1 Background

In recent years, to reduce weight, increase fuel efficiency, and lower emissions, lightweight materials with high specific strength and stiffness have been required for aerospace applications. This has led to the innovation of alloy materials with improved microstructural and mechanical properties to meet the challenging requirements for aerospace components [1,2,3]. Alloying has been employed to give materials desirable qualities. It entails mixing a major element with a minor amount of a secondary element. To generate novel materials known as high-entropy alloys (HEAs), a new alloying technique that combines many principal elements in optimized concentrations has gained research interest [4, 5]. Although only a small part of the multi-dimensional compositional space has been studied, it is nearly infinite. However, a few high-entropy alloys have already been shown to have amazing qualities that surpass those of conventional alloys. The AlCoCrFeNiCu HEAs compositional design has six elements with a configurational entropy of 1.79 R and atomic concentrations between 5 and 35%, so the high-entropy alloy (HEA) system is thermodynamically favorable. Boltzmann’s theory shows the role of entropy on alloys; while, its core effects are one of the factors that denote stability [6,7,8,9]. The sluggish diffusion effect contributes to the growth of amorphous structures or monocrystalline structures [10]. The lattice distortion effect dictates the properties of the alloy; while, the high entropy effect contributes to the merging of alloys having compatible elements in a composition and comprising solid solution phases or single phases [11, 12]. The cocktail effect relies on the contribution of each principal element in the composition, resulting in the bulk property of the alloy composition. These core effects improve the electrochemical, tribological, thermal, and mechanical properties of HEA materials, especially. However, most of the AlCoCrFeNiCu high-entropy alloy compositions contain single-structured or predominant Body-Centered Cubic (BCC) phases and/or intermetallic compounds, which may make the alloy brittle and difficult to machine or test for tension, especially via advanced manufacturing techniques [5]. The aerospace industry has a lot of components built via the three-dimensional (3D) additive manufacturing (AM) technique, which builds parts layer by layer and has the potential to generate parts that are stronger, lighter, and better engineered than traditional parts [13, 14]. Additionally, AM technology can produce unique or replacement parts from any location, at any time, and more quickly than previous procedures [15, 16]. However, the high contact area between layers causes worse performance under tension if the part's consecutive layers lack cohesiveness. As with many design methodologies, AM optimization is traditionally thought of as a process parameter optimization, especially when directed toward topology optimization in the generation of the geometry through a computer-aided design (CAD) model [17,18,19]. To determine the stress–strain mechanism of laser-deposited high-entropy alloy test samples, the amalgams are subjected to tensile testing. In many industries, such as construction, aerospace, automotive, machinery, biomedical implants, and consumer goods, tensile strength is essential. For structural integrity, cables, steel beams, and concrete reinforcement bars are needed in buildings and bridges. For structural integrity, the fuselage and wings of an aircraft need to have extraordinary tensile strength. For safety, airbags and seatbelts must be able to withstand extremely high tensile forces. Using the phase-field method, Li et al. [30] simulated coherent BCC/B2 microstructures in Al–Ni–Co-Fe–Cr HEAs based on BCC using COMSOL. Precipitate morphology was influenced by the lattice misfit and Young's moduli anisotropy difference, as demonstrated by the simulations' results, which were in agreement with the experimental findings. Additionally, the study covered the coarsening behavior of precipitates and offered a method for forecasting composition design and microstructural evolution. FeCoNi(Mn–Si)x HEAs were created by Sahu et al. [31] using mechanical alloying. X-ray diffraction (XRD), Scanning Electron Microscope (SEM), and Transmission Electron Microscope (TEM) were used to examine how the Mn and Si content affected the alloy's magnetic behavior and structure. The surface morphology was found to be affected by high Mn and Si content, resulting in a shift from the BCC phase to the Face Centered Cubic (FCC) phase. The magnetic flux density on the transformer core was measured with the finite element method (FEM) and COMSOL Multiphysics software to evaluate the performance of the proposed HEAs. The magnetic flux density on the transformer core was measured with the FEM and COMSOL Multiphysics software to evaluate the performance of the proposed HEAs. For laser cladding AlCrNiTiNb high-entropy alloy (HEA) coatings on titanium alloy (Ti6Al4V) substrates, a three-dimensional FEA model was created by Kayane et al. [32]. Because of its great strength but low hardness, Ti6Al4V is used in the automotive, aerospace, and marine industries. Using a laser cladding technique, the study synthesized AlCrNiTiNb HEA hard coating with an equi-atomic ratio. Heat transfer in solids modules was modeled using COMSOL Multiphysics 5.3a, which revealed a dendritic and interdendritic structure in the AlCrNiTiNb alloy coating. Solid solution phases were found to be both ordered and disordered by X-ray diffraction, and intermetallic compounds were absent. The combination of the FCC and BCC solid solution phases greatly enhanced the coating's microhardness. The coating exhibits strong metallurgical bonding and a dendritic and interdendritic structure. The coating has a higher microhardness than the substrate.

