 Original Paper
 Open access
 Published:
Combined effects of pores and cracks on the effective thermal conductivity of materials: a numerical study
Journal of Materials Science: Materials in Engineering volume 19, Article number: 25 (2024)
Abstract
Porous solids are commonplace in engineering structures and in nature. Material properties are inevitably affected by the internal inhomogeneity. The effective thermal conductivity of porous materials has been and remains to be a subject of extensive research. Less attention has been devoted to thermal conductivity impacted by internal cracks. This study is devoted to theoretical analyses of the combined effects of pores and cracks on the effective thermal conductivity. Systematic numerical simulations using the finite element method are performed based on twodimensional models, with periodic distributions of internal pores and cracks. The parametric investigations seek to address how individual geometric layout can influence the overall thermal conduction behavior. In addition to circular pores and isolated cracks, angular pores with cracks extending from their sharp corners are also considered. It is found that both isolated cracks and cracks connected to existing pores can significantly reduce the effective thermal conductivity in porous materials. Since it is much easier to microscopically detect internal pores than thin cracks, care should be taken in using the apparent porosity from microscopic images and density measurements to estimate the overall thermal conductivity. Quantitative analyses of the detailed geometric effects are reported in this paper.
Introduction
Heterogeneous materials are ubiquitous in nature and in engineered structures. Their physical properties can be significantly influenced by local heterogeneities and defects. One form of heterogeneity is porosity, and in such materials, distinct features of the solid phase and void space exist. Conductive heat flow inside the material is thus impeded and diverted by porosity, causing a reduction of effective thermal conductivity compared to its pure solid counterpart (Kaviany 1991; Bauer 1993; Smith et al. 2013; Pietrak and Wisniewski 2015). The thermal conduction behavior plays a critical role in a wide range of engineering applications including nuclear fuel components (Bakker et al. 1995; Bai et al. 2016; Hawkes et al. 2017; Iasir et al. 2018; Wang and Zhao 2021); energy storage systems (Dinesh and Bhattacharya 2019); thermal barrier coatings (Gu et al. 2001; Renteria et al. 2006; Zhu et al. 2019); thermal insulation (Lu et al. 1998; Amran et al. 2015; Dou et al. 2019); electronic packaging (Ogushi et al. 2004; Zhao et al. 2019; Sun et al. 2024); and compact heat exchangers (Yeap et al. 2013). In additively manufactured materials, defects such as voids normally exist or they may be designed for physical media flow. Heat conduction for additively manufactured materials is essential due to the local elevatedtemperature processes and the resulting porosity in the fabricated products (Zhang et al. 2018; Ibrahim et al. 2020; Ghorbani et al. 2022; Yap et al. 2023).
Internal macroscopic or microscopic cracks may be considered a special form of porosity in solids. Although there is essentially no bulk void volume associated with an internal crack, each crack still represents a discontinuity for thermal conduction. Compared to the pure solid counterpart, the effective thermal conductivity of a crackcontaining media will be reduced (Hasselman 1978; Hoenig 1983; Benveniste and Miloh 1989; Kushch and Sangani 2000; Sevostianov 2006; Nguyen et al. 2017; Rangasamy Mahendren et al. 2020). Various analytical expressions, either modelbased or empirical, describe the effect of porosity on the effective thermal conductivity (Maxwell 1873; Rayleigh 1892; Landauer 1952; Loeb 1954; Hashin and Shtrikman 1962; Schulz 1981; Chiew and Glandt 1983; Bauer 1993; OrdonezMiranda and AlvaradoGil 2012; Yang et al. 2013; Smith et al. 2013; Pietrak and Wisniewski 2015; Kiradjiev et al. 2019). A large spread of predictions however exists, with some models being specific to certain porous configurations. Computational approaches such as the finite element method (Bakker et al. 1995; Iasir et al. 2018; Wang and Zhao 2021; He et al. 2019; Xu et al. 2021) can also be employed for such prediction. The numerical results were generally used for obtaining trends, elucidating theoretical and analytical solutions, and assisting in two and threedimensional conversions, etc. While analytical and numerical studies of effective thermal conductivity provided insights for porous and crackcontaining materials (separately), one scenario has not been systematically studied: materials with coexisting internal pores and cracks. Typical porous structures may also contain distributed cracks, which is expected to further reduce the effective thermal conductivity. Examples include nuclear fuels (Kubota et al. 1964), sintered materials (Qin et al. 2020), and geological materials (Rutqvist et al. 2002). Furthermore, in additively manufactured materials, both voids and cracks caused by lack of fusion are commonplace (Bidare et al. 2022; Hashemi et al. 2022; Byun et al. 2024). When individual voids take an angular rather than rounded shape, cracks may also be expected to form at the sharp corners. Note that, compared to regular porosity, cracks are much more difficult to detect in a microscopic image or from density measurement. How these “hidden” cracks may further degrade the thermal conductivity of a porous material requires fundamental studies. The present work provides such a systematic numerical assessment under the condition of periodic distributions of cracks in addition to pores.
