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Theory of Plasticity for Engineers: Concepts and Applications

By Prof. Amirtham Rajagopal   |   IIT Hyderabad
Learners enrolled: 74
ABOUT THE COURSE:

This NPTEL course provides a comprehensive understanding of plasticity, damage mechanics, and high strain rate behavior, focusing on the fundamentals, applications, and computational aspects of plasticity and its interaction with damage in materials. The course is structured into four modules: Module 1 covers the basics of plasticity, including stress-strain relations, yield criteria, and plastic stress-strain relations. Module 2 explores the application of plasticity in metals, reinforced concrete, and composites, including failure criteria and fracture models. Module 3 delves into plasticity-based damage models, covering material damage mechanisms, continuum damage mechanics, and inelastic damage theories. Module 4 focuses on the plasticity for high strain rate behavior, covering the fundamentals of blast and impact models, rate effects during crack growth, and computational aspects of plasticity. This course integrates theoretical knowledge with practical applications, aimed at professionals and researchers in the fields of material science and structural engineering.

INTENDED AUDIENCE:
Post-graduate M.Tech ,MS students
Core course for Ph.D students
3rd or 4th Year B.tech Students as Elective Course or Honors Course. Of the following branches
Civil / Mechanical/ Aerospace/ Applied Mechanics/ Computational Engineering Course

PREREQUISITES:
Engineering Mechanics
Mechanics of Solids
Theory of Elasticity

INDUSTRY SUPPORT:
General Electric
General Motors
Larsen and Tubro
FEM software companies like ABAQUS/ANSYS/ COMSOL

Summary
Course Status : Upcoming
Course Type : Core
Language for course content : English
Duration : 12 weeks
Category :
  • Civil Engineering
  • Structural Analysis
Credit Points : 3
Level : Undergraduate/Postgraduate
Start Date : 19 Jan 2026
End Date : 10 Apr 2026
Enrollment Ends : 26 Jan 2026
Exam Registration Ends : 13 Feb 2026
Exam Date : 17 Apr 2026 IST
NCrF Level   : 4.5 — 8.0

Note: This exam date is subject to change based on seat availability. You can check final exam date on your hall ticket.


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Course layout

Module-1: Basics of Plasticity

Week 1:  
Introduction to Plasticity and Elastic Stress-Strain Relations
Lec-01 : Introduction to plasticity: Plastic behavior of materials, elastic and plastic strains, elastic, perfectly plastic, and hardening behavior. Definition of Elastic range, Yield stress, Yield surface, and admissible stress state. Introduction to Hardening and Softening Behavior. Introduction to Haigh Westergard coordinates.
Lec-02 : Stress-Strain Relations: Elastic constants, isotropy, homogeneity, and Mohr's Circle. Material Types: Introduction to anisotropic, monoclinic, and orthotropic materials.

Week 2:  Fundamentals of Yield and Failure Criteria
Lec-03 : Failure Criterion for Pressure-Independent Materials - Tresca, Von mises
Lec-04 : Failure Criterion for Pressure-Dependent Materials- Mohr Columb, Rankine, and Drucker-Prager, Failure criteria for anisotropic materials

Week 3: Foundations of Plasticity and Plastic Stress-Strain Relations
Lec-05 : Kuhn-Tucker loading and unloading Conditions, State of stress in a plastic process.
Lec-06 : Stress-Strain Relations for Perfectly Plastic Materials Flow rule/normality condition- Associated and Non-Associative flow rules.
Lec-07 : Stress-Strain Relations for Work-Hardening Materials Hardening Rule, Work Hardening, and Strain Hardening Rules. Hardening Parameter. Internal Hardening variables and Potential function. Consistency Conditions, Incremental elastic stress-strain relations, Simple 1D model for rate-independent plasticity for isotropic, kinematic, and Mixed hardening cases.

