Bending transition in the penetration of a flexible intruder in a 2D dense granular medium
We study the quasi-static penetration of a flexible beam in a two-dimensional dense granular medium lying on an horizontal plate. Rather than a buckling-like behavior we observe a transition between a regime of crack-like penetration in which the fiber only shows small fluctuations around a stable straight geometry and a bending regime in which the fiber fully bends and advances through series of loading/unloading steps. We show that the shape reconfiguration of the fiber is controlled by a single non dimensional parameter: , the ratio of the length of the flexible beam to , a bending elasto-granular length scale that depends on the rigidity of the fiber and on the departure from the jamming packing fraction of the granular medium. We show moreover that the dynamics of the bending transition in the course of the penetration experiment is gradual and is accompanied by a symmetry breaking of the granular packing fraction in the vicinity of the fiber. Together with the progressive bending of the fiber, a cavity grows downstream of the fiber and the accumulation of grains upstream of the fiber leads to the development of a jammed cluster of grains. We discuss our experimental results in the framework of a simple model of bending-induced compaction and we show that the rate of the bending transition only depends on the control parameter .
Slender structures are extremely flexible and get easily unstable. The ratio of the bending stiffness of a beam to its axial stiffness in tension or compression scales with , the square of the ratio of the thickness to the length of the beam Audoly and Pomeau (2010). Similarly, over a threshold compressive strain of order , the beam undergoes a buckling instability Bazant and Cedolin (2010). As a consequence of their high geometrical aspect ratio, slender structures can be sensitive to low forces of various origins and exhibit complex mechanical behaviors due to the associated couplings. This is in particular the case of the elasto-capillary problems (interactions between slender structures and capillary forces) which have raised a growing interest in the recent years Roman and Bico (2010).
In the same spirit, the effect of flexibility on fluid-structure interactions has recently motivated a growing number of studies at low Shelley and Zhang (2011); Lindner et al. (2017) or high Reynolds number Ramananarivo et al. (2011). Model experiments of reconfigurations have been performed with flexible fibers (1D), plates (2D) or assembly of plates organized in a circular pattern (3D) that were placed initially perpendicular to the incoming flow of air Gosselin et al. (2010), water Schouveiler and Boudaoud (2006) or 2D soap film Alben et al. (2002). In all these cases, the flexible body experiences shape reconfiguration with self-streamlining and reduction of the surface area exposed to the flow, thus resulting in a drastic reduction of the drag force exerted on the object Vogel (1994). An immediate field of application of these problems can be found in bio-physical domains, when a passive or an active elastic part of a body interacts with the flow Coq et al. (2008); Ramananarivo et al. (2011); Xiao and Chen (2013); Jawed et al. (2015). In particular, in animal locomotion, the flexibility of wings or fins intervene in the flying of birds and insects or the swimming of fishes or eels. Even in micro-organisms the propulsion through flagella takes advantage of the flexibility. Numerous examples can also be found in the plant domain de Langre et al. (2012) where the flexibility of cereal stems, tree’s trunk, branches or leaves can be beneficial under wind flow for reducing lodging. Even marine algae and plants in aquatic canopies adapt their shape under water current Luhar and Nepf (2011); Barsu et al. (2016).
Here we consider an original case of fluid structure interaction between a slender structure and a granular medium. More specifically, we study the quasi-static penetration of a flexible beam in a two-dimensional dense bidisperse mixture of discs lying on a plate. Applications of this problem range from nuclear engineering Buster et al. (2016) to biology: growth of roots Kolb et al. (2012); Wendell et al. (2012) in structured soils or locomotion of worms or sandfish lizards in granular media Hosoi and Goldman (2015).
