Fracture toughness of porous materials – Experimental methods and data

We provide numerical experimental data and detailed information about the sample preparation and the experimental methods, used by different research groups for measuring the fracture toughness of porous materials. These data are supplemental information to the publication “A Geometric Model for the Fracture Toughness of Porous Materials,” [1], which is based on experimental data of ceramic and polymer materials. For the sake of completeness, we provide here also data from fracturing metallic foams. The corresponding theoretical curves, which are based on the model described in the reference, are given additionally in the diagrams. The utilized publications are not a comprehensive compilation of all corresponding measurements concerning porous materials, but should be seen as a typical set of respective experiments with the focus on the fracture toughness of porous materials. The discussion and interpretation are provided in the above-mentioned reference.


Data
The main data are the normalized fracture toughness and corresponding errors (if available in the references) as a function of the normalized porosity. These data are carefully extracted graphically from figures in the given references [3e17]. Errors of a few percent like 1%e5% are possible, but they should be mostly below 2%. The only exception is Ref. [2], at which one of the authors sent us the numerical results. The graphical extraction is described in more detail in Ref. [1]. All of the data are provided in the Tables 2e5 and visualized in Figs. 1e5. After a major literature research and to the best of our knowledge, the given numerical data are not provided elsewhere. From the given references, the data of (mostly) isotropic materials and tests under quasi-static loading conditions are considered, in accordance to the model presented in Ref. [1]. For the sake of completeness, the data of the Young's moduli, used in Ref. [1], are provided in the Tables 3 and 4. The information and data comprise: Brief description of experimental design, materials, and methods Overview of the experimental information in Table 1 Numerical experimental data in the Tables 2e5 Visualization of the main experimental and theoretical data in Figs. 1e5

Experimental design, materials, and methods
In this section, we provide a brief description of the experimental details (if available) for each research group. This allows for a better understanding of the performed measurements, the results, and their reliability. The experimental methods, specified here, are mainly those, which are relevant for Ref. [1]. An abbreviated overview of the materials and methods concerning the fracture toughness measurements is given in Table 1. In each case, the sample preparation was more complicated than described here. Additional details are provided in the corresponding references. For an easier visualization, the results, shown in Fig 7 of Ref. [1], are displayed here in three Figs. (1e3) with different scales. This article covers a typical set of porous ceramics and polymers.
In order to complete the overview, results from porous metals are added. Although we checked about 20 publications concerning metallic foams, not much respective data of the toughness and fracture toughness exist. We found two papers in which the toughness (initiation toughness), the fracture toughness, and the Young's modulus of metallic foams are published. These three quantities are exactly those, which can be calculated by the model propounded in Ref. [1]. For detailed information concerning the results for ceramics and polymers, we refer to the latter reference.

Value of the data
We provide a compilation of mechanical data, especially of fracture toughness and Young's modulus, concerning porous materials from several research groups.
These experimental data allow for an easy check of theoretical models, including Ref. [1], which describes the fracture toughness of porous materials like advanced ceramics, polymers, and metals.
With the given experimental details of the measurements, the validity of the data can be estimated, particularly with regard to their application in Ref. [1]. Table 1 Brief overview of the experimental methods concerning the fracture toughness of porous materials, measured by different research groups. The given "microstructural scale" is a rough estimate and corresponds to the average pore size (if not otherwise specified) or to the grain size. For more information, like the pore or grain size as a function of the porosity, see the respective references.

Reference
Material Compact tension method (ASTM E1820-08) [19], crack length by image processing and compliance method 500 … 800 (size of space holders) McCullough et al. [17] Aluminum alloy, AleMg1eSi0.6, specimens from sandwich panels of Alulight foam [17] Compact tension method (ASTM E813-89) [20], crack extension by d.c. potential drop method, "initiation toughness"~5 00 … 1500 (depending on the relative density) Table 2 Measured normalized fracture toughness and their errors as a function of the porosity. The data were graphically taken from the references. In four cases, only the absolute K IC -values were given. So, these data were normalized by dividing them by K IC (dense) of 5.44 [5], 4.16 [8], 3.21/4.18 [9], and 3.63 MPa√m [10], respectively. The latter numbers were obtained by extrapolation as described in Ref. [1]. The absolute errors, DK IC (error bars), were also determined graphically from the references.

