Cellular Ceramics / 4
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4.1 Mechanical Properties 299
the simulation this evaluation used random cross sections of actual specimens and hence of their three-dimensional pore character (elongated spheroids in this case) from which to generate a pore structure for analysis of the mechanical effects of the pores. However, they used a 2D analysis which is much less demanding of computational capacity. Thus, the pore structure generated for their 2D analysis is tubular, which is clearly different from the starting 3D pore structure. The generated tubular pores also have varying diameters whose size distribution is not that of the actual pores themselves, but of the intersections of the original pores with the plane on which their evaluation took place. The results of this evaluation were consistent with that for tubular pores, which is of some use, but caution is indicated in use of this technique, since many of the 2D structures and issues are amenable to 2D analytical solutions, but are of limited applicability to 3D modeling [19].
4.1.3
Porosity Effects on Mechanical Properties of Cellular Ceramics
4.1.3.1
Honeycomb Structures
Honeycomb structures are almost exclusively anisotropic in their mechanical properties. The Young’s modulus for honeycombs with straight tubular pores aligned parallel with an applied uniaxial stress is found readily by several methods (e.g. MSA and others) to simply be:
E/E0 = 1 – P, |
(4) |
that is, Eq. (1) with n = 1, which holds for any straight tubular pores, regardless of the (constant) cross sectional geometry, and any packing of the pores. Note that this dependence is also the upper bound for porosity dependence of elastic properties (e.g., this represents the highest stiffness attainable for any given amount of porosity). This 1–P dependence is also the upper limit for all but some atypical toughness behavior and most, if not all, strength behavior [1]. Solutions for the same porosities stressed normal to the direction of aligned tubular pores are all substantially lower in property levels and have lower PC values depending on the stacking of the aligned pores (Fig. 3). The properties and their porosity dependence also depend on pore cross-sectional shape. Those of circular cross section are apparently isotropic in any plane normal to the aligned pores, while pores of other basic geometrical cross-sec- tional shapes can range from isotropic, or nearly so, E values, to quite anisotropic modulus in a plane normal to that of alignment. Thus, for stressing normal to aligned close-packed pores of square cross section but parallel to the square pore walls gives half the values of E as for stressing parallel to the aligned pores, but the same dependence on P (i.e., varying as 1–P), while stressing along diagonals of the square pore cross section results in greater porosity dependence. This and other variations noted by Gulati [22] are shown in Fig. 4 for aligned tubular pores of trian-
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Fig. 3 Semilog plot of relative Young’s modulus or other pertinent mechanical property for aligned cylindrical pores versus volume fraction porosity P for MSA and computer [20, 21] models of such cubically packed pores aligned with the stress axis (which is the upper limit of properties obtained with pores present). Results for simple cubic, random, and hexagonal stackings with the stress axis normal to
that of the tubular pores are shown. (Only cubic packing was treated by MSA models, while all three stackings were calculated by 2D computer modeling.) Solid curves are for small specimen to pore size ratios (ca. 3/1); dashed lines and arrows are respectively for increasing and large specimen to pore size ratios. After Rice [1], published with permission of Marcel Dekker, Inc.
gular and square cross section. Note that the basic difference in behavior parallel and perpendicular to that of aligned tubular pores of circular cross section is corroborated by evaluation by Francl and Kingery [23] for thermal conductivity data for alumina (determined by conduction through the solid phase rather than radiation or convection through the pore phase).
Strength dependence on porosity is often similar to that of E, but often somewhat greater (i.e., lower strengths) since it is more affected by extremes of the porosity in the body. However, close-packed aligned pores of square cross section stressed in tension (flexure) show a linear dependence on cell wall strength rf and on 1–P, while tensile strength for stressing normal to one set of cell walls and parallel to the others shows the same dependences on rf and P, but with only about one-half the tensile strength. Much more limited information is available (mainly or only for close-packed hexagonal cells) for bodies under biaxial flexure (tension) [24].
