ball-on-three-balls test

Initial work on the ball-on-three-balls (B3B) test began in the 1980s and was subsequently expanded by Austrian materials scientist Andreas Börger et al.{{Sfn|Staudacher et al.|2021|page=3}}{{Sfn|Börger|Supancic|Danzer|2002|pp=1425-1436}} Earlier methods, such as the ring-on-ring (RoR) and ball-on-ring (BoR) tests, were already in use but presented limitations related to stress uniformity, specimen preparation, and the sensitivity to geometric imperfections like surface curvature or waviness.{{Sfn|Börger|Supancic|Danzer|2002|p=1425}} The B3B test configuration was introduced as a modification of the ball-on-ring setup, which supports the specimen with a continuous ring.{{Sfn|Fett et al.|2007|pp=2-4}}

The B3B test is a biaxial flexural strength testing method developed for brittle materials,{{Sfn|Danzer et al.|2007|p=1481}} such as ceramics and cementitious materials,{{Sfn|Fu|Weiss|2020|p=128}} in the form of discs or plates.{{Sfn|Lube|Rasche|Nindhia|2016|p=249}} The test is intended to replicate multiaxial stress states, providing strength measurements that more closely reflect real-world loading conditions compared to conventional uniaxial testing methods.{{Sfn|Fett et al.|2007|p=1}} In the B3B test, the specimen—typically a circular disc or a rectangular plate{{Sfn|Lube|Rasche|Nindhia|2016|p=249}}—is supported on its underside by three equally spaced spherical balls that form an equilateral triangle.{{Sfn|Danzer et al.|2007|p=1482}} The specimen is then loaded on the opposite side by a fourth ball, which is positioned centrally to the three supporting balls,{{Sfn|Danzer et al.|2007|p=1482}}{{Sfn|Lube|Rasche|Nindhia|2016|p=249}} applies a compressive force to the centre of the opposite face.{{Sfn|Börger|Supancic|Danzer|2004|pp=283}} The four balls are typically of the same size.{{Sfn|Danzer et al.|2007|p=1482}}{{Sfn|Lube|Rasche|Nindhia|2016|p=249}} This symmetric configuration results in a biaxial stress field in the specimen, with the maximum tensile stress occurring at the centre of the supported face.{{Sfn|Danzer et al.|2007|p=1482}}{{Sfn|Börger|Supancic|Danzer|2002|pp=283}}

The test is generally conducted by gradually increasing the load until the specimen fractures.{{Sfn|Lube|Rasche|Nindhia|2016|p=250}} Fixture designs often include radial grooves to position the support balls and centring pins or screws to align the specimen.{{Sfn|Fett et al.|2007|p=3}} A small preload (typically 10% of the expected fracture load) may be applied initially to stabilise the setup before loading begins.{{Sfn|Harrer et al.|2008|p=1271}} The B3B test is often compared to the Small Punch Test (SPT), as both use miniature disc specimens. While the SPT involves clamping the specimen and deforming it with a small punch, the B3B test offers a contact geometry that is more suitable for testing unmachined or as-sintered surfaces.{{Sfn|Rasche et al.|2014|p=962}}

The test setup often includes precision alignment aids such as centring rings and a punch mechanism that transmits load from a mechanical testing machine. Components in contact with the specimen are usually made from high-purity aluminium oxide ({{chem2|Al2O3}}), ensuring chemical and mechanical stability during testing.{{Sfn|Aneziris|Biermann|2024|pp=595-596}}

Testing is typically performed under displacement control at a constant rate until the specimen fails. The load is recorded continuously using a load cell. From the critical load at fracture, the maximum tensile stress can be calculated using analytical equations or finite element models.{{Sfn|Aneziris|Biermann|2024|p=595}}

High-temperature testing is possible using specialized furnaces, with reported extensions up to {{convert|1400|-|1600|C|F}}.{{Sfn|Aneziris|Biermann|2024|p=595}} An argon gas environment is often employed to prevent oxidation or decarburization during elevated-temperature tests.{{Sfn|Aneziris|Biermann|2024|p=586}}

