Mécanique non linéaireMéthodes numériques et expérimentales (GSI)

This course is offered in 2 main parts:

The first part concerns the calculation of finite element structures taking into account nonlinear material behaviour, then in the presence of geometric nonlinearities and taking into account large transformations.

The second part deals with the concept of non-linear vibrations and consists of:

• To understand the influence of geometric nonlinearities on vibrations of simple mechanical systems in free and forced regimes.
• To understand the concepts of linear stability and self-sustaining oscillations.
• To initiate so-called disruptive methods for the resolution of nonlinear vibrating systems.
• Implement these methods using Python programs.

Programme of the course:

Part 1: Calculation of nonlinear structures by finite elements

I. Nonlinear behaviour in small disturbance

1. Reminder of finite elements
2. Reminder of plasticity, visco-elasticity and isotropic damage.
3. Reminder of Newton's Schema
4. Integration of a law of behavior - Jacobian.– "return mapping" algorithm
5. Balance: Global step and local step in non-linear
6. One-dimensional example in Matlab
7. Application in Standard Abaqus: Elastoplastic case with contacts.

II. Geometric linearities and major transformations

1. Introduction based on impact calculation
2. Basics for major transformations
3. Deformity measures
4. Coercive measures
5. Simple shear case; analytical approach and applications in Abaqus
6. Location limiter and mesh dependency; Application to 1D beam - Comparison with ABAQUS results.

Part 2: Nonlinear vibrations

1. Introduction and preliminary results. General information on non-linear mechanical systems. Introductory examples.
2. Methods of analysis. Stability of dynamic systems. Disturbing methods.
3. Harmonic forcing of Duffing's oscillator.
4. Self-maintained oscillations.

• "Computational Inelasticity", J.C. Simo, T.J.R. Hughes, Mechanics and materials, Springer,
• Documentation Abacus: Integration of a law of behaviour
• Abacus Documentation: Deformation, Strain and Strain rates; Stress measurements, stress rates.
• "Mechanics of deformable Solids", Linear and Nonlinear, Analytical and Computational Aspects, Issam Doghri, Springer, ISBN 3-540-66960-4.
• A.H. Nahfeh and D.T. Mook: Nonlinear Oscillations. John Wiley & Sons, Ltd, 2007.
• A.H. Nayfeh: Introduction to Technical Disruption. Wiley Classics Library. John Wiley & Sons, Ltd, 2011.
• D.H. Hodges and G.A. Pierce: Introduction to Structural Dynamics and Aeroelasticity. Cambridge, University Press, 2011.