Supplementary Information

Compression Testing Spherical Particles for Strength: Theory of the Meridian Crack Test and Implementation for Microscopic Fused Quartz

Václav Pejchal*, Goran Žagar, Raphaël Charvet, Cyril Dénéréaz and Andreas Mortensen

Laboratory of Mechanical Metallurgy, Institute of Materials, École Polytechnique Fédérale de Lausanne

(EPFL), Station 12, CH-1015 Lausanne, Switzerland

*Corresponding author: Václav Pejchal, vaclav.pejchal@epfl.ch

Supplementary Section 1. Finite element analysis of a spherical particle crushing test

The stress field obtained by Hiramatsu and Oka equations, Eqs. (3a-d), for compressed spherical particle, is here briefly compared with results obtained via a basic finite element model of the problem at hand, which was implemented in the Abaqus FEA v 6.11 (Dassault Systèmes Simulia Corp., Providence, RI, USA) software.

We model the particle as an isotropic, linear-elastic solid of elastic modulus of Young’s modulus 72 GPa and Poisson ratio 0.17 (these values are characteristic of fused quartz). The platen material is considered to be an isotropic, elastic-perfectly-plastic steel of Young’s modulus 200 GPa, Poisson’s ratio 0.3 and yield strength 1.3 GPa (these properties are characteristic of one of the platen materials used here, namely HV450, Table 1 of the main text, and were measured by tensile testing in our laboratory). The finite element mesh used for this problem, is shown in Figure S1a; due to symmetry we only account one-half of the axisymmetric model in the problem. Elements used to discretize the particle and the platen parts are quadratic (cax8) quadrilaterals. Contact between particle and platen in the model is explicitly included; this includes hard contact formulation in the normal direction of the contact surface and Coulomb friction characterized by a constant friction coefficient μ in the tangential direction. Loading is by prescribing vertical displacement uy of the top surface of the platen. Preliminary runs were performed to optimize the mesh size and the size of the platen domain in order not to affect the predicted first principal stresses in the center and at the equator of the particle. During simulations we monitor the contact pressure distribution over the entire particle surface; from this we calculate the contact radius a as the x-coordinate of the point on the particle surface at which the contact pressure drops to zero.

(a) (b)

Figure S1. (a) Sketch of an axisymmetric finite element mesh used for modeling the crushing test of spherical particles by elastic-perfectly-plastic platen. Prescribed vertical boundary displacement uy along the top platen surface was used to load the system. (b) Stress field contour map for particle compressed up to contact radius a /R ≈ 0.85 for case of frictionless contact between the sphere and the platen.

In Figure S2a we compare the evolution of the normalized first principal stress, σ 1 π R2/F , in the center and along the equator of a spherical particle as calculated via the Hiramatsu and Oka solution for a particle Poisson’s ratio equal to 0.17 (black) and via FEM (colored lines). Due to the unknown experimental contact friction coefficient, in Figure S2a we explore FEM calculations over a range of μ values. Although the Hiramatsu and Oka solution generally follows the same trend as the FEM results, the Hiramatsu and Oka solution seems to increasingly underestimate the first principal stress as the relative contact radius increases; up to a/R ~0.8, the first principal stress at the particle equator is underestimated by a factor of ~1.3. The main reason for this discrepancy could originate from the assumption that the contact pressure distribution in the Hiramatsu and Oka treatment is considered to be uniform, while the FEM calculations (Figure S2b), although somewhat noisy, indicate that this assumption is approximately valid only for a /R < 0.3. Still, as can be seen, given the complexity of the problem and the presence of unknown boundary conditions (of friction notably), one can see that the Hiramatsu and Oka solution is a relatively faithful descriptor of the stress state in hard elastic spheres as they are compressed between two elastoplastic platens.(a) (b)

Figure S2. (a) Normalized first principal stress σ 1 π R2/F as a function of the relative contact radius a /R in the center of the sphere (dashed lines) and on the surface equator (solid lines). Solution by Hiramatsu and Oka, Eqs. (3a-d, main text), is shown with black lines and results of FEM calculations with colored lines. FEM calculations are carried out for three values of the contact friction coefficient: 0 (red), 0.1 (green) and 0.3 (blue). (b) Distribution of the contact pressure at contact radii a /R=¿ 0.2, 0.4 0.6 and 0.8 and for the case of contact friction μ = 0 (red) and 0.3 (blue).

Supplementary Section 2. Elastic contact with a particle representing an inhomogeneity along the platen-particle contact area

Let us imagine a situation where, during the compression test (as described in the main text of the present paper), a debris particle gets trapped between the fused quartz particle (here considered to have radius R=20µm) and the steel platen (see Fig. S3a). The debris particle is located at some point A on that portion of the surface of a fused quartz particle that is in direct contact with the platen. Let the perimeter of the contact between the compressed fused quartz particle and the platen be designated as the ensemble of Points B. The debris particle may be a graphite particle from the colloidal paint used in this study or from other sources, e.g. visible debris that is covering the tested particles (Fig. 7b, main text). The presence of a debris particle has the potential to locally redistribute the stress; we examine here the question in detail.

