Connecting the macro- and microstrain responses in technical porous ceramics: modeling and experimental validations
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DOI: 10.1007/s10853-010-4899-0
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- Bruno, G., Efremov, A.M., Levandovskyi, A.N. et al. J Mater Sci (2011) 46: 161. doi:10.1007/s10853-010-4899-0
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Abstract
The relation between the macroscopic and the microscopic (lattice) strain response to external uniaxial stress has been investigated for porous ceramics. Analytical and finite element modeling (FEM) have been performed and neutron diffraction data on porous sintered alumina and extruded honeycomb SiC have been used to validate the theoretical approach. By FEM simulations, it is shown that in spite of the complex pore microstructure, shear stresses are small during uniaxial compression. Analytical modeling shows that while the average microscopic stress depends on the applied macroscopic stress only through the porosity p, the average microscopic strain depends on the macroscopic stress through the pore morphology factor m, as well. Novel relationships are proposed to describe this dependence. Analytical calculations and numerical modeling perfectly agree with each other, and both show good consistency with experiments. As predicted, it has been observed that the microscopic (diffraction) Young’s modulus does not depend on the pore morphology factor, and follows the rule-of-mixtures, while the microscopic Poisson’s ratio does not even depend on porosity, but is equal to the value for the dense material property. A practical implication of these findings is that it is not possible to attach a pore morphology factor to a material, unless the processing conditions are tailored to vary p without varying m. In fact, the different values of m found for the different porosities explain why many models can be used to rationalize the experimental data. With the proposed method, the factor m can be independently evaluated by the use of macro- and micro-elastic properties of the porous body. Analogously, the macroscopic elastic properties of the dense material can be obtained by macroscopic and microscopic values measured on the correspondent porous material.
Introduction
A very large amount of data is available in the literature about the macrostrain response to applied stress of porous materials (see [1, 2, 3, 4, 5, 6] for some examples). In particular, the works of Knudsen [2], Dean and Lopez [5], and Munro [6] on alumina outline that several models (exponential, polynomial, and even linear) can be fitted to the experimental data, especially if porosities p < 0.5 are considered. In his book, Green reports that linear Young’s modulus reduction as a function of porosity can be interpreted in the frame of a percolation theory [7]. Further, Wang [8] and Sudduth [9] discuss the limits of validity of different hypotheses. Ramakrishnan and Arunachalam [10] point out that one should refrain from using a simplistic composite model with the void having zero mechanical properties. A useful approach is also given by Roberts and Garboczi [11]: these authors derive analytical formulae (power laws) for the cases of spherical and ellipsoidal pores, as well as for overlapping matrix spheroid particles. They use different percolation levels (i.e., porosity values at which the Young’s modulus drops to zero) in the three cases.
Thus, whereas all literature models agree that the macroscopic elastic modulus of a porous material is proportional to the elastic properties of the solid domain, they disagree about the dependence on porosity. Indeed, the pore structure (shape, size distribution or connectivity) significantly influences mechanical properties as well as porosity. This implies that a universal dependence of mechanical properties on porosity must include at least two parameters: one accounting for porosity (scalar) and one for pore morphology (which might be a tensor [12]).
To our knowledge, only a few papers tackle the question of the microscopic (lattice) response [13, 14] of porous media to external loads. It has already been found experimentally, using neutron diffraction (ND), that the microscopic stress–strain behavior of natural (porous) rocks is very different from the macroscopic. This behavior is commonly explained by the influence of porosity, but little insight has been given to clarify the mechanisms. Even further, very little modeling work is available on the lattice strain behavior [15], so that data are not yet rationalized in a widely accepted coherent frame.
