1 Motivation

It has been widely recognized for a long time, at the latest since the materials genome initiative, that optimized materials are a key enabler for solving various challenges regarding structural durability, resource efficiency and sustainable product/material life cycle management. This makes accelerating materials development a meaningful worthwhile target across various disciplines. Especially simulation-based workflows from the field of computational mechanics offer a great potential in speeding up the design process by enabling digital workflows. This is known as the inverse computational materials engineering concept or data-centric design [1, 2]. At the heart of these approaches lies the relation between the manufacturing process, the resulting microstructure and the associated properties. This relation is to be established and inverted or optimized for designing materials, whereby the associated cost and time can be minimized by relying on a large proportion of digital, simulation-based data. With the microstructure as a mediator between the process and properties, the process-structure–property linkages can be divided into process-structure and structure–property linkages. The latter are of interest in this work.

The structure–property linkage is nothing but a function that maps a material microstructure to its effective property. While the property, like an effective Young’s modulus or crack resistance, is trivial to quantify, the same cannot be said about the microstructure: Because it is a random heterogeneous medium, different realizations of a material, e.g., microscopy images taken at different locations of the same specimen, might have the same overall microstructure, but the values of individual pixels is not comparable. It is therefore reasonable to quantify the random microstructure in terms of translation-invariant, statistical descriptors like volume fractions and spatial correlations. Subsequently, the structure–property linkage is defined as a function that maps from descriptors to properties [3].

Thus, given multiple, segmented micrographs of different microstructures, a (slightly oversimplified) simulation-based workflow for establishing structure–property linkages for crack resistance might be as follows: First, the micrographs are meshed and the crack growth is simulated using a phase-field method [4,5,6]. Secondly, the microstructure descriptors and effective crack resistances are computed. Finally, a regression is fitted to map the former to the latter. However, considerations of data scarcity and microstructural stochasticity invalidate such a simple approach, motivating the present work as discussed in the following.

Fig. 1
figure 1

Phase-field simulation of crack growth in a heterogeneous medium. The microstructure (blue and yellow) of a recent TiFe system [7] is taken from [8], where it is published under the Creative Commons license [9]. The structure is loaded vertically and crack patterns (red) evolve over time (left) and reach a final state (center). Therein, the yellow phase has a ten-fold larger crack resistance than the blue phase and an initial crack is prescribed on the left side of the structure. Furthermore, a diffuse interface with locally refined mesh is used (right). It can be seen that the emerging crack patterns are highly dependent on the specific positions and shapes of microstructural features, which fluctuate randomly, leading to stochastic variations

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This work is motivated by observations from Fig. 1, where the crack evolution in a heterogeneous medium is simulated: Although the domain is certainly large enough for accurately estimating the effective elastic properties, the same cannot be said about the crack resistance. If the initial crack did not happen to coincide with a long horizontal region of the weaker phase, the initiation of crack growth might have occurred at substantially higher loads. Similarly, if the specific morphology of this realization did not happen to force the crack to develop two branches shortly before failure, the cracks from the left and upper right might have met earlier. Although very small elements are used in Fig. 1, it is clear that a finer mesh or a more sophisticated crack phase-field formulation are not sufficient for obtaining more accurate and reliable predictions if the morphologically induced variations are the main source of uncertainty. Instead, a statistical analysis over multiple realizations of the random heterogeneous microstructure of a given material is required. We aim at making such an analysis possible by leveraging microstructure reconstruction algorithms.

This work presents a digital workflow for investigating the linkages between the microstructure morphology and the effective crack resistance. Particularly noteworthy in this regard is the statistical analysis of the emerging crack patterns: It is assumed that real data of newly developed materials are expensive and only a single or very few micrographs are available. For this reason, microstructure reconstruction is used to generate multiple microstructure realizations from each reference structure. For the sake of simplicity and computational efficiency, the present study is conducted in 2D only. Although the computational cost of applying the same methods in 3D should by no means be underestimated, all methods (including the microstructure reconstruction) are available in 3D, which is required for applying the presented ideas to the development of real materials.

The required methods range from statistical microstructure descriptors in Sect. 2.1 and reconstruction algorithms in Sect. 2.2 to phase-field models in Sect. 2.3. Implementational details are given in Sect. 2.4 and numerical experiments are carried out in Sect. 3. Finally, a conclusion is drawn in Sect. 4.

The following notation is used: Tensors of first, second and fourth orders are denoted as \(\vec n\), \(\varvec{\varepsilon }\) and \(\mathbb C\), respectively. In contrast, arrays \(\textbf{A}\) are represented by bold, non-italic letters regardless of their number of dimensions. Scalars as well as scalar entries of arrays or tensor coordinates are denoted as \(\psi \), \(A_{ij}\) or \(n_k\), respectively.

2 Methods

The used statistical descriptors and a discussion of their necessity are given in Sect. 2.1. From these descriptors, multiple realizations of a material are generated by microstructure reconstruction as outlined in Sect. 2.2. The numerical framework for simulating crack growth in heterogenous media is presented in Sect. 2.3. Implementational aspects are given in Sect. 2.4.

