in volume
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Linear Elasticity Problem 3. For and calculate the normal strains for and
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Things to remember : strain energy density : isotropic medium
: Young’s modulus : Poisson’s ratio : shear modulus : bulk modulus
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