Parcours Geomechanics, civil engineering and risks 2e année

• Kinematics : Physical space ‐ frames ‐ continuous media ‐ deformation function ‐ Lagrange and Euler variables
• Strains : Deformation gradient ‐ metric tensor ‐ Green deformation tensor ‐ small strains ‐ strain rate
• Material derivative  and conservation laws
• Mass conservation ‐ volume change ‐ balance of momentum
• Stresses : Fundamental principle of dynamics ‐ Cauchy stress tensor‐equation of motion and boundary conditions ‐ Piola‐Kirchhoff stress tensors ‐ virtual power formulation ‐ linearization of equation of motion
• Examples of formulations of problems of continuum mechanics
• Frame invariance

Constitutive equations: Large elasticity ‐ elastoplasticity incremental constitutive equations, generalized continuous media, Non local, second grade, Cosserat and micromorphic continuum mechanics ‐ an introduction

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• From physics to numerical models: continuum mechanics problems, variational formulations, Rayleigh Ritz methods, Finite element one dimensional example
• Introduction to solid mechanics problems : elastostatics virtual work theorem : finite element discretization, the example of simple finite elements (constant strain triangle), Comments about Stiffness matrices
• Variational formulation of an initial boundary value problem: change of configuration, introduction to different stress and deformation tensors, the so called small strain approximation
• Time discretization and incremental problem: Newton method, residual computations, auxiliary linear system computations, boundary condition issues
• Space discretization : finite element method, projection on to a finite dimensional space, isoparametric finite element numerical integration Gauss method
• Constitutive equations integrations : consistent tangent stiffness matrix: numerical approach, Hardening plasticity,  integration algorithms, consistent tangent stiffness matrix : analytical approach, Locking ant related topics
• Miscellaneous : coupling problems, the rate problem and uniqueness issues

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This course aims to provide the fundamental elements necessary to understand and perform an experiment in Geomechanics, as well as to provide an overview of some of the key “traditional” and emerging approaches and tools available.

The course is structured in 6 blocks covering:

1. Introduction to experimental mechanics
2. Classical tests in Geomechanics
3. Physical modeling
4. Full-field techniques (e.g., x-ray tomography)
5. Waves for and in geomaterials
6. Rapid dynamics

1.1 Course structure
Lectures: 30h
Practical sessions: 12h

Block one: Introduction to experimental mechanics
The theoretical classes will cover:
• Working principles of the components of experiments (e.g., motors, encoders, transducers, cameras ...)
• Elements of calibration and metrology
Practical sessions: Construction of a simple testing apparatus from scratch.

Block two: Classical tests in Geomechanics
• Traditional tests in geomechanics (e.g., Triaxial, oedometer ...)
• Stress/strain paths and practical aspects
Practical sessions: the practical session will be either one four-hour session or two two-hour sessions.

Block three: Physical modeling
To be defined

Block four: Full-field techniques
The theoretical classes will cover:
• Working principle of cameras and elements of optics
• Fundamentals of generation, transport and detection of ionising radiation (x-ray, neutrons..)
• Tomography, reconstruction and its applications in geomechanics
• Diffraction and Small Angle Scattering
Practical sessions: Study of a practical 2D full-field test with the 1γ2ε apparatus

Block five: Waves for and in geomaterials
The theoretical classes will cover: To be defined
Practical sessions
Measurement of stress waves in a slender bar (Hopkinson bar). Different loading scenarios. Retrieving elastic properties of the bar thanks to the strain measurement, build the Lagrange diagram of the different tests.

Block six: Rapid dynamics
The theoretical classes will cover:
• The specificity of the instrumentation in the transient dynamic loading experiments.
• The use of ultra high speed imaging sensors
• The adaptation of the full field measurement techniques to this environment

Practical sessions
Continuity of the practical realised in block five. A test will be done on a geomaterial using Hopkinson bar, ultra high speed sensor, advanced instrumentation. The test will have to be processed based on the experience aquired in the practical of block five.

