EN0175 10 / 19 / 06 Mechanical Behavior of SolidsLinear Elastic solidsεσE=(1D) ⇒klijklijCεσ=or klijklijSσε=(3D) where Cis sometimes called the stiffness tensor and Sis sometimes called the compliance tensor. Both of them are 4thorder elastic moduli tensors. Symmetry of elastic moduli tensors: jiklijklCC=, (minor symmetry) ijlkijklCC=The minor symmetries reduce the independent elastic constants from 81 to 36. There is also a major symmetr in elastic moduli klijijklCC=, which reduces the number of independent elastic constants from 36 to 21. We use the concept of energy & work to demonstrate the major symmetry of Cand S: uF,Assume an increment of displacement at the bar end, uuuδ+→The work done by the applied load should equal to the stored energy in the material, ()εσδεσδεδσδδVAllAuFW====εσδδδ==VWwshould be the stored elastic energy per unit volume, which is also called the strain energy density. ( )εww=, εσ∂∂=w, ( )∫=εεσε0dw1
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