A Newtonian liquid is incompressible and transfers only compressive stresses with no tensile strength. The pressure in a liquid spread is isotropic, such that the relationship (λ) between the horizontal and vertical pressure is one.
Stress ratio (λ) = horizontal pressure(σh) / vertical pressure(σv)
An ideally stiff solid however transfers pressure and tension. It deforms under the influence of reversibly but doesn't flow; hence, λ=0.
A bulk solid transmits small tensile strengths, compressive and shear stresses and flows under the influence of shear stresses if large enough. In a fluidized state a bulk material may behave like a liquid or in the other limiting case like a solid. Therefore, stress ratios between the horizontal pressure (σh) and vertical pressure (σv) are possible in the following range: 0 <λ <1.
|Plastic bulk materials(1)||φi=0||Τc>0|
|Cohesive bulk materials (2)||φi>0||Τc>0|
|Cohesion less bulk materials (3)||φi>0||Τc=0|
|Fluidized bulk materials (4)||φi=0||Τc=0|
Ideal plastic powders are mostly saturated and have very high cohesion. It is difficult for the individual particles to change position. Because of the saturation there is no influence from the compression load and the packing density. Consequently, due to incompressibility, the change in pressure cannot change the number of contacts and the shear stresses are only dependent on cohesion.
This materials with high cohesion are difficult to handle. Other samples only show plastic behaviour at higher stresses. By measuring at different loads this phenomenon can be demonstrated.
For cohesive powders, the inter-particular forces are frictional and cohesive.
The rearrangement of particles is limited by cohesion, such that previous compression from the normal and shear stresses have an influence on later behaviour. The slope angle depending on previous history can take different positions. To get qualified physical properties, measurements must be performed in shear testers where the flow behaviour is measured under load. This bulk material can be described by the equation: Τ = σ * tan(φi) + Τc.
Cohesive powder has a stationary value, which is independent of the previous normal stresses. The line through stationary points is the same for all measured data at all different yield loci for consolidation stresses. The powder in a flowing condition is nearly cohesionless. Consolidations can therefore be undone.
In practice, this group of bulk materials is the most common. Depending on the flow properties, these materials are difficult to handle. However, determining the physical properties under normal operating conditions is essential. The compressive strength can cause bridging, if the hopper angle or outlet are not adequate. Time consolidation should be considered as well.
For free flowing materials only frictional forces between particles exist. The number of contacts can be changed through changing the pressure. Because there is no cohesion, a complete rearrangement of the particles is possible. Free flowing bulk materials can be described by the linear equation τ=σ*tan(φi). The yield locus of such materials passes through the origin of the σ-τ diagram and the internal friction angle (φi) corresponds to the angle of repose of the bulk material.
These bulk materials are typically easy to handle. Due to the absence of compressive strength, bridging does not occur and preloads are reversible. The bulk density at normal stresses and the stress ratio should be determined.
Fluidized powders are cohesionless and internal friction become zero due to high particle distances. In the fluidized state, the bulk solids behave more like a liquid.
This can occur during the filling of silos or the collapsing of a bridge or rate hole. In such cases many problems are caused, like flooding out of a silo.
By measuring the yield locus, several physical properties can describe the behavior of the bulk material. For the classification of powders, and to express the flowability in a single parameter, the flowability factor (FL = ffc = σ1 / σc) was developed. The flowability factor provides a quantitative characterization of the flowability of a powder. This factor is dimensionless and enables the classification of bulk materials into classes:
|25 >||FL||> 15||slightly cohesive|
|15 >||FL||> 5||cohesive|
|5 >||FL||> 2||very cohesive|
|2 >||FL||> 1||plastic|
For materials which very different densities, an extended calculation basis for the flow factor was developed:
Relative flow factor: FLR = ( σ1 - σ2 ) / σc
Absolute flow factor: FLA = FLR * ρb0
Each measured yield locus is based on the individual consolidation stress or the major principal stress (σ1). It is only possible to compare materials using the same reference stress. Usually, better flowability will be obtained at higher consolidation stress. By measuring several yield loci at different consolidation stresses the flow function is obtained, which describes the flow factor over an operational range.
The flow function describes the dependence of the unconfined compressive strength (σc) to the principal stress (σ1). This is necessary in order to calculate the critical outlet diameter in the hopper. In simplified terms, the load on the bulk material bridge must be greater than the unconfined compressive strength so that the material can flow.