In the literature, the microstructure, magnetic flux density, and heat transfer of HEAs have been simulated using COMSOL, with limited studies on the tensile properties of the HEAs fabricated via additive manufacturing. Hence, due to geometrical restrictions and difficulties experienced in building specific dog-bone shapes of alloy samples with brittle dominant BCC structures via additive manufacturing, computational simulation can be applied as a possible solution to determining the tensile properties of these alloys. The complexity, material properties, validation, computational efficiency, software, and user-friendliness of a proposed tensile loading model for laser-deposited HEAs using COMSOL Multiphysics are taken into account to assess the model's uniqueness and high performance. To improve prediction accuracy, temperature-dependent material properties and the unique composition of HEA were taken into consideration. COMSOL's features were used for a more approachable interface, simpler customization, and wider research community adoption.

There are limited reports in the literature on laser-deposited high entropy alloys available on this subject. In this study, the COMSOL finite element model was used to simulate the mechanical behavior of an AlCoCrFeNiCu high-entropy alloy sample to solve the structural mechanics challenge of the additive manufacturing process. Computational analysis of AlCoCrFeNiCu HEA composition is currently understudied in the literature; hence, this study may aid in the creation of a more successful modeling strategy for HEA systems. From this study, new functionalities within COMSOL Multiphysics for tensile loading simulations, such as user-defined material properties or coupling with other physical phenomena like heat transfer, may also be reviewed and explored for other HEA compositions.

2 Methods

The composition of AlCoCrFeNiCu HEA powder was acquired from F.J. Brodmann & CO., L.L.C, New Orleans, USA. The laser engineering net shaping (LENS) system was used to fabricate the HEA powder on a preheated A301 steel substrate using optimized parameters [20]. The beam diameter of 2 mm, the powder feed rate of 2 rpm, the 50% overlap, and the layer thickness of 0.5 mm were used for the direct energy deposition (DED). The laser power ranges from 1200 to 1600 W, and the scan speed ranges from 8 mm/s to 12 mm/s. The elastic modulus was measured using Anton Paar TTX-NHT3 Nanoindentation tester. With geometry created in SolidWorks and imported via the COMSOL CAD Import Module, a 3D Solid Mechanics physics model was created using COMSOL Multiphysics 5.4. The AlCoCrFeNiCu HEA dog-bone sample was modeled as a homogenous, linearly elastic component with the dog-bone’s left end designated as a fixed constraint boundary. The right-hand side of the dog-bone’s sample received point boundary loads in the y direction. Using the solver's default settings, a stationary analysis was carried out on a fine physics-controlled mesh and an analysis with the stationary study was performed using the intended load range of 10 kN–20 kN in the default solver setting.

2.1 Equations

Three equations such as equilibrium balance, a constitutive relation between stress and strain, and a kinematic relation between displacement and strain were needed to model the high-entropy alloy sample. The equilibrium equation, which is Newton's second law in tensor form, is represented in Eq. 1 [23]:

$$ \Delta \cdot \sigma + F_{v} = \rho {\text{\"u }} $$
(1)

where \({F}_{v}\) is the force per volume, \(\sigma \) is the stress, ü is the acceleration and \(\rho \) is the density. For static analysis, the right side of this equation equals 0. The constitutive equation that links the stress tensor to strain is the generalized Hooke's law stated as Eq. 2 and expanded as Eq. 3 [23]:

$$\sigma =C :\epsilon $$
(2)
$$\sigma -{\sigma }_{0}=C:(\epsilon -{\epsilon }_{0}-{\epsilon }_{\text{inel}})$$
(3)

where the semicolon is the double dot tensor, \({\sigma }_{0} and {\epsilon }_{0}\) is the initial stress and strain, respectively. C is the elasticity tensor in fourth order and \({\epsilon }_{\text{inel}}\) is the inelastic strain all equal to zero. The elastic tensor is reduced to a 6 × 6 elastic matrix for isotropic high entropy alloys using Eq. 4 [23]:

