Shear jamming, discontinuous shear thickening, and fragile state in dry granular materials under oscillatory shear
The linear response of two-dimensional frictional granular materials under an oscillatory shear is numerically investigated. It is confirmed that the shear storage modulus and the loss modulus depend on the initial amplitude of the oscillation to prepare the system before the measurement. For sufficiently large initial strain amplitude, the shear jammed state satisfying is observed even if the packing fraction is below the jamming point. The fragile state is also identified as a long-lived metastable state where depends on the phase of the oscillatory shear. The dynamic viscosity evaluated from the shear loss modulus exhibits a sudden jump similar to the discontinuous shear thickening in the fragile state.
Introduction.– Amorphous materials consisting of repulsive and dissipative particles including randomness such as granular materials, colloidal suspensions, foams, and emulsions can form solid-like jammed states. Since Liu and Nagel suggested that jammed states exist only above a critical packing fraction (the jamming point) Liu (), the jamming transition has attracted much attention among physicists Hecke (); Behringer18 (). Several numerical simulations of frictionless grains support this picture and reveal various critical behaviors near the jamming point, where the pressure and the coordination number exhibit continuous and discontinuous transitions, respectively OHern02 (); OHern03 (); Wyart05 (). Continuous transitions are also observed for rheology of frictionless particles under steady shear Olsson (); Hatano07 (); Hatano08 (); Tighe (); Hatano10 (); Otsuki08 (); Otsuki09 (); Otsuki10 (); Nordstrom (); Olsson11 (); Vagberg (); Otsuki12 (); Ikeda (); Olsson12 (); DeGiuli (); Vagberg16 (); Boyer (); Trulsson (); Andreotti (); Lerner (); Vagberg14 (); Kawasaki15 (); Suzuki (); Rahbari () and oscillatory shear Tighe11 (); Otsuki14 ().
Nevertheless, granular particles cannot be free from mutual frictions between grains, which play crucial roles in the dynamics of granular materials. Indeed, recent experiments suggest frictional grains follow a different scenario, i.e. jammed states for frictional grains are induced by shear deformation even below the critical fraction , which depends on the mutual friction between grains Bi11 (). Such a transition, known as shear jamming, has been studied experimentally Zhang08 (); Zhang10 (); Wang18 () and numerically Sarkar13 (); Sarkar16 (). In Ref. Bi11 (), the shear jammed state is characterized by the percolation of an isotropic force network, while the fragile state is characterized by an anisotropic network.
It is also known that mutual frictions between grains cause drastic changes in rheological properties such as the discontinuous shear thickening (DST) Otsuki11 (); Chialvo (); Brown (); Seto (); Fernandez (); Heussinger (); Bandi (); Ciamarra (); Mari (); Grob (); Kawasaki14 (); Wyart14 (); Grob16 (); Hayakawa16 (); Hayakawa17 (); Peters (); Fall (); Sarkar (); Singh (); Kawasaki18 (); Thomas (); Somfai (); Magnanimo (); Otsuki17 (), which is applied to flexible protective gears, robotic manipulators, and traction controls Brown14 (); Brown10 (). There have been several studies focusing on the relationship between the DST and the shear jamming in suspensions of frictional grains under steady shear Peters (); Fall (); Sarkar (); Singh (). The definitions of the shear jamming and the fragile state, however, are inconsistent with each other, and the conclusion is still controversial Bi11 (); Peters (). Therefore, we have to clarify the relationship between the mechanical response and the shear jamming or the fragile state in granular materials.
To resolve the above puzzled situation, we numerically study the mechanical response of two-dimensional frictional grains near the jamming transition under oscillatory shear. We find that the linear response exhibits the shear jamming and the DST depending on the amplitude of the oscillatory shear before measuring it. The shear jammed state satisfying the storage modulus can be observed for the packing fraction above a critical strain amplitude. We also confirm that the observable region for the DST-like behavior is basically identical to that of a fragile state: a long-lived metastable state depending on the phase of the oscillatory shear.
Setup of Simulation.– Let us consider a two-dimensional assembly of frictional granular particles. They interact according to the Cundall-Strack model with an identical mass density in a square periodic box of linear size Cundall (). The normal repulsive interaction force between the grain and the grain is given by , where and with the normal spring constant , the normal viscous constant , , , , , and . Here, the position, the velocity, and the diameter of the grain are denoted as , , and , respectively. is the Heaviside step function defined by for and otherwise. The tangential contact force is given by , where selects the smaller one between and , for and otherwise, is given by , and with the -component of . Here, and are the elastic and the viscous constants in the tangential direction, respectively. The tangential displacement is given by with the tangential velocity , where “stick” on the integral indicates that the integral is performed when the condition is satisfied. Here, the angular velocity of the grain is denoted as . To avoid crystallization, we use a bi-disperse system which includes equal number of grains of the diameters and , respectively.