The relationship linking an applied macroscopic stress to the macroscopic strain and to the lattice response is, however, of great importance in order to: (a) understand and quantify the role of materials properties in the specimen or component response, and (b) infer the macroscopic behavior from microscopic data (or vice versa), as in the case of a crack initiation or propagation criterion. Commonly, the average microscopic and macroscopic moduli are assumed equal for a dense material. However, they do differentiate in porous ceramic materials and it is not obvious that the properties of solid skeleton are the same as the properties of the analogous dense material. In fact, while for dense materials analytical models exist to calculate the polycrystalline average microscopic (crystallographic plane specific) elastic properties [16, 17], this is not the case when pores represent one of the phases (see e.g., [18], where a mean field approach is used, but not crystal plane-specific).
Recently, new microstructural information has allowed further progress. Indeed, inputs to stress–strain modeling have been expanded to microscopic data such as the three-dimensional (3D) pore structure, retrievable for example by means of synchrotron or table X-ray tomography [19]. In this framework, microcrack behavior under applied stress has also been used, as measured by SEM and TEM [20]. Other powerful instruments, such as focused ion beam (FIB) chambers are being widely used to acquire microstructural information. These techniques, however, do not yield any quantitative information about the material’s mechanical properties.
In this work, the relationship between macroscopic and microscopic properties has been analytically modeled using stress balance considerations. We started from a simple empirical relationship (introduced by Gibson and Ashby [21]) between the Young’s modulus E_{p} of a porous body and its dense material properties (E_{d}), assumed uniform and isotropic. Indeed, a one-parameter model cannot cope with the anisotropy of the pore structure, since porosity is a scalar quantity. At least one more model parameter is required for an adequate description of the influence of the pore structure on elastic properties. We chose this second parameter to be the pore morphology factor m. While studying porous solids with open and closed cells, Gibson and Ashby used the parameter m to characterize the cells and derived m = 2 and m = 3, respectively, for the two cases above.
The approach has proved promising and deserving extension to other porous ceramics. Therefore, we have made use of the parameter m in a broader sense, to define open (continuous) porosity. This parameter would be analogous to the pore shape for closed porosity. A relationship has been derived, linking the average microscopic lattice strain (microstrain) measured by diffraction methods, to the macroscopic strain (macrostrain) measured by extensometry or strain gauges. Some direct measurements of the microscopic and macroscopic strain response under external applied stress in porous honeycomb silicon carbide and porous compact alumina have been carried out as examples. We will show that the combined experiment plus modeling approach and the addition of microscopic (diffraction) data to macroscopic tests, both cast new light on the micromechanics of porous ceramics. We will also see that the different values of m found for the different porosities explain why many models could and can be used to rationalize the experimental data.
Model description and predictions
Analytical relations for micro- to macrostrain conversion
(Here and below angle brackets denote averaging over the solid domain and the component normal to the cross-section has been taken.)
In other words, Eq. 4 implies that the average microstress measured by ND along the load direction equals the applied stress normalized by the fraction of the solid domain.
While Eq. 5 has been mentioned in the work of Kachanov et al. [12], to the best of our knowledge, Eq. 6 has been derived in this context for the first time. They imply that the average microstress is higher than the macrostress in porous media and does not depend on pore morphology. The average microstrain, e.g., measured by ND, is lower than the macroscopic value and does depend on pore morphology. Therefore, the model allows the estimation of microscale strain and stress, responsible for crack initiation and, ultimately, for failure, as a function of the applied macroscopic load, as long as m, E_{d}, and p are known. Conversely, if the properties of the porous material are available (i.e., p, ε_{Macro}, σ_{Macro}, and 〈ε_{micro}〉 are measured) the dense material properties and pore morphology (m) can be evaluated.
Finite element modeling calculation of microstrain
In order to validate the analytical formulae (Eqs. 5 and 6) of conversion between micro- and macrostress and strain, we ran a comparison with FEM calculations, thereby using a completely independent approach.
We considered several artificial structures of various porosities and used two different pore morphologies: overlapping spherical pores (OSP) and overlapping solid spheres (OSS). This is very similar to what Roberts and Garboczi [11] or Rossi [25] did in their pioneering works. The aim of this approach was not to simulate real structures, but rather to set boundaries where the behavior of bi-continuous porous materials may fall. It is to be noted that porous microcracked materials could possibly fall in these boundaries. Similarly, the input material properties do not necessarily correspond to a real case, although we used typical Young’s moduli and Poisson’s ratios close to those of dense alumina.
