Strain-Energy-Density Guided Design of Functionally Graded Beams

Abstract

Functionally graded materials (FGMs) are revolutionizing various industries with their customizable properties, a key advantage over traditional composites. The rise of voxel-based 3D printing has furthered the development of FGMs with complex microstructures. Despite these advances, current design methods for FGMs often use abstract mathematical functions with limited relevance to actual performance. Furthermore, conventional micromechanics models for the analysis of FGMs tend to oversimplify, leading to inaccuracies in effective property predictions. To address these fundamental deficiencies, this paper introduces new gradation functions for functionally graded beams (FGBs) based on bending strain energy density, coupled with a voxel-based design and analysis approach. For the first time, these new gradation functions directly relate to structural performance and have proven to be more effective than conventional ones in improving beam performance, particularly under complex bending moments influenced by various loading and boundary conditions. This study reveals the significant role of primary and secondary gradation indices in material composition and distribution, both along the beam axis and across sections. It identifies optimal combinations of these indices for enhanced FGB performance. This research not only fills gaps in FGB design and analysis but also opens possibilities for applying these concepts to other strain energy density types, like shearing and torsion, and to different structural components such as plates and shells.

1. Introduction

Functionally graded materials (FGMs) represent a significant advancement over traditional composites, offering tailored properties for specific application needs [1,2,3,4]. They enable strategic material distribution, allowing for lighter structures without compromising strength [5], a benefit critical in the aerospace and automotive industries where weight reduction directly enhances fuel efficiency and performance. By varying material composition, FGMs provide superior mechanical properties such as increased toughness, strength, and fracture resistance [6]. Their gradient transition minimizes stress concentrations found at interfaces in conventional composites, enhancing durability. Additionally, FGMs can have customized thermal conductivity gradients [7], ideal for managing extreme temperatures, thereby reducing thermal stress and extending product life. Tailoring surface properties allows FGMs to provide increased wear resistance [8], essential in tooling, mining, and drilling. Furthermore, by adjusting material composition, FGMs improve corrosion and chemical degradation resistance [9], crucial for durability in harsh environments. This adaptability ensures component longevity and reliability while reducing maintenance costs.

The applications of FGMs are rapidly increasing in various industrial sectors such as aerospace, automotive, biomedical, energy, and electrical engineering sectors, mainly driven by the continuous advances in additive manufacturing technologies [4,10,11,12,13,14,15,16,17]. In particular, the emerging multi-material voxel-based 3D printing technology can not only produce microstructures of complicated geometry but can also precisely distribute chemical composition in FGMs [18,19,20]. In voxel-based 3D printing, the material body is voxelized with a resolution that can be as small as microns; each voxel represents a basic unit of material [20], which can consist of a single phase material or even a mixture of several phase materials. Nevertheless, advancements in the design and analysis of FGMs have not kept pace with the developments in advanced manufacturing technologies, a disparity that significantly hinders the application of FGMs in industry and engineering. There are two major deficiencies in the design and analysis of FGMs [4,20,21,22]: firstly, the currently used gradation functions are not directly related to the overall performance of FGMs, and secondly, the methods used to determine the effective properties of FGMs are inaccurate. Gradation functions play a fundamental and crucial role in the regulation of FGMs’ properties and overall behavior because they govern the distribution of phase materials and the variation in microstructure. Gradation functions currently used in the design of FGMs include linear [23,24], power law [25,26,27,28,29,30,31], sigmoid [32,33,34,35], exponential [36,37], and other mathematical functions [20,38,39]. Among these, power law functions are the most commonly used due to the additional degrees of freedom for design provided by the power indices or gradation indices [21,22,40,41,42,43]. These functions are constructed purely based on the mathematical requirements of gradation functions without considering structural functionalities and performance, which may significantly compromise the effectiveness of FGMs in achieving the desired performance. Regarding the determination of FGMs’ effective properties from given gradation functions and phase material properties, there are two main deficiencies in the existing methods. One is the use of the representative volume element (RVE), and the other is the application of analytical or semi-analytical equations derived from conventional micromechanics models. Unlike conventional composites, FGMs are inherently heterogeneous and anisotropic, regardless of the size of the material body considered. This means their properties are always dependent on location and orientation. This nature of FGMs originates from the inhomogeneous and anisotropic variations in their phase materials and microstructure, which are dictated by the gradation functions. The concept of the RVE, widely used in homogenizing and characterizing conventional composites, is theoretically not applicable to FGMs.

