Plane-Strain Slip-Line Fields for Metal-Deformation Processes

Plane-Strain Slip-Line Fields for Metal-Deformation Processes

A Source Book and Bibliography

1st Edition - January 1, 1982

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  • Authors: W. Johnson, R. Sowerby, R. D. Venter
  • eBook ISBN: 9781483278490

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Plane-Strain Slip-Line Fields for Metal-Deformation Processes: A Source Book and Bibliography provides information pertinent to the theory and application of plain-train slip fields to metal-working problems. This book discusses the industrial importance of axial symmetry. Organized into seven chapters, this book begins with an overview of the oldest processes of metal forming, including forging, coining, hammering, drifting, cutting, or parting. This text then examines the basic aspects of the basic theory of classical plasticity. Other chapters consider the governing equations of the plane plastic flow of a rigid-perfectly plastic solid. This book discusses as well the methods for the solution of problems of plane plastic flow of a rigid-perfectly plastic solid. The final chapter deals with the application of the theory of plasticity to the quasi-static plane-strain deformation of an isotropic rigid-perfectly plastic, rate insensitive material. This book is a valuable resource for mechanical engineers, materials scientists, teachers, and research workers.

Table of Contents

  • Chapter 1. Introduction

    The Metal-Forming Process in History

    Plane-Strain Slip-Line Fields: Historical Note

    Physical Observations

    (i) Surface Coatings

    (ii) Surface Markings on the Metal

    (iii) Grids or Nets of Lines

    (iv) Heat Lines and Zones

    (v) The Use of Plasticine and Wax

    (vi) Measurement of Friction Coefficient

    (vii) PVC : Chromoplasticity

    (viii) Defects: Voids or Cavities, Cracks, De-Densification, Piping

    (ix) Anisotropy

    (X) Thermography

    (xi) Load-Punch Travel Diagrams

    (xii) Hardness Traverses

    (xiii) Photoplasticity

    (xiv) Photostress

    (XV) Residual Stresses

    (xvi) Photoelasticity

    (xvii) Temperature-Sensitive Paints

    (xviii) Thermocouples

    (xix) Inserts


    Chapter 2. General Plasticity Theory


    Strain Rate

    Equilibrium and Virtual Work Equations

    Surfaces of Stress Discontinuity

    Surfaces of Velocity Discontinuity

    Maximum Work Principle


    Extremum Principles


    Tensor Calculus and Suffix Notation

    Surfaces of Stress and Velocity Discontinuities

    Plastic Potential and Flow Rule

    Representative Stress and Representative Plastic Strain Increment

    The Extremum Principles


    Chapter 3. Basic Theory of Plane Plastic Flow


    Stress Equations

    Velocity Equations

    Hencky's First Theorem

    Hencky's Second Theorem


    Stress Plane

    Requirements for a Complete Solution


    Chapter 4. Application to Specific Problems


    Construction of Slip-Line Fields

    Kinds of Slip-Line Fields

    Stress Boundary Conditions

    (i) Stress-Free Surface

    (ii) Frictionless Interface

    (iii) Interface with Coulomb Friction

    (iv) Full Shearing along an Interface

    Constructing the Hodograph

    Solutions to Problems Using the Upper-Bound Method


    Chapter 5. Applications to Specific Processes: Bibliography

    Centered-Fan Fields

    Pressure Vessels


    (i) Perfectly Rough Parallel Dies

    (ii) Perfectly Rough Inclined Dies

    (iii) Frictionless Parallel Dies

    (iv) Compression of Other than Slabs

    Indentation of a Semi-Infinite Medium


    Sheet Drawing

    Extrusion and Extrusion Forging

    (i) Inverted Extrusion

    (ii) Extrusion Forging

    (iii) Grid Distortion

    (iv) Energy-Dissipation Rate



    Orthogonal Machining


    Yielding of Notched Bars in Tension

    Crack Initiation and Fracture


    (i) Cantilevers Under Concentrated Loading

    (ii) Pure Bending of Notched Bars


    Blanking and Shearing



    Bibliographies and Reviews



    Pressure Vessels









    Notched Bars Pulled in Tension

    Crack Initiation and Fracture



    Blanking, Shearing and Surface Asperities

    Chapter 6. Matrix-Operator Methods for Solving Plane-Strain Slip-Line Field Problems


