The continuum approach for isotropic materials

1 The continaum approach for isotropic materials In order to appreciate the development of the continuum approach for oriented polymers it is helpful to discuss briefly the progress made in dealing with plasticity of isotropic polymers. 

Most attempts at dealing with the yield criteria of isotropic polymers derive from the criteria due to Hencky-von Mises or Tresca (see Ref. 5, Chapter 2 for an excellent introduction to this topic). The Hencky-von Mises criterion can be written in the alternative forms 

These equations state that yield will occur when the function of the stress components represented by the left-hand side of these equations reach a critical value, 6k. If we consider an isotropic material with (Tj = — da = fc and 0-3 = 0, it is clear that this stress configuration satisfies 2 and so we can identify k with the yield stress in pure shear. 

Note that both (1) and (2) are symmetrical with respect to the subscripts X, y, z and 1, 2, 3 respectively, reflecting the isotropy of the material. Note further, that both of these equations are unchanged by increasing each normal stress by a constant amount, p, Le. if we write Gx- Gx+p,Gy- Gy+p,Gf- Gi+p, then eqn. (1) is unchanged. This feature, implying that hydrostatic pressure does not affect yield, is a necessary ingredient for a yield criterion for metals which deform mainly by slip processes at constant volume. It however serves only as a first approximation to the yield behaviour of isotropic polymers (see Ref. 6), for 

Such a dependence on hydrostatic pressure naturally implies a difference between tensile and compressive yield stresses as is observed for most polymers. 

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