II. Basic Concepts
PERFECTLY PLASTIC
MATERIAL
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 strainrate sensitive. That is, the strength of the
material increases with increased deformation levels. Above the recrystallization
temperature, the material is strainrate 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 strainrate 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:

The material is incompressible.

There exists a material constant K with dimension
of stress.

The stress deviator (s_{ij})
depends on the strain rate (
) in a quasilinear way:
(1)
where f is a scalar invariant
function of the principal strain rates,
(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)
Moreover, the ideal materials defined above have a yield
criteria of the general form
(4)
where s_{1}, s_{2} and s_{3}
denote the principal stress deviators. The same constitutive function F
thus enters in the definition of the constitutive relations and of the
yield criterion.
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.
FLOW
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)
WORK AND
POWER OF
DEFORMATIONS
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.
FRICTION,
LUBRICATION, AND
WEAR
In all of the processes described in Figs. 13, 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)
where t is the friction resistance
to sliding, and D v is the relative sliding
speed.
