1
Fundamentals: The Tensor Structures Induced by Contact Friction

This chapter details the tensor structures induced by contact friction, whose prominent characteristics are summarized in the synoptic Figure 1.1 – from the scale of an elementary contact to the scale of macroscopic equivalent pseudo-continuum – displaying how energy dissipation by contact friction induces the structures in the eigenvalues of internal actions at all scales.

These structures are shown to result in energy dissipation equations operating on internal action invariants, at every scale, integrating population effects from the mesoscopic scale to the macroscopic one: the “internal feedback” effect resulting from interactions between adjacent inter-granular contacts in motion, which is a kind of micro-mechanical mixed arching and domino effect.

These structures are shown to result from energy dissipation by contact friction associated with the “minimum dissipation rule” detailed in the Introduction to this book. At the mesoscopic scale, the minimum dissipation solutions, i.e. the distributions of elementary contact actions achieving the minimum dissipation, are shown, in general, to present high polarization of internal contact action orientations. Under plane strain conditions, the mesoscopic minimum dissipation solution results in the polarization of elementary contact sliding motion corresponding to Rankine’s slip lines.

The last part of this chapter is focused on the correspondence between the discontinuous granular mass (mesoscopic scale) and its equivalent pseudo-continuum (macroscopic scale), leading to the macroscopic equation of energy dissipation by contact friction near minimum energy dissipation.

Figure 1.1. Synopsis of multiscale tensor structures induced by contact friction. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip

From the author’s point of view, these tensor structures and their material expressions in the polarized distributions of internal actions can be seen as dissipative structures induced by a specific form of energy dissipation by contact friction.

1.1. Microscopic scale: the elementary inter-granular contact

1.1.1. Vector formulation of energy dissipation

Consider a simple contact c between two grains a and b, sliding with a relative velocity v(a/b) under a contact force f(a/b), with an elementary friction angle at contact φ_μ (Figure 1.1). The elementary laws of friction result in the following relation between the two vectors:

[1.1]

It may be noted that the above-mentioned vector equation still holds even when the movement stops (i.e. v(a/b) becomes null) or when the contact disappears as the grains separate in the motion (i.e. f(a/b) becomes null). Equation [1.1] expresses the equality between the work rate of contact forces on its left-hand side, and an always positive function – then a dissipation function – on its right-hand side. The vector equation then corresponds to the energy dissipation during sliding.

1.1.2. Tensor formulation of energy dissipation

These two vectors may be considered as the internal movement and internal force of our contact c. From their symmetrical product, the “tensor of elementary contact actions” p(c), a symmetrical second-order tensor, whose trace is the mechanical work rate produced by the contact force f(a/b) during sliding, can be defined as follows:

[1.2]

It may be noted that, by its definition, this tensor is also independent of the order affected by the considered grains or particles, either the contact of grain a on grain b (earlier denoted as a/b) or the reverse, b/a, because the relative velocity and exerted force in the b/a case are opposite to the ones in the case a/b. This justifies the notation p(c) that now relates this variable to the contact c, independent of the way we consider it, either contact of grain a on grain b or the reverse b/a.

This tensor of elementary contact actions p(c) can be easily diagonalized in its natural basis formed by the two bisecting lines in the directions of v(a / b) and f (a /b) (eigendirections numbered 1 and 3), and their common normal (eigendirection numbered 2). In this natural basis

[1.3]

From the three eigenvalues of p(c), we can define the symmetrical function as follows:

[1.4]

This function, which is a tensor norm of p(c)¹, named “octahedral norm” in the following, is related to the Euclidian norms of the two vectors v(a/b) and f(a/b) by the following relation, resulting from the diagonalized expression [1.3]:

[1.5]

Merging equations [1.1], [1.2], and [1.5], we can now express the dissipation relation resulting from the elementary laws of friction by a relation between the eigenvalues of p(c), which corresponds to the tensor equation of the energy dissipation by friction at a single contact point

[1.6]

It may be noted that the tensor equation [1.6], which has the same validity domain as that of the vector equation [1.1], is linear in its first member and piece-wise linear in its second member. Equation [1.6] is also simpler than the vector equation [1.1] because only one physical quantity appears as an argument, which turns out to be additive, unlike our two vectors.

1.1.3. Physical significance – algebraic and geometrical representations

Now, on considering our contact as an elementary physical system, various quantities related to tensor p(c), whose components may be interpreted as mechanical energy fluxes exchanged with the outside (with both normal and tangential components), will take on remarkable significance:

• – the positive eigenvalue of p(c) may be interpreted as an “input power” p⁺;
• – the negative eigenvalue of p(c) may be interpreted as an “output power” p⁻;
• – the required objectivity of these notions is provided by the following symmetric formulations:

[1.7]

In this framework, the trace Tr{p(c)} may be interpreted as an energy balance between the input and output powers as it results from their algebraic sum, while the norm N{p(c)}, which is a global measure of the intensity of all energy exchanges (either positive or negative), appears as a norm of these energy fluxes.

