1 Introduction

Traditional homogeneous materials find it difficult to meet performance needs as high-tech industries like nuclear energy, defense, astronautics, and aeronautics evolve quickly. Introduced by Japanese researchers in 1984, functionally graded materials (FGMs) provide a solution by gradually changing the material properties of individual components, usually combining ceramics and metals [1, 2]. By guaranteeing a smooth transition in material composition, FGMs reduce stress singularities at material interfaces, which are a problem for typical composites, particularly when exposed to high temperatures. This improves bonding strength and lowers thermal residual stresses [2]. Thermal barriers, nuclear reactors, optical devices, implants, and energy systems are just a few of the many industries that have used FGMs [3,4,5]. Although the theory and application of FGMs in plates and shells have advanced significantly, there are still few studies that are especially concerned with functionally graded beams (FGBs) [6].

2 Literature review

Li [6] Present a new unified approach for analyzing the static and dynamic be- haviour of FGM beam. Forth order governing differential equation is derived, for a cantilever static result of stress distribution and deflection is presented. Frequency equation is given and free vibration of FGM beam analyzed. Numerical results are given by power law for FGM beam. Mesut Simsek [7] Present 2D FGMs beam to find out the buckling of beam for different boundary conditions. He assumed that material properties vary in two directions according to power law. Critical buckling load of 2D FGM beam is obtained by using Timoshenko beam theory (TBT). Clamped–clamped (CC), Clamped-simple (CS), simple-simple (SS) and Clamped-Free (CF) boundary conditions are analyzed. For comparison buckling load of Euler–Bernoulli beam theory is also calculated. Li and Batra [8]. Analysis analytical relation between critical buck- ling load for FGM Timoshenko beam and homogeneous Euler Bernoulli beam which is subjected to axial compressive load, this relation derived for CC, SS and CF edge condition. Transcendental equation is also derived to find critical buckling load for Timoshenko beam. Here Youngs modulus, E, and Poisson’s ratio, ν, are assumed vary in thickness direction. Simsek and Yurtcu [9]. In this paper the object is based on static bending and buckling of a functionally graded nanobeam are examined based on the nonlocal Timoshenko and Euler Bernoulli beam theory when this type of work would be done the conclusion found that the new nonlocal beam model produces large deflection and smaller buckling load than the classical (local)beam model. Giunta et al. [10] proposed several axiomatic re- fined theories for the linear static analysis of beams made of material whose properties are graded along one or two direction. After many types of searches the result show that the proposed unified formulation yields the complete three-dimensional displacement and stress field as long as the appropriate approximation is considered. The accuracy of the solution depends upon the geometrical parameters of the beam and the loading condition. Park and Gao [11] proposed a new model for the bending of the Bernoulli Euler beam is developed using modified couple stress theory. A variational formulation based on the principal of minimum total potential energy is employed. When this project would be done the result show that the bending rigidity of the two model is very significant when the beam thickness is small, but is diminishing with the increase of the beam thickness. Nateghi et al. [12] present that buckling analysis of functionally graded micro beam based on modified couple stress theory is presented. Three different beam theories i.e. classical the conclusion found that size dependency of FG micro beams differs from isotropic homogeneous micro beams as it is a function of power index of material distribution. Filippi et al. [13] show that 1D Carrera Unified Formulation (CUF) is here used to perform static analysis of functionally graded (FG)structure. The result compared with 1, 2 and 3-D solution both in term of displacement and stress distribution. The comparison conforms that the 1-D CUF element are valuable tools for the study of FG structure. Chen et al. [14] present that the elastic buckling and static bending analysis of shear deformable functionally graded (FG) porous beams based on the Timoshenko beam theory. Then conclusion show that influence of varying properties distribution on the structural performance is highlighted to shed important insights into the porosity design to achieve buckling behavior. Eroglu [15] present that arbitrarily large in-plane deflections of planar curved beams made of Functionally Graded Materials (FGM) are examined. The result found that the arbitrarily large in-plane deflections of planar curved beams made of FGM. Benatta et al. [16] High-order flexural theories for short functionally graded symmetric beams under three-point bending are presented. The material properties of the plate are assumed to be graded continuously in the direction of thickness. The variation of the material properties follows a simple power-law distribution in terms of the volume fractions of constituents within a symmetric laminated beam to create a functionally graded material (FGM). The formulation allows for warping of the cross-section of the FGM beam and eliminates the need for using arbitrary shear correction coefficients as in other theories. Based on higher-order shear deformation theories, the governing equations are obtained using the principle of virtual work (PVW). The justification for use of higher-order shear deformation theories is established for short and FGM beams where cross-sectional warping is predominant (Table 1).