Although largerscale numerical simulations incorporating many pores or actual imagebased pore geometries may be employed (Bakker et al. 1995; Carson et al. 2003; Kaddouri et al. 2016; Iasir et al. 2018; Bowen et al. 2020; Wang and Zhao 2021; Zhang et al. 2021; Marr et al. 2022), the results generally pertain to the specific regular or irregular geometries and may not offer sufficient insight into how local arrangement of pores actually dictates the overall behavior. The “unitcell” approach assuming a periodic arrangement of microconstituents is an alternative (Hsu et al. 1995; Ogushi et al. 2004; Yvonnet et al. 2008; Xiong et al. 2022). Our previous analyses thus used this approach and considered simple geometric models with easy tuning of geometric parameters, in order to systematically pinpoint the connection between microscopic (shape, size and spatial distribution of pores) and macroscopic (effective composite behavior) features (Shen et al. 2022a). It was found that the effective thermal conductivity is insensitive to the spatial arrangement of circular and spherical pores, and the same insensitivity also holds true for clustering and a mixture of pore sizes. In a subsequent study, the effects of internal cracks (no pores) using similar types of geometric arrangement were analyzed (Shen et al. 2022b). The effective conductivity was found to be insensitive to the actual crack arrangement and can be uniquely represented by the ratio of crack size to the unitcell size of the model. The periodic crack configurations were also seen to be representative of random sizes, orientations, and spatial distributions of cracks, thus exhibiting a high degree of generality. The present work builds on the previous efforts mentioned above, to incorporate both pores and cracks into the analyses, and further explore the effects of cracks extended from the existing pores.
The remainder of this paper is organized into the following sections. The numerical model descriptions and simulation approach are introduced in the “Numerical model description” section. The “Preliminary results: separate effects of periodic pores and periodic cracks” section provides representative results from the two previous studies mentioned above (one figure replotted from each study (Shen et al. 2022a, 2022b), for laying the foundation for the new analyses in subsequent presentations. The “Combined effects of periodic pores and cracks” section reports new findings from the combined pore/crack models. To test the generality of the periodic geometric arrangements, more random pore/crack distributions are considered in the “More random distributions” section. The “Cracks extended from angular pores” section focuses on several scenarios of pores with sharp corners where cracks exist. General conclusions are given in the “Conclusions” section. Attention in this work is devoted to twodimensional models only. Threedimensional simulations will involve exceedingly complex geometric possibilities and will be left as future work.
Numerical model description
The current study focuses on the coexistence of pores and cracks in a solid material. The geometric models are built upon 2D distributions of pores (without cracks) and cracks (without pores), so these reference configurations are first introduced. Representative results for these separate analyses will be shown in the “Preliminary results: separate effects of periodic pores and periodic cracks” section, with data taken from our previous numerical studies for pores only (Shen et al. 2022a) and cracks only (Shen et al. 2022b).