2.1 Statistical descriptors

The microstructure of real materials is stochastic in nature. Considering a manufacturing process that is repeated multiple times: Even with perfect control of the process parameters, microscopy images or computed tomography scans of the resulting structure differ between multiple realizations. To be more specific, although the morphology is statistically equivalent throughout the realizations, the exact location of specific features like grains differs. It is clear that the similarity between structures cannot be computed directly by comparing pixel values. This makes pixel values (or any other realization-specific parametrization like nodal positions of a surface mesh) an unsuitable input to structure–property linkages.

In order to quantify the morphology of a microstructure in a statistical and translation invariant manner, the characterization function

$$\begin{aligned} {\textbf {f}}_C: {\textbf {M}} \mapsto \{ {\textbf {D}}_i \}_{i=1}^{n_\text {d}} \end{aligned}$$
(1)

is introduced, which maps a microstructure realization \({\textbf {M}}\) (hereafter simply referred to as microstructure) to a set of \(n_\text {d}\) different descriptors \({\textbf {D}}_i\). Herein, the microstructure is represented as an indicator function on a 2D grid of \(I \times J\) pixels

$$\begin{aligned} {\textbf {M}} \in \mathcal M \subsetneq \mathbb R^{I \times J} \;, \end{aligned}$$
(2)

which can take real numbers between 0 and 1

$$\begin{aligned} 0 \le M_{ij} \le 1 \; \end{aligned}$$
(3)

for all \(i = 1, \dots ,I\) and \(j=1, \dots , J\). The descriptors can be scalars or vectors, where the number of entries is not necessarily related to I and J. Instead of directly linking microstructure to properties, a sensible structure–property linkage takes the descriptors or a reduced-dimensional version of the same as input.

As a trivial example, the volume fraction

$$\begin{aligned} \phi = \dfrac{1}{IJ} \sum _{ij} M_{ij} \end{aligned}$$
(4)

is a statistical descriptor. Regarding analytical homogenization rules as structure–property-linkages for the elastic behavior, it can be understood that \(\phi \) is highly relevant and that descriptors are an incomparably better suited for fitting structure–property linkages than individual pixel values or node coordinates. Localized phenomena might be influenced by the amount of phase boundary per unit area, which is quantified by the normalized variation

$$\begin{aligned} \mathcal {V} = \dfrac{1}{IJ} \sum _{ij} \vert M_{ij} - M_{\tilde{i}j} \vert + \vert M_{ij} - x_{i\tilde{j}} \vert \;, \end{aligned}$$
(5)

where \(\tilde{i}\) and \(\tilde{j}\) denote the periodic next neighbors of i and j. Using the modulo operator mod, this is expressed as

$$\begin{aligned} \tilde{i} = (i + 1) \; \textrm{ mod } \; I \quad \textrm{and} \quad \tilde{j} = (j + 1) \; \textrm{ mod } \; J \;. \end{aligned}$$
(6)

Beyond scalar descriptors, the anisotropy of the material structure as well as the characteristic length scales of the structure can be described by the spatial two-point autocorrelation

$$\begin{aligned} {\textbf {S}}_2 = \dfrac{1}{IJ} \mathcal {F}^{-1} \left( \mathcal {F}({\textbf {M}}) \odot \mathcal {F}^*({\textbf {M}}) \right) \;, \end{aligned}$$
(7)

where \(\mathcal {F}\) denotes the two-dimensional (fast) Fourier transform, \(\mathcal {F}^{-1}\) is the inverse transform and \(\odot \) is the Hadamard product, i.e., an element-wise multiplication. It should be mentioned that the indices of each entry of \({\textbf {S}}_2\) represent a spatial vector for which the corresponding entry contains the correlation. If the microstructure is a binary indicator function, the probability that the start and end of the vector both fall in the corresponding phase if it is placed randomly in the structure. This interpretation makes it easy to comprehend that the maximal entry of \({\textbf {S}}_2\) corresponds to the zero-vector and is identical to the volume fraction.

For the simple microstructure morphologies considered in this work, \(\mathcal {V}\) and \({\textbf {S}}_2\) are sufficient to capture all relevant information, but in the general case, more complex structures require higher-order information. Although the three-point correlations are used, e.g., in [8, 10], higher-order spatial correlations are extremely expensive regarding computational effort, memory as well as required data for accurate estimation. One solution is to limit higher-order correlations to short-range information, as it is done in the multipoint statistics method [11,12,13]. A small subset can also be defined by restricting higher-order correlations to polytope functions [14]. As an alternative, restricting the descriptors to two-point information, the simple correlations can be replaced by a measure that comprises connectivity information. For example, the lineal path and cluster correlation function are two well-known candidates [15, 16]. Integral measures like Minkowski functionals [17, 18] or entropic descriptors [19,20,21] are also possible. Recently, it was shown that the latent representation of an image in a pre-trained convolutional neural network like VGG-19 [22] can be harnessed to extract statistical descriptors. For this purpose, Gram matrices are computed from the feature maps in order to render them approximately translation-invariant [23,24,25]. To eliminate the arbitrariness of the network pretraining, a conceptually similar descriptor is recently proposed [26] based on the scattering transform [27, 28].