With these definitions, the tensor equation [1.6] simply states that in the dissipation process that takes place inside the elementary contact, this contact appears as a physical system (see Figure 1.2(a)) which:

• – on the one hand, receives mechanical energy from the outside, as a normal flux along its eigendirection no. 1, where the input power received in that direction is its positive eigenvalue p⁺;
• – on the other hand, returns the mechanical energy to the outside, as a normal flux along its eigendirection no. 3, where the output power returned in that direction is its negative eigenvalue p⁻;
• – the balance between the input and output powers, which is the work rate developed from the contact forces, is dissipated by friction, where this dissipation rate turns out to be proportional to the norm of these energy fluxes;
• – the coefficient of proportionality is set by the contact friction.

As a consequence, p⁺ and p⁻ also remain proportional during the dissipation process, and their coefficient of proportionality is also fixed by the contact friction

[1.8]

Tensor p(c) is a plane tensor, and its representation by Mohr circles is more simple (Figure 1.2(b)), which provides us with a geometrical expression of tensor equation [1.6]. From this geometric representation, it may be noted that in the plane of eigendirections 1 and 3, there are two directions, which are the directions of our two vectors v(a/b) and f(a/b), of facets for which the energy fluxes are purely tangential.

Figure 1.2. Elementary contact tensor p(c) – physical interpretation and representation by Mohr circles. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip

The above-mentioned relations, together with the definition of tensor p(c), condense the elementary tensor structure induced by the contact friction, and their properties will form the backbone of the developments in section 1.2. These relations, which directly result from the energy considerations, describe an apparently perfect symmetry between the forces and movements. However, the fundamental laws of dry friction include a condition that breaks the apparent symmetry at the scale of elementary contact: the consequence of unilaterality of contacts which rules that the normal component of contact force shall always be in compression (or null).

In the multiscale transposition, it will be necessary to maintain a condition bearing this symmetry breaking between internal forces and internal movements: it will be the role of macroscopic non-traction condition on normal stresses stated in the basic assumptions, which will take on importance in the description of the equivalent pseudo-continuum.

1.2. Mesoscopic scale: the discontinuous granular mass

The granular assembly that experiences strain motion under the action of external forces may be considered as constituted by moving continuous parts (the grains and the interstitial fluid that fills the voids between the grains), separated by discontinuities (the contacts and the interfaces grains/interstitial fluid). The mechanics of continuous media with surfaces of discontinuity (see [GER 80]) show that the work rate of internal forces is the sum of two terms: the work rate of internal forces inside the continuous parts and the work rate developed from the contact forces on the discontinuities.

Provided that the grains can be considered as rigid bodies in slow motion, the work rate of internal forces will reduce to the work rate of discontinuities: here, the work rate of internal forces in our granular assembly is the sum of the work rates developed at all elementary contacts.

This leads us to consider the granular mass to be in motion as a population of moving elementary contacts featured by their internal actions (Figure 1.3).

1.2.1. Vector formulation of energy dissipation

In our granular mass, the vector formulation of total energy dissipation is the sum of the elementary dissipations at all contacts; thus, by numbering the grains from 1 to N and applying equation [1.1], we obtain

[1.9]

The property of symmetry outlined in section 1.1.2 ensures that this sum is independent of the numbering of grain order (provided that each contact is taken into account only once; the reason for the sum for n < m ≤ N), thus this sum is objectively attached to the population of contacts.

Figure 1.3. The granular mass in motion, represented by a set of moving contacts with their internal actions. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip

1.2.2. Tensor aspects of energy dissipation

The vector formulation of energy dissipation addressed in section 1.2.1 is expressed with local vector quantities, contact forces, and sliding velocities, which have no direct physical signification for the whole granular mass, as these quantities are neither extensive nor intensive. To reformulate the equation with extensive quantities, it may be noted that the work rate of internal forces and the dissipated energy rate are both extensive quantities over our population of contacts.

Note that the left-hand side of equation [1.9] is the trace of a tensor, equal to the summation over the whole granular mass A, of all tensors of the elementary contact actions, and this leads us to consider a tensor of “internal actions in the granular assembly” defined as the resultant of elementary contact actions

[1.10]

The second-order symmetrical tensor possesses a natural basis (built on its eigendirections) in which the eigenvalues can again be considered as fluxes of energy exchanged with the outside: fluxes coming from the outside or “input power in the granular mass” for the positive eigenvalues, fluxes returning to the outside or “output power for the granular mass” for the negative eigenvalues, defined by the same relation [1.7].

The left-hand side of equation [1.9] is linked to P⁺ and P⁻ by relation [1.7]. Furthermore, the properties of tensor norm N in relation [1.7] link P⁺ and P⁻ to the input and output powers of elementary contacts by the inequalities as follows:

[1.11]

The dissipation equation for the granular mass [1.9] may be written as

[1.12]

Thus, because of inequalities [1.11], relations [1.10]–[1.12] are not sufficient to directly express the energy dissipation inside the granular mass, only in function of global quantities defined from the global tensor P(A). Passing from an individual behavior (the elementary contact) to a collective behavior (the granular mass) implies some physical indetermination: inequalities [1.11].

