Linear elastic solid

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Chapter V

• Linear elastic solid

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Linear elastic solid

• THERMODYNAMICS
• 1st law
• K Kinetic energy introduced into the system
• E Internal energy introduced into the system
• Q Heat flow introduced into the system
• PE Power introduced into the system

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Linear elastic solid

• THERMODYNAMICS
• 2nd law
• dSint gt 0 irreversible process
• dSint 0 reversible process

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Linear elastic solid

• THERMODYNAMICS
• Gibbs theorem
• ISOLATED SYSTEM
• in equilibrium
• internal energy MINIMUM
• with respect to
• any other adjacent state
• of the system with the same entropy S

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Linear elastic solid

• Application of 1st law quasi-static loading
• ?

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Linear elastic solid
free energy per unit of mass
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Linear elastic solid

• 2nd law reversible behaviour
• ? 1st 2nd laws

entropy p.u. mass
strain energyp.u. mass
internal energyp.u. mass
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Linear elastic solid

• Gibbs theorem
• natural stable state unloaded solid
• adjacent state loaded and deformed solid

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Linear elastic solid

• Unicity of the solution
• Lets assume two solutions s and s " for one
  loading

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Linear elastic solid

• Linear isotropic elastic material

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Linear elastic solid

• Isotropic linearly elastic material

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Linear elastic solid

• Constitutive law

21 independent components remain
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Linear elastic solid
END
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Linear elastic solid

• Orthotropic linearly elastic material

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Linear elastic solid

• Transversaly isotropic linearly elastic material

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Linear elastic solid

• Influence of temperature
• Thermal expansion
• Elastic modulus is a function of temperature E(T)
• Natural state T T0
• sij 0
• DT T T0 (small variation)
• Free energy

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Linear elastic solid

• Influence of temperature

reference statenatural state
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Linear elastic solid

• Influence of temperature

reference statenatural state
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Linear elastic solid

• Influence of temperature

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Linear elastic solid

• Applied to isotropic linearly elastic materials