Course Content:

Fundamentals of plasticity, yield and flow, anisotropy, instability, limit analysis, slipline field theory. Applications to forging, wire and tube drawing, deep drawing, extrusion and rolling. High velocity forming. 

Lecturewise Breakup (based on 75min per lecture)

I. Introduction (1 Lecture)

• Introduction to the course: different metal forming processes, importance of plasticity in the course. [1 Lecture]

II. Fundamentals of Plasticity (17 Lectures)

• Review-Analysis of stress: transformation relations, principal stresses and directions, maximum normal and shear stresses, invariants, hydrostatic and deviatoric parts; Analysis of (infinitesimal) strain: transformation relations, principal strains, invariants, hydrostatic and deviatoric parts; (Infinitesimal) rotation, Stress-strain relations for isotropic, linearly elastic material. [4.5 Lectures]

• Experimental observations on plasticity: yielding, strain-hardening, viscoplasticity, temperature softening, Baushinger effect, hysteresis, incompressibility of plastic deformation, anisotropy, plastic instability. [1 Lecture]

• Yield criterion for isotropic materials: von Mises and Tresca yield criterion, their geometric interpretation, convexity of the yield surfaces, experimental validation. [2 Lectures]

• Incremental and rate forms of the measures of plastic deformation: linear incremental strain tensor, strain rate (i.e. the rate of deformation) tensor and their relation, incremental rotation tensor and spin tensor. [2 Lectures]

• Change in yield criteria due to isotropic hardening: strain hardening and work hardening hypotheses, experimental validation of the hypotheses. [2 Lectures]

• Plastic stress-strain relations for isotropic materials: plastic potential and associated flow rule, incremental and rate forms of elasto-plastic stress-strain relations, simplifications for non-hardening and rigid-plastic materials (Prandtl- Reuss and Levy-Mises relations),Objective measures of stress rate and incremental stress. [3.5 Lectures]

• Incremental and flow formulations of plasticity: updated Lagrangian and Eulerian formulations, boundary and initial conditions, examples. [2 Lectures]

• Anisotropy: strain rate ratio, normal and planer anisotropies, Hill’s anisotropic yield criterion, example. [2 Lectures]

III. Approximate Methods of Plastic Analysis (6 Lectures)

• Approximate methods of solving plasticity problems: upper and lower bound theorems, upper and lower bound methods, slip line field equations, different boundary value problems of slip line method, one example of all 3 methods.

IV. Metal Forming Processes (14 Lectures)

• Slab method for sheet and wire drawing processes for predicting drawing force and die pressure, comparison with Wistreich experimental results, optimum die angle, maximum reduction for non-hardening material, upper bound method for sheet/wire drawing, slip-line method for sheet drawing, correction for hardening effect in upper bound and slip line  methods. [4 Lectures]

• Slab method for extrusion of rod and sheets for predicting extrusion pressure, upper bound method for sheet/wire extrusion, dead-metal zone for square die, slip-line method for sheet extrusion. [2 Lectures]

• Slab method for plane strain rolling for predicting roll force and roll torque, limiting reduction, roll diameter to sheet thickness ratio and friction coefficient. [2 Lectures]

• Slab method for sheet and disc forging processes for predicting forging force, sticking radius, slab method for hollow disc forging, neutral radius, upper bound method for sheet and disc forging with and without bulge, slip-line method for sheet forging. [4 Lectures]

• Slab method for the flange analysis in deep drawing, limiting drawing ratio for non-hardening materials. [2 Lectures]

Term Paper:

It should be based on the analysis of a different forming process (other than drawing, extrusion, rolling, forging and deep drawing) using either the upper bound method or the lower bound method or the slip line method. .

References:

1. The Mathematical Theory of Plasticity by R. Hill, Oxford University Press, 1950

2. Engineering Plasticity by W. Johnson and P.B. Mellor, von Nostrand Co. Ltd, 1972

3. Theory of Plasticity by J. Chakrabarty, McGraw-Hill Book Co., International Edition, 1987

4. Metal Forming: Processes and Analysis by B. Avitzur, McGraw-Hill Book Co., 1968

5. Continuum Theory of Plasticity by A.S. Khan and S. Huang, John Wiley and Sons Inc., 1995.