1.2.3. A key population effect in energy exchanges: the internal feedback interaction

Now, we show that this physical indetermination is due to a certain kind of mechanical interaction, which may occur between two adjacent moving contacts. Next, we show that the global extent of these interactions:

• – set the intensity of dissipation in the granular mass between two limits (minimum and maximum dissipation);
• – is directly linked to the degree of disorder inside the spatial distribution of contact movements orientations.

In fact, in the granular mass in motion, the movement of one particle may have repercussions on the movements of its neighbors (in short, the stones are pushing each other) and so on, with secondary repercussions cascading inside the whole granular mass. However, this chain of effects, pertaining to some extent to arching and domino effects, is likely to have limited extension, because of dissipation in each elementary movement, which quickly attenuates this cascade effect during its propagation.

This can be expressed simply with the elementary contact action tensors (Figure 1.4):

• – the output power returned from one moving contact p⁻(c₁) (i.e. a/b in Figure 1.4) can feedback the input power of another nearby moving contact p⁺(c₂) (i.e. b/c in Figure 1.4);
• – when these quantities are oriented, such a feedback will be more significant as the direction bearing p⁻(c₁) is close to the direction bearing p⁺(c₂), as shown in Figure 1.4;
• – in the global energy balance for these two adjacent contacts, the global input power will be less than the sum of both elementary input powers, because of this internal feedback (and symmetrically, for the output power), and similarly for the global balance at the scale of whole granular mass;
• – thus, inequalities [1.11] are the algebraic consequence on the resultant of elementary contact tensors of this physical interaction effect inside the granular mass, named by the author as “internal feedback” interaction².

It may be noted that if we consider that the whole granular mass is partitioned into sub-masses, this notion of internal feedback also applies to the interactions between the sub-masses forming the partition, as the tensor of internal actions in the whole granular mass is the sum of the tensors of internal actions of the sub-masses forming the partition.

Furthermore, as the global output power and the input power of the whole granular mass always verify the condition of conservation of the trace, i.e. , we can globally assess the intensity of this internal feedback by a sum of interaction components over the whole granular mass Ė_IF(A) as follows:

[1.13]

Figure 1.4. Example of internal feedback configuration in the granular mass in motion. For a color version of the figure, please see www.iste.co.uk/frossard/geomaterials.zip

This global intensity of internal feedback effects Ė_IF (A) may be usefully scaled by the sum of the norms of all elementary contact actions tensors, and with a normalization coefficient, we can always define an “internal feedback rate” R(A) by the following relation:

[1.14]

Relations [1.7], [1.13] and [1.14] then lead to the expression of the internal feedback rate R(A) as a function of the whole distribution of elementary contact actions tensors in the granular mass, which is always a positive number, and turns out to be less than or equal to 1

[1.15]

It may be noted that the above expressions are the most basic and global forms of this internal feedback interaction, resulting only from the satisfaction of energy balance considerations without any particular assumption on the distribution of these interactions.

1.2.4. The mesoscopic equation of energy dissipation by contact friction

Then, taking into account the internal feedback interaction quantified by its global rate R(A), we can now rewrite the dissipation equation for the granular mass, only in the function of global physical quantities related to the whole granular mass P(A) and R(A), from relations [1.9] and [1.15], as follows:

[1.16]

Compared to the basic form of the dissipation equation at the elementary contact [1.6], as observed here, in relation to the dissipation process within the granular mass in motion, the quantity R(A) plays the role of state function, bearing the population effect. In fact, from its definition [1.15], this function is irreducibly linked to statistical aspects of motion inside the granular mass: if the population reduces to only one moving contact, then R(A) becomes null, the dissipation equation [1.16] reduces to its basic form at elementary contact [1.6], and the population effect then disappears from equation [1.16]. Other detailed correspondences with state functions have been outlined in [FRO 01].

Furthermore, we will see that:

• – the function R(A) grows with the grade of disorder in the statistical distribution of moving contacts, and it can be interpreted as a measure of this grade of disorder, specifically relevant for this specific dissipative process;
• – its minimum R(A) = 0 is associated, in a bijective correspondence, with a set of ordered structures in the statistical distribution of moving contacts orientations within the granular mass;
• – these ordered structures that minimize the function R(A) are in limited number and do constitute the minimum dissipation structures for this specific dissipation process.

With the above definition of internal feedback, the indetermination of inequalities [1.11] may be resolved into a function of the quantity R(A), using relations [1.13] and [1.14]

[1.17]

1.2.5. Minimal dissipation and ordered structures

It may be noted that from the dissipation equation [1.16], the dissipation rate relative to the norm of mechanical energy exchanges between the granular mass and the outside, which may be defined as , is a monotonous function of the quantity R(A). Thus, this dissipation rate is minimal for the lowest possible value of R(A), i.e. for R(A) = 0.

The situations that verify this condition R(A) = 0 are the ones for which

[1.18]

The distributions of elementary contact action tensors, which are the solutions of condition [1.18], are determined in Appendix A.1.1 and constitute the minimal dissipation structures. For these minimal dissipation solutions, the granular assembly follows the same dissipation equation as the elementary contact (see relation [1.6] in section 1.1.2), this particular form becomes the minimal dissipation equation for the granular mass.

