II. Basic Concepts

Metal deformations are introduced through the application of external forces to the workpiece, these forces being in equilibrium. With the application of load to the workpiece, internal stress and displacements are generated causing shape distortions. If the loads are low, then with the release of the loads, the internal stresses will disappear and the workpiece will be restored to its original shape. It is then said that the applied loads were elastic and so were the stresses and strains. Elastic strains are recoverable on release of the loads. When the loads are high enough, the changes in shape will not disappear after the load is released. The changes in shape and the strain, those that did not disappear, namely, the permanent ones, are called plastic deformations. The loads causing plastic deformations are said to have surpassed the elastic limit. During metal forming by bulk plastic deformations (to be defined), the plastic deformations are much larger than the elastic deformations, which in general are ignored. Thus, only plastic deformations are considered. It is also recognized that plastic deformations do not involve volumetric changes. Thus, volume constancy is maintained. In metals, the load and the intensity of the internal stresses at which (plastic) flow initiates are functions of the structure, the temperature, the deformation history, and the rate of straining. It is fair to assume that at temperatures below the recrystallization temperatures at which new crystal structures emerge the material strain hardens but is not strain-rate sensitive. That is, the strength of the material increases with increased deformation levels. Above the recrystallization temperature, the material is strain-rate sensitive. That is, its strength is higher with higher rates of straining, but it does not strain harden. The point in loading where incipient plastic flow commences is called the "yield point," to be defined mathematically in Eq. (7).

A perfectly plastic material is such that it does not strain harden and is not strain-rate sensitive. The strength is not a function of strains nor of strain rates. A deformable material is said to be a perfectly plastic material (Talbert, 1984), when:

1. The material is incompressible.
2. There exists a material constant K with dimension of stress.
3. The stress deviator (s_ij) depends on the strain rate (

(1)

(2)

These assumptions have been shown to imply that there exists a function , homogeneous of degree –1 in the principal strain rates , such that

(3)

(4)

Dr. Talbert (1984) specifies the function F for four popular and different perfectly plastic materials named after Tresca, Mises, and others. For the Mises perfectly plastic material

(5)

which in an arbitrary Cartesian coordinate system reads

(6)

so that the yield criteria becomes

or

(7)

where s_ij are the components of the stress deviator and K is a constant property of the material to be determined experimentally. As long as the scalar quantity on the left of Eq. (7) does not reach the value of K, the material does not deform. On reaching the value of K, plastic flow commences, The tensile test is a most common test to evaluate K. Thus, if the load stress in tension at the yield point is s ₀, then K = ± s ₀/Ö 3.

A pattern of deformations is known when specified by a velocity field. A velocity field is defined through the vector ?_{i in space, confined to the workpiece. In a Cartesian coordinate system of X1, X2, and X3, at any instant of time,}

_{the velocity vector is} ?_{i(X1, X2, X3) (where i =1,2,3). The strain–rates components can be derived from the velocity vector as follows:}

(8)

and by Eqs. (3) and (8), the components of the stress deviator can be determined from the velocity field and the strength constant K.

(9)

When the stress deviator components and the strain rates components are known, the internal power of deformations per unit volume can be derived as follows:

(10)

and since s_ij = s _ij - sd _ij where s = (s ₁₁+s ₂₂+s ₃₃)/3 and d _ij =1 if i = j and d _ij =0 if i ¹ j, by Eq. (10)

(11)

Thus, the internal power of deformations is determined from the strain rates components as derived by Eq. (8) from the velocity field vector and from the material constant K.

Although the characteristics of metals are more complex than those described here by a perfectly plastic metal, we will not expand further. The treatment becomes too complex, and the bulk of the study of metal forming is served well with the simpler model.

In all of the processes described in Figs. 1-3, there exits a sliding motion along the interfaces between the workpiece and the tools. Whenever sliding occurs between solids, a resistance to the sliding motion is observed. This resistance is called friction. Friction resistance is accompanied by damage to the surfaces, which is mostly manifested by the wearing of the surfaces. The resistance to sliding, measured as a shear stress per unit surface area of contact (t ), is a complex function of many parameters, including workpiece and tool materials, surface smoothness of the tool, and speed of sliding. Friction and wear are also controlled by the introduction of lubricants between the interfacing surfaces. The lubricant serves not only to minimize friction and wear but also to cool the surfaces by removing the heat generated through sliding. Most effective lubrication methods may provide a thin film of lubricant separating the two surfaces completely. When full liquid film separation is created, a condition of hydrostatic or hydrodynamic lubrication prevails, minimizing the friction and wear. The energy generated through sliding can be calculated by

(12)