The complex behavior of granular flows has recently been studied through its interaction with rigid intruders Albert et al. (1999, 2001); Stone et al. (2004); Candelier and Dauchot (2009, 2010); Fiege et al. (2012); Kolb et al. (2013, 2014); Seguin et al. (2013, 2016); Clark et al. (2016); de la Cruz and Caballero-Robledo (2016); Merceron et al. (2016). The high sensitivity of flexible intruders to low forces opens here a promising new way of probing the mechanical behavior of granular media Mojdehi et al. (2016); Schunter Jr. et al. (2017); Algarra et al. (2017). The complexity of such a question of elastogranular mechanics (as recently coined in Ref. Schunter Jr. et al. (2017)) will naturally arise from the coupling of the low rigidity of the elastic beam with the discrete nature of the granular flow, its non-local rheology Bouzid et al. (2015) and the emergence of turbulent-like fluctuations Radjai and Roux (2002); Combe et al. (2013); Saitoh and Mizuno (2016) in the vicinity of the jamming transition.
In contrast with two recent works focusing on buckling Mojdehi et al. (2016); Schunter Jr. et al. (2017), the present study will highlight a bending transition of the flexible intruder and an associated transition in the structure of the granular medium.
In the following we first give in section II a brief description of the experimental set-up and tools of analysis; we then discuss in section III the phenomenology of the interaction between an elastic beam and a granular flow, we report our experimental observation of bistability between two different regimes (straight or bent fiber), we identify the control parameters (rigidity of the fiber and packing fraction of the granular medium) and propose a simple model ; in section IV we give a specific focus on the development of the bending transition and show that it is associated with a clear symmetry breaking of both the geometry of the fiber and the density of the granular material in the vicinity of the fiber; we then propose in section V a model of bending induced compaction that allows us to reasonably reproduce the evolution of the fiber deflection in the course of the bending transition; a summary of our main results is finally given in section VI.
Ii Experimental set-up
ii.1 The granular medium
The 2D granular material is a bidisperse mixture of brass cylinders, in equal mass proportion, forming a dense and disordered assembly of rigid disks Kolb et al. (2013, 2014)). The two kinds of cylinders of respective outer diameters = 4 mm (in number ) and = 5 mm (in number ) lie on a horizontal glass plate delimited by four brass walls forming a rectangular frame of width = 269.5 mm (54 along the axis) and adjustable length along the axis (Fig. 1). The total number of cylinders is kept constant around 6800, but as we varied in the range 457.5 mm (91 ) to 470.5 mm (94 ), we could adjust the total available cell surface and therefore the packing fraction . This packing fraction has been chosen in a small range just below the assumed jamming packing fraction for our 2D granular medium, =83.56% Kolb et al. (2013).
ii.2 The flexible beam
The flexible beam is cut into a mylar sheet of thickness =350 m, Young modulus =3.8 GPa and Poisson ratio =0.4 Boudaoud et al. (2000). The flexible part of the intruder is a long beam, that we will call further fiber, of length with a rectangular cross-section defined by the sheet thickness and a height = 3 mm corresponding to the height of the cylinders of the granular material. One of the intruder extremity is clamped into a rigid metallic part supported by a transverse arm along the plane (Fig. 1).
The anchoring point of the intruder is fixed in the laboratory frame and is located at an equal distance of the two lateral walls delimiting the space for the granular layer. The basis of the fiber is fixed slightly above the bottom glass plate, so that there is no friction with the bottom. The measurement of the grain-grain friction coefficient gives =0.32, while the one for the frictional contact of grains with the glass bottom leads to =0.490.09. At the beginning of the experiment, the fiber is straight along the axis with no contact with neighboring grains. The anchoring point of the fiber is initially located at a distance of 80 mm=16 (or 120 mm=24 depending on the batch of experiments) from the back wall, to avoid boundary effects.
ii.3 Principle of the experiment
A typical experiment consists in translating the granular material supported by the glass plate along the axis against the free extremity of the fiber. The plate velocity is held constant. In the plate frame this is equivalent of having a fiber penetrating the granular material at a constant velocity . The experiment is stopped after a plate displacement of typically 260 mm (52 ).