These minimal dissipation solutions correspond to ordered patterns in the distribution of contact motion orientations within the granular mass (see Figure 1.5) and consist of two three-dimensional (3D) modes, named Mode I and Mode II, separated by a border mode in plane strain:

• – In 3D Mode I, with signature (+,−,−) in tensor P(A), the eigendirection no. 1 of tensor P(A) bearing P⁺(A) is also the eigendirection no. 1 for all the elementary tensors p(c) in the granular mass; hence, the sliding motions at contacts are all inclined at on this eigendirection no. 1 of tensor P(A), and the directions of these sliding motions admit an envelope that is a cone with eigendirection no. 1 as axis. At the macroscopic scale, the usual testing conditions known as “triaxial compression test” correspond to the kind of signature of Mode I.
• – In 3D Mode II, with signature (+,+,−) in tensor P(A), the eigendirection no. 3 of tensor P(A) bearing P⁻(A) is also the eigendirection no. 3 for all the elementary tensors p(c) in the granular mass, so the sliding motions at contacts are all inclined at on this eigendirection no. 3 of tensor P(A), and the directions of these sliding motions admit an envelope that is a cone with the eigendirection no. 3 as axis; at the macroscopic scale, testing conditions known as “triaxial extension test” correspond to the kind of signature of Mode II.
• – In plane strain border mode between 3D Modes I and II, with signature (+,0,−) in tensor P(A), the orientations of sliding motions at contacts satisfy both conditions of Modes I and II and are inclined both at on the eigendirection no. 1 of tensor P(A) and at on the eigendirection no. 3, hence only two sliding motion directions are allowed (the envelope reduces to the pair of straight lines common to both cones of 3D Modes I and II), and all eigendirections of elementary tensors p(c) coincide with the corresponding eigendirections of tensor P(A); in macroscopic plane strain conditions, the pattern of sliding motion orientation coincides with the well-known Rankine’s slip lines [RAN 57], used in the theory of failure by “limit equilibrium,” still used commonly in professional practice, because of its widely proven efficiency.

These theoretical solutions imply a complete polarization of the orientation distribution of sliding motions at contacts within the granular assembly in motion.

Figure 1.5 summarizes the prominent characteristics of these theoretical minimal dissipation solutions, including the patterns of sliding motion distributions with their polarization characteristics, the signatures in the tensor P(A), the layout of their representations by Mohr circles, and the corresponding macroscopic strain motion patterns.

Various theoretical and experimental results have suggested the key assumption of the “minimum dissipation rule”, related to thermodynamics of dissipative systems near equilibrium (see section I.2), describing the fact that under specific regular boundary conditions, the experimental behavior appears naturally prone to tend to some kind of minimum dissipation characteristics.

It may be noted from the above results that:

• – if the behavior tends to minimum dissipation, then the statistical distribution of active contacts necessarily tends to one of the above-described ordered structures associated with minimum dissipation, and simultaneously the function R(A) tends to 0;
• – such an evolution toward minimum dissipation is then necessarily associated with a progressive polarization of the statistical distribution of active contacts;
• – a measure of the degree of this polarization is then provided by the function [1 − R(A)], which once again outlines the role of state function played by R(A).

Another key feature to be noted for these minimal dissipation solutions is the fact that, in true 3D motions, Modes I and II exclude each other: if a minimal dissipation motion is in Mode I in one part of the granular mass, it cannot be in Mode II in another part of the same granular mass (on the other hand, one part of the granular mass can be in plane strain border mode at the border of Mode I domain, and another part can be inactive, and similarly if one part of the granular mass is in Mode II).

The plane strain border mode achieves a noticeable property of internal similitude inside the granular mass in motion: the internal action tensors of any sub-part of the granular mass have the same orientations, are homothetic, and verify the same conditions (plane tensor) as elementary contacts. This internal similitude is not well developed in the 3D Modes I and II, where only the property of mode signature conservation is maintained: the diagonals of internal action tensors of any sub-part of the granular mass maintain a fixed signature.

Finally, the minimal dissipation solutions may be observed either in biaxial motion (the border mode in plane strain) or in 3D motion (Modes I and II), where no monoaxial motion exists in the set of minimal dissipation solutions.

1.2.6. Maximal dissipation and disordered structures

At the other extremity of the range of possible internal feedback intensity, there are regimes of maximal dissipation with a maximal value of R(A) = 1.

Here, relation [1.17] states that high output power is not returned to the outside (as P⁻ becomes null): all eigenvalues of P(A) are positive or null, the material receives mechanical energy (in the broadest sense) in all directions, and any mechanical energy fed into the granular mass in motion is completely dissipated inside, resulting from the dissipation equation [1.16] with R(A) = 1.

Figure 1.5. Theoretical minimal dissipation solutions

It has been shown (section 1.2.3) that internal feedback interaction is made possible by a wide range of arrangements of mutual orientations of adjacent elementary contact action tensors. If the distribution of the elementary contact tensors p(c) is sufficiently disordered, some internal feedback will appear unavoidably and will have incidence of the summation forming the tensor P(A); here appears the link between internal feedback intensity, and the grade of disorder in the distribution of the orientations of moving contacts is determined.

Several examples of such distributions verifying R(A) = 1 have been detailed in [FRO 01], including monoaxial motions as in the experimental “odometer test” or in the geostatic consolidation equilibrium, biaxial motions, or 3D ones.