Poresonly and cracksonly models
Figure 1a shows the baseline porous model with a square array of circular pores. The shaded circles represent the empty space surrounded by a solid matrix. Because of the existence of symmetry and periodicity, the highlighted squares (unit cells) are the actual domains needed for the numerical analyses. Temperature gradient can be prescribed along the 0/90° directions (same results for these two directions) and along the 45° direction, as represented by the small arrows. Figure 1a also shows a typical micromechanical unit cell. With appropriate boundary conditions imposed, a single cell effectively represents an infinite array of microstructures. In the absence of heat source, the governing equation is
where K_{m} is the thermal conductivity of the solid matrix and T is temperature. For simulating the steadystate heat conduction along the ydirection, for instance, constant temperatures of T_{0} and T_{1} (with T_{1} > T_{0}) are imposed on the horizontal boundaries y = 0 and y = L, respectively. The Neumann adiabatic boundary conditions are enforced on the side boundaries x = 0 and x = L with temperature gradient in the xdirection prohibited, due to symmetry. The effective thermal conductivity of the porecontaining structure, K_{eff}, is then
where J is the total reaction heat flux obtained from the simulation. Note that when there is no pore and no crack in the model, K_{eff} = K_{m}. It is worth mentioning that empty space may be used in the model to represent the pore, or one may treat the pore as a secondphase particle with a zero thermal conductivity (which essentially represents an air bubble). Both approaches generate the same numerical results. In other engineered and geological materials, internal pores may also contain various fluids (Atabaki and Baliga 2007; Abdulagatova et al. 2010; Zhou et al. 2016). While these material systems are beyond the scope of the present work, the same modeling methodology can be employed for analyzing their thermal conduction behavior.
For examining the model generality, an alternative hexagonal arrangement is considered in Fig. 1b. Two rectangular unit cells are highlighted, where all their side boundaries are symmetry lines of the global periodic structure. The case with small horizontal arrows signifies not just the xdirection conduction but also along the ± 60° directions, meaning that this particular unitcell model encompasses all three orientations with the nearestneighbor pores. The rectangular unit cell with small vertical arrows, on the other hand, represents conductions along the ydirection as well as the ± 30° directions, namely all three orientations with pores having the secondnearest neighbors. As a consequence, the hexagonal arrangement in Fig. 1b effectively features every possible 30° direction in the entire 360° span.
For studying the effect of internal cracks, numerical models of periodic geometric arrays are also utilized. Figure 2a shows equal numbers of cracks oriented along the x and ydirections in an alternate fashion, with the crack centers placed in a square alignment. For easy visualization, cracks are depicted as sharp ellipses. In Fig. 2b, the crack centers form a staggered array. Both cases will lead to the same effective thermal conductivity along the x and ydirections. The squares highlighted in Figs. 2a and b have all their side boundaries being the symmetry lines, so they are the unit cells needed in actual simulations as shown in Fig. 2c. The same methodology is used to calculate the effective thermal conductivity as described above. The side lengths of the cells are different for different models, for the purpose of keeping one full crack per area of L^{2} in all cases thus ensuring valid comparisons of the effective thermal conductivity. For example, in the middle unit cell in Fig. 2c, there is a net “halfcrack” within an area of L^{2}/2 so the crack density is 1/L^{2}. When presenting the numerical results, the quantity a/L is used to represent the normalized crack size, where “a” denotes the half crack length (Shen et al. 2022b).