Crushing strengths are often important (e.g., for automotive exhaust catalyst supports) and have received some evaluation for densely packed aligned tubular pores stressed parallel to the axis of the aligned pores, or in two directions normal to the axis of alignment (one normal to half of the cell walls and parallel to the remaining
Fig. 4 Linear plot of the Young’s modulus normal to densely packed aligned tubular pores with square or triangular cross sections showing, respectively, high anisotropy and near isotropy with high E. Note that densely packed hexagonal pores result in even more isotropic
4.1 Mechanical Properties 301
behavior, but at very low levels of E (slightly greater than the minimum for tubular pores with square cross sections) due to inherently low stiffness of such shaped pores. Data after Gulati [1, 22], published with permission of Marcel Dekker, Inc.
half of the cell walls, and the other parallel with the a diagonal direction of the square cross sections of the cells). While all three tests show crushing strengths linearly dependent on cell wall strength, they have greater dependence on P [e.g., as (1–P)2] [24].
Thermal stress and shock failure has been investigated, especially for automotive exhaust catalyst supports, for which both closed-form and finite-element methods have been used [22].
4.1.3.2
Foams and Related Structures
More experimental and modeling evaluations are available for foams and related structures, especially along the lines of Gibson and Ashby and others. Modeling of open-cell foams is easier and results in simpler equations such as:
E/E0 ~ (1–P)2 |
(5) |
G/G0 ~ 3/8 (1–P)2 |
(6) |
where G is the shear modulus at any P, G0 is its value at P = 0, and the symbol ~ reflects the fact that constants of proportionality have been determined empirically.
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(Note that the exponents are dependent on the packing and interconnection structure of the cells; the value of 2 is for one simple, commonly used structure [1, 2, 14].) Poisson’s ratio v is approximately constant regardless of P (i.e., the value at P = 0, commonly ca. 1/3). Equations for several other mechanical properties such as fracture toughness (normalized), tensile (flexure) strength, crushing (compressive) strength, and hardness all again involve proportionality constants (ca. 0.65 for each of these cases), depend linearly on strut strength, and all have a porosity dependence of (1–P)1.5. (Note that while direct measurement of strut strengths is often not feasible, it can be obtained in some cases where fracture mirrors on fractured struts are clear enough to correlate with their fracture strengths, as discussed below. Also the indicated equivalence of tensile and compressive strengths is generally not born out by experiment, consistent with compressive strength being significant higher than tensile strength at low porosity [1].)
Modeling the properties of closed-cell foams and the transition from opento closed-cell structure is done by incorporating a parameter j, the fraction of solid material in the cell edges (struts), and the portion of solid material in the cell walls 1–f, which are straightforward in definition but difficult to determine in real bodies. However, much closed-cell behavior is similar to that of open-cell foams [2, 11].
The above modeling results provide useful, at least general, guidance, as shown by the more limited data on ceramics and by the applicable data for plastic and metal foams, but there are still uncertainties. For example, data of Zwissler and Adams [25] for foamed glasses is cited as being consistent with the G–A open-cell model for Young’s modulus, fracture toughness, and flexural strength [2]. This is true in a broad sense, but there is some uncertainty (Fig. 5). Thus, while most of the limited data fall within the scatter band of other foam data, some is near or outside this band, and there are some indications that the slopes (i.e., exponent values) may increase as P decreases (see also Fig. 8), and the average trends may differ measurably from those predicted by the applied model.
One source of the uncertainty are the measured values themselves and their extrapolation to P = 0. Linear extrapolation of the Young’s modulus data to P = 0 gives E0 » 35 GPa, about half the expected value. Doubling the measured values to be consistent with values at P = 0 still leaves the data in rough agreement with the model. However, even greater discrepancy is indicated in the extrapolation of tensile (flexure) strengths to P =0, which gives a value of about 11 MPa, which may be as much as 5–10 times lower, but still be consistent with the model if so corrected. On the other hand, the fracture toughness values extrapolate to a typical value of about 0.8 MPa m–1/2, consistent with values for bulk silicate glasses. However, this apparent consistency could be misleading, since there can be significant discrepancies in extrapolating fracture toughness values of some porous bodies owing to incompletely understood effects of porosity and some methods of toughness measurements [1, 26]. This issue of measurement accuracy will be addressed further below.
