Validation of fixture used for ROR equibiaxial flexural strength test

The flexural strength measured by the ring-on-ring configuration can be biased when test pieces are not sufficiently parallel/flat, when there is a friction between the fixture and the test-piece or when rings are not concentric. To validate the strength results obtained with the custom fixture, a set of 40 alumina samples was ordered from Coorstek Inc. for direct comparison with a published Army Research Laboratory (ARL) report from Wereszczak, Swab and Kraft [123].

The set comprised 99.5 % alumina discs (AD-995) of 20.0 ± 0.1 mm diameter and 1.00 ± 0.05 mm thickness. Specific properties of AD-995 are listed in Table 4-5. Tolerance for parallelism and flatness was specified as 0.01 mm (according to ASTM standard C1499-15 [16]).

Table 4-5 Reported properties of AD-995 alumina ceramics [124]

Property Testing method Reported value

Density (g∙cm⁻³) ASTM C20 3.90

Average grain size (µm) Thin section 6.0

Flexural strength (MPa) ASTM F417 379

Elastic modulus (GPa) ASTM C848 370

Poisson’s ratio ASTM C848 0.22

Hardness Rockwell 45N 83

Fracture toughness (MPa∙m^1/2) Notched beam 4−5

Generally, the strength of specific material measured by different testing methods can be scaled using an effective area (SE) or effective volume (VE) [15, 125], which represent the portion of sample volume/surface under tensile stress that has the same probability of failure as a specimen of corresponding size tested in a simple tension. Both parameters were previously described in section 2.1.1.

It is implied that the effective surface should be used when fracture originates from surface flaws and the effective volume should be used when volume flaws are the strength-limiting

features. To satisfy this criterion, every fracture origin must be properly determined, and results must be strictly censored. In the case of the ARL report, effective surface was used despite some of the critical flaws being volume type, as it would be too laborious to characterize the fracture origin of every test piece. Moreover, some of the critical flaws such as machining damage and porosity seemed to be combined. Since alumina discs used in this work presumably contained the same flaws, a similar approach was taken, and effective surface was used for estimated strength scaling.

The effective area of a specimen tested in 4-point flexure can be calculated using the following equation: specimen and m is the Weibull modulus [15].

For multiaxial stresses, equation (2-3) must be changed using one of the available multiaxial failure models. These include the Principle of Independent Action (PIA), the Batdorf model and the Multiaxial Elemental Strength model [15, 126-128]. The principle of independent action states that the 3 principle stresses operate independently in their respective directions. It doesn’t consider shear/compressive stresses, combined local principle stresses or the nature of defects.

Therefore, the PIA failure model is phenomenological. The Batdorf model incorporates the effect of multiaxial stresses into the weakest-link theory based on linear elastic fracture mechanics (LEFM). Flaws are randomly oriented and non-interacting, shear and normal forces are incorporated [128, 129]. Lamon argued in his paper that the Batdorf theory does not consider compressive components [128]. These components were considered in the multiaxial

elemental strength model, which he developed. There are no closed-form expressions of effective surface or volume available for his model.

The Batdorf theory was used to calculate the effective area of biaxially stressed discs in this work as it was previously successful in predicting biaxial strength from 4-point bend test results [127, 130]. This theory was also used in the ARL report [123].

According to Batdorf [125], the effective area of a specimen tested in a ring-on-ring equibiaxial flexure configuration can be approximated using the following expression:

𝑆 𝑅𝑂𝑅 2𝜋 𝐷

2 𝑚 ^. , (4-13)

where DL is the load ring diameter and m is the Weibull modulus.

Calculated effective surface of specimen 1 (SE,1) and its Weibull characteristic strength (σ0,1) can be subsequently used to predict the Weibull characteristic strength of specimen 2 (σ0,2) with effective surface S^E,2 using the following expression:

𝜎 _,

The mechanical test results are summarized in Table 4-6. Results from the current work are compared to the ARL report results of AD-995 alumina specimens tested in 4-point bend (chamfered rectangular section blocks of 3 × 4 × 50 mm, where 3 mm is the height). Surfaces of both batches were rotary ground to 80-grit finish. Calculated effective surfaces of the two configurations were very similar. Using equation (4-14), the predicted strength of specimens tested in current work was the same as measured in 4-point bend, 294 MPa. ROR strength was 2.3 % lower (287 MPa). This can be interpreted as a good match considering that both materials were produced at different locations and 12 years apart. There is also some inaccuracy present

in the Batdorf scaling model itself and its approximated form that has been used. Weibull moduli can be compared directly and show a good agreement, since their values are well within their respective 95 % confidence intervals. Based on these observations, it can be concluded that the custom-made jig was well designed and gives satisfactory results.

Table 4-6 Mechanical test results of rotary ground AD-995 alumina (80-grit finish)

Source Test type

† [± 95% confidence intervals]

Figure 4-15 shows the Weibull plot of the alumina pieces tested in this work and from the ARL report. It appears that points measured in this work could be fitted with multiple lines of different slopes which suggests a presence of multiple strength-limiting flaw populations. A general discussion of deviations from Weibull distribution can be find later in section 8.4.4.

The results from the ARL report do not exhibit such behaviour. The discrepancy might be caused by differences in stress conditions, character of flaws, or by unknown testing issues.

Fractography revealed that most strength-limiting flaws were volume type – porosity, large grains and agglomerates, some also hybridized with machining damage, similar to findings in the ARL report. Unfortunately, no distinct strength-limiting flaw populations were identified.

A few examples of strength-limiting flaws are shown in Figure 4-16.

Figure 4-15 Weibull plot of AD-995 alumina tested in the ring-on-ring biaxial flexural strength test compared with the results of alumina tested in 4-point bend from the ARL report [123].

Figure 4-16 SEM micrographs showing examples of fracture origins found in tested Al2O3

specimens: a) Cluster of large grains, b) agglomerate likely coupled with machining damage on the surface.

220 240 260 280 300 320

1 5 10 50 90 99.999

Probability of failure, Pf (%)

Failure stress (MPa) Current results

ARL report