Stochastic mechanical mesoscale modeling of cylindrical cavities in metallic foams

Abstract

Aluminum foam is a critical component of state-of-the-art micrometeoroid and orbital debris shielding. However, its randomly oriented constituent ligaments create significant variability in properties at different length scales. Previous efforts have thoroughly characterized the statistical implications of micro- to macro-scale-length transitions on the homogenized elastic behavior of damage-free, pre–micrometeroid and orbital debris (MMOD) strike structures. In this work, the properties of the residual material post MMOD impact are assessed by introducing cylindrical cavities representing the damage track. It was observed that configuring the cavity parallel to the through-the-thickness direction transformed a nominally isotropic foam into an anisotropic (transverse isotropic) material, weaker in-plane than through-the-thickness. Furthermore, by calculating similarity indices of computed probability distribution functions, we observe that in a substantial fraction of cases local variability arising from the ligamented structure itself is large enough that the elastic moduli of structures with and without cavities are statistically indistinguishable. These findings are evidence that we are able to draw statistics-based conclusions regarding the effect of MMOD impact-inspired defects on the mechanical properties of aluminum foam structures.

1. Introduction

Micrometeroid and orbital debris (MMOD) poses a significant danger to both robotic and crewed spacecraft operating beyond Earth’s atmosphere and necessitate carefully designed shielding materials [1–6]. The impact of even a sub-millimeter projectile traveling at hypervelocity can cause devastating damage [1]. A number of studies have focused on simulating the strike event through both experimental and computational approaches [7–10]. The natural continuation of this work is an evaluation of the mechanical behavior of the resulting damaged materials, incorporating the configuration of the damage track as a change to the material structure. This is critical for assessing the potential response not only to future MMOD strikes but also to any mechanical stress the shielding will experience for the duration of its lifetime.

Open cell metallic foam sandwich panel structures have proven effective at mitigating MMOD impact damage [3–5]. The foam core layer is composed of open, node-ligament networks of randomly oriented polycrystalline metallic ligaments. Ligament lengths and pore sizes vary over some distribution characteristic of the specific material, but typically have lengths on the order of microns. Impacting particles penetrate the outer sacrificial facesheet of the sandwich structure and suffer repeated impacts with ligaments in the foam—deflecting, fragmenting, and vaporizing in the process. Any residual impactor fragments arriving at the rear facesheet have sufficiently reduced kinetic energy that they can be stopped without danger of perforation.

Predicting the properties of such a material in conditions involving hypervelocity impacts is difficult on two primary fronts. First, while metallic foams enable the “multiple-impact” energy absorption mechanism, the stochastic ligamented structure results in non-singular materials properties at smaller length scales. Since the component ligaments have nominally random orientations, the metallic foams behave inhomogeneously at millimeter and centimeter length scales, exhibiting homogeneous materials properties only at large length scales. This local or mesoscale inhomogeneity leads to materials properties that present as distributions rather than singular values at length scales relevant for MMOD damage. Second, spacecraft designers must consider the properties of previously damaged shielding components and/or the effects of repeated strikes. In this context, knowledge of the properties of pristine material is not sufficient for accurate prediction of material’s behavior over long-duration missions. Models for the overall behavior of shielding materials must account for the effects of previous damage, represented here as cylindrical cavities similar to what is observed experimentally (e.g., Figure 1).

Figure 1.

Examples of MMOD impacts: (a) Surface of an ISS radiator panel [14] and (b) MMOD impact on satellite component retrieved during the STS-41C Solar Max repair mission [15].