Table 1 Analysis of latest research work on functionally graded material-based beam
Full size table

Comparing functionally graded beams (FGBs) to plates and shells, which have been thoroughly studied in elasticity, vibration, and thermal analysis, highlights the need for more thorough work to support their application in real-world structures. Advanced numerical simulations and detailed parametric studies on FGBs are still scarce, and the few analytical models that exist for FGBs—such as Sankar's Euler–Bernoulli beam formulation [6]—mostly address simplified loading and boundary conditions. The main objective of the work is to simulate of “Alumina-Aluminium” beam and find out total point displacement, total surface displacement, von mises stress and principal stresses. This work is more advance because analysis for total point displacement, total surface displacement, von mises stress and principal stress for length to thickness ratio l/h = 10 is done for functionally graded materials in this work which was not done previously. Prior work on FGM beam analysis is done for displacement and stress analysis but they did not do this analysis for clamp-simple, simple-simple and clamp-free supported condition, in this work total point displacements at four corner points as C1, C2, C3 and C4, total surface displacement, Von mises stress and principal stresses I, II and III is simulated for clamp-simple, simple-simple and clamp-free supported condition. This study uses COMSOL to provide precise numerical insights, even though the reported trends are consistent with known FGM behavior. It draws attention to how boundary constraints and material gradation affect mechanical reaction. A methodical simulation methodology that is helpful for engineering design and optimization is provided by the work.

Innovative contributions in fields such as advanced material gradation, multiphysics coupling, and design optimization can be made by the numerical analysis of Functionally Graded Material (FGM) beams using COMSOL. It offers information on dynamic responses, nonlinear analysis, and thermal–mechanical behaviors. There is substantial publication value added by research on FGM beam optimization for practical uses in industries like aerospace and biomedicine. Furthermore, combining experimental validation with multiscale modeling improves computational accuracy and usefulness. Computational engineering, structural design, and material science are all advanced by these developments.

3 Material and methods

3.1 Model and drawing

There is a beam of uniform rectangular cross-section which have depth h, length l, and made of an isotropic and linear elastic functionally graded material (FGM), with rectangular Cartesian coordinate axes x-axis along the geometric centroidal axis and the z-axis in the thickness direction and it described in Fig. 1. Furthermore, we assume that Youngs modulus (E) and Poisson’s ratio (ν) continuously vary in the thickness direction z according to the power-law [8]. Power for material property is given in Eq. (1).

Fig. 1
figure 1

Schematic sketch of a FGM beam with coordinate axes

Full size image
$$p\left(z\right)=\left({p}_{c}-{p}_{m}\right){\left(\frac{z}{h}+\frac{1}{2}\right)}^{n}+({p}_{m})$$
(1)

Exponential distribution

$$E(z)={E}_{0}{e}^{kz},\nu (z)={v}_{0}{e}^{lz}$$
(2)

k, l: Gradient coefficients (positive or negative depending on whether properties increase or decrease with z).

Sigmoid function distribution

$$E(z)={E}_{1}+\frac{{E}_{2}-{E}_{1}}{1+{e}^{-a(z-\frac{h}{z})}},v(z)={v}_{1}+\frac{{v}_{2}-{v}_{1}}{1+{e}^{-b(z-\frac{h}{z})}}$$
(3)

This form models smooth transitions between two materials (e.g., metal–ceramic). a, b: Controls steepness of the transition. E1, E2, ν1, ν2: Material properties at bottom and top surfaces.

Piecewise linear distribution (for laminated composites)

$$E\left(z\right)=\left\{\begin{array}{c}{E}_{1}, 0\le z<{h}_{1}\\ {E}_{2}, {h}_{1}\le z<{h}_{2}\\ .\\ .\\ .\\ {E}_{n}, {h}_{n-1}\le z<h\end{array}\right\}$$
(4)
$$v\left(z\right)=\left\{\begin{array}{c}{v}_{1 }, 0\le z<{h}_{1}\\ {v}_{2}, {h}_{1}\le z<{h}_{2}\\ .\\ .\\ .\\ {v}_{n}, {h}_{n-1}\le z<h\end{array}\right\}$$
(5)

Suitable for discrete layers with distinct properties.

