Isotropic hardening

Sohd rocket propellants represent a very special case of a particulate composite ia which inorganic propellant particles, about 75% by volume, are bound ia an organic matrix such as polyurethane. An essential requirement is that the composite be uniform to promote a steady burning reaction (1). Further examples of particulate composites are those with metal matrices and iaclude cermets, which consist of ceramic particles ia a metal matrix, and dispersion hardened alloys, ia which the particles may be metal oxides or intermetallic compounds with smaller diameters and lower volume fractions than those ia cermets (1). The general nature of particulate reinforcement is such that the resulting composite material is macroscopicaHy isotropic. 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

In the classical theory of plasticity, constitutive equations for the evolution of the isotropic and kinematic hardening parameters are usually expressed as... 

Atluri, S.N., On Constitutive Relations at Finite Strain Hypo-Elasticity and Elasto-Plasticity with Isotropic or Kinematic Hardening, Comput. Methods Appl. Mech. Engrg. 43, 137-171 (1984). 

The present review shows how the microhardness technique can be used to elucidate the dependence of a variety of local deformational processes upon polymer texture and morphology. Microhardness is a rather elusive quantity, that is really a combination of other mechanical properties. It is most suitably defined in terms of the pyramid indentation test. Hardness is primarily taken as a measure of the irreversible deformation mechanisms which characterize a polymeric material, though it also involves elastic and time dependent effects which depend on microstructural details. In isotropic lamellar polymers a hardness depression from ideal values, due to the finite crystal thickness, occurs. The interlamellar non-crystalline layer introduces an additional weak component which contributes further to a lowering of the hardness value. Annealing effects and chemical etching are shown to produce, on the contrary, a significant hardening of the material. The prevalent mechanisms for plastic deformation are proposed. Anisotropy behaviour for several oriented materials is critically discussed. 

The more rigorous stress/strain nonlinear material model, oflen referred to as the plastic zone method, is theoretically capable of handling any general cross section Both isotropic and kinematic hardening rules are usually available. This method is... 

Powder Yield Loci For a given shear step, as the applied shear stress is increased, the powder will reach a maximum sustainable shear stress "U, at which point it yields or flows. The functional relationship between this limit of shear stress "U and applied normal load a is referred to as a yield locus, i.e., a locus of yield stresses that may result in powder failure beyond its elastic limit. This functional relationship can be quite complex for powders, as illustrated in both principal stress space and shear versus normal stress in Fig. 21-36. See Nadia (loc. cit.), Stanley-Wood (loc. cit.), and Nedderman (loc. cit.) for details. Only the most basic features for isotropic hardening of the yield surface are mentioned here. 

The stress-strain relationship for FGMs is assumed to be a bilinear form and can be described by the isotropic hardening model and kinematic hardening model as ... 

Strains and stresses were computed for the joined specimen cooled uniformly to room temperature from an assumed stress-free elevated temperature using numerical models described in detail previously [19, 20]. The coordinate system and an example of the finite element mesh utilized are shown in Figure 3. Elastie-plastic response was permitted in both the Ni and Al203-Ni composite materials a von Mises yield condition and isotropic hardening were assumed. 

In FCC materials, there are 12 different slip systems, which can contribute to the deformation process. Dislocation density histories at a peak stress of 4.5 GPa for [001], [111] and [Oil] orientations and isotropic case with [001] orientation are calculated and plotted as shown in Fig. 13. It is clear that the dislocation density is very sensitive to crystal orientation with the highest density exhibited by [111] orientation followed by the isotropic media, [011] and [001] orientations respectively. This may be attributed to the number of slip systems activated and to the way in which these systems interact. The [001] orientation has the highest symmetry among all orientations with four possible slip planes 111 that have identical Schmid factor of 0.4082, which leads to immediate work hardening. The [011] orientation is also exhibits symmetry with 2 possible slip planes that have Schmid factor of 0.4082. 

During isotropic loading, plastic deformation takes place when the isotropic stress p reaches the preconsolidation pressure pf. The pressure pf is a measure of the size of the yield surface on the isotropic axis and can be viewed as an hardening/softening parameter (the specific shape of the yield surface is described in the next section). An essential feature of the proposed model is the decrease of pf with respect to an increase in contaminant concentration. This can be expressed as... 

The constant X represents the slope of the isotropic mechanical response plotted in the (v,ln p lp f)-plane. By differentiating (4) and using (6) one can deduce the incremental hardening/softening law... 

This relation contains two competing terms the first term represents plastic hardening as a function of the volumetric part of plastic strain, the second term describes chemical softening due to an increase in contaminant concentration. Let us consider the plastic response to an increase in contaminant concentration at constant isotropic stress. The condition p =pc =0 in equation (7) implies... 

In the manufacture of wood chipboard, which represents one of the largest applications of UF resins, wood chips are mixed with about 10% of a resin-hardener solution and pressed in a multidaylight press at 150°C for about 8 min. Since some formaldehyde is released during the opening of the press, it is necessary to use a resin with a low formaldehyde content. Because it has no grain, a wood chipboard is nearly isotropic in its behavior and so does not warp or crack. However, the water resistance of chipboard is poor. 

It is often convenient to stiffen or harden a material, commonly a polymer, by the incorporation of particulate inclusions. The shape ofthe particles is important [see Christensen, 1979]. In isotropic systems, stiff platelet (or flake) inclusions are the most effective in creating a stiff composite, followed by fibers and the least effective geometry for stiff inclusions is the spherical particle, as shown in Figure 41.3. A dilute concentration of spherical particulate inclusions of stiffness , and volume fraction Vj, in a matrix (with Poisson s ratio assumed to be 0.5) denoted by the subscript m, gives rise to a composite with a stiffness E ... 

Usually, the material starts with a kinematic behavior. With increasing strain level, the kinematic behavior saturates and isotropic hardening is taken place. The problem now is to describe the behavior of a as a function of the strain and strain-rate level. Several models are available but only the models from Armstrong and Frederick (Armstrong and Frederick 1966) and the Yoshida model (Yoshida and Uemori 2002) are described in the following because of their popularity in the last decades. 

---