# FFT-based interface decohesion modelling by a nonlocal interphase

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## Abstract

In this paper, two nonlocal approaches to incorporate interface damage in fast Fourier transform (FFT) based spectral methods are analysed. In FFT based methods, the discretisation is generally non-conforming to the interfaces and hence interface elements cannot be used. This limitation is remedied using the interfacial band concept, i.e., an interphase region of a finite thickness is used to capture the response of a physical sharp interface. Mesh dependency due to localisation in the softening interphase is avoided by applying established regularisation strategies, integral based nonlocal averaging or gradient based nonlocal damage, which render the interphase nonlocal. Application of these regularisation techniques within the interphase sub-domain in a one dimensional FFT framework is explored. The effectiveness of both approaches in terms of capturing the physical fracture energy, computational aspects and ease of implementation is evaluated. The integral model is found to give more regularised solutions and thus a better approximation of the fracture energy.

## Keywords

FFT-based spectral methods Interfaces Damage Nonlocality## Abbreviations

- FFT
fast Fourier transform

- FE
finite element

- FEM
finite element method

- TSL
traction separation law

## Introduction

Polycrystalline materials at a microscopic level show clear heterogeneous deformation patterns. This heterogeneity arises from the locally fluctuating mechanical properties of different phases and differences in lattice orientations between different grains.

FFT based spectral solver was originally introduced to model the mechanical behaviour of composite microstructures [1]. Since then it has emerged as a promising tool for modelling the micromechanical response of polycrystalline materials [2, 3]. A comparison of different FFT formulations and solution approaches in a crystal plasticity constitutive framework [4] was presented in [5]. Recently, an FEM perspective on an FFT based spectral formulation for small strain non-linear material behaviour was given in [6] and its extension to a finite strain setting was presented in [7]. Alongside such improvements, much effort has gone into making the method suitable for various applications. The computational efficiency of FFT methods makes them attractive to solve multi-field problems, for e.g. a nonlocal crystal plasticity formulation [8], ferroelectric switching [9], etc.

In finite element (FE) based solution schemes, such problems are generally tackled using cohesive zones [12, 13], see Fig. 1 (\(a_1,a_2,a_3\)). Interfaces are identified a priori and interface elements are introduced at these physical locations. This is possible since the FE discretisation is made to conform to the sharp interfaces. The opening behaviour of these interface elements is governed by a traction-separation law (TSL) that is assigned to them. FFT based spectral methods, however, rely on a regular grid and a method to include sub-dimensional (e.g. planar in 3D) elements cannot be used. Composite pixel based approaches, for example [14], which make use of the interface normal to assign homogenised mechanical properties to grid points near the interfaces, are a step in that direction. But at this point it is not clear how they will perform in situations of damaging interfaces that are inclined or curved. The high mechanical contrast combined with inherent pixelation effects of FFT may cause stresses relayed across damaging interfaces, which is undesirable for the modelling of cracks. This limitation is dealt with using the idea of an interphase (volumetric) band, as depicted in Fig. 1 (\(b_1,b_2,b_3\)). The material points in the vicinity of the sharp interface are identified and furnished with a damage constitutive behaviour (softening) and corresponding kinematics—in addition to the deformation mechanisms attributed to them as a part of the respective grains they belong to. We aim to have multiple points across the thickness of the interphase. We expect this to reduce the pixelation effect and ensure that the two crack faces are fully decoupled. This approach allows capturing the interfacial mechanics and still benefits from the computational (memory and speed) and implementation related advantages of FFT based methods. Such an approach to represent interfaces as interphase has been used previously in the literature. Hsueh-Hung et al. [15] used it to assign different material behaviour to grain boundary regions than the bulk. Their approach was physically motivated and directed towards understanding metal plasticity in the nanocrystalline regime. Clayton et al. [16] used this approach within a finite element implementation of phase field damage to understand the competition between intergranular and intragranular damage.

In volumetric dissipation based models, one specifies a dissipation density and an internal length scale to the model—which for the present case is the band thickness, *l*. The total dissipation resulting from the volumetric damage process must equal the physical dissipation of the real sharp interface (or cohesive zone model). Accordingly, the dissipation density has to scale inversely with the band thickness. A straightforward inversely proportional relationship results if the entire interphase band damages uniformly. This is not guaranteed in a conventional local softening model due to localisation and lacking objectivity with respect to the grid spacing. Various regularisation strategies have been used in the literature to remedy this. They can broadly be classified into two categories—integral based averaging [17] and gradient damage based regularisation [18, 19, 20, 21, 22].