The effective design of FGMs relies on accurately characterizing their effective properties, essential for predicting behavior under various loading conditions. The main task in FGM analysis and characterization is to predict the effective properties such as stiffness and strength, using the gradation functions and phase material properties. Currently, analytical formulas developed from conventional micromechanics models are widely applied for this purpose, e.g., [44,45,46,47,48,49,50,51,52,53,54], among many others. The rule of mixtures is the one most widely used [21,38,55,56,57,58,59,60,61,62,63,64], mainly because of its simplicity. However, micromechanics models are generally based on special assumptions, which limits the scope of their applicability. The rule of mixtures only provides an upper bound estimate, rather than the actual value, of the effective property. Previous studies [65,66] have shown that micromechanics models, including popular ones like the rule of mixtures, the generalized self-consistent scheme, and Mori–Tanaka method, have significantly reduced accuracy when applied to composites that have a high volume fraction of inclusion and a large contrast of phase properties, which are both present in FGMs. This is because these models explicitly or implicitly assume dilute dispersion, which is apparently invalid for FGMs. In addition, none of the existing micromechanics models can handle the anisotropy present in FGMs. This limitation further compromises the accuracy of micromechanics models if they are applied to FGMs. In recent studies [67,68], two voxel-based approaches were developed for the design and characterization of composite and functionally graded materials: a micromechanics-based approach and a statistics-based approach. It was demonstrated that the statistics-based voxel approach is more reliable and accurate, primarily because it avoids the assumptions inherent in micromechanics models. Furthermore, the simulation results from this statistics-based voxel approach showed excellent agreement with the experimental data reported for quasi-homogeneous and quasi-isotropic particulate composites [65,68].

In this paper, these deficiencies in functionally graded beams (FGBs) with three phases are addressed by introducing a set of innovative gradation functions derived from the bending strain energy density within the beam. These new gradation functions are, for the first time, directly and explicitly related to the structural performance of the beam. Examples of three-phase composites in engineering include materials such as concrete and polymer–ceramic–metal composites. Subsequently, a voxel-based approach is applied to implement the design and analysis of FGBs. The superiority of the new gradation functions over the conventional ones is demonstrated by comparing their effectiveness in reducing the maximum deflection of the beams.

The remainder of this paper is structured as follows: Section 2 delves into the methodologies employed, including the development of new gradation functions and the voxel-based design approach, providing a foundation for the subsequent analysis. Section 3 presents the results of our study, showcasing the performance of functionally graded beams (FGBs) under various loading and boundary conditions and illustrating the effectiveness of our proposed gradation functions compared to conventional ones. Section 4 discusses the implications of our findings, elaborating on the significance of gradation indices in the design and performance of FGBs and highlighting potential areas for future research. Finally, Section 5 concludes this paper by summarizing the main findings and contributions of our work to the field of mechanical sciences and engineering.

2. Methods

In this section, a set of new gradation functions for FGBs is first derived based on strain energy density distribution in the beam. Then, the implementation of the new gradation functions in designing FGBs is described, utilizing the previously developed voxel-based approach. This is followed by the introduction of a method for the fair comparison of the performance of FGBs designed using both the new and conventional gradation functions.