Any linear combination with positive coefficients of some of these particular distributions, whose orientations in space would be distributed at random, will also result in tensors P(A) verifying P⁻(A) = 0 (because a sum of symmetric tensors with positive eigenvalues, even oriented at random, always provide a symmetric resultant tensor with also positive eigenvalues). This is the property of convexity of the set of solutions for maximal dissipation.

Detailed analysis of this set of solutions in the cited reference has shown that the maximum dissipation solutions correspond to a wide set of statistical distributions of moving contacts orientations, without predefined order, with structures covering all intermediaries between two kinds:

• – either disordered distributions of the elementary tensors spread over the whole granular mass: in any direction either some p⁺ or some p⁻ will be found; the corresponding motion in the granular mass displays an internal feedback distributed over the whole granular mass;
• – or composite structures including within the granular mass, the juxtaposition of ordered sub-parts under minimal dissipation motion, but with different distributions from one sub-part to the other; these structures are characterized by internal feedback interactions concentrated at the borders between these sub-parts.

It may be noted that these disordered or composite structures deprive these maximal dissipation regimes of the property of internal similarity: there is neither a common active eigendirection nor a dissipation equation verified at all scales in any subpart.

1.2.7. General solutions of dissipation equation with 0 ≤ R(A) ≤ 1 – some key properties and geometrical representation

In [FRO 01], as cited above, general solutions of dissipation equation [1.16] have been detailed quite extensively. The prominent characteristics include two key properties: allowable solutions include motions with the partition of the granular mass into active sub-parts and inactive sub-parts, and the decomposition of tensor P(A) into two subcomponents, one in minimal dissipation and the other in maximal dissipation.

The first of these properties is due to the fact that the quantities involved in the general dissipation equation are all related to sliding contacts. So, if the granular mass A includes an active part A1, and another totally inactive part (i.e. a “dead zone” without any sliding contact inside, whose global motion is that of a rigid body), the global tensor P(A) is indeed that of the active part A1. Symmetrically, an inactive granular mass may be considered together with an active one, the tensor for the resulting granular mass will remain equal to the tensor of the active part, and verify equation [1.16] under the same conditions. Furthermore, the number of active and inactive parts is not limited: a motion verifying the dissipation [1.16] may include the juxtaposition of a large number of “active cells” and “inactive cells” inside the global granular mass, and these “cells” may be ordered in organized structures. In these allowed solutions, the following can be found:

• – motions in which large masses behave like rigid bodies, separated by smaller zones concentrating strain motions, well known in geomechanics as “strain localization zones” or “shear zones” or “sliding surfaces”;
• – motions in which a large granular mass in the movement may carry along some lumps which behave like solid bodies, this kind of motion is commonly observed in large landslides.

The second key property is due to the fact that in a situation with 0 < R(A) < 1, on the one hand, the tensor P(A) displays an output power P⁻(A) ≠ 0, and, on the other hand, its input power P⁺(A) is larger than the value required in a minimal dissipation mode: .

These particularities always allow splitting of the tensor P(A) (with 0 < R(A) < 1) into two coaxial tensors:

• – one tensor in minimal dissipation P_m(A) bearing the totality of the output power of P(A), and with an input power tuned in proportion;
• – the other tensor in maximal dissipation P_M(A) bearing only the complement of input power of P(A):

A geometrical representation of the set of general solutions of the tensor dissipation equation [1.16] may be provided in the natural basis, with figuration of normalized tensors , on the unit ball of the norm N, which is an octahedron (Figure 1.6):

• – the set of minimal dissipation solutions (case with R(A) = 0, section 1.2.5) is a closed polygonal line, resulting from the intersection of the octahedron with the plane Tr{p} = sinφ_μ , this polygon is a hexagon, symmetric but not regular, whose small sides correspond to Mode I, and the large sides correspond to Mode II, the vertices represent the border modes in plane strain;
• – the set of maximal dissipation solutions (the case with R(A) = 1, section 1.2.6) is the face (+,+,+) of the octahedron, including the edges and vertices on its border;
• – the set of intermediate solutions (cases with 0 < R(A) < 1, section 1.2.7) is the part of the octahedron comprised between the line of minimal dissipation and the face of maximal dissipation;
• – all the parts of the octahedron situated below the plane Tr{p} = sinφ_μ (below the minimal dissipation line) are excluded;
• – the property of splitting any solution into two tensor components, provided above, is a result of the fact that the set of all solutions of general dissipation equation with 0 ≤ R(A) ≤ 1 is a convex set, whose minimal dissipation solutions (for R(A) = 0) constitute the edge.

Other particularities of the set of general solutions of equation [1.16] may be found in [FRO 01].

