1st Edition

Plasticity Fundamentals and Applications

By P.M. Dixit, U.S. Dixit Copyright 2015
    602 Pages 115 B/W Illustrations
    by CRC Press

    602 Pages 115 B/W Illustrations
    by CRC Press

    Explores the Principles of Plasticity

    Most undergraduate programs lack an undergraduate plasticity theory course, and many graduate programs in design and manufacturing lack a course on plasticity—leaving a number of engineering students without adequate information on the subject. Emphasizing stresses generated in the material and its effect, Plasticity: Fundamentals and Applications effectively addresses this need. This book fills a void by introducing the basic fundamentals of solid mechanics of deformable bodies. It provides a thorough understanding of plasticity theory, introduces the concepts of plasticity, and discusses relevant applications.

    Studies the Effects of Forces and Motions on Solids

    The authors make a point of highlighting the importance of plastic deformation, and also discuss the concepts of elasticity (for a clear understanding of plasticity, the elasticity theory must also be understood). In addition, they present information on updated Lagrangian and Eulerian formulations for the modeling of metal forming and machining.

    Topics covered include:

    • Stress
    • Strain
    • Constitutive relations
    • Fracture
    • Anisotropy
    • Contact problems

    Plasticity: Fundamentals and Applications enables students to understand the basic fundamentals of plasticity theory, effectively use commercial finite-element (FE) software, and eventually develop their own code. It also provides suitable reference material for mechanical/civil/aerospace engineers, material processing engineers, applied mechanics researchers, mathematicians, and other industry professionals.

    Solid Mechanics and Its Applications


    Continuum Hypothesis

    Elasto-Plastic Solids

    Applications of Solid Mechanics

    Scope of this Textbook

    Review of Algebra and Calculus of Vectors and Tensors


    Index Notations

    Kronecker Delta and Levy-Civita Symbols


    Transformation Rules for Vector Components under the Rotation of Cartesian Coordinate System


    Tensors and Vectors in Curvilinear Coordinates




    Stress at a Point

    Surface Forces and Body Forces

    Momentum Balance Laws

    Theorem of Virtual Work

    Cauchy’s Theorem

    Transformation of Stress Components

    Stresses on an Oblique Plane

    Principal Stresses

    Maximum Shear Stress

    Octahedral Stresses

    Hydrostatic and Deviatoric Stresses

    Mohr’s Circle


    Measures of Deformation and Rate of Deformation



    Linear Strain Tensor

    Infinitesimal Rotation Tensor

    Deformation Gradient

    Green Strain Tensor

    Almansi Strain Tensor

    Logarithmic Strain Tensor

    Strain–Displacement Relation in Curvilinear Coordinate

    Transformation of Strain Components

    Principal Strains

    Maximum Shear Strain

    Octahedral Strain

    Volumetric Strain

    Mean and Deviatoric Strain

    Mohr’s Circle for Strain

    Incremental Strain Tensor

    Material and Local Time Derivative

    Rate of Deformation Tensor

    Spin Tensor

    On Relation between Incremental Strain and Strain Rate Tensors

    Compatibility Conditions


    Incremental and Rate Type of Elastic–Plastic Constitutive Relations for Isotropic Materials, Objective Incremental Stress and Stress Rate Measures


    Elastic Stress–Strain Relations for Small Deformation

    Experimental Observations on Elastic–Plastic Behavior

    Criteria for Initial Yielding of Isotropic Materials

    Modeling of Isotropic Hardening or Criterion for Subsequent Isotropic Yielding

    Elastic–Plastic Stress–Strain and Stress–Strain Rate Relations for Isotropic Materials

    Objective Incremental Stress and Objective Stress Rate Tensors

    Unloading Criterion


    Eulerian and Updated Lagrangian Formulations


    Equation of Motion in Terms of Velocity Derivatives

    Incremental Equation of Motion

    Eulerian Formulation

    Example of Eulerian Formulation: A Wire Drawing Problem

    Updated Lagrangian Formulation

    Example on Updated Lagrangian Formulation: Forging of a Cylindrical Block


    Calculus of Variations and Extremum Principles



    Extremization of a Functional

    Solution of Extremization Problems Using δ Operator

    Obtaining Variational Form from a Differential Equation

    Principle of Virtual Work

    Principle of Minimum Potential Energy

    Solution of Variational Problems by Ritz Method


    Two-Dimensional and Axisymmetric Elasto-Plastic Problems


    Symmetric Beam Bending of a Perfectly Plastic Material (-D Problem)

