Microstructure-Properties - Materials Science and Engineering

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34 Objective Linear Ferromagnets Non-linear properties Electric. Conduct. Tensors Elasticity Symmetry Non-Linear properties, example • Another important example of non-linear properties is plasticity, i.e. the irreversible deformation of solids. • A typical description of the response at plastic yield (what happens when you load a material to its yield stress) is elastic-perfectly plastic. In other words, the material responds elastically until the yield stress is reached, at which point the stress remains constant (strain rate unlimited). • A more realistic description is a power-law with a large exponent, n~50. The stress is scaled by the crss, <strong>and</strong> be expressed as either shear stressshear strain rate [graph], or tensile stress-tensile strain [equation]. ˙ ! = # " & % ( $ " yield ' n
35 Objective Linear Ferromagnets Non-linear properties Electric. Conduct. Tensors Elasticity Symmetry Einstein Convention • The Einstein Convention, or summation rule for suffixes looks like this: A i = B ij C j where “i” <strong>and</strong> “j” both are integer indexes whose range is {1,2,3}. So, to find each “i th ” component of A on the LHS, we sum up over the repeated index, “j”, on the RHS: A 1 = B 11 C 1 + B 12 C 2 + B 13 C 3 A 2 = B 21 C 1 + B 22 C 2 + B 23 C 3 A 3 = B 31 C 1 + B 32 C 2 + B 33 C 3
Page 1 and 2: 1 Objective Linear Ferromagnets Non
Page 11 and 12: 11 Objective Linear Ferromagnets No
Page 17 and 18: 17 Effect of symmetry on stiffness
Page 33: 33 Objective Linear Ferromagnets No
Page 41: 41 Objective Linear Ferromagnets No

Microstructure-Properties - Materials Science and Engineering

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