11
Concluding Remarks

11.1. Concluding remarks on features resulting from energy dissipation by friction

Basic friction laws have been in the past sometimes described as incomplete, or suffering imperfections, and somewhat unsound elementary laws, although of great practical utility. In the author’s point of view, the revealing of the multi-scale tensor structures induced by contact friction in granular media behavior developed in this book, with all the properties resulting directly from these dissipative structures, is a clear proof of the global soundness of these basic friction laws – and by the way, perhaps not so “basic.”

In this regard, note that the wide set of results accumulated up to Chapter 8 with their analytical and experimental proofs is the direct consequence of the approach selected from the beginning: to cope head-on with irreversibility alone, without any support from a more regular and reversible background (such as elasticity in standard elastoplasticity). The resulting simplicity allowed us to achieve a great deal with analytic description.

Finally, this multi-scale dissipative approach also provides clear links between macroscopic pseudo-continuum behavior features and micro-scale polarization features of contacts motion distributions throughout Chapters 3–8, holding a close connection between macroscopic properties and underlying micromechanical structures, shedding a new light on these macroscopic behavior features.

11.1.1. Tensor structures induced by contact friction on internal actions

This book precisely displays how the energy dissipation by contact friction at micro-scale deeply structures at macro-scale the quasi-static mechanical behavior of granular media, with its so-typical specificities:

• – the process and specific features of strain localization and shear banding from the orientation and internal structures of shear bands, up to the whole process of failure lines generation (Chapter 3), failure lines which are still in the core of stability analysis methods [SAL 83];
• – the 3D Coulomb Failure Criterion at a critical state with its attached strain regimes and the failure criterion in the neighborhood of a critical state, but with some remaining deviations, appearing as a kind of “smoothed Coulomb Criterion” (Chapter 4);
• – the generalized 3D stress–dilatancy relationships, simplifying into Rowe’s stress–dilatancy equations in particular situations of axisymmetric stresses and plane strain, and their consequences on peak shear strength criterion (Chapter 5);
• – the existence of a “Characteristic State,” which occurs to coincide with the critical state failure criterion (Chapter 5);
• – the compaction under alternate shear motion with a ratcheting effect, typical of irreversible processes (Chapter 7);
• – the finely intricate relationships between friction, shear strength, and volume changes, typically the behavior of these granular materials, emerge naturally from our multi-scale approach of energy dissipation by contact friction, without requiring any “added ingredient.”

These finely intricate theoretical relations between friction, shear strength, and volume changes, associated with micro-scale polarization patterns, turn out well, confirmed by a very large set of experimental data, measured on all the available kinds of testing apparatuses (Chapter 6).

The purely frictional roots of this wide set of properties deserve to be outlined, as in the developments presented above:

• – energy dissipation is considered as acting alone from Chapter 1 to Chapter 8, as outlined in Introduction;
• – there are no other “ingredients” here than pure contact friction laws, multi-scale transposition through energy balances, and minimum dissipation rule;
• – thus, it is the transposition to macroscopic pseudo-continuum scale that makes appear, as “built inside” the dissipation relation, the set of all these specific macroscopic properties, evidencing this intrinsic relation between friction, shear strength and volume changes;
• – this is also why, when friction vanishes, all these properties also vanish and the macroscopic behavior reduces to “perfect incompressible fluid” mechanics (section 5.9): without friction, as there is no more energy dissipation in our dissipative approach, there are no more shear strength nor volume changes, and the granular quasi-solid becomes a fluid;
• – even the question of coaxiality vanishes when friction vanishes, as it does not matter any longer under the isotropic stress states of perfect fluid mechanics;
• – the same happens with features of polarization patterns, which disappear as the norm N vanishes in key equations, when friction vanishes;
• – so, the whole induced dissipative structure vanishes when friction vanishes.

11.1.2. Relevance of minimum dissipation rule

The approach presented in this book makes an extensive use of the minimum dissipation rule stated in the Introduction, which underlies most of the key results displayed within their specific conditions of motion – and boundary conditions – including the localization criterion and the process of generation of failure lines (Chapter 3). Note that in the experimental validations of Chapter 6, the various situations investigated are all associated with their corresponding micro-scale polarization patterns; the key to this correspondence is again the minimum dissipation rule.

In their perceptive and pioneering work, Rowe and Horne used a similar argument to prove their stress–dilatancy relation [ROW 62, HOR 65, 69], which is a restricted form of our far more general dissipation relation. However, it was later strongly refuted by De Josselin de Jong [DEJ 76] on the grounds of lack of objectivity of input and output power concepts, and other related arguments; a thorough analysis shows that the arguments of De Josselin de Jong in his conclusions were flawed [FRO 01]. In the proposed dissipative approach, these concepts have valid and clear objective tensor definitions under general 3D conditions and the minimum dissipation rule has been clearly related to the thermodynamics of dissipative processes; some of the analogies could also be traced with the Principle of Minimum Plastic Dissipation in the classical mechanics of solids [BOW 10].