In this system, we apply an oscillatory shear along the direction under the Lees-Edwards boundary condition with the SLLOD algorithm which stabilizes uniform shear flows Evans (). As an initial state, the disks are randomly placed in the system with the initial packing fraction , and we slowly compress the system until the packing fraction reaches a given value as shown in Ref. Otsuki17 (). After the compression, the shear strain is applied as , where , , and are the strain amplitude, the angular frequency, and the initial phase, respectively. For the initial cycles, we use with the initial strain amplitude . After the initial shear, we apply the oscillatory shear with sufficiently small strain amplitude for cycles, and measure the storage and the loss moduli in the final cycle defined by Doi ()
The shear stress is given by
where is the component of . Here, we have ignored the kinetic part of because it is significantly smaller than the potential part for highly dissipative grains. Note that and the dynamic viscosity with are almost independent of and for and , where is the characteristic time of the stiffness with the mass for a grain of diameter Otsuki17 (). We, thus, focus on the dependence of the shear modulus only on , , and for and . Here, the dynamic viscosity is almost identical to the shear viscosity Doi (). We mainly use , , , , and . This set of parameters corresponds to the constant restitution coefficient . Note that we have estimated the isotropic jamming point . See Supplement Material Supple () for the determination of and its -dependence. We have also confirmed that is almost independent of , , and for , , and . We adopt the leapfrog algorithm with the time step .
Mechanical response.– In Fig. 1, we plot the force chain network after the initial oscillatory shear for and with and . For small initial strain amplitude (), the system stays in a liquid-like state without percolating force chain networks. For and , however, the systems have percolating force chain networks, where the network might be more anisotropic for .
Figure 2 exhibits the transition from a liquid-like state to a solid-like shear jammed state, where we plot against for and with . changes from to a finite value at a critical strain around . The value of the critical strain for , however, depends on , and the solid-like state with and the liquid-like state with coexist in the shaded region of Fig. 2 as a metastable state. The inset of Fig. 2 exhibits the storage modulus against for at , which indicates that in the metastable state has peaks at and becomes near with an integer . The stress-strain curve in the metastable state exhibits both the liquid-like and the solid-like behaviors depending on the phase of the oscillatory shear, which leads to the -dependence of . See Supplement Material for the stress-strain curves showing the onset of the shear jamming and explaining the -dependence in the metastable state Supple ().
In Fig. 3, we plot against for various with . For , is finite for , but depends on . The decrease of for is similar to softening observed in glassy materials under steady shear Fan (), but takes a minimum value in the intermediate for . We confirm the existence of the shear jamming for with . See Supplement Material Supple () for the determination of . We also find the reentrant transition from finite to zero value for . This reentrant transition might be related to the shear jamming for frictionless grains caused by a cyclic compression Kumar ().
Figure 4 plots the dimensionless viscosity against for with various . For , is almost independent of , while for exhibits a sudden increase from a negligibly small value to a larger value at a critical strain amplitude . The sudden increase of is similar to the DST under steady shear.
Phase diagram.– Figure 5 illustrates the phase diagram on the plain of and . Here, we have introduced the shear storage modulus without the initial oscillatory shear: . Then, we define the jammed state (J) as the region where and for any with a sufficiently small threshold . Note that the phase diagram is unchanged if we use . The unjammed state (U) is defined as for any . The shear jammed state (SJ) is defined as and for any . Finally, we define the fragile state (F) as the solid-like state with and the liquid-like state with coexist as shown in the shaded region of Fig. 2. In Fig. 5, SJ exists basically for and . It is remarkable that the unjammed phase exists even for , which must be related to the yielding transition, and the jammed state for intermediate with located above a bay-like unjammed state might be regarded as a SJ. The unjammed state for and its relation to the yielding transition will be studied elsewhere. We have also confirmed that F exists between U and SJ. See Supplement Material Supple () for the -dependence of SJ and the -dependence of F.
In Fig. 5, we also plot the critical strain estimated from the DST-like behavior, where exceeds a threshold . Note that also changes from to a finite value at . The critical strain for exists on the boundary between U and F, while for other lies in the fragile state. This suggests that the region of the fragile state is almost identical to that for the DST-like behavior, at least, for not extremely large . We should note that the DST is originally defined by a jump of the viscosity against the shear rate Brown (), while our DST-like behavior is the discontinuous jump when we control the initial strain amplitude . Therefore, it is dangerous to identify DST-like behavior we have observed with the standard DST.
Discussion and concluding remarks.– Let us discuss our results. Recent numerical simulations Kumar (); Jin (); Urbani (); Jin18 (); Bertrand (); Baity (); Chen18 (); VinuthaN (); VinuthaJ (); VinuthaA () indicate that the shear jamming can be observed even in frictionless systems. However, the observation of SJ in frictionless systems may need special protocols such as the swap Monte Carlo algorithm Kumar (); Jin (); Urbani (); Jin18 (), small system sizes Bertrand (); Baity (); Chen18 (), or the modification of the contact between grains VinuthaN (); VinuthaJ (); VinuthaA (). These results suggest that SJ for frictionless systems in low density regime might be unstable VinuthaN (); VinuthaJ (); VinuthaA () and disappear in the thermodynamics limit Bertrand (); Baity (); Chen18 (). This is consistent with the -dependence of our results shown in Supplement Material Supple (). Nevertheless, some of SJ for frictionless systems might be stable only for high density regime Kumar (); Jin (); Urbani (); Jin18 () and related to the reentrant from the unjammed state to the jammed state for in our system. Further investigation is needed for frictionless SJ.