Input data for FEM simulations
d | Element size, micron | 1 |
E_{d} | Element Young’s Modulus, GPa | 395 |
ν_{d} | Element Poisson ratio | 0.23 |
n | Structure cubic sample size in elements | 100^{3} |
eps_y | Macroscopic uniaxial strain applied (y), ppm | 1000 |
Stress corresponding to a uniaxial strain of 1000 με (or ppm) was applied along the y-axis of Fig. 2b and the (linear) response of the body along the principal axes x, y, z was calculated. The microscopic properties were calculated for each voxel and then averaged over all elements across the structure.
Results of FEM simulations and derived micro/macro relationships as a function of porosity and pore shape
ID | Parameter\\structure | Overlapping pores | Overlapping spheres | ||||||
---|---|---|---|---|---|---|---|---|---|
Porosity (p) | 0.30 | 0.40 | 0.50 | 0.60 | 0.63 | 0.51 | 0.40 | ||
Average Microscopic | SX | Tang. stress (x), MPa | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
SY | Axial stress (y), MPa | 273 | 230 | 193 | 152 | 17 | 46 | 106 | |
SZ | Tang. stress (z), MPa | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
SXY | Shear stress (xy), MPa | 2 | 2 | −1 | 0 | 0 | 0 | 0 | |
SYZ | Shear stress (yz), MPa | −1 | −1 | 0 | −1 | 0 | 0 | 0 | |
SXZ | Shear stress (xz), MPa | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
EPSX | Tang. strain (x), ppm | −159 | −134 | −112 | −89 | −10 | −27 | −62 | |
EPSY | Axial strain (y), ppm | 690 | 583 | 488 | 386 | 43 | 116 | 269 | |
EPSZ | Tang. strain (z), ppm | −159 | −134 | −112 | −89 | −10 | −27 | −62 | |
EPSXY | Shear strain (xy), ppm | 10 | 11 | −5 | 0 | −3 | −2 | 0 | |
EPSYZ | Shear strain (yz), ppm | −5 | −6 | −1 | −5 | −1 | 1 | −3 | |
EPSXZ | Shear strain (xz), ppm | 0 | −2 | −1 | −1 | −1 | 2 | −2 | |
Macroscopic | S_x | Tang. stress (x), MPa | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
S_y | Axial stress (y), MPa | 191 | 138 | 96 | 61 | 6 | 23 | 64 | |
S_z | Tang. stress (z), MPa | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
eps_x | Tang. strain (x), ppm | −219 | −213 | −220 | −215 | −161 | −154 | −174 | |
eps_y | Axial strain (y), ppm | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | |
eps_z | Tang. strain (z), ppm | −218 | −221 | −219 | −224 | −170 | −164 | −173 | |
Derived values | Axial Young’s modulus (E_{p}), GPa | 191 | 138 | 96 | 61 | 6 | 23 | 64 | |
Pore morphology factor (m) | 2.04 | 2.05 | 2.03 | 2.04 | 4.15 | 4.05 | 3.57 | ||
Poisson ratio (ν_{p}) | 0.22 | 0.22 | 0.22 | 0.22 | 0.17 | 0.16 | 0.17 | ||
〈σ_{micro}〉/σ_{Macro} | 1.43 | 1.67 | 2.00 | 2.50 | 2.71 | 2.03 | 1.67 | ||
〈ε_{micro}〉/ε_{Macro} | 0.69 | 0.58 | 0.49 | 0.39 | 0.04 | 0.12 | 0.27 | ||
Diffraction modulus, GPa | 276 | 237 | 197 | 158 | 146 | 195 | 237 | ||
〈ε_{micro}〉x,z/〈ε_{micro}〉y | 0.23 | 0.23 | 0.23 | 0.23 | 0.23 | 0.23 | 0.23 |
Interestingly, the microscopic Poisson’s ratio 〈ε_{x}〉/〈ε_{y}〉 = 0.23 turns out to be invariant on porosity and pore morphology, and corresponds exactly to the input value ν_{d} = 0.23.