For the author, this minimum dissipation rule is deeply rooted in the collective nature, i.e. statistical, of the global dissipative process within the granular mass with a large number of degrees of freedom (section 1.2), in slow motion remaining close to static equilibrium. It is the reason why, when this granular mass is reduced to one single contact in motion, this collective nature disappears, and the dissipation relation becomes determined (section 1.1). Note that when friction vanishes, as the mechanical behavior tends to “perfect incompressible fluid” behavior, it becomes completely determined by the two conditions of constant specific volume and isotropy of stresses, and as there is no more dissipation, this minimum dissipation rule becomes meaningless.

The idea that an extremum principle is at the basis of the physical properties of these materials is quite ancient: it is already present in the essay of Coulomb [COU 73]¹, incidentally titled On an application of rules of Maximis and Minimis to some problems of Statics concerning Architecture…

The combination of the principle of virtual work together with the method of forces minimization used by Coulomb, on a failure mechanism “at incipient movement,” would link his results to the least energy principle. Incidentally, note that in Chapter 4, the least shear resistance criterion occurs to be the envelope of the least dissipation criterions for the sets of boundary conditions analyzed, both criterions becoming identical under conditions of plane strain.

11.1.3. Compatibility with heterogeneity

The wide compatibility with mechanical heterogeneity explored in Chapter 2 is induced by the presence of the octahedral norm N, intrinsically attached to the expression of contact friction laws with the internal actions tensors.

For the author, it constitutes a key property of these tensor structures induced by contact friction, as this compatibility with mechanical heterogeneity is the pass that allows the entry of macroscopic properties with simple reasoning on average quantities.

11.1.4. Localization and shear banding

The present dissipative approach, thanks to this property of compatibility with mechanical heterogeneity, allows juggling simply between the discontinuous granular mass and its equivalent pseudo-continuum, providing a renewed vision of localization and shear banding, up to a detailed internal structure of shear bands.

Note that, in this new approach, no macroscopic discontinuity, whether kinematic, static, nor bifurcation in the mechanical behavior, has been required to set the results of Chapter 3. Furthermore, the statistical process leading to the formation of failure lines from the growth of local fluctuations constitutes a quite general process, as it just needs:

• – to satisfy the minimum dissipation rule;
• – that the mechanical behavior includes a key relevant deformation as an internal variable, such as in Chapter 3; and
• – that this mechanical behavior, during monotonous shearing, be such that , which is the case in most of the geomaterials.

Under such conditions, the material will evolve toward motion patterns that maximize Var(), i.e. under plane strain and monotonic movement, toward motions presenting strong stationary heterogeneities in shear, maximizing , and the kinematic compatibility requiring that such shear concentrations be ordered into linear structures, these quite general conditions will lead to shear banding.

This evolution process corresponds precisely to numerical results published by Gudehus and Nübel [GUD 04] which have modeled the generation of shear bands in finely meshed FEM models by associating the numerical field of local variables with statistical random distributions of local fluctuations.

11.1.5. Failure criterion

11.1.5.1. Shape near apexes

The developments in Chapter 3 show that shear localization within shear bands initiate early in motion, in the case of monotonic (fixed directions) shear strains. Therefore, strain regimes found in Chapter 4 attached to failure criterion at critical state, reached after sustained monotonic shear strains, are considered as localized, unavoidably leading to localized plane strain regimes.

Thus, the question of apex shapes is not physically relevant for this critical state failure criterion, which represents a kind of “ultimate” shear strength, after large monotonous shear strains.

Before reaching this critical state, Figures 4.8 and 5.1 display smooth apex failure criterion shapes, corresponding to allowable solutions, including some small to moderate deviations from the true minimum solution, corresponding to transient states before tending asymptotically to the critical state.

11.1.5.2. Incidence on design methods based on plasticity

The strain regime found here naturally attached to the Coulomb Failure Criterion, under stationary volume conditions, is generally plane strain . Therefore, plasticity models based on this Coulomb Criterion should also incorporate these attached features for consistency: here, the corresponding attached plastic potential would be a non-associated Tresca potential. This feature could lead to some upgrades in kinematic methods in plasticity [MIC 95, COO 13].

11.1.6. Experimental validations

The wide set of experimental data displayed in Chapter 6 has been restricted to a limited set of selected examples.

In [PRA 89], the outstanding experimental results of Pradhan, Tatsuoka, and Sato are outlined and include a lot of other results, with which the present dissipative approach matches with a similar quality of fit.

11.1.7. Coaxiality assumption in macroscopic properties

Under the large monotonic strains near critical state considered here, the simple coaxiality assumption appears reasonable and supported by experimental data.

Note that for the failure criterion, within all simple (disordered) coaxial situations considered, and within all possible near-minimum solutions, the true minimum dissipation solutions do achieve both ordered coaxiality and the convexity of failure criterion.

Far from the above conditions, the rotation of the principal axis of stresses and oriented structures within the material may induce departures from coaxiality [ODA 75, SPE 97]: the basic dissipation relation [1.30a], still valid without coaxiality assumption, would apply anyway, but would need to be completed by a kind of “non-coaxiality rule” defining this deviation from coaxiality.