The fragile state is originally defined by the anisotropic percolation of the force network under quasi-static pure shear process Bi11 (). Note that there is neither specific compression direction nor quasi-static operations in our system. Therefore, the anisotropy of the force chain network in our fragile state (Fig. 1(b)) is not clear. Nevertheless, we have confirmed that stress anisotropy , which also characterizes the onset of the shear jamming Sarkar16 (); Thomas (); Chen18 (), exhibits the maximum in the fragile state and keep constant in SJ as shown in Fig. 6, where and with the maximum and the minimum principal stresses and , respectively. This behavior is qualitatively similar to that observed in an experiment Sarkar16 (). Further careful study on the mutual relationship should be necessary.
In conclusion, we have numerically studied the frictional granular systems under oscillatory shear. Controlling the strain amplitude of the oscillatory shear before the measurement, we find that the shear jamming is regarded as a memory effect of the initial shear. This can be used to detect the DST-like behavior, where the viscosity exhibits a discontinuous jump against the initial strain amplitude. The region we observe the DST-like behavior is almost identical to that of the fragile state. Our results clarify properties of shear induced exotic states in granular materials.
Acknowledgements.The authors thank R. Behringer, B. Chakraborty, T. Kawasaki, C. Maloney, C. S. O’Hern, K. Saito, S. Sastry, S. Takada, and H. A. Vinutha for fruitful discussions. We would like to dedicate this paper to the memory of R. Behringer who has passed away in July, 2018. This work is partially supported by the Grant-in-Aid of MEXT for Scientific Research (Grant No. 16H04025 and No. 17H05420). One of the authors (M.O.) appreciates the warm hospitality of Yukawa Institute for Theoretical Physics at Kyoto University during his stay there supported by the Program No. YITP-W-15-19.
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Appendix A Introduction
In this Supplemental Material, we present the stress-strain curves in the shear jammed and the fragile states, the dependence of the transition points on the friction coefficient , and the -dependence of the fragile state.
Appendix B Stress-strain curves in the shear jammed and the fragile states
In this section, we show the stress-strain curves to illustrate how the shear jamming takes place in the initial oscillatory shear. We also explain the origin of the -dependent in the fragile state from the stress-strain curves. In Fig. S1, we plot the shear stress against the strain for with and . Note that corresponds to the shear jammed states. Until reaches a certain value around , remains . This means that the system stays in a liquid-like state because is zero in the linear response regime. On the other hand, follows a stress-strain loop once exceeds . After the initial oscillatory shear, the residual stress remains as shown in the solid square in Fig. S1, and the gradient of around indicates corresponding to the shear jamming.
Figure S2 illustrates the shear stress against the strain in the final cycle of the initial oscillatory with and for and . For , exhibits a linear dependence on near the maximum and the minimum values of , though remains for . Thus, the linear response after the reduction of the strain amplitude is solid-like i.e. near . For , the stress-strain curve of the initial oscillation is shifted without changing its shape. The linear response for after the reduction of the strain amplitude is liquid-like near i.e. because of the shift of the stress-strain curve. These behaviors explain the dependence of on in Fig. 2 of the main text.
Appendix C Dependence of transition points on
In this section, we discuss the dependence of transition points and . In Fig. S3, we plot the storage modulus against for with various . For every , exhibits an almost discontinuous transition. Note that the transition point depends on and . We, thus, introduce the critical fraction as a packing fraction where exceeds . Then, we define the jamming transition point for isotropic packings as
which does not depend on . We also define the minimum critical fraction for solid-like states as
Note that has the minimum at , and decreases with increasing . We, thus, evaluate by the extrapolation of in the limit with using the data for .
In the main text, we have discussed the case only for , but we investigate the -dependence of the critical points and in Fig. S4. Note that the shear jamming is observed for . As shown in Fig. S4, the difference between and decreases to as decreases, which indicates that the shear jamming we have observed disappears in the frictionless limit.
Appendix D Order parameter for the fragile state
As shown in Fig. 5 of the main text, the fragile state disappears as approaches the fraction at the tricritical point, which is almost equivalent to . Here, we introduce the order parameter of the fragile state for a given as
Here, represents the boundary between SJ and F, while denotes the boundary between F and U. In Fig. S5, we plot against for . rapidly increases as decreases from . This suggests that the behavior at the critical point is singular, which might be helpful to construct a mean field theory of the shear jamming transition.