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This basic seismology course is a physics based course, which covers the breadth of traditional seismology, developed from the seismic wave equation, and the physics of the earthquake source. The course aims to provide an understanding of the dynamics of the solid Earth from theoretical and observational seismology and seismotectonics in relation to earthquake hazard and mitigation.

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This course is dedicated to constitutive models for structural failure. If plasticity is suitable to represent permanent deformation due to sliding of the microstructure, damage mechanics and fracture mechanics are oriented towards the modelling of decohesion and failure of cohesive materials. This course is two fold.

1. Damage mechanics is based on continuum mechanics taking into account a progressive degradation of the material stiffness without permanent strain. Starting from thermodynamic principles, the damage theory is derived and applied to some examples.
2. Fracture mechanics is based on the presence of a crack and propagation criteria are proposed to estimate the direction, the length and the stability of the crack propagation. Around the crack, the material behaviour can be considered as linear elastic to yield the Linear Elastic Fracture Mechanics approach based on stress intensity factor or strain energy release rate. In most of engineering materials, the fracture process zone (FPZ) around the crack tip cannot be neglected and fictitious crack models are derived.

The course is illustrated with some numerical examples showing the relative pros and cons of both approaches depending on the type of material, the size of the structure and the quantities of interest.

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The first part of the course is devoted to the study of geotechnical structures such as retaining structures, shallow and deep foundations. The aim of this course is to show how the behaviour of geotechnical structures can be explained from the fundamental mechanisms governing the elemental behaviour of soils and soil‐structure interfaces. A special attention is given to the effects of non linearities, stress level and dilatancy. The fundaments of the behaviour of these structures are first presented through the results of experiments performed in Laboratory, with specific similitude conditions (Physical modelling), and large scale field tests or case studies. Classical analytical design methods (bearing capacity, settlements) or using the results of field testing are presented and justified by the mechanisms of soil‐structure interaction. The numerical approach using the Finite Element Method (FEM) is presented, with emphasis on the role of the constitutive model used in the code on the results.A personal project involving the analysis of a case study has to be presented by the students.

The second part is concerned by the behaviour and design of geotechnical structures under earthquake loading: Elements of Engineering seismology, dynamic behaviour of soils, seismic liquefaction of soils and prediction methods, simplified approach of the seismic design of retaining walls, shallow and deep foundations, seismic calculation of embankment dams elements of dynamic soil‐structure interaction, this second part will be evaluated by a written exam.

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Different aspects combine to bring about safety of civil engineering works: rules used for their design, actions they are submitted during time (classical or accidental actions, effects of time) and/or generated by environment (climate actions, natural hazards,..). The objective of this course is to treat the generality of these problems and to study the specific case of concrete structure particularly in the context of seismic risks.

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The modern numerical methods now currently used in geomechanics allow to take into account advanced constitutive relations for geomaterials. The objective of these lectures is to present the most used constitutive models at an international level : isotropic and anisotropic non‐linear elasticity, associated and non‐associated elasto‐plasticity, visco‐elasto‐plasticity. First the most important mechanical properties to be described for soils are presented (compressibility, dilatancy, drained/undrained shear failure, ...). Then the general framework to build constitutive relations is given : definition of the constitutive functional, the incremental formalism for constitutive relations. The next chapter is devoted to non‐linear elasticity with examples like "hyperbolic" model. Then it is dealt with the elasto‐plasticity and the visco‐elasto‐plasticity with examples like "Cam‐Clay" model. Finally we introduce the incrementally non‐linear relations with applications like liquefaction phenomenon, instabilities and bifurcations in natural media, landslides and rockfalls modelling.

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Physical properties of rock (origin, types of rock, porosity, anisotropy, etc.)

