Cellular Ceramics / 4

291

4.1

Mechanical Properties

Roy Rice

4.1.1

Introduction

Mechanical properties of cellular ceramics play an important role in their various uses. This is clearly so for the more limited cases in which the cellular component serves primarily or exclusively a mechanical function. However, mechanical properties are also typically quite important for most, if not all, cases where the primary or exclusive function(s) of a cellular component is nonmechanical such as thermal insulation, filtration, catalysis, or controlled combustion (e.g., burner). This arises since such applications commonly entail various mechanical stresses which must be survived for the cellular component to continue to satisfactorily serve its desired nonmechanical functions.

The mechanical properties addressed are, in order of treatment, primarily elastic moduli (mainly Young’s modulus) and Poisson’s ratio, fracture toughness, tensile (flexure) strength, compressive (crushing) strength, and thermal stress/shock resistance. Hardness, erosion, and wear, for which there is limited or no literature, are addressed little or not at all. (Sonic velocities are not directly covered, but their dependence on porosity is directly derivable from that of the elastic properties [1].) The focus of this survey is on nominally room temperature behavior, but thermal stress/shock resistance of cellular materials is addressed. Note that while average property trends are addressed as a function of average porosity and its average character, local porosity extremes often play a role. The extent of this role of such porosity variations varies with the property, generally being lesser for elastic properties and greater for fracture properties, especially tensile fracture [1].

Besides the basic properties of the material of which the cellular body is made, the primary determinant of the mechanical, and many of the pertinent nonmechanical, properties is the amount and character of the porosity. Thus, the focus is on the porosity dependence of mechanical properties, primarily the reduction of most mechanical properties as the amount (i.e., volume fraction) of porosity P increases and as a function of pore character. Differences due to basic material properties at theoretical density (i.e., at P = 0) are addressed by normalizing (i.e., dividing) property values for a given material at any value of P by those at P = 0. However, effects of pore/cell size are noted later, as are property differences due to the character of

Cellular Ceramics: Structure, Manufacturing, Properties and Applications.

Michael Scheffler, Paolo Colombo (Eds.)

Copyright 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-31320-6

292 Part 4 Properties

the porosity and some of the important trade-offs between mechanical and nonmechanical properties needed for important types of applications. The important and developing topic of porous sandwich components consisting of a porous core between two denser layers of the same or other material is noted here (see refs. [2, 3] for more information), but not addressed. However, much of what is presented here is valuable input for development of such sandwich bodies.

Turning to the porosity dependence of mechanical properties, this is addressed by models for such dependence based on the amount and character of the porosity, which also reflects aspects of component fabrication. These models are summarized in the next section, where the common distinctions between the two main types of cellular solids are followed. The first are honeycomb structures, which consist of tubular pores, often aligned with one another, as are commonly derived by green-body extrusion of ceramics. There are other methods of fabricating tubular porous structures of the same or similar tubular pore structures as obtained from extrusion, such as tape processing and, more recently and more diverse, rapid prototyping/solid free-form fabrication (SFF) [4, 5]. However, these structures would be included under honeycomb structures. The other basic type of cellular ceramics are foam structures. Again, while these are commonly made by foaming, hence their name, related structures are made by other fabrication routes that produce similar pore structures and are thus included under the heading of foamed materials.

Before proceeding to modeling of the porosity dependence of mechanical properties two general related issues should be noted. First, such modeling involves simplifying the pore structure of given bodies to a more uniform, idealized one. Second, there are challenges in characterizing the porosity of real bodies due to variations in the type, character, and amounts of porosity in a given body. Thus, the focus, as noted earlier, is on basic property trends as a function of basic porosity trends. Comparison of such modeling with measured results is basic to improving understanding and modeling, as well as porosity characterization. Other important factors in such improvements are comparison of modeling and property results of differing fabrication/processing factors for the same as well as differing properties whose porosity dependences may or may not be similar. However, note that where electrical and thermal conductivity of bodies are determined mainly or totally by conduction through the solid as opposed to the pore phase, the dependences of the conductive property on porosity are the same as for the elastic moduli, as shown theoretically [1, 6] and experimentally [1].

4.1.2

Modeling the Porosity Dependence of Mechanical Properties of Cellular Ceramics

4.1.2.1

Earlier Models

While there are a number of models and approaches [1, 2, 7–15], four prominent models are considered here to provide perspective as well as both practical and functional utility. The first is an older, lesser used, model originally derived for elastic

4.1 Mechanical Properties 293

moduli, mainly Young’s modulus E. The first model has both empirical and some analytical origins, with the latter being more rigorous in terms of the mechanics, and much less so in terms of the microstructure, which is generally taken as consisting of essentially homogeneously distributed, often spherical, pores to give homogenous properties [1]. This approach commonly results in the form:

where E/E0 is the value of E normalized by that at P = 0, the density at any P and,0 the value at P = 0, and n an empirically determined exponent that is generally in the range of >0.5 to about 4, most commonly around 1–3. This expression has also been used for other mechanical properties, especially strengths, based on the rationale that their values are often determined by the porosity dependence of E [1].

