Internal friction of bulk solids: measurement and parameters
The yield locus describes the yield limit of a consolidated bulk solid: the stress state at which it starts to flow under shear stress. It depends on the consolidation – which is why it is measured at defined consolidation stresses.
From the yield loci you obtain angles of friction, cohesion, unconfined compressive strength and flowability – the basis for silo design and for comparing bulk solids.
Parameters from the yield locus measurement
Each yield locus provides a complete set of parameters.
| Parameter | Symbol | in EN 1991-4 | Use |
|---|---|---|---|
| Effective angle of internal friction | φe | φi | silo loads, mass flow, lateral pressure ratio |
| Angle of internal friction | φi | φc | slope of the yield locus |
| Cohesion | τc | c | silo loads, flow behaviour |
| Unconfined compressive strength | σc | – | flow function, arching |
| Major principal stress (consolidation stress) | σ1 | – | reference value of the flow function |
| Minor principal stress | σ2 | – | stress state at consolidation |
| Flowability | ffc = σ1 / σc | – | comparing bulk solids |
| Lateral pressure ratio | K = 1.1 · (1 − sin φe) | K | silo loads to EN 1991-4 |
| Bulk density at steady-state flow | ρb(σr) | – | bulk unit weight, density function |
Carrying out the yield locus measurement
Measurements are carried out on standardised shear testers to ASTM D6682-A. The tester reproduces, in a defined way, the loading to which the bulk solid is subjected in the silo or the process.
- Consolidate The sample is consolidated at the chosen normal stress (reference stress) and pre-sheared – the pre-shear point.
- Shear At a reduced normal stress the shear stress is measured until incipient flow – the shear points.
- Yield locus The measuring points give the yield locus for this consolidation.
- Yield loci Repeated at several consolidation stresses – the basis of the flow function.
Consolidation stresses
The position of the yield locus depends on the packing density: the more strongly the sample is consolidated, the higher it lies. To compare bulk solids, one yield locus at one consolidation stress is sufficient. For the flow function and silo design we measure at least three yield loci at different consolidation stresses, which we derive from the silo geometry or the application.
Duplicate measurement and time consolidation
Each yield locus is measured at least in duplicate. On request we also determine the time yield locus: the sample remains at rest under load for a defined time before it is sheared. This shows whether the bulk solid gains strength during storage.
Test report and customer portal
The test report contains the measured values, all parameters and the σ-τ diagram for each yield locus. In the customer portal, several yield loci and samples can be evaluated together.
Effective yield locus and flow function
The effective angle of internal friction φe is determined as the angle of the straight line through the origin and the pre-shear point – it contains no contribution of cohesion and is approximately the same for yield loci at different consolidation stresses. Among other things, it gives the lateral pressure ratio K for the silo loads.
Several yield loci at different consolidation stresses give the flow function σc(σ1). With the effective angle of internal friction, the stress states between the measured yield loci can also be calculated – the following diagram shows the relationship.
Stress states between the measured yield loci
Principal stresses σ1(φe) and σ2(φe) from the effective angle of internal friction, unconfined compressive strength from the flow function σc(σ1). Move the mouse pointer over the diagram.
Stresses in the bulk solid
Normal and shear stresses
A force on a plane in the bulk solid divides into a normal stress σ perpendicular to the plane and a shear stress τ parallel to it. The shear stresses move the particles against each other – if they exceed the yield limit, the bulk solid flows.
Lateral pressure ratio
A vertical load produces a smaller stress in the horizontal direction in the bulk solid. Their ratio λ = σh / σv lies between the limiting cases of a solid (λ = 0) and a liquid (λ = 1); in EN 1991-4 it is denoted K.
Principal stresses and Mohr stress circle
The principal stresses act in the planes without shear stress: σ1 is the major, σ2 the minor principal stress. The Mohr stress circle shows the stresses of all planes in between. If it touches the yield locus, the bulk solid starts to flow.
The circle through σ = 0 that touches the yield locus gives the unconfined compressive strength σc; the ratio σ1 / σc is the flowability ffc.
In detail: fundamentals of bulk solids mechanics.
Frequently asked questions about yield locus measurement
- How many yield loci are measured?
- To compare flowability, one yield locus at a defined consolidation stress is sufficient. For the flow function and silo design we measure yield loci at a minimum of three different consolidation stresses, which we define on the basis of the application or the silo geometry. Each measurement is carried out at least in duplicate.
- What is the difference between the angle of internal friction and the effective angle of internal friction?
- The angle of internal friction φi is the slope of the yield locus. The effective angle of internal friction φe describes the straight line through the origin and the pre-shear point – without the contribution of cohesion. In EN 1991-4 the effective angle of internal friction is denoted φi and the angle of internal friction φc; calculations to the standard use our φe.
- How much sample material is needed?
- About 0.5 litres is sufficient for a flowability measurement alone; for the flow function for silo design about 1 to 2 litres per sample are needed. For a standard-compliant measurement the particle size should not exceed 2 mm.
- How long does the measurement take?
- Results are usually available about two weeks after sample receipt – depending on laboratory capacity and the number of samples. Time consolidation tests can extend the standard lead time accordingly.
Have the yield locus measured
Describe your bulk solid and the application – we define the consolidation stresses and prepare a non-binding quote for you.
- Yield loci at least in duplicate, time consolidation on request
- All parameters, also with the symbols of EN 1991-4
- Results online in the customer portal and as a test report