Indetermination in Granular Mechanics

Transcription

1 Indetermination in Granular Mechanics J. J. Moreau Laboratoire de Mécanique et Génie Civil CNRS-Université Montpellier II, France Workshop: Discrete numerical simulations of granular materials Institut Henri Poincaré, Paris, March 16th 18th 2005.

2 Basic example A rigid rod is inserted with friction between two fixed walls slightly converging upwards. Depending on the intensity of wedging, it may remain at rest or fall down under gravity.

3 In case of equilibrium under gravity an infinity of values are possible for the contact forces. They don t make an affine manifold in R 4, i.e. the standard discussion of hyperstaticity does not apply.

4 In case of equilibrium under gravity an infinity of values are possible for the contact forces. They don t make an affine manifold in R 4, i.e. the standard discussion of hyperstaticity does not apply. Instead of linear spaces, Unilateral Mechanics is dominated by convex cones For x and x in C, for λ and λ in ]0, + [ : λx + λ x C (but not necessarily if λ or λ are negative)

5 The natural measure of the intensity of wedging consists of the values of the contact forces. Knowing these initial values here allows one to predict motion unambiguously.

6 The natural measure of the intensity of wedging consists of the values of the contact forces. Knowing these initial values here allows one to predict motion unambiguously. In contrast with the 19th century conception of mechanical determinism, according to which initial data should consist only of positions and velocities, we propose to consider contact forces as part of the description of each state.

7 Using a richer model, viz. taking into account some microscopic longitudinal elasticity of the rod, doesn t essentially change the scheme of information processing as far as only instant t 0 is considered.

8 Using a richer model, viz. taking into account some microscopic longitudinal elasticity of the rod, doesn t essentially change the scheme of information processing as far as only instant t 0 is considered. In fact, the longitudinal deformation then becomes part of the description of each position. And the given elasticity law connects it with contact forces.

9 But the microelastic model may also be used to handle hysteretic information. Assume that a recording of the the active forces (gravity and action of an operator) exerted to the rod prior to t 0 is available. By taking into account infinitesimal motions of contacts, this allows one to reconstruct the evolution of contacts forces and finally determine their values at instant t 0.

10 But the microelastic model may also be used to handle hysteretic information. Assume that a recording of the the active forces (gravity and action of an operator) exerted to the rod prior to t 0 is available. By taking into account infinitesimal motions of contacts, this allows one to reconstruct the evolution of contacts forces and finally determine their values at instant t 0. The possibility of hysteretic analysis does not preclude the existence of variables which, at time t 0, convey a summary of history sufficient for predicting further evolution.

11 What about computation? Two time-stepping numerical strategies are commonly used in the statics or the dynamics of multicontact systems.

12 Molecular Dynamics Non-interpenetrability is approximated by steep repulsion forces which enter into action when two bodies come close to each other. Hence the interaction forces (also the frictional ones) are connected with fictitious deviations passed from step to step. Such a numerical scheme complies with the proposition of including these forces in the description of each state.

13 Contact Dynamics Here the essential nonsmoothness of unilateral constraints and dry friction is faced without any regularizing alteration. The velocity function t u R d is evaluated from step to step through an implicit integration scheme. The position function t q is simply updated at each step.

14 Contact Dynamics Here the essential nonsmoothness of unilateral constraints and dry friction is faced without any regularizing alteration. The velocity function t u R d is evaluated from step to step through an implicit integration scheme. The position function t q is simply updated at each step. Numerical stability is automatically secured, allowing for much larger time-steps than in MD. But each step of the implicit scheme requires solving a heavy nonsmooth Core Problem, the unknowns of which are the contact forces (or impulsions) and the update of u (zero in case of equilibrium).

15 Various strategies are available to solve the Core Problem.

16 Various strategies are available to solve the Core Problem. The most common method consists of an iterative procedure à la Gauss-Seidel. One ranges cyclically through the contacts detected as active. For each of them, one solves a single-contact dynamical problem with all other contact forces treated as known. This yields some updates of the velocity u R d and of the local contact force. Cyclic review is continued until a convergence criterion is fulfilled.

17 The iterative procedure needs to start from an initial guess of the contact forces. This may consist of zero values, but, when computing the evolution of dense assemblies, one considerably accelerates convergence by taking as initial guess the values calculated for contact forces at the antecedent time-step (for the contacts which were already active).

18 The iterative procedure needs to start from an initial guess of the contact forces. This may consist of zero values, but, when computing the evolution of dense assemblies, one considerably accelerates convergence by taking as initial guess the values calculated for contact forces at the antecedent time-step (for the contacts which were already active). If this is done, CD computation complies with the recommendation of treating contact forces as part of state description.

19 Otherwise... the iterative procedure allows one to explore indetermination by surveying the whole set of the solutions to the Core Problem of a selected time-step. 1st technique: Repeat the iterative procedure a large number of times, with initial guesses drawn at random from a plausible set. 2nd technique: From some fixed initial guess, repeat the iterative procedure by changing at random the ordering of the cyclic review of contacts.

20 Equilibrium of a system of b two-dimensional bodies exhibiting κ points of frictional contact. The collection of contact forces evaluates as a point r in R 2κ

21 Equilibrium of a system of b two-dimensional bodies exhibiting κ points of frictional contact. The collection of contact forces evaluates as a point r in R 2κ which has to verify: The conditions of static Coulomb friction r C C: a polyhedral cone in R 2κ, the Cartesian product of κ angular regions of the respective copies of R 2. The 3b conditions of equilibrium under the specified loads, defining an affine manifold A with dimension 2κ 3b. The set of the solutions equals the (hyper)polyhedron C A

22 Collection of circular objects at rest under gravity.

23 Four other contacts

24 Two ways of repeating the Gauss-Seidel procedure: On the left: The ordering of contact review is changed at random. On the right: Fixed ordering, but initial guess chosen at random. Same 2D sets, but different ghost views of edges of the 2κ-dimensional polyhedron.