Nevertheless, it should be outlined that all theoretic lines calculated and traced with experimental data in Chapter 6 are based on this assumption of coaxiality, even in the interpretation of cyclic tests data, with quite good results.

11.1.8. Tracks for further developments

In the author’s opinion, the full inventory of all the relevant properties “built inside” these dissipation relations [1.30] is still to be completed, particularly in non-coaxial situations.

In that inventory, other boundary conditions could be usefully explored (e.g. considering c = const. trajectories instead of b = const. trajectories, etc.). Some original experimental works could be considered on such particular boundary conditions, with systematic determination of the terms of internal actions, allowing to compute the ratio of internal work to the norm N of internal actions, which is in this approach a material parameter (equal to ), assumed in this book to be independent of boundary conditions, to be checked.

The basic simplifying assumptions (Introduction and Chapter 2) could also be revised, an efficient way for that residing with the possibilities offered today by the numerical simulations involving discrete particles with realistic shapes, and non-smooth contact conditions, allowing detailed micromechanical data out-of-reach for other experimental means to be acquired, such as in the work of Nouguier et al. [NOU 03, 05a, 05b]. These results, highlighted in Chapter 6 (section 6.5), show that, with realistic particles, numerical simulations by Non-Smooth Contact Dynamics methods can be fairly well representative of the key physical experimental features.

The dissipation relation, in its phenomenological form published in 1983 [FRO 83] has been perceived as a step forward in the development toward relevant constitutive models [ZIE 87, PAS 90, CHA 07, GUO 04, 09]. However, its eventual implementation in numerical algorithms seems to be restrained by some mathematical difficulties attached to the octahedral norm N (presence of absolute values).

Through this book, this dissipation relation [1.30] is now clearly identified as a macroscopic consequence of dissipative structure induced by elementary contact friction and is also proven to include as “built inside” a very wide set of properties characteristic of the quasi-static mechanical behavior of granular media, with sound experimental validation. This provides a sound base to this dissipation relation [1.30], strengthening its relevance, which could motivate a renewed interest.

In the author’s opinion, in order to go further into numerical implementations, some tracks to overcome the mathematical difficulties mentioned above could be utilized:

• – circumvent the difficulties attached to the octahedral norm N by using, for numerical applications with another norm, N′ belonging to the same family of p-norms, as close as required to the octahedral norm N, but fully continuous to the nth order (see Appendix A.11).
• – involve specific algorithms based on a convex analysis which have already proven their ability to handle the complex non-regular mechanics induced by dry friction in granular materials, as in the Non-Smooth Contact Dynamics method [JEA 99, MOR 03].

This multi-scale dissipative approach of structures induced by friction was made possible thanks to the concept of internal actions. It could be possible to extend the concept and method to:

• – external actions (resulting from the tensor product of external forces by the velocities of their material points of application – see Appendix A.2.2), which could bring a renewed vision of static or quasi-static equilibrium rules, principles of virtual work and mutual actions;
• – dynamic actions (resulting from the tensor product of inertia forces by the velocities of their material points of application), which could bring a renewed vision of dynamics of these materials.

11.2. Concluding remarks on features resulting from grain breakage

Grain breakage constitutes the second main dissipative process in granular geomaterials, after contact friction, and is particularly relevant in rockfill behavior. Surprisingly, it has been possible to enter into the macroscopic consequences without investigating in detail the energy-dissipation process, reasoning only on the statistical features of mineral grain breakage, resulting from fracture mechanics.

The resulting “Scale Effect Rule” developed in Chapter 9, which operates on shear strength envelopes of granular materials, is now readily accepted [ALO 12], its validity has been extended on general 3D stress-paths [XIA 14], and it provides an efficient key for the evaluation of shear strength of materials that are too coarse to enter in the usual laboratory testing apparatuses.

This same “Scale Effect Rule” allows consideration of the impact on rockfill slopes stability, up to their explicit incidences on slope stability safety factors vis-à-vis the usual shear failure mechanisms, as well as deformation and settlement features, which are key engineering concerns.

11.3. Final conclusions

Although incomplete, as they do not provide a complete constitutive law, the approaches developed in this book nevertheless provide numerous relevant results in key aspects of granular geomaterials mechanics, of practical use in civil engineering.

In the author’s opinion, these results show the major relevance of a better representation of the true micro-scale physics of these materials within macro-scale constitutive models, with the aim of developing simple, clear, and efficient representation of some kind of “ideal granular material,” capturing the key features of mechanical behavior, with very few relevant physical parameters.

Such a representation would be in the spirit of these typical models of standard material behaviors in Engineering Sciences which have proved in the past to be so useful in hydraulics, physics of gases, mechanics of materials in structural analysis, etc.

¹ A century later, Rayleigh [RAY 73], in his contribution to the developments of thermodynamics, has suggested that irreversibility was ruled by a principle of least energy dissipation.