Invariance of neck formation to material strength and strain rate for power-law materials
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Invariance of Neck Formation to Material Strength and Strain Rate for Power-Law Materials
~ = effective strain = s
f(deo)[2] -path
e = effective strain rate = __d(~) dt = effective stress = g(tro)
A finite element formulation derived for analysis of largestrain forming problems has been used to show that plastic strain distributions are independent of material strength and deformation velocity under certain common conditions. These conditions include displacement boundary conditions (or mixed displacement and traction-free boundary condition) and a power-law type of strain-rate sensitivity. This result has clear implications for the interpretation of a range of common mechanical tests, including standard tensile tests, in which external displacements are imposed rather than external forces. One of the simplest and most widely used constitutive equations employed for the mechanical behavior of metals and alloys may be written ~ in one-dimensional form as follows:
[3] [4]
The functionsf and g follow from the plastic yield function by application of "normality ''4 and "equivalent plastic work "principles. Strictly speaking, the strains should be plastic only. We will restrict our attention to rigid-plastic material models, equivalent to examination of large-strain problems where the elastic modulus is much larger than the plastic modulus. The only other restriction on the material model is thatf depends only on de 0 and g depends only on o'ij. Isotropic hardening rules meet this criterion, as do certain 6 non-isotropic hardening formulations. A rigid-plastic finite-element formulation has been derived 3'7 to analyze in-plane sheet deformation 2'3's using the incremental, or flow, theory of plasticity. This formulation may be written as follows: J
[11
where ~r, e, and k are true axial stress, strain, and strain rate, respectively, and k, n, and m are material constants often called the strength coefficient, work-hardening rate, and strain-rate sensitivity, respectively. Examination of Eq. [1] reveals that a change in constant strain rate (by a factor of 10, e.g. ) is equivalent to a change in k (by 10m, e.g. ), and that both result in a material law change tantamount to a simple change in units of stress. This result intuitively suggests that a change in the strength of a real material (as manifested by k) or a different tensile testing rate (such that k changes by a constant factor) should not alter test results in any way except for a constant stress factor. Such a result is clearly true during uniform extension at constant rate (both external and internal) where strains and strain rates are uniform throughout the specimen and a change in external extension rate simply imposes a new rate throughout the specimen. Application of this principle to nonuniform strain and strain-rate fields and to multi-axial deformation problems is K. CHUNG, Postdoctoral Researcher, and R.H. WAGONER, Professor, are with the Department of Metallurgical Engineering, The Ohio State University, Columbus, OH 43210. Manuscript submitted
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