The stochasticity of the material microstructure and the associated effective properties motivate the reconstruction of multiple microstructures from descriptors. This is described in the following section.

2.2 Microstructure reconstruction

Descriptor-based microstructure reconstruction can be regarded as the inverse to the characterization in Eq. (1): The goal is to find a microstructure realization that satisfies a given set of reference microstructure descriptors

$$\begin{aligned} {\textbf {f}}_R: \{ {\textbf {D}}_i^\textrm{ref} \}_{i=1}^{n_\text {d}} \mapsto {\textbf {M}}^\text {rec} \quad \mathrm {s.t.} \quad {\textbf {D}}_i^\textrm{ref} \approx {\textbf {D}}_i({\textbf {M}}^\text {rec}) \; \forall \; i \;. \end{aligned}$$
(8)

This can be naturally formulated as an optimization problem

$$\begin{aligned} {\textbf {M}}^\text {rec} = \underset{\textbf{M}\, \in \, \mathcal M}{\arg \min }\ \; \mathcal {L}({\textbf {M}}) \;, \end{aligned}$$
(9)

where the space of possible microstructures \(\mathcal M\) is searched for a reconstruction \({\textbf {M}}^\textrm{rec}\) that minimizes an objective function \(\mathcal {L}\). Usually, \(\mathcal {L}\) is chosen as a weighted sum over the mean squared error (MSE) in the microstructure descriptor spaces

$$\begin{aligned} \mathcal {L}({\textbf {M}}) = \sum _{i=1}^{n_\textrm{d}} \lambda _i \vert {\textbf {D}}_i^\textrm{ref} - {\textbf {D}}_i({\textbf {M}}) \vert _\textrm{MSE} \;, \end{aligned}$$
(10)

where the weights \(\lambda _i\) compensate for different orders of magnitude of descriptors and introduce a weighting according to the relative relevance of each descriptor.

The most common method for solving the optimization problem given in Eq. (9) is the Yeong–Torquato algorithm [15, 29]. Therein, the objective function is minimized using an adapted version of simulated annealing, which is a stochastic, gradient-free algorithm. As a first step, the microstructure is initialized randomly such that the volume fraction matches the reference. Then, in each iteration, two pixels of different phases are chosen at random and swapped in order to change the microstructure morphology while keeping the volume fractions constant. To avoid local minima, swaps are not only accepted if they reduce the objective function, but occasionally also in the opposite case. In combination with further improvements on efficiency, the Yeong–Torquato algorithm has been applied successfully to various soils [30], sandstone [31], chalk [32] batteries [33] and more. These improvements include multigrid schemes [34,35,36,37], different-phase neighbor sampling rules [38], efficient descriptor updates [33, 39] and optimized directional weighing of correlation functions [40]. Furthermore, efficient, lower-fidelity reconstructions based on Gaussian random field-based methods have been used as initialization [32, 41]. More information is given in [42]. Regardless, in practical applications, the high number of iterations that is required because of the stochastic and pixel-based optimization procedure limits the feasible wallclock time per iteration and hence the realizable fidelity of microstructure descriptors [43].

As an alternative, differentiable microstructure characterization and reconstruction (DMCR) reduces the number of iterations by leveraging gradient-based optimizers [10]. This requires the microstructure descriptors to be formulated in a differentiable manner and there is currently no theoretically sound strategy for avoiding local minima other than changing the used microstructure descriptors and associated weights. However, in practice, local minima play a subordinate role for most materials and many microstructure descriptors are naturally differentiable or can easily be approximated by differentiable functions. As an example, spatial correlations can be differentiated if they are defined in Fourier space or, especially for higher-order correlations, if a regularized convolve-threshold-reduce pipeline is used [10]. Furthermore, volume fractions, surface fractions and Gram matrices are naturally differentiable and a differentiable approximation to the lineal path function is presented in [44]. The method is presented in [10], extended to 3D in [45] and validated in [8]. Because of its high efficiency, DMCR is chosen as a microstructure reconstruction algorithm for the present study.

