Research Journal of Applied Sciences, Engineering and Technology 6(5): 761767, 2013
DOI:10.19026/rjaset.6.4116
ISSN: 20407459; eISSN: 20407467
© 2013 Maxwell Scientific Organization Corp.
Submitted: August 07, 2012 Accepted: September 03, 2012 Published: June 25, 2013
Mahir EsSaheb, Department of Mechanical Engineering, College of Engineering, King Saud
University, P.O. Box 800, Riyadh 11241, Saudi Arabia
This work is licensed under a Creative Commons Attribution 4.0 International License (URL: http://creativecommons.org/licenses/by/4.0/).
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Mahir EsSaheb, Ali AlWitry and Abdulmohsen Albedah
Department of Mechanical Engineering, College of Engineering, King Saud
University, P.O. Box 800, Riyadh 11241, Saudi Arabia
Traditional upper bound analyses for planestrain compression leads to low load prediction values, for a
large ratio range of contact Length (L) to thickness (h). These predicted load values are based on deformation fields
consisted of rigid regions separated by planes upon which discrete shear occurs. In this study, the relatively simple
deformation fields consisted of odd number of triangles such like, 1, 3 and 5, are used. A general minimum solution
for this class of upper bounds is derived and found to occur when the base of the center triangle (w) [= 2L/ (n+1)],
where n is an odd integer ≥3. Consequently, whether still lower values would occur with this class of field when the
ratio(R), of the field triangles side lengths is varied, has been systematically investigated. Thus, the upper bound
loads are calculated over a wide range of the ratio (R) and the deformation field parameters. These parameters
include the thickness (h) the contact Length (L) and the base of the center triangle (w). It is found that, all the
minimum load values occur at unity R ratio and the minimum values of h, L and w. The minimum values obtained
for hopt, Lopt and wopt are 1.466, 3.266 and 2.0 units, respectively. Also, the corresponding overall minimum
optimal upper bound value found is P/2k = 1.93.
Fields, metal forming, planestrain compression, plasticity, upper bound method
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In metal forming, engineering plasticity, offers
various methods recommended for predicting the loads
needed to effect the shape change desired. These
methods include the four distinct approaches namely:
uniform work or energy; slab or equilibrium force
balance; upper and lower bounds; slip line field.
The calculation of the exact loads or forces to
cause plastic flow of metals is often difficult, if not
impossible. Exact solutions require that both stress
equilibrium and a geometrically selfconsistent pattern
of flow are satisfied simultaneously everywhere
throughout the deforming body and on its surface.
Fortunately, limit theorems permit force calculations,
which provide values that are known to be either lower
or higher than the actual forces. These calculations
provide lower or upper bounds. A lowerbound solution
will give a load prediction that is less than or equal to
the exact load needed to cause a body to experience full
plastic deformation. Several texts (Johnson and Mellor,
1973; Calladine, 1969) may be consulted for greater
detail on lower bound However, in metalforming
operations, it is of greater interest to predict a force that
will surely cause the body to deform plastically to
produce the desired shape change. This can be achieved
through the use of the upper bound approach. Actually,
an upperbound analysis predicts a load that is at least
equal to or greater than the exact load needed to cause
plastic flow. Upperbound analyses focus upon
satisfying a yield criterion and assuring that shape
changes are geometrically selfconsistent. Hence, to
avoid timeconsuming complexities in calculating the
load that is at least large enough to cause plastic flow; it
is often resorted to the upper bound approach. Johnson
and Mellor (1973) discuss the use of the upper bound
theorem in detail as it applies to plane strain operations
while (Avitzur, 1968; Caddell and Hosford, 1980,
1983) use an upper bound approach in analyzing a
number of ax symmetric operations.
Over the last few decades, since the early works on
forming operations by Kudo (1960, 1961) and Rowe
(1965), the upper bound method has been in use by a
large number of investigators. Amongst those are: Fox
and Lee (1989), Na and Cho (1989), Cho and Kim
(1990), Mulki and Mizuno (1996), Kimura and Childs
(1999), Pater (1999), Garmestani (2001) and Chai
(2003). Later more related works on the use of upper
bound method in some tenacious patterns are reported,
notably, the work of Bona (2004), EsSaheb (2004) and
Moller . (2004), as well as Ebrahimi and
Najafizadeh (2004). Also, in the civil engineering field,
the upper bound method is widely used in predicting
and estimating the loads in the foundations and footings
of different shapes, as reported in the last few years, by
Zhu and Michalowski (2005), Merifield and Nguyen