Multiscale modeling of the mechanical behavior of brazed Ni-based superalloy sheet metals

Ni-based brazing metals are commonly used for joining and repairing Ni-based superalloys used in high-temperature applications. Besides providing a joint that is resistant to high temperatures, the brazing layers formed in the joining process can also deteriorate the mechanical properties of the material composite. An example where the brazing process has a particularly detrimental effect is abradable sealing systems in a turbomachinery. Honeycomb liners are typically brazed to a carrier plate and form the stator of the sealing system. Sealing fins attached to the rotor rub into the honeycomb liners either in order to achieve a minimal leakage air flow or due to a change of the operation conditions [1]. Due to capillary action, the brazing material can rise from the carrier plate up to the rubbing surface and form a layered composite consisting of the base metal and the braze filler metal (see Fig. 1, left). Brittle phases that form during brazing pose a high risk of rotor damage as they show strong mechanical resistance up to high temperatures. Additionally, brittle phases can act as an accelerator for critical stator damage such as cracking and large breakouts in the honeycomb liners, which can significantly increase the leakage flow and can also damage other parts of the turbomachinery.

The microstructure of the brazing layer between the joining partners shows different characteristics depending on the duration of brazing and the braze gap size [2‐4]. A common feature is the coexistence of brittle and ductile phases in the brazing zone. Ulan kyzy et al. [5] identified three major microstructural constituents in the double-foiled segments of a Hastelloy X honeycomb liner brazed with BNi-5 filler metal. Those were a matrix of \(\gamma \)-nickel solid solution (1), NiSi (2) and several silicides of Ni\(_{\textrm{y}}\)(Cr,Mo)\(_{\textrm{x}}\)Si type (3). A similar microstructure could be identified in Haynes 214 sheet metals brazed with BNi-5 (see contrast of back-scattered electrons in Fig. 1, right). Previous studies compared Hastelloy X with Haynes 214 as honeycomb liner materials [6, 7]. It was shown that the latter alloy leads to higher mechanical and thermal loads on the rotor during a rubbing event. For this reason, we focus on Haynes 214 instead of Hastelloy X in this study.

According to Hawk et al. [3], the matrix (1) has a microstructure similar to the base metal Haynes 214 consisting of \(\gamma \)-nickel and nanometer-sized \(\gamma '\)-precipitates. Measurements of microhardness showed a strong local increase in hardness in the Si-rich phase (2) and the silicides (3) [3, 5]. Table 1 shows the chemical composition of two base metals typically used for the honeycomb foils as well as the braze metal BNi-5.

Since in the literature mainly macroscopic mechanical testing has been performed as, e.g., by Hawk et al. [3] as well as by Li and Schulson [8], the effect of the microstructure on the mechanical behavior of brazed joints is not fully understood. Isolated mechanical testing of the identified phases is cumbersome due to the difficulty of capturing the conditions in the brazing layer that stem from the process of brazing. Numerical simulations can help to overcome this challenge. However, current modeling approaches treat the brazing zone as either purely ductile or purely brittle depicted by an elastic material behavior while neglecting the multiphase microstructure in the brazing layer [7, 9].

In this work, numerical simulations are conducted to identify the most influential effects on the elastic–plastic behavior of the layered composite on the macroscale as well as on the microscale. The Ni-based superalloy Haynes 214 is chosen as base metal and BNi-5 as braze filler metal. Virtual high-temperature tensile tests at different strain rates will serve as a model basis for the simulative studies. They depict the characteristics of the loading conditions during a rubbing event in a controlled environment that will allow a quantitative comparison between the simulations and future experiments.

The goal of this work is twofold. First, the necessary depth of modeling and crucial parameters for experimental characterization of brazed Ni-based sheet metals are identified and validated. Second, guidelines for a beneficial microstructure as well as a beneficial macroscopic geometry of base metal-braze composites in honeycomb liners will be derived. In particular, this means This work is organized as follows: In Sect. 2, the constitutive relations employed on each scale are described. A sequential multiscale simulation setup is presented in Sect. 3. The employed micromechanical model is validated with experimental results from the literature in Sect. 4. Sensitivity studies on both scales are discussed in Sect. 5. The work is closed in Sect. 6 with the most important conclusions as well as an outlook on further studies.