The relative difference of the Poisson’s ratio and other quantities, as calculated with the analytical and with the numerical approaches, is astoundingly small, of the order of a few ppm: e.g., Δν/ν ~ 10^{−6}. This confirms the validity of the analytical model set forth above.
Experimental validation
Samples
Geometrical and physical parameters of the SiC sample, a cellular bar of cross-section b × w, cell size L_{1}, L_{2}, and wall thickness t (see Fig. 5a)
Sample | Cell | A_{eff} (mm^{2}) | MOR (MPa) | p | d_{50} (μm) | |||
---|---|---|---|---|---|---|---|---|
b (mm) | w (mm) | t (mm) | L_{1} (mm) | L_{2} (mm) | ||||
15.4 | 23.2 | 0.40 | 1.60 | 2.35 | 130.5 | 268 | 0.38 | 6.7 |
Geometrical and physical parameters of the alumina samples investigated
Sample | ∅ (mm) | H (mm) | A_{eff} (mm^{2}) | MOR (MPa) | p | d_{50} (μm) |
---|---|---|---|---|---|---|
Bulk | 12.6 | 19.4 | 125.5 | 1050 | 0 | – |
17% | 12.2 | 18.5 | 116.3 | – | 0.17 | – |
33% | 10.5 | 18.5 | 87.5 | 446 | 0.33 | 7.3 |
43% | 11.3 | 19.3 | 100.5 | 116 | 0.44 | 8.0 |
ND in situ experiments
Diffraction experiments on the SiC samples were carried out on the instruments Strain Analyzer for Large and Small Engineering Applications (SALSA) at the ILL, Institut Laue-Langevin, Grenoble, France [30]. The stress rig was mounted with the scattering vector q oriented along the axial and the radial sample direction. The axial mount is sketched in Fig. 5b, the radial mount implies a rotation of the rig by 90°. For the compressive tests we took 12 points upon loading and 6 upon unloading, using a load rate of ~10 MPa/min. Diffraction spectra were acquired at fixed load for about 5 min. In order to measure the microstrain in the transverse direction, a sister sample had to be measured. In spite of the use of a position-sensitive detector with about 4° width in 2θ, only the hexagonal SiC-116 peak could be used. This proved to be at 2θ ~ 72° for the wavelength used (λ = 0.1549 nm). The diffraction peaks were fitted with simple Gaussian functions and the errors on the fit parameters (peak position, integral width, and area) were taken as the statistical errors. Figure 5 also shows a photo (c) of the setup used with the rig mount on SALSA’s hexapod.
A hydraulic Instron test rig was used (see Fig. 6b), with two load cells: 10 kN for the 43% porosity Al_{2}O_{3} and 250 kN for the 0%, 17 and 33% porosity Al_{2}O_{3} samples. Steel compression platens with cylindrical shape were especially built and a paper foil was always interposed between the platens and the sample ends. A high resolution Instron extensometer was used to measure the macroscopic strain (Fig. 6b), attached to the samples using rubber bands.
Experimental results
The microstrain behavior is almost perfectly linear within the data scatter. More data scatter occurs in the transverse direction, because of the smaller amount of deformation.
The macrostress strain curve is also linear, with a slightly higher slope at low applied stress. The choice of the initial slope would avoid introducing the possible effect of damage on the Young’s modulus values [26], but the average slope has been used for the sake of consistency with the ND data treatment. A small but visible hysteresis was indeed noticed for the macroscopic curve, which suggests the presence of some mechanical microcracking.
For all but the 17% porosity sample, the a- and c-axis strains (ε_{a} and ε_{c}) are coincident in the longitudinal direction. On the contrary, for all samples, ε_{a} and ε_{c} differ slightly in the transverse direction.