The detailed derivation process of the new gradation functions can be found in Appendix A, and the resulting expressions are provided in Equation (1) within the coordinate system depicted in Figure A1.

$$V_1=1-{\left({\displaystyle \frac{M\left(x\right)}{M_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\right)}^{2m},\hspace{1em}{V}_2={\left({\displaystyle \frac{M\left(x\right)}{M_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\right)}^{2m}\left(1-{\left({\displaystyle \frac{2y-h}{h}}\right)}^{2n}\right),\hspace{1em}{V}_3={\left({\displaystyle \frac{M\left(x\right)}{M_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\right)}^{2m}{\left({\displaystyle \frac{2y-h}{h}}\right)}^{2n}.$$

(1)

In the above gradation functions, $M\left(x\right)$ represents the resulting bending moment at the cross-section located at $x$. Meanwhile, $M_{\mathrm{m}\mathrm{a}\mathrm{x}}$ denotes the maximum bending moment experienced by the beam. Note that the beam has a rectangular cross-section with height $h$.

In designing FGBs, the statistics-based voxel approach [67] is utilized, the rationale for which is detailed in the Section 1. A key characteristic of this approach is that the gradation functions are realized through a collective group of voxels, in contrast to the micromechanics-based approach, which uses a single voxel. Figure 1 illustrates this design process using the statistics-based voxel approach, and a detailed description is provided in Appendix B.

To facilitate a fair comparison between the newly developed gradation functions and the conventional ones, conventional gradation functions in Equation (2) are also utilized in the design of FGBs. These functions are widely adopted in the design of three-phase two-dimensional FGBs, e.g., [69,70,71], among many others. Note that Equation (2a) is applied in the design of cantilever FGBs, whereas Equation (2b) is utilized for designing simply supported or clamped–clamped FGBs.

$$U_1={\left({\displaystyle \frac{x}{L}}\right)}^m, U_2=\left(1-{\left({\displaystyle \frac{x}{L}}\right)}^m\right)\left(1-{\left({\displaystyle \frac{\left|2y-h\right|}{h}}\right)}^n\right), U_3=\left(1-{\left({\displaystyle \frac{x}{L}}\right)}^m\right){\left({\displaystyle \frac{\left|2y-h\right|}{h}}\right)}^n.$$

(2a)

$$U_1={\left({\displaystyle \frac{\left|2x-L\right|}{L}}\right)}^m, U_2=\left(1-{\left({\displaystyle \frac{\left|2x-L\right|}{L}}\right)}^m\right)\left(1-{\left({\displaystyle \frac{\left|2y-h\right|}{h}}\right)}^n\right), U_3=\left(1-{\left({\displaystyle \frac{\left|2x-L\right|}{L}}\right)}^m\right){\left({\displaystyle \frac{\left|2y-h\right|}{h}}\right)}^n.$$

(2b)

Equations (1) and (2) will be employed in the design of FGBs corresponding to the three loading and boundary conditions depicted in Figure 2.

The internal moment $M\left(x\right)$ and its absolute maximum value $M_{\mathrm{m}\mathrm{a}\mathrm{x}}$ for each of the three beams in Figure 2 are provided below.

$$M\left(x\right)=P\cdot (L-x),\hspace{1em}\hspace{1em}\hspace{1em}{M}_{\mathrm{m}\mathrm{a}\mathrm{x}}=P\cdot L.$$

(3a)

$$M\left(x\right)={\displaystyle \frac{1}{2}}q\cdot x\cdot (L-x),\hspace{1em}\hspace{1em}\hspace{1em}{M}_{\mathrm{m}\mathrm{a}\mathrm{x}}={\displaystyle \frac{1}{8}}q\cdot L^2.$$

(3b)

$$M\left(x\right)={\displaystyle \frac{1}{12}}q\cdot L^2\cdot \left(6{\displaystyle \frac{x}{L}}-6{\left({\displaystyle \frac{x}{L}}\right)}^2-1\right),\hspace{1em}\hspace{1em}\hspace{1em}{M}_{\mathrm{m}\mathrm{a}\mathrm{x}}={\displaystyle \frac{1}{12}}q\cdot L^2.$$

(3c)

where $P$ is a force, $q$ is the intensity of uniform pressure, and $L$ is the length of the beam.