Models with mixed pores and cracks
Numerical results for the poresonly and cracksonly models are presented in in the preliminary results section below, and it will be shown that the reduction in effective thermal conductivity is insensitive to the spatial distribution of pores and cracks. Therefore, to study the combined effect of pores and cracks, the simple square array is used as a basis to construct the geometric models. Figure 3a shows the first distribution which is a combination of Figs. 1a and 2a. The highlighted squareshaped unit cell is the actual domain needed for the simulations, with global symmetry across all the unit cell’s side boundaries. One can define the crack density D_{c} and pore density D_{p}, and in this case, their relation is D_{c} = D_{p} since there is one full pore and one net crack in each unit cell. The second configuration is shown in Fig. 3b, which is a combination of Figs. 1a and 2b. Note that, with the staggered arrangement of cracks, D_{c} = 0.5 D_{p} in this case. Highlighted in Fig. 3b are two unit cells with all their boundaries being symmetry lines. Figure 3c includes all three unit cells. The side lengths of the cells are chosen such that there is one circular pore per area of L^{2} in all cases while keeping one full crack and onehalf crack per L^{2} for the configurations in, respectively, Figs. 3a and b.
For the purpose of checking if the periodic unitcell approach in Fig. 3 is representative of realistic microstructures, random distributions of pores and cracks are also considered. More random models with the pore area fractions, f_{p}, of 0, 0.1, and 0.2 are constructed, with either D_{c} = D_{p} or D_{c} = 0.5 D_{p}. An example of D_{c} = 0.5 D_{p} is shown in Fig. 4, where six pores and three cracks of various sizes exist in a 3L × 3L domain, with the total f_{p} = 0.2 and average normalized crack size of a/L = 0.29. Note that the effective conductivity values along the x and ydirections for such types of random models are essentially the same. The resulting effective thermal conductivity will be compared with the periodic unitcell analyses.
Cracks extended from pores
In this study, another form of coexisting pores and cracks is considered, where cracks are initiated from the sharp corners of each pore. Figures 5a and b show the model geometries based on diamondshaped (rotated square) pores and square pores, respectively. Note that the general configuration in Fig. 5a is frequently observed in polymers additively manufactured by fused filament fabrication, where many visible aligned voids exist due to poor filling of the filament rasters (Sun et al. 2008; Rankouhi et al. 2016; Ghorbani et al. 2022). Such materials may also exhibit a relatively weak interfilament strength causing crack formation. In each of Figs. 5a and b, two separate unitcell models of the aligned (0/90°) and staggered (45°) arrangements can be identified and used in actual simulations. The aligned diamond shape (Fig. 5a), staggered diamond shape (Fig. 5a), aligned square (Fig. 5b), and staggered square (Fig. 5b) are henceforth referred to as models of AD90, SD45, AS90, and SS45, respectively. The primary objective is to quantify (i) how the pores and associated cracks degrade the effective thermal conductivity from one extreme (foursided pores only) to the other extreme (crossshaped cracks only), and (ii) the effect of the relative sizes of the pore/crack combination.
All simulations described in the “Poresonly and cracksonly models,” “Models with mixed pores and cracks,” and “Cracks extended from pores” sections were carried out using the finite element software ABAQUS (VIAS3D ACADEMIA, Houston, TX, USA). The threenoded and fournoded linear heat transfer elements were employed. Mesh convergence was checked through various degrees of refinement. Verification of the computational model was demonstrated in our previous work, by comparing the simulation results with analytical solutions for select cases of pure pores and pure cracks (Shen et al. 2022a, 2022b). In the current study of steadystate heat transfer, the only material property needed is K_{m}. Values of the effective thermal conductivity are all presented in a normalized form, K_{eff}/K_{m}, so the results are applicable to all possible values of K_{m} at any temperature where K_{m} is valid.
Preliminary results: separate effects of periodic pores and periodic cracks
In this section, the numerical results based on the models in Figs. 1 and 2 are presented. The data were obtained from our previous work (Shen et al. 2022a, 2022b), and they are redrawn here to serve as the basis for the studies of the combined effects of pores and cracks presented in the following sections. The results for the poresonly case are presented in Fig. 6, with the effective thermal conductivity (normalized by the conductivity of the matrix, K_{eff}/K_{m}) as a function of the area fraction of the pores (f_{p}), for the four models of uniform square array (Fig. 1a) and hexagonal array (Fig. 1b). It is evident that the reduction of thermal conductivity follows the same curve as the pore fraction increases. The primary conduction direction, as well as the spatial arrangement of periodic pores, do not cause any appreciable difference in the overall behavior. This observation is consistent with other numerical findings (Iasir et al. 2018; Wang and Zhao 2021). From our earlier study (Shen et al. 2022a), geometric clustering and coexistence of pores of different sizes were also found to have insignificant influence on the overall thermal conduction. The effective conductivity is thus dictated by the pore fraction.
