Population of chain particles and average chain distance inside the conical volume

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Experimental investigations

Experimental investigations of boulder-medium interaction are usually conducted in small scales, half scales, or in few cases in real scales, associated with small or big boulder incident kinetic energies. Impact experiments deal with various impact problems and impact conditions. For instance, boulder impacts on natural or man-made slopes, rockfall shelters, embankments of simple and complex configurations (e.g. Labiouse et al., 1996; Stoffel, 1998; Calvetti et al., 2005; Pichler et al., 2005; Peila et al., 2007; Peila, 2011). Mea-surements are set in experiments to record the acceleration of boulder, the penetration depth of the boulder, the impact time, the stresses within the granular medium. The obtained experimental results were used for improving rockfall protection structure design.
Most part of the work concerns the response of a granular layer covering a rigid structure, for the case of a boulder impacting a rock shed. Labiouse et al. (1996) conducted half-scale impact experiments of a boulder impacting a granular layer covering a concrete slab. The size of the slab is 3.2 m × 3.2 m × 0.2 m (Figure 2.1). Four essential parameters influencing the design of rock shed structures were investigated: weight of blocks (100, 500, 1000 kg), falling height (0.25 to 10 m), cushion thickness (0.35, 0.5, 1.0 m) and material of the soil cushion (three different soils). Empirical expressions are proposed based on experimental results:
In addition, Stoffel (1998) conducted similar experiments as Labiouse et al. (1996) and proposed expressions to calculate impact force, transmitted force and penetration depth during a boulder impacting a granular medium:
where Pint is the maximum transmitted force on the bottom, Epot is the potential energy of the boulder.
In case of a boulder impacting a medium resting on a rigid wall, the expression is:
Calvetti et al. (2005) investigated experimentally the phenomenon of boulder impact-ing granular soils typically used to reduce loads on shelters. Series of full scale tests were performed to investigate the influence of geometrical and mechanical properties of the medium, as well as the effects of previous impacts (so as Stoffel (1998)). Experimental results of Calvetti et al. (2005) show that relative density of the superficial layers of the stratum is the key factor that determines the peak value of impact force, while the inclination of the stratum plays a less significant role.
Both the experimental results of Stoffel (1998) and Calvetti et al. (2005) proved that the impact force transmitted to the rigid bottom from the impacted granular medium is much larger than the impact force on the boulder, with a ratio of 1 to 3. This ratio is inversely proportional to the layer thickness. The existence of a larger value of transmitted force on the bottom than the impact force is called dynamic effects. This phenomenon is also observed in numerical simulations (Calvetti et al., 2005). In addition, the propagation of compression waves within the medium is governed by the mechanical properties of the granular material (Calvetti et al., 2005). For a given impact, the impact force and the speed of compression waves in the granular soil are mainly affected by the elastic properties of the soil.
Pichler et al. (2005) designed real-scale experiments based on analytical relations, the corresponding results were used to conduct back-analysis to obtain the indentation resistance of gravel. This permits to estimate of penetration depths caused by rockfall events which are beyond the experimental acceleration measurements. In addition, small scale laboratory experiments of Degago et al. (2008) showed that the hemispherical impactor induced higher impact force as well as less penetration as compared to the pyramidal impact boulder.
Gerber and Volkwein (2010) conducted 54 tests with two different concrete boulders (800 kg and 4000 kg) dropped on two ground layers (0.5 m and 1.3 m). The deceleration curves show either two significant local maximum accelerations or one maximum and a plateau-like behaviour. The observations illustrated the importance of compaction in the ground layer, which is consistent with the conclusion drawn by Lorentz et al. (2006) that layer compactness has a major effect in increasing the peak value of the transmitted impact force. The results of the 4000 kg block tests show a correlation between the penetration Z and the maximum deceleration amax, which can be shown simply as:

Numerical modelling

An alternative approach for studying the impact mechanisms is through numerical modelling. Plassiard and Donzé (2009) modelled the impact process of a boulder on a granular medium with a discrete element method. Both quasistatic and dynamic behaviours of soil have been examined and additional contact laws were used to be rep-resentative. The impact corresponded to a 500 kg boulder hitting a 50 cm thick layer of a compacted soil, with a falling height of 10 meters. The impact force was well reproduced, while the transmitted force through the granular medium was less well modelled even with the introduction of additional dissipation laws (Figure 2.2).
Figure 2.2: Time evolution of (a) the deceleration of the boulder, (b) the force transmitted to the bottom (Plassiard and Donzé, 2009) Calvetti et al. (2005) performed numerical simulations with the aim of investigat-ing loading conditions characterised by different values of stratum thickness under a wide range of impact energy. A simplified DEM model of the experimental set-up is established, the stratum is represented by a 2 m thick assembly of about 10000 spheres with uniform diameter distribution between 0.1 and 0.3 m. The results of the numerical simulations show that the peak value of impact force may be conveniently correlated to impact energy (Figure 2.3(a)), although a more refined approach should consider the individual effects of boulder mass and falling height. In addition, for a given impact force, the stresses on the plate decrease with the thickness of the stratum. A linear correlation between impact force and stresses on the plate was proposed to describe the dynamic amplification effect that characterised the wave propagation within the stratum (Figure 2.3(b)).

