Shear banding in geomechanics
29/11/08 08:25 Filed in: Geomechanics
Shear bands a bands of high shear strains within a
material. They are common features of certain
geotechnical problems such as slope stability
failures. Simple numerical models, including FLAC
analyses can simulate the development of shear
bands when modeled correctly, and the development
of shear bands within the model can have a
significant influence on the results of the model.
In order to consider the formation of shear bands within numerical models, we will consider a simple model of a shear box test on a Mohr-Coulomb material.
When the two sides of the box slide apart, the shearing occurs across a very thin band as the soil grains slide across each other. In this case the shear band that is formed is very thin. We can simulate the shear box test relatively simply in most general numerical analysis packages, such as FLAC. In order to do this we need to discretise the problem, i.e. divide the problem into small elements. Typically there is a preference for a coarse discretisation as it gives the most efficient and simplest models.
In this model the shear box has been divided into a 4 x 4 grid giving 16 elements/zones. The base of the box has been fixed with roller boundary conditions on the three sides, as has the top of the box. A constant force is applied to the side of the box to model the shear force applied during the test.
We need to consider how this model will behave when the force applied to cause the shearing starts. The top section of the shear box has started to elastically deform under the shearing force.
The block of soil in the top section of the ground will behave roughly as a cantilever beam giving a linear variation in the stress shear stress through the model. Based on this stress distribution, we can obtain a basic stress distribution for elements within the model. This stress profile assumes that the elements behave as constant stress elements, a common assumption among general numerical analysis programs.
The stresses in elements 1 and 2 is therefore roughly an average of the theoretical stress and can only be correct at a few locations. The stress profile calculated by the model is therefore only an approximation of the actual theoretical stress profile. In order to consider the implications of this averaging we can increase the shearing in the theoretical model further.
The load, and hence the stress, in the section has been increased to the point where theoretical plastic shear failure will occur as indicated by the blue line. It can also be seen that the average stress in element 2 is not high enough to cause plastic failure, therefore the model will not exhibit plasticity at this load. Assuming a linear stress profile and the stress in the elements is the average theoretical stress across the element, the stress in element 2 will only be 75% of the limiting shear stress. In order to cause shear failure in element 2 to occur, the load on the model needs to be further increased.
In this case the theoretical stress has been increased to 33% above the shear stress limit so that the stress in element 2 will achieve the shear stress limit. We can now plot the distorted form of the model under additional load with the elements undergoing plastic deformation highlighted in blue.
In this case the numerical model is overestimating the strength of the ground by 33%. This is obviously not a conservative solution and indicates the problems that shear bands and other similar plasticity based problems can cause when carrying out numerical analyses.
A possible solution to this problem is to re-mesh the problem.
Now consider the case where the grid density has been increased by a factor of 2 in each direction. This results in a revised stress profile.
In this case the overstress required to cause plasticity is reduced to 14% and so the revised grid has significantly improved the performance of the model. We can plot how the accuracy of a model varies with increasing mesh density.Based on the mesh regenerating approach used above a density of 11 in each direction zones is required to get results with less than 5% overstress. Increasing the density of the zones does however reduce the speed of the solution of a typical model by an amount between the increase in density of the zones, or the square of the increase in the density of the zones. In this case then the models solution speed would decrease by between 7 times and 57 times. These could be large reductions in solution speeds for large models.
This is a very simplified way of looking at shear banding but the principals and logic will help understand some of the more complex problems. Lack of shear banding within certain types of geomechanical models is a critical reason why many models fail to produce credible results. Critically models without the ability to model shear banding will often overestimate soil strength. A modeller therefore needs to understand how shear banding forms and how to model it to create an effective and credible model.
In order to consider the formation of shear bands within numerical models, we will consider a simple model of a shear box test on a Mohr-Coulomb material.
When the two sides of the box slide apart, the shearing occurs across a very thin band as the soil grains slide across each other. In this case the shear band that is formed is very thin. We can simulate the shear box test relatively simply in most general numerical analysis packages, such as FLAC. In order to do this we need to discretise the problem, i.e. divide the problem into small elements. Typically there is a preference for a coarse discretisation as it gives the most efficient and simplest models.
In this model the shear box has been divided into a 4 x 4 grid giving 16 elements/zones. The base of the box has been fixed with roller boundary conditions on the three sides, as has the top of the box. A constant force is applied to the side of the box to model the shear force applied during the test.
We need to consider how this model will behave when the force applied to cause the shearing starts. The top section of the shear box has started to elastically deform under the shearing force.
The block of soil in the top section of the ground will behave roughly as a cantilever beam giving a linear variation in the stress shear stress through the model. Based on this stress distribution, we can obtain a basic stress distribution for elements within the model. This stress profile assumes that the elements behave as constant stress elements, a common assumption among general numerical analysis programs.
The stresses in elements 1 and 2 is therefore roughly an average of the theoretical stress and can only be correct at a few locations. The stress profile calculated by the model is therefore only an approximation of the actual theoretical stress profile. In order to consider the implications of this averaging we can increase the shearing in the theoretical model further.
The load, and hence the stress, in the section has been increased to the point where theoretical plastic shear failure will occur as indicated by the blue line. It can also be seen that the average stress in element 2 is not high enough to cause plastic failure, therefore the model will not exhibit plasticity at this load. Assuming a linear stress profile and the stress in the elements is the average theoretical stress across the element, the stress in element 2 will only be 75% of the limiting shear stress. In order to cause shear failure in element 2 to occur, the load on the model needs to be further increased.
In this case the theoretical stress has been increased to 33% above the shear stress limit so that the stress in element 2 will achieve the shear stress limit. We can now plot the distorted form of the model under additional load with the elements undergoing plastic deformation highlighted in blue.
In this case the numerical model is overestimating the strength of the ground by 33%. This is obviously not a conservative solution and indicates the problems that shear bands and other similar plasticity based problems can cause when carrying out numerical analyses.
A possible solution to this problem is to re-mesh the problem.
Now consider the case where the grid density has been increased by a factor of 2 in each direction. This results in a revised stress profile.
In this case the overstress required to cause plasticity is reduced to 14% and so the revised grid has significantly improved the performance of the model. We can plot how the accuracy of a model varies with increasing mesh density.Based on the mesh regenerating approach used above a density of 11 in each direction zones is required to get results with less than 5% overstress. Increasing the density of the zones does however reduce the speed of the solution of a typical model by an amount between the increase in density of the zones, or the square of the increase in the density of the zones. In this case then the models solution speed would decrease by between 7 times and 57 times. These could be large reductions in solution speeds for large models.
This is a very simplified way of looking at shear banding but the principals and logic will help understand some of the more complex problems. Lack of shear banding within certain types of geomechanical models is a critical reason why many models fail to produce credible results. Critically models without the ability to model shear banding will often overestimate soil strength. A modeller therefore needs to understand how shear banding forms and how to model it to create an effective and credible model.