1 Introduction

Composites reinforced with discontinuous fibers play a significant role in various industries, such as in the mass production of automotive parts (Ishikawa et al. 2018). Predicting the mechanical behavior of composite structures made of such materials requires consideration of the orientation distribution of fibers (ODF) since this factor significantly influences the effective material properties (Mortazavian and Fatemi 2015; Ogierman and Kokot 2016). Due to the possibility of manufacturing parts of complex shapes, for instance, by an injection molding process, the orientation distribution typically varies throughout the part’s volume and may change in a range from unidirectional to random. The fibers are distributed by the flow of the material during the injection molding process. The orientation distribution of fibers is then influenced by multiple factors including flow pattern depending on the geometrical features of the manufactured part, parameters of the manufacturing process and rheological properties of the material (Laspalas et al. 2008). Therefore, the resulting orientation distribution of reinforcing fibers is hard to predict precisely, and some uncertainties arise. One way to describe the orientation distribution is to use advanced software for the simulation of the process of injection molding such as Moldex3D (Tseng et al. 2017), Moldflow (Żurawik et al. 2022) or Cadmould (Albrecht et al. 2017). The result related to the orientation state obtained throughout such software is typically a distribution of the second-order orientation tensor. The tensorial description of orientation state (Advani and Tucker 1987) has concise form convenient to deal with practical engineering problems and therefore it is widely used in engineering science. Many literature studies show good agreement between the orientation distribution obtained by using simulation software and experimental observations, however, since the orientation distribution of fibers depends on plenty of parameters it is not so straightforward to obtain accurate predictions. Some limitations of the fiber orientation models have been discussed in the work of Kugler et al. (2020). Moreover, the outcome of the simulation is typically the second-order tensor only, while for the stiffness predictions the fourth-order tensor is required either. For the determination of the fourth-order orientation tensor the closure approximations (Dray et al. 2007) are used, which adds another uncertainty from the standpoint of prediction of mechanical properties of such materials. The ambiguity of the fourth-order tensor approximation based on the second-order orientation tensor has been discussed in the work of Ogierman (2022). Müller and Böhlke (2016) concluded that describing the microstructure of a composite by using only the second-order orientation tensor is generally insufficient. Another way of reflecting the orientation state of fibers throughout the manufactured part is the application of experimental methods. The experimental investigation can be carried out by using a destructive approach based on microscopic analysis of the material specimens (Lee et al. 2003; Sharma et al. 2018; Zhao et al. 2023). However, in recent years, the most popular experimental approach is the usage of X-ray micro-computed tomography (µCT) (Bernasconi et al. 2012; Nguyen Thi et al. 2015; Shen et al. 2004; Sietins et al. 2021). It enables non-destructive investigations and extends its applicability to a broader region of interest compared to microscopic analysis enabling even analysis of a whole volume of parts (Zwanenburg et al. 2023). Nonetheless, the reconstruction of the ODF based on the µCT scans may also be affected by some uncertainties related to the applied resolution and accuracy of the reconstruction of the geometry of fibers. In general, two groups of methods of description of orientation distribution can be pointed out. The first one is based on a reconstruction of each individual fiber. It typically demands high-resolution scans and substantial computational resources, also the segmenting of closely spaced fibers may be challenging. The second one, the voxel-wise orientation approach, allows to estimate the orientation distribution through global measurements when the orientation of each individual fiber is not known (Karamov et al. 2020; Pinter et al. 2018). This group of methods has the advantage of not requiring the separation of fibers, making it less prone to both under- and over-segmentation. (Hessman et al. 2019). The limitation of this method is a lack of a complete description of orientation distribution, for instance only the second-order orientation tensor can be reconstructed. Hessman et al. (2019) presented a robust and accurate algorithm that relies on an iterative single fiber segmentation and merging procedure to obtain the fiber characteristics. In this study, the authors investigated the influence of scan resolutions and achieved fiber detection rates ranging between 85% and 88% for the composite PA66-GF35. Additionally, they demonstrated a dependency of the components of the second-order orientation tensor on the resolutions.

