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EN0175-14 - EN0175 10 19 06 Mechanical Behavior of Solids...

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EN0175 10 / 19 / 06 Mechanical Behavior of Solids Linear Elastic solids ε σ E = (1D) kl ijkl ij C ε σ = or kl ijkl ij S σ ε = (3D) where C is sometimes called the stiffness tensor and S is sometimes called the compliance tensor. Both of them are 4 th order elastic moduli tensors. Symmetry of elastic moduli tensors: jikl ijkl C C = , (minor symmetry) ijlk ijkl C C = The minor symmetries reduce the independent elastic constants from 81 to 36. There is also a major symmetr in elastic moduli klij ijkl C C = , 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 C and S : u F , Assume an increment of displacement at the bar end, u u u δ + The work done by the applied load should equal to the stored energy in the material, ( ) ε σδ ε σδ ε δ σ δ δ V Al l A u F W = = = = ε σδ δ δ = = V W w should be the stored elastic energy per unit volume, which is also called the strain energy density. ( ) ε w w = , ε σ = w , ( ) = ε ε σ ε 0 d w 1
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