Also shown as insets in Fig. 6 are the temperature distributions in the four models with a pore fraction of 0.15, with the blue color representing T_{0} and the red color representing T_{1} at the opposite boundaries. Note that in each model the temperature contours are independent of the actual temperature values used in the simulations (same bluetored distribution when different T_{0} and T_{1} values are assigned in the simulations). The calculated effective conductivity is based on Eq. (2) so a greater (T_{1} − T_{0}) value will give rise to a proportionally higher reaction heat flux, which in turn leads to the same K_{eff}. It can be seen that the existence of pores disturbs the temperature field which would have been simple parallel color stripes for a homogeneous porefree solid.
The effect of cracks is shown in Fig. 7 where the normalized effective thermal conductivity is plotted against the normalized crack size (a/L), for the three crack models of aligned square array with 0/90° direction (Fig. 2a) and staggered array with 0/90° and 45° directions (Fig. 2b). It is evident that the three models in Fig. 2, with evenly distributed and oriented cracks, lead to the same outcome. The insensitivity of effective thermal conduction to spatial distribution of cracks is not limited to the 0/90° and 45° directions in the aligned or staggered square arrays. Models with more random cracks, not just orientation distribution but also size variation, were also found to follow basically the same quantitative response (Shen et al. 2022b). It is the average normalized crack size (a/L) that determines the reduction in overall thermal conductivity. Figure 7 also includes four temperature contour maps obtained from the 45° staggered array simulations with a/L equal to 0.1, 0.2, 0.3, and 0.4. The blue and red colors at the boundaries again represent prescribed T_{0} and T_{1}, respectively. The disturbance of temperature field due to the crackinduced discontinuity is evident, and its extent is greater if the crack is longer.
Combined effects of periodic pores and cracks
The “Preliminary results: separate effects of periodic pores and periodic cracks” section established the baseline relationship between thermal conductivity and periodic pores or cracks. The modeling results with combined pores and cracks as depicted in Fig. 3 are now presented. The crack configurations in Figs. 3a and b correspond to those of Figs. 2a and b, respectively. This results in the same crack density and pore density (D_{c} = D_{p}) in Fig. 3a and only onehalf the crack density (D_{c} = 0.5D_{p}) in Fig. 3b. Before showing the systematic numerical results, the representative temperature fields are first illustrated, and the resulting effective thermal conductivities based on the two forms of unit cells (0/90° vs. 45°) as highlighted in Fig. 3b are compared. The three temperature fields in Figs. 8a and b correspond to the one unitcell model in Fig. 3a and two unitcell models in Fig. 3b, respectively. The images shown are for the pore fraction f_{p} = 0.2 and normalized crack size a/L = 0.4. The primary heat conduction direction considered in these images is vertical. The disturbance of temperature field caused by the pores and cracks is evident. The simulated effective thermal conductivity as a function of pore fraction, for the two configurations in Fig. 8b, is shown in Fig. 8c. The 0/90° and 45° cases generate essentially the same K_{eff}, affirming the insensitivity to conduction direction with the combined periodic pores and cracks.