The B3B test induces a multiaxial stress state, which is more representative of real-world conditions such as thermal shock or contact loading. The maximum principal tensile stress at the specimen centre is used to define the material's biaxial strength. For linear elastic discs, this stress ($\backslash sigma\_\{max\}$) is calculated using the equation:{{Sfn|Börger|Supancic|Danzer|2002|p=1429}}{{Sfn|Harrer et al.|2008|p=1272}}

$\backslash sigma\_\{max\}\; =\; \backslash frac\{F\}\{t^2\}\; f$

where $\{F\}$ is the applied force, $\{\{t\}\}$ is the thickness of the specimen, and $\{f\}$ is a dimensionless geometry factor determined by finite element method (FEM). This factor depends on specimen and support geometry as well as the material's Poisson's ratio ({{mvar|ν}}).{{Sfn|Börger|Supancic|Danzer|2002|p=1429}}{{Sfn|Harrer et al.|2008|p=1272}} FEM has been extensively used to derive empirical formulae for stress evaluation across a wide range of materials and geometries.{{Sfn|Börger|Supancic|Danzer|2002|pp=1429-1430}}

The B3B test lends itself to statistical strength analysis, often based on the Weibull distribution. This accounts for the size effect in brittle fracture, where larger specimens typically exhibit lower average strength due to the increased likelihood of critical flaws. Because the stress field in the B3B test is non-uniform, equivalent stress criteria—such as the principle of independent action (PIA) or the maximum principal stress criterion—are used to interpret the data.{{Sfn|Rasche et al.|2014|p=964}}

The B3B test can be adapted for fracture toughness ($K\_\backslash text\{Ic\}$) testing by introducing a specimen that is supported by three equidistant balls and loaded on the opposite face by a fourth, centrally positioned ball. A semielliptical surface crack on the loaded face of the specimen. After applying a small preload, the load is increased until fracture, and the crack dimensions are analysed post-failure.{{Sfn|Lube|Rasche|Nindhia|2016|p=249}}

Conventional fracture toughness tests require relatively large specimens, which may be unsuitable for materials processed in small quantities or in thin geometries. The B3B-$K\_\backslash text\{Ic\}$ method addresses this limitation by using circular or rectangular plate specimens and integrating a precrack to enable linear elastic fracture mechanics (LEFM)-based evaluation.{{Sfn|Lube|Rasche|Nindhia|2016|p=250}} When combined with simulations (e.g., FEM or cohesive zone model), B3B results can be used to extract fracture toughness values and fit cohesive parameters by comparing simulated and experimental load–deflection curves.{{Sfn|Aneziris|Biermann|2024|pp=595-596}}

The fracture toughness $K\_\backslash text\{Ic\}$ is calculated using the Griffith–Irwin equation:{{Sfn|Lube|Rasche|Nindhia|2016|p=250}}

$K\_\backslash text\{Ic\}\; =\; \backslash sigma\_\{\backslash mathrm\{B3B\}\}\; \backslash cdot\; \{Y\}\; \backslash cdot\; \backslash sqrt\backslash pi\{a\}$

Here, $\backslash sigma\_\{\backslash mathrm\{B3B\}\}$ is the maximum tensile stress from the B3B configuration, $a$ is the crack depth, and $Y$ is a dimensionless geometric factor derived via FEM. Because $Y$ varies and depends on specimen and crack geometry as well as the Poisson's ratio {{mvar|ν}}. For semicircular or semielliptical surface cracks, $Y$ varies along the crack front, and the maximum value $Y\_\{\backslash mathrm\{max\}\}$ is used for evaluating toughness. Calculated values for $Y\_\{\backslash mathrm\{max\}\}$​ at various parameter combinations are available in the literature for {{mvar|ν}} = 0.3.{{Sfn|Lube|Rasche|Nindhia|2016|p=250}}

$\backslash sigma\_\{\backslash mathrm\{B3B\}\}$ can be expressed by empirical formula:{{Sfn|Lube|Rasche|Nindhia|2016|p=250}}

