Professors:
Mario Spagnuolo, Researcher, DICAAR, Università degli Studi di Cagliari
Hours: 30 hours
Dates: [From Wednesday 4 March 2026 (18:00) and Thursday 5 March 2026 (15:00)]
Contents
Tensor algebra and vector analysis: definition and properties of tensors; fundamental tensor operations; coordinate systems and transformations; vector and tensor differential analysis applied to problems in mechanics.
Continuum kinematics: deformation; infinitesimal strain tensor and approaches to finite deformations; linear and nonlinear strain measures; motion and Eulerian and Lagrangian descriptions.
Stress analysis: stress tensors; equilibrium equations; general formulations and specific applications.
Constitutive relations: linear isotropic elastic materials; elastic moduli; generalized Hooke’s law; anisotropic and orthotropic materials; elasticity criteria and practical applications.
Principle of virtual work and energy methods: virtual work; elastic potential energy; energy-based methods for solving elasticity problems.
Duration: 30 hours distributed within the second semester
Extended description
| Lecture | Date | Topic | Detailed topic | Pages |
| 1 | 04.03.2026 | Introduction to the course | Simple examples of nonlinear structural behavior: Cantilever; Column. | 1–18 |
| 2 | 05.03.2026 | Introduction | 1D nonlinear strain measures. Directional derivative, linearization and equation solution. | 1–18 |
| 3 | 11.03.2026 | Mathematical Preliminaries (1) | Vector and Tensor algebra. | 22–46 |
| 4 | 12.03.2026 | Mathematical Preliminaries (2) | Vector and Tensor algebra. | |
| 5 | 18.03.2026 | Numerical Application | ||
| 6 | 19.03.2026 | Mathematical Preliminaries (3) | Linearization and the directional derivative. Tensor analysis. | 47–62 |
| 7 | 25.03.2026 | Kinematics (1) | The motion. Material and spatial descriptions. | 94–104 |
| 8 | 26.03.2026 | Kinematics (2) | Deformation gradient. Strain. | 94–104 |
| 9 | 15.04.2026 | Numerical Application | ||
| 10 | 16.03.2026 | Kinematics (3) | Polar decomposition. Volume change. Distortional component of the deformation gradient. Area change. Linearized kinematics. | 105–118 |
| 11 | 22.04.2026 | Kinematics (3) | Velocity and material time derivatives. | 118–133 |
| 12 | 23.04.2026 | Kinematics (4) | Rate of deformation. Spin tensor. Rate of change of volume. Superimposed rigid body motions and objectivity. | 118–133 |
| 13 | 29.04.2026 | Numerical Application | ||
| 14 | 30.04.2026 | Stress and Equilibrium (1) | Cauchy stress tensor. Equilibrium. | 134–143 |
| 15 | 06.05.2026 | Stress and Equilibrium (2) | Principle of Virtual Work. | 134–143 |
| 16 | 07.05.2026 | Stress and Equilibrium (3) | Alternate stress representations. | 144–154 |
| 17 | 13.05.2026 | Numerical Application | ||
| 18 | 14.05.2026 | Hyperelasticity (1) | Hyperelasticity. Elasticity Tensor. Isotropic Hyperelasticity. Incompressible and nearly incompressible materials. | 155–173 |
| 19 | 20.05.2026 | Hyperelasticity (2) | Isotropic elasticity in principal directions. | 174–184 |
| 20 | 21.05.2026 | Linearized Equilibrium Equations | Linearization and Newton-Raphson process. Lagrangian linearized internal Virtual Work. Eulerian linearized internal Virtual Work. Linearized external Virtual Work. Variational methods and incompressibility. | 216–231 |
Contact and registration
Mario Spagnuolo, mario.spagnuolo@unica.it
The certificate is issued for participants that attend more than 80% of the lectures and successfully pass a written exam.
Bibliography
- Bonet, R. D. Wood “Nonlinear continuum mechanics for finite element analysis”. Cambridge University Press, 2008.
- E. Malvern “Introduction to the Mechanics of a Continuous Medium”. Prentice-Hall, 1969.
- C. Fung “A First Course in Continuum Mechanics: for Physical and Biological Engineers and Scientists”. Prentice-Hall, 1994.
- Constantinescu, A., Korsunsky, A. “Elasticity with Mathematica®: An Introduction to Continuum Mechanics and Linear Elasticity”. Cambridge University Press, 2012.