Systematic results for the models in Fig. 3 are now presented. The case of D_{c} = D_{p} (Fig. 3a) is considered first, and its results are shown in Fig. 9a where the normalized effective thermal conductivity is plotted against the normalized crack size, under various pore fractions. When f_{p} is zero or small, the decrease in thermal conductivity caused by cracks is significant. For instance, at f_{p} = 0.1, the existence of cracks of size a/L = 0.5 will cause a reduction in K_{eff}/K_{m} from about 0.81 to 0.57. At larger f_{p}, the pore effect becomes more dominant so only a moderate decrease in K_{eff}/K_{m} caused by cracks can be observed (e.g., the curve of f_{p} = 0.5 is much less steep). The same thermal conductivity data in Fig. 9a can also be plotted as a function of pore fraction for various specified normalized crack sizes, shown in Fig. 9b. Note that the curve of a/L = 0 (no crack) is essentially the same as those shown in Fig. 6. With an increasing crack size (a/L), the thermal conductivity decreases further so the curves in Fig. 9b become lower. When the crack size is very large, the heat conduction pathways are much narrowed already so the variation of K_{eff}/K_{m} with pore fraction tends to be more moderate.
The results presented in Fig. 9 are for the case of D_{c} = D_{p} (Fig. 3a). The same set of graphs can be shown for the case of D_{c} = 0.5D_{p,} as plotted in Figs. 10a and b, which correspond to the model configuration in Fig. 3b. Due to the smaller crack density, the thermal conductivity curves in Fig. 10 are all at a higher position than those in Fig. 9, except for the nocrack case of a/L = 0 in Fig. 10b which is the same as the topmost curve in Fig. 9b. However, the qualitative trends in Figs. 9 and 10 are the same.
The numerical results for the case of equal crack density and pore density, D_{c} = D_{p}, may be used to shed light on the relative influence of pores vs. cracks. One may consider, for instance, f_{p} = 0.5 (pores only) and use the pore’s radius as the imaginary crack’s halflength (for cracks only), which leads to a/L ≈ 0.4. Then from Figs. 9a and b, f_{p} from 0 to 0.5 (no crack) results in a greater reduction in K_{eff}/K_{m} than a/L from 0 to 0.4 (no pore). Therefore, pores inside the material exert a greater influence than cracks in reducing the effective thermal conductivity.
More random distributions
To examine the generality of the periodic unitcell approach, larger models with more randomly distributed pores and cracks are also analyzed. The results are shown in Figs. 11a and b, for the cases of D_{c} = D_{p} and D_{c} = 0.5D_{p}, respectively. In the figures, the K_{eff}/K_{m} values obtained from the unitcell models of f_{p} = 0, 0.1, and 0.2, taken from the data in Figs. 9 and 10, are plotted as a function of crack size (a/L) as dashed curves. Discrete points in Figs. 11a and b represent simulation results of more random pores and cracks for the corresponding pore fractions and crack sizes. Some pore/crack configurations and their temperature fields are shown as insets. Note that in a given model, the pores and cracks are not necessarily uniform in size; the discrete points shown in Fig. 11 are at positions corresponding to the average pore fractions and average crack sizes. It can be seen that the results of the random models largely follow those of the periodic unit cells. Slight deviations occur as the crack size and pore fraction become greater. Figures 11a and b thus attest to the generality of the simple unitcell approach. It is also a manifestation that pores in the model have a greater influence than cracks in lowering the effective thermal conductivity, as discussed in the previous section.
The results presented in Figs. 9, 10, and 11 demonstrate that the coexistence of pores and cracks leads to further reduction in the effective thermal conductivity, compared to the cases of pores only and cracks only. It is worth noting that in actual experimental characterization of porous materials, porosity can be easily revealed by microscopic images, but the possible existence of cracks may be harder to discern. As a consequence, the overall thermal conductivity of the material may actually be lowered than what can be predicted from the apparent porosity. This may partially explain why there is frequently a large discrepancy in measured thermal conductivity values for materials with a given porosity (Smith et al. 2013; Kumar et al. 2019; Bowen et al. 2020).