2.3 Phase-field model with diffuse interface

Crack phase-field models (hereafter referred to as phase-field models) approximate sharp cracks by a smeared crack indicator function \(d(\vec x)\), which is often called the phase-field variable [4,5,6]. The evolution of a crack driving force leads to an increase in d, which manifests itself in terms of a locally degraded stiffness. To be more specific, the formation of mechanical stresses as well as the initiation, propagation and branching of cracks in a domain \(\Omega \) is described by a regularized energy functional

$$\begin{aligned} \Pi =&\int _\Omega g(d(\vec x)) \psi ^+(\varvec{\varepsilon }( \vec x) ) + \psi ^-( \varvec{\varepsilon }(\vec x)) \textrm{d}V \nonumber \; \\&\quad + \int _\Omega G_c \left[ \dfrac{d(\vec x)^2}{4 l_c} + l_c \vert \nabla d(\vec x) \vert ^2 \right] \textrm{d}V \; , \end{aligned}$$
(11)

where the length parameter \(l_c\) influences the width of the regularization zone and \(G_c\) is akin to the critical energy release rate in the Griffith theory. The stiffness degradation is given by

$$\begin{aligned} g(d) = (1 - \alpha )(1 - d)^2 + \alpha \;, \end{aligned}$$
(12)

where \(\alpha \) is a small number

$$\begin{aligned} 0 < \alpha \ll 1 \end{aligned}$$
(13)

that regulates the residual stiffness of damaged material in order to avoid numerical problems. Furthermore, the degradation only applies to and is only driven by the tensile part \(\psi ^+\) of the strain energy density

$$\begin{aligned} \psi = \dfrac{1}{2}\varvec{\varepsilon }: \mathbb C: \varvec{\varepsilon } \;, \end{aligned}$$
(14)

where \(\varvec{\varepsilon }\) and \(\mathbb {C}\) denote the strain and stiffness tensor of second and fourth order, respectively. The additive decomposition of

$$\begin{aligned} \psi = \psi ^+ + \psi ^- \end{aligned}$$
(15)

is defined by the Amor split [46, 47], where the compressive part of the hydrostatic part of \(\varvec{\varepsilon }\) is omitted in \(\psi ^+\). Crack healing is avoided by requiring

$$\begin{aligned} \partial _t d \ge 0 \quad \forall \; \vec x \in \Omega , \; t \ge 0 \;, \end{aligned}$$
(16)

where t denotes timeFootnote 1 and \(\partial _t\) is a temporal derivative. A set of algebraic equations is derived by requiring the total potential in Eq. (11) to be minimal, yielding both the balance of linear momentum with Neumann boundary conditions on \(\partial \Omega ^\textrm{N}\)

$$\begin{aligned} \nabla \cdot \varvec{\sigma } = \vec {0} \text { in } \Omega , \quad \varvec{\sigma }\cdot \vec {n} = \vec {{t}} \text { on } \partial \Omega ^\textrm{N} \end{aligned}$$
(17)

and the evolution equation of the phase-field

$$\begin{aligned} 4l_c^2\Delta d - d - g'(d) \frac{2l_c}{G_c}\psi ^+ = 0 \text { in } \Omega , \quad \nabla d\cdot \vec {n} = 0 \text { on } \partial \Omega ^\textrm{N}. \end{aligned}$$
(18)

Besides the evolving crack phase-field parameter \(d(\vec x)\), an additional, static order parameter \(p(\vec x)\) is introduced as a smeared approximation to the geometry of the reconstructed microstructure realizations [48]. It is computed by convolving the binary indicator function with a Gaussian filter

$$\begin{aligned} f_g(\vec x) = \dfrac{l_p^2}{\pi } \textrm{exp} \left( -l_p^2 \, \vec x \cdot \vec x\right) \;, \end{aligned}$$
(19)

where \(l_p\) is the characteristic length scale. This is shown in Fig. 2. The convolution is expressed as

$$\begin{aligned} {\textbf {p}} = {\textbf {f}}_g * \textbf{M}^\textrm{rec} \;, \end{aligned}$$
(20)

where \({\textbf {p}}\) and \({\textbf {f}}_g\) are voxel discretizations of \(p(\vec x)\) and \(f_g(\vec x)\), respectively. While the crack phase-field changes over time and is a part of the solution to the physical problem, the static order parameter remains unchanged. The different regions defined by \(p(\vec x)\) differ in terms of the micro-material properties, i.e., \(\mathbb C\) and \(G_c\). For the transition region where \(0< p < 1\), a linear mixing rule is used to interpolate the Helmholtz free energy

$$\begin{aligned} \psi ^\pm = (1-p) \cdot \psi ^\pm _0 + p \psi ^\pm _1 \;. \end{aligned}$$
(21)
Fig. 2
figure 2

Visualization of the sharp and smeared static order parameter across an interface

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Finally, it should be mentioned that in the presented workflow, the specific choice of phase-field model is non-binding, i.e., it can be replaced modularly. For example, interface debonding might be considered by an appropriate phase-field formulation [49]. Similarly, multiple static order parameters can be handled by a suitable choice from the phase-field literature in combination with the multiphase capabilities of MCRpy [44]Footnote 2.

2.4 Implementation

The implementation is divided into the microstructure reconstruction, the diffuse modeling and the phase-field simulation.

Reconstruction The computation of microstructure descriptors and the reconstruction from the same is carried out in MCRpy. This free, open-source software package has been developed by the authors and is described in an earlier publication [44]. It implements a number of different descriptors and reconstruction algorithms, among others the Yeong–Torquato algorithm and DMCR.