In this work, we used different packing fractions ranging from =79.0% to 83.1% and different fiber lengths between =1 cm and =5 cm. The velocity of the plate was always kept constant at =5/6 mm/s. This value corresponds to a quasi-static regime in which the forces are dominated by frictional contact forces. Actually we observe that the drag force experienced by a rigid cylindrical intruder does not depend on velocity in these ranges of values. A CCD camera of 1600*1200 pixels placed above the set-up records images at a frequency of 2 Hz, corresponding to a plate displacement of =5/12 mm= between 2 successive images. Ten similar experiments are performed for each set of parameters and , and grains are carefully remixed between two consecutive experiments and fiber checked and replaced if necessary.
ii.4 Segmentation and image processing
For all images of each batch of experiments, a technique of correlation on gray levels has been used for determining the grain centers with a sub-pixel accuracy. Then the grains displacements from one image to the next one have been computed. Furthermore, the full deflected shape of the fiber is obtained for most images. The delicate challenge for image analysis was to locate the very thin and elongated fiber amongst the circular grains surrounding it. For this purpose, the gray-level image has been convoluted with different filters to enhance the contrast of the fiber and disconnect it from the neighboring grains. This first step gave a skeleton of the fiber, which was further fully reconstructed by starting from the fiber anchoring point. From the fiber shape, different informations can be obtained like the free end lateral deflection or the local slope of the fiber relative to the axis. This local slope can be computed as a function of the curvilinear abscissa (normalized by the length of the fiber), with =0 at the clamped extremity and =1 at the free extremity.
Iii Phenomenology: fluid-structure interaction in a granular flow
When immersed in a fluid, a flexible structure interacts with the flow. In particular, depending on the rigidity of the structure and the flow conditions, it can switch between two stable states: the initial undeformed configuration and a deformed configuration that reduces the drag force. Alternatively, for a given bending rigidity, there exists a critical length of the structure placed across the flow above which reconfiguration occurs.
A very similar phenomenology can be observed in the present experiment which can be regarded as an interaction between a flexible beam and a granular flow. In Fig. 2a we show a series of configurations of a short (rigid) fiber ( cm) facing the granular flow. Small fluctuations are observed around the stable straight configuration, as if the fiber was jiggling around its initial position. The slight deflections are here caused by collisions with incoming grains.
In Fig. 2b we show the contrasting case of a slightly longer fiber ( cm) for the same granular packing fraction as before. In this case, the fiber experiences reconfiguration due to the flow and enters into a regime of bending.
When the fiber has been highly tilted, fluctuations of the fiber positions are again observed but of a very different nature: in Fig. 3a we show the gradual bending of the deformed fiber due to an accumulation of incoming grains and in Fig. 3b its sudden partial recovery. In Fig. 4, we represent the corresponding grain displacements in the plate frame (by subtracting the plate displacement between 2 consecutive images) as if the fiber was penetrating the granular material. Fig. 4a shows the displacement field between the two images just before the unloading event. It is characterized by longe range displacements in front of the fiber and small recirculations on both sides. Fig. 4b shows the displacement field corresponding to the unloading phase, with the fiber returning from position B to position C. In contrast with the loading phase, the elastic return of the fiber is associated to huge recirculations on both sides. Therefore the sudden change of the fiber conformation can be associated to a large reconfiguration of the grains, in other words a burst of the velocity field of the granular flow. Such an alternation of smooth and regular loading stages interspersed with sudden unloading events is typical of granular avalanches and can also be associated to the turbulent-like velocity fluctuations reported in dense granular flows Radjai and Roux (2002); Combe et al. (2013); Saitoh and Mizuno (2016).
In Fig. 5 we summarize our observations of the straight (jiggling regime) or deformed conformation (bending regime) of the fiber when varying its length as well as the packing fraction of the granular medium.
In the present experiment, the flexible structure is initially facing the flow in a geometry of penetration. Therefore one would naturally introduce the critical force for fiber’s buckling. In the case of a non embedded fiber, this force is given by the Euler force with clamped-free extremities. It drastically depends on the fiber length as:
where is the quadratic moment of the fiber corresponding to a non embedded fiber buckling in the plane. Taking into account the rectangular cross section of the fiber yields . Thus varying is a way to vary the maximum acceptable loading force (or equivalently the maximum penetrative force) in a case of a pure axial loading. For example, the typical bending rigidity for a beam of length =3 cm and thickness =350 m is and the corresponding Euler force would be 0.11 N. For comparison, the bending force produced by the fiber tip displacement on one grain size (the typical size of obstacle encountered by the fiber) is:
Thus the bending force will be N for =3 cm, which is much smaller than the corresponding buckling force for the same geometrical parameters of the fiber. In any cases, the thickness of the fiber is small compared with the diameter of a grain with , such that the probability of a force purely acting along the axis of the beam will be negligible. Therefore the main mechanism producing reconfiguration of the fiber will be attributed to bending and not buckling, contrary to the recent experiment of Mojdehi et al. (2016).
