Shear testing

In order to determine the unconfined yield strength of the materials, shear testing was carried out using the cylindrical shear cell attachment of the FT4 Powder Rheometer (Freeman Technology, UK).

0 20 40 60 80 100 120 0 50 100 150 200 C u mula ti ve V o lu me (%) Particle Size (μm) Maize starch Maltitol Pea protein

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The powder was poured through a funnel into the split cell chamber of 50 mm diameter to a bed height of approximately 50 mm. The 48 mm diameter FT4 blade was rotated clockwise and driven downwards into the bed at a helix angle of 5o _{and tip speed of -60 mm/s until it was 10 mm vertically}

above the base, after which the helix angle was reduced to 2o _{and the tip speed to -40 mm/s until it}

was 1 mm vertically above the base. The impeller was then driven clockwise upwards through the powder bed at a helix angle of -5o _{and tip speed of 60 mm/s until it exited the powder bed and was}

clear of the chamber. This step serves the purpose of preconditioning the powder bed, bringing it to a reproducible state, as mentioned in section 2.3.4.2.

Then, the blade was removed and replaced by the 48 mm vented piston, which is used to compact the powder at the desired pre-shear normal stress, allowing the entrained air to escape. The piston was driven downwards at a speed of 0.5 mm/s until the surface of the powder bed was detected and a selected small force, depending on the chosen pre-shear normal stress, was registered. Following this, the piston’s speed was reduced to 0.08 mm/s until the desired consolidation stress was reached. Once the desired stress was reached, the compaction force was held for 60 s before it was unloaded. The piston was then removed, and the chamber was split horizontally prior to shearing, leaving a bed height of approximately 43 mm.

After that, the 48 mm diameter shear head, with 18 radially aligned vanes of 3 mm depth, was attached to the device and driven downwards to the powder bed in exactly the same manner as the vented piston. The normal stress was maintained in order to remove any disturbances caused by the split, and to ensure that the powder bed surface was suitably consolidated, whilst the shear head was rotated at 0.05 rpm until steady-state flow was achieved. The pre-shearing protocol that was followed is the standard FT4 approach, which involves aspects of both the Jenike and Peschl protocols. According to this, multiple pre-shear steps are used in order to bring the powder to a critically consolidated state, with a maximum number of pre-shear steps being specified; between 2 and 16 steps were used in this work. Each pre-shear step is considered complete 20 seconds after the maximum shear stress has been established with no subsequent increase in shear stress. The pre- shear phase is considered complete when two consecutive pre-shear steps have maximum shear stresses which show a difference of less than 1 %. If this criterion is not achieved before the maximum number of pre-shears has been reached, then the test proceeds to shearing, regardless of the similarity between the pre-shear shear stresses, with the user not receiving any notification about this, therefore the user has to be cautious and manually discard the test results if this happens.

Once steady-state flow was attained, shearing was stopped, the normal stress was reduced to the normal stress intended for the actual measurement, and shearing at the same rotational speed was

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performed until a point of incipient failure was obtained. Finally, shearing was stopped, and the shear stress was returned to zero by rotating the shear head in the reverse direction. This procedure of pre- shear followed by incipient failure measurement was repeated at different ranges of decreasing applied normal stress. At each pre-shear normal stress, the shear cell software takes the measured shear stress at each normal stress to generate a linear yield locus for this packing state. Application of Mohr circles, as described in section 2.1.1, allows the major principal stress, σ1, and unconfined yield strength, σc, to be determined for each pre-shear normal stress. In addition to this, the flow factor, ffc, is determined at each stress. For each material, three repeats were made at each pre-shear normal stress, and the average results are reported throughout the thesis, with the error bars indicating the standard deviation of the measurement. The overall procedure of the FT4 shear test is depicted in Figure 3.29.

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3.2.2 Ball indentation

The ball indentation method was applied to all materials assessed in the FT4 shear cell. The criteria for sample, die and indenter dimensions established by Zafar et al. (2017) were adhered to for this work. A 20 mm diameter stainless steel die, which was attached to a metal plate extending beyond the outer wall of the die, was filled by passing the powder through a sieve with an aperture approximately five times greater than d50 in the case of glass beads, and an aperture large enough to be able to break the large agglomerates and fill the die in a reasonable time (1 - 2 minutes) in the case of all other cohesive materials. The sieve was placed directly above a funnel, above the die. The die height was 20 mm, with a bed height of 15 - 20 mm generated in all cases, and the powder mass weighed. The die was placed below a stainless steel piston of 19.8 mm diameter attached to an Instron 1175 mechanical testing machine (Instron, USA) by a 1 N load cell, which has a resolution of 0.25 mN. Before each test was started, the metal plate to which the die was attached was driven towards the piston while the force was recorded (with the die offset to prevent contact with the die walls) until contact was made in order to determine the distance between the base of the die and the piston. After that, the plate was returned to its starting position, and the die was centred below the piston. At the start of the actual test, the die was driven upwards, towards the piston, at a vertical speed of 1 mm/min, therefore testing in the quasi-static regime, until the desired consolidation stress was reached. The final displacement of consolidation, zf, was recorded and used along with the distance between the base of the die and the piston at the starting point, z0, to determine the bed height, and consequently determine the packing fraction, χ, using Equation 3.1:

𝜒 =𝜌𝑏 𝜌_𝑡 = 𝑀/𝑉 𝜌_𝑡 = 4𝑀 𝜋(𝑧0− 𝑧𝑓)𝐷𝑑2 𝜌_𝑡 (3.1)

where ρt is the true density, V is the volume of the powder and Dd is the die diameter.

The sample was then unloaded at the same speed, and the piston was replaced by a 4 mm diameter, spherical, stainless steel indenter aligned centrally above the powder bed. The die was then driven upwards, towards the indenter, at the same speed as the consolidation step, until contact was detected, which was considered to be when a force of 2 or 3 mN (depending on the material) is registered. Following this, the penetration was continued until the desired penetration depth was reached, and the sample was then unloaded. The ball indentation setup is shown in Figure 3.30.

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Figure 3.30. Ball indentation setup (a: consolidation, b: penetration)

The bed hardness was calculated using Equation 2.16, with the projected area of the impression of the indenter determined using Equation 2.17. Equation 2.17 is valid as long as the penetration depth is equal to or less than the indenter radius. The penetration depth was non-dimensionalised using Equation 3.2, and dimensionless penetration depths, hd, in the range of 0.1 - 0.7 were applied to each powder at consolidation stresses of 0.1 and 1 kPa.

86 ℎ_𝑑=2ℎ𝑐

𝑑𝑏 (3.2)

If unloading has negligible effect on the material’s recovery, the penetration depth at maximum indentation load can be used in place of hc in Equation 3.2.

Hardness measurements have to be independent of the indentation load/penetration depth in order

to represent plastic yield stress (Zafar, 2013), therefore a dimensionless penetration depth determined to be in the stable hardness range was then applied in all experiments for the remaining consolidation stresses for a given powder. The ball indentation method was applied at the major principal stresses determined in the shear cell experiments at moderate to high pre-shear normal stresses, where it was assumed that the normal stress in the indentation process is equal to the major principal stress. The constraint factor was then determined at these major principal stresses. In addition, the ball indentation method was applied at low consolidation stresses of 0.1, 0.2, 0.4, 0.6, 0.8 and 1 kPa. The unconfined yield strength was then inferred at these stresses using Equation 2.18 and the established constraint factor for the powder. For each material, five repeats were made at each pre-shear normal stress, and the average results are reported throughout the thesis, with the error bars indicating the standard deviation of the measurement.

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