3.2 Used material properties

Young’s modulus and Poisson’s ratio are two important physical properties of functionally graded materials (FGM) shown in the table; however, how they vary depends on the volume percentage of the constituent materials, such as metals and ceramics. How these properties vary over the beam thickness is determined by the volume fraction, which is usually expressed as a function of location (e.g., power-law distribution). Power low is given by Eq. (1) “Alumina-Aluminium” is used as FGM in this simulation study. The mechanical properties of Li&Batra are given in Table 2 [8].

Table 2 Mechanical properties of FGMs
Full size table

3.3 COMSOL multiphysics 4.2

This paper used FEM software COMSOL Multiphysics because, in FGM beam material properties vary across the beam thickness and it becomes difficult to model the problem analytically but FEM can easily model any difficult irregular geometry so I choose this software. COMSOL Multiphysics is a cross-platform finite element analysis, solver and Multiphysics simulation software. It allows conventional physics-based user interfaces and coupled systems of partial differential equations (PDEs). COMSOL provides an IDE and unified workflow for electrical, mechanical, fluid and chemical applications. An API for Java and Livelink for MATLAB may be used to control the software externally, and the same API is also used via the Method Editor. COMSOL contains an App Builder which can be used to develop independent domain-specific apps with custom user- interface. Users may use drag-and-drop tools (Form Editor) or programming (Method Editor). Specific features may be included from the model or new features may be introduced through programming. It also contains a Physics Builder to create custom physics-interfaces accessible from the COMSOL Desktop with the same look-and-feel as the built-in physics interfaces. COMSOL Server is the software and engine for running simulation apps and the platform for controlling their deployment and distribution. User developed apps can be run in COMSOL Server through web browsers or a Windows- installed client. COMSOL was started in July 1986 by Svante Littmarck and Farhad Saeidi at the Royal Institute of Technology (KTH) in Stockholm, Sweden, and was then known as FEMLAB.

3.4 Modeling procedure

Set-up Model environment: In new window we start Model Wizard in COMSOL Multi-physics. Now we select space Dimension like 1-D, 2-D and 3-D after it we select physics, there are structure mechanics, heat transfer, fluid flow and AC/DC etc. available. We select structure mechanics physics and linear elastics for study. Create Geometric Object: -After set-up Model environment we create geometric object. There are two options to create Geometry, first we can use anyone COMSOL Multiphysics Drawing second, we can import geometry from external source like AutoCAD, SolidWorks, Inventor etc. Specify Material Properties: After creating object, we specify material properties. COMSOL material browser have several Build-in materials available to use. In the addition of material library have many materials with several properties. Define Physics and Boundary conditions: After specify material properties next step would be to define physics. We define physics for structure mechanics problem. Create the Mesh: After define physics we create mesh. There are several options to create the mesh like User controlled Mesh and Physics Controlled Mesh. Running Simulation: In simulation we select study type, if select structure mechanics problem then stress will generate. Post-processing of Result: Post processing of structure mechanics problem.

4 Geometry and type of beam

Fig (a) is representing the geometry of the beam here C1, C2, C3 and C4 are the corner points of the beam. Fig(b) showing meshing of the beam generated in the software and Fig(c), (d) and (e) are showing three conditions of the that are beam clamp-simple, simple-simple and clamp-free respectively. The FGM beam geometry in this study was discretized using a free triangular mesh approach, which provides flexibility and accuracy in capturing the stress gradient, especially close to boundaries and interfaces. Unlike linear elements, Lagrange quadrilateral finite elements were used because they provide second-order interpolation functions for increased accuracy in the stress and displacement fields. To guarantee the accuracy and convergence of the solution, mesh refinement checks were performed as shown in Fig. 2b. Mesh sensitivity was examined by fine-tuning the mesh and tracking variations in von Mises stress and maximum displacement. When additional refinement resulted in less than 2% variation in these key outputs, the solution was considered mesh-independent.

Fig. 2
figure 2

Geometry of beam and type of beam with loading condition

Full size image

Meshing ensured accurate stress and displacement capture, especially in high-gradient regions. Boundary conditions realistically represented physical constraints and loads. Together, they ensured the reliability and accuracy of the simulation results.