Since uncertainties related to the reconstruction of the ODF are often unavoidable, this paper investigates the impact of some of the uncertainties on the effective elastic constants of composite through Monte Carlo (MC) simulations. The primary objective of this study is to address how the ratio of fiber detection and the measurement error of individual fiber orientation influence the orientation tensors and effective elastic constants. It should be emphasized that in the parts manufactured in the injection molding process, the elastic constants of the composite locally vary throughout the part’s volume (ODF depends on the spatial coordinates). Following the two-scale analysis paradigm, this study focuses on a single material point from the perspective of the macroscale level. From the microscale viewpoint, this point is associated with a system of fibers (that express local orientation distribution) embedded in a matrix. Three cases of different fiber orientation distributions have been analyzed.

2 Materials and methods

2.1 Micromechanical analysis

One of the most popular approaches to the prediction of the elastic constants of composite with misaligned fibers is the application of the orientation averaging procedure. In such a case, the elastic stiffness tensor of a composite with misaligned fibers may be determined in terms of the elastic constants of a unidirectional composite weighted by the fiber orientation distribution. In this scenario, the elastic constants of the unidirectional composite can be determined by using well-known micromechanical models such as Mori-Tanaka (Benveniste 1987; Pierard et al. 2004) or numerical homogenization (Böhm and Rasool 2016; Burczyński et al. 2010; Kouznetsova et al. 2001; Segurado and Llorca 2002). Advani and Tucker (1987) proposed to use the orientation tensors aij and aijkl to describe the orientation distribution of fibers and provided closed-form solution for the stiffness tensor of composite with misaligned fibers:

$$\begin{aligned}{{\mathbf{C}}_{ijkl}}=&{B_1}{{\mathbf{a}}_{ijkl}}+{B_2}({{\mathbf{a}}_{ij}}{\delta _{kl}}+{{\mathbf{a}}_{kl}}{\delta _{ij}})\\&+{B_3}({{\mathbf{a}}_{ik}}{\delta _{jl}}+{{\mathbf{a}}_{il}}{\delta _{jk}}+{{\mathbf{a}}_{jl}}{\delta _{ik}}+{{\mathbf{a}}_{jk}}{\delta _{il}})\\&+{B_4}({\delta _{ij}}{\delta _{kl}})+{B_5}({\delta _{ik}}{\delta _{jl}}+{\delta _{il}}{\delta _{jk}}),\end{aligned}$$
(1)

where constants B are related to the components of the stiffness tensor of the unidirectional composite (transversally isotropic):

$$\begin{gathered} {B_1}=C_{{1111}}^{U}+C_{{2222}}^{U} - 2C_{{1122}}^{U} - 4C_{{1212}}^{U} \hfill \\ {B_2}=C_{{1122}}^{U} - C_{{2233}}^{U} \hfill \\ {B_3}=C_{{1212}}^{U}+\frac{1}{2}(C_{{2233}}^{U} - C_{{2222}}^{U}) \hfill \\ {B_4}=C_{{2233}}^{U} \hfill \\ {B_5}=\frac{1}{2}(C_{{2222}}^{U} - C_{{2233}}^{U}) \hfill \\ \end{gathered}$$
(2)

The second and fourth-order orientation tensors ai and aijkl are constituted by forming the dyadic products of orientation vector p and then integrating with the distribution function ψ(p) over all possible directions (Advani and Tucker 1987):

$${{\mathbf{a}}_{ij}}=\oint {{{\mathbf{p}}_{i}}{{\mathbf{p}}_{j}}{{{\Psi}}}({\mathbf{p}})} {\text{d}}{\mathbf{p}},$$
(3)
$${{\mathbf{a}}_{ijkl}}=\oint {{{\mathbf{p}}_{i}}{{\mathbf{p}}_{j}}{{\mathbf{p}}_{k}}{{\mathbf{p}}_{l}}{{{\Psi}}}({\mathbf{p}})} {\text{d}}{\mathbf{p}},$$
(4)

These tensors can be straightforwardly computed on the basis of a finite number of fibers; when the n fibers have got the same aspect ratio and volume the formulas (3) and (4) can be expressed in the following form:

$${{\mathbf{a}}_{ij}}\; = \;{1 \over n}\sum\limits_{Z = 1}^n {{\mathbf{p}}_{Z,i}{\mathbf{p}}_{Z,j}\;{\rm{for}}\;i,\;j\; = \;1,\;2,\;3,}$$
(5)
$${{\mathbf{a}}_{ijkl}}\; = \;{1 \over n}\sum\limits_{Z = 1}^n {{\mathbf{p}}_{Z,i}{\mathbf{p}}_{Z,j}{\mathbf{p}}_{Z,k}{\mathbf{p}}_{Z,l}\;{\rm{for}}\;i,\;j,\;k,\;l\; = \;1,\;2,\;3,}$$
(6)