    Governing Equations in Slip-Line-Field Theory

    (i) Radii of Curvature

    (ii) Moving Coordinates

    (iii) Velocity Equations

    Series Expansion for Radii of Curvature

    (i) Definitions: Radius of Curvature, Base Point, Intrinsic Direction

    (ii) Hencky's Second Theorem

    (iii) Series Representation of the Radius of Curvature

    (iv) Series Representation of a Circular arc

    (v) Generalized Radius of Curvature at a Point

    Centered Fans and Regular Nets

    (i) Centered Fans

    (ii) Regular Nets

    (iii) Construction on the Concave Side of the Base Line

    Matrix Operators

    (i) Reversion Operator, R

    (ii) Operators P and Q: the Centered Fan: the Regular Net

    (iii) Shift Operator, S

    (iv) Numerical Application of the Operators Rφ and Sφ

    (v) Frictionless Boundary Operator, T

    (vi) Straight Rough Doundary Operator, G

    (vii) Stress-Free Surface Boundary Operator, F

    (viii) Matrix Operator Identities

    (ix) Matrix Operator Subroutines

    Determination of the Coordinates of Slip Lines

    Determination of Forces Acting on a Slip Line

    Method of solution

    (i) Direct-Type Solutions

    (ii) Indirect-Type Solutions

    Application of the Matrix Operator Method to the Solution of Direct-Type Problems

    (i) Extrusion/Drawing through a Frictionless Wedge-Shaped Die

    (ii) Extrusion at High Reduction

    (iii) Symmetrical and Asymmetrical Piercing Examples

    (iv) Slip-Line Fields for Drawing and Wall-Ironing

    Application of the Matrix Operator Method to the Solution of Indirect-Type Problems

    (i) Range of Slip-Line-Field Solutions for Extrusion/Drawing through Frictionless Wedge-Shaped Dies

    (ii) Drawing through Rough Wedge-Shaped Dies

    (iii) Slip-Line-Field Solutions for the Hot Rolling of Strip


    Summary of Matrix Operators


    Appendix 1

    Matrix Operators

    Appendix 2

    Mikhlin and Cartesian Coordinates of a Slip Line

    Determination of Expressions for the Mikhlin Coordinates

    Appendix 3

    Computer Listing for Indentation Problem

    Appendix 4

    The Determination of the Forces Acting on a Slip Line: Subroutine SLFORC

    Chapter 7. Plasticity Problems for Other than Plane-Strain Conditions


    The Method of Characteristics for Other than Plane Strain Processes

    (i) Plane Stress

    (ii) Axial Symmetry

    (iii) Soils


    (i) Crystallographic Approach

    (ii) Macroscopic Theories of Anisotropy

    Slip-Line Fields for Anisotropie Materials

    Analogies with Metal-Working Operations

    (i) Minimum Weight Frames

    (ii) The Transverse Plastic Bending of Rigid-Perfectly Plastic Plates

    (iii) The Force-Plane Diagram for Plane-Strain Slip-Line Fields



    5. Suffix Notation and the Summation Convention

    6. Characteristics of Partial Differential Equations

    7. Centered-Fan Fields : Nodal Points

    Additional References

    Author Index

    Subject Index

Product details

  • No. of pages: 376
  • Language: English
  • Copyright: © Pergamon 1982
  • Published: January 1, 1982
  • Imprint: Pergamon
  • eBook ISBN: 9781483278490

About the Authors

W. Johnson

R. Sowerby

R. D. Venter

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