A phenomenological elastic–viscoplastic material law considering the face-centered cubic (fcc) crystal structure is used for the ductile nickel solid solution phase (1). The generalized Hooke’s law is used to calculate the second Piola–Kirchhoff stress tensor in the plastic configuration \(\textbf{S}_{\textrm{p}}\) caused by elastic deformation \(\textbf{F}_{\textrm{e}}\) depicted by the Green–Lagrange strain tensor \(\textbf{E} = 1/2 (\textbf{F}_{\textrm{e}}^{\textrm{T}}\textbf{F}_{\textrm{e}}-\textbf{I})\) with \(\textbf{I}\) being the second rank identity tensor,In Eq. (1), \(\mathbb {C}\) is the fourth-order elastic stiffness tensor inheriting the symmetries of the underlying crystal lattice. For the cubic crystal of the nickel solid solution phase, three independent elastic constants are required (\(C_{\textrm{11}}\), \(C_{\textrm{12}}\) and \(C_{\textrm{44}}\)) [11]. In the framework of large deformations, the total deformation gradient \(\textbf{F}\) is multiplicatively split into a lattice-preserving plastic deformation step followed by a lattice distorting elastic deformation step, \(\textbf{F} = \textbf{F}_{\textrm{e}}\textbf{F}_{\textrm{p}}\). Plastic deformation is depicted by the plastic deformation gradient \(\textbf{F}_{\textrm{p}}\). Its evolution is given by the single dot product of the plastic velocity gradient \(\textbf{L}_{\textrm{p}}\) and the plastic deformation gradientFor crystal plasticity, it is assumed that the plastic deformation follows the slip systems of the underlying crystal lattice. A suitable flow rule for a fcc crystal lattice can be formulated as:with the slip direction \(\textbf{m}^{\alpha }\) and the normal of the slip plane \(\textbf{n}^{\alpha }\) defining the slip system \(\alpha \) [12]. The shear rate of plastic deformation \(\dot{\gamma }^{\alpha }\) in the slip system \(\alpha \) caused by the Schmid shear stress \(\tau ^{\alpha } = \mathbf {S_{\textrm{p}}} : \textbf{m}^{\alpha } \otimes \textbf{n}^{\alpha }\) is calculated using a phenomenological power law by Rice, Hutchinson and Peirce [13]Therein a reference shear rate \(\dot{\gamma }^{\alpha }_{\textrm{0}}\) acts as a proportional factor on the ratio of the Schmid shear stress acting on the slip system \(\alpha \) and the corresponding critical resolved shear stress \(\tau ^{\alpha }_{\textrm{c}}\). The evolution of the critical resolved shear stress is given as:where \(h_{\alpha \alpha '}\) is a hardening matrix accounting for active and latent hardeningIn the equation above \(h_{\textrm{0}}\), a and \(\tau _{\mathrm {\infty }}\) are hardening parameters and \(q_{\alpha \alpha '}\) assigns a factor of 1.0 for coplanar slip systems and 1.4 for systems with distinct slip planes. For the fcc crystal structure, the hardening parameters are assumed to be the same for all 12 slip systems [12].

The corresponding material parameters for Haynes 214 have been identified by Fischer et al. [6] by comparing experimental tensile tests at different strain rates with virtual tensile tests on a polycrystalline representative volume element (RVE). According to Jamaloei et al. [4], Ni\(_{\textrm{3}}\)Si precipitates can be found in the ductile phase of the brazing layer. These precipitates show the same crystal structure as the \(\gamma '\) phase of Haynes 214 and have a similar size as the \(\gamma '\) precipitates. Therefore, the already identified material parameters of Haynes 214 should be suitable for the ductile phase in the brazing layer. The crystal elastic constants were chosen to match not only the effective Young’s modulus E and Poisson’s ratio \(\nu \) but also to achieve a single crystal anisotropy factor typical for a Ni-based alloy (Zener ratio of \(A = 2.4\)) [14]. This is an enhancement to [6] in which the elastic anisotropy of the grains is neglected.

