Attention Powder Nerds - How to Interpret Shear Cell Test Results

If you are fortunate to have an RST, PFT, or FT4 shear cell tester to measure flow properties of powders, and if you have too much time on your hands, please take a moment to read this article. If you want to waste even more time, you can download my textbook on powder handling and bin design from my website www.mehos.net.

Let’s begin with some definitions: FC is the unconfined yield strength, which is a fancy name for cohesive strength. FC depends on the major principal stress σ?, which is the solids stress on the plane that gives it its maximum value. The flow function, abbreviated FF, is the relationship between the unconfined yield strength Fc and the major principal stress, σ?, i.e., the relationship between the cohesive strength and solids stress. The flowability coefficient FFC is the ratio of the major principal stress to the unconfined yield strength, but it is often erroneously called the flow function or denoted FF by people who should know better. Just remember that FFC is a ratio, and FF is a function. And just to confuse you a little more, the flow factor, denoted ff (lower case), is the ratio of the major principle stress to the stress on the abutments of an arch that forms at the hopper outlet. So FFS, be careful how you use FFC, FF, Fc, and ff. Many investigators use these terms interchangeably. Sad!

A shear cell tester measures the unconfined yield strength Fc, but it measures it indirectly.  It would be awfully convenient if we could measure Fc directly, say, in a uniaxial tester where we place a sample of powder in a cylinder, briefly apply a normal load, and then remove the load and cylinder walls. If the powder is cohesive, it retains its shape. We then incrementally apply a load until the specimen falls apart. The initial load can be considered the major principal stress σ?, and the load that caused the specimen to fail can be considered the unconfined yield strength Fc. It’s unconfined after all, because we removed the walls of the cylinder. Unfortunately, to get the right result, we would need a frictionless cylinder, plus the results will likely be variable as the failure will begin at a local weak spot and then propagate along some ill-defined yield plane. 

So instead of a uniaxial tester, we use a shear cell tester. Shear cell testing allows us to calculate the major principal stress and unconfined yield strength indirectly, a bit of an inconvenience, but worth the effort because we get the right answer. To measure a powder’s cohesive strength using a shear cell tester, we place a sample inside a cell and affix a lid. The cell is designed so that when the cell rotates, a moving layer of powder slides along a stationary layer. The tester applies a normal load and then records the steady-state shear stress that develops when a layer of powder slides past a stationary layer of powder.

Next, the stresses are removed, and a lighter load is applied. The cell is again rotated, but at first, the powder will not slide past itself because it has been packed together. The measured shear stress increases until the powder slides, that is, it yields or fails, and the peak stress is recorded. The steps in which the powder is consolidated and sheared until steady and then sheared under a lighter load until failure are repeated over a range of lighter loads. A plot of the shear stresses against the normal loads is called the yield locus.

Now the fun begins. We have plenty of normal stress and shear stress data. We need to be able to analyze the data in a way that will allow us to calculate the major principal stress and the unconfined yield strength. 

To understand the analysis, we need to know a little bit about Mohr’s circles. Unlike fluids, powders are anisotropic, that is, their properties, including stress, depend on direction. For example, if we were to measure the solids stress inside a cylinder full of powder, the measured stress would change as we rotated the probe. At the centerline, the measured stress is at a maximum when the probe is directed vertically and at a minimum when the probe is horizontal. This is different from fluids where the pressure is the same no matter what direction we direct the probe.

To determine how the solids stress varies with direction, a set of equations that have a lot of trigonometric functions can be solved, no big deal now, but a bit of a pain when engineers did not have calculators and instead relied on slide rules. Fortunately, in the late 1800s, an engineer named Otto Mohr toyed around with enough trigonometric identities to derive an equation of a circle that described the magnitudes of normal stresses and shear stresses acting on various planes. He was able to ditch his slide rule as now all he needed was a ruler and a compass. That’s a good thing, because during a shear cell test, we are actually measuring forces, but need to know stresses, which are equal to the force divided by the area, and the areas are different for different planes. That’s why it’s not a matter of vector addition, and that’s why we should be thankful that Mohr came up with his circles.

With the help of Mohr’s circles, the major principal stress and the unconfined yield strength can be calculated from the yield locus. The major principal stress is found by constructing a Mohr’s circle through the steady-state point of the yield locus and tangent to the yield locus. It turns out that the major principal stress is equal to approximately twice the normal load used during the shear cell test. That makes sense, because not only are you applying a normal load during the test, you are also creating a shear stress (its magnitude depends on internal friction), and both the normal stress and shear stress contribute to the magnitude of the major principal stress.

Drawing a Mohr’s circle through the origin and tangent to the yield locus gives us the unconfined yield strength. Remember that a Mohr’s circle tells us the magnitude of the stress acting on any plane. So consider a state of stress where the powder is unconfined and we apply just enough stress to cause the powder to yield. It makes sense that the Mohr’s circle that passes through the origin (zero stress) and tangent to the yield locus (the powder has yielded) is the circle that we want. Where it intersects the horizontal axis is a major principal stress and is equal to the unconfined yield strength. The effective angle of friction δ can be found by drawing a line that is tangent to the larger Mohr’s circle and passes through the origin. Where the yield locus hits the vertical axis is sometimes called the cohesion c. Unlike the unconfined yield strength, the cohesion is not particularly useful.

