# Shear Bands: the Role of Friction and Dilation Angles

### Roberto Weinberg, Monash University, Australia

 Copyright 2005-2009 by Roberto Weinberg. All rights reserved. Unlimited permission to copy or use is hereby granted for non-profit driven enterprise subject to inclusion of this copyright notice and my World Wide Web URL: users.monash.edu.au/~weinberg. We would very much appreciate an email stating how this material will be used. Thanks, RW.

This section illustrates the role of the friction F and dilation angles Y in the development of shear bands during pure shear deformation (non-associated plasticity). It also illustrates the role of grid resolution and the role of the dimensionless parameter B defined by Poliakov et al. (1994) for these models.

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MODEL SET UP
The model is a horizontal plane at a fixed lithostatic pressure (p), fluid absent and is deformed in pure shear with a horizontal shortening axis. The bulk modulus, K, and shear modulus G are such that they yield a Poisson ratio n = 0.25 and l = G. Input file

Figure 1. Comparisson between results with different values of friction for dilation of zero and other identical parameters. An increase in the value of the friction angle leads to increased localization of the shear bands and increased irregularity. Note also the change in the angle between the conjugate pair in relation to the horizontal shortening axis. The angle becomes more acute with increasing frictional angle. Remember the acute angle is 2*{45o- (f+Y)/4}.    Figure 2. Constant friction and changing dilation angle.   Figure 3. A) and B) demonstrate that for the same difference between F and Y the results are different but both result in thick and regular shear bands. Grid size 100*100

A) F -Y=20o  B) F -Y=40o  DIMENSIONLESS PARAMETER B

Dimensionless parameters is an effective way of simplifying and summarizing the physics of numerical models. The following section is an example of its use.

RW's notes: For calculations using fluids another dimensionless number is necessary which uses the variables related to fluid flow such as permeability and pore pressure gradient. For this, the dimensionless fluid velocity D was created. This is the ratio between the Darcy velocity and the sound velocity:
D= KFLAC* Grad pp / c.

The following models were inspired by and repeat the models described in Poliakov et al. (1994, Fractals, 2, p. 567-581). These authors found that the shear band patterns depend solely on a dimensionless parameter B and the ratio between the size of the mesh and and the system size, L, for a fixed value of the friction angle F. The parameter B, defined as the ratio between two intrinsic time scales, is: B= pc/ m V, where p is lithostatic pressure, c is sound velocity, m=G is shear modulus and V is deformation velocity. Thus for a similar grid size (L), B controls the pattern of shear bands. A series of models was run in order to illustrate not only the role of B, but also that of grid size and that of dilation and friction angles. In contrast to Poliakov et al.'s models, the models presented here are controlled also by a second dimensionless parameter, denominated here T', which arises from the use here of a cohesion value (coh) different than zero and consequently by the existence of a tensile strength. Poliakov et al. (1994) used coh=0 and consequently the rocks did not yield in tension. T' is defined here as the ratio between the applied lithostatic pressure p and tensile strength T. The role of T' is only important when p is small. In all models presented here, unless stated otherwise, coh=3.00e7 Pa and tens=1.50e7 Pa.

Figure 4 A) Two models with constant B and with the same friction and dilation angles. Pressure and m are varied but keeping the value of B constant. For the tensile strength used of tens=1.50e7 Pa, and the applied of 2.e8 Pa and 4.e7, the value of T' is 12.5 and 2.67 respectively.

The results are different but the style of shear bands developed are similar. The model on the right-hand-side has a fifth of the pressure and shear modulus of the model on the left-hand-side. Because our expectation was to find similar results (Poliakov et al. 1994, Fractal 2), we increased the grid from 100*100, as in the previous models, to 150*150.

The two embedded movies (click on figures) illustrate the evolution of the models as they are strained. In both models the width of the shear bands tend to remain constant throughout, but the number of shear bands decreases with time. Velocity here is half that of the models above.

B) At higher resolution (300*300) the results for the two models converge. Note in the two movies that after the first 30000 to 40000 steps the distribution of shear bands is reasonably stable. Poliakov et al. (1994) concluded, and these calculations confirm, that the non-dimensional parameter B controls the evolution of the system.

 M 10, 150*150, T'=12.5, p=2.e8 Pa, G=5.e9 Pa M 8, 150*150, T'=2.67, p=4.e7 Pa, G=1.e9 Pa M 10b, 300*300, T'=12.5, p=2.e8 Pa, G=5.e9 Pa M 8b, 300*300, T'=2.67, p=4.e7 Pa, G=1.e9 Pa Figure 4 (Cont) Different B-values. In c) the value of B is five times smaller than that of D) due to a decrease in the value of the shear modulus G. An important difference is that with the decrease in the value of the shear modulus in D) yielding starts at a much later stage, only after 40000 step at the same deformation velocity (compared with the yilding in C) between 5000 and 10000 steps. This had been predicted by Poliakov et al. (1994) which determined that the time at which the yield limit is reached is proportional to the inverse of G, so five times longer for D) given the same boundary velocity.

 C) M 10b, 300*300, T'=12.5, p=2.e8 Pa, G=5.e9 Pa D) M 15, 300*300, T'=2.67, p=2.e8 Pa, G=1.e9 Pa Figure 5. Another high resolution model (300*300) was run with even lower lithostatic pressure (1.e7 Pa) and a decrease in T'=1.e7/1.5e7=0.67. B is kept constant by a decrease both the shear and bulk modulus. While in the models in Fig. 4 the role of T' is not visible, the decrease in T' here to a value below unity has lead to the initiation of tensile failure and the emergence of a different pattern.

 M 14b, 300*300, T'=0.67, p=1.e7, G=0.25e9 Pa Figure 6. Changes in shear band distribution and width as a function of resolution for model 8 above. At 100*100 fewer shear zones developed. At higher resolution the results converge. Poliakov et al. (1994) , using FLAC, found also that . Compare the results here with those in Poliakov et al.'s Fig. 10.

 Resolution 100*100 Resolution 150*150 Resolution 300*300 Figure 7. Changes in model 8 as a function of deformation velocity only. Deformation velocity changes the value of B and we should therefore expect some changes in the shear bands. Model on B) was deformed an order of magnitude slower than model on A). Diagram on the right-hand-side shows the history of unbalanced forces as the model develops. When yielding first takes place there is a peak in unbalanced forces which gradually decreases as the model progresses. The histories in both cases are not particularly good.

 A) Resolution 150X150 History of unbalanced forces B) Resolution 150X150,V/10 History of unbalanced forces CONCLUSIONS

 The models illustrate the role played by the friction and dilation angles in the evolution of shear zones during pure shear. This exercise illustrates the use of the dimensionless parameters as an effective way of simplifying and summarizing the physics of numerical models. It illustrates also issues of resolution and that the yielding history is dependent on the amount of straining.