The remaining phases (phases 2 and 3 in Fig.  1) are assumed as purely brittle. As a first approach, a purely isotropic linear elastic material behavior that matches the effective elastic properties of polycrystalline Haynes 214 [15] is assigned to the brittle phases. Table 2 summarizes the material parameters used for the ductile and brittle phases at room temperature (\(T = 20 \, ^{\circ }\textrm{C}\)) as well as at \(T = 1100 \, ^{\circ }\textrm{C}\) which is close to the melting point of Haynes 214 (\(T_{\textrm{melt}} = 1355 \, ^{\circ }\textrm{C}\)).

The most important effects on the macroscale are expected to be the temperature and rate dependencies of the elastic–plastic material behavior. A preliminary study considering fully resolved grains assigned with crystal elastic–plastic material behavior has been conducted to analyze the influence of varying crystal orientations on the base metal layers. It was shown that the stress heterogeneity due to the contrast between ductile and brittle layers is a magnitude of order larger than the stress heterogeneity due to the contrast between differently orientated grains in the base metal. Therefore, the crystalline structure of the base metal can be regarded as a second-order effect on the fluctuations of mechanical fields in the layered composite. Consequently, the base metal can be regarded as homogeneous and isotropic. A further study on the propagation of microstructural variation to the macroscopic material behavior showed that also the brazing layer can be regarded as homogeneous and isotropic (see Sect. 3.1). These considerations lead to the macroscopic material behavior of the base metal-braze layered composite being modeled as follows:All necessary material parameters for the base metal are taken from tensile tests conducted at different strain rates by Ulan kyzy [10]. For the brazing layer, a homogenization approach using a dual-phase RVE is employed (see Sect.  3.1). The uniaxial yield limit at different levels of plastic equivalent strain and strain rate is given as tabular data to the FE-solver Abaqus [16].

On the microscale the focus lies on capturing the relevant effects of the multiphase microstructure consisting of brittle and ductile phases. To model the effect of the anisotropic flow behavior of the grains in the ductile phase, the above-described crystal elastic–plastic constitutive law is employed. Compared to modeling isotropic phases (as done by, for example, [17, 18]), additional contributions to the stress and strain fluctuations due to crystallographic misorientation can be depicted. This is potentially important for assessing the potential risk of micro-damage.

