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Unformatted text preview: Section 5.4 Solid Mechanics Part I Kelly 196 5.4 Strain Energy Potentials 5.4.1 The Linear Elastic Strain Energy Potential The strain energy u was introduced in §5.2 1 . From Eqn 5.2.19, the strain energy can be regarded as a function of the strains: ( ) ( ) ( ) [ ] ( ) 2 2 2 2 2 2 2 2 ) 1 ( 2 1 zx yz xy xx zz zz yy yy xx zz yy xx ij u u ε ε ε μ ε ε ε ε ε ε ν ε ε ε ν ν μ ε + + + + + + + + − − = = (5.4.1) Differentiating with respect to xx ε (holding the other strains constant), ( ) ( ) [ ] zz yy xx xx u ε ε ν ε ν ν μ ε + + − − = ∂ ∂ ) 1 ( 2 1 2 (5.4.2) From Hooke’s law, Eqn 4.2.9, with Eqn 4.2.5, ( ) [ ] ν μ + = 1 2 / E , the expression on the right is simply xx σ . The strain energy can also be differentiated with respect to the other normal strain components and one has zz zz yy yy xx xx u u u σ ε σ ε σ ε = ∂ ∂ = ∂ ∂ = ∂ ∂ , , (5.4.3) The strain energy is a potential , meaning that it provides information through a differentiation. Note the similarity between these equations and the equation relating a conservative force and the potential energy seen in §5.1: F dx dU = / . Differentiating Eqn. 5.4.1 with respect to the shear stresses results in zx zx yz yz xy xy u u u σ ε σ ε σ ε 2 , 2 , 2 = ∂ ∂ = ∂ ∂ = ∂ ∂ (5.4.4) The fact that Eqns. 5.4.4 has the factor of 2 on the right hands side but Eqns. 5.4.3 do not The fact that Eqns....
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 1 taught by Professor Staff during the Fall '11 term at Auckland.
- Fall '11