2016 Theory of Elasticity and Plasticity

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Academic unit or major
Undergraduate major in Mechanical Engineering
Inoue Hirotsugu  Mizutani Yoshihiro 
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Course description and aims

In this course, the instructor will introduce the relationship between stress and strain in an elastic body, the two-dimensional theory of elasticity, applications to problems of rod-torsion and plate-bending, handling of anisotropic materials, approaches to elastic-plastic problems, the bending and torsion of elastic-plastic materials, and applications to a thick-walled cylinder.
Students will learn a method for analytically dealing with the elastic deformation and elastic-plastic deformation of homogeneous isotropic materials (typified by metal materials) and anisotropic materials (typified by fiber reinforced plastics).

Student learning outcomes

By the end of this course, students will be able to:
1) Gain knowledge of basic concepts and analytically approach the strength and deformation of machines and structures.
2) Deal with elastic deformation problems for homogeneous isotropic materials and anisotropic materials which are the basis of mechanical design.
3) Also mechanically handle the transition to plastic deformation from elastic deformation.


Two-dimensional problems in elasticity, Equilibrium Equations for Stresses, Compatibility Condition of Strain, Hooke's Law, Stress Function, Elastic-plastic problems, Composite Materials

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Towards the end of class, students are given exercise problems related to the lecture given that day to solve. To prepare for class, students should read the course schedule section and check what topics will be covered. Required learning should be completed outside of the classroom for preparation and review purposes.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Stress and strain (definition/component/transformation of stress) Understand the definition, expression of stress components, transformation of stress.
Class 2 Stress and strain (principle stress, equilibrium equations for stresses, definition of strain) Derivation of principle stress, equilibrium equations for stresses, definition of strain.
Class 3 Stress and strain (transformation/compatibility condition of strain, Hooke's law) Understand the transformation of strain and Hooke's law. Derivation of compatibility condition of strain.
Class 4 Stress and strain (polar display, Saint-Venant's principle, boundary condition) Understand the contents of pages 22–29 of the textbook.
Class 5 Two-dimensional problems in elasticity (stress function) Derivation of stress function.
Class 6 Two-dimensional problems in elasticity (thick-walled cylinder problem) Derivation of stress distribution in pressure vessels.
Class 7 Two-dimensional problems in elasticity (stress concentration) Derivation of stress distribution around a hole.
Class 8 Torsion of rods Understand the contents of pages 66–82 of the textbook.
Class 9 Bending of plates Understand the contents of pages 84–98 of the textbook.
Class 10 Thermal stress Understand the contents of pages 102–106 of the textbook.
Class 11 Anisotropic materials Understand the law of mixture, stress-strain curve and stress transformation for anisotropic materials, and lamination theory.
Class 12 Composite materials Learn the application examples of composite materials.
Class 13 Elastic-plastic problems (yield criteria) Understand the Tresca yield criterion and von Mises yield criterion.
Class 14 Elastic-plastic problem (residual stress of a beam and a rod) Understand the residual stress caused in a beam and a rod.
Class 15 Elastic-plastic problem (elastic-plastic deformation of a thick-walled cylinder) Understand the elastic-plastic problem for a thick-walled cylinder.


Kobayashi, Hideo and Todoroki, Akira. Elastic-plastic Solid Mechanics. Tokyo: Suurikougakusha; ISBN978-4-901683-51-7. (Japanese)

Reference books, course materials, etc.

None required

Assessment criteria and methods

Students' knowledge of Stress and Strain, Two-dimensional problems in elasticity, and their ability to apply them to problems will be assessed.
Final exams 80%, exercise problems 20%.

Related courses

  • MEC.A201 : Engineering Mechanics
  • MEC.C201 : Mechanics of Materials
  • MEC.H212 : Fundamentals of Machine Design and Drawing
  • MEC.K332 : Finite Element Analysis
  • MEC.G211 : Mechanical Materials
  • MEC.C331 : Strength and Fracture of Materials (Mechanical Engineering)

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students must have successfully completed Mechanics of Materials (MEC.C201.R) or have equivalent knowledge.

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