5 Model validation

Below figures show simulated model results and graph for dimensionless critical buckling load from model and Li and Batra [8]. By comparing these it is explained that both the graphs (model and Li and Batra) match each other with a small error. It is found that the RMSE is varying between 0.396 and 1.224 and average value is 1.013 for CS beam in Fig. 3d (see Table 3), the RMSE varying between 0.0521 and 0.510 and average value is 0.330 for SS beam in Fig. 3e (see Table 3), the RMSE varying between 0.040 to 0.328 and average value is 0.146 for CF beam in Fig. 3f (see Table 3). Both the results are matched with the small acceptable error and hence the model is validated which is used for further analysis. The validation of critical load predictions for functionally graded material beams yielded RMSE values in the range of 0.04–1.224, consistent with benchmark studies reported by Reddy [20], Praveen and Reddy (1998), and Yang and Chen (2008), confirming the accuracy of the proposed COMSOL-based analysis.

Fig. 3
figure 3

Simulated model and model validation of FGM beam and with Li and Batra

Full size image
Table 3 Model validation of FGM beam for dimensionless critical buckling load
Full size table

6 Simulation study of alumina-aluminium beam for different conditions

Model is validated for critical buckling load with Li and Batra. Now total point displacement, total surface displacement, Von Mises Stress and principal stresses are simulated using FEM software COMSOL Multiphysics in the following section with the above model.

6.1 Total point displacement (TPD)

Because the CS, SS, and CF beams have different boundary conditions and beam designs, the Total Point Displacement (TPD) results show significant variances in the structural behavior under stress. With readings for point C3 ranging from 0.4399 to 0.5754 (average: 0.5128) and point C4 ranging from 0.3072 to 0.466 (average: 0.3725), the TPD values for the CS beam are modest. These numbers imply that the beam has a fairly balanced stiffness despite a certain amount of flexibility. The coupled supports' limitations, which prevent excessive movement while permitting certain flexural action, are responsible for the comparatively low to moderate displacement values. With TPD values ranging from 0.6409 to 0.8193 (average: 0.7399), the SS beam exhibits higher fluctuation, particularly at point C3. This suggests better flexibility. Point C4, on the other hand, exhibits much lower displacement values (0.0145–0.2122, average: 0.1242), which may indicate localized stiffness or a support situation that limits mobility there. According to this contrast, the basic support condition permits higher overall deflection, especially at mid-span regions (like C3), but it may also have limitations at the ends or close to C4 that limit movement. On the other hand, the TPD values of the CF beam are consistently high at C3 (2.8042–2.8092, average: 2.8070) and C4 (2.7979–2.8051, average: 2.8009). These extraordinarily high scores imply a high degree of adaptability and perhaps little restriction. The CF arrangement might be an example of a cantilevered or virtually free condition, in which significant displacements occur under load due to insufficient support. A uniform reaction across the measured sites is also suggested by the range's little change, which is probably the result of stable loading or structural characteristics. Graph is shown in Fig. 4c (see Table 4).

Fig. 4
figure 4

Total Point Displacement (TPD) for a CS beam, b SS beam and c CF beam

Full size image
Table 4 Total point displacement (TPD) for CS beam, SS beam, and CF beam
Full size table

The TPD analysis shows that beam deformation behavior is mostly controlled by structural support conditions. CS beams are appropriate for applications needing a compromise between strength and deformability because of their well-distributed stiffness and moderate flexibility. Despite being more flexible, SS beams exhibit localized stiffness changes that could compromise durability and stress distribution. Without additional support mechanisms, CF beams may not be ideal for load-bearing applications due to their high flexibility and low stiffness, as indicated by their maximum displacement values. To guarantee structural efficiency and safety, it is crucial to comprehend these displacement characteristics in order to make well-informed design decisions.

6.2 Total surface displacement (TSD)

Because of balanced support circumstances, the CS beam's TSD ranges from 12.8713 to 12.8938 (average 12.8841), showing mild and uniform displacement. With TSD values ranging from 13.7787 to 13.8827 (average 13.8362), the SS beam exhibits greater flexibility and less constraint. As a result of its more limited support, the CF beam exhibits the lowest TSD, ranging from 10.1886 to 10.2005 (average 10.1965), indicating great stiffness and strong resistance to deformation in Fig. 5c (Table 5).

Fig. 5
figure 5

Total surface displacement (TSD) for a CS beam, b SS beam and c CF beam

Full size image
Table 5 Total surface displacement (TSD) (m3) for CS beam, SS beam, and CF beam
Full size table

According to TSD research, CS beams have a balanced performance, SS beams are the most flexible, and CF beams are the stiffest and most stable. Support conditions should direct beam selection based on structural requirements since they have a substantial impact on displacement behavior.

6.3 Von Mises stress (VMS)

Below figures show the Von Mises stress (Pa) from model. It is found that the VMS is varying between 0.02 and 194 MPa for CS beam 0.012 and 59.1 MPa for SS beam 0.004–13.8 MPa for CF beam in Fig. 6 (Table 6).