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Bouncing of a boulder on a granular medium

Among the motions of boulder (free falling, sliding, rolling, and bouncing) along the trajectory down the slope, bouncing, occurring when the falling block collides with the slope surface, is the most difficult one to predict (Dorren, 2003; Labiouse and Heidenreich, 2009). Theoretically, the motion of boulder during the interaction can be calculated according to Newton’s second law. However, the interaction is a very complex process, the force on the boulder, the contact points or surfaces evolves with time depending on the substrate properties, rock properties and the kinematics of the boulder before the impact. The use of computer programs in rockfall simulations is one of the most popular approaches for accounting for the bouncing of a boulder on a granular medium (Prisco and Vecchiotti, 2006; Bourrier et al., 2009; Lambert et al., 2013).

Experimental investigations

Most computer codes model the bouncing simply by adopting one or two resti-tution coefficients which are obtained by analytical, back-analysis, experimental and numerical modelling results (Wong et al., 2000; Nouguier-Lehon et al., 2003; Heidenreich, 2004; Oger et al., 2005, 2008; Bourrier and Hungr, 2011; Vijayakumar et al., 2011). The trajectory simulations are very sensitive to these restitution coefficients, and the accu-racy of the simulations depends to a large extent on these parameters (Dorren et al., 2011).

Table of contents :

1 General introduction 
1.1 Rockfall hazards
1.2 Engineering expectations
1.2.1 Trajectory predictions
1.2.2 Protection structure design
1.3 The role of impact in rockfall predictions and mitigations
1.4 Objectives and contents of the thesis
2 State of the art 
2.1 Interaction between a boulder and a granular medium
2.1.1 Engineering practices
2.1.2 Experimental investigations
2.1.3 Numerical modelling
2.2 Bouncing of a boulder on a granular medium
2.2.1 Experimental investigations
2.2.2 Numerical modelling
2.3 Micromechanical properties of granular materials
2.3.1 Bimodal character of load transmission in granular materials
2.3.2 The role of force chains in granular materials
2.4 Numerical modelling based on a discrete element method
2.4.1 Introduction of discrete element method
2.4.2 Modelling of particle shape
2.4.3 Energy dissipation in granular materials
2.5 Conclusion
3 DEM modelling of the impact process 
3.1 Introduction
3.2 Contact model description
3.2.1 Description of Cundall-Strack contact law
3.2.2 Rolling resistance
3.2.3 Calculation of energy items
3.3 Contact law calibration and validation
3.3.1 Triaxial experimental test
3.3.2 Contact law calibration
3.3.3 Contact law validation
3.4 Impact simulation
3.4.1 Modelling of a falling boulder interacting with a granular medium
3.4.2 Validation of the impact modelling
3.5 Conclusion
4 Global bouncing occurrence of the boulder 
4.1 Introduction
4.2 Variability of the results with respect to impact locations
4.2.1 Introduction
4.2.2 Methodology
4.2.3 Results
4.3 Energy propagation inside the granular medium
4.3.1 Layer division
4.3.2 Evolution of kinetic energy Ek and elastic strain energy Es
4.3.3 Evolution of coordination numbers
4.3.4 Evolution of incremental energy dissipation
4.3.5 Discussion
4.4 Boulder’s global bouncing occurrence
4.4.1 Impact simulations
4.4.2 Definition of bouncing
4.4.3 Bouncing occurrence diagrams
4.4.4 Discussion
4.5 Conclusion
5 Micromechanical behaviour of the impacted medium
5.1 Introduction
5.2 Characterization of force chains inside an impacted medium
5.2.1 Major principal stress s1 of a particle
5.2.2 Algorithms for selecting force chain particles
5.2.3 Procedures for identifying force chain network
5.2.4 Average major principal stress s1
5.2.5 Number of chain particles and average chain length
5.3 Parametric analysis of the effects of grain sizes
5.3.1 Effect of grain sizes on impact force
5.3.2 Effect of grain sizes on chain length
5.3.3 Effect of grain sizes on chain age
5.4 Spatial and temporal evolution of force chains
5.4.1 Spatial structure of force chains
5.4.2 Chain particle distances
5.4.3 Population of chain particles and average chain distance inside the conical volume
5.4.4 Critical length lHDCD
5.5 Force chain buckling mechanisms
5.5.1 Buckling angle and buckling number
5.5.2 Triggering and energy dissipation of force chain buckling
5.5.3 Relations between buckling, impact force and energy items
5.6 Potential of force chain micromechanisms for rockfall engineering
5.7 Conclusion
6 Conclusions and perspectives 
6.1 Conclusions
6.2 Perspectives


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