If we approximate the yield locus as a straight line, we can use analytical expressions that give the major principle stress and the unconfined yield strength. The calculated value of the unconfined yield strength is a bit higher than what we had calculated using a curved yield locus, but it’s always a good idea to be conservative.

If we could some how conduct a test in which we pull the powder while shearing rather compact it, we could calculate the tensile strength σ?, which is the stress required to cause a consolidated powder to fail if we were to pull on it rather than push on it. Extending the yield locus would look like this:

Tensile stresses don’t develop in hoppers and most other powder handling processes, so it’s not really important to know the tensile strength. The cohesion is even more meaningless, because it represents an unlikely state of stress where there are both compressive stresses and tensile stresses. People like c because it appears less mysterious because there are no Mohr’s circles involved. Unlike Fc, however, c is pretty useless.

In addition, because the yield locus becomes very curvy at low compressive or tensile stresses, a linear representation of the yield locus will not allow an honest value of the cohesion and tensile strength to be calculated.

A nice thing about automated shear cell testers is that they are easy to operate. Place the powder sample inside the cell, affix the lid, input the normal load targets, and have a cup of coffee while the computer operates the tester.  Repeat the test at least two more times using different steady-state normal loads, and then after two more cups of java you can plot the flow function, i.e., a plot of the major principal stress σ? vs. unconfined yield strength FC.

With the major principal stress and the unconfined yield strength, we can calculate the flowability coefficient FFC. FFC again is the ratio of the major principal stress to the unconfined yield strength. Because the cohesive strength term is in the denominator, it makes sense that you would prefer to handle a powder with a large flowability coefficient. It’s tempting to use FFC to define the flowability of a powder. Powders are frequently described as "very cohesive", "cohesive", "easy-flowing", and "free-flowing" based on their FFC values:

But not so fast! If we were to determine a powder’s flow function by measuring its cohesive strength over a range of consolidation stresses and then plot the flow function along with lines of constant flow function coefficient, we see that the FFC of the powder depends on the magnitude of the major principal stress. A powder may be considered easy-flowing at high stresses but very cohesive at low stresses. Investigators are tempted to tabulate FFC values determined at high values of the major principal stress because the flow functions of different materials tend to be more distinguishable at higher stress levels. However, if our goal is to determine whether or not a powder will flow in a hopper, we are more interested in the strength of the powder at low solids stresses. In fact, if the size of the hopper outlet is the minimum required to prevent a cohesive arch from developing, FFC is equal to the flow factor ff, which typically ranges between 1.1 and 1.7.

Again, the flow factor is the ratio of the major principal stress to the stress on the abutments of an arch.  If we have a hopper that isn’t too crazy, e.g, it’s not super steep when the wall friction is extremely low, then the relationship between the flow factor ff and the effective angle of friction δ that Jerry Johanson published is convenient to use:

Provided that the arch supporting stress is greater than the powder’s unconfined yield strength, the arch will fail and powder will flow from the hopper outlet. This external stress is proportional to the diameter or width of the outlet and inversely proportional to the powder’s bulk density. This is another reason we shouldn’t rely on FFC as a metric for flowability. Two materials may have the same FFC, but the one with a higher bulk density will be able to flow unhindered from a hopper with a smaller opening.

We can calculate the critical outlet dimension by plotting the flow factor ff and the flow factor FF together on the same graph. If the lines intersect, we can locate the critical stress, where the powder's strength is equal to the stress imparted on it. From the critical stress we can then calculate the size of the hopper outlet that is the minimum required to prevent a cohesive arch. The critical outlet dimension is equal to the critical stress times a geometry factor, divided by the weight bulk density of the powder. The geometry factor is approximately equal to 2.3 for a conical hopper and approximately 1.15 for a planar hopper, i.e., a hopper with straight walls and a slotted outlet.

If the flow function FF is always above the flow factor, we are S.OL., since the stress on the arch will never be greater than the cohesive strength of the powder, and the arch will not fail. And if the flow function FF is always below the flow factor, congratulations! We don't have to worry about a cohesive arch developing.

So which is a better metric for flowability, a powder with a high FFC measured at an inappropriate solids stress or the minimum size of the outlet of a hopper that will not bridge? Perhaps we are not using the test results to predict flow in a hopper, but it is still better to tabulate values of a powder's unconfined yield strength Fc rather than its cohesion c or flowability coefficient FFC. We should also measure wall friction and permeability. With these test results, we can determine the recommended hopper angle for mass flow and the size of a hopper outlet necessary to obtain a desired solids discharge rate. It's all explained in my textbook.

If you have a shear cell tester, you can download the Excel workbook from my website www.mehos.net that I use to analyze cohesive strength, wall friction, and permeability test results to predict powder flow behavior or design bins and hoppers. You can use the analysis to determine if your powder is a appropriate for a hopper or if you had better reformulate or find another hopper. FFC may tell you if one powder is likely to flow better than another, but using fundamental principles to analyze the test results will allow you to predict flow behavior.

 Greg Mehos, Ph.D., P.E. ? [email protected] ? [email protected] ? 978-799-7311