25 In dynamical evolutions the unknowns of the Core Problem are r R 2κ and the velocity update u R 3b. In the event of a sliding contact, Coulomb s law comes to restrain the contact force more tightly than in static situations. The solution set is not expected to be convex anymore.

26 Accelerated granular flow on a slope Slope 26 ; grain to grain friction 0.3; ground friction 0.5; restitution 0.

27 Dispersion of acceleration vectors

28 Detail of the dispersion of acceleration

29 Dispersion of contact forces

30 2D model of a grain pile

31 Pile is created by pouring grains from a localized source onto some rough horizontal ground. It has been experimentally found that the distribution of ground pressure doesn t relate to the height of material above. A local minimum of pressure may even be found at the vertical of the apex.

32 This necessarily means that some central portion of the pile has part of its weight supported by arching actions from the surrounding bank. In this 2D simulation, a vertical cut is drawn through the granular matter. The programme calculates the total force transmitted across, i.e. the resultant vector of the contact forces exerted by grains with centers on one side upon antagonists on the other side.

33 At each time-step, when pile construction is numerically simulated, the message from history consists of two parts: the geometric arrangement of grains the contact forces, as calculated in the antecedent step If the latter are not used to initialize Gauss-Seidel iterations, the calculated transmitted force is affected by the undeterminacy shown above as a cloud.

34 This example was meant to show that, in situations of this sort, the main part of the message from history is conveyed by the geometric arrangement. The small size of the cloud implies that the indeterminacy arising from ignoring antecedent contact forces doesn t gravely impair the demonstration of arching.

35 This example was meant to show that, in situations of this sort, the main part of the message from history is conveyed by the geometric arrangement. The small size of the cloud implies that the indeterminacy arising from ignoring antecedent contact forces doesn t gravely impair the demonstration of arching. Actually varying slightly the cut position may produce changes in transmitted force of larger amplitude than the above cloud. In the example only 10 contacts are involved in the force transmission across the cut.

36 All this refers to the time-stepping approximation. of evolution problems.

37 All this refers to the time-stepping approximation. of evolution problems. Coming to the theoretical formulation of these problems, it doesn t make sense to distinguish input and output values of contact forces. What the preceding reflects is an incremental version of the law of dry friction.

38 What if friction is zero? A conjecture (J.-N. Roux, C. Moukarzel): Generically, a collection of N frictionless round objects in equilibrium is isostatic.

39 What if friction is zero? A conjecture (J.-N. Roux, C. Moukarzel): Generically, a collection of N frictionless round objects in equilibrium is isostatic. In 2D models, a frictionless disk has two effective degrees of freedom. Hence a collection of N disks with κ contacts obeys 2N independent equilibrium equations involving κ unknown normal contact forces. Isostaticity requires κ = 2N

40 A misleading example: Hierarchical deposition, a geometric procedure for constructing sequentially, in 2D, an assembly of contacting disks. Each added disk touches two precedingly deposited ones, hence κ = 2N. Values of frictionless contact forces securing the equilibrium of the whole under given loads are uniquely determined through sequential calculation. But they have not necessarily the directions allowed by unilaterality.

41

42 hierachical deposit (31 disks, 62 contacts) frictionless equilibrium under gravity (same)

43 In view of translational invariance, the effective number of DOF equals 67

44 As a rigid body in space, the table possesses 6 DOF; its position relative to ground is invariant under displacements parallel to this plane. Hence the effective number of DOF equals 3. The above case of the table having exactly 4 contacts with ground is exceptional.

45 Proposed explanation:

46

47 However there may exist loose grains which make equality κ = 2N not valid anymore. This is related to some of the functions f i being non-strictly convex. Should the exact orthogonality of gravity to ground be considered as exceptional?

48 Indetermination due to dry friction Halsey, T. C. & Erta, D., A ball in a groove, Phys. Rev. Letters, 83, , Moreau, J. J., Indétermination liée au frottement sec dans le calcul des granulats, in (M. Potier-Ferry, M. Bonnet, A. Bignonnet, coordonnateurs), Actes du 6ème Colloque National en Calcul des Structures, 2003, vol. 3, pp Unger, T., Kertész, J. & Wolf, D.E., Force indeterminacy in the jammed state of hard disks, ArXiv:cond-mat/ , 2004.

49 Moreau J. J., Indetermination due to dry friction in multibody dynamics. In (P. Neittaanmäki et al., eds.) European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004), Jyväskylä, Finland, July 2004, vol. 2, CD ROM, ISBN

50 On the CD method (and the example of grain piles) J. J. Moreau, Modélisation et simulation de matériaux granulaires, in: (B. Mohammadi, ed.) Actes du 35e Congrès National d Analyse Numérique, Montpellier, 2-6 juin 2003, 30 pages (CD ROM)., An introduction to unilateral dynamics, in: (M. Frémond & F. Maceri, eds.) Novel approaches in Civil Engineering, Lecture Notes in Applied and Computational Mechanics, vol. 14, Springer-Verlag, 2004, pp

51 Isostaticity in frictionless assemblies Moukarzel, C., Response functions in isostatic packings, in (H. Hinrichsen, D. Wolf eds)the physics of granular media, Wiley-Vch, Berlin, 2005 Roux, J.-N., Geometric origin of mechanical properties of granular materials, Phys. Rev. E, 61 (6), , 2000.