where two distinct types of indices can be identified: the iterator within a set of discrete values {1,2,…,n} is denoted by the uppercase letter Z, while the indices of tensor components in three-dimensional space are expressed by the lowercase letters i, j, k, l. For the prediction of the stiffness tensor CU, the FE-based homogenization associated with the analysis of the RVE (representative volume element) model representing statistically the microstructure of the unidirectional composite has been carried out. The geometrical model of the representative volume element and mesh consisting of tetrahedral finite elements with quadratic shape function has been visualized in Fig. 1, 30 fibers with aspect ratio 20 (ratio between length and diameter) were randomly placed with an assumption of interpenetration avoidance. The volume fraction of fibers 0.15 has been considered.

Fig. 1
figure 1

Representative volume element representing analyzed unidirectional composite: a) geometrical model, b) FE mesh

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Isotropic matrix and transversally isotropic fibers have been taken into account according to data collected in Table 1 (Miyagawa et al. 2006). A standard procedure for obtaining the effective stiffness tensor involving six FE analyses addressing six unit macro strains has been carried out (Zheng and Fish 2008). Each of the FE analyses is associated with the application of unit macro strains through the periodic boundary conditions. The effective stiffness tensor obtained through the FE-based numerical homogenization has the following form:

$${{\mathbf{C}}^*}=\left[ {\begin{array}{*{20}{r}} {25.222}&{4.470}&{4.466}&{0.002}&{0.003}&{0.001} \\ {4.470}&{7.037}&{4.440}&{ - 0.001}&{ - 0.042}&{0.000} \\ {4.467}&{4.440}&{7.033}&{0.000}&{ - 0.001}&{0.000} \\ {0.002}&{ - 0.001}&{0.000}&{1.297}&{0.000}&{0.003} \\ {0.003}&{ - 0.042}&{ - 0.001}&{0.000}&{1.441}&{0.000} \\ {0.001}&{0.000}&{0.000}&{0.003}&{0.000}&{1.435} \end{array}} \right],{\text{ GPa}}$$
(7)
Table 1 Elastic properties of the composite constituents: isotropic properties of epoxy resin and transversally isotropic properties of fibers
Full size table

Although this tensor exhibits evident transversally isotropic symmetry, the symmetry is not perfect due to random fiber placement and the approximate character of the method. Therefore, for obtaining the stiffness tensor with transversally isotropic symmetry CU a procedure for finding the best approximation of such symmetry has been used (Sevostianov and Kachanov 2008). The nonzero components of the tensor CU have been computed in terms of stiffness tensor obtained directly through the FE-based numerical homogenization C* in the following way:

$$\begin{gathered} C_{{1111}}^{U}={C}_{{1111}}^{*},{\text{ }}C_{{2222}}^{U}=\frac{{3({C}_{{2222}}^{*}+{C}_{{3333}}^{*})+2{C}_{{2233}}^{*}+4{C}_{{2323}}^{*}}}{8}, \hfill \\ C_{{2233}}^{U}=\frac{{{C}_{{2222}}^{*}+{C}_{{3333}}^{*}+6{C}_{{2233}}^{*} - 4{C}_{{2323}}^{*}}}{8},{\text{ }}C_{{1133}}^{U}=\frac{{{C}_{{1133}}^{*}+{C}_{{1122}}^{*}}}{2}, \hfill \\ C_{{2323}}^{U}=\frac{{C_{{2222}}^{U} - C_{{2323}}^{U}}}{2},{\text{ }}C_{{1212}}^{U}=\frac{{{C}_{{1212}}^{*}+{C}_{{1313}}^{*}}}{2}. \hfill \\ \end{gathered}$$
(8)

Finally, the stiffness tensor used in further analyses associated with the Eq. (1) has the following form:

$${{\mathbf{C}}^U}=\left[ {\begin{array}{*{20}{r}} {25.222}&{4.468}&{4.468}&{0.000}&{0.000}&{0.000} \\ {4.468}&{7.035}&{4.440}&{0.000}&{0.000}&{0.000} \\ {4.468}&{4.440}&{7.035}&{0.000}&{0.000}&{0.000} \\ {0.000}&{0.000}&{0.000}&{1.298}&{0.000}&{0.000} \\ {0.000}&{0.000}&{0.000}&{0.000}&{1.438}&{0.000} \\ {0.000}&{0.000}&{0.000}&{0.000}&{0.000}&{1.438} \end{array}} \right],{\text{ GPa}}.$$
(9)

It has to be emphasized that in this scenario the time-consuming FE computations are performed only once (for unidirectional composite) and subsequent MC simulations use the time-efficient analytical expression (1). Such an approach enables taking advantage of the FE-based homogenization but does not require conducting time-consuming computations for each analyzed ODF. A combination of the orientation averaging procedure with the finite element-based homogenization has already been discussed for example in the works of Modniks and Andersons (2010), Ogierman and Kokot (2017) and Mirkhalaf et al. (2020).