Mechanical behaviour (experimental observations)

Constitutive modeling (general principles, anisotropic elasticity, elastoplasticity)

Failure, rupture, damage, localized deformation

Hydromechanics of rock, poroelasticity

Wave propagation in rock

Discontinuities, joints, interfaces

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This course aims to let students in the research masters discover the  main aspects of Strain localization in geomaterials on both experimental and modelling side. The course is based on past and current research work performed in 3SR laboratory. It brings a multi-decennal view of how progresses in experimental and imaging techniques, as well as in theoretical concepts and modelling approaches, serves the understanding of this still challenging phenomenon.

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This course concerns the modeling of physical phenomena occurring in porous media using an upscaling approach. Theoretical developments are illustrated by numerical examples and experimental results :
(i) General issues about porous media.
(ii) Modeling of porous media by using upscaling approaches,
(iii) Fluid flow in porous media, (iv) Diffusion/ dispersion in saturated porous media,
(v) hydro-mechanical behaviour of porous media.

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The scope of this course is to introduce students into the importance of dynamic loading and to familiarize them with the different techniques used for the dynamic modeling and analysis of structures.

The beginning of the course concentrates on the study of wave propagation phenomena in elastic solids (essentially one dimensional progressive plane waves). Emphasis is placed on the split Hopkinson pressure bar experiment.

The second part of the course focuses on direct numerical integration methods applicable to linear and nonlinear systems.

The last part of the course concentrates on the study of multiple‐degree‐of‐freedom linear systems and the calculation of their response by modal superposition.

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natural petroleum systems and of petroleum production operations / tectonics and structural geology / geomechanical drivers and mechanical property consequences / natural fluid flow systems in the subsurface / natural thermal systems in the subsurface / processes at the geological and the reservoir production timescales

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Fundamentals: Soil behaviour under earthquake loading: Fundamentals of soil behaviour under cyclic and dynamic loading. Basic mechanical models. Non-linear stiffness, damping, cyclic strength, liquefaction.

Geotechnical characterisatio: Techniques and examples of characterization of layered soils by in situ tests: borehole and surface methods. Techniques and examples of results of laboratory tests: cyclic triaxial, simple shear, torsional shear, resonant column. Factors influencing stiffness and damping. Linear equivalent models for dynamic analyses.

Liquefaction: Case histories of liquefaction worldwide; Evaluation of liquefaction potential by empirical to analytical methods. Zonation for liquefaction.

Slope stability: Case histories of slope failures worldwide; Analysis: pseudo-static methods, displacement analysis, advanced analyses. Examples of seismic analysis of natural slopes, earth dams and embankments. Zonation of slope stability.

Tutorials (with laptop): 1D site response analysis and assessment of liquefaction potential. Slope displacements by simplified dynamic analysis

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In this class we teach the different operations that can be done on 3D tomographic images in order to handle noise and measure geometrical and topological quantities of the material's microstructure.

We also teach how we can measure kinematic fields from a set of images using digital volume correlation.
Finally examples of Finite Element simulations using information from the images are proposed.

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The modelling of mechanical properties of materials and structures is a complex subject. In some applications, it is sufficient to assume that the material remains elastic. However, such a simplified assumption is appropriate only within a limited range, and in general must be replaced by a more realistic approach that considers the inelastic processes such as plastic yielding. The course aims to provide the students with the basic concepts of plastic analysis of structures: limit analysis, multiaxial stress-strain relations, limit state theorems, concept of plastic hinges and yield line of plates, calculation of collapse load and displacement at incipient collapse, application to engineering materials.

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The course provides the students with the basic concepts of geomechanics, that is the theoretical and applied science dealing with the mechanical behavior of geomaterials (soil and rock).

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Each student is required to work on a research project, to be carried out through a 5-month internship.  This may be undertaken in France or abroad, also depending on where the student originates from (internships cannot be completed in the country of nationality of the student). Topics range from fundamental geomechanics to applied civil engineering and geotechnics. Projects are designed around the student and can therefore be theoretical, numerical or experimental in nature, depending on students' skills and inclinations.  A written dissertation will be required at the end of this period, to be then defended in front of a jury.

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