6
Experimental Validations

In this chapter, beyond the few previously discussed partial validations (Chapter 3), a wide set of experimental validations is presented, resulting mainly from various past independent experimental data, re-interpreted through the present approach of dissipative structures induced by friction, and covering all kinds of experimental tests. These experimental data cover the situations of:

• – triaxial compression tests data (i.e. in Mode I Direct), with various initial compacities and same confining pressure, or with various confining pressures and same initial compacity;
• – triaxial cyclic tests data, cycling between compression and extension conditions (i.e. between Modes I and II, involving both Direct and Reverse motions);
• – simple shear apparatus data with some rotations of the principal axis (i.e. plane strain mode);
• – cyclic hollow cylinder torsional shear tests data (interpreted as alternate motions in plane strain modes);
• – true 3D compression apparatus data with large amplitude strain cycles; and
• – some numerical simulations with discrete particles.

One of these large amplitude strain cycles experimental data in a true 3D compression apparatus is re-interpreted in detail, showing that not only the general layout but also the detailed characteristics of these experimental results are deeply shaped and densely structured by the pattern set by energy dissipation relations.

This chapter concludes with a section on typical values of material parameters.

6.1. Validations from classical “triaxial” test results

The principle and testing procedures of the “triaxial” test (in fact under axisymmetric stresses) may be found in reference books such as Bishop and Henkel [BIS 62] or Lade [LAD 16]. Triaxial compression means that the cylindrical sample is strained by axial compression; triaxial extension is the same motion, reversed.

6.1.1. Triaxial compression

Figures 6.1(a) and (b) show an experimental validation of the dilatancy rule (equation [5.4]) for a triaxial compression test on crushed limestone with very angular particles of irregular shape at three different initial densities and under the same confining pressure [FRO 79]. The stress–strain curves on the lower left-hand diagram and the volume change on the upper left-hand diagram show the influence of the initial density on the material behavior, especially at peak strength. The dilatancy diagram on the right-hand side shows a linear relationship between the stress ratio σ₁/σ₃ and the dilatancy rate, which is in agreement with equation [5.4].

Figure 6.1. Experimental validation of the energy dissipation relation based on friction through triaxial compression tests. (a) Stress–strain data. (b) Stress–dilatancy diagram. (c) Peak data. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip

The slope of this straight line gives a value of the apparent inter-granular friction . This straight line is independent of the initial density, in agreement with the minimum dissipation rule. We observe that the three samples exhibit the same volumetric contraction rate at the beginning of the loading (see Figure 6.1(a)). This is also in agreement with equation [5.4] in the vicinity of an isotropic stress state.

In Figure 6.1(c), we report the results of triaxial compression tests on crushed basaltic-reduced rockfills published by Charles and Watts [CHA 80]. The granular material is well graded with grain sizes varying from sand to gravel with 38 mm maximum size. The samples, of 230 mm diameter, were tested at the same initial density and with confining pressures varying from 27 to 700 kPa. The diagram shows the evolution of the internal friction angle at peak strength (defined by sinΦ = max((σ₁ – σ₃)/(σ₁ + σ₃)), with the rate of volume change. From the experimental data, the dissipation relation allows fitting a theoretical curve, (see Appendix A.6.1), which gives a value for the apparent inter-granular friction of 44° with dispersion close to ±5%. In these tests, the effect of grain breakage reduces the dilatancy rate at peak when the confining stress increases. Despite the phenomenon of grain breakage, the experimental values remain close to the curve predicted by the equation of dissipation by friction, even if the results obtained at peak are more dispersed than the ones deduced from a fitting over the whole test data.

Figure 6.2. Shear strength envelope for reduced-basalt rockfill. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip

If these results are plotted in the plane of Mohr circles (see Figure 6.2), we can observe the effect of the dilatancy on the shear strength by comparing two circles at the same confinement. The decrease of the dilatancy at elevated stresses due to grain breakage reduces the internal friction. As a consequence, the maximum strength envelope is no longer a straight line but has a curvature with a reduced slope when the stresses increase, which can be fitted by a power law with a coefficient approximately equal to 0.77.

6.1.2. Triaxial extension and cyclic triaxial

In early works, Barden and Khayatt [BAR 66] detail the triaxial tests in both compression (i.e. our strain Mode I) and extension (i.e. our strain Mode II), in the classical dilatancy diagram with coordinates . Their results are distributed along two conjugate straight lines, as predicted by section 5.6 (Chapter 5) [PRA 89].