A fully periodic RVE is used to model the microstructure of the brazing layer in a highly resolved manner. A necessary condition in the concept of the RVE is the separation of scales [19], which can be formulated asIn the microstructure of the brazing layer, the characteristic length of the heterogeneity \(L_{\textrm{h}}\) is the diameter of the smallest silicides (3 in Fig.  1). The macroscopic length scale \(L_{\textrm{macro}}\) is set to be the brazing layer thickness \(t_{\textrm{braze}}\), which lies in the range of \(50 \, \mathrm {\upmu m}\) to \(150 \, \mathrm {\upmu m}\). For the generation of the geometry of the RVE, the software Neper is employed [20]. The implemented grain growth algorithm is used which is based on a Laguerre–Voronoï tessellation and lets the user introduce statistics for the grains’ equivalent diameter (diameter d of a sphere with same volume as the grain) and shape. In order to account for differently sized grains, the grain diameter is set to lie in the range \(L_{\textrm{h}} = [0.5--2.0] \, \mathrm {\upmu m}\). The grains’ equivalent diameter follows a logarithmic normal distribution (mean value of 0.87 \(\mathrm {\upmu m}\) and standard deviation of 0.3 \(\mathrm {\upmu m}\)). This logarithmic normal distribution was chosen as it is generally able to describe measured grain size distributions as shown by, for example, [21] and allows for a few larger precipitates that were also observed, even though those were of small quantity. Through a "coloring" process in which one out of two phases is assigned to each grain the RVE is divided into a brittle and a ductile phase. Thereby, the grains can be grouped to form a bigger microstructural cluster. The coloring process is random with the fraction of the ductile phase being the only condition that has to be satisfied. In combination with the varying grain diameter, a wide range of microstructural cluster sizes and shapes can be achieved. Compared to a multilevel tessellation [22], this approach has the advantage that the internal boundaries in a single phase are conserved. This is relevant for the ductile phase wherein varying crystallographic orientations depict a possible source of heterogeneity inside a single phase. In this approach, the grain size dependency of the flow stress (Hall–Petch relation) is not transferred to the grains individually as suggested by Meier et al. [23] or Fischer et al. [6]. Instead, the critical resolved shear stress \(\tau _{\textrm{c}}\) and the saturation stress \(\tau _{\mathrm {\infty }}\) are chosen to be the same for all grains in the ductile phase, and their values are set according to the mean equivalent grain diameter using the Hall–Petch constants \(k_{\textrm{c}}\) and \(k_{\mathrm {\infty }}\) from [7]. It is assumed that the geometrically necessary plastic deformation at the grain boundaries depicted by a spatial jump in the crystallographic orientation in combination with the hardening behavior following the equations (5) and (6) already accounts for a varying resistance against dislocation slip for grains with distinct sizes. Smaller grains have a higher proportion of areas of geometrically necessary deformation than bigger grains and therefore exhibit a higher mean flow stress. A shortcoming of this approach is that the contrast of the flow stress between two differently sized grains only builds up gradually with progressing deformation but is not present at the initial plastic deformation. As the mechanical contrast between the brittle and ductile phases is expected to be much higher than the contrast between grains of the ductile phase due to distinct grain sizes, the above-mentioned shortcoming is accepted in order to reduce the implementation effort. At this point, it should also be noted that the grain size hardening is only applicable to polycrystals up the recrystallization temperature (\(T/T_{\textrm{melt}} \, < \,\)0.3–0.5) [24]. For the elevated temperatures (\(T = 1100 \, ^{\circ }\textrm{C}\)) considered in this work, the grain size effect is assumed to be insignificant and is therefore neglected (see Table 2).

The geometric tessellation consists of 2000 grains. This is a rather large number of grains compared to studies by other authors [9, 17] who employed RVEs consisting of only 150 grains. However, regarding the complex microstructural clusters showing features of different size this is necessary to ensure the representativeness of the volume element. Figure 2 shows the Laguerre–Voronoï tessellation with varying grain sizes as well as the colored RVE showing the dual-phase microstructure with a fraction of ductile phases of \(P_{\textrm{ductile}} = 0.50\).

Due to its superior computational performance for fully periodic solution fields, a spectral method for solving mechanical equilibrium is employed on the microscale. The RVE is discretized by 128 voxels in each spatial direction. A convergence study that justifies this numerical resolution can be found in “Appendix B”. The spectral method implemented in the simulation toolkit DAMASK [12] was verified against a FEM simulation. This was done for the case of the high contrast in the numerical tangential stiffness between the purely elastic brittle and the yielding ductile phases. This characteristic of the microstructure in the brazing layer is challenging for the spectral solver due to jumps in the stress field at phase boundaries. Both solution methods showed good agreement, and the spectral solver is considered to be suitable for the microstructures that are observed in the honeycomb brazing layer. The micromechanical model itself was validated with an experimental analysis of a similar microstructure taken from the literature (see Sect. 4).

