Fundamentals of bulk solids mechanics
A bulk solid lies between a liquid and a solid body: it transmits small tensile stresses, compressive and shear stresses at rest, and flows when the shear stresses become sufficiently large.
This page explains the terms used to describe yield locus, flowability and silo loads – from the continuum to the Mohr stress circle.
Stresses and flow in the bulk solid
Bulk solid as a continuum
A bulk solid consists of many non-uniform particles. For bulk solids characterisation, this assembly is regarded as a continuum in order to make statements about the behaviour of the bulk solid as a whole rather than about individual particles. Many influencing factors act together: moisture, temperature, particle size and shape, surface structure, mechanical properties such as elastic, viscoelastic, plastic or brittle behaviour, the chemical composition, gravitational forces and interparticle forces such as electrostatic and van der Waals forces.
Yield limit and consolidation
Under stress, the particles first deform elastically and reversibly; the bulk solid transmits forces without the particles moving against each other. If the stresses exceed a limiting stress, the particles start to move relative to each other – the bulk solid deforms plastically, it flows.
The resistance to this flow depends essentially on the packing density and thus on the preconsolidation. A bulk solid therefore has a yield limit that depends on the stress state, and it largely retains a packing once reached after the load is removed. It transmits small tensile stresses, compressive and shear stresses at rest, and flows when the shear stresses become sufficiently large.
Normal and shear stresses
A force F on a boundary surface A of a volume element can be divided into the normal force FN perpendicular to the surface and the shear force FS parallel to the surface. Relative to the surface, this gives the normal stress σ = FN / A and the shear stress τ = FS / A. The shear stresses are responsible for the relative movement of the particles, i.e. for flow. In bulk solids mechanics – unlike in classical mechanics – compressive stresses are defined as positive.
Lateral pressure ratio
A load on a bulk solid is transmitted via the particle contacts with different strength depending on the direction. A vertical compressive stress therefore produces a smaller horizontal stress. The ratio λ = σh / σv is the lateral pressure ratio (in EN 1991-4: K).
Two limiting cases illustrate the range: in an ideal liquid no shear stresses occur, and the pressure spreads equally in all directions (λ = 1). An ideally rigid solid transmits stresses of any magnitude without flowing (λ = 0). A bulk solid lies in between (0 < λ < 1) – in the fluidised state it approaches the behaviour of a liquid.
Principal stresses
If a volume element is cut so that no shear stresses act on its boundary surfaces, the normal stresses there are the principal stresses. For bulk solids technology, the major principal stress σ1 and the minor principal stress σ2 are sufficient (plane stress state). In sections rotated relative to the principal stress planes, by contrast, normal and shear stresses occur at the same time.
The Mohr stress circle
The Mohr stress circle represents the normal and shear stresses in all planes of a volume element graphically. It intersects the σ-axis at the principal stresses σ2 and σ1, where the shear stress vanishes, and is completely described by these two values. It is the most important tool for illustration and evaluation in bulk solids mechanics.
With increasing consolidation stress σ1 the circle grows, and σ2 shifts accordingly via the lateral pressure ratio. If a stress circle touches the yield locus, the bulk solid is about to flow; stress states below the yield locus are stable, those above it physically impossible. The major principal stress σ1 of the circle that touches the yield locus at the pre-shear point causes the consolidation of the bulk solid and serves as the reference value for the bulk solid functions.
A stress circle through σ = 0 that touches the yield locus gives the unconfined compressive strength σc of the consolidated bulk solid. The ratio ffc = σ1 / σc describes the flowability; the unconfined compressive strength is at the same time the basis for assessing arching in the silo.
Shear testers in bulk solids technology
Shear testers originate from soil mechanics, where they are used to determine the shear strength of soils. A. W. Jenike transferred these principles to the bulk solids of process engineering and derived from them material properties for designing silos with trouble-free discharge. Since then, the behaviour of bulk solids relevant to process engineering can be predicted from shear tests.
What can be derived from flowability
The measured parameters answer different questions – from comparing two batches to dimensioning the silo.
| Question | Parameters required |
|---|---|
| Comparing bulk solids, batches and formulations | ffc, FLR and FLA at the same consolidation stress |
| Arching at the outlet | flow function σc(σ1), effective angle of internal friction φe, wall friction, density |
| Ratholing in funnel flow | flow function, angle of internal friction φi, effective angle of internal friction φe, density |
| Mass flow or funnel flow | effective angle of internal friction φe, wall friction on the hopper material |
| Silo loads to EN 1991-4 | φe, K, wall friction coefficient μ, bulk unit weight γ, cohesion |
| Flow behaviour after storage time | time yield loci, time flow function σc,t(σ1), wall friction after storage time |
| Effect of moisture | yield locus and wall friction at different moisture contents |
| Cause of a silo problem | depending on the symptom – from yield locus to wall friction |
Frequently asked questions about flowability
- What is the Mohr stress circle used for?
- It represents the normal and shear stresses of all planes of a volume element and is completely described by the major and minor principal stresses σ1 and σ2. If it touches the yield locus, the bulk solid starts to flow – this gives consolidation stress, unconfined compressive strength and flowability.
- How is the flowability of a bulk solid measured?
- The measurement in the shear tester is meaningful: the yield locus shows at which stresses the consolidated bulk solid starts to flow. This gives internal friction, cohesion, unconfined compressive strength and the flowability factor ffc. Simple methods on the loose bulk solid such as the angle of repose only give an indication.
- Why is a measurement on the loose bulk solid not sufficient?
- For cohesive bulk solids, the flow behaviour depends on the consolidation and the history – the angle of repose can also take different values depending on the preloading. Measurements at or near the bulk density ρb0 therefore allow no conclusions about the storage and flow behaviour under load.
Have your bulk solid measured
Describe your bulk solid and your question – we choose the right measurements and prepare a non-binding quote for you.
- Yield locus, flowability ffc and flow function
- Wall friction, density, time consolidation and moisture as required
- Results online in the customer portal and as a test report