The outstanding publication of Pradhan, Tatsuoka, and Sato [PRA 89] reported cyclic triaxial tests data conducted on Toyoura sand at the Institute of Industrial Science of University of Tokyo. These authors reported their data on special dilatancy diagrams, whose coordinates may be expressed in our notations:

[6.1]

Appendix A.6.2 details our theoretical envelopes of data foreseen by the dissipation equation in these coordinates on strain Mode I (triaxial compression) and strain Mode II (triaxial extension), which are shown in Figure 6.3; again, the experimental points distribute along two conjugate trajectories corresponding to Modes I and II. The theoretical envelopes, adjusted to a material parameter S = 0.445 (i.e. ), are in good agreement with the data.

Figure 6.3 is related to dense Toyoura sand. In the same publication, Pradhan et al. also reported on cyclic triaxial tests on loose Toyoura sand; the results are quite similar to those for dense sand, and similar theoretical envelopes may be adjusted, providing the same material parameter, with similar quality of agreement with the data.

Figure 6.3. Theoretical envelopes on dilatancy diagrams of cyclic triaxial tests (adapted after Pradhan et al. [PRA 89]). (a) Cyclic motion and stress-path in the octahedral plane. (b) Stress–strain trajectories. (c) Specific dilatancy diagram. For a color version of this figure, please see www.iste.co.uk/frossard/geomaterials.zip

6.2. Validations from simple shear experimental results

Early works on stress–dilatancy relation in simple shear published by Oda [ODA 75] report simple shear tests performed at Cambridge University on Leighton Buzzard sand in the development of the simple shear apparatus, by Cole [COL 67]. Here, the data are displayed in another set of special coordinates as a dilatancy diagram, which are expressed in our notation as:

[6.2]

Appendix A.6.3 details our theoretical envelope of data foreseen by the dissipation equation in these coordinates as shown in Figure 6.4: the experimental data distribute well on the theoretical envelope, with the same material parameter for dense or medium to loose sand, here S = 0.542 (i.e. ).

Figure 6.4. Theoretical envelopes on dilatancy diagrams of simple shear tests (adapted after Oda [ODA 75], on data from Cole [COL 67]). (a) Motion, stresses, and micro-scale polarization pattern. (b) Specific dilatancy diagrams. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip

6.3. Validations from true 3D compression apparatus results

Figure 6.5 shows the results of a large amplitude cyclic drained three-dimensional test performed in the 3D apparatus (with coordinated rigid platens) at Institut de Mécanique de Grenoble by Lanier [LAN 84] (see also [LAN 14]), on a granular material used in construction of a large dam under constant axisymmetric confining stress σ_x = σ_y = 0.5 MPa, starting from the isotropic stress state.

These results, which display successive motions in four Modes (Mode I Direct → Mode II Reverse → Mode II Direct → Mode I Reverse → return to Mode I Direct), showcase in one single experiment numerous features predicted by the energy dissipation relation. These features are summarized below.

The untreated laboratory recordings are displayed in Figure 6.5(a)–(c), and the features corresponding to the energy dissipation relation are superimposed on the widened stress–strain recordings [Figure 6.5(d)]:

• – At motion reversal (1) or (2) on diagrams, the previously ordered coaxiality is reversed [e.g. at motion reversal (1), Mode I D → Mode II R] then the principal stress ratio σ₁/σ₃ decreases toward isotropy, crosses the isotropic line, and increases again on another side, restoring the ordered coaxiality.
• – The energy diagram [Figure 6.5(c)] displays the total mechanical work versus input mechanical work, i.e. ∫Tr{π} dt versus ; in this diagram, the energy dissipation relation has been shown [FRO 86] to set a straight line trajectory passing through the origin, the slope of which is 2S/(1 + S); this is very well verified on the diagram, and the material parameter of apparent inter-granular friction is S ≈ 0.485 or .
• – This apparent friction parameter allows us to define the “characteristic state” which will be crossed during the cycles; for the Mode I motion, it is σ_z = σ_Ca(I) = (1 + S/1 - S), σ_x denoting this value at σ_Ca(I) = 1.44 MPa; and for Mode II motion, it is σ_z = σ_Ca(II) = (1 − S/1 + S), σ_x denoting this value at σ_Ca(II) = 0.173 MPa [Figure 6.5(d)].
• – We observe that each time the evolving stress σ_z crosses the “characteristic state” under Mode I or II Direct, the volume evolution passes precisely through a minimum, such that the material is always contracting inside the “characteristic state” limits, and is dilatant outside these limits, but only for Modes I Direct and II Direct, as set by the energy dissipation relation (see section 5.7.2).
• – Similarly, each time the evolving stress σ_z crosses the isotropic line, the slope of the volume evolution computed by the contraction relation [5.1] (see Appendix A.6.4) agrees quite well with the data, although with some deviation on the side of Mode II.

