Shear Transformation Zones: State Determined or Protocol Dependent?
Oleg Gendelman, Prabhat K. Jaiswal, Itamar Procaccia, Bhaskar Sen Gupta, Jacques Zylberg
aa r X i v : . [ c ond - m a t . s o f t ] A ug Shear Transformation Zones: State Determined or Protocol Dependent?
Oleg Gendelman , Prabhat K. Jaiswal , Itamar Procaccia , Bhaskar Sen Gupta , and Jacques Zylberg Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel Weizmann Institute of Science, Rehovot 76100, Israel
The concept of a Shear Transformation Zone (STZ) refers to a region in an amorphous solid thatundergoes a plastic event when the material is put under an external mechanical load. An importantquestion that had accompanied the development of the theory of plasticity in amorphous solids formany years now is whether an STZ is a region existing in the material (which can be predicted byanalyzing the unloaded material), or is it an event that depends on the loading protocol (i.e., theevent cannot be predicted without following the protocol itself). In this Letter we present strongevidence that the latter is the case. Infinitesimal changes of protocol result in macroscopically bigjumps in the positions of plastic events, meaning that these can never be predicted from consideringthe unloaded material.
The origin of plastic responses to external mechani-cal loads in crystalline solids is understood: topologicaldefects, and in particular dislocations, glide under theaction of external stresses or strains, and this glide isirreversible, dissipating energy as it is taking place [1–3]. Of course, when the density of such defects increases,the situation becomes hairy, and proper theories are stillunder active research. The fundamental mechanism ofplasticity in amorphous solids is, on the other hand, stillnot fully resolved. In essence there are two schools ofthought. The first considers plasticity resulting from theexistence of some regions in the material that are moresensitive to external load. These regions are referred toas Shear Transformation Zones (STZ) and their intro-duction to rheological models of amorphous solids goesback to the work of Argon, Spaepen, and Langer [4–7].The second school considers plasticity as an instability ofthe amorphous solids [8–10] resulting from a protocol ofan increase in the external load. This instability can beunderstood by focusing on the Hessian matrix of the ma-terial (and see below for more details) with an eigenvaluethat goes to zero following a saddle-node bifurcation [11–13]. Both schools of thought agree that until the appear-ance of system spanning plastic events (shear bands) athigh values of the external load, the plastic events thatone is discussing are localized. In the instability way ofthinking this is explained by the localization of the eigen-function associated with the eigenvalue that is going tozero.The difference in thought is not only in choosing wordsto describe plasticity in amorphous solids. If the STZ ap-proach is valid, one should be able to predict, by a judi-cious analysis of the unstrained system, where a plasticevent is likely to take place. If indeed there are someregions that are more sensitive than others to externalloads, they should be identifiable and marked prior toexercising the external load. On the other hand, if theprotocol dependence of an instability is the right wayof thinking, then one should be able to show that even minute changes in protocol will result in a major changein the plastic event that may take place. Then it wouldbe argued that it were not possible to predict where plas-ticity should appear. The aim of the present Letter is topropose simple numerical simulations that can decide be-tween the two possibility, with the proposed result thatthe second way of thinking should prevail.In our simulations we construct a 2-dimensional glassforming system in the usual way [14], selecting a binarymixture of N particles, 50% particles A and 50% par-ticles B, interacting via Lennard-Jones potentials. Thedifference between the particles is in the positions andthe depths of the minima of the potentials; we choosethe positions of the minima such that σ AA = 1 . σ AB = 1 .
0, and σ BB = 0 . ǫ AA = ǫ BB = 0 . ǫ AB = 1 .
0. Belowlengths and energies are measured in units of σ AB and ǫ AB . The potential is truncated at r co = 2 . R at a high temperature ( T = 0 .
8) with periodicboundary conditions. Secondly, the system is quenchedto temperature T = 0 .
001 at constant volume by molec-ular dynamics. Lastly, the system is energy minimizedto T = 0. At this point we build from the given config-uration a sub-system with circular symmetry using thefollowing protocol: we discard all the particles outsidea circle of radius R, fixing the positions of particles inan annulus (wall) of width dR = 2 r co = 5 .