Diffuse model and mesh An in-house Python tool is used to read the reconstruction result and create a non-conformal, locally refined mesh using GMSH [50]. This free, open-source tool allows the user to define local values for the element size h in terms of an external process function. For this purpose, the reconstruction result is smeared by a Gaussian filter with the width \(l_m\). Given a bulk and interface mesh size \(h^\textrm{max}\) and \(h^\textrm{min}\), the scaled gradient norm of the convolution result is used to interpolate between these extreme values. The static order parameter is regularized using the same Gaussian filter, but with a different length scale \(l_p\) as shown in Eq. (19). A parameter study reveals that high-quality meshes and results are achieved if \(l_m \approx 5 \cdot l_p\). Furthermore, \(h^\textrm{min} \le l_p / 2\) is required to adequately resolve the diffuse interface. Although periodic structures can be reconstructed seamlessly with MCRpy, creating periodic meshes for applying periodic boundary conditions is more challenging. Hence, the periodicity of the reconstructed microstructures is neglected in the present study, which merely aims at providing a first implementation of the envisioned workflow.

Simulation A user element (UEL) is implemented in ABAQUS for the phase-field simulation. The stiffness degradation and the evolution of the phase-field variable are implemented in the material model according to the equations given in Sect. 2.3. Triangular plane strain elements with linear shape functions are chosen for the displacements and the phase-field variable. The static order parameter is interpolated using the same shape functions, whereby the GMSH output is included as an initial condition. In contrast, the initial crack is considered as a Dirichlet boundary condition for d, since the irreversibility constraint in Eq. (16) is implemented in terms of the crack driving force and not d itself [51]. In order to decelerate the rapidly evolving crack caused by the brittle nature of the fracture problem, causing numerical difficulties, a small numerical viscosity with the viscosity parameter \(\eta \) is introduced. Moreover, a one-way staggered scheme is chosen for solving the coupled problem.

3 Numerical experiments

First, the numerical simulation and extraction of effective properties is outlined for different synthetic materials in Sect. 3.1. A statistical analysis based on microstructure reconstruction is then carried out in Sect. 3.2.

3.1 Simulation and properties of reference structures

As a numerical experiment, six reference microstructures of two-phase materials are created from Gaussian random fields using the free open-source tool pyMKS [52]. As shown in Fig. 3, the structures are isotropic, vertically and horizontally elongated and are each generated with two volume fractions, \(\phi \approx 25 \, \%\) and \(\phi \approx 35 \, \%\), respectively.

Fig. 3
figure 3

Original structures (top), central crop of the corresponding two-point autocorrelations (bottom) considered in this study. Two horizontally elongated (left), isotropic (center) and vertically elongated structures are generated using pyMKS [52] with different volume fractions. Note that the different volume fractions are achieved by thresholding the same realizations of the Gaussian random field at a different value

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As discussed in Sect. 2.4, the structures are smeared by a Gaussian filter in order to create a non-conformal mesh that is refined at the interface. The generated diffuse structures and corresponding meshes are shown exemplarily in Fig. 4. It can be seen that the structure and the interface are sufficiently well resolved. Although the mesh is not as fine as in Fig. 1, an h-convergence study confirms that the emerging crack patterns are largely unaffected by further mesh refinement (not shown here). All meshes contain approximately 25, 000 nodes, where the exact number depends on the amount of phase boundary of each specific realization. Approximately 500 increments are used and the authors have validated by means of some representative realizations that doubling the number of increments does not affect the result in this case. Depending on the number of cracks, each simulation takes between 45 min and 1:15 hon six Intel(R) Xeon(R) processors with 2.5 GHz.

Fig. 4
figure 4

Exemplary mesh of a reconstructed structure considered in the present study. Three zoom levels are shown with boxes indicating the magnified region. At the coarsest level, the mesh is omitted for clarity. The first reconstruction of the vertically elongated structures with \(\phi \approx 25\%\) is shown, which can also be seen in Fig. 9b in the first row

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Based on the micro-scale material parameters given in Table 1, the crack growth is simulated using the phase-field model from Sect. 2.3. For this purpose, an initial crack is introduced at the top of the structure and a displacement-controlled horizontal elongation is applied.Footnote 3 The emerging crack paths are shown in Fig. 5. As expected [48], the crack avoids the stiff yellow phase whenever possible, accepting deflections whenever the direct path is blocked by an inclusion. Independent of the inclusion orientations, a higher volume fraction increases the probability of these events, leading to more deflection. The second structure in Fig. 5 is an extreme case, as the inclusions are perpendicular to the crack and the volume fraction is very high. In this case, the first crack is stopped by the inclusions and a second crack evolves.