Figure 1.6. Geometrical representation of the set of tensor solutions of general dissipation equation

1.2.8. Practical situations: theoretical and practical minimum dissipation rule

The theoretical minimal dissipation solutions found above with no internal feedback interaction between neighboring moving contacts and consequently R(A) = 0 (section 1.2.5) mean a complete and high polarization of sliding contact motions distribution. However, experimental results, physical and numerical, suggest that in real granular geomaterials, a small part of disorder is necessary to secure kinematic compatibility of movements, resulting in a somewhat fuzzy polarization of active contacts, and some internal feedback interaction between neighboring moving contacts, and consequently R(A) that is small to moderate but not null. The theoretical minimal dissipation solutions found above then appear rather ideal or asymptotic situations that the dissipative process tends to reach, but being prevented to achieve completely the corresponding features, because of internal constraints within the material. The minimum dissipation rule then becomes, in practical situations, a minimum internal feedback rule, and these more real situations, with R(A) being small and not null, deserve to be analyzed as “near minimum solutions”.

1.2.9. Practical situations: the apparent inter-granular friction

In section 6.2 of Chapter 6, it is shown that under regular boundary conditions and cautious experimental procedures, the ratio of dissipation rate relative to the norm of mechanical energy exchanges between the granular mass and the outside, remains remarkably constant, under a large spectrum of testing conditions, independent of material compacity.

From the dissipation equation [1.16], this ratio appears as follows:

Therefore, the “apparent inter-granular friction” is denoted by and the apparent inter-granular friction coefficient is defined as follows:

[1.19]

The dissipation equation [1.16] for the granular mass in motion now becomes

[1.20]

This relation is formally identical to the one corresponding to the elementary contact [1.6]; however, the mineral inter-particle contact friction φ_μ is now replaced by the apparent inter-granular friction φ_µ^*, which includes the population effect R(A), resulting from the internal feedback interactions between neighboring active contacts within the granular mass in motion.

In Chapters 2 to 8, except otherwise stated, this apparent inter-granular friction coefficient will be considered as a material constant parameter attached to “near minimum solutions”, as shown by a wide set of experimental data in Chapter 6.

1.3. Macroscopic scale: the equivalent pseudo-continuum

1.3.1. Previous works on a tensor formulation of energy dissipation

Experimental results have shown that the macroscopic behavior of granular media verifies an energy-dissipation relation, provided that boundary conditions are sufficiently regular, such as the ones applied during triaxial or plane strain tests, e.g. [FRO 79, FRO 83, FRO 86]. This experimental relation links the eigenvalues of the experimental average Eulerian effective stress and strain rate tensors,³ σ and usually assumed to be coaxial – together with a material constant, interpreted as an apparent friction ψ^*

[1.21]

In this experimental energy-dissipation relation, the coefficient sin ψ^* appears as a material constant parameter, intrinsic to the material tested, quite independent of experimental conditions and, in particular, independent of variations in compacity; for these reasons, it has been interpreted as a coefficient of apparent friction between particles.

This experimental energy-dissipation relation is remarkably similar to the dissipation relations derived in previous sections for the discontinuous granular mass with a work rate of internal forces on the left-hand side and a sum of absolute values on the right-hand side – which suggests the presence of our octahedral norm N – multiplied by a material coefficient. Thus, the experimental relation [1.21] appears as an objective constitutive relation linked with an intrinsic material parameter.

It has also been shown [FRO 83] that equation [1.21] can be written with the sole eigenvalues of a certain tensor π, resulting from the contracted symmetric product of the pseudo-continuum internal forces σ and internal movements . Tensor π is a second-order symmetrical tensor, representing the internal actions for the equivalent continuous medium, and its trace corresponds to the work rate of the internal forces in the pseudo-continuum (see Figure 1.1); moreover, if σ and are coaxial, their common eigendirections are also eigendirections for tensor π

[1.22]

By this definition, the experimental dissipation relation [1.21], in its local form, can be written as follows:

[1.23]

The striking formal identity between above relation [1.23] and dissipation equation [1.20] may be observed, with the expression of energy dissipation within the discontinuous granular mass.

1.3.2. Correspondence between equivalent pseudo-continuum and discontinuous granular mass

By comparing the discontinuous granular mass A with its representation by an equivalent pseudo-continuum V(A), it appears that, under our assumptions, the work rate produced by the internal forces within the pseudo-continuum is equal to the work rate developed within the discontinuous medium it represents

[1.24]

Comparing the internal actions P(A) and π, it must be verified that the internal forces and movements, which have been taken into account within the discontinuous medium, correspond to all the observable macroscopic internal forces and movements when considering the equivalent pseudo-continuum.

For the internal forces, the Love–Weber relation [LOV 27, WEB 66] links, without ambiguity, the internal forces and geometrical arrangement of the discontinuous granular mass to the Eulerian stress tensor σ.

For the internal movements, the subject is not so straightforward, as there is no general relation such as the Love–Weber relation for internal forces. From the author’s point of view, the identity between internal movements requires the following two conditions:

• – the macroscopic deformations in the pseudo-continuum are actually due to inter-granular contact sliding within the discontinuous granular mass;
• – the component of the macroscopic movements due to eventual rolling or spinning movements at the grain scale can be neglected, i.e. there is no significant “roller bearing” motions of the grains within the granular mass, which would add a significant component to macroscopic strains, with a negligible contribution to dissipation.