Previous related results [11] described the variation in mechanical properties exhibited by pristine foam materials over a range of length scales—from the feature-scale, where individual structural features like ligaments, pores, and nodes dictate properties, to the mesoscale, where ensembles of features dictate mechanical behavior. This previous work also indicated that macroscale behavior, where mechanical properties are single valued, no longer exhibiting variation due to the underlying stochastic structure, arises only for volumes orders of magnitude larger than the typical diameters of MMOD damage tracks. Here, we extend previous work on damage-free foam structures by computing the effects of cylindrical cavities representing MMOD damage tracks on mechanical properties. Using the Kentucky Random Structures Toolkit (KRaSTk) [12, 13] we quantify the reduction in mechanical stiffness due to prior damage and assess the variation in damage effects as both the relative size of the damage path and the relative orientation of the damage path vary. We find that the presence of a cylindrical damage tracks alters otherwise isotropic metallic foams, yielding regions with transverse isotropic behavior, and that the effects of damage tracks cannot simply be represented as changes in foam reduced densities. Overall, we find that the effects of MMOD damage can be larger than the intrinsic variability in foam properties due to their stochastic structure, demonstrating that presence of damage in foam-based shielding cannot be ignored in modeling system-level p roperties and performance.

2. Methods

2.1. RVE generation

KRaSTk was used to procedurally generate large numbers of stochastic model representative volume elements (mRVEs) based on a geometric seed description characterizing the structure of a material. Individual mRVEs are volumes that stochastically sample arrangements of structural features possible based on the seed geometry. Sets of mRVEs constructed with common parameters (e.g., sizes or lengths of characteristic features) can then be used to quantitatively predict properties of real materials whose structure is described by the seed geometry. Previous investigation [11] has shown that a node-ligament geometry accurately represents the mechanical behavior of Duocel, a commercially available Al foam utilized in MMOD shielding.

For this study, mRVEs were produced with a node-ligament seed geometry. Input parameters for mRVE generation with this seed geometry include the range of ligament diameters, the volume density of nodes at which ligaments meet (functionally equivalent to an average, or characteristic, ligament length, ℓ), and the ligament coordination number at each node ( $N_C$ ). Length parameters are all relative to the size of the computational volume (measured in terms of an mRVE side length, s). Resulting mRVEs can also be characterized by computing their reduced density (ϕ), distribution of ligament aspect ratios, and an effective size, $\lambda \equiv s∕\ell$ , which expresses the mRVE size relative to typical structural features—here, the average ligament length. To represent properties of the foam material itself (as opposed to representing properties of individual ligaments and/or pores), mRVEs must have some minimum size relative to structural features—that is, they must be constructed with λ greater than some ${\lambda }_{\min }$ . The mRVEs constructed for this study all have $\lambda \approx 10-11$ , a relative size that has been previously shown to be large enough to effectively capture properties of metallic foams at the mesoscale [11]. The structural parameters used to generate each mRVE set are summarized in Table 2 in Appendix 1.

To accurately model distributions of properties on the mesoscale, sets of mRVEs must contain enough individual mRVEs to meaningfully sample the full range of possible microstructural configurations. Based on previous convergence testing of the node-ligament seed geometry [12], sets of 60 mRVEs were used here to determine distributions of properties for given input structural parameters. To probe the effects of MMOD damage tracks on distributions of mesoscale properties, three sets of undamaged mRVEs were first constructed expressing three different target reduced densities¹. Three initial sets of mRVEs were generated with “low”, “medium”, and “high” reduced densities of approximately ${\varphi }_0=$ 5.9%, 10.7%, and 15.1%, respectively. Reduced densities of individual mRVEs in each set vary by a few percent of the target value (e.g., low reduce densities mRVEs range from 5.8% to 6.0%)—exactly as local density in a metallic foam materials varies due to the stochastic nature of the ligamented structure.