Table 1 Parameters of phase-field model as well as the mesh generation and the boundary conditions
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Fig. 5
figure 5

Simulated crack paths for the microstructures shown in Fig. 3 using the material parameters from Table 1. It can be seen that with a high volume fraction, the crack is blocked by inclusions that are perpendicular to it

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To objectively quantify this visual impression, the effective fracture work is extracted from the force–displacement curve as shown in Fig. 6. As shown in the visualization, the slower decay of the reaction force increases the area under the curve, which is identified as the total external work \(W^\textrm{ext}\) applied to the structure. Naturally, \(W^\textrm{ext}\) must equal the energy functional \(\Pi \) defined in Eq. (11), which in turn comprises the stored elastic energy \(\Pi ^\textrm{EL}\) and the dissipated energy \(\Pi ^\textrm{D}\)

$$\begin{aligned} \int F \, \textrm{d} u = W^\textrm{ext} = \Pi ^\textrm{EL} + \Pi ^\textrm{D} \;. \end{aligned}$$
(22)

Even in a completely cracked microstructure, \(\Pi ^\textrm{EL}\) is not zero because of the residual stiffness \(\alpha \), which is introduced to the degradation function Eq. (12) for numerical stability. \(\Pi ^\textrm{EL}\) is given by the triangular area under the unloading path in the force displacement curve, which is not simulated in ABAQUS but is assumed linear. This is visualized in Fig. 6. Subtracting \(\Pi ^\textrm{EL}\) from \(W^\textrm{ext}\) yields \(\Pi ^\textrm{D}\) which summarizes all dissipated energy, such as the crack formation energy and the numerical dissipation due to the numerical viscosity and other effects. It is assumed that the numerical dissipation can be neglected.

Fig. 6
figure 6

Two qualitatively different force–displacement curves to visualize the extraction of effective properties. The slope at the origin and the maximum yield the Young’s modulus and the tensile strength, respectively. The area under the curve represents the external work. The remaining elastic energy (blue) is subtracted from the total work (orange) to estimate the dissipated energy. The final solution is shown for each structure to provide an intuitive understanding of how the reference node response relates to the underlying crack pattern

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With the effective response quantified by \(\Pi ^\textrm{D}\), the characterization of the microstructure morphology by statistical descriptors remains as a final prerequisite for establishing a structure–property linkage. Because the two-point correlation used for reconstructing the microstructures is very high-dimensional, it cannot be used for establishing a structure–property linkage with such a small data set. A common alternative is to reduce the dimensionality of microstructure descriptors by a principal component analysis [3]. However, the direction of the obtained basis vectors depends on the data set at hand, making the structure–property linkages less comparable and understandable. Moreover, a linear dimensionality reduction implicitly assumes that the high-dimensional descriptors can be interpolated linearly. As an alternative, two scalar quantities are defined manually. The first is the volume fraction \(\phi \) which is known to dominate the elastic response as well as the order of magnitude of the two-point correlation. Because it is isotropic, an additional quantity \(\mathcal {A}\) needs to be defined to measure the anisotropy, which clearly matters in the present case. It is extracted from the two-point correlation \(S_2(\vec r)\) by integrating over all vectors that are aligned with the loading direction and subtracting the integral over all orthogonal vectors

$$\begin{aligned} \mathcal {A} = \int _{\Omega _\parallel ^{\vec r}} S_2(\vec r) \, \textrm{d} \vec r \; - \int _{\Omega _\perp ^{\vec r}} S_2(\vec r) \, \textrm{d} \vec r \;. \end{aligned}$$
(23)

Herein, \(\Omega _\parallel ^{\vec r}\) and \(\Omega _\perp ^{\vec r}\) define all correlation vectors that are parallel and perpendicular to the loading direction, respectively. As an alternative, Minkowski functionals or inertia could be used.

With this quantification of the morphology and response, Fig. 7 shows the relation between them. A preliminary analysis on these results is given in the following, whereas a reliable statistical analysis is carried out later. For the horizontally elongated structures, the larger volume fraction is related to an increased energy dissipation in this case. This meets the expectations that are discussed in the context of the energy computation in Fig. 6: The cracks are blocked by the inclusions and the subsequent deflection or growth of additional cracks requires more energy. For the isotropic and vertically elongated structures, however, the opposite is the case. An increased volume fraction of the stiff phase reduces the dissipated energy. This seemingly counter-intuitive result might be explained by local weakening for these structures. Because the prescribed displacement is distributed to smaller regions of the compliant material, stress peaks might be reached faster, leading to an increased crack growth.

Fig. 7
figure 7

Relation between the volume fraction \(\phi \) (a), the anisotropy \(\mathcal {A}\) (b) and the dissipated energy. Spherical markers denote isotropic structures, while left and upward facing triangles represent horizontal and vertical elongations, respectively

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Finally, the role of crack deflection is discussed by means of a known example from the phase-field literature [48]. A structure with four spherical inclusions is simulated and shown in Fig. 8. Although the inclusion shapes do not change, their spatial arrangement has a massive effect on the crack path. As shown in Fig. 8a, the crack naturally grows directly through the structure in an open configuration where no structure is in its path. In contrast, Fig. 8b shows a closed arrangement where the crack is deflected. Therein, the large distance between the inclusion and the crack path is due to the smeared but unweakened interface. Regardless, the example illustrates the role of spatial arrangement. Whether or not an inclusion blocks a crack does not only depend on the inclusion shape but also on their arrangement and position relative to the crack tip. This latter aspect can introduce tremendous variations to the observed behavior, as revealed in the following section by statistical analyses.