Figure 6.5. Validation of features set by the energy dissipation relation: large amplitude cycling testing on 3D apparatus (adapted from [LAN 84]). (a) Scheme of apparatus, stress-path, and micro-scale polarization patterns. (b) Untreated stress–strain and volume change recordings. (c) Energy diagram results. (d) Detailed interpretation of stress–strain and volume change records. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip

• – At each maximum of the evolving stress σ_z, it corresponds a maximum for the slope of the volume evolution which, computed by the stress–dilatancy relation [5.4] (see Appendix A.6.4), agrees quite well with the data.
• – Just before motion reversals, the slope of volume variation versus strain can be evaluated from the stress σ_z and the stress–dilatancy relation [5.4], and the same slope just after the motion reversal can be foreseen by the “reversal relation” [5.6] (see Appendix A.6.4). These slopes are outlined on the volume recordings (Figure 6.5(d)): on reversal (1), under σ_z = 1.05 σ_Ca(I), all slope predictions are fairly good, except discrepancy on the contraction foreseen just after the reversal (1). This effect is attributed by the author to some elastic restitution that can be seen on the energy diagram. On reversal (2) under σ_z = 0.70 σ_Ca(II), this effect is no longer visible, likely due to the lower level of stresses, and the fit is excellent.

Thus, not only the global layout of dilatancy diagrams (sections 6.1, 6.2, and 6.4) but also detailed features of these experimental results reveal to be deeply shaped and densely structured by the pattern set by the energy dissipation relation [1.30].

6.4. Validation from cyclic torsional shear tests data

In [PRA 89] cited in section 6.1.2, a wide set of detailed cyclic torsional shear tests performed on hollow cylinder apparatus were presented, on the same material Toyoura sand as in the cyclic triaxial tests. Assuming that this kind of solicitation with alternate motion is basically similar to simple shear in plane strain, i.e. , the theoretical envelopes of experimental data are detailed in Appendix A.6.5 in this interpretation of that particular coordinate system’s dilatancy diagram

[6.3]

These theoretical envelopes foreseen by the dissipation equation are shown in Figure 6.6 for the same material parameter as already adjusted in the above cyclic triaxial tests results (Figure 6.3), together with experimental data. Despite the approximate character of this interpretation (approximate because the torsional hollow cylindrical motion is compared directly with plane strain simple shear, without corrections), the theoretical envelopes fit quite well with the experimental data.

Note that the same material parameter appears valid either for cyclic triaxial or for cyclic torsional shear (here on dense Toyoura sand). The same publication also displays the results for loose material; the results are again quite similar to that of dense sand, and similar theoretical envelopes may be drawn with the same material parameter, with similar quality of agreement with experimental data.

Figure 6.6. Theoretical envelopes on dilatancy diagrams of cyclic torsional shear tests (adapted after Pradhan et al. [PRA 89]). (a) Cyclic motion, stresses, and micro-scale polarization patterns. (b) Stress–strain diagram. (c) Specific dilatancy diagram. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip

6.5. Validations from detailed numerical simulations with realistic discrete particles

Remarkable detailed numerical simulations of 2D biaxial tests have been performed by Nouguier and co-workers at the LTDS laboratory at Ecole Centrale Lyon [NOU 03, 05a, b], with a non-smooth contact dynamics method. This method uses realistic discrete particles, randomly shaped irregular convex polygons with various elongation ratios [Figure 6.7(a)] to analyze the statistics of their behavior, particularly the features of the internal kinematics involved, and to compare it with the behavior of the circular particles that are more frequently used in simulations.

Figure 6.7. Rotations in biaxial tests numerical simulations with discrete particles of different shapes (from [NOU 05a]). (a) Typical randomly-shaped irregular convex polygons with various elongation ratios R_a. (b) Mean rotations developing with strains during monotonic biaxial compression. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip

Among the numerous results obtained, a comparison of the mean absolute values of rotations, as shown in Figure 6.7(b), shows that:

• – Every sample made of polygonal-shaped particles displays average rotations approximately identical to the rotation simply resulting from shear kinematics, .
• – The behavior of the sample with circular particles appears clearly different, their rotations being about three times this kinematic shear rotation (Figure 6.7(b)).
• – These features sustain the assumption (see Introduction and section 1.3.2) that inside the macroscopic strains in the granular geomaterials in motion (made of irregularly-shaped particles), there is no significant global component in pure rolling, which is definitely not the case for 2D granular media with circular particles.
• – A detailed analysis of energy dissipation led to an evaluation of the rolling proportion in macroscopic strains between 30% and 40% (see [NOU 05a]).