In studies of interactions between fluids and flexible structures it is customary to design a non dimensional number that compares drag and elastic restoring forces in order to determine the stability of the flexible object Lindner et al. (2017); Gosselin et al. (2010). In the present context, a first naive scaling analysis may consist in building such a Cauchy-like number from the comparison between the friction force associated to the contact between the bottom plate and an individual grain and the bending force associated to a deflection of the tip of the fiber equivalent to one grain size , i.e. :
In this analysis that corresponds to a very dilute gas-like regime where the flexible beam only experiences collisions with isolated grains, the transition between straight and deformed configurations of the beam would occur at for a critical length:
With the experimental parameters used in the present set-up, we get cm, i.e. below this length scale the flexible intruder should not experience bending reconfiguration.
However, as shown in Figs. 2-3, fully bent configurations were actually observed for much shorter fibers (L = 3 cm) than . This diluted regime argument thus underestimates the drag force exerted by the granular flow. In order to account for the effect of a denser liquid-like flow, one can consider the friction force associated to the grains present in the area that a slightly bent beam should sweep over to recover its initial straight shape. For a tip deflection of the swept area is of the order of and the corresponding number of grains that are pushed back is where is the surface occupied by one grain on the bottom plate. The associated friction force will be . Comparing this friction force to the elastic restoring force for an homogeneous and perpendicular loading on the whole fiber leads to:
which gives now a slightly lower value of the critical length, cm, but which remains much larger than the experimental observations despite the change of scaling exponent. This means in particular that the friction forces exerted by the grains are underestimated in the present argument.
Close to the jamming packing fraction, interactions of the flexible beam with the flow are actually expected to strongly alter the structure of the granular medium Mills et al. (2008). Transient jammed clusters can thus form in the vicinity of a wall Rognon et al. (2015) or a rigid intruder Kolb et al. (2013, 2014), and more generally by the application of shear stress Bi et al. (2011).
In the present context, the elastic return of the deflected fiber to its straight conformation not only induces the displacement of the grains present in the swept over area but also the building of a jammed cluster. The area of the latter can be estimated by a simple conservation argument. Assume an initial grain packing fraction , the area of the cluster that reaches the jamming packing fraction after it has absorbed the grains swept over by the elastic fiber is:
In this close-to-jamming regime, we can thus build a Cauchy number with a drag force that results from the friction force of the jammed cluster . Here, the vicinity of the jamming density limit thus simply induces an amplification of the friction force by a factor so that we get for the Cauchy number:
and for the critical length:
The stability of the flexible beam thus appears to be controlled by two parameters: the rigidity (here the length) and the packing fraction of the granular medium. Using the value determined in Ref. Kolb et al. (2013), this leads, for an initial density to cm, a value that is more consistent with our experimental observations. In the phase diagram of Fig. 5 we see that the expression of the elasto-granular length from eq. (9) gives indeed a reasonable account of the transition between the straight or deformed conformation of the fiber.
Iv Development of the transition
In contrast to buckling instability, the switching of the flexible beam from its initial straight conformation to a bent one is gradual and requires reorganization of the granular medium. We focus here on the development of this transition.
