This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
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
 Staff
 Strain

Click to edit the document details