Module-2: Application of Plasticity in Metals, Reinforced Concrete & Composites

Week 4:  Metal Plasticity, Concrete Plasticity and Elasto-Plastic Metal Matrix Composites
Lec-08Implementation in Metals: Formulation of the elastic-plastic matrix, finite element formulation, numerical algorithms for solving nonlinear equations, and numerical implementation of elastic-plastic incremental constitutive relations. Bounding surface theory and its extension to anisotropic cases.
Lec-09Implementation in Concretes: Introduction, Failure Criteria, Plasticity Modeling: Hardening Behaviour and Softening Behavior
Lec-10 : Elasto-Plastic Metal Matrix Composites: Effective stresses, strains, and yield function, constitutive relations, stresses in the damage composite system, damage evolution, and both elastic and elasto-plastic constitutive relations in damaged composite systems.

Week 5:  Concrete Elasticity and Failure Criteria
Lec-11 : Linear-Elastic Brittle-Fracture Models: Linear-elastic isotropic stress-strain relations for uncracked concrete, transversely isotropic stress-strain relations for cracked concrete, linear-elastic fracture analyses of undersea pressure-resistant concrete structures, fracture analyses of beams, and inelastic analysis of reinforced concrete panels.
Lec-12Nonlinear-Elastic Fracture Models: Isotropic nonlinear-elastic stress-strain relations, general formulation of hyperelastic models, and the formulation of a third-order hyperelastic constitutive model. Incremental stress-strain relations, hypoelastic models, and an orthotropic hypoelastic constitutive model for concrete.
Lec-13Failure Criteria of Concrete: Stress and strain invariants, and characteristics of the failure surface of concrete. One-parameter, two-parameter, three-parameter, four-parameter, and a five-parameter model, along with a fracture model for concrete.

Week 6: Fracture Models for Concrete
Lec-14 : Elastic Perfectly Plastic Fracture Models: criteria of loading and unloading, elastic-strain-increment tensor, and plastic-strain-increment tensor, elastic perfectly plastic concrete models, Prandtl-Reuss material (J2 theory), Drucker-Prager material, Mohr-Coulomb material with a tension cutoff, and William-Warnke material.
Lec-15 Limit Analysis of Perfect Plasticity: theorems of limit analysis, a plasticity model for concrete, splitting tests on cylinders, shear resistance of joints, shear in beams with vertical stirrups, punching shear of reinforced concrete slabs, and the load-carrying capacity of concrete pavements.
Lec-16Elastic-Hardening Plastic-Fracture Models: loading function, concept of effective stress and strain, hardening rule, flow rule, and Drucker's stability postulate, incremental stress-strain relations, Drucker-Prager material with isotropic hardening and softening, von Mises material with mixed hardening, and three-parameter models for concrete displaying isotropic and independent hardening in tension and compression.

Module-3: Plasticity based damage model for Concrete & Inelastic response of Composite Materials


Week 7: Mechanisms and Representation of Material Damage Models
Lec-17Material Damage and Continuum Damage Mechanics microscopic mechanisms of damage, including scales of damage phenomena, physical mechanisms, and damage in fracture problems, concept of the representative volume element and continuum damage mechanics, including the notion of continuum damage mechanics. 
Lec-18 Mechanical Representation of Damage and Damage Variables modeling techniques of damage, including effective area reduction, variations in elastic modulus, and void volume fraction, mechanical representation of damage states using scalar, vector, and higher-order damage variables, along with effective stress tensors, hypotheses of mechanical equivalence, and elastic constitutive equations for damaged materials.
Lec-19Thermodynamics of Damaged Material fundamentals of thermodynamics, including the state variables, the first and second laws of thermodynamics, and Gibbs relations, thermodynamic constitutive theory of inelasticity with internal variables, dissipation potentials, and the evolution equations of internal variables, generalized standard materials and the quasi-standard thermodynamic approach.

Week 8: Inelastic Damage Mechanics: Constitutive Models, Energy Criteria, and Anisotropic Damage Behavior
Lec-20 Inelastic Constitutive Equations for Materials with Isotropic Damage One-dimensional and three-dimensional inelastic constitutive equations, including elastic-plastic, viscoplastic deformation, internal variables, thermodynamic potentials, and damage evolution.
Lec-21 : Strain Energy Release and Stress Criteria in Damage Development
Strain energy release, energy dissipation in elastic-plastic damage, stress triaxiality, stress sign effects, and stress criterion for ductile damage.
Lec-22 : Inelastic Damage Theory and Anisotropic Damage Models
Inelastic damage theory based on total energy equivalence, thermodynamic potentials, dissipation potentials, and anisotropic damage theory using second-order symmetric and brittle damage models.