Cracks extended from angular pores
Attention is now turned to angular pores with diamond and square shapes, with possible cracks extending from the sharp corners as depicted in Figs. 5a and b. Four unitcell geometries, aligned diamond shape (AD90), staggered diamond (SD45), aligned square (AS90), and staggered square (SS45), were identified in Fig. 5. In the analyses presented below, equal lengths of the cracks from all corners of the pore along the two mutually perpendicular directions in each model are considered. The crack size is expressed as a normalized quantity of actual crack extension divided by the unitcell span along the crack directions (horizontal, vertical, or diagonal). The reference cases with pores only are characterized first.
Figure 12 shows the effective normalized thermal conductivity as a function of pore fraction, for the four model configurations with pores only (no crack). For reference purposes, the result from the models of circular pores, taken from Fig. 6, is also included in the figure. It can be seen that the AD90 and SD45 models generate essentially identical thermal conductivity values for a given pore fraction. AS90 and SS45 display a similar type of orientationindependent behavior. When f_{p} is smaller than about 0.2, all four models show very close thermal conductivities; beyond the 0.2 pore fraction, the diamondshaped pores lead to lower conductivities. This is due to the fact that the diamond shape possesses a wider span (diagonal direction) perpendicular to the primary heat conduction direction, thus imposing greater blockage of conductive pathways. Note also that all four models with diamond or squareshaped pores result in moderately lower K_{eff}/K_{m} compared to that of circular pores. This trend is consistent with analyses reported in other studies (Yang et al. 2013).
With the reference cases of poresonly established, the presentation now focuses on how the extension of cracks will further degrade the thermal conduction behavior. Figure 13 shows the normalized effective thermal conductivity as a function of the normalized crack extension, for the models of (a) AD90, (b) SD45, (c) AS90, and (d) SS45. In all cases, the pore fractions of f_{p} = 0.1, 0.2, and 0.3 are included. In each model, a representative temperature contour plot with the corresponding pore/crack configuration is included as an inset. Note that the last points in all curves are shown in Figs. 13a–d correspond to an overall crack/pore span (from the crack tip through the center pore to the other crack tip) of 80% of the unitcell span along the same direction. As a consequence, in each plot the curves of f_{p} = 0.1, 0.2, and 0.3, all have different lengths (the larger pore has a longer diagonal and thus the maximum crack extension will be smaller). It can be seen that the existence of cracks further decreases the effective conductivity. Care should thus be taken when deducing thermal conductivity from the apparent porosity obtained microscopically, since cracks evolving from the pore corners may exist especially in additively manufactured materials due to relatively poor adhesion. Also observed in Figs. 13a–d is that, in each figure, the last point of f_{p} = 0.3 has the lowest K_{eff}/K_{m} compared to the last points of f_{p} = 0.2 and then 0.1. This suggests that the embedded pores lead to more significant reduction of thermal conductivity than the extended cracks since the last points in all the curves in Fig. 13 correspond to the same normalized span of 0.8. Further illustrations of the tendency are presented in Fig. 14.
Comparing the different models shown in Fig. 13, it is also evident that, as the crack length increases to reach the 80% tiptotip crack/pore span, AS90 and SS45 (Figs. 13c and d) display a greater reduction in thermal conductivity than AD90 and SD45 (Figs. 13a and b). This observation implies different geometric effects of the cracks in blocking the heat flow for the two sets of models. Figure 14 compares the poresonly (no crack) results with those of cracksonly (no pore). Due to the essentially identical behavior between AD90 and SD45 and between AS90 and SS45, only the two models AD90 and AS90 are included, and the tiptotip (diagonal) span is used to represent the sizes of the pore and crossshaped cracks in a unified manner. In each configuration of AD90 and AS90, the pore model (no crack) is seen to result in a lower curve than the crack model (no pore), thus illustrating again the larger effect of the pores on overall thermal conductivity. AS90 tends to reduce the conductivity more, which is consistent with the general trend observed in Fig. 13. This is due to the fact that, at any given normalized tiptotip span, there is either a crack parallel to the primary heat flow direction (thus not contributing to blocking the primary conduction path) or a pore with a smaller area fraction, for the AD90 model compared to AS90 (see the geometric layout in Fig. 5).