The above data on glass foam of Zwissler and Adams [25] has been shown to agree as well or better with the MSA model for cubically stacked spherical bubbles than with the Gibson and Ashby model, regardless of whether the data are corrected for differences between extrapolated values at P = 0 or not [1]. The more extensive
Fig. 5 Log-log plot of relative (i.e., normalized) Young’s modulus E, fracture toughness KIC, and flexure strength rf, that is, values for porous bodies divided by their values at
P = 0 (data on foam silicate glass of Zwissler and Adams [25]) versus log (1–P) (volume fraction porosity). Note again the equivalence of/ 0, for example, as shown in Eq. (1). Approximate values of the data for each property extrapolated to P = 0, given in parentheses, shows some consistencies and inconsistencies
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of extrapolated and known values at P = 0. Note the mean and approximate bounds of values (solid lines) from surveys of Gibson and Ashby [2], and that the data are near to or within these limits (but less so if they are low due to possible test effects). While the approximate slopes of the 1–P dependence are not grossly inconsistent with model values, they are probably of lower slope (with changes to higher slopes at lower P values, similar to those shown in Fig. 8).
range of data for sintered SiO2 foam of Harris and Welsh [27] also shows good agreement for the same MSA model (Fig. 6), and data for various porous SiO2 bodies from sol–gel processing show agreement with and transitions between various MSA models pertinent to different stages of pore character and amount [1] (Fig. 7).
More recent and detailed studies on polycrystalline ceramic foams are overall consistent with the basic trends of foam models and the data trends noted above for the glass-foam results of Zwissler and Adams [25] (Fig. 5). Thus, Hagiwara and Green [28] found in their study on some alumina open-cell foams that the exponent for the porosity dependence of elastic moduli was about 2, as predicted in Eq. (5), but that the proportionality constants between the elastic properties and their porosity dependences were often lower than those given in the literature (e.g., a value of ca. 0.14 was found for G instead of the expected value of ca. 0.4). These differences were attributed to variations in cell-strut microstructure. Subsequently, Brezny and
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Fig. 6 Semilog plot of relative Young’s modulus of sintered SiO2 foam of Harris and Welsh versus the volume fraction porosity P and the MSA model for cubic stacking of ideal spherical pores (see also Fig. 8), which is a reasonable approximation for random packing of pores [1]. Note the general agreement of the
model and the data. Some increasing deviation of the data below model values may reflect possible changes in cell structure and especially somewhat lower PC values for real as opposed to ideal materials [1]. After Rice [8], published with permission of J. Mater. Sci.
Green [29], in a study on bending strength, fracture toughness, and compressive strength of some similar and other alumina-based open-cell foams, showed that the observed exponent for the porosity dependences was about 1.5, as expected, but the proportionality factor was found to be 0.13–0.23 instead of 0.65. More recently Vedula et al. [30] summarized the exponents for the porosity dependences to typically be about 2 and 1.5, respectively, for elastic moduli and fracture toughness, but that the exponent for compressive and tensile strengths could range from slightly less than the predicted value of 1.5 to greater than 2. Colombo and Modesti [31] reported that the porosity exponents for flexural strength and elastic moduli were each about 1 for quite open cell foams of SiOC (produced from foamed preceramic polymers), that is, lower than the respective expected values of 1.5 and 2.0. On the other hand, foams with less open cells gave respective values of 2 and 3.5 (i.e., somewhat to substantially higher than expected). This shows that general trends of Gibso- n–Ashby-type models may differ somewhat to substantially from data.