The microstrain behaviors are almost perfectly linear. Some data scatter can be seen in the 43% porosity sample, possibly due to the poorer diffraction signal, leading also to larger error bars.
The macroscopic stress–strain curves are also linear, with the exception of the bulk sample (0% porosity), which seems to have a higher Young’s modulus at low applied load. This implies a possible effect of damage on the Young’s modulus values [31], analogous to what has been found on SiC. Indeed, the macroscopic curves seem to display some hysteresis, especially at very high porosities.
The (macroscopic) Young’s modulus of the dense material (E = 356 GPa) is lower than the values of 400 GPa quoted in literature [4, 34]. However, note that the initial slope of the macroscopic stress–strain curve is 395 GPa. This confirms that some damage is occurring in the dense alumina sample during the test.
All microscopic strains show negligible hysteresis.
The slope difference between the macro- and the micro-response increases as porosity increases.
As mentioned earlier, another important parameter can be extracted from the peak fitting procedure: the integral peak width. This gives the measure of the intra-granular strain and of the grain strain distribution. For steady-state reactor measurements, this is just one fitting parameters of the Gaussian fit function we used. In the case of pulsed sources, a model is required to fit all the integral peak widths at the same time, as shown below.
Interestingly, the integral width in the longitudinal direction increases with the applied stress for all samples, again indicating a larger strain distribution at high applied stresses. This increase is more prominent for high porosities, and has the same explanation as in the case of SiC.
On the contrary, the transverse integral width decreases for the bulk alumina sample, indicating that Poisson’s effect induces a narrower strain distribution inside and among the grains. This phenomenon has two main origins: (a) the residual microstrain in the transverse direction changes in the opposite way to the longitudinal direction; (b) the possible sample texture causes grains to ‘see’ a different stress field and therefore behaves differently in the two directions. At high porosities, the integral width begins decreasing with the applied load but then stabilizes or increases above a certain applied stress, analogous to what has been observed for SiC. Eventually, for high porosity and high applied load, the increase of the stress distribution width is similar to that in the axial and transverse directions.
Discussion
Elastic constants
Indices i,j | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
1 | 501 | 111 | 52 | 0 | 0 | 0 |
2 | 111 | 501 | 52 | 0 | 0 | 0 |
3 | 52 | 52 | 553 | 0 | 0 | 0 |
4 | 0 | 0 | 0 | 163 | 0 | 0 |
5 | 0 | 0 | 0 | 0 | 163 | 0 |
6 | 0 | 0 | 0 | 0 | 0 | 195 |
We can therefore calculate the macroscopic Young’s modulus of the dense material from the measureddiffraction Young’s modulus of the porous material (221 GPa): E_{d} = E_{diff(116)}/(1 − p) = 356 GPa.
The obtained E_{d} value falls well into the literature data range (350–375 GPa) for reaction-sintered silicon carbide ceramics http://www.memsnet.org/material/siliconcarbidesic/.
α-Al_{2}O_{3} stiffness matrix C_{ij} (GPa) [38]
Indices i,j | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
1 | 495 | 160 | 115 | 0 | 0 | 0 |
2 | 160 | 495 | 115 | 0 | 0 | 0 |
3 | 115 | 115 | 497 | 0 | 0 | 0 |
4 | 0 | 0 | 0 | 146 | 0 | 0 |
5 | 0 | 0 | 0 | 0 | 146 | 0 |
6 | 0 | 0 | 0 | 0 | 0 | 167.5 |
Experimental results for the macroscopic and microscopic Young’s moduli, as well as for the Poisson’s ratio
Alumina | SiC | ||||
---|---|---|---|---|---|
Porosity % | 43% | 33% | 17% | 0% | 38% |
E_{diff,long} (GPa) | 252 | 296 | 346 | 403 | 221 |
E_{diff,trans} (GPa) | −1120 | −1212 | −1634 | −1725 | −1842 |
E_{M} (GPa) | 50 | 132 | 233 | 352 | 91 |
〈−ε_{micro,trans}〉/〈ε_{micro,long}〉 | 0.22 | 0.24 | 0.21 | 0.23 | 0.12 |
One further comment can be made: the slight nonlinearity of the macroscopic stress–strain curve in Figs. 7 and 8 implies a degradation of the sample stiffness at high load. This could be related to local damage of the structure at the points of high stress. Indeed, we have seen that the peak width in the longitudinal direction shows an increase at high applied load (Figs. 9, 10). At macroscopic level (see Eq. 1) the nonlinearity should be interpreted as an increase of the pore morphology factor m due to microdamage.