Another result of these numerical simulations on cyclic 2D biaxial tests with discrete realistic particles is shown in the Figure 6.8 stress–dilatancy diagrams; the data appear distributed over a set of two well-defined conjugate straight lines, corresponding to the theoretical stress–dilatancy relationship for both plane strain Modes Direct (loadings) and Reverse (unloadings).

Figure 6.8. Dilatancy diagrams for cyclic biaxial numerical simulations with realistic discrete particles (from [NOU 05a]). For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip

6.6. Measurement of apparent inter-granular friction – typical values of the parameters

The apparent inter-granular friction coefficient , being intrinsically a mesoscopic to macroscopic material quantity, has an unavoidable statistical nature, which is integrated into its definition through the internal feedback rate R (see Chapter 1, sections 1.2 and 1.3). Thus, the rational way to assess this material parameter is through experiments involving a representative sample of the whole material, including a sufficient number of grains. This can be achieved through cautious “triaxial” compression tests associated with adequate volume change measurements, allowing us to draw stress–dilatancy diagrams of good quality, as shown in Figure 6.1(a) and (b). Experimental data, plotted in the same kind of dilatancy diagram allow us, through linear regression, to evaluate the material parameter .

Through such experiments, a wide set of granular materials of the sand size, cautiously selected, has been tested systematically to display the combined effect of mineralogy and morphology of grains. This methodology is detailed in [FRO 78] and summarized with the results in [FRO 79].

The granular materials tested involved in particular limestone particles, from very angular and rough to well-rounded and smooth, and siliceous particles with a similar morphological range. These are displayed in Figure 6.9: top, the typical morphology of limestone particles; bottom the typical morphology of the siliceous particles, on the left the angular and rough particles, and on the right the rounded and smooth particles.

The morphology was defined for each tested material, by averaging measurements over at least 100 particles, using a quantitative numerical method of morphology analysis, detailed in the above-cited publications.

The results¹ show that:

• – no clear correlation appeared with the elongation of particles (however, the tested real materials were not so well contrasted as shown in Figures 6.7 and 6.8);
• – a strong correlation appeared with the roughness of particles;
• – the structure of this correlation suggested that the apparent inter-granular friction appeared as the sum of two components: a “mineralogical” component independent of morphology, and a “morphological” component independent of mineralogy;
• – the “mineralogical” component was found to be equal to the friction which could be measured on contact between two mineral blocks, polished and under hygrometric equilibrium, and it is in fact the “mineral friction” of Rowe’s theory ϕ_μ; and

  

  Figure 6.9. Typical particle morphology of tested granular materials

• – the “morphological” component obtained by the difference was found in a range of 10°; it was minimal for materials made of well-rounded and smooth particles, and maximal for very angular and rough particles. This component includes both the effect of contact roughness and the increase in internal movements disorder (the internal feedback effects) due to the irregularity of the morphology of the angular and rough particles.

These results, complemented with data from the literature (see [LAM 69]), are be summarized in Table 6.1.

Table 6.1. Apparent inter-granular friction for mono-mineral granular media under low confining stresses, for narrow gradations

Mineralogy morphology	Micas	Quartz	Calcite	Feldspars
Smooth and well-rounded particles	8–13	26	34	34–37
Rough and angular particles	18–23	36	44	44–47

These results were achieved under low confining stresses (about 50 KPa). For larger confining stresses, the crushing resistance of the minerals constituting the particles plays a key role; as shown by McDowell and Bolton [MDO 98], there is no more dilatancy in shear motion when the confining stress exceeds about 10% of the crushing strength of the particle.

For well-graded granular materials with multi-mineral particles, systematic investigations are still to be developed, although some valuable data are now available in the literature, as in [CHA 80], providing values as high as 47° for a strong basalt rockfill or even 50° for a high-strength slate under low-confining stresses.

These experimental results also provide indirect estimates of the internal feedback rate values, R:

• – between 0.05 and 0.2 according to grain angularity, roughness and elongation in triaxial compression tests;
• – between 0.15 and 0.35 according to grain angularity, roughness and elongation under plane strain conditions.

¹ The present theoretical developments not being known at the time of these experiments, the interpretation of measured apparent inter-particle friction was identified in the referenced papers as a kind of average mineral contact friction. The present interpretation shows the difference between ϕ_μ and , the first one being the elementary mineral contact friction quantity, and the second involving the statistical internal feedback effects explained in Chapter 1.