    Hole Expansion in an Infinite Plate (Plane Stress and Axisymmetric Problem)

    Analysis of Plastic Deformation in the Flange of Circular Cup during Deep Drawing Process (Plane Stress and Axisymmetric Problem)

    Necking of a Cylindrical Rod


    Appendix A

    Appendix B

    Contact Mechanics


    Hertz Theory

    Elastic–Plastic Indentation

    Cavity Model

    Sliding of Elastic–Plastic Solids

    Rolling Contact

    Principle of Virtual Work and Discretization of Contact Problems


    Dynamic Elasto-Plastic Problems


    Longitudinal Stress Wave Propagation in a Rod (-D Problem)

    Taylor Rod Problem (Impact of Cylindrical Rod against Flat Rigid Surface, -D Problem)


    Continuum Damage Mechanics and Ductile Fracture



    Objective and Plan of the Chapter

    Classification of Fracture

    Global and Local Approaches to Fracture

    Limitations of Global and Local Approaches to Fracture

    Ductile Fracture

    Models of Fracture Initiation

    Thermodynamics of Continuum

    Continuum Damage Mechanics

    Techniques for Damage Measurement

    Application of a CDM Model


    Plastic Anisotropy


    Normal and Planar Anisotropy

    Hill’s Anisotropic Yield Criteria

    Plane Stress Anisotropic Yield Criterion of Barlat and Lian

    Three-Dimensional Anisotropic Yield Criteria of Barlat and Coworkers

    Plane Strain Anisotropic Yield Criterion

    Constitution Relations for Anisotropic Materials

    Kinematic Hardening




    Dr. P.M. Dixit obtained a bachelor’s degree in aeronautical engineering from the Indian Institute of Technology (IIT) Kharagpur in 1974 and a PhD in mechanics in 1979 from the University of Minnesota, Minneapolis, USA. He joined the Department of Mechanical Engineering at the IIT Kanpur in 1984, where he is currently a professor. For the past 25 years, he has been working in the area of computational plasticity with applications for metal-forming processes and ductile fracture in impact problems using finite element method as a computational tool. He has published approximately 50 journal papers, 25 conference papers, and two books.

    Dr. U.S. Dixit obtained a bachelor’s degree in mechanical engineering from the University of Roorkee (now Indian Institute of Technology Roorkee) in 1987, an MTech in mechanical engineering from Indian Institute of Technology (IIT) Kanpur in 1993, and a PhD in mechanical engineering from IIT Kanpur in 1998. A professor for the department of mechanical engineering, Indian Institute of Technology Guwahati, Dr. Dixit has published numerous papers and three books. He has also edited a book on metal forming, guest-edited a number of special journal issues, and is an associate editor for the Journal of Institution of Engineers Series C.

    "This book has been written in a way that a plasticity course can be offered to graduate students without previous solid mechanics background. The concept of Cartesian vectors and tensors in index notation is discussed in chapter 2 to prepare students for understanding the topics presented in subsequent chapters. ... This book emphasizes the application of plasticity in solving engineering problems. Eulerian and updated Lagrangian formulations, calculus of variations and extreme principles are discussed in chapters 6 and 7 to prepare students for numerical calculation."
    —Han-Chin Wu, University of Iowa, Iowa City, USA

    "The book is successful in presenting a modern treatment of plasticity theories without sacrificing details both at the conceptual and the applied level. The breadth of applications covered is unique and includes a wide range of
    disciplines ranging from contact mechanics to fracture. In this the book will find no parallels in the modern literature on plasticity."
    —Prof. Anurag Gupta, Indian Institute of Technology Kanpur

    "Comprehensive coverage from mathematic tools to constitutive formulations, from application examples to computational aspects."
    —Tongxi Yu, Hong Kong University of Science and Technology (HKUST)