The results presented in Figs.13 and 14, as well as those in previous sections, illustrated that internal cracks can significantly impact the effective thermal conductivity, regardless if the cracks are isolated or connected to the existing pore.
Although the current study is devoted to the periodic arrays of pores and cracks, qualitative comparisons with experiments with realistic microstructures can still be attempted. Figure 15 includes some measured effective thermal conductivities as a function of pore fraction, taken from the literature, for various material systems (Koh and Fortini 1973; Nosewicz et al. 2022; Simmons et al. 2020; Smith et al. 2003; Živcová et al. 2009). The figure also includes representative simulations from the current study, based on circular pores and squareshaped pores (AS90 with no crack), as well as squareshaped pores with extended cracks (AS90 with extended cracks of normalized crack extension of 0.2). Note that the simulation curves as shown have been adjusted using a 2D/3D conversion relation (Bakker et al. 1995), which was validated in Shen et al. 2022a,
It is evident from Fig. 15 that a great extent of scatter exists in the measurements. While there may not be a direct correspondence between simulations and actual measurements in Fig. 15, the relative positions of the simulation curves and experimental distributions nevertheless point to the possibility of internal cracks as an important factor in lowering the effective thermal conductivity of porous media.
Conclusions
Systematic numerical finiteelement analyses were conducted, to study the combined effects of internal pores and cracks on the effective thermal conductivity of twodimensional solids. Periodic models of intermixed circular pores and sharp cracks, as well as square or diamondshaped pores with cracks extending from the sharp corners, were studied. Salient findings are summarized below.

Both the isolated cracks and cracks connected to existing pores can significantly reduce the effective thermal conductivity in porous materials. Since bulky voids are much easier to detect, microscopically or using density measurement, than sharp cracks, care should be taken in using the apparent porosity to estimate the overall thermal conductivity.

For models containing only circular pores, the effective thermal conductivity is dictated by the pore fraction and is insensitive to the geometric distribution and orientation of the pores. For models containing only cracks, the effective thermal conductivity is also insensitive to the crack layout and is largely controlled by the normalized crack size and density. The coexistence of pores and cracks results in similar features, with the crack density relative to the pore density playing an additional role.

More randomly distributed pores and cracks showed quantitatively similar results to those of the periodic counterparts, demonstrating the generality of the regular unitcell approach considered in this study.

With cracks extending from the sharp corners of pores, the unitcell models considered in this study also showed a high degree of generality, as the aligned and staggered arrangements resulted in very close conductivity values for a given diamond or square pore configuration. Thermal conductivity is not solely determined by the overall tiptotip crack spans; the pore at the crack center also contributes to lowering the effective thermal conductivity.
The simulations conducted in this study are valid for uniform arrays of pores and/or cracks covering the entire twodimensional body. If there is significant clustering of pores and/or cracks with a large area of material otherwise free of defects, the problem will be less well defined and a casebycase analysis will be necessary. Future developments may also include direct simulations of the complex threedimensional geometries.
Availability of data and materials
All data generated and analyzed during this study are included in this published article.
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Acknowledgements
YLS acknowledges the endowment support from PNM Resources Foundation, as the PNM Chair for Renewable Energy Research.
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This work was supported by the US Department of Energy, award #DENA0004113, under the NNSA Minority Serving Institution Partnership Program, through a subaward contract from Florida International University.
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YLS conceived the idea, supervised the research, and wrote the original draft. YLS and GR performed the numerical simulations and analyses, and contributed to finalizing the manuscript.
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Shen, YL., Rose, G. Combined effects of pores and cracks on the effective thermal conductivity of materials: a numerical study. J Mater. Sci: Mater Eng. 19, 25 (2024). https://doi.org/10.1186/s4071202400172w
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DOI: https://doi.org/10.1186/s4071202400172w