0. An exam-ple of the resulting system with N = 20 ,
000 is shown inFig. 1. Needless to say, once we fix the wall the periodicboundary conditions are lost.Having constructed a system with circular symmetrywe can now follow its response to external loading. Weload the system athermally and quasi-statically, pullingalong the x -axis and compressing along the y -axis suchas to conserve the area. Thus, the circular system that FIG. 1: The system with circular symmetry constructed asdescribed in the text. In this example the total number ofparticles is N = 20 , begins with L x = L y = R deforms to an ellipse with prin-cipal axes L x = L y . The affine step is area preserving,written as x ′ = x (1 + δγ ) , (1) y ′ = y δγ . (2)Note that in this affine steps also the wall particles areparticipating, to hold the system as desired. After everyaffine step of loading we annul the forces between thebulk particles (excluding the wall particles) using gradi-ent energy minimization. The system then undergoes anon-affine step that brings the system back to mechan-ical equilibrium. This quasi-static loading is continuedas long as the system responses reversibly. The mechan-ical stability of the system is determined by the Hessianmatrix H : H ij ≡ ∂ U∂ r i ∂ r j , (3)where U ( r , r , · · · r N ) is the total potential energy ofthe system as a function of the particle positions { r i } Ni =1 .The Hessian matrix is real, symmetric, and positive def-inite as long as the system is mechanically stable, thefirst plastic event occurs when the lowest eigenvalue of H approaches zero. It is well known that this happensvia a saddle-node bifurcation, meaning that as a func-tion of γ = P δγ there exists a value γ = γ P where thelowest eigenvalue λ P approaches zero via a square-rootsingularity λ P ∼ √ γ P − γ . (4)An example of this scaling law is presented in Fig. 2. -4 -3.8 -3.6 -3.4 γ P - γ -1.7-1.6-1.5-1.4 λ P FIG. 2: A log-log plot of λ P vs. γ P − γ . The measured slopeis 0 . ± . −80 −60 −40 −20 0 20 40 60 80−80−60−40−20020406080 N = 20000 , θ = 31 ◦ FIG. 3: The first plastic event that occurs as a result of choos-ing the x -axis to be in 31 o with respect to the horizontal direc-tion of the original square box. In this example N = 20 , In the unloaded state all the eigenfunctions of the Hes-sian matrix which are associated with low lying eigenval-ues are delocalized. Upon the approach of the lowesteigenvalue to zero, the associated wave-function Ψ P lo-calizes on a typical quadrupolar structure which is iden-tical with the non-affine irreversible displacement asso-ciated with the plastic instability. An example of thisphenomenon is shown in Fig. 3 which is obtained by se-lecting the x -axis in Eq. (2) to be at 31 o with respect tothe horizontal direction of the original square box fromwhich we constructed the circularly symmetric system.The plastic event is shown as the quadrupolar displace- −80 −60 −40 −20 0 20 40 60 80−80−60−40−20020406080 N = 20000 , θ = 32 ◦ FIG. 4: Same as in Fig. 3 but with choosing the x -axis to bein 32 o with respect to the horizontal direction of the originalsquare box. ment field near the bottom of the system. One can at willcall it an STZ, but consider what happens if we changethe x -axis to be at 32 o with respect to the horizontal di-rection. This is shown in Fig. 4. We see that a relativelysmall change in the chosen strain protocol, in this casein 1 o in the chosen direction of the principal stress axes,results in a huge change in the position of the first plasticevent. The aim of the rest of this Letter is to explain thatthis sensitivity increases indefinitely with the system size,such that for macroscopic systems, i.e., in the thermody-namic limit, any arbitrarily small change in protocol willresult in a macroscopic change in the position of the firstplastic event.To this aim we prepare between 30 to 100 different real-izations of our system for each system size, changing thenumber of particles in the range N = 5 , − , x -axis to coincide with the orig-inal x -axis of the square box. For each realization wedetermine what