Studies (e.g., by Green et al.) show that much of the above variations in open-cell, mainly alumina-based, foams were due to effects of cell struts. Thus, the major source of variations of elastic moduli in their open-cell alumina foams was the microstructure of the cell struts, not the varying limited contents of closed cells [28]. (They also confirmed that Poisson’s ratio was nominally constant at about 0.2, reasonably consistent with predictions of its being about 0.3 given the data scatter.) Strengths of the foams were found to correlate with those of the struts, which commonly
Fig. 7 Plot of absolute or relative mechanical properties versus volume fraction porosity P for bodies made from partially to fully sintered SiO2 gels [38–42]. These bodies reflect transitions from stacked sintered particles to arrays of bubble-type pores, as shown by the various
4.1 Mechanical Properties 305
models. See also Fig. 8 for the same data in a different plot. Note the limited magnitude and modest change of Poisson’s ratio v, reasonably consistent with expectations from G–A models. After Rice [8], published with permission of
J. Mater. Sci.
had low Weibull moduli (e.g., 1–3) [29, 32]. These trends were also shown to be true for glassy carbon foams, which gave particularly clear fractographic results [33].
Thermal shock resistance has also been addressed with more complex and less defined results. Tests with alumina-based open-cell foams showed that neither the macro thermal stresses across the complete foam specimen nor the local stresses across individual struts were the source of failure. This was shown to be due to heating up of quench fluid as it infiltrated the specimen, which strongly increased with increasing cell size [34]. Cell-size effects and limited dependence on density were confirmed in further experiments. Correlations between maximum thermal strains and degree of thermal stress damage (measured nondestructively) were shown [30].
Measurements of compressive (crushing) strength, performed on a few ceramic foams, gave some more complex and different results. Tests by Dam et al. [35] on open-cell alumina foams gave differing results according to whether the loading rams had stiff or compliant loading surfaces, with the former leading to significant discrepancies with G–A models, and the latter to better agreement. The use of compliant loading surfaces appeared to reduce general damage accumulation with
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increased loading. An increase in both Young’s modulus E and crush strength with decreasing cell size was observed, contrary to G–A models. Later strength measurements on glassy carbon foam [36] showed both bending (tensile) and compressive (crushing) strengths that increased with decreasing cell size, but not E or fracture toughness. These results are more consistent with those of Morgan et al. [37] on foam glass, which showed a pronounced increase in compressive, tensile, and flexure strengths with increasing inverse square root of cell size (fracture toughness and E were essentially independent of cell size).
Structures the same as or similar to those generated by foaming can be produced by other fabrication methods and give added insight to the mechanical behavior as a function of porosity. SiO2 bodies produced by sol–gel processing that range from zero to various high degrees of porosity are an example of this [38–42]. Figure 7 shows agreement with MSA models and indicates changes in pore structure with increasing porosity based on comparison of their property data (especially E data of Ashkin et al [38, 39]). Thus, the data are consistent with the MSA model for pores
Fig. 8 Log-log plot of relative Young’s modulus of SiO2 made by sol–gel processing by Ashkin et al. [38, 39] (see also Fig. 7) and by sintering of fused quartz by Harris and Welsh [27] versus 1–P (see also Fig. 6). Note that both sets of data, while approximately consistent with G–A-type and related models using a
constant value of the exponent n (e.g., ca. 2), shows some variation to higher n values at low P, consistent with the major change in pore structure indicated in Fig. 7. Note also for reference E and v data for 3D finite-element analysis by Agarwal et al. [16] for spherical pores in glass.
4.1 Mechanical Properties 307
between cubic packings of uniform spherical pores at low porosity, with a transition to one of cubic packing of uniform spherical pores at higher porosity. Such pore transition is consistent with the basic gel structure being one of strings of fine beads which pack closely at low porosity (high solids contents) but transition to outlining nominally spherical pores at lower densities [1]. For comparison of MSA and G–A models some of the previously shown Young’s modulus data for SiO2 are replotted as log E versus log (1–P) in Fig. 8. While the sintered-foam data [27] show an essentially linear log-log plot, the data for the sintered sol–gel shows a bior trilinear plot consistent with the transitions in pore structure noted in conjunction with Fig. 7. Further, the two indicated changes in slope of the data for the sintered sol–gel are consistent with the physical requirements that the lower limit of P = 0 should give E/E0 = 1 and that at the upper limit of P (i.e., P = PC) the plot should give E/E0 = 0. Note also that data of Agarwal et al. [16] for a silicate glass from 3D computer modeling show reasonable agreement with those of the sintered SiO2 foam.