At each target reduced density, three additional sets of “damaged” mRVEs were generated by adding cylindrical cavities with ${\varphi }_{\mathrm{cav}}$ equal to 5%, 10%, and 20%. These cylindrical cavities represent residual MMOD damage tracks with different diameters relative to the feature sizes (ligament and pore sizes) in the mRVE and penetrate through the entire mRVE. Effective reduced density ( ${\varphi }^{\ast }$ ) is defined as the final reduced density after the fibers within the cavity, denoted ${\varphi }_{\mathrm{fc}}$ , has been subtracted from the original damage-free structure, such that ${\varphi }^{\ast }={\varphi }_0-{\varphi }_{\mathrm{fc}}$ and ${\varphi }^{\ast }={\varphi }_0$ in the absence of a cavity. Figure 2 provides a visual representation of ${\varphi }_0,{\varphi }_{\mathrm{cav}},{\varphi }_{\mathrm{fc}},$ and ${\varphi }^{\ast }$ . Collectively, 12 sets (3 undamaged and 9 damaged), each comprising 60 mRVEs, were generated, resulting in 720 total mRVEs. Throughout this study, mRVE sets are referenced by their target reduced volume fraction (low, medium, or high, as defined above) and ${\varphi }_{\mathrm{cav}}$ , with undamaged mRVE also referred to as “damage-free.” For example, the low reduced density mRVE set with ${\varphi }_{\mathrm{cav}}=$ 20% is designated “L-20.” A singular, representative mRVE for each set is shown in Figure 3. In discussing damaged structures, the direction along the axis of the damage track is the through-the-thickness (TTT) direction, while directions in the plane perpendicular to the damage track cylinder are referred to as in-plane (IP) directions, as shown in Figure 8 in Appendix 2.

Figure 2.

A representation of the variables referenced in this study. The quantity $V_0$ is the aggregate volume of all fibers in the mRVE volume. The quantity $V_{\mathrm{fc}}$ is the volume of all fibers within the cavity, itself designated $V_{\mathrm{cav}}$ and calculated as $\mathrm{\pi s}{(d∕2)}^2$ . The quantity $V^{\ast }$ is the volume of the remaining fibers once those within the cavity are removed, such that $V^{\ast }=V_0-V_{\mathrm{fc}}$ . Throughout this paper, the reduced densities ${\varphi }_0,{\varphi }_{\mathrm{cav}},{\varphi }_{\mathrm{fc}},$ and ${\varphi }^{\ast }$ refer to $V_0,V_{\mathrm{cav}},V_{\mathrm{fc}},$ and $V^{\ast }$ normalized by the mRVE volume, $s^3$ , then converted to a percentage.

Figure 3.

A map of each of the 12 structure sets utilized in this study. It is important to note that the structure shown for each set of conditions is a single example; in reality, each set contains 60 mRVEs.

2.2. Computing mRVE Properties via FEM

The orthorhombic stiffness tensor for each mRVE was computed using an FEM-based approach, extending a previously developed method [12]. To extract the full stiffness tensor, total elastic energies were computed for each mRVE in nine independent strain states established using combinations of displacement and frictionless support boundary conditions. The nine strain states included the three independent uniaxial compressions ( ${ϵ}_i=\overline{ϵ}\ne 0,{ϵ}_j={ϵ}_k=0$ ), three biaxial compressions ( ${ϵ}_i={ϵ}_j=\overline{ϵ}\ne 0,{ϵ}_k=0$ ), and three pure shear strains. The subscripts $i,j$ and k refer to the three principle directions orthogonal to the faces of the cuboid mRVE. All applied strains had magnitudes of $\overline{ϵ}=0.5$ %. Computed total strain energies for the nine strain states yield a system of equations of the form:

$$\begin{array}{cccc}\multicolumn{1}{c}{U_{\mathrm{ij}}}& \multicolumn{1}{c}{=\frac{1}{2}{C}_{\mathrm{ij}}{ϵ}_i{ϵ}_j,}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}$$

(1)

allowing direct calculation of $C_{11}$ , $C_{22}$ , $C_{33}$ , $C_{12}$ , $C_{13}$ , $C_{23}$ , $C_{44}$ , $C_{55}$ , and $C_{66}$ in terms of the nine computed $U_{\mathrm{ij}}$ values for each mRVE. Stiffness constants were then converted to elastic moduli ( ${\mathrm{E}}_1$ , ${\mathrm{E}}_2$ , ${\mathrm{E}}_3$ ) Poisson’s ratios ( ${\nu }_{12},{\nu }_{13},{\nu }_{32}$ ), and shear moduli ( $G_{23},G_{13},G_{12}$ ), as described in Appendix 3 and Ref. [16].

