Plasticity: Fundamentals and Applications, 1st Edition (Paperback) book cover


Fundamentals and Applications, 1st Edition

By P.M. Dixit, U.S. Dixit

CRC Press

602 pages | 115 B/W Illus.

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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.


"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)

Table of Contents

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



About the Authors

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.

Subject Categories

BISAC Subject Codes/Headings:
SCIENCE / Mechanics / General