A typical sequence (zoomed on the fiber) is shown in Fig. 6 for an intermediate fiber length cm and a granular packing fraction . The initial location of the fiber is along the vertical of the figure (-axis). In a first stage, the fiber experiences only small lateral deflections and fluctuates around its straight initial position. For example, in Fig. 6a, the fiber is slightly deflected on the left side and the lateral displacement of the free extremity is almost not visible. The granular material is still homogeneous, while grains are contacting the fiber on both sides. After a given traveling distance, the fiber bends irreversibly on one side with no possibility of returning back to its straight shape. While progressing inside the granular material, the fiber continues to bend (Fig. 6b) with a clear lateral deflection on the right side, indicated by the horizontal (red) arrow along the -axis. Associated to this deflection, we clearly observe the appearance of a cavity downstream of the fiber. As the whole granular material still flows along the axis, the fiber continues to bend and the free extremity might reach its maximum lateral deflection (Fig. 6c) when most of the fiber axis is placed perpendicularly to the initial flow. At this stage, the area of the cavity is larger. In the following we will focus on this bifurcation in the penetration behavior by introducing some quantitative descriptors.
iv.1 Deflection of the fiber tip
One way to quantitatively characterize the penetration of the fiber relative to the granular medium is to compute the lateral deflection (along the axis) of the free extremity of the fiber as a function of the distance traveled by the plate (or equivalently the displacement of the fiber anchoring point relative to the granular medium). The evolution of as a function of associated to the experiment shown in Figs. 6abc is shown in Fig. 6d.
In order to give a more systematic account of this phenomenon, we report in Fig. 7 the evolution of the deflection as a function of the penetration distance expressed in diameters of a large grain () for ten independent experiments performed for a fiber length cm and a granular packing fraction . At the end of an experiment, the fiber is observed to bend either toward the right (along ) or toward the left side (along ). In order to compare experiments, the values corresponding to experiments where the fiber eventually bends toward the left side have been inverted in Fig. 7, such that the final value of is always positive.
- In a first regime (I), the lateral deflection is observed to fluctuate around zero. The fiber continuously penetrates the granular medium and keeps in average its initially straight shape. When the fiber is contacting grains at its tip, it is slightly deflected on one side or the other but the maximal amplitude of does not exceed the radius of a grain. Thus the extremity of the fiber experiences an erratic and fluctuating motion, depending on the local arrangement of the grains encountered by the fiber. There is only few interaction between the fiber and individual grains, and the wake left behind the fiber is reduced to a small elongated cavity almost not visible.
- After a penetration distance (indicated by the vertical blue dotted line in Fig. 6d) that strongly depends on the realization, the fiber irreversibly bends towards one side (left or right). We checked the absence of bias in the direction of deflection. This regime II is characterized by a progressive and continuous bending of the fiber, shown by the regular increase of with the pentration distance . This increase appears to be quite similar for the different experiments of Fig. 7.
- A third regime appears when the extremity of the fiber roughly lies perpendicular to the direction of the plate translation , ie . Then the corresponding lateral deflection reaches its maximal value . For the examples of Fig. 7, the fibre then fluctuates around this new bended configuration. In the case of a pinned rigid fiber, the maximum possible value of the deflection would be =. In our case, the anchoring point is clamped and the fiber is flexible. Then the observed is slightly smaller than , but has a constant value independent of . For example, the average experimental values are = (0.9520.005) for =3 cm.
As it appears clearly in Fig. 7a, we observe two fluctuating regimes separated by a transition that in contrast appears to be very deterministic. This deterministic aspect appears even more clearly in Fig. 7b after a resetting of the abscissa. Starting from the observation that the maximal deflection only depends on the length of the fiber we could indeed locate the traveling distance corresponding to the deflection value , the half of the average maximal deflection for each experiment . Then we could plot in Fig. 7b the deflection as a function of the resetted penetration distance distance . The curves collapse rather well such that the average over the ten shifted experiments can be computed (Fig. 8a). Interestingly the first part of the bending regime can be fitted by an exponential growth (orange curve superimposed on the averaged experimental deflection in Fig. 8a). The same plot is presented in semi-log scales in Fig. 8b. This fit provides a characteristic length for the kinetics of bending according to the formula . For the example of Fig. 8, =(4.890.04). The length has been obtained for all batches of experiments where bending occurred. Its value normalized by the value from eq. (9) associated to the packing fraction is plotted in Fig. 13 as a function of the rescaled fiber length, that is . The value can be viewed as an order parameter: For , no irreversible bending occurs, the fiber stays in the jiggling regime, while for , bending occurs.
iv.2 Asymmetry of the packing fraction
As clearly visible in Fig. 6, when the fiber enters the deflected regime and that the lateral deflection is larger than a grain size, we observe the formation of a cavity empty of grains downstream of the fiber. A similar trend was observed in the framework of the penetration of a rigid intruder in a granular layer Kolb et al. (2013).