## Problem statement

*L*as shown in Fig. 2. Let us assume a sharp interface at the center (\(x\,=0\)) of the bar. To capture the response of this interface using an FFT based spectral scheme, the sharp interface is substituted by a volumetric interphase band of thickness

*l*spanning the region from \(x_1\,=-\frac{l}{2}\) to \(x_2\,=\frac{l}{2}\). The elastic properties are taken to be homogeneous, with Young’s modulus defined as

*E*. To model the kinematics of decohesion, a damage strain field, \(\varepsilon _d\,(x)\), is introduced for \(x\in \varOmega _i\) where \(\varOmega _i=(x_1,x_2)\) is the sub-domain representing the interphase region in the total domain \(\varOmega \,=\,(-L/2,L/2)\). Since only the interphase region accommodates the damage strain, the damage constitutive model is only used in \(\varOmega _i\). In \(\varOmega _i\) it is superimposed on the elastic strain \(\varepsilon _e\) (which exists on \(\varOmega \)) as,

### Constitutive model

### Analytical solution for the rate-independent case

*l*, in order to recover the strength and dissipation of a cohesive zone, for arbitrary

*l*:

## Gradient based nonlocal damage

Phase field approaches also provide a differential equation based continuum solution for evolution of damage. In principle, a phase field model based on a Ginzburg–Landau approach can be used as well, for example [21, 23]. The resulting governing equations are similar to models based on the micromorphic approach, with an evolving length scale [24]. In fact, these approaches were introduced to avoid spurious spreading of the damage field (away from the nucleation point), which was observed when using a linear phase field potential in the former and a decreasing length scale in the latter.

In the present approach, based on the classical gradient based damage, the issue of compact support of damage field is avoided by using the length scale contrast to restrict the damage field to the interphase band. In order to recover physically meaningful results, which are objective with respect to the thickness of the interphase band, the interphase band needs to damage uniformly, even for bands that are not extremely thin. From this perspective, the classical gradient damage model provides better regularisation characteristics than these phase field approaches. If a phase field approach would be considered instead, models based on quadratic or double-well potential would suit best for the good regularisation characteristics needed in our application.

The numerical solution of the boundary value problem requires solving Eqs. (3) and (16) simultaneously. Note that these equations are coupled through the constitutive Eq. (5). Both differential equations are discretised using the FFT scheme. The coupled system of equations is solved using a staggered iterative scheme as detailed in an FEM context in [21]. The discretisation and residual evaluation of the mechanical equilibrium differential equation is well documented in [5]. Here, we only outline the residual formulation of the damage Eq. (16).

## Integral based nonlocal damage

*y*is the position of the infinitesimal volume \(d\varOmega \,\in \,\varOmega _i\). The weighting function \(\varPsi \left( y;x\right) \) that is commonly used is the Gaussian distribution, which reads

*x*is the position vector of the point at which the distribution is centred and \(\lambda \) is the characteristic length, which determines the distance along which \(\varPsi \) decays to zero. The denominator in Eq. (24) normalizes the weighting function, which ensures that for a homogeneous \({\phi _{l}}(x)\), the nonlocal field \({\phi _{nl}}\) calculated using Eq. (24) coincides with its local counterpart.

*i*and

*j*represent material points in the interphase sub-domain and \(\varDelta \,x\) is the grid spacing. Equation (26) can be implemented by looping over all the points in \(\varOmega _i\). A more efficient way is to store the normalised weights in Eq. (26) in a matrix and implement the convolution via a matrix-vector product.

## Results and discussion

In this section, we present and discuss the results of two numerical studies. The first is on the effect of the length scale contrast method for the gradient damage model. Next, we compare this method with the integral averaging model. The comparison is made based on their performance in providing a delocalised damage field and thus giving a numerically calculated fracture energy \(\mathcal {G}_c^n\) in close agreement with two ideal solutions: the rate-independent analytical solution derived above (denoted by ‘Rate indep.’) and a rate-dependent uniform damage strain solution denoted by ‘Rate dep.’. The latter is easy to obtain in our 1D numerical setting: it only requires running simulations of the underlying local damage model (\({\phi _{nl}}={\phi _{l}}\)) without imperfection. Given the piecewise uniform nature of this problem, the numerical solution for its calculation does not show localisation and thus does not need any regularisation.

### Model parameters used

The numerical studies are performed on the 1D bar of Fig. 2, discretised by 1000 uniformly spaced Fourier grid points. Young’s modulus of the material is \(E=100\,\text {GPa}\) and a critical stress \(\sigma _c\,=1\,\text {MPa}\) is used in all the simulations performed. The elastic properties throughout the bar are assumed to be same. The bar is loaded at an average strain rate \(\dot{\bar{\varepsilon }} = 10^{-5}\,\text {s}^{-1}\) for 5 s in time steps of \(\varDelta t=0.01\,\text {s}\). Since the considered properties are uniform, we introduce a small imperfection by means of a \(1\%\) reduction of the critical stress \(\sigma _c\) for the grid points in the interval \(x\in [-l/20,l/20]\) within the interphase band, in order to trigger localisation.