Pore morphology factor
- a.
Using the ratio between micro- and macroscopic strain, i.e., using Eq. 6, we get
The ratio of the micro- to macrostrain was shown to be independent of the solid phase elastic constants, and therefore of literature values, which may not correspond to the processing route of the investigated materials.$$ m_{\text{micro}} = 1 + {\frac{{\ln \left( {{\frac{{\left\langle {\varepsilon_{\text{micro,long}} } \right\rangle }}{{\varepsilon_{\text{Macro}} }}}} \right)}}{{\ln \left( {1 - p} \right)}}} $$(13) - b.Using the classical macroscopic approach, i.e., taking for the Young’s modulus the measured macroscopic (E_{Macro}) and the literature value (e.g., E_{d} = 400 GPa for dense alumina) we have from Eq. 1.$$ m_{\text{Macro}} = {\frac{{\ln \left( {{\frac{{E_{\text{M}} }}{{E_{\text{d}} }}}} \right)}}{{\ln \left( {1 - p} \right)}}} $$(14)
Derived values for the pore morphology factor
Porosity % | Alumina | SiC | ||
---|---|---|---|---|
43% | 33% | 17% | 38% | |
m_{Macro} | 3.47 | 2.46 | 2.21 | 2.85 |
m_{micro} | 3.77 | 3.01 | 3.49 | 2.83 |
All the m values are intermediate between the OSP and the OSS structures, if we compare them with FEM results; this is also suggested by the SiC micrograph in Fig. 4. This consistency is further evidence for the model validity.
For alumina, some further comments must be made: the plot of the macroscopic and microscopic Young’s moduli versus (1 − p) (Fig. 11) allows one to fit an average exponent m for the macroscopic structure as m = 2.71 ± 0.35. This indicates that, expectedly, alumina possesses neither spherical (m = 2) nor star-like pores, m = 4 (i.e., its porosity is generated by overlapping spherical particles). However, from Fig. 11 we notice that the macroscopic Young’s modulus of the 43% porosity alumina lies off the fitting curve more than the other points do. This can be explained by the fact that the sintering parameters (temperature and time), required to obtain lower porosity materials from initial powders, played an important role in defining the pore morphology factor.
It is not the goal of this paper to go into the details of the sintering process (see [41, 42] as examples). While the curvature of the pores is constant for this kind of ceramics in the intermediate stage of sintering and theoretically tends to zero [43] due to surface energy minimization, the pore morphology factor changes. This can be formalized if we take the different m_{Macro} values listed in Table 9 and compare them with the average m quoted above. The m_{Macro} value for low porosity is close to 2, while it increases at higher p. This would introduce a dependency of the pore shape factor m on the porosity itself. However, materials with very different porosities may have similar pore morphology factors or vice versa. Therefore, the connection between p and m is fictitious, since both truly depend on the processing route. In fact, as Green reports in [12], a linear dependence of the macroscopic Young’s modulus as a function of porosity could also be attached to our data, getting a percolation value of p_{0} = 0.49. This is represented in Fig. 11 (dashed curve) and implies that strictly speaking the pore morphology factor used in this work is not directly linked to the curvature of the pores. This also can explain why in the literature all the models proposed (see the introduction: exponential, rational, linear, power) equally fit the experimental data. We further notice that whichever model is used, the compression MOR seems to follow the elastic coefficient law as a function of porosity [44] (i.e., it has for instance the same percolation level in the linear model).