Week 9: Elastic-Plastic Damage and Ductile Fracture Mechanics
Lec-23Constitutive and Evolution Equations of Elastic-Plastic Damage: Constitutive and evolution equations of elastic-plastic isotropic damage, ductile damage, brittle damage, and quasi-brittle damage, Elaboration of the two-scale damage model, threshold values for damage initiation, and critical fracture values.
Lec-24Ductile Damage, Fracture and Application to Metal Forming: Ductile damage analysis approaches, Lemaitre’s ductile damage model, extension of the model, and finite element analysis of ductile fracture, Application to metal forming processes, including fracture limits of sheet metal forming, forging, blanking processes, and fatigue life assessment of cold working tools.

Week 10:  Coupled Damage and Plasticity Models for metals and Composites
Lec-25 : Damage and Plasticity in Metals stress and strain rate transformations between damaged and undamaged states, effective stress tensors, backstress tensors, elastic strain, and plastic strain rates, damage effect tensor, constitutive models, damage evolution, plastic deformation, coupling of damage and plastic deformation, application to void growth (Gurson’s model), effective spin tensor, example of ductile fracture.
Lec-26A Coupled Anisotropic Damage Model for the Inelastic Response of Composite Materials theoretical formulation, constitutive equations for composite materials, computational aspects, covering program flow, plastic corrector algorithm, damage corrector algorithm, implementation of the viscoplastic damage model, including flow of the program, viscoplastic corrector algorithm, and results for viscoplastic damage analysis.

Module-4: Plasticity for high-strain-rate behaviour of concrete

Week 11: Fundamentals of Blast and Impact and Crack Growth: Plasticity and Rate Effects
Lec-27 : Basics of Blast and Impact Models
Lec-28Plasticity and Rate Effects During Crack Growth covers viscoelastic crack growth, steady crack growth in elastic-plastic materials, high strain rate crack growth in a plastic solid, fracture mode transition due to rate effects, ductile void growth, and microcracking and fragmentation. It includes topics on plastic strain on the crack line, growth criteria, toughness-speed relationships, crack arrest, and time-dependent strength under pulse loading.

Week 12: Computational Aspects of Plasticity
Lec-29 : Fundamentals of Return Mapping Algorithms
Lec-30 : Introduction to hyper elastic, Viscoelastic, and Visco plastic Materials

Books and references

1. Chen, W.F., and Han, D.J., Plasticity for Structural Engineers,” Springer-Verlag, New York, 1988
2. Voyiadjis, G. Z., & Kattan, P. I. Advances in Damage Mechanics: Metals and Metal Matrix Composites with an Introduction to Fabric Tensors. Elsevier, 2nd edition, 2006.
3. Chen, W.-F. (2007). Plasticity in Reinforced Concrete. J. Ross Publishing.
4. Murakami, Sumio. Continuum Damage Mechanics: Theory and Applications. Springer, 1999.
5. Freund, L. B. Dynamic Fracture Mechanics. Cambridge University Press, 1998.

Instructor bio

Prof. Amirtham Rajagopal

IIT Hyderabad
Professor Amirtham Rajagopal has been a full professor in the Department of Civil Engineering at IIT Hyderabad since August 2010. He previously worked as a postdoctoral researcher at the University of Erlangen Neurenberg, Germany, from 2007 to 2010. He obtained his PhD from IIT Madras. He also obtained his M.Tech from SJCE Mysore, VT University, in 2000 and B.Tech from Bangalore University in 1997. He has over 18 years of Teaching and research experience.

Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: April 17, 2026 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

Please note that assignments encompass all types (including quizzes, programming tasks, and essay submissions) available in the specific week.

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Hyderabad. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team
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