Finite element calculations were conducted using the FEniCSx general PDE solver package [17, 18]. A Krylov solver incorporating the generalized minimal residual method and a successive over-relaxation pre-conditioner was implemented. The absolute tolerance was set to $1\times 10^{-7}$ and the relative tolerance to $1\times 10^{-5}$ . Displacement boundary conditions were used to apply strains appropriate to each of the nine considered strain states. The material within each ligament was chosen to be isotropic, with bulk intrinsic mechanical properties of E = 69 GPa and ν = 0.33, similar to reported values for polycrystalline Al. Geometries were meshed utilizing mshr, the mesh generation component of FEniCSx. Mshr builds simplicial DOLFIN meshes in 3D, utilizing CGAL and Tetgen as mesh generation backends. For each structure, a tetrahedral mesh with a resolution of 150.0 was generated. The resolution quantity is defined as the inverse of the cell size h, the longest edge in any mesh element. Reduced densities were computed as the total mesh volume.

2.3. Determining materials properties from mRVE properties

As stated above, each of the 12 sets of mRVEs contains unique 60 mRVEs ( $N=60$ ). FEM calculations yield distinct stiffness tensors for each mRVE, resulting in a data set that includes 720 $E_1$ values, 720 $E_2$ values, and 720 $E_3$ values. The properties of an individual mRVE are not particularly useful; however, when viewed in aggregate, the mesoscale properties of the represented material emerge. The properties of the material represented by the chosen seed geometry are characterized by the distribution of computed properties for a sufficiently large set of mRVEs, found in a previous investigation to be a minimum of $\mathrm{\sim }$ 30 mRVEs [12]. As a value of $N=60$ mRVEs were used here, each distribution presented in the following sections is thus derived from a minimum of 60 values. Results are represented as Γ-distributions, selected because they are positive definite (as elastic modulus must be) and converge to normal distributions for expectation values far from zero. Expectation values for Γ-distributions of E values are ${\mu }_E=\mathrm{ab}$ , and standard deviations of E values within mRVE sets are ${\sigma }_E={(a{b}^2)}^{1∕2}$ . The quantities a and b are the shape and scale parameters, respectively, that describe the Γ-distribution.

The standard deviation ( ${\sigma }_E$ ) here represents more than a statistical parameter. Rather, it is a measure of how the stochastic nature of the material’s structure results in a range of values for a given property. The quantity $\pm \sigma$ is designated the standard variability. It is crucial to emphasize that the standard variability is not a computational error, but a property of the material at some size scale designated by λ. At this size scale ( $\lambda \approx 10-11$ ), each set will have non-zero σ values. For mRVEs at extremely large λ values, the standard variability will fall to zero, representing the transition to the macroscale—the size scale at which the material exhibits singular bulk properties. While $[\pm {\sigma }_E]$ will fall to zero in the macroscopic limit, the decay with increasing λ is expected to be extremely slow for the range of λ values considered in this study–all of which are expected to fall within, or just below, the mesoscale.

While individual distributions describe the properties of individual sets of mRVEs, comparing multiple distributions allows for an examination of the relative effects of the cavities and structural stochasticity. Here, we assess the similarity of pairs of distributions by quantifying the degree to which they overlap. The overlap coefficient, a similarity measure that determines the overlap between two finite sets $(X,Y)$ was defined as the size of the intersection divided by the smaller of the size of the two sets:

$$\begin{array}{cccc}\multicolumn{1}{c}{\mathrm{overlap}(X,Y)=\frac{|X\cap Y|}{\min (|X|,|Y|)}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}$$

(2)

Because the total area of each distribution (i.e. the probability distribution function) generated by KRaSTk is always equal to 1, Eq. 2 can be rewritten in terms of distribution area A:

$$\begin{array}{cccc}\multicolumn{1}{c}{A_{\mathrm{overlap}(X,Y)}=A_{|X\cap Y|}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}$$

(3)