Besides the serious neglect of specific microstructural effects in both its derivation and use (i.e., pore character and values of n are not correlated), it suffers two other deficiencies. First, it does not recognize the intrinsic limitation that different types of porosity have on the upper limit of the amount of that type of porosity that can exist in a body that is still a coherent solid (i.e., one that can sustain at least some stress) as opposed to simply being a pile or clump of individual powder grains (see Fig. 1 and discussion of Minimum Solid Area models below). Thus, though often not adequately recognized, each porous body that behaves as a solid body has a critical value PC of the volume-fraction porosity P at which key physical properties, such as elastic moduli, of the body must go to zero, so that it no longer acts as a solid body [1, 7–10]. The values of PC depend directly on the character of the porosity, ranging from slightly less than 1 for bodies with regular tubular pores (aligned with the stress direction) or spherical pores to typical values of 0.6–0.25 and potentially as low as about 0.1 in partially densified powder compacts. The PC values can be readily calculated for Minimum Solid Area porosity models discussed below (and possibly from percolation models), or estimated by extrapolatiing key properties such as elastic moduli for porous bodies to zero. For porous bodies of partially densified powder, PC values are the true green densities (i.e., without intentional or unintentional binder materials, e.g., water, after binder burnout).

A potential solution to inadequate recognition of PC values is to normalize the values of P for a given body by dividing them by their PC values (which then effects the value of n in fitting data to Eq. (1)), but this has thus far shown at best limited consolidation or improved coherence of individual real or model porous structures [10]. The other basic deficiency in use of Eq. (1) is that it by itself does not recognize that n = 1 is the upper limit of achievable properties [1], and thus that allowable values are n < 1.

Though there are serious questions about this model, it is included since it has been used a fair amount, and its use, limitations, and questions are important factors in the field of porosity effects on properties. Further, though it has been used primarily for partially sintered rather than cellular bodies, this model may have application to the latter (e.g., since it has a common form of some cellular models discussed next) and thus possibly aid in evaluating the transition in porosity depen-

4.1 Mechanical Properties 295

to key microstructural parameters, with the constants of proportionality determined by fitting property data to their appropriate models. An important factor in this work of Gibson and Ashby [2] and others is that, while there can be important differences for different honeycomb or foamed materials, there are many close similarities within each of these types of cellular bodies for various polymer, metal, and ceramic materials, which reinforces trends for bodies for which there are fewer data (i.e., ceramics). A limitation of these cellular models is that there is less basis for relating the properties at high P to those at low P, and the transition in structure and behavior as P progresses from low to high values or the opposite. Thus, both the foregoing model and the following two modeling approaches may be of use in addressing such transitions and extrapolations of properties at high P to those at low P and vice versa. However, other than noted in this chapter, little consideration has been given to such cross-correlation of these models.

4.1.2.3

Minimum Solid Area (MSA) Models

Minimum Solid Area (MSA) models cover a broader range of pore structures ranging from low porosity to the medium and high porosity of cellular materials. However, much of MSA modeling is complementary to the above cellular modeling of Gibson and Ashby and others in that it also focuses on similar microstructures and their evolution as a given type of pores increases or decreases over the allowable range of P. A particular type of porosity reflecting idealizations of the pore structure resulting from each of several various basic processing parameters is assumed for a given model. Primary types of pores in MSA models are tubular pores from green body extrusion or tape casting and lamination, spherical pores from gas bubbles generated in sintering or melting, cubic or other polyhedral pores formed as intragranular pores in polycrystalline bodies (or single crystals), and pores between packed spherical particles that are being bonded to one another, especially by sintering. As with the preceding cellular models, MSA modeling assumes first that the body can be represented as a dense uniform packing of identical cells such that the effects of the pore in a cell reflect the effects of the porosity in the entire body (Fig. 1). Each cell consists of either a pore whose center is also the center of the cell surrounded by the solid phase or a spherical particle whose center is also the center of the cell surrounded by the pore phase (Fig. 1B). The cell geometry is determined by the dense (i.e., space-filling) packing or stacking of the individual cells (note that only two dimensional cells are needed for tubular pores). Most commonly, simple cubic packing is used for both its simplicity and as a good approximation for random packing of the solid and pore phases, but some denser packings have been used, and others offer possibly important extensions of this method. Once a representative cell structure has been selected based on the type and scale of the porosity being modeled, a complete evaluation of the range of feasible porosity can be made by incrementally changing the size of the pore or spherical particle to the cell size. This is basically a simulation of the porosity changes in a variety of foam and sin-

296 Part 4 Properties

tered bodies, but over their complete range of porosity, thus providing extrapolation from one degree or type of porosity to another.