2.2 Monte Carlo simulations

The Monte Carlo simulations were carried out with the aid of the pseudorandom number generator nested in Matlab software. The base for the optimization was the distribution of fibers generated by the software Digimat-FE which is typically used for the generation of representative volume elements (RVE) representing the microstructure of composite materials. The input data for the Digimat software were the second-order orientation tensors. In this paper three different cases of fibers orientation distributions have been studied, each fiber system comprises 5000 fibers. The cases were chosen in a way to analyze the orientation distributions that significantly differ from one another. Consequently, the generated data contains a fully described orientation distribution of fibers and therefore the second and fourth-order orientation tensors can be determined directly (using Eqs. 5 and 6). The mentioned orientation tensors corresponding to the distributions OD1-OD3 have the following form:

$${\mathbf{a}}_{{ij}}^{{OD1}}=\left[ {\begin{array}{*{20}{r}} {0.76599}&{0.16060}&{0.07025} \\ {0.16060}&{0.14601}&{0.05415} \\ {0.07025}&{0.05415}&{0.08800} \end{array}} \right],{\text{ }}$$
(10)
$${\mathbf{a}}_{{ijkl}}^{{OD1}}=\left[ {\begin{array}{*{20}{r}} {0.65197}&{0.07101}&{0.04300}&{0.02263}&{0.05174}&{0.12064} \\ {0.07101}&{0.05503}&{0.01997}&{0.01986}&{0.00879}&{0.03042} \\ {0.04300}&{0.01997}&{0.02503}&{0.01166}&{0.00972}&{0.00954} \\ {0.02263}&{0.01986}&{0.01166}&{0.01997}&{0.00954}&{0.00879} \\ {0.05174}&{0.00879}&{0.00972}&{0.00954}&{0.04300}&{0.02263} \\ {0.12064}&{0.03042}&{0.00954}&{0.00879}&{0.02263}&{0.07101} \end{array}} \right].$$
(11)
$${{\mathbf{a}}_{ij}}^{{OD2}}=\left[ {\begin{array}{*{20}{r}} {0.33350}&{ - 0.01166}&{ - 0.00244} \\ { - 0.01166}&{0.33820}&{0.00826} \\ { - 0.00244}&{0.00826}&{0.32830} \end{array}} \right],{\text{ }}$$
(12)
$${{\mathbf{a}}_{ijkl}}^{{OD2}}=\left[ {\begin{array}{*{20}{r}} {0.20589}&{0.06438}&{0.06323}&{ - 0.00007}&{ - 0.00079}&{ - 0.00532} \\ {0.06438}&{0.20873}&{0.06509}&{0.00561}&{ - 0.00087}&{ - 0.00653} \\ {0.06323}&{0.06509}&{0.19998}&{0.00272}&{ - 0.00078}&{0.00019} \\ { - 0.00007}&{0.00561}&{0.00272}&{0.06509}&{0.00019}&{ - 0.00087} \\ { - 0.00079}&{ - 0.00087}&{ - 0.00078}&{0.00019}&{0.06323}&{ - 0.00007} \\ { - 0.00532}&{ - 0.00653}&{0.00019}&{ - 0.00087}&{ - 0.00007}&{0.06438} \end{array}} \right].$$
(13)
$${{\mathbf{a}}_{ij}}^{{OD3}}=\left[ {\begin{array}{*{20}{r}} {0.88434}&{0.12669}&{0.06800} \\ {0.12669}&{0.06119}&{0.03668} \\ {0.06800}&{0.03668}&{0.05447} \end{array}} \right],{\text{ }}$$
(14)
$${{\mathbf{a}}_{ijkl}}^{{OD3}}=\left[ {\begin{array}{*{20}{r}} {0.81384}&{0.03918}&{0.03132}&{0.02105}&{0.05681}&{0.11041} \\ {0.03918}&{0.01246}&{0.00955}&{0.00747}&{0.00602}&{0.01126} \\ {0.03132}&{0.00955}&{0.01360}&{0.00816}&{0.00517}&{0.00501} \\ {0.02105}&{0.00747}&{0.00816}&{0.00955}&{0.00501}&{0.00602} \\ {0.05681}&{0.00602}&{0.00517}&{0.00501}&{0.03132}&{0.02105} \\ {0.11041}&{0.01126}&{0.00501}&{0.00602}&{0.02105}&{0.03918} \end{array}} \right].$$
(15)