Fig. 8
figure 8

Two different configuration of spherical inclusions and their corresponding crack patterns, reproducing the results from [48]. The open configuration (a) is associated with a lower dissipated energy than the closed configuration (b), which deflects the crack. The distance between the crack and the interface could be eliminated by weakening the interface as in [48]

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3.2 Statistical analysis by microstructure reconstruction

In the following, the findings obtained and discussed based on the six reference structures are repeated for a higher number of microstructure realizations. From each of the six reference structures, five statistically equivalent microstructures are reconstructed using MCRpy as outlined in Sect. 2.2 using the parameters presented in Table 2. Although MCRpy is generally built for computing complex descriptors on the GPU, the simple morphologies considered in this work can be captured sufficiently well using \(\phi \), \(\mathcal {V}\) and \({\textbf {S}}_2\), which can be computed efficiently on a regular CPU. On a laptop with an Intel(r) Core(TM) i7 processor with 14 cores at 2.4 GHz, the reconstructions take approximately 2 minutes per realization and have a negligible memory consumption.

Table 2 Parameters of microstructure reconstruction using MCRpy version 0.2.0.
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The reconstructed structures are shown in Fig. 9. It can be seen that the morphology is captured very well, as the original structures are indistinguishable from the reconstructed realizations to the untrained eye.

Fig. 9
figure 9

From each reference microstructure (a), five reconstructions are carried out to obtain multiple realizations of the same material (b–f). A central crop of the original structures’ autocorrelations is given in Fig. 3

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In analogy to the previous section, the crack paths are simulated using the material parameters from Table 1. The result is overlaid on the structure and displayed in Fig. 10. It can be seen that the crack patterns vary strongly within the same material class (row), even if the morphology is statistically equivalent. The results indicate that two cracks can not only form with inclusions that are elongated perpendicular to the crack path, but also in the other cases and even with lower volume fractions. This illustrates the massive role of the specific location of features relative to the crack tip. Although this is intuitively clear to an engineer, it is surprisingly difficult to make reliable predictions. The interested reader may chose a column in Fig. 9, e.g., column (d), and try to estimate the number of cracks for each structure. The difficulty of this task turns out to seriously impede the establishment of structure–property linkages, as discussed in the following.

Fig. 10
figure 10

Simulated crack patterns for the original (a) and reconstructed (b–f) microstructures of each material class (row). It can be seen that the patterns vary greatly throughout the different realizations. The structures without overlaid crack paths are shown in Fig. 9

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Like in the previous section, the effective properties are extracted based on the force–displacement curve of the reference point which is used for applying the displacement boundary condition. These force–displacement curves are shown in Fig. 11. It can be seen that for each reference structure, the five reconstructed microstructures differ and vary significantly in terms of their force–displacement curve, both, qualitatively and quantitatively. For example, in some cases the reference structure fails completely shortly after the force maximum is reached. Even for the same material, this need not be the case in the reconstructed structures: If a specific realization of a material happens to cause the formation of additional cracks, this can postpone the unification of different crack paths, which manifests itself as a slower decay of the reaction force.

Fig. 11
figure 11

Force–displacement curves for each of the considered structures. The rows correspond to \(\phi = 25 \%\) (top) and \(\phi = 35 \%\) (bottom) and the columns (a–c) represent horizontal, isotropic and vertical elongation, respectively. For each reference structure, all five reconstructions are given in the same plot. It can be seen that the curves vary quantitatively and qualitatively

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Figure 12 shows the relation between the microstructure descriptors and the dissipated energy for all structures from Fig. 10, including both, original and reconstructed realizations. The results confirm the bad premonition: In fact, the scatter throughout different realizations of the same material is much larger than the trends in the effective property. Consequently, no reliable statement can be made regarding the influence of structural descriptors. The only conclusion that Fig. 12 allows for is that an increased volume fraction induces more scatter, which reduces the effective crack resistance based on a weakest link criterion.

