Yield isotropic material

From the discussion by Tomida et al. (1999), it is naturally concluded that HDDR processing of lanthanide-transition-metal alloys without boron, such as Sm2Fei7, should always yield isotropic materials because of the absence of the Fe-B phases, in which the nuclei of the main phase continuously exist during HDDR process without changing their initial crystal orientations. 

Perhaps the most dramatic exception to the perfectly elastic, perfectly plastic materials response is encountered in several brittle, refractory materials that show behaviors indicative of an isotropic compression state above their Hugoniot elastic limits. Upon yielding, these materials exhibit a loss of shear strength. Such behavior was first observed from piezoelectric response measurements of quartz by Neilson and Benedick [62N01]. The electrical response observations were later confirmed in mechanical response measurements of Waekerle [62W01] and Fowles [61F01]. 

Re-entrant foam provides a counter-intuitive demonstration of processing (5). Polyurethane can be isotropically compressed in a mold and heated to about 170 °C. The microstructure of the resulting solid yields a material that bulges in cross section when stretched More information on polymers will be available from John Droske s complementary NSF-funded project (described in the preceding section). 

Microfibrillar structure in isotropic materials makes asymmetrical peaks, because microfibrils are materials with linear disorder. Steep is the increase from small scattering angle. The peak shape can be quantitatively analyzed (S tribeck [106]) yielding extra information on the lateral extension of the microfibrils. 

Warning. For isotropic materials the ID projection /, and the Lorentz correction yield different ID intensities. Both are related by... 

If the structural entities are lamellae, Eq. (8.80) describes an ensemble of perfectly oriented but uncorrelated layers. Inversion of the Lorentz correction yields the scattering curve of the isotropic material I (5) = I (s) / (2ns2). On the other hand, a scattering pattern of highly oriented lamellae or cylinders is readily converted into the ID scattering intensity /, (53) by ID projection onto the fiber direction (p. 136, Eq. (8.56)). The model for the ID intensity, Eq. (8.80), has three parameters Ap, dc, and <7C. For the nonlinear regression it is important to transform to a parameter set with little parameter-parameter correlation Ap, dc, and oc/dc. When applied to raw scattering data, additionally the deviation of the real from the ideal two-phase system must be considered in an extended model function (cf. p. 124). 

The yield criterion first suggested for metals was Tresca s criterion, which proposes that in isotropic materials yield occurs when the maximum shear stress X reaches a critical value (10). ... 

Grinding. An important aspect is often whether such a material can be ground, and how small then are the particles obtained. It can be derived from the theory that the thickness of the zone near a crack in which plastic deformation (yielding) occurs in a homogeneous isotropic material is given by... 

Consider an isotropic material that exhibits elastic-plastic fracture. Will its yield stress be higher or lower than its fracture stress, or is the difference unimportant ... 

The number of dislocations in a unit volume of crystal is the dislocation density. Since it is taken to be the total length of dislocations per unit volume, its units are cm"2. A less precise, but more readily measurable, quantity is the number of dislocations intersecting the unit area of the crystal, and this is also expressed in miits of cm 2. For isotropic materials the two methods of defining dislocation density are almost equivalent (to within a factor of 2). But for anisotropic materials, e.g., layer structures, the two methods may yield vastly different values (factors of up to 10 ). 

Tresca and R. von Mises criteria are for isotropic materials. In 1948, Rodney Hill provided a quadratic yield criterion for anisotropic materials. A special case of this criterion is von Mises criterion. In 1979, Hill proposed a non-quadratic yield criterion. Later on several other criteria were proposed including Hill s 1993 criterion. Rodney HiU (1921-2011) was bom in Yorkshire, England and has tremendous contribution in the theory of plasticity. 

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. 

There is a formal similarity between (10) and the Coulomb theory of yield mentioned above for isotropic materials. On the latter theory yield occurs on the plane which makes an angle a with the tensile axis, for which the shear component of the applied stress reaches a critical value. This shear component at yield decreases linearly with the normal component of stress on that plane. For an axial stress [Pg.379]

It is important to appreciate that plasticity is different in kind from elasticity, where there is a unique relationship between stress and strain defined by a modulus or stiffness constant. Once we achieve the combination of stresses required to produce yield in an idealized rigid plastic material, deformation can proceed without altering stresses and is determined by the movements of the external constraints, e.g. the displacement of the jaws of the tensometer in a tensile test. This means that there is no unique relationship between the stresses and the total plastic deformation. Instead, the relationships that do exist relate the stresses and the incremental plastic deformation, as was first recognized by St Venant, who proposed that for an isotropic material the principal axes of the strain increment are parallel to the principal axes of stress. 

In isotropic materials, the yield surface must not depend on the orientation of the load. Thus, the function / describing the yield surface can only contain those parts of the deviatoric stress tensor that do not change during coordinate transformations. This is already ensured if the principal stresses are used because the hydrostatic stress (Thyd is also coordinate invariant. 

It was stated in section 6.2.3 that a dislocation will start to move if a sufficiently large shear stress acts on the slip system. This stress value is called critical resolved shear stress Tcrit- It is not equal to the yield strength rp of an isotropic material under shear loading because in the latter case different slip systems have to be activated that are usually not parallel to the shear stress. For a single crystal, the yield criterion (cf. section 3.3.1) is... 

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