It should be noted that the fourth-order tensors (Eqs. 11, 13 and 15) are presented by using contracted (Voigt) notation. Fig. 2 presents the geometrical representations of the analyzed fiber orientation distributions.

Fig. 2
figure 2

Geometrical representation of the analyzed fiber orientation distributions: a) OD1, b) OD2, c) OD3

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The MC approach has been used to simulate the effects of:

  1. a)

    various fiber detection rates (DR),

  2. b)

    error of measurement of individual fiber’s orientations (OE).

The fiber detection rate DR was simulated by random generation of z integer numbers in the range from 1 to n where n is the number of fibers in the system:

$$DR=1 - \frac{z}{n}\, ,$$
(16)

afterwards, fibers with numbers corresponding to the generated z numbers were deleted. The fiber system without the selected z fibers was then used as the basis for computing orientation tensors and the effective stiffness tensor. The effect of the fiber detection rate on the orientation tensors and effective elastic constants has been tested by applying three different values of DR = 0.7, 0.8, 0.9.

The error of measurement of fiber’s orientations OE was included by modifying the components of the orientation unit vector p of each individual fiber in the following way:

$${\mathbf{\tilde {p}}}=\left[ {\begin{array}{*{20}{c}} {{{\text{p}}_1}+{R_1}} \\ {{{\text{p}}_2}+{R_2}} \\ {{{\text{p}}_3}+{R_3}} \end{array}} \right]\, ,$$
(17)

where R is a vector whose components are selected randomly with respect to the probability defined by normal distribution. For each individual fiber, a different R vector is computed. While focusing on the random errors, a mean of zero and three different standard deviation values are assumed during the investigation σ = 0.1, 0.05, 0.02. The orientation of each fiber is modified, and afterwards the orientation vector is normalized to maintain its unit length. Described procedures have been repeated 10,000 times for all analyzed DRs and OEs in order to give an insight into the distributions of components of orientation tensors and elastic constants computed on the basis of effective stiffness tensor (Eq. 1).

Another objective of the paper is to provide a quantitative description of how the level of error of the second-order orientation tensor representation corresponds to the error related to the effective stiffness tensor. The mentioned error can be regarded as for instance the discrepancy between the reconstructed tensor considered for analysis and the tensor representing an actual fiber distribution. The idea is to collect the results of MC simulations conducted for different levels of uncertainty and find the relationship between the following norms:

$$\left\| {a-\bar {a}} \right\|=\sqrt {\sum\limits_{{i=1}}^{3} {\sum\limits_{{j=1}}^{3} {{{\left( {{{\mathbf{a}}_{ij}} - {{{\mathbf{\bar {a}}}}_{ij}}} \right)}^2}} } }\, ,$$
(18)
$$\left\| {C-\bar {C}} \right\|=\sqrt {\sum\limits_{{i=1}}^{6} {\sum\limits_{{j=1}}^{6} {{{\left( {{{\mathbf{C}}_{ij}} - {{{\mathbf{\bar {C}}}}_{ij}}} \right)}^2}} } }\, ,$$
(19)

where: \({{\mathbf{a}}_{ij}}\) is the actual second-order orientation tensor representing the prescribed orientation distribution of fibers, while \({{\mathbf{\bar {a}}}_{ij}}\) is the second-order orientation tensor obtained for the orientation distribution including uncertainties. Similarly, the \({{\mathbf{C}}_{ij}}\) is the actual stiffness tensor obtained for the prescribed orientation distribution of fibers while \({{\mathbf{\bar {C}}}_{ij}}\)incorporates considered uncertainties (Voigt’s notation has been applied in this case). For each considered orientation distribution, 20,000 simulations with randomly selected DR (uniform probability within a range of 0.7-1) and OE (normal probability involving zero mean, and standard deviation selected randomly with uniform probability within a range of 0-0.15) have been carried out.