In order to assess the significance of microstructural variation influencing the mechanical behavior on the macroscale, virtual uniaxial tensile tests were performed on different microstructures. Concerning the mechanical model on the macroscale (see Sect.  3.2), it is important to evaluate whether the effective material parameters show significant uncertainties leading to a potentially heterogeneous and anisotropic material behavior in the brazing layer on the macroscale. The tests were carried out at room temperature and under quasi-static conditions. As a measure for the uncertainties on the macroscale due to variations in the microstructure, the coefficient of variation \(\textrm{CV}\) is introduced as the ratio of the standard deviation to the mean value of the mechanical response of the RVETwo types of microstructural variation were analyzed. The first one was the variation due to different phase morphology. A random grain-to-phase-assignment process was applied to generate 10 different microstructures with a fraction of the ductile phase of \(P_{\textrm{ductile}} = 0.50\). The second type of microstructural variation concerns the crystallographic orientation of the grains of the ductile phase. As extreme case all grains of the ductile phase were given the same crystallographic orientation. This represents a single-crystalline ductile phase. The loading direction was then successively orientated parallel to three different crystallographic directions, the first one being [100], the second one [110] and the third one [111]. This was realized by first aligning the crystal coordinate system with the laboratory system and then either no further rotation (1), a rotation of \(45^\circ \) about the y-axis (2) or a rotation of \(45^\circ \) about the y-axis followed by a rotation of \(35.3^\circ \) about the z-axis (3). For the two types of microstructural variation, the \(\textrm{CV}\) of the mechanical response at 5% volume-averaged strain is shown in Fig.  3.

The stress measures evaluated are the stress component in the load direction \(\sigma _{\textrm{xx}}\) as well as the 90% quantiles of the first principal stress in the brittle phases \(P_{\textrm{90}}\{\sigma _{\textrm{I, brittle}}\}\) and the hydrostatic stress in the ductile phase \(P_{\textrm{90}}\{\sigma _{\textrm{m, ductile}}\}\). The \(\textrm{CV}\) is small for both types of microstructural variation. For a varying phase morphology, the mechanical response of the RVE varies stronger than for the varied crystallographic texture, but the \(\textrm{CV}\) is still less than 7%. The chosen fraction of the ductile phase of \(P_{\textrm{ductile}} = 0.50\) is expected to form the most complex grain clusters and subsequently the strongest possible morphological variation. Regarding the crystallographic variation, it is expected that an increasing fraction of the ductile phase could increase the microstructural variation further to a certain degree. However, since the observed effect of crystallographic variation on the stress measures of interest is significantly smaller than the effect of the morphological variation, any potential increase in the CV is expected to be negligible. Therefore, an assumption of isotropic and homogeneous material behavior in the brazing layer on the macroscale is reasonable as long as the volume fraction of the ductile phase is constant.

The macroscopic tensile sample is modeled as a layered composite of thickness \(t_{\textrm{comp}}\) with a brazing layer that shows spatial thickness fluctuations. This is motivated by the observation that the depth of interdiffusion varies along the in-plane directions forming a non-flat interface between the brazing layer and the base sheet metal. The layers are generated by coloring a Voronoï tessellation with 2000 equally sized cells on the macroscale according to the cell centroid positions, which leads to an interface roughness in the order of magnitude of the Voronoï cell diameter. Figure 4 shows the procedure schematically. As a model input parameter, the mean thickness of the brazing layer \(t_{\textrm{braze}}\) is used.

The interface roughness is smaller than the sample’s in-plane dimensions by a factor of approximately 12. Therefore, the condition of the separation of scales is satisfied, and a semi-periodic macroscopic RVE with periodic boundary conditions applied in the x- and z-directions is employed to reduce the computational time significantly. For the y-direction force free surface boundary conditions are applied. The uniaxial tensile load is introduced over a multipoint constraint that prescribes a volume-averaged deformation in the x-direction. Details of the implementation of periodic boundary conditions can be found in, for example, [25]. The cube-shaped computational domain is discretized by 65\(^3\) brick finite elements with linear shape functions (C3D8R). An example of the layered composite model with a mean thickness of the brazing layer of \(t_{\textrm{braze}} = 0.5 \, t_{\textrm{comp}}\) is provided in Fig.  5 together with the sources of the material behavior for each layer that are described in Sect. 2. To be able to deal with the free surfaces in the y-direction of the tensile samples, the finite element method is chosen as the solution method for mechanical equilibrium on the macroscale. This modeling approach still accounts for an accurate representation of the interface between the base metal and the brazing layer while neglecting second-order effects caused by anisotropic material behavior of the oligocrystalline base sheet metals. A summary of the reference configuration in terms of geometry and loading for the simulations on the microscale as well as the macroscale is given in “Appendix C”. The values reported there are used if not stated differently.