Therefore, the ratio of overlap to total unique area beneath the two distributions, here referred to as the similarity index (η), is:

$$\begin{array}{cccc}\multicolumn{1}{c}{\eta =\frac{A_{\mathrm{overlap}(X,Y)}}{(A_X+A_Y)-A_{\mathrm{overlap}(X,Y)}}=\frac{A_{\mathrm{overlap}(X,Y)}}{2-A_{\mathrm{overlap}(X,Y)}}}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}& \multicolumn{1}{c}{}\end{array}$$

(4)

3. Results and discussion

3.1. Deviation from elastic isotropy

Figure 4 (and Appendix 4) summarizes both expected bulk stiffnesses and distributions of local stiffnesses computed for each set of undamaged and damaged foam mRVEs. Expected bulk values of stiffness for each mRVE set are indicated with points, and the standard variabilities in stiffness are shown as whiskers. Blue, green, and red colors indicate results for low-, medium-, and high-volume fraction mRVEs respectively, while thick whiskers and darker points are for mechanical responses in “in-plane” directions and lighter whiskers and points for the “through-the-thickness” (TTT) direction. In Figure 4, the five panels show results for (from top to bottom) ${\varphi }_{\mathrm{cav}}=0\%$ (damage-free), 5%, 10%, 20%, and 30% mRVEs. As expected, for all reduced densities, cylindrical voids representing damage tracks reduce foam stiffness, with larger reductions resulting from larger voids (seen by comparing same color data from top to bottom panels in Figure 4).

Figure 4.

The expected bulk stiffnesses (represented as points) and the standard deviations (represented as whiskers) for both the TTT and IP directions for each mRVE set.

With respect to defect free, ${\varphi }_{\mathrm{cav}}=0\%$ structures (top panel), the expected stiffness and standard variability in the IP and TTT directions are the same for each set, L-0, M-0, and H-0, confirming that the damage-free material is isotropic. As expected, stiffnesses vary significantly with ${\varphi }^{\ast }$ ; in fact, the ranges of expected stiffness for the L-0, M-0, and H-0 sets, as characterized by the standard variability, do not overlap. Therefore, individual local measurements of stiffness in damage-free Duocel would be sufficient to distinguish between foams with reduced densities of 5.9%, 10.7%, and 15.1%, even accounting for local variation in observed properties.

In contrast to damage-free sets, medium and high reduced density sets damaged exhibit disparities in stiffness for both the TTT and IP directions. In these foams the mechanical response in the IP direction (orthogonal to the damage track) is more strongly affected (specifically, stiffnesses are more greatly reduced) by damage than in the TTT direction (along the damage track). However, the ranges of expected local stiffnesses (as characterized by the standard variability) in the TTT and IP directions in damaged structure sets overlap significantly. This implies that at individual points along the damage track, local structural variability will result in (local) regions where loading in the IP and TTT directions still yields similar responses (or even responses where the IP stiffness is greater than the TTT stiffness). In a similar vein, damaged structure sets of different reduced densities no longer exhibit fully distinct stiffness ranges–they overlap significantly. That is, in contrast to damage-free foams, individual measurements of the stiffness of damaged foams are not sufficient to distinguish between foams of different densities—or, equivalently, the effect of MMOD damage on mechanical properties can be larger than the effect of changes in overall foam reduced density, depending on local structural configuration details. As ${\varphi }_{\mathrm{cav}}$ increases, both of these effects—disparity between IP and TTT mechanical response and blurring of the effect of foam volume fraction—increase.

Low reduced density sets appear to behave differently, remaining isotropic and exhibiting minor overall effects due to the cylindrical damage tracks. This may be because the already relatively open ligament networks in low reduced density foams are comparatively less impacted by the presence of a damage track. Some evidence supporting this hypothesis can be gleaned from considering the change in overall reduced density due to the removal of material within the cavity. Where the addition of ${\varphi }_{\mathrm{cav}}=$ 20% cavities to the defect free ${\varphi }_{\mathrm{cav}}=$ 0% structures in low reduced density sets reduces ${\varphi }^{\ast }$ from 5.9% to 4.6% (Δ = 1.3%), the same addition to high reduced density sets reduces ${\varphi }^{\ast }$ from 15.1% to 12.0% (Δ = 3.1%). Thus, the overall change in reduced density is more than twice for the high reduced density sets than the low. Another possibility originates from the number of contiguous paths of connected ligaments broken by the addition of a cavity in a low reduced density foam structure, which contains widely spaced contiguous paths, compared to a high reduced density structure with much more closely spaced contiguous paths.