### Effect of length scale contrast in the gradient based model

Figure 3 depicts the distribution of nonlocal damage \({\phi _{nl}}\) inside the interphase band and the region outside, to which it has spread significantly. Due to symmetry, only half of the band (and its surroundings) is shown. The profiles are plotted for the cases \(({\lambda _{in}}/{\lambda _{out}})^2 = 10^1,10^2,10^3,10^4\). The nonlocal damage variable field extends more than 1.5 times the interphase band thickness into the bulk region on either side of the interphase for \(({\lambda _{in}}/{\lambda _{out}})^2=10^1\). This is due to the poor approximation of Eq. (17) by Eq. (18) with this value of length scale contrast. However, this extension of the interphase damage into the bulk drops systematically for higher contrasts. The maximum contrast that the numerical solver for the damage equation could handle was \((\lambda _{in}/\lambda _{out})^2=10^4\). For this contrast, we observe that the zone of spread of interphase damage into the bulk has reduced to less than \(0.2\,l\).

The effect of this slowing down is observable in the averaged stress–strain response, see Fig. 4. As a result of it, the initial softening slope is less steep. This in turn results in an over-prediction of the amount of dissipation during the damage process as compared with that for the cases with a sufficient contrast, i.e. \((\lambda _{in}/\lambda _{out})^2=10^3\)–\(10^4\).

### Comparison of gradient and integral nonlocal damage models

The analysis in “Analytical solution for the rate-independent case” section showed that for a uniform damage strain distribution in the rate-independent limit, choosing \(\varepsilon _{f}\) according to Eq. (15), a constant fracture energy can be obtained, irrespective of the interphase band thickness *l*. In this section, we want to test the same hypothesis but in a numerical setting, where localisation is triggered by the presence of an imperfection (as discussed above). This is a test of the effectiveness of the two types of nonlocality in regularizing the damage (and damage strain rates) and thus facilitating the use of simple scaling Eq. (15) for the interphase band.

Different interphase band thicknesses (\(l/L=0.010,0.015,0.020,0.025,0.030\)) are studied. \(\dot{\varepsilon }_{d0}=1.0\,\text {s}^{-1}\) and \(\varepsilon _f=5\times 10^{-5}\,L/l\) in combination with the parameters in “Model parameters used” section are used. For both models, three different values of the nonlocal length scale are used: \({\lambda }/{l}=0.5,\,1.0,\,\text {and}\,1.5\), where for the gradient damage based approach \(\lambda _{in}=\lambda \) and \(\lambda _{out}\) is varied such that a constant length scale contrast \((\lambda _{in}/\lambda _{out})^2=10^4\) is obtained.

*l*. However, a slight gradient remains even for \(\lambda /l=1.5\). This is due to the imperfect insulation of the interphase damage from the surrounding (undamaged) bulk, as discussed in “Effect of length scale contrast in the gradient based model” section. The integral nonlocal model, in Fig. 7, shows a nearly constant damage inside the band for \(\lambda /l\ge \,1.0\)—again for all

*l*/

*L*. For the smallest length scale considered, \(\lambda /l=0.5\), some of the non-uniformity introduced by the imperfection remains.

*l*/

*L*considered, indicating that the nonlocal averaging is more effective for a large nonlocal length scale \(\lambda \). In the gradient model a significant degree of localisation remains, even for the largest \(\lambda \) considered, i.e \(\lambda /l=1.5\).

*l*/

*L*shown, the computed stress-average strain responses for \(l/L=1.0\) and 1.5 practically coincide. They furthermore deviate only slightly from the analytical solution. This shows that for these values of the nonlocal length scale both regularisation approaches are effective in rendering the global response objective with respect to the interphase band thickness, despite the non-uniformity observed in \({\phi _{nl}}\) (Figs. 6, 7) and, particularly, the damage strain rate (Figs. 8, 9). The smallest value of \(\lambda ,\lambda =0.5\,l\), clearly is insufficient, as the softening responses computed for it are systematically steeper than for larger values (and than the analytical solution).

For a more quantitative assessment of the objectivity of the computed response with respect to the interphase band thickness adopted, the variation in the calculated fracture energy \(\mathcal {G}_c^{n}\), normalised by the ideal input fracture energy \(\mathcal {G}_c\), is shown for the gradient damage method and the integral averaging method in Fig. 12. In the rate-independent limit, one would like to observe this ratio to be constant at unity. Given the fact that the numerical model is rate-dependent, however, a slight deviation should be anticipated—which ideally would be constant. To illustrate the effect of the rate dependence, numerical solutions for the rate dependent case with piecewise constant damage are included in both diagrams as black dashed curves. Note that these are slightly above \(\mathcal {G}_c^n/\mathcal {G}_c=1\) and show a downward trend, which is due to the fact that post-peak the strain rate in the interphase band varies with the interphase band thickness.