$$ \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {2\mu + \lambda } & \lambda & \lambda \\ \lambda & {2\mu + \lambda } & \lambda \\ \lambda & \lambda & {2\mu + \lambda } \\ \end{array} ~~\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {~~~~\begin{array}{*{20}c} 0 & {~~~~~~~~0} & {~~~~~~~0~~~~~~~} \\ 0 & {~~~~~~~~0} & 0 \\ 0 & {~~~~~~~0} & 0 \\ \end{array} ~~~~~\begin{array}{*{20}c} \mu & 0 & 0 \\ 0 & \mu & 0 \\ 0 & 0 & \mu \\ \end{array} } \\ \end{array} } \right] $$
(4)

where the \(\mu \) and \(\gimel \) are the Lamé constants, \(\upsilon \) is the Poisson’s ratio and E is the elastic modulus of the AlCoCrFeNiCu high-entropy alloy. Table 1 lists the material properties; while, the kinematic relationship between the strain \(\epsilon \) and the displacement \(\mathcalligra{u} \) in tensor format are expressed in Eq. 5 [23]:

$$\epsilon =\frac{1}{2}\left[\nabla \mathcalligra{u}+(\nabla \mathcalligra{u}{)}^{T}\right]$$
(5)

where the T is the tensor transpose; while, small deformations in a very high order terminology are negligible but the strain reduces to a Cauchy infinitesimal strain tensor \({\epsilon }_{ij}\) shown in Eq. 6 [23]:

Table 1 Material Parameters of the AlCoCrFeBiCu High Entropy Alloys used for the Simulation [20]
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$${\in }_{ij}=\frac{1}{2}\left[\frac{\delta {\mathcalligra{u}}_{j}}{\delta {x}_{i}}+\frac{\delta {\mathcalligra{u}}_{i}}{\delta {x}_{j}}\right]$$
(6)

3 Results

The AlCoCrFeNiCu High-Entropy alloy dog-bone geometry with dimensions in meters is shown in Fig. 1a; the influence of the boundary load of 10 kN is displayed in Fig. 1b.

Fig. 1
figure 1

a Geometry b Effect of the 10 kN Boundary Load

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Using the solver's default settings, a stationary analysis was carried out on a fine physics-controlled mesh in Fig. 2. The displacement magnitude at 2.76 m and the resulting energy per volume or energy density at 2.31 × 107 J/m3 are represented in Fig. 3a and 3b, respectively. The test region experiences uniform stress, as anticipated. The total displacement is zero because the sample is held in place without any movement [24]. The ratio of the change in volume to the original volume represented as the volumetric strain, is represented in Fig. 4a. The Von Mises stress in Fig. 4b at 8.83 N/m2 shows when the alloy will yield or fracture. Hence, if the value is greater than the yield limit under tension, the alloy is expected to yield [25]. A contour plot of the first and second major strains is shown in Fig. 5.

Fig. 2
figure 2

COMSOL Multiphysics Fine Mesh of the AlCoCrFeNiCu High Entropy Alloy Test Sample

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Fig. 3
figure 3

a Displacement Magnitude (m) b Stored Energy Density (J/m.3)

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Fig. 4
figure 4

a Volumetric Strain b Von Mises Stress

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Fig. 5
figure 5

First and Second Principal Strain Contour Plot for the AlCoCrFeNiCu High Entropy Alloy with A Load of 10 N, Showing the Complexity of Strains near the Sample Grip Ends and the Test Region with a Uniform Center

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Temperature dependence of the HEA is relevant because it aids in the prediction of failure mechanisms, residual stresses, and the strength and ductility of the coating. Comprehending these variables aids in guaranteeing that the coating can tolerate anticipated loads, forecasting the progression of residual stresses, and projecting the coating's lifespan and modes of failure.

A probe plot showing the total displacement of the deformed shape of the alloy at a certain load is shown in Fig. 6a, where the correlation efficiency is 1. The measured first principal stress and volumetric strain are shown in Fig. 6b for the sample.

Fig. 6
figure 6

Line Graph Showing a Probe Plot b The First Principal Stress (N/m2) Vs the Volumetric Strain Curve

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The AlCoCrFeNiCu HEA solid sample in Fig. 7 was accurately weighed to 1.5 g and placed into the sample chamber and the lid was closed shown.

Fig. 7
figure 7

Laser-Deposited AlCoCrFeNiCu High Entropy Alloy Sample

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From Table 2, the gas pycnometer determined the volume at 0.2135 cm3 and the density was given as 7.026 g/cm3. Which is less than stainless steels, Inconel 738LC and other high temperature materials with densities between 7.3 g/cm3 and 9 g/cm3 [21, 22].