Conversely, we expect an accumulation of grains, more specifically the formation of a jammed cluster upstream of the fiber. Such an evolution is indeed observable in Fig. 9 where we present three successive snapshots of an experiment of penetration of a fiber of length cm in granular medium of packing fraction .
In the present experiment, we recall that the anchoring point of the flexible intruder is fixed in the laboratory frame while the granular medium lies on a glass plate that is displaced at a constant velocity with , the typical displacement of the plate between 2 successive images. The grains that accumulate upstream of the fiber are thus expected to be slowed down with respect to the mean flow. In Fig. 9, we show the distribution of the slowest (red-labeled) grains, here selected on a simple threshold displacement criterion: We labeled the grains whose amplitudes of displacement between 2 successive images in the laboratory frame are . As expected, we observe a strong clusterization of these slow grains upstream of the flexible beam. Although only qualitative, this observation supports the idea of the development of a jammed cluster of grains ahead of the fiber.
The penetration of the flexible beam in the granular medium thus leads to a clear symmetry breaking in the local packing fraction. In order to quantify this effect, we computed the coarse-grained packing fraction in the vicinity of the fiber.
Here we adapted the computation of the coarse-grained packing fraction Goldenberg et al. (2007) to define over a coarse-grained length scale a left and a right packing fraction of grains on either side of the fiber. As sketched in Fig. 10a, we define two domains and from either side of the fiber such that the distance of any point of a domain to the fiber remains below a cut-off value . The two packing fractions and are then computed as:
where is the area occupied by the grain and is its distance to the fiber. Here the parameter is the coarse-graining length and we use . The normalization factor is obtained by the expression:
where is the distance to the fiber of a point .
In Fig. 10b, we show the evolution of the lateral packing fractions and obtained for a coarse-graining length after resetting the abscissa as in Fig. 7 by using the distance and then averaging over 10 experiments. The bending transition appears very clearly to be accompanied by i) a strong decrease of the downstream packing fraction down to values approaching zero and ii) a slight increase of the upstream up to a plateau value about .
Note that this value is far lower than the jamming packing fraction of the present granular medium estimated in Ref. Kolb et al. (2013). However, the value of the maximum packing fraction of a granular cluster is known to be significantly altered (decreased) in presence of a wall Desmond and Weeks (2009). A clear illustration of this effect is given in Figs. 10c and 10d where we show the respective evolution of the upstream and downstream packing fractions and with increasing values of the coarse-graining length . We see that the larger the coarse-graining length, the larger the maximum upstream packing fraction , which for reaches a plateau value close to the expected jamming packing fraction.
By comparing Fig. 10b with Fig. 8, we observe that the symmetry breaking in the packing appears when the fiber deflection typically exceeds the half diameter of a grain. These results thus give a quantitative support to our phenomenological observations: the bending transition of the flexible beam is associated to a structural transition of the flowing granular medium with the formation of jammed cluster upstream of the fiber and the development of a cavity downstream of the fiber.
V A simple model of bending induced compaction
v.1 Bending induced compaction
As discussed in the previous section, the bending transition of the flexible beam is associated with an accumulation of grains upstream of the fiber, leading to the growth of a cluster at the jamming density as well as the development of a cavity empty of grains downstream of the fiber (see the schematic diagram of Fig. 11). The gradual bending of the flexible beam in the course of the penetration of the granular medium tends to increase the section of the intruder across the granular flow. We make the simple assumption that a larger intruder cross-section collects a larger number of grains in the cluster. Thus the number of grains entering the cluster for an infinitesimal penetration distance of is assumed to be proportional to the deflection along . This accumulation of grains in the cluster, which acts as a solid block connected to the fiber, increases the force acting against the elastic restoring force on the fiber.