$\backslash sigma\_\{\backslash mathrm\{B3B\}\}\; =\; \backslash frac\{F\}\{t^2\}\; \backslash cdot\; f\backslash biggl(\backslash frac\{t\}\{R\},\backslash frac\{R\_\backslash text\{a\}\}\{R\},\backslash nu\backslash biggl)$

where the dimensionless factor $\{f\}$ accounts for the specimen's geometry through three dimensionless parameters: the thickness-to-radius ratio ($\backslash frac\{t\}\{R\}$), the loading radius-to-specimen radius ratio ($\backslash frac\{R\_\backslash text\{a\}\}\{R\}$), and the Poisson's ratio ({{mvar|ν}}). Although originally intended for circular discs, an effective specimen radius can also be calibrated for rectangular plates. The loading radius ($\{R\_\backslash text\{a\}\}$) is determined by the ball diameter ($\{D\_\backslash text\{B\}\}$) of the four balls, given as $\{R\_\backslash text\{a\}\}\; =\; \backslash frac\{D\_\backslash text\{B\}\}\backslash sqrt3$ .{{Sfn|Lube|Rasche|Nindhia|2016|p=250}}

Accurate positioning of the crack is essential, as lateral misalignment can lead to significant overestimation of $K\_\backslash text\{Ic\}$. However, rotational misalignment has negligible effect on test results.{{Sfn|Lube|Rasche|Nindhia|2016|p=254}}

A modified three-balls-on-three-balls (3B3B) version uses three balls for both the loading and supporting forces. This variant of the test was patented in 2004 to expand the effective high-stress region on the specimen surface, addressing one of the limitations of the original B3B setup.{{Sfn|Fett et al.|2007|p=5}}

For a specific case where the loading and supporting balls are aligned and the disk geometry has a radius ratio $\backslash frac\{R\}\{R\_1\}\; =\; 2.25$, the maximum principal stress $\backslash sigma\_\{\backslash text\{max\},1\}$​ can be estimated using the following empirical relation derived from finite element analysis:{{Sfn|Fett et al.|2007|pp=14-15}}

$\backslash sigma\_\{\backslash text\{max\},1\}\; \backslash frac\{t^2\}\{F\}\; =\; 0.656\; \backslash left(\; \backslash frac\{t\}\{R\_1\}\; \backslash right)^\{-0.196\}\; +\; 0.274\; \backslash left(\; \backslash frac\{t\}\{R\_1\}\; \backslash right)^\{-0.448\}\backslash nu$

Here, $\{F\}$ is the applied force, $t$ the specimen thickness, $R$ the specimen radius, and {{mvar|ν}} the Poisson's ratio. The expressions $\backslash left(\; \backslash frac\{t\}\{R\_1\}\; \backslash right)$ serve as a dimensionless geometric factor specific to the 3B3B arrangement and highlights the reduced stress concentration resulting from load distribution across three points.{{Sfn|Fett et al.|2007|pp=14-15}}

The three-on-three configuration may also vary in terms of alignment, with some designs positioning the loading and supporting balls directly in line and others offset by angular displacements such as 60 degrees.{{Sfn|Fett et al.|2007|p=5}} FEM has been used to analyse stress states in this variation, which has shown significantly reduced maximum principal stresses for the same total load compared to the one-on-three B3B test.{{Sfn|Fett et al.|2007|pp=9}}

The B3B test and its variations are widely used for evaluating the strength and toughness of brittle materials, including:

• Ceramic engineering (e.g., alumina, silicon nitride){{Sfn|Danzer et al.|2007|p=1483}}
• Electroceramics (e.g., zinc oxide varistors){{Sfn|Danzer et al.|2007|p=1483}}
• Dental materials (e.g., CAD/CAM restorative blocks){{Sfn|Wendler et al.|2017|p=99}}

Its tolerance for surface roughness{{Sfn|Börger|Supancic|Danzer|2002|p=1434}} and geometric imperfections{{Sfn|Harrer et al.|2008|p=1271}} makes it well-suited for as-sintered samples,{{Sfn|Rasche et al.|2014|p=966}} providing a practical solution for industrial quality assurance and rapid screening.{{Sfn|Harrer et al.|2008|p=1271}}

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{{refend}}

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