Fig. 6
figure 6

Von Mises stress (VMS) for CS beam, SS beam and CF beam

Full size image
Table 6 Von Mises stress (N) for a CS beam, b SS beam, and c CF beam and l/h = 10
Full size table

Actual significant value of von mises stress for FGM is vary from 2.05E + 04 to 1.94E + 08 according to power law, where value of power index varies from 0 to 10, which is more significant for FGM beam as properties vary metal to ceramic as shown in Table 6. Value of von mises stress at power index 100 is just numerical representation because at this value FGM do not show desire properties only single material exist, another material has negligible volume to form composite.

6.4 Principal stress (PS)

As shown in Fig. 7a–c value of principal stress is vary between − 46.6 and 168 MPa for CS beam − 12.2 to 53 MPa for SS beam and − 10.1 to 5.52 MPa for CF beam for power index value 0–10.

Fig. 7
figure 7

Principal Stress (PS) for a CS beam, b SS beam and c CF beam

Full size image

Actual significant value of principal stress for FGM is vary from 1.14E + 04 to 1.68E + 08 according to power law, where value of power index varies from 0 to 10, which is more significant for FGM beam as properties vary metal to ceramic (Table 7). Value of principal stress at power index 100 is just numerical representation because at this value FGM do not show desire properties only single material exist, another material has negligible volume to form composite. By integrating principal stress analysis into the design and evaluation process, engineers can enhance the reliability and efficiency of FGM beams in a variety of applications like- Failure Analysis and Prevention, Material Optimization, Design of Biomedical Implants and Thermal Stress Management etc.

Table 7 Principal stress (PS) for a CS beam, b SS beam, and c CF beam and l/h = 10
Full size table

7 Results

The CS beam experienced a 30.8% increase in TPD at C3 and a 34.1% decrease at C4; the SS beam experienced a 27.9% increase at C3 and a 1363.4% decrease at C4; the CF beam experienced negligible changes (≤ 0.2%). TSD increased marginally as the power index increased from 0 to 10—0.09% (CS), 0.59% (SS), and 0.03% (CF).

8 Conclusion and future scope

The study demonstrated good agreement with Li and Batra and validated the accuracy of critical buckling load estimates for FGM beams under point load. With boundary circumstances, structural reactions like TPD, TSD, VMS, and PS differed significantly. Through thickness, the power-law gradation successfully represented material behavior. In summary, the study discovered that when the power index increased from 0 to 10, the Von Mises Stress (VMS) for the CS beam become by 946 × and 476 × for the SS beam, while the Total Point Displacement (TPD) for the CS beam increased by 30.8% at C3 and reduced by 34.1% at C4. The analysis shows that the sensitivity of stress behavior to material gradation in FGM beams is highlighted by the conclusion that increasing the power index greatly amplifies stress responses, especially Von Mises and primary stresses, while displacement variations remain low.

The simulation findings, which include primary stresses, von Mises stress, total point displacement, and total surface displacement, provide important information for real-world applications in biomedical and aerospace engineering. For aerospace constructions to maintain structural integrity under severe loading circumstances, displacement must be kept to a minimum and stress distribution must be optimized. In order to avoid yielding failure, the von Mises stress results can help guide the cross-sectional design and material selection of aircraft components. Similar to this, managing fundamental stresses and deformation is crucial for biomedical applications like dental prosthesis and orthopedic implants in order to guarantee long-term dependability and compatibility with biological tissues. Engineers can improve implant performance and patient safety by customizing designs that more closely resemble the mechanical response and stress distribution of natural tissues by examining these parameters in functionally graded materials.

A solid numerical basis was ensured by validating our model against the critical load standard of Li and Batra. Although experimentally validating the displacements and stresses was beyond the scope of this work, our findings are consistent with the patterns found in the literature. The theoretical basis for upcoming empirical research is provided by this study. In future studies, we intend to include experimental comparisons and create more straightforward analytical models to go along with the simulations. FGMs are extensively utilized in engineering, biomedical, aerospace, and automotive applications due to their exceptional directional and high-temperature characteristics. Even though FGM beams have been the subject of much research, there are still important areas that need attention. Future research can examine strain, volumetric strain, and stress invariants across various forms of FGM, as well as the impacts of changing density.

There are various FGMs combination of metal and ceramic to improve the mechanical properties. Some of them possible FGMs named FGM1, FGM2 and FGM3 are analyzing and will be completed shortly.