Figure 5 replots data to better highlight how damage tracks can have as large an impact on expected bulk (and local) mechanical properties as otherwise significant changes in reduced density. The H-30 set has a ${\varphi }^{\ast }$ similar to that of the M-0 and M-5 sets, all within the purple box in Figure 5(a). However, both stiffness expectation values and standard variabilities differ. Even though the ${\varphi }^{\ast }$ of H-30 is less than M-0, its expected TTT stiffness is higher than that of M-0. In contrast, the expected IP stiffness of H-30 was lower than M-5, despite the latter having the lower ${\varphi }^{\ast }$ . This series of results signifies that stiffnesses of potentially damaged ligamented-based materials cannot be modeled based on reduced density alone. Critically, for modeling/predicting the performance of foams where damage may be present, reduced density cannot be used as a proxy for uniform mechanical properties. Because regions of similar density may or may not behave similarly (or even isotropically) depending on whether damage is present, a more nuanced model accounting for local variations in properties is required.

Figure 5.

Elastic modulus distributions, in both IP and TTT orientations, for all mRVE sets. In (a), the elastic modulus values are plotted as points and whiskers. In (b), only the expectation values are included.

These results also suggest more fundamental differences in underlying effects of MMOD damage on IP versus TTT mechanical responses. This is highlighted in Figure 5(b), which plots only the expectation values of stiffness. As the diameter of the damage track increases, stiffness in the TTT direction falls in proportion to the fraction of the mRVE cross-sectional area occupied by the damage track. In contrast, in the IP direction, increasingly large diameter damage tracks lead to a more rapid decrease in stiffness than in the TTT direction. In the IP direction, we observe that E reaches 0 while ${\varphi }^{\ast }$ remains greater than 0 because of the geometry of the cavity. At ${\varphi }_{\mathrm{cav}}=78.5\%$ (precisely – $\pi ∕4\ast 100\%$ ), the cylindrical cavity will be tangent to the sides of the mRVE. Thus, it will no long represent a single solid, but rather a set of four corner pillars (see Figure 9 in Appendix 5). In this limiting case, there will be no mechanical stiffness IP (rigid body motion will occur), while finite TTT stiffness can still be expected proportional to the area fraction of remaining solid. These trends can be explained by considering the importance of “load-bearing paths”–sequences of contiguous ligaments that can be traced from one load-bearing face to another–in determining mechanical response. For TTT loads, these paths are generally oriented along the damage track, and therefore are only broken if they fall within the cylindrical damage track. In this case, stiffness would generally decrease as the cross-sectional area of the damage track increased, and therefore be proportional to the residual reduced density. In contrast, IP damage will disrupt any load-bearing path passing through the two-dimensional projection of the track along its length, the cross section of the cavity perpendicular to the IP plane. Therefore, TTT stiffness will decrease proportional to the square root of the radius of the cavity, while IP stiffness will decrease proportional to the radius. Regardless, damage tracks in these structures lead to anisotropy in mechanical response, and to non-linear changes in mechanical properties that depend on more than simply the reduced density.