To perform a numerical comparison of beam performance, the material properties listed in

Table 1. Properties of phase materials.

Phase #	Young’s Modulus (MPa)	Poisson’s Ratio	Color	Gradation Functions
1	6000.0	0.30	Green	$V_1$$, U_1$
2	80.0	0.45	Blue	$V_2$$, U_2$
3	12,000.0	0.15	Red	$V_3$$, U_3$

are used. Note that the phase materials exhibit high contrast in their properties, presenting challenges for conventional micromechanics models to accurately predict the effective properties [65]. The colors in the table represent how phase materials are visualized in the color figures of FGBs presented in the Section 3, and the gradation functions ($V_i$ and $U_i$, where $i=1, 2, 3$) listed in the table correspond to those defined in Equations (1) and (2).

For all FGBs to be designed subsequently, the beam is specified to have a length of 160 mm and a rectangular cross-section measuring 10 mm in width and 20 mm in height. The variation among the designed FGBs arises from different loading and boundary conditions, as depicted in Figure 2.

To investigate the influence of gradation indices on FGB performance, specifically the primary index ($m$) and the secondary index ($n$), a range of indices were incorporated into our FGB designs, and their efficacy was tested. These indices, expanded from those adopted in the literature, are detailed in

Table 2. Gradation indices m and n.

FGBs based on Equation (1)	$2m$	0.2	0.5	1.0	2.0	3.0	4.0	5.0	6.0	7.0	8.0
	$2n$	0.2	0.5	1.0	2.0	3.0	4.0	5.0	6.0	7.0	8.0
FGBs based on Equation (2)	$m$	0.2	0.5	1.0	2.0	3.0	4.0	5.0	6.0	7.0	8.0
	$n$	0.2	0.5	1.0	2.0	3.0	4.0	5.0	6.0	7.0	8.0

. Considering the distinct mathematical characteristics of the two sets of gradation functions, the values of $m$ and $n$ in Equations (1) and (2) are intentionally matched. The aim of this exploration is to identify the optimal combinations of $m$ and $n$ to achieve the maximum reduction in beam deflection for specific types of FGBs.

Directly comparing the performance of different FGBs is challenging due to their varying compositions of phase materials. Theoretically, a fair comparison of performance is only possible if two FGBs have identical phase material compositions. However, these compositions vary in FGBs when different gradation functions or indices are used. To enable a ‘fair’ comparison, the maximum deflection of each FGB is assessed relative to that of a corresponding homogeneous beam. This homogeneous beam shares the same phase material content as the FGB but has a uniform distribution. A greater reduction in the maximum deflection indicates superior performance in an FGB, and this performance metric is quantified using the following parameter:

$$\eta ={\displaystyle \frac{{\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{F}\mathrm{G}\mathrm{B}}-{\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{H}\mathrm{O}\mathrm{M}}}{{\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{H}\mathrm{O}\mathrm{M}}}}\times 100\%.$$

(4)

Here, ${\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{F}\mathrm{G}\mathrm{B}}$ and ${\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{H}\mathrm{O}\mathrm{M}}$ represent the maximum deflections of the FGB and the homogeneous beam, respectively. A higher value of this parameter indicates a better performance of the FGB; conversely, a negative value signifies that the FGB underperforms compared to its homogeneous counterpart.

The values of gradation indices ($m$ and $n$) in Table 2 were permuted and combined, resulting in a total of 1200 beam designs for the three loading and boundary conditions in Figure 2. This includes 300 FGBs based on Equation (1), another 300 FGBs based on Equation (2), and 600 homogeneous beams, with each corresponding to an FGB. Given the extensive number of finite element analyses required, understanding the effect of voxel size is crucial. As reported in a previous study [67], voxel size significantly impacts the prediction accuracy of the voxel-based approach. However, this effect remains consistent across models with the same voxel size, indicating that voxel size does not alter the trends in predictions made by this approach. Due to the substantial computational workload involved, a larger voxel size (1 mm) was opted for in predicting the maximum deflection of the FGBs.