The fracture energy computed by the gradient damage approach show a discrepancy with both reference values (i.e. rate-independent and rate-dependent) on the order of \(10\%\). For \(\lambda /l=0.5\), the fracture energy is under-estimated, whereas the larger values of \(\lambda \) result in an over-estimation. In each of these cases, the deviation furthermore varies with *l* / *L*. The integral nonlocal model performs much better. For \(\lambda /l=0.5\), which we already observed to be too small to obtain a uniform damage in Fig. 7, it under-estimates \(\mathcal {G}_c\) by up to \(10\%\) (Fig. 12). But for \(\lambda /l\ge 1.0\), it over-estimates \(\mathcal {G}_c\) by \(\le \,5\%\), in a way which is consistent with the rate-dependent reference solution: the difference between the regularised solutions for \(\lambda /l=1.0\) and 1.5 and the piecewise constant reference solution is less than \(1\%\). For practical purposes, this is more than satisfactory and the slight trend with *l* / *L*, due to the rate-sensitivity, should also pose no problem.

## Conclusion and outlook

In this paper, the issue of nonlocality associated with a method to incorporate interface decohesion at polycrystalline interfaces approximated by an FFT based spectral method was discussed. Interfaces were approximated as interphase bands. The softening nature of decohesion required the use of nonlocality within the interphase domain. The applicability and performance of integral and gradient damage based nonlocal averaging methods have been discussed.

For the gradient damage approach, it was found that the mechanical response and dissipation depends on the accuracy with which the flux boundary condition at the edges of the interphase domain can be enforced. In order to restrict the damage to the interphase only, a flux free condition at the interphase boundaries was used. Since FFT solvers do not allow solving an equation on an irregular domain, in the gradient approach, a contrast in the nonlocal length scale was used to approximate the flux free boundary condition at the interphase edges. The error in the fracture energy reduces upon increasing this length scale contrast. However, a high contrast renders the gradient based approach highly heterogeneous and entails numerical problems.

The implementation of the integral approach on the other hand was straightforward and effective. It only requires storage of the nonlocal weights in a matrix that can be optimized using sparse storage. Regularisation length scale values equal to or greater than the interphase band thickness were found to give accurate predictions for the fracture energy, largely independent of the (arbitrary) interphase band thickness *l*. The slight (\(\le \,1\%\)) variation which remains is due to the fact that the scaling of \(\varepsilon _f\) according to Eq. (15) does not take into account the strain rate sensitivity of the damage model. From the current study it is very clear that integral approach offers more advantage from the computational efficiency point of view. We expect the same advantages to carry over in multidimensional cases on periodic microstructures. Nevertheless, this still remains to be tested.

We wish to emphasize that, although it enables a rigorous and transparent comparison, the 1D problem considered here may not reveal all complexities that could be encountered in two or three dimensions. For instance, care should be taken that the nonlocality introduced to homogenise the damage across the interphase band thickness does not affect the propagation of damage along the band in an unrealistic manner. One possible way to avoid this is by introducing anisotropy in the nonlocal averaging. In the gradient damage approach this can be achieved by having a tensorial form for the nonlocal length scale, while for the integral approach an orientation dependent averaging kernel can be used. Modelling decohesion of polycrystalline interfaces will require a proper treatment of triple junctions to avoid the issue of non-unique interface normals. Furthermore, a method to couple interface damage with the bulk damage in voxel based models [28] can also be explored to model more complex crack patterns—kinking and branching of cracks into the bulk. These issues are currently under investigation and will comprise future works.

## Notes

### Author's contributions

LS carried out most of the study and drafted the manuscript. LS and PS developed the methodology and implemented it. RHJP, FR and MGDG conceived of the study and also participated in its design and coordination in addition to critically reviewing the manuscript. All authors read and approved the final manuscript.

## Acknowledgements

The authors would like to acknowledge the fruitful discussions with Prof. Bob Svendsen of RWTH Aachen University and Max Planck Institut für Eisenforschung, Düsseldorf, Germany. The work presented is part of the research done under the project S22.2.1349b in the framework of the research program Materials innovation institute M2i.

## Competing interests

The authors declare that they have no competing interests.

## Availability of data and materials

Not applicable.

## Consent for publication

Not applicable.

## Ethics approval and consent to participate

Not applicable.

## Funding

This research is supported by Tata Steel Europe through the Materials innovation institute (M2i) and Netherlands Organisation for Scientific Research (NWO), under the grant number STW 13358.

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