Understanding the Bulk Strength of Particulate Solids

by Lyn Bates

‘Critical State’

The strength of a bulk solid in a static condition, i.e, its resistance to shear, depends on two features, - how firmly it is compacted and the stresses acting on it, the degree of compaction being defined by its bulk density. In steady state flow, these two factors are no longer independent, but become directly related. The ‘critical state’ of a bulk material is an important concept in bulk technology as it describes its condition when it is in steady flow. This ‘state’ is uniquely related to the bulk density of the material whilst it is deforming in shear and is determined by the stresses acting on the flow channel. To measure the forces acting in such a situation it is necessary to attain a condition of shear without the density changing. Consider a sample of the material that has been consolidated from a loose condition by a particular normal stress. Applying shear to the material with the consolidating load in place will increase the stresses acting on the bulk due to the combined loads, so the bulk material will compact further and increase in bulk density. On the other hand, shearing the sample with the compacting load removed will allow the unconfined material to expand, at least in the shearing region. It follows that there is a normal load lower than the one that caused the compaction that will hold the bulk material in its initial compacted state of density when shearing takes place. The shearing of a bulk material in a steady density condition is precisely what is taking place during flow, so the density condition of a bulk material in steady flow is known as its ‘critical state’. In practice, it is necessary to perform a range of shear tests on similarly compacted static samples with differing normal loads to establish what load will sustain its original bulk density when it is sheared. Plotting the various normal loads, , against the shear load, , produces a ‘shear locus’ for that condition of density, with an end point to the line being where the material shears without changing density, but any further increase in the normal load increases the bulk density of the sample when it is sheared. This test procedure has to be conducted on samples compacted under a range of different normal loads to produce a family of shear locus, from which a line drawn through all the end points indicates the ‘critical state line’ for that bulk material, where the bulk material is in a state of failure and will deform to flow under the application of the shear and normal loads that are acting at differing density condition. Should the nature of the material be such that the density of the bulk material tends to increase in time under sustained loading, it is necessary to conduct a similar series of ‘time consolidation’ tests to determine the gain in strength that take place. The test samples should be held under continued loading for the period that reflects the storage time that is likely to occur.

‘Principle Stresses’

The effect of combined shear and normal stresses acting on a bulk sample can be translated by way of a ‘Mohr`s circle’, to a plane where two simple normal stresses act at right angles to each other reflect the maximum and minimum principle stresses present in the material. (Fig.1). Such a circle can be drawn tangential to the yield locus at any point, to indicate the largest and smallest principle stresses, 1, and 2, at this shear value, where the circle cuts the axis of zero shear stress, 0.

A Mohr’s circle of major interest is that touching the yield locus nearest to the end point of the line, as 1 on this Mohr’s circle represents maximum principle stress, 1Max, that the sample can bear without increasing in density if sheared and is also the value of the actual normal stress that consolidated the sample. The actual load used to consolidate the sample will indicate a higher stress, but some work is absorbed in the consolidation process by wall friction. Another important Mohr’s circle is that touching the yield locus with one end passing through the origin, 0, where there is no normal load acting on the sample. (Fig.2). The stress value at the opposing diameter of the Mohr’s circle represents the magnitude of the principle stress required to cause an arch to fail, because the under-surface of an arch is unconfined, i.e, has zero stress acting at right angles to stresses acting along the surface of the arch. This ‘unconfined failure stress’ is designated Fc. The ratio between 1Max. and Fc is an important measure of the flow properties of the material.

Critical State Line

The family of yield loci that represents the condition of compaction at all normal loads from zero to that which reduced the volume of the voids to an absolute minimum, form a Hvorslev surface in a three dimensional graph of Shear Stress, Normal Stress and Bulk Density. (Fig. 3). This surface denotes how a bulk material over the whole range of possible density conditions will fail under different values of shear stress and normal load. The shear stress and normal stress are zero when dilatation eliminates particle-to-particle friction and the bulk material exhibits conditions of fluidity. A line through the end points of the yield loci at differing density conditions is called the ‘critical state line’ and shows the stress conditions that are acting on the bulk material when in flow at the given density.