3 Results and discussion

The results of the MC simulations have been presented in the form of histograms showcasing the resulting distributions of effective elastic constants and components of second-order orientation tensors. Although all components of anisotropic stiffness tensor have been determined, for the sake of clarity and concise presentation of the results only the distributions for three effective Young moduli E1, E2, E3 have been presented (corresponding to the directions of the principal coordinate system). To maintain a concise form of presentation of the second-order orientation tensor either, the distributions of its two eigenvalues have been visualized a(1) and a(2) (due to the normalization condition a11 + a22 + a33=1 the third eigenvalue is not unique). Figures 3 and 4 present results for the orientation distribution OD1 concerning different detection rates (DR). Figures 5 and 6 also illustrate results for OD1, but they pertain to different orientation measurement errors (OE) with respect to various standard deviations (σ). Figures 7 and 8 depict results for OD2 concerning different DR. Figures 9 and 10 similarly display results for OD2, but they address varying OE. Figures 11 and 12 illustrate the outcomes for OD3, reflecting different DR values. Figures 13 and 14, on the other hand, demonstrate the effects of varying OE for the same OD3. Moreover, the results have been summarized in Tables 2, 3 and 4 where for each analyzed case the estimates of normal distribution parameters are presented. Finally, the relationship between the norms expressed by Eqs. (18) and (19) for all analyzed orientation distributions has been visualized in Fig. 15.

Fig. 3
figure 3

The distributions of the eigenvalues of the second-order orientation tensor (OD1) concerning different detection rates (DR): a) the first eigenvalue a(1), b) the second eigenvalue a(2)

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Fig. 4
figure 4

The distributions of Young’s moduli (OD1) concerning different detection rates (DR): a) E1, b) E2, c) E3

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Fig. 5
figure 5

The distributions of the eigenvalues of the second-order orientation tensor (OD1) concerning different orientation measurement errors (OE) expressed by different standard deviations σ: a) the first eigenvalue a(1), b) the second eigenvalue a(2)

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Fig. 6
figure 6

The distributions of Young’s moduli (OD1) concerning different orientation measurement errors (OE) expressed by different standard deviations σ: (a) E1, (b) E2, (c) E3

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Fig. 7
figure 7

The distributions of the eigenvalues of the second-order orientation tensor (OD2) concerning different detection rates (DR): a) the first eigenvalue a(1), b) the second eigenvalue a(2)

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Fig. 8
figure 8

The distributions of Young’s moduli (OD2) concerning different detection rates (DR): a) E1, b) E2, c) E3

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Fig. 9
figure 9

The distributions of the eigenvalues of the second-order orientation tensor (OD2) concerning different orientation measurement errors (OE) expressed by different standard deviations σ: a) the first eigenvalue a(1), b) the second eigenvalue a(2)

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Table 2 Summary of the results obtained for the OD1, for each analyzed case the estimates of normal distribution parameters are presented: mean and standard deviation (in brackets)
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Fig. 10
figure 10

The distributions of Young’s moduli (OD2) concerning different orientation measurement errors (OE) expressed by different standard deviations σ: a) E1, b) E2, c) E3

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Fig. 11
figure 11

The distributions of the eigenvalues of the second-order orientation tensor (OD3) concerning different detection rates (DR): a) the first eigenvalue a(1), b) the second eigenvalue a(2)

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Fig. 12
figure 12

The distributions of Young’s moduli (OD3) concerning different detection rates (DR): a) E1, b) E2, c) E3

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Table 3 Summary of the results obtained for the OD2, for each analyzed case the estimates of normal distribution parameters are presented: mean and standard deviation (in brackets)
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Fig. 13
figure 13

The distributions of the eigenvalues of the second-order orientation tensor (OD2) concerning different orientation measurement errors (OE) expressed by different standard deviations σ: a) the first eigenvalue a(1), b) the second eigenvalue a(2)

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Table 4 Summary of the results obtained for the OD3, for each analyzed case the estimates of normal distribution parameters are presented: mean and standard deviation (in brackets)
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Fig. 14
figure 14

The distributions of Young’s moduli (OD3) concerning different orientation measurement errors (OE) expressed by different standard deviations σ: a) E1, b) E2, c) E3

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Fig. 15
figure 15

Relationship between the norms \(\left\| {a - \bar {a}} \right\|\)and \(\left\| {C - \bar {C}} \right\|\)for the analyzed orientation distributions OD1-OD3