Table 2 Density Measurements and Processing Parameters for the As-Built AlCoCrFeNiCu High Entropy Alloy Sample
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4 Discussion

The modulus of the AlCoCrFeNiCu High-Entropy alloy sample was derived via Nanoindentation tests in previous studies [20]. The density measurements using an Anton Paar Single Station Gas Pycnometer for True Density; Ultrapyc 1200e were made after laser metal deposition assisted with preheat (LMDAP) was used. Then helium was purged through the chamber for 9 or more minutes to remove moisture and air from the chamber. The expansion and sample chambers were closed for helium to be introduced again into the sample chamber at a preset pressure of \(\sim \) 17 Pa. The system used about 250 to 300 s to stabilize before recording the chamber pressure, P1, then the valve was opened again to let out the pressurized helium from the sample chamber to flow into the expansion chamber to create another pressure P2 in about 300 s. The volume of the sample is calculated from Eq. 7 [26].

$${V}_{\text{sample}}={V}_{\text{Cell}}+ \frac{{V}_{\text{exp}}}{\frac{{P}_{1}-{P}_{a}}{{P}_{2}-{P}_{a}}-1}$$
(7)

The sample cell volume is \({V}_{\text{Cell}}\), the sample volume is \({V}_{\text{sample}}\), the expansion cell volume is \({V}_{\text{exp}}\), and the ambient pressure is Pa, respectively. The density is calculated from the measured volume and the sample weight. The characterization of the density measurements of the as-built AlCoCrFeNiCu HEAs was based on pressure–volume relationships using Boyle’s and Mariotte’s law. The elastic modulus and density that were extracted through experimental analysis demonstrated that this simulated method via COMSOL is capable of measuring material properties with accuracy for any high-entropy alloy sample, which was predicted by the model results.

A fixed constraint boundary was applied to one end of the sample, which restricted the degree of freedom over all the assigned objects. This constraint models the geometry connected to a rigid body or a boundary load. Stress concentrations are expected to occur in regions very close to the assigned face of load application [27]. Hence, a mesh was applied to smoothen the stress gradients and extract the exact stress values at different concentration points. While a boundary load of 10 kN was applied to the other end using a parametric sweep while keeping the alloy linearly elastic. The maximum ultimate tensile strength of the model was approximately given as 8.46 N/m2, which is the maximum stress the AlCoCrfeNiCu HEA alloy can withstand before failure. Typically, an alloy with a high ultimate tensile strength (UTS) can withstand a lot of force per unit area. This is a weak and potentially broken HEA due to low stress, possibly due to brittleness [28]. Energy loss was not accounted for, which is one of the limitations of this study because the temperature distribution, microstructure, and mechanical properties of the laser-deposited HEA may have been enhanced by accounting for energy loss in the model. This could lead to larger grain sizes and residual stresses, which could affect the performance of the coating under load. Hence, further studies are recommended because gaining insight into the reasons behind this particular composition's low strength could help to enhance HEA production techniques and comprehend how element ratios impact mechanical properties.

Nonetheless, this work investigates the under researched topic of using computational analysis of AlCoCrFeNiCu HEA composition in literature using COMSOL Multiphysics. From the contour map, it was obvious that there was a uniform center test region, inches long, where strain gages may be installed for precise strain measurements, despite the complicated local stresses near the grips (ends). The measured strain was linear under the full-applied load range of 10 kN showing the deformation progression of the AlCoCrFeNiCu HEA. The strain curve showed the tendency of the AlCoCrFeNiCu HEA to deform when uniformly loaded in all directions. The first principal stress expressed the maximum tensile stress induced by boundary load conditions; while, the volumetric strain showed the ratio of the change in the volume of the alloy to the original volume when an external load was applied [29].

5 Conclusion

Despite its simplicity, the AlCoCrFeNiCu high-entropy alloy sample's maximum tensile stress was estimated using COMSOL Multiphysics. The model was useful in studying the stress concentrations between the grab regions at the test sample ends. With a tensile strength of 0.847 Pa, the weak HEA is unlikely to support a large load and could fracture at low stress at its current state and may be suitable for lightweight fillers or dampening vibration applications. The high sensitivity correlates with the brittleness of the alloy which limits its application However, optimizing the alloy composition, applying heat treatment or severe plastic deformation may enhance the tensile properties of this alloy. Hence, the computational model made it possible to visualize phenomena like Saint–Venant’s principle in high entropy alloys, a topic that is now of research interest.