3. Results

In this section, FGBs designed using Equation (1), representing the new gradation functions, and Equation (2), associated with conventional functions, are initially presented. This facilitates a visual comparison based on their adaptation to various loading and boundary conditions. Subsequently, numerical results are provided to demonstrate the effectiveness of the newly developed gradation functions in comparison to conventional ones, with a focus on their impact on beam deflection and the optimal combination of primary and secondary indices. Following this, the reasons for the superiority of the newly developed gradation functions over the conventional ones are explained, as evidenced by their enhanced ability to adapt to variations in the bending moment along the beam’s axis. Finally, the results presented reveal the influence of the primary and secondary gradation indices on the allocation of phase materials in the beams.

Figure 3 visualizes the distributions of bending strain energy density (SED) within the beam under the three distinct loading and boundary conditions shown in Figure 2. It is important to note that these SED distributions can vary significantly due to both loading and boundary conditions, even for beams with identical geometry.

Figure 4 presents three exemplar FGBs, designed in accordance with the gradation functions delineated in Equation (1). A noteworthy observation is the similarity in the distribution patterns: each phase material distribution pattern in Figure 4 closely resembles its SED profile in Figure 3.

Similarly, Figure 5 presents three FGBs designed using Equation (2). For the simpler bending moment or SED variations, as depicted in Figure 3a,b, the phase material distributions in Figure 5a,b show notable similarities. However, in the case of more complex SED variation, such as that shown in Figure 3c, the phase material distribution in Figure 5c exhibits significant differences.

Figure 6 displays a sample homogeneous beam, which has the same phase material composition as its corresponding FGB counterpart. This homogeneous beam serves as a benchmark for evaluating the performance of the FGB.

The influence of gradation indices on the performance of FGBs under the three distinct loading and boundary conditions is presented in Figure 7, Figure 8 and Figure 9. These figures highlight the optimal values of gradation indices $m$ and $n$ that yield the maximum improvement in FGB performance. In these figures, the red curves, labeled ‘SED Guided’, represent FGBs designed according to Equation (1). Conversely, the blue curves, marked ‘Conventional’, correspond to designs based on Equation (2). Notably, under cantilever loading/boundary conditions, the SED-guided and conventional FGBs demonstrate similar behavior in minimizing the maximum deflection, as shown in Figure 7. However, this similarity diminishes under simply supported loading/boundary conditions (Figure 8), with the disparity becoming even more pronounced in the clamped–clamped loading/boundary scenario (Figure 9).

To understand the performance differences between FGBs designed using Equations (1) and (2), the variations in the gradation functions $V_1$ and $U_1$ with the primary index ($m$) in relation to the normalized bending moments are examined. These variations are depicted in Figure 10, Figure 11 and Figure 12. A critical observation from the variations is the remarkable adaptability of the new gradation function $V_1$ in responding to varying bending moments caused by loading/boundary conditions, as clearly demonstrated in Figure 10a, Figure 11a and Figure 12a. The consistent accommodation of the changes in bending moment by this function is showcased, highlighting its versatility across different loading conditions. The conventional gradation function $U_1$, in contrast, exhibits a limited range of adaptability. Simpler variations in bending moment are effectively accommodated, as seen in Figure 10b and Figure 11b. However, in scenarios involving more complex bending moment variations, such as the one illustrated in Figure 12b, its efficacy is notably diminished. This distinction suggests that while $U_1$ is suitable for simpler loading scenarios, its application may be less effective in more complex situations where bending moments exhibit intricate variations.

To gain an insight into the influence of gradation indices on the composition of phase materials in FGBs designed from Equation (1), the relationship between these indices and the average volume fractions of the phase materials was explored. Figure 13 presents this relationship, demonstrating how variations in the gradation indices $m$ and $n$ impact the average volume fractions within the FGBs. Specifically, the average volume fraction of each phase is calculated as the ratio of the volume of that phase to the total volume of the beam.