Fig. 12
figure 12

Relation between the volume fraction \(\phi \) (a), the anisotropy \(\mathcal {A}\) (b) and the dissipated energy for the reference and reconstructed structures. Spherical markers denote isotropic structures, while left and upward facing triangles represent horizontal and vertical elongations, respectively

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This stands in contrast to the Young’s modulus, which is clearly related to the microstructure descriptors as shown in Fig. 13. Therein, the scatter of \(\overline{E}\) over \(\phi \) can be explained by \(\mathcal {A}\) and vice-versa. Especially Fig. 13b displays a clear clustering of the six material classes, where the reference has a higher Young’s modulus than the average of all reconstructions. This is due to a discrepancy in \(\phi \), which is slightly reduced in the reconstructed structures due a final rounding step from \(0 \le M_{ij}\le 1\) to \(M_{ij} \in \{0, 1\}\). The phenomenon is discussed in more detail in [8] and could be compensated for by a post-processing step as proposed therein. However, in the present work, there is no need to carry out this step. In summary, Fig. 13 illustrates how easily the elastic properties can be predicted in comparison with the effective fracture behavior.

Fig. 13
figure 13

Relation between the volume fraction \(\phi \) (a), the anisotropy \(\mathcal {A}\) (b) and the dissipated energy for the reference and reconstructed structures. Spherical markers denote isotropic structures, while left and upward facing triangles represent horizontal and vertical elongations, respectively

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Figure 12 clearly invalidates the possible interpretations from Sect. 3.1. No statements can be made about how the influence of the volume fraction on the crack pattern changes with the inclusion orientation if the underlying data are nothing but a random number. One might ask why the false interpretations are given if they are later discarded. This leads to the central message of the present work. Localized properties of random heterogeneous media are inherently random, making statistical analyses indispensable. Individual results can be quickly over-analyzed and over-interpreted as long as the results are plausible. However, a publication on microstructures must not end after analyzing individual results. The existence of trends in the data as well as the sufficiency of the domain size and modeling assumptions need to be assessed statistically, for example, leveraging microstructure reconstruction.

4 Conclusion and outlook

A workflow for statistical analyses of the effective fracture behavior of random heterogeneous media is given, where microstructure reconstruction plays a central role in generating the required data. The underlying reconstruction algorithm and phase-field model are introduced in depth and the workflow is applied in a numerical experiment. The key difficulty therein is that beyond the estimation of elastic properties, the effective crack resistance of a material depends strongly on localized features in the microstructure. Because the microstructure is a random heterogeneous medium, the emerging crack paths and associated energies fluctuate strongly, making structure–property linkages extremely challenging. By deliberately over-analyzing results that are statistically not verified, it is demonstrated how quickly misleading conclusions can be drawn from insufficient data. Because data is usually expensive in materials engineering, multiple synthetic realizations are generated from each reference sample using a microstructure reconstruction algorithm. This allows for a statistical evaluation that invalidates the previously made, premature conclusions.

Besides confirming the relevance of statistical investigations, these results show that further developments are required for establishing structure–property linkages based on the presented idea. First and foremost, significantly larger domain sizes might be required for reducing the scatter, which in turn needs to be quantified appropriately by a much larger number of realizations per descriptor. Furthermore, the effective crack properties may be quantified by far-field J-integrals instead of the total dissipated energy as in [48]. However, this goes beyond the scope of the present work, which mainly aims at illustrating the relevance of statistical evaluations and pointing out the utility of microstructure reconstruction algorithms in this context.

In the more distant future, application to real data with complex morphologies first and foremost requires to account for the three-dimensional microstructure. This necessitates improvements on the computational efficiency of the differentiable microstructure characterization and reconstruction (DMCR) algorithm as well as the phase-field simulation. As far as the former are concerned, the differentiability of the underlying statistical descriptors not only enables the use of highly efficient gradient-based optimizers in DMCR, but also facilitates a further speedup if significantly more structures per descriptor should be reconstructed: As shown in [53], a neural cellular automaton can be trained to learn systems of partial differential equations (PDEs) that generate structures with given descriptors. Because the learned PDEs can be solved very efficiently, the expensive training phase is compensated for if many microstructure have to be generated. As discussed in the present work, this would enable an identification of the underlying statistical distribution of the properties as well as a statistically significant estimation of the variance or equivalent parameters of non-Gaussian distributions.

From a computational standpoint, the phase-field simulations are the major bottleneck. Depending on the complexity of the microstructure, they are approximately one order of magnitude slower than the reconstruction. Much current research on the phase-field method is focused on extended capabilities like anisotropy [54,55,56] and fatigue [57] rather than computational efficiency. A promising solution might be to use Fourier-based methods, which also make the incorporation of periodic boundary conditions trivial and the need for non-conformal meshes obsolete. In this context, the crack path can be assumed as globally optimal, recasting crack growth as a continuous flow maximization problem [58]. This is very efficient, however, crack trapping might be predicted differently as in the present formulation [48, 49]. As an alternative, extending composite voxel [59] or boxel [60] formulations to crack growth might be promising.

With an extension of the presented workflow to three dimensions, it can be applied to real material structures, which may be obtained from 3D computed tomography scans or 2D microscopy images on three orthogonal planes. In the latter case, microstructure reconstruction can be used to reconstruct 3D domains for numerical simulations [8]. With a properly parametrized micro-scale material model, true insights into the structure–property linkages and their uncertainty may be obtained in order to speed up computational materials engineering.