By Ismail C. Noyan; Jerome B. Cohen
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Aluminum, on the other hand, undergoes a gradual transition from elastic to plastic deformation. Most of the engineering materials with BCC structures behave in a fashion similar to mild steel, while metals with FCC or HCP structures exhibit flow curves similar to aluminum. 40 2 Fundamental Concepts in Stress Analysis b Fig. 16. Schematic flow curves for (a) grey cast iron, (b) mild steel and (c) pure aluminum The actual flow curves for ductile materials shown above are hard to model mathematically and can not be incorporated easily in plasticity theories.
63) may be obtained through the cyclic permutation of the suffixes in these equations. 64 ) where epkj is the permutation tensor. It is defined as . for even permutation of pki, for off permutation of pki, for all other combinations. For example, eI23·~1, e132=e321 = -1, e112=e313=0. The compatibility conditions restrict the types of displacement functions that may exist in a material for a given type of loading. For a homogeneous elastic stress distribution, for example, the displacement components U j must be continuous, single valued functions of the coordinates Xj.
28) define the principal stresses, they are independent of the stresses O";j, defined in the arbitrary coordinate system Xi' These quantities are called the stress invariants. 7 Elastic Stress-Strain Relations for Isotropic Materials 25 stress tensor, it is seen that the sum of the normal components in any coordinate system is invariant and equal to the sum of the principal stresses. An equivalent discussion can be also given for the strain tensor. In this case Eq. 31 ) I~ = E~ 1E~2 + E~2E~3 +E~3E'11 -El2 -E'/3 -E~23 I~ = E'l1 E~2E~3 + 2E'12E~3E'13 - E'l1 El3 - E~2El3 - E~3El2 .