Ceramics
Yang et al. Ohji [4]: Two types of silicon nitride were tested: isotropic and anisotropic porous Si 3 N 4 . In the latter case, the elongated grains were mainly aligned in one direction, but as already said, only tests of the isotropic material are considered here. Different porosities between 0% and 30% were obtained by using the PHP process. The porosity was controlled by the configuration of the carbon mold and the amount of powder. The other parameters were fixed, like powder mixture and sintering temperature of 1800 C in nitrogen atmosphere for 2 hours. The fracture energy g eff was determined by the chevron-notched beam (CNB) test and the fracture toughness K IC was converted from the relation: (1 e n 2 )/(2E) with n and E being the Poisson's ratio and the Young's modulus.
Deng et al. [5]: Pure Al 2 O 3 and a mixture of Al 2 O 3 þ Al(OH) 3 were used as starting powder, having the designations A and AH. The relative amount of Al(OH) 3 was 60% and 90%, indicated by AH60 and AH90, respectively. Green bodies were prepared by cold pressing and then sintered in air. Various porosities were obtained at different sintering temperatures from 1100 C to 1450 C for 30 minutes each. The specimens with dimensions of 3 Â 4 Â 40 mm 3 were used for strength and toughness measurements.  The preparation procedure of the RBAO precursor powder includes 40 vol% Al, 30 vol% fine grained Al 2 O 3 , 10 vol% coarse grained Al 2 O 3 , and 20 vol% fine grained TZ-3Y (yttria-stabilized zirconia). Additionally, alumina fibers were incorporated in order to get a full scale all-oxide composite with fiber volume fractions of 35e40%. Green bodies were obtained by uniaxial and subsequent cold isostatic pressing. However, for the measurement of the fracture toughness at different porosities, also monolithic (fiber-free) samples were prepared with dimensions of 3 Â 4 Â 40 mm 3 . Table 3 Measured Young's moduli as a function of the porosity. For a diagram of these numbers, see Fig. 10 in Ref. [1].

Reference (Material) Relative density E [MPa]
Deng et al. [ Table 4 Measured relative Young's modulus of nanoporous gold [2]. The first two columns were provided from one of the authors of the given reference. We normalized the data by dividing through the Young's modulus of the dense gold (81 GPa), specified in Ref. [2]. They were tested by the single-edge V-notched beam (SEVNB) method in 4-point bending with a notch tip radius of 10 mm and support distances of 10 and 20 mm. The porosities were controlled by different sintering temperatures between 1100 C and 1500 C. The grain size increased with increasing sintering temperature. Table 5 Measured Young's modulus, fracture toughness, and toughness of porous titanium [15] and porous aluminum alloy [17]. The quantities were extracted graphically from Fig. 4 in Ref. [15] and Fig. 10a) in Ref. [17]. The marked numbers at the bottom of the table are not provided in this reference: * taken from Ref. [22], ** by adapting the "ligament parameter" n of the model equations to the experimental data of Ref. [17], *** calculated from E and J IC as before. In this case, the numbers of J IC 1. Overview of the measured normalized fracture toughness of different ceramic and polymer materials. As additional information, the main corresponding theoretical data from the model, described in Ref. [1], are provided.
Hong et al. [9]: After appropriate preparation, TiB 2 powder with a mean particle size of 2 mm was prepressed under uniaxial pressure and further compacted by (cold) isostatic pressing. Two different material variants were obtained by using 10 MPa and 50 MPa isostatic pressure. Afterwards, the samples were sintered under vacuum at temperatures from 1800 C to 2000 C. The fracture toughness was measured in 3-point bending with the singe-edge-notched bend (SENB) method. The beam dimensions were 2 Â 4 Â 22 mm 3 , the notch depth 2 mm, the notch tip radius 0.2 mm, and the support distance 16 mm. Each data point represents an average of three tested specimens. Samborski and Sadowski [10]: Porous ceramics of Al 2 O 3 and MgO were prepared by powder sintering technique at the Institute of Electronic Materials Technology (Warsaw, Poland). The sintered bodies were cut to create specimens with a cross section of 3 Â 4 mm 2 and with a length of 40 mm (Al 2 O 3 ) and 50 mm (MgO), respectively. Both types of materials were loaded quasi-statically and dynamically. For comparison with the model in Ref. [1], only the quasi-static tests of alumina were used. With the magnesia data, the extrapolation of K IC to the density of the solid material, in order to normalize the fracture toughness, included too much uncertainty. The alumina tests were performed with the SEVNB method in 3-point bending with a support distance of 20 mm and a notch depth between 0.7 mm and 1.3 mm.