In the spirit of the compaction front observed in the work of Waitukaitis and al. Waitukaitis et al. (2013) and the three phase-model we developed for the penetration of a rigid intruder in a granular medium Kolb et al. (2013) we propose that the cluster grows faster for a packing fraction approaching the jamming transition. The compaction front (see Fig. 11) delimits the interface between the cluster at and the rest of the granular layer at in front of the fiber. For each penetration step , the compaction front advances over a distance of greater than . Indeed there is an amplification factor linked to the approach to the jamming transition: , thus resulting in = for a one dimensional growth of the cluster as illustrated in Fig. 11. Therefore the number of grains arriving in the cluster will evolve like: .
However the compaction of a cluster of grains upstream of the fibre induces long range effects in the granular medium Reddy et al. (2011); Bouzid et al. (2015). In particular, it is accompanied here by recirculation flows on both sides of the intruder that tend to erode the cluster and convect grains downstream of the fibre. These recirculations are visible in the map of displacement fields of Fig. 4 in the plate frame, when the displacement of the plate has been subtracted from each grain displacement. In our previous work on the penetration of rigid intruders Kolb et al. (2014), it was observed that the lateral extent of recirculations was proportional to the diameter of the cylindrical intruder. Adapting this argument to the flexible intruder of spanwise length leads to a grain flux recirculating on both sides of the fiber which is proportional to . The number of grains eroded from the cluster for each penetration step is thus expected to scale like . Hence the evolution of the number of grains in the cluster is the balance of two terms, an accretion term and an erosion term that both depend on with .
We assume that the friction force resulting from contacts between grains and between grains and fiber remains negligible with respect to the friction force resulting from the contact grains and the supporting plate. We also assume that during bending the accretion term () is larger than the erosion term (), such that the cluster grows and induces an increment of the friction force with . During this transition regime we expect a quasi-static loading where the elastic force due to the bending compensates the friction force induced by the jammed cluster of grains upstream of the fiber. This leads to the following evolution law:
Note that is a phenomenological parameter which has the dimension of a pressure.
Given the knowledge of the constitutive relation that relates the bending force of the flexible beam with its lateral deflection, Eq. (12) should thus give us an evolution law for the lateral deflection with respect to the penetration of the fiber .
In the limit of small deflection and for a uniform perpendicular force density along the fiber, we can use the linear elasticity result and we get:
We thus recover an exponential growth of the deflection of the fiber with respect to the penetration. This was observed experimentally in Fig. 8 where for the early stage of the bending. The identification of the exponential fit for the experimental values of with the evolution law of eq. 13 gives for the length characteristics of the kinetics of bending:
However the experimental evolution of the deflection with penetration eventually shows a saturation which can not be reproduced in the framework of the linear approximation of the elastic bending of the fiber, only valid for small deflection.
In order to account for the large deflection regime and for the saturation behavior, we set up a nonlinear numerical model of our experimental system. The fiber is modeled by a unidimensional inextensible centerline parameterized in a space by the rotation angle along the arclength between the local tangent of the deformed and undeformed curve. Our model is geometrically exact in the sense that there is no restriction on the amount of deflection the inextensible fiber can take (the geometrical limit is that self-interactions are not allowed in our model). The equilibrium configuration of the weigthless fiber under an external density of force along its arclength is given by a ’s Kirchhoff equation Audoly and Pomeau (2010). To model the contacts of the grain on the fiber, we choose a density of forces that remains orthogonal to the fiber upon deformation (note that because of this non-conservative positional loading, the equilibrium equation cannot be derived from a mechanical energy formulation). Finally, we ensure a free boundary condition at one end and a torsional spring with a given stiffness on the other end to account for the possible imperfections of the experimental clamping.
Using finite differences Lazarus et al. (2013), the second order partial differential equilibrium equation is discretized in a set of nonlinear algebraic equations with unknowns that are the rotational degrees of freedom of the discretized structure. Here, is the number of discrete elements modelling the fiber that has been set to to ensure numerical convergence. For a given discrete density of forces applied on the nodes of the discrete fibers, the equilibrium configuration is computed thanks to a classic Newton-Raphson algorithm. Note that the intensity of the density of forces has to be applied gradually from to the desired intensity to ensure proper convergence of the Newton-Raphson algorithm. This numerical process gives us the nonlinear relation that we were seeking.