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## Bulk Strength of Particulate Solids

Understanding the Bulk Strength of Particulate Solidsby Lyn Bates‘Critical State’The strength of a bulk solid in a static condition, i.e, its resistance to shear, depends on two features, - how firmly it is compacted and the stresses acting on it, the degree of compaction being defined by its bulk density. In steady state flow, these two factors are no longer independent, but become directly related. The ‘critical state’ of a bulk material is an important concept in bulk technology as it describes its condition when it is in steady flow. This ‘state’ is uniquely related to the bulk density of the material whilst it is deforming in shear and is determined by the stresses acting on the flow channel. To measure the forces acting in such a situation it is necessary to attain a condition of shear without the density changing. Consider a sample of the material that has been consolidated from a loose condition by a particular normal stress. Applying shear to the material with the consolidating load in place will increase the stresses acting on the bulk due to the combined loads, so the bulk material will compact further and increase in bulk density. On the other hand, shearing the sample with the compacting load removed will allow the unconfined material to expand, at least in the shearing region. It follows that there is a normal load lower than the one that caused the compaction that will hold the bulk material in its initial compacted state of density when shearing takes place. The shearing of a bulk material in a steady density condition is precisely what is taking place during flow, so the density condition of a bulk material in steady flow is known as its ‘critical state’. In practice, it is necessary to perform a range of shear tests on similarly compacted static samples with differing normal loads to establish what load will sustain its original bulk density when it is sheared. Plotting the various normal loads, , against the shear load, , produces a ‘shear locus’ for that condition of density, with an end point to the line being where the material shears without changing density, but any further increase in the normal load increases the bulk density of the sample when it is sheared. This test procedure has to be conducted on samples compacted under a range of different normal loads to produce a family of shear locus, from which a line drawn through all the end points indicates the ‘critical state line’ for that bulk material, where the bulk material is in a state of failure and will deform to flow under the application of the shear and normal loads that are acting at differing density condition. Should the nature of the material be such that the density of the bulk material tends to increase in time under sustained loading, it is necessary to conduct a similar series of ‘time consolidation’ tests to determine the gain in strength that take place. The test samples should be held under continued loading for the period that reflects the storage time that is likely to occur.

‘Principle Stresses’The effect of combined shear and normal stresses acting on a bulk sample can be translated by way of a

, to a plane where two simple normal stresses act at right angles to each other reflect the maximum and minimum principle stresses present in the material.‘Mohr`s circle’(Fig.1). Such a circle can be drawn tangential to the yield locus at any point, to indicate the largest and smallest principle stresses, 1, and 2, at this shear value, where the circle cuts the axis of zero shear stress, 0.A Mohr’s circle of major interest is that touching the yield locus nearest to the end point of the line, as 1 on this Mohr’s circle represents maximum principle stress, 1Max, that the sample can bear without increasing in density if sheared and is also the value of the actual normal stress that consolidated the sample. The actual load used to consolidate the sample will indicate a higher stress, but some work is absorbed in the consolidation process by wall friction. Another important Mohr’s circle is that touching the yield locus with one end passing through the origin, 0, where there is no normal load acting on the sample.

(Fig.2). The stress value at the opposing diameter of the Mohr’s circle represents the magnitude of the principle stress required to cause an arch to fail, because the under-surface of an arch is unconfined, i.e, has zero stress acting at right angles to stresses acting along the surface of the arch. Thisis designated Fc. The ratio between 1Max. and Fc is an important measure of the flow properties of the material.‘unconfined failure stress’Critical State LineThe family of yield loci that represents the condition of compaction at all normal loads from zero to that which reduced the volume of the voids to an absolute minimum, form a

in a three dimensional graph of Shear Stress, Normal Stress and Bulk Density.Hvorslev surface(Fig. 3). This surface denotes how a bulk material over the whole range of possible density conditions will fail under different values of shear stress and normal load. The shear stress and normal stress are zero when dilatation eliminates particle-to-particle friction and the bulk material exhibits conditions of fluidity. A line through the end points of the yield loci at differing density conditions is called theand shows the stress conditions that are acting on the bulk material when in flow at the given density.‘critical state line’For any problem not covered, contact Ajaxat:http://www.ajax.co.uk/

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