Polymers
Maiti et al. [11]: Commercial foamed polymethacrylimid specimens ("Rohacell", Rohm GmbH, Germany) were tested in 3-point bending (ASTM E-399-81) at room temperature. On the basis of the SEVNB method, the specimen dimensions were 25 Â 50 Â 250 mm 3 with a relative notch depth of 0.5. The term "foamed" suggests that the material had closed porosity e at least partially e like many other polymer foams. This was not explicitly confirmed by the authors but, nevertheless, this seems obvious from their arguments on pages 213 and 215 in Ref. [11]. The pore size of approximately 300 mm was almost independent of the porosity. In their publication, Maiti et al. used also data from Refs. [12e14]. McIntyre and Anderton [12]: (Quotation) "Rigid polyurethane foams were prepared from a commercial two component system, Propocon MR49 and Isocon M supplied by Lankro Chemicals Ltd." Different porosities were obtained by different molds, containing the mixed raw material. The mold was open so that the material could expand freely, or it had a defined closed volume restricting the expansion to increase the density. The average pore size became larger with increasing porosity. The samples with dimensions of 5 Â 35 Â 150 mm 3 were tested in tensile configuration (single-edgecracked specimen) with a notch of different lengths up to 10 mm [13]. The amount of closed cells was determined with the ASTM Test-Method D1940-62T, which showed that the foams had predominantly closed porosity. (Remark: The used material was slightly anisotropic, as also in Ref. [14]. However, the data are well in the range of the data from Maiti et al. and so, this slight anisotropy was neglected by them [11] as also in Ref. [1].) Fowlkes [14]: Only one type of polyurethane foam with a relative porosity of approximately 92.5% was fabricated. Therefore, the toughness G Ic was measured by using various methods: 1. compliance method, 2. double cantilever beam (DCB), 3. tensile tests with three different specimen geometries: a) center-cracked plate, b) double-edge-cracked plate, and c) single-edge-cracked tension specimen. The corresponding results agreed relatively well. The proportions of the specimens for the tensile tests correspond closely to the ASTM norm [18]. The relative density and the calculated normalized fracture toughness e applied in Ref. [1] e were taken from Ref. [11], where the respective result of Ref. [14] was used.

