A generalized quadratic flow law for sheet metals
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I.
INTRODUCTION
THE forming limit diagram
in its current configuration is based upon the work of Keeler ~ and of Goodwin.: For a particular sheet metal it represents limiting strain states in principal strain coordinates. It provides information that is very useful in press shop operations and has also attracted wide theoretical interest. Forming limit diagrams are determined experimentally. According to Kleemola and Kumpulainen, 3 the limit strains for a given sheet metal have been shown to depend upon the strain rate, 4 the strain gradient, 5 the strain path, 6-9 the strain measurement practice, 1~ and other effects. To the extent possible, in view of the mathematical difficulties, these effects have been incorporated into various theoretical analyses; however, a wide gap remains between theory and experiment. There has been a great deal of discussion in the literature and privately to the effect that a more accurate relationship between effective stress, effective strain, and effective strain rate, i . e . , a more accurate material constitutive relation is required to improve the agreement of theory with experiment. One purpose of this note is to suggest that a more fundamental theoretical deficiency may cause the discrepancy, namely, the use of an inappropriate anisotropic yield (or flow) criterion in these analyses along with its associated flow rule.
a flow criterion. Let this relationship be denoted in quadratic form as:
= f(0-0, 0-90,
[1]
An effective strain ~ is defined in terms of the corresponding in-plane strain components e0, e9o, and y by a relationship that is generally incremental in form and which derives from the material flow rule. To provide an initial characterization of the material, tension tests are commonly performed on specimens taken at orientations of 0 and 90. Suppose two such specimens are homogeneously deformed at equivalent strain rates to equal values of effective plastic strain. Denote the flow stresses of the two specimens at these equivalent test points by So and $90, respectively. An accurate flow criterion must show the effective stress to be equal for each of these stress states, i . e . ,
f(s0, 0, 0) = f(0, s90, 0)
[2]
Flow anisotropy is traditionally described for sheet metals by the plastic anisotropy ratio, or r-value, defined as the ratio of width-to-thickness strains for a specimen pulled longitudinally in simple tension. The following material parameters are determined: ro = - e 9 0 / ( e o + E90)
0"90 ~-- '7" ~- 0
[3]
r90 = -e0/(e0 + e90)
0"0 = r = 0
[4]
For simplicity we introduce the related constants: II.
BACKGROUND
Before a stress, strain relation can be expected reasonably to describe the results of complex biaxial stressing programs in which necking occurs, it should be able to describe adequately the results of simple tension tests during homogeneous deformation. Consider a sheet metal that will be treated under the theory of generalized plane stress. Let the subscript 0 denote the rolling direction of the sheet and subscript 90 the in-plane d
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