The basic assumption of the relation of the cell structure to mechanical properties (and others, e.g., thermal and electrical conductivity) is that the property scales directly with the ratio of the MSA of a cell normal to the stress (or conductive flux) to the cross-sectional area of the cell in the same plane of the MSA in the cell. For example, the MSA for sintering particles is the area of a sintered neck with an adjacent particle, while that for a spherical pore is the area of surrounding solid material in the equatorial plane of the cell normal to the stress direction. Dividing such MSA values by the cross-sectional area of the cell in the same plane as the MSA gives the relative MSA, with which properties are assumed to directly correlate. While this assumption may be modified by further (e.g., finite-element) analysis, it has proven to be suitable for substantial useful evaluations [1, 7–9].

The output of MSA modeling over the entire allowable range of porosity is a curve of the relative property at any P up to the value of PC for that porosity type, which is defined, at least approximately, if not exactly, by each specific model. Some MSA models, mainly those for tubular pores aligned parallel or perpendicular to the stress, result in simple equations of the form of Eq. (1), commonly with n = 1. However, most MSA models are generally not presented as an equation, since the individual calculations of each MSA value are complex due to physical requirements such as mass conservation in sintering and changes at key values of P (e.g., where the transition between open and closed porosity begins and the PC value, both of which are valuable and often unique outputs of MSA modeling). Instead, the output of MSA modeling is generally a plot of the MSA values for the particular porosity type selected for analysis, normalized by the corresponding cell cross-sectional area, usually on a log scale, versus P (usually on a linear scale; Fig. 2).

Various earlier investigators used the MSA concept for a particular pore type, then tried to generalize results for the one pore structure to more general pore structures. It was later recognized that the MSA approach provided a variety of models for various common individual pore structures that collectively give a much more accurate picture of the porosity dependence of mechanical and related properties [1]. Thus, though only a limited number of MSA models have been developed, they cover a broad range of pore character and property behavior. Thus, they show that bodies with tubular or spherical pores have greater property retention and extend to higher P values, as opposed to the much more rapid decrease of properties and more limited range of allowable P values of bodies with porosity from partial sintering of particulate/powder compacts (Fig. 2). The intervening middle region of P values is where curves for bodies with combinations of these two types of porosity should occur. Linear combinations of the limited data for bodies with such mixed porosities support this, as does the recent demonstration that the curves of Fig. 2 can be normalized to one master curve by plotting them versus normalized P (i.e., versus P/PC).

While, as noted above, the results of MSA modeling are generally not given as an equation or a family of equation, but as one or more plots, there are clearly three stages to the decrease in relative MSA, hence property, values on the typical semilog plot as P increases. These are first an approximately linear decrease for about one-

298 Part 4 Properties

PC values.) The values obtained for the various slopes of the different MSA models (Fig. 2) are consistent with slopes for the same plots of experimental data for real bodies in which the dominant pore character is similar to that of a particular MSA model. This agreement between extensive experimental data and MSA models supports the use of such models [1, 7–10].

4.1.2.4

Computer Models

The fourth, and potentially very diverse, method of modeling uses computers. A promising example is extension of MSA models, which is in the development stage, but the focus is on the more established use of the finite-element method (FEM) since there has been more work on this. While there are may possibilities, there are also challenges, an important one being whether the analysis is a simpler twodimensional (2D) or a more rigorous three-dimensional (3D) analysis. Agarwal et al. [16] demonstrated that such 3D analysis of spherical pores in a glass matrix produced reasonable results by idealizing the structure as consisting of identical spherical pores uniformly spaced in the matrix and limited in total pore volume fraction to P < 0.5 (to limit interactions with adjacent pores), so the analysis could be done by solving the asymmetric problem of a representative cell consisting of a pore imbedded in a section of the matrix. Their results are in reasonable agreement with the MSA model for simple cubic packing of identical spherical pores [1] (see Figs. 2 and 8).

Recently, Roberts and Garboczi [17] reported FEM analyses of ceramics with pores between randomly packed identical spherical particles in various stages of sintering or of randomly packed overlapping spherical or ellipsoidal pores in a densifying ceramic matrix for values of P £ 0.5. They also used a basic cubic cell structure of nominally identical particles, but consisting of substantially more than one randomly packed particle or pore per cell. Results were found to depend on the size of the cell and the pixels relative to one another. However, it was found that results from three suitable sizes could be extrapolated to an essentially “true” value for each degree of a given porosity. Results were shown to be reasonably consistent with some typical literature results and with the equation

with n = 2.23 and an approximate PC of 0.65. The value of PC is only approximate since computational time increases rapidly as PC is approached, so only an approximation can be made. However, the PC values obtained are too high for the type of porosity involved to an extent which indicates modeling problems that need further attention [10].

More recently, the computer program OOF (Object-Oriented Finite Element Analysis of Real Material Microstructures) for simulating the effects of microstructure on physical properties has been used to evaluate effects of porosity on mechanical properties of ceramics [18]. In an attempt to introduce some real pore character into