Fig. S3. (a) Sketch of a spherical particle of radius R compressed between two elasto-plastic platens under load F with a far smaller debris particle present in-between the compressed particle and platen at Spot A. The contact perimeter between the compressed particle and steel platen is indicated by the letter B. (b) The debris particle of Young’s modulus E’, Poisson’s ratio ν’ and radius ρ (red) indents the larger tested particle (grey) under uniaxial compression with a force P, creating a local elastic contact of radius b. The local cylindrical coordinate system x , z , α is at the debris particle plane of symmetry taken to be homothetic with the compressed sphere coordinate system θ , r ,φ.

For simplicity, let us consider that both the compressed fused quartz particle and the debris particle are elastic solids of Young’s modulus and Poisson’s ratio E, ν and E’, ν’, respectively. Also, let us assume that the debris particle is spherical with radius ρ considerably smaller than that of a fused quartz particle (ρ ≪ R). In the limit of elastic contact between the two particles, the local situation under loading can be approximated as the indentation of a debris sphere into the fused quartz half-space, as described by Hertz’s theory (see sketch in Fig S3b).

The stress field equations for the Hertzian indentation can be found for example in (Louapre and Breder, “Hertzian Indentation Stress Field Equations”, Int. J. Appl. Ceram. Technol. 12, 1071–1079 (2015)); in the cylindrical coordinate system (x, z, α) it has four non-zero components; σ x , σz , σ α , σ zx. The highest tensile stress attained due to indentation of a debris particle will be located on the fused quartz particle surface where the shear component σ zx vanishes; the remaining three normal stress components on the surface then automatically become the principal stresses, of which only σ x is positive with its peak value located at the perimeter of the contact characterized by the radius b (see Figure S3b). Because the debris particle is located in the contact zone between the steel platen and the fused quartz particle (point A, Figure S3a), and is compressed by pressure q, the force P with which the debris is pressed against the fused quartz is approximated here as P ≈ qπ ρ2 (somewhat higher values may obtain; however, this should give the right order of magnitude). With this, the only relevant (positive) Herzian stress component σ x as a function of a coordinate x can be

easily calculated. A few examples of normalized ~σx ( x )=σ x (x) ∙ π R2/ F for different debris particle sizes and their elastic parameters at arbitrary point A are shown in Fig S4 with colored lines.

For a fused quartz particle compressed by the platens, this local σ xwill superimpose onto the Hiramatsu and Oka stress field. Because the shear component σ rθ of the Hiramatsu and Oka solution also vanishes on the surface of a fused quartz particle, from the remaining three principal components, σ θ , σ r and σ φ, only σ θ is worth considering, because only σ θ can become significantly positive (tensile) after superposition with σ x. Note that, at Point A of the contact in the plane of symmetry of the debris particle, the local cylindrical system (z, x, α) is homothetically related to the global coordinate system (r , θ ,ϕ) (see Fig S3a), which means that the stress components σ z and σ r, σ x and σ θ and σ α and σ ϕ, are directly additive; however, because σ r, σ ϕ, σ z and σ α<0, only the addition of σ x and σ θ can give a positive (tensile) value. Normalized Hiramatsu and Oka stress component ~σθ=σθ ∙ π R2/F particle with radius R=20µm compressed with relative contact radius (a/R) = 0.7 and for Meyer’s law describing HV600 platen is traced in Fig S4 as the black line.

From Figure S4 it is evident that, if the debris particle at point A is located far away from the perimeter point B (the distance between points A and B is |AB|>ρ), then the compressive stress ~σθ is so high that even indentation by a very stiff debris particle (with properties similar to alumina) can not locally raise the total stress, ~σθ+

~σ x, along the fused quartz surface to make it positive (tensile). High tensile stress on the fused quartz particle surface is only possible when the debris particle is located very near the contact perimeter at B, i.e. |AB|≈ ρ, where ~σθ is steeply rising to a value that is close to zero (Figure S4, black line); only at this relative position will the local stress ~σx induce the total stress on the fused quartz surface to be tensile. If this happens, we find that for any debris particle size or debris elastic properties, the total stress at point B is ~σθ+

~σ x ≈ 0.3, which is comparable to the peak stress at the equatorial plane for the investigated relative contact radius (a/R) = 0.7 (see Fig. 2, main text). The volume of the stressed material around the Hertzian indentation is also much smaller than the volume of material exposed to tension around the equator. Hence, failure from tensile stresses along the equator will be a far more probable outcome than failure induced by a small debris particle.

Fig. S4. Distribution of the normalized stress σ θ along the surface of a compressed fused quartz particle due to (black) uniaxial compression with relative contact radius (a/R)=0.7 and platen with Meyer’s constants representing HV600 steel platen, (green) a Hertzian indentation stress along the surface of fused quartz particle by a debris particle with relative radius ρ=0.2 µm, elastic modulus of 72 GPa and Poisson’s ratio 0.17, (red) a Hertzian indentation stress along the surface of fused quartz particle by a debris particle with relative radius

ρ=1µm, elastic modulus of 400 GPa and Poisson’s ratio 0.25 and (blue) a Hertzian indentation stress along the surface of fused quartz particle by a debris particle with relative radius ρ=1µm, elastic modulus of 72 GPa and Poisson’s ratio 0.17. The dashed vertical line represents the contact perimeter defined by the compressed particle-platen contact radius a.