Detailed kinematic analyses of 2D numerical simulations by the discrete element method [NOU 05] (see Chapter 6, section 6.5) have shown that these assumptions are realistic as long as the grain shapes are sufficiently irregular, such as random irregular shaped convex polygons, but no longer realistic if the grain shapes are perfectly regular and smooth, such as circular disks. For the latter perfectly regular and smooth grain shapes, it was shown that rolling motions with negligible participation to energy dissipation could represent as much as about 40% of the macroscopic strains. However, random irregularly shaped convex shapes do correspond to the granular media used in civil engineering, which is not the case of perfectly smooth, regular, and symmetric shapes such as 2D disks or spheres in 3D, often used in numerical simulations but not truly representative of granular geomaterials used in civil engineering.

Therefore, the conditions that correspond to our key assumptions result in a complete equality between the average values of the internal actions within the discontinuous granular mass, and its corresponding equivalent pseudo-continuum is as follows:

[1.25]

As a result, the internal actions within the equivalent pseudo-continuous medium follow, on an average, the same dissipation relation as the internal actions within the discontinuous medium. Therefore, the phenomenological relations [1.21] and [1.23] found for the equivalent pseudo-continuous medium can be seen as a direct explicit consequence at macroscale of the energy dissipation due to friction within the discontinuous granular mass.

In this framework, other key elements of this correspondence between the equivalent pseudo-continuum and the granular mass do link the stress tensor and the strain rate tensor to local contact forces, sliding velocities, and the internal geometry of the granular mass, following the notations for the granular mass in section 1.2.1:

• – The average tensor of internal actions is
  [1.26]
• – The stress tensor, by considering an arbitrary material point Gn in the grain number n as the centroid to fix ideas and the corresponding “branch vectors” GnGm , is provided by the Love–Weber relation [LOV 27, WEB 66], and here provided in its symmetric form

[1.27]

For the strain rate tensor, the subject needs to be elaborated in the background of our assumptions. First, the local tensors π,σ, remain linked by relation [1.22]. Moreover, provided that the local covariances of stress and strain rates fluctuations may be neglected relative to the average value, relation [1.22] also holds for the average values over volume V (provided this volume is large enough). Then, as the stress tensor is always regular and invertible because of the non-traction condition, except near a free surface, relation [1.22] generally becomes invertible, i.e. allows us to determine , by knowing and (see Appendix A.1.2).

However, it may be noted that the validity of this expression of strain rates presented in Appendix A.1.2 is subordinate to the validity of the assumption that local covariances of stress and strain rates fluctuations may be neglected relative to the average value, as detailed in Appendix A.1.2. This assumption is stronger than the assumption strictly required for the global validity of the approach developed in this book (see Chapter 2, section 2.2.2.2), which is rather widely confirmed by experimental data.

1.3.3. The macroscopic equation of energy dissipation by contact friction

As long as the material corresponds to the above-mentioned requirements regarding internal movements, the macroscopic equation of energy dissipation by contact friction, in the equivalent pseudo-continuum representing the discontinuous granular mass, is then, in terms of average values of internal actions, represented as follows:

[1.28]

All the properties already displayed by the solutions of the energy dissipation equation for the granular mass (sections 1.2.4–1.2.9) are verified by the average values of the internal action tensor .

It may be noted that equation [1.28] is now completely independent of eventual coaxiality between stress and strain rate tensors σ and .

However, a last key issue remains to be clarified:

• – the experimental dissipation relationship [1.23] holds for tensor π computed as the symmetric product of the average stress and strain rates calculated over a volume of material equivalent to the experimental test body;
• – relationship [1.28] resulting from the transposition of granular mass dissipation holds for the average value of tensor π calculated as the symmetric product of local stress and strain rates;
• – in other words, relationship [1.23] holds for , while relationship [1.28] holds for
• – are relationships [1.23] and [1.28] actually equivalent?

In the description of pseudo-continuum with the concept of representative elementary volume (REV), if the boundary conditions are sufficiently regular as to allow the granular body to evolve close to the minimal dissipation situations, relation [1.28] will also hold for “local” values of tensor π, defining this “local” value as the average over the REV. In this framework, if the size of this REV is chosen sufficiently large, under our main assumptions (section I.2, assumption on local covariances of stress and strain rates fluctuations relative to the average value), the difference between and remains negligible in terms of dissipation equation, and the apparent friction coefficients can be considered as identical

[1.29]

Thus, the dissipation equation for the pseudo-continuum becomes

[1.30a]

This macroscopic equation of energy dissipation in the equivalent pseudo-continuum, now written with the “local” tensor values (see note in section 1.3.1), will be widely used in the following chapters. As we will focus on the motion near minimal dissipation, the material constant will be considered as < 1.

Except otherwise stated, this macroscopic equation of energy dissipation in the equivalent pseudo-continuum will be generally used in Chapters 2 to 8, with the assumption of coaxiality between stress and strain rate tensors (and then also with internal actions tensor π). In these situations, the macroscopic equation of energy dissipation is represented in the natural basis as follows:

[1.30b]

1.3.4. Coaxial situations: the six allowed strain modes and their physical meaning

In Chapters 2 to 7, considering macroscopic 3D motions near minimal dissipation in the equivalent pseudo-continuum framework with coaxiality, we will methodically investigate the solutions allowed by the dissipation equation.