As shown by the blue curve in Fig. 12, this computation now allows us to reasonably reproduce the full bending transition (regime II) of the fiber induced by the gradual compaction of grains upstream of the fiber. By tuning the value of to a given value, it is possible to reproduce the experimental evolution of deflection till its maximum value. The model does not allow to reproduce the behavior beyond (regime III), in particular the plateau observed for this fiber length =3 cm, as the elastica calculations are based on the same type of loading whatever the deflection, i.e. an orthogonal repartition of forces along the whole fiber length. This strong assumption is certainly no more valid in regime III when the fiber adopts a hook shape, that is when the angle exceeds .
v.3 Discussion of the model
The values of that match the curves of deflection like in Fig. 12 have been obtained for the different experiments performed with various fiber lengths or packing fractions . For getting a physical meaning of in terms of length, we derive the corresponding values from Eq. 14. The value characterizes the typical penetration distance necessary to bend the fiber in regime II once the bifurcation occurs. In Fig. 13 these values (blue triangles) are plotted as a function of the relative excess length above , that is . The values of (red circles) obtained from the direct fit of Fig. 8 are also plotted on the same curve for comparison. For both derivations, the values are of the order of but both curves also exhibit the same complex and non-monotonous evolution with . For large , that is for long fiber lengths, the value seems to be governed by the excess length . However for , that is for fiber length approaching the critical length for bending, increases by a factor of 2. For a fiber length just at the transition () one would expect a divergence of , as provides the limit for the non-bending case. The increase of observed for smaller might be interpreted in this framework. In any cases, the Fig. 13 provides a intriguing relationship between , the length scale characterizing the rate of bending, and defining the critical fiber length for bending.
The elastica-derived model, though based on simple and minimal arguments, well captures the full range of the bending transition (regime II) (Fig. 12), even if it can not describe the deflection in regime III. More refined and complex loadings of the fiber (like non orthogonal forces or non homogeneous amplitudes or locations of point forces…) might be investigated to improve the matching of the model with experimental deflections.
Moreover, even in the bending regime II, the observed non-monotonous evolution of with probably indicates that there are two competing effects with increasing fiber lengths. Following the work of Mojdehi et al. (2016) on the buckling of elastic beams in granular media, it is tempting to consider that the fiber can be decoupled in two parts: one part of effective length (probably related to ) that might supports the force exerted by the cluster, while the complementary part of length would be simply led by the granular flow like a pinned rod. Direct measurements of forces and torques exerted on the fiber are needed to clearly identify the way the fiber is loaded. Complementary elastica simulations would provide a systematic way of quantifying the role of the fluctuating and discrete nature of force transmissions specific of granular material at the grain-fiber contacts. The approach to jamming introduces a further complex step in this new and intriguing fluid-structure interaction between and a flexible fiber and a dense granular flow.
This work presented a new fluid/structure interaction between a granular flow close to the jamming transition and a flexible fiber in a geometry of penetration. We identified a bending transition occuring for fiber longer than a characteristic length, that we called elasto-granular length as in the very recent work of Schunter Jr. et al. (2017). In our case, the reconfiguration of the fiber shape was due to bending and not buckling but one can imagine to promote mechanisms of buckling by progressively increasing the aspect ratio between the fiber thickness and the grain diameter.
The bending transition was associated with a symmetry breaking in the packing fraction, with the formation of a cluster upstream of the fiber and a cavity empty of grains downstream of the fiber. We proposed an expression for that combines both rigidities of the flexible fiber and granular material approaching the jamming transition. The elasto-granular length controlled the transition between the bending and jiggling regimes for the fibers embedded in the granular medium. But surprisingly was also observed to determine the kinetics of the bending once the transition started.
We acknowledge the crucial technical help of T. Darnige as well as enlightning discussions with P. Gondret and A. Seguin. We also acknowledge all the graduated students who have contributed to this work, Marguerite Leang, Yuka Takehara (from the group of K. Okumura), Francois Postic as well as Katherin Luginbuhl, Dawn Wendell and Nadia Cheng from the group of A. Hosoi through the MIT-France Seed Fund project on ”Flexible Objects in Granular Media”.
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