Metals
Huber et al. [2]: Nanoporous gold was prepared by melting an Ag 75 Au 25 alloy, homogenizing by vacuum annealing at 750 C, cutting cylindrical samples with a wire saw after cooling, electro-chemical dealloying, and cleaning (rinsing) the sample with ultrapure water. The achieved porosity was 74 ± 1% Fig. 3. On the right side of the figure, the normalized fracture toughness of the tested polymer materials is shown [11e14], which is again a scale magnification of the same data in Figs. 1 and 2. Additionally, in the lower part of the diagram slightly left, we provide the normalized Young's moduli of nanoporous gold [2], together with results of model C, n ¼ 1 … 7 [1]. The NPG was mechanically and successively loaded and unloaded. and the mean ligament diameter 63 ± 6 nm. During the test, the sample was successively compressed and unloaded until the volume of the sample was reduced to half of its initial value. With these compression tests, only the Young's modulus was measured by evaluating the linear parts of the loading/unloading curves.
Kashef et al. [15]: Porous titanium was produced by sintering of compacted mixtures of commercially available titanium powder (purity 99.9%, average particle size 45 mm) and space holder material (ammonium bicarbonate, NH 4 HCO 3 , particle size 500e800 mm). The achieved relative densities were 0.30 and 0.40. Green bodies were made by pressing with 200 MPa at room temperature. The space holders were removed by heat treatment at 100 C for 10 hours and sintering took place at 1120 C for 7 hours in a vacuum furnace. Then CT-specimens were wire cut to a size of 16 Â 15.36 Â 6.4 mm 3 and fracture toughness testing was performed in accordance to ASTM E1820-08 [19]. The crack length was determined by the resistance curve procedure. From the J IC -curves, measured at stable crack growth, a conditional value of J IC could be derived. The Young's modulus was determined by the elastic unloading compliance technique. As in Ref. [16], the fracture toughness was calculated by K IC ¼ √(J IC ,E 0 ) with E' ¼ E/(1 en 2 ) and n ¼ 0.3 being the Poisson ratio. Corresponding data for the relative density 0.65, used in Ref. [15], had been taken from Teoh et al. [16].
McCullough et al. [17]: Closed cell aluminum-based foams (trade-name "Alulight") with the composition AL-Mg1-Si0.6 and AleMg1eSi10 (wt%) were tested by compact tension for relative densities between 0.1 and 0.4. The CT-specimen geometry was 50 mm, measured from the center of the holes to the back side, and a thickness of 7.5 mm. The J-integral test procedure, according to ASTM E813-89 [20], allowed for the determination of the initiation toughness, J IC , being equal to the strain energy release rate (toughness), G C . The Young's modulus, E, was determined from the elastic unloading compliance as specified in the afore mentioned ASTM norm. Finally, K IC could be calculated Fig. 4. Young's modulus, fracture toughness, and initiation toughness of porous titanium as a function of the relative density. The presentation of the data in this figure is identical to Fig. 4 in the original reference [15] so that the figures can be compared easily. The theoretical curves are calculated with Eqs. (13), (16), and (21) of Ref. [1] and are multiplied by the measured values of the dense material. from J IC and E according to the equations, already used by Kashef et al. (see above). Measurement of J IC was performed using the single specimen technique. For the crack length, the DC potential drop method was applied and checked twice by compliance methods using back face clip gauge on the one hand and the displacement transducer on the other hand.

Additional information for the data visualization
For the calculation of the theoretical absolute values in Fig. 5 from the normalized curves, we need the quantities for the solid material. As these are not provided numerically in Ref. [17], we found the Young's modulus of 70 MPa for the material AleMg1eSi0.6 in Ref. [22]. The toughness J IC was obtained by adapting the "ligament parameter" n in the model equations to the measured J IC -data. This is like a nonlinear extrapolation of the toughness to the dense material on the basis of the given model. Then, K IC of the solid material was calculated as before.
For open porosity (model C), Eqs. (13), (16), and (21) of Ref. [1] are applied. For closed porosity, an extended version of model A [21] was used. This means that Eqs. (7), (14), and (19) of Ref. [1] are equipped with the additional factor (1eP) n , which is described in detail in Ref. [21]. Here, P represents the normalized porosity. For the calculation of the theoretical K IC in Figs. 4 and 5, E was replaced by E/ (1en 2 ) with n ¼ 0.3 in order to be compliant with the plane strain condition, assumed in Refs. [15,17].
Note that for the equations of E, J IC , and K IC (Figs. 4 and 5) the same parameter, n, is used. The two model versions (open and closed porosity) in Fig. 5 yield similar data, but the "ligament parameters" n are different. Young's modulus, fracture toughness, and initiation toughness of aluminum alloy (AleMg1eSi0.6) as a function of the relative density. As in Fig. 4, the presentation of the data is identical to that one in the original reference [17]. The theoretical colored curves (open porosity, model C) are calculated with Eqs. (13), (16), and (21) of Ref. [1]. The black dashed lines correspond to the extended model A (closed porosity) [21].