3.2. Implications for statistically significant effects of damage

Additional implications arise by considering the overlap of distributions for damaged and undamaged sets of structures. These overlaps essentially measure the probability that the effects of any particular damage track in any particular local volume are so limited that they cannot be distinguished from the background, intrinsic variations in local foam properties. Figure 6 presents the ${\varphi }_{\mathrm{cav}}=$ 0% and 20% Γ-distributions of both orientations (TTT and IP) for L, M, and H density samples, with the overlapping area shaded. Table 1 quantifies the overlaps and similarity indices and Figure 7 provides a visual example comparing the H-0 and H-20 sets. For L, M, and H density samples, $A_{\mathrm{overlap}}$ values steadily decrease as reduced density increases for IP properties. This is evidence that as ${\varphi }_{\mathrm{cav}}$ grows relative to the sizes of features in all density foams, the effects of damage tracks on IP properties become more and more distinct from intrinsic variations in local properties. In contrast, for the TTT mechanical response, low and medium reduced density sets exhibit comparable $A_{\mathrm{overlap}}$ values, with a moderate reduction for H density samples. This suggests that the effects of damage on TTT properties fall significantly within the range of intrinsic local variations.

Figure 6.

Plots showing the overlap in distributions of ${\varphi }_{\mathrm{cav}}=$ 0% and 20% elastic modulus values for all three target reduced densities for both the IP and TTT orientations. (a) L-IP; (b) Low-TTT; (c) M-IP; (d) M-TTT; (e) H-IP; (f) H-TTT.

Table 1.

Similarity measures of the ${\varphi }_{\mathrm{cav}}=$ 0% and 20% Γ -distributions of each reduced density class and both IP and TTT orientations.

Distributions $(X,Y)$	Orientation	$A_{\mathrm{overlap}(X,Y)}$	η
(L-0, L-20)	IP	0.760	0.613
(M-0, M-20)	IP	0.480	0.316
(H-0, H-20)	IP	0.293	0.172
(L-0, L-20)	TTT	0.758	0.610
(M-0, M-20)	TTT	0.780	0.639
(H-0, H-20)	TTT	0.613	0.442

Figure 7.

A representation of the physical implications of the overlapping Γ-distributions in Figure 6(e) and (f).

The computed distributions of expected stiffness values for both damaged and undamaged structure sets suggest that the present stochastic approach to predicting properties of complex materials and the effects of defects may have impactful applications in the context of uncertainty quantification. Comparing distributions predicts both the magnitude and frequency of changes in mechanical properties due to significant structural changes (here, the addition of a cylindrical void mimicking an MMOD impact damage track). Of particular interest is the ability to distinguish between background variations in local materials properties–present in as-fabricated samples of all stochastic materials–and “above the noise” effects of damage events.

4. Summary and outlook

We have analyzed the effect of cylindrical cavities on statistically significant sets of mRVEs built to represent Duocel foam, a commonly employed component of MMOD shielding, using techniques that are easily transferable to other complex or randomly structured materials. Cylindrical cavity defects were selected to mimic the structure of MMOD damage tracks and were varied to quantify the effects of relative volume and load orientation. Utilizing FEM, elastic modulus values in both the IP and TTT orientations were computed for all mRVEs representing both damaged and undamaged structures. While it was shown that in all cases increasing the relative size of the cavity defect decreases the stiffness of the structure, this effect is not necessarily proportional to the relative volume of the removed material and is not symmetric with respect to the orientation of the loading relative to the damage track. Critically, larger reductions in stiffness are seen for IP loading (that is, perpendicular to the damage track). This implies that system- or component-level models intended to account for potential MMOD damage must account for not only changes in stiffness in damaged materials but also increases in structural anisotropy in damaged regions. More fundamentally, the trends for reductions of stiffness in both IP and TTT loading conditions suggest that damage to the ensemble of load-bearing paths present in a material influence the magnitude of changes in stiffness. For the relative sizes of damage tracks considered here, the computed changes in mechanical stiffness due to damage track formation were frequently similar to the magnitude of intrinsic local variations in stiffness arising purely from the stochastic nature of the material structure itself. Despite this, the results provide strong evidence that there is always a meaningful probability that a particular damage track will alter mechanical properties well beyond expected intrinsic variations. Overall, the present findings suggest that the approach applied here could be used to quantify uncertainty in system or component performance, particularly for cases where defects or accumulations of defects are possible. The approach utilized here can and should also be extended to address thermal properties alongside mechanical properties, and can be applied to any materials system well described by identifiable sets of geometric structural primitives (including both porous and, e.g. polycrystalline, fully dense solids).