Another informative case of foamlike structures made by techniques other than foaming is that of bodies made by bonding ceramic beads, or especially balloons, together. Ceramic balloons can be made with relatively thin walls, but with more uniform wall thickness and resultant properties than are common in foaming, for which significant tapering to thinner walls with increasing distance from foam cell struts is observed [43]. This greater uniformity of wall thickness of ceramic balloons translates into substantially greater strengths of low-density foamlike bodies made by bonding such balloons together [1, 44, 45]. The strength to mass ratio of bodies formed by bonding balloons together can be improved by limiting bonding mainly to the contact areas between adjacent balloons.
Green [46] has made foamlike bodies by sintering compacts of glass balloons. Green and Hoagland [47] modeled the strength of these bodies.
4.1.4
Discussion
4.1.4.1
Measurement–Characterization Issues
Adequate measurement of pertinent properties and characterization of microstructure are necessary for a sound understanding of the mechanical behavior of cellular materials. This is particularly true of ceramics because of their brittle behavior and resultant sensitivity to stress concentrations. This need is shown by Dam et al. [35], whose tests on open-cell alumina foams showed more consistent results using compressive loading rams with compliant versus stiff loading surfaces. This is supported by Green’s observation [46] that elastic moduli determined by compression loading of bodies of sintered glass balloons showed progressive deviations from values from ultrasonic or tensile measurements. Brezny and Green [48] concluded from their evaluation of scale effects on mechanical measurements on glassy carbon foams that specimen dimensions should be at least 15–20 times the cell size. Huang and
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Gibson [12] concluded that valid fracture toughness measurements on brittle foams requires cracks whose size is greater than 10 times the cell size. Again note that the Young’s modulus and tensile (flexure) strengths of Zwissler and Adams [25] for foamed glasses extrapolated to only about half the values expected at zero porosity, while the fracture toughness extrapolated to about the value expected for zero porosity. While extrapolation of properties to P = 0 is best done with models known to fit the type of porosity involved, linear extrapolations can raise questions for more detailed evaluation. Brezny and Green [49] have suggested procedures for uniaxial tensile and compressive strength testing.
Both the above examples and the basic nature of many porous bodies, including most, if not all, cellular ones, strongly suggest that much more work is needed to properly assess their mechanical, especially failure behavior. A key component of this which has been paid even less attention are effects of surface finishing; most surface finishing is presumably by machining, but is of limited or no specification. There has been limited or no comparison of, for example, as-fired versus asmachined specimens, which might be significant, especially for specimens of glassbased bodies.
Another basic limit of much data is the incomplete characterization of the microstructure, especially that of the porosity. Values of the volume fraction porosity P are usually given, but many reflect only the dominant source of porosity, and no indication of the statistical or spatial variations of P values are given. However, of greater concern is the almost total lack of characterization of the pore structure, such as the shape of pores and how they are spatially arrayed relative to each other and the solid structure.
4.1.4.2
Impact of Fabrication on Microstructure
The fabrication method used to make a cellular body obviously plays a major role in the amount, type, and character of the pore structures introduced, as shown for example by comparing foams with bodies made by bonding balloons together. The latter have more uniform pore wall thickness and a broader range of achievable wall thicknesses, but they are more limited in achieving thin and tapered wall thickness. However, the characterization of bonded balloon structures is probably simpler and more accurate than that of foams (and they can have significant advantages in the size, shape, and practicality of making porous bodies [1]). Similarly, there are tradeoffs between extrusion and tape lamination, and to some extent with solid free-form fabrication, that impact costs, sizes, and shapes of bodies, as well as of resultant pore structures [43]. The limited development of foaming of glass and other ceramics, while producing reasonable foam cellular bodies, produces cell variations and often some overall anisotropy of the cellular body and its structure due to differing expansions in the rise direction versus the lateral direction during foaming. (Such anisotropies in foaming are common to forming many materials, including foamed foods such as cakes, pastries, and breads. Property effects of such anisotropies on properties have been modeled [1, 2].)