Concepciones del género entre las mujeres indígenas líderes 37

There have been many attempts to formulate the stress-strain relationship which governs the plastic deformation of porous materials. These attempts can be classified into two groups. The first is purely mathematical and based on a number of assumptions and conditions. The second is based on the findings of some basic experimental relationships which describe the effect of densification on the stress-strain relationship known from conventional plasticity theories.

Mathematical solutions are developed from the attempts to analyse a uni-pore element which represents a single cell in a porous body. Often many assumptions have to be made to simplify the solution. Storozhevskii (37) presented a number of possible models and selected a parallelepiped cell with an ellipsoidal pore for his solution. He assumed

that the volume of the porous body is made up of regular shaped elements or by their deformation (cubic, hexagonal, prism, rhombic dodeccahedron, cubic octahedron ...) which fill the space without any gap or super-impositions. Also he assumed that all the pores have the same shape and size and are evenly distributed. Furthermore, model and pore have the same centre of mass (central single-pore model) and the cross-sections for which a solution by the relative relaxation method (38) was determined, are parallel to the co-ordinate planes. Kreher and Schopf (39) studied the relationship between the hydrostatic pressure and the mean pore volume of a porous body of a homogeneous non strain- hardening material with spherical holes. Applying Tresca’s yield condition to the solid part with a yield stress in tension at a quasistatic loading condition and by using a self consistent method, results comparable to experiments were obtained. This method considers the interaction

between different pores unlike the work carried out by Chu and Hashin (40) in which they suggested a composite sphere model where the pressure is acting at infinity and the

material is composed of hollow spheres of constant material/ pore volume ratio. On the basis of plasticity theory and using a Polytropic force relationship, Dorofeev (41) calculated the

density distribution in powder blanks during dynamic hot free upsetting. He assumed the cylindrical blank to be a

composite of concentric cylindrical shells and claimed that the formula obtained gave good agreement with experimental data. Further description of the densification of a

porous body during plastic deformation with allowance for work hardening was attempted by Skorokhod and Martynova (42) on two basis:--

a) The mechanics of plasticity using a spherically symmetrical matrix model for a porous body and

b) The root mean square viscous strains and stresses using a statistical model for a porous body.

The values of stresses obtained using the first method were higher than the second one. Mori, et al, (43) applied the rigid plastic finite element method for the analysis of

forming of porous materials to predict the density distribution in axi-symmetric upsetting and plane-strain distribution.

Equivalent stress and strain rate values given by empirical equations obtained•from experiments were used in the solution. Haynes (44) investigated the effect of porosity on tensile

strength of sintered materials in an attempt to establish a yield criterion for a minimum acceptable ductility where the least-highly stressed parts of a material should not exceed their yield point before the most highly stressed

parts reach their fracture stress. He derived a quantitative expression relating the yield stress/tensile strength ratio of the matrix material to the limiting porosity content. Applying the upper bound approach on a porous matrix of

idealized geometry, Kahlow and Avitzur (45) investigated analytically (and experimentally) the effects of voids and their characteristics on the minimum effective pressure to

initiate a permanent void volume change. Also, Shima, et al, (46) derived an upper bound solution and utilized it to

estimate an approximate extrusion pressure and a final density ratio in a plane-strain condition.

The second group of solutions for the plastic deformation of porous materials ignores the geometrical analysis of cells and voids and instead is based on the empirical

relationships between porosity and stress-strain character­ istics of the sintered material during plastic deformation. The yield stress of porous materials (Yo) is less than the yield stress of the solid matrix (Y) and with densification

it increases until full density is reached. Hence, Yo is a function of Y. Also, densification is usually accompanied by deformation of the solid matrix and consequently work hardening takes place which adds to the strength of the material. Furthermore, densification indicates that the hydrostatic stress component has an effect on yielding and

is a function of porosity too. Therefore, any yield criterion for porous materials should consider the aforementioned

parameters. The conventional yield criterion by Von Mises states that 'yielding is a function of the second invariant of stress tensor only', and by including porosity functions, a modified relationship can be obtained. Along these lines investigators have recently proposed yield criteria for

sintered materials. These criteria will be reviewed in some detail in a separate chapter and will be compared with the experimental results of the present investigation in an attempt to testify their applicability.