It is fundamental to control the sintering conditions in such a way to have comparable pore morphology factor, if we want to assess the universal dependence from porosity of the elastic properties of a material in the classical manner used by Gibson and Ashby [21]. This is also one of the most important findings of the investigations of Wang [40]. In fact, the method proposed in this work (see Eq. 13) is a way to evaluate the pore morphology factor deconvoluting the pore shape from the porosity value. Tomography work would obviously helps to assess the influence of the sintering parameters on the pore morphology, but this is again beyond the scope of this work.
One of the main limitations of the analytical model is its scalar nature, which contrasts with the tensorial character of the stress state in a porous material. The bridge between the two worlds is laid down by the FEM calculations. Although based on a few different model microstructures, they yield insights into the general features of the stress state in the material. These general features are independent of the detailed microstructure of the solid, since the latter is even more complicated.
In conclusion, we can state that the agreement between literature, calculated and measured data allows a reliable use of the latter to extract the properties of the dense material, which would be inaccessible to solely macroscopic techniques.
Summary and conclusions
In this work, analytical modeling of the macro- to microstrain and stress conversion has been carried out for porous ceramics. This approach has been based on stress balance in elastically isotropic media. It has been shown that the macro-stress to average micro-stress ratio depends linearly on porosity and does not depend on pore morphology factor m, while the ratio of macro-strain to average micro-strain does. The factor m and the Young’s modulus of the dense material, both can be extracted from uniaxial compression testing, if the porosity is known. Thus, the model suggests a way to evaluate the elastic constants of the dense material using experiments on the corresponding porous material.
FEM calculations have allowed cross-checking the analytical calculation for different model microstructures. The agreement between the two theoretical approaches proved to be astounding. In addition, the FEM approach allowed calculating shear strains arising during uniaxial testing and visualizing the behavior of the pore structure under load.
ND in situ compression testing carried out on porous honeycomb SiC bars and on compact porous alumina cylinders has fully validated the model and the FEM calculations. Experimental results show that the macroscopic and microscopic strain responses to external stress are not equal: they reflect the peculiarities of the whole body deformation (including pores) and the grain response, respectively. For alumina, we have shown that, knowing the macroscopic Young’s modulus of the dense material, values of the pore morphology factor m could be extracted in two different ways: as a common value to all samples, following a power law, or as individual values. The mismatch between the two approaches gives a cross-check of the assumption that pores have the same morphology factor for a material with different porosities. Moreover, it has been shown that using the microscopic strain response, a novel method to estimate the pore morphology factor could be proposed, which is independent of the literature’s Young’s modulus values.
FEM confirmed the experimental finding that the microscopic Poisson’s ratio does not vary as a function of porosity and pore morphology factor, and it is always equal to the macroscopic value for the dense material. The average measured value for alumina was 0.23 ± 0.02, which agrees very well with the calculated and literature value of 0.23.
The approach has proven to be self-consistent. Through the use of ND, the link found here between average microscopic and macroscopic strains and stresses in porous media opens new opportunities for the estimation of macroscopic and microscopic crystalline mechanical properties of the correspondent dense material.
Acknowledgements
Thomas Glasson and Cedric LeGoff (Corning SAS, CETC, Avon, France), Angela Graefe, Andy Schermerhorn, James E. Webb and Lisa Noni (Corning Inc, Painted Post, NY, USA.), Irina Pozdnyakova (CNRS, Orleans, France), and Darren J Hughes (ILL, Grenoble, France), Donald W. Brown and Thomas A. Sisneros (MST-8, LANL, Los Alamos, NM, USA) are kindly acknowledged. This work has benefited from beam time from the Institut Laue-Langevin (ILL), Grenoble, France, as well as the use of the Lujan Neutron Scattering Center at LANSCE, which is funded by the Office of Basic Energy Sciences (DOE). Los Alamos National Laboratory is operated by Los Alamos National Security LLC under DOE Contract DE AC52 06NA25396.