Here, we consider simple coaxiality (or un-ordered coaxiality) between stress and strain rates tensors, meaning that these tensors have the same set of eigendirections, but the respective order of corresponding eigenvalues sets do not necessarily coincide. By numbering the principal directions according to principal stress ranking, σ₁ ≥ σ₂ ≥ σ₃ > 0 (all normal stress shall be compressive), the analysis of the solutions of dissipation equation [1.30b] begins by the relevant nomenclature of this set of allowed solutions, based on the signature of the strain rates eigenvalues .

Theoretically, this signature has 2³ = 8 possibilities.

However, both combinations (+,+,+) and (−,−,−) will be discarded – the first one, because , and the second one, because of the positive dissipation rate condition (the trace of internal actions remains positive). Therefore, the remaining six allowed strain modes, corresponding to six tensor zones of strain rates signature, are listed in Table 1.1 and represented in Figure 1.7 in the octahedral plane, together with their micro-scale polarization patterns in the corresponding contact motion distributions. These six allowed 3D strain modes are separated by border modes in plane strain (see Figure 1.7).

Table 1.1. The six allowed strain mode situations

Mode designation	Signature of eigenvalues set	Comments
Mode I Direct	+,−,−	Ordered coaxiality (in part)
Mode I Transverse	−,+,−	—
Mode I Reverse	−,−,+	—
Mode II Direct	+,+,−	Ordered coaxiality (in part)
Mode II Transverse	+,−,+	—
Mode II Reverse	−,+,+	—

These six allowed strain modes cover the set of all the solutions to the dissipation equation [1.30b], independent of other specific restrictions such as boundary conditions. However, boundary conditions may prevent the occurrence of certain strain modes, according to their particularities, as will be seen later.

It may be noted that in Table 1.1, only Mode I Direct and Mode II Direct can achieve a fully ordered coaxial 3D situation between stress and strain rates, i.e. complying simultaneously with both conditions σ₁ ≥ σ₂ ≥ σ₃ and (with our conventions of the sign, compressive stress and contraction strain were noted as positive). The other allowed strain modes, complying only with un-ordered coaxiality, will be shown in Chapter 5 to correspond to what can be observed after a reversal in motion, with a change in the sign of part of the principal strain rates.

This means that the dissipation equation [1.30] provides a unified 3D description of what is usually called the “loading–unloading” behavior in elastoplastic approaches of the granular material behavior, without requiring a different behavior formulation for “loading” motion and for “unloading” motion.

Here, a reversal in motion is seen as a simple switch between different allowed strain modes obeying the same behavior equation, because their characteristics result from the same energy dissipation by contact friction, although their corresponding algebraic formulation of dissipation relation will change from one strain mode to another because of the absolute values in equation [1.30]. Experimental validations detailed in Chapter 6, including cyclic shear tests with motion reversals under various configurations (“triaxial” axisymmetric stress, torsional shear on hollow cylinder tests, and true 3D compression apparatus) do clearly confirm this point of view.

This particular feature can be linked to the concept of “tensor zone”, developed earlier and implemented on a phenomenological basis in “incremental constitutive models” of the mechanical behavior [DAR 85], with formulations depending on the signature of strain rates. However, these “tensor zones” represented by our six allowed strain modes, do appear here naturally as a specific consequence of tensor structures induced by energy dissipation by contact friction.

Of course, as this energy dissipation remains irreversible, the stress–strain path forecast by relation [1.30] after motion reversal may not be the exact reverse of the stress–strain path before the motion reversal, the difference is provided by the absolute values in relation [1.30] brought by our norm N from basic friction dissipation.

Figure 1.7. Mapping of Table 1.1 strain modes in the octahedral plane. (a) Layout in the octahedral plane. (b) Layout in the angular positioning diagram, with corresponding micro-scale polarization characteristics (contact motion hodographs)

Specific characteristics attached to motion reversals provided by the dissipation equation [1.30] will be detailed in Chapters 5 and 7. Moreover, experimental data for cyclic shear tests will be discussed in Chapter 6, including large amplitude strain cycles in a true 3D compression apparatus corresponding to cycling between motions in Mode I Direct and Mode II Reverse of Table 1.1, evidencing that the specific characteristics forecast by equation [1.30] for such motion reversals, do effectively show up.

¹ This norm, also known as “Manhattan or Taxicab norm”, belongs to the mathematical family of p-norms including also the Euclidian norm and the Supremum norm. However, unlike the Euclidian norm, our octahedral norm is a piece-wise linear function, each linearity domain corresponds to one face of its unit ball, being a regular octahedron. This piece-wise linearity will turn out to be a key property when dealing with the pseudo-continuum heterogeneous mechanical behavior like shear banding (Chapters 2 and 3).

² Note that Figure 1.3 shows that this internal feedback is a mutual interaction: if (c₁) is feeding back (c₂), then (c₂) is also feeding back (c₁), although with a different intensity. It can be shown that this apparent reciprocity in this interaction is verified in two dimensions, as shown in Figure 1.3, but it is not true in three dimensions.

³ Here, we define the “local” values of stress and strain rate tensors, as the average values over a volume of material equivalent to the experimental test body, so equation [1.21] holds for such “local” values of stress and strain rate tensors σ and .