### 2024　Theory of Elasticity and Plasticity

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Instructor(s)
Inoue Hirotsugu  Mizutani Yoshihiro
Class Format
Lecture
Media-enhanced courses
Day/Period(Room No.)
-
Group
-
Course number
MEC.C211
Credits
2
2024
Offered quarter
3Q
Syllabus updated
2024/3/14
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

The topics taught in this course are sterss / strain (the basic concept related to the strength and deformation of machines and structures), the basic theory of linear elasticity for isotropic and anisotropic bodies, and basic theory of elasto-plasticity. More specific topics are as follows.
1. Stress and strain
2. Two dimensional elasticity (Stress function)
3. Torsion of shafts and bending of plates
4. Theory of elasticity for anisotropic bodies
5. Basic theory of elasto-plasticity (Yielding criteria)
6. Elasto-plastic bending of beams and elasto-plastic torsion of shafts
7. Elasto-plastic deformation of thick-walled cylinder

### Student learning outcomes

By the end of this course, students will be able to:
1. Understand the definition of stress, stress components and their transformation of coordinates, principal stress. Also derive the equillibrium equation of stress components.
2. Understand the definition of strain, strain components and their transformation of coordinates. Also derive the compatibility equation of strain components.
3. Understand Hooke's law. Also understand the basic framework of elastic problem including physical quantities, basic equations, and boundary conditions.
4. Understand the basic analytical methods (obtain the stress distribution) for several two dimensional elastic problems.
5. Understand the methods for analyzing stress and deformation for torsion of shafts and bending of plates.
6. Understand the rule of mixture and the stress-strain relationship for anisotropic materials.
7. Understand the yielding criteria: Tresca and von Mises.
8. Understand the methods of elasto-plastic analysis for bending of beams and torsion of shafts. Also obtain the residual stresses.
9. Understand the elasto-plastic analysis of thick-walled cylinders.

### Keywords

Two-dimensional elasticity, Equilibrium equations for stress components, Compatibility condition for strain ccmponents, Hooke's law, Stress function, Composite materials, Elasto-plastic problems, Yielding criteria

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills ✔ Practical and/or problem-solving skills ✔ The ability to organise and analyse, advanced specialist knowledge in mechanical engineering

### 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, principal stress, equilibrium equations for stresses, definition of strain) Understand the definition, expression of stress components, transformation of stress, Derivation of principle stress, equilibrium equations for stresses, definition of strain.
Class 2 Stress and strain (transformation/compatibility condition of strain, Hooke's law, polar coordinate, Saint-Venant's principle, boundary condition) Understand the transformation of strain and Hooke's law. Derivation of compatibility. Understand the contents of pages 22–29 of the textbook.
Class 3 Two-dimensional problems in elasticity (stress function, thick-wall cylinder) Derivation of stress function. Derivation of stress distribution in pressure vessels.
Class 4 Two-dimensional problems in elasticity (stress concentration), Torsion of rods (1) Derivation of stress distribution around a hole. Understand the contents of pages 66–82 of the textbook.
Class 5 Torsion of rods (2), Bending of plates, Thermal stress Understand the contents of pages 73–98 of the textbook.
Class 6 Thermal stress, Anisotropic materials, Composite materials, Elastic-plastic problems (initial stress) Understand the contents of pages 102–106 of the textbook. Understand the law of mixture, stress-strain curve and stress transformation for anisotropic materials. Learn the application examples of composite materials. Understand the contents of pages 132–135 of the textbook.
Class 7 Elastic-plastic problems (yield criteria, residual stress of a beam and a rod, elastic-plastic deformation of a thick-walled cylinder) Understand the Tresca yield criterion and von Mises yield criterion. Understand the residual stress caused in a beam and a rod. Understand the elastic-plastic problem for a thick-walled cylinder.

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

### Textbook(s)

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

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．