Elastoplastic material, stress-strain

Some plastic materials have different tensile and compressive characteristics. For example, polystyrene is tough under compressive load but very brittle in tension. However, for most elastoplastic materials, the stress-strain curves in compression are the same as in tension. Hence, the deformation properties of these materials in tension may also be applied to those in compression, which is of great interest to gas-solid flows. 

In contrast to the simplicity of elastic deformation, plastic deformation occurs in diverse ways. Figure 1.9 illustrates the stress-strain curves for two typical elastoplastic materials (hardened metal and polymer). Both materials show similar linear relationships between stress and strain for the elastic deformation (i.e., before yield strength) but quite different correlations in the yielding processes before fracture. 

Figure 1.9. Deformation of typical elastoplastic materials (hardened metal and polymer) in the stress-strain diagram (after Guy, 1976).

Elastoplastic materials Elastoplastic materials deform elastically for small strains, but start to deform plastically (permanently) for larger ones. In the small-strain regime, this behavior may be captured by writing the total strain as the sum of elastic and plastic parts (i.e., e = e -I- gP, where e and gP are the elastic and plastic strains, respectively). The stress in the material is generally assumed to depend on the elastic strain only (not on the plastic strain or the strain rate), and hence, no unique functional relationship exists between stress and strain. This fact also implies that energy is dissipated during plastic deformation. The point at which the material starts to deform plastically (the yield locus) is usually specified via a yield condition, which for one-dimensional plasticity may be stated as (38)... 

Viscoplastic materials, therefore, share many features with elastoplastic materials but, in addition, exhibit a dependence on the rate of straining. Again, a decomposition of the total strain is convenient, this time into elastic and viscoplastic parts (i.e., e = where e and are the elastic and viscoplastic strains, respectively). In analogy with the elastoplastic case, the boundary of the elastic region may be specified in terms of a yield function. However, whereas the region outside the yield surface was inadmissible in the elastoplastic case, the stress is allowed to lie outside the yield surface in the viscoplastic one. Hence, straining beyond the yield point generally results in the creation of an excess (or extra) stress o that decays toward zero with time, typically as (38)... 

Fig. 2.14 Schematic view of the stress-strain relationship for an elastoplastic material. The symbols Yp and yr denote the yield strain, the plastic strain and the failure strain, respectively. The same notation is used for the stress a.

Besides linear viscoelastic behavior, elastoplastic behavior is also often encountered for food products. In Fig. 2.14, the stress-strain behavior of an elastoplastic material is shown schematically. Because the stress-strain relationship is not linear and the strain does not recover if the yield stress is exceeded, the equations to describe this behavior are much more complicated. 

In the linear elastic region, constitutive material models are able to correlate strain to stress without the need to consider the history of stress and strain events a specific object or assembly has been subjected to previously. If an elastoplastic material is subjected to mechanical stress it will also respond with instantaneous elastic deformation but above the yield strength deform plastically at a deformation rate governed by the process of deformation. Again, the behavior is not considered to be time dependent. 

For the yield stress in compression, deviations from Tabor s relation giving values of 2Yc are found. This is presumably due to the elastic strain of the indented material. A detailed analysis of the H/Yc ratio on the basis of mechanical models of elastoplastic indentation reveals that H/Yc linearly increases with ln[(tan/3Ec)/Yc]. Compression-moulded (chain-folded) PE samples, which present the lowest crystallinity of all the samples investigated, also show the lowest H/Yc ratio as a consequence of the comparatively large elastic strains. 

Figure 5 Schematic illustration of the behavior of elastic, elastoplastic, and viscoplastic materials. A loading-unloading sequence is shown in (A) and (B), whereas stress relaxation following straining past the yield point is depicted in (C). (A) Elastic. (B) Elastoplastic. (C) Viscoplastic.

The in-plane extensional strain and curvature in the film-substrate system arises as a consequence of thermal expansion mismatch between the film and substrate materials during temperature excursion from the reference temperature. Conditions are assumed to be such that the temperature is uniform throughout the film and substrate at all times during thermal cycling. Some particular temperatures at which distinct transitions occur in the elastoplastic deformation of the film material are first identified by adopting the assumption that, over the range of temperature, the properties of the film and substrate materials remain essentially unchanged effects of temperature dependence of plastic yield or flow behavior on the evolution of film stress and substrate curvature are examined subsequently. 

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