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The conducted MC simulations enabled the presentation of a quantitative description of the uncertainty associated with the reconstruction of the orientation distribution of fibers, including the prediction of the effective elastic constants. The distributions of orientation tensors and elastic constants obtained for different analyzed orientation distributions possess some common features. It can be observed that as the fiber detection rate decreases, the distributions associated with the orientation tensors as well as the Young moduli maintain similar mean values in close agreement with the reference (accurate) value. However, the standard deviation increases. Different observations can be made when analyzing various orientation measurement errors expressed by different standard deviations. Here, as the standard deviation of error increases, both the mean values and standard deviations related to the resulting distributions of the second-order orientation tensor’s eigenvalues and Young’s moduli change. It can be observed that as the standard deviation of error increases for each analyzed orientation distribution, the first eigenvalue of the second-order orientation tensor decreases, and similarly, the highest value of the Young modulus also decreases. It could be interpreted that the random errors in orientation measurement decrease the anisotropy index of the system. There are also variations in the results obtained for different analyzed orientation distributions. The mean values associated with the resulting distributions of the second-order orientation tensor’s eigenvalues and Young’s moduli change differently for each analyzed case. The differences are most pronounced in the case of OD3, corresponding to the highest anisotropy index (the highest value of a(1)). The smallest difference can be observed for OD2, associated with the lowest anisotropy index. The differences in uncertainty predictions between the analyzed orientation distributions are vividly illustrated in Fig. 15. It could be noticed that for OD2, which is quite close to the random orientation, the discrepancies between the actual second-order orientation tensor and the reference one described by the Eq. (18) are relatively low. In this case, the assumed levels of DR and OE have got relatively low impact on the effective stiffness tensor. Another scenario is evident in the cases of OD1 and OD3, where the norm associated with the effective stiffness tensor undergoes more pronounced changes. It could be concluded that the cases of higher anisotropy index are more prone to obtaining higher discrepancies expressed by the Eqs. (18) and (19).

4 Conclusions

The performed MC simulations enabled the depiction of distributions of orientation tensors and elastic constants for three different analyzed orientation distributions of fibers. These resulting distributions quantitatively describe the uncertainty in the reconstruction of the orientation distribution of fibers, as well as the prediction of elastic constants based on it. The following main conclusions can be drawn:

  • as the fiber detection rate decreases, the distributions associated with the orientation tensors and Young’s moduli retain similar mean values, closely aligning with the reference (accurate) value while the standard deviation increases,

  • as the standard deviation of orientation measurement error increases the mean value of the first eigenvalue of the second-order orientation tensor decreases, and similarly, the mean value of the highest Young modulus also decreases,

  • the differences in the mean values of resulting distributions are most pronounced for the case with the highest anisotropy index,

  • the uncertainty level is case-sensitive and should be predicted separately for each orientation distribution of fibers, orientation distributions with a high anisotropy index are prone to higher errors of stiffness prediction.

The obtained results provide an insight into the influence of the fiber detection rate and orientation measurement errors on the predicted elastic properties, as well as highlight discrepancies in this influence between different fiber orientation distributions. The presented insights can be used in practical cases to estimate errors of stiffness predictions for instance in terms of known limitations of measurement device accuracy or data segmentation procedure. Moreover, the proposed MC simulations may aid in selecting the optimal resolution of the measurement, balancing the required measurement time with the desired level of accuracy.

It should be noted that this investigation has certain limitations. Firstly, the fiber detection rate is calculated under the assumption that each fiber has an equal probability of being excluded. However, during the analysis of actual µCT scans or series of microscopic images, this assumption may not reflect reality properly. For instance, unidentified fibers (or parts of fibers) may be associated with regions of fiber intersection or connection, which can pose challenges to segmentation. It would be important to examine in future work whether such regions are independent of the orientation of fibers that intersect or connect within them. When considering the modeling of errors in individual fiber’s orientation measurements, three different levels of error were arbitrarily chosen to illustrate a general trend in the resulting distributions. Nevertheless, this approach can be applied across various expected measurement error levels, including, for example, information pertaining to the resolution of µCT scans. The influence of systematic errors has not been considered in this study, however, it may be important to include them in further work. It should also be emphasized that the proposed simulations are applicable only to approaches that assume the segmentation of individual fibers. To evaluate the uncertainty of stiffness predictions for voxel-wise approaches, alternative simulation assumptions need to be proposed.