is the first plastic event and what is thevalue of γ P where it appears. In a second step of thisexercise we change the x -axis to have an angle θ withrespect to the original horizontal direction. We then de-termine, for each realization, the first angle θ for which the first plastic event is different , as seen in Figs. 3 and4. Finally, we average the angle θ over the 100 realizationto get h θ i as a function of the system size N . The cen-tral result of this exercise is that h θ i ( N ) is a decreasingfunction of N as seen in Fig. 5. A fit to the numerical -4.8 -4.4 -4.0 -3.6 log (1/N) -0.45-0.40-0.35-0.30 l og 〈 θ 〉 FIG. 5: The average angle required to observe a major changein the position of the first plastic event as a function of thesystem size. Note the logarithmic scale used that supportsthe power-law dependence Eq. (5). The systems studied herespanned the sizes N = 5 ,
000 to N = 100 , − . data shown in Fig. 5 supports a power law of the form h θ i ( N ) ∼ N − α , α ≈ . ± . . (5)Clearly, in the thermodynamic limit N → ∞ , this resultstrongly indicates that indeed any infinitesimal changein angle should result in a macroscopic change in the po-sition of the plastic event. This evidently refutes anypossibility to predict the position of the plastic eventfrom the analysis of the system’s state in equilibrium,before straining. We should note that the similar datatop those shown in Fig. 5 were also obtained with otherquench rates with identical conclusions.In summary, we have presented very simple tests to de-cide between two deeply contrasting views of the natureof plastic events in amorphous solids. The evidence pro-vided above indicates that in the thermodynamic limitit is impossible to predict where the first plastic eventshould appear in a stressed amorphous solid. The plasticevents are protocol dependent, and any minute changein the protocol should result in a macroscopic change inthe position of the first plastic event. We conclude thatit would be futile to predict the position of the first plas-tic event from analyzing the structure of the amorphoussolid at equilibrium, be the method of analysis as sophis-ticated as one might think of. It is important to stressat this point that our analysis also indicate that laterplastic events are even more sensitive to the change inprotocol, and the system size dependence of their sen-sitivity is more steep than the findings reported in Eq.(5). This and related findings are however beyond thescope of this Letter which aims specifically to sharpenthe difference in the current approaches to plasticity inamorphous solids. Acknowledgments
PKJ is supported by a PBC outstanding postdoctoralfellowship from the Council of Higher Education (Israel)for researchers from India. This work was supported byan “ideas” STANZAS grant from the ERC. [1] “Mechanics of Solids”, Ed. C. Truesdell (Encyclopediaof Physics. Chief Editor S. Flugge, Vol. YIa/1): J.F. Bell, “The Experimental Foundations of Solid Me-chanics” (Springer-Verlag, Berlin-Heidelberg-New York,1973).[2] A. H. Cottrell, “Dislocations and plastic flow in crystals”,(Oxford University Press, 1953).[3] J.-C. Toldano, “Physical Basis of Plasticity in Solids”,(World Scientific, 2011). [4] A. S. Argon and H. Y. Kuo, Mater. Sci. Eng. , 101(1979); A. S. Argon, Acta Metall. , 47 (1979); A. S.Argon and L. T. Shi, Philos. Mag. A , 275 (1982).[5] A.I Taub and F Spaepen, Acta Metall. , 1781 (1980).[6] M. L. Falk and J. S. Langer, Phys. Rev. E 57 , 7192-7205(1998).[7] E. Bouchbinder, J.S. Langer and I. Procaccia, Phys. Rev.E, , 036107 (2007); , 036108 (2007).[8] D. L. Malandro and D. J. Lacks, Phys. Rev. Lett. ,5576 (1998).[9] C. Maloney and A. Lemaˆıtre, Phys. Rev. Lett. , 195501(2004).[10] S. Karmakar, A. Lemaˆıtre, E. Lerner, I. Procaccia, Phys.Rev. Lett. , 215502 (2010).[11] E. Lerner and I. Procaccia Phys Rev E, ,066109 (2009).[12] R. Dasgupta, S. Karmakar and I. Procaccia, Phys. Rev.Lett. , 075701 (2012).[13] H.G.E. Hentschel, S. Karmakar, E. Lerner and I. Procac-cia, Phys. Rev. E , 061101 (2011).[14] W. Kob and H. C. Andersen, Phys. Rev. Lett.73