4. Discussion

In the engineering design of FGBs for load-bearing purpose, the primary goal is to enhance their mechanical performance [21,72,73,74,75,76,77], including minimizing static deflection [78,79,80], by optimizing the distribution of phase materials [81]. To minimize static deflection, the strategy involves placing stiffer phase materials in areas experiencing higher strains and less stiff materials in areas with lower strains. Central to this strategy are gradation functions and gradation indices, which determine the amount and distribution of phase materials in FGBs [26,82]. This study compares the effectiveness of two distinct sets of gradation functions, represented by Equations (1) and (2), using the performance of homogeneous beams as a standard for comparison. The gradation functions in Equation (1) are based on the variation in strain energy density (SED) within the beam, reflecting an adaptive approach as they account for the effects of both loading and boundary conditions. As illustrated in Figure 3 and Figure 4, this results in a material distribution that aligns with the SED patterns. The red color in Figure 3 represents the highest SED, which is associated with high strains (if the beam is homogeneous); the corresponding regions (red) in Figure 4 are assigned the stiffest phase material listed in Table 1. On the other hand, the conventional gradation functions in Equation (2) consider only mathematical requirements, not structural performance. This is clearly demonstrated in Figure 5b,c, where FGBs designed from Equation (2) are shown to be unresponsive to boundary conditions. Theoretically, the gradation functions derived from SED variations offer a more effective approach to material allocation in FGBs, leading to enhanced structural performance, particularly in terms of reducing beam deflection under varying conditions.

The results presented in Figure 7, Figure 8 and Figure 9 confirm the theoretical expectation. Compared with the conventional gradation functions in Equation (2), the gradation functions in Equation (1) consistently achieved the maximum reduction in beam deflection under the three loading/boundary conditions. Gradation functions $V_1$ and $U_1$, which are functions of the beam axial coordinate x, play a key role in allocating phase materials along the beam axis to resist the bending moment, thus having a significant effect on beam deflection. The effectiveness of the SED-derived gradation function $V_1$ is rooted in its capability to adapt to the variations in bending moments caused by loading and boundary conditions, as shown in Figure 10a, Figure 11a and Figure 12a. In contrast, the conventional gradation function $U_1$ is only capable of adapting to simple variations, as exemplified in Figure 10b and Figure 11b, and not the complex variation shown in Figure 12b.

In addition to the direct relationship between gradation functions and structural performance, the gradation indices, $m$ and $n$, also play a crucial role in regulating the distribution of phase materials, thereby significantly impacting FGB performance. First, there exists an optimal value for the primary index in both SED-derived and conventional gradation functions that leads to the maximum reduction in beam deflection, as illustrated in Figure 7a, Figure 8a and Figure 9a. This optimal value appears to be dependent on the loading and boundary conditions. For instance, in the case of a cantilever beam subject to an end force, leading to a simple bending moment variation, the primary index of $2m=1.0$ in function $V_1$ or $m=1.0$ in function $U_1$ produced the maximum reduction in beam deflection. This setting aligns the gradation function $V_1$ or $U_1$ precisely with the normalized bending moment in the beam, as shown in Figure 10, and an increasing secondary index further enhances structural performance. However, this phenomenon is not evident under more complex loading and boundary conditions, leading to a more intricate variation in the bending moment, as demonstrated in Figure 11 and Figure 12. An interplay or optimal combination of the primary and secondary indices is crucial to achieve the maximum reduction in beam deflection. According to the mathematical expressions of the gradation functions defined in Equations (1) and (2), the primary gradation index ($m$) is associated with the $x$-coordinate and governs the longitudinal distribution of phase materials along the beam axis. Conversely, the secondary gradation index ($n$), influencing the gradation functions related to the y-coordinate, predominantly affects the distribution of phase materials across the beam’s cross-section. A larger $m$ tends to allocate more stiff phase material close to the top and bottom surfaces of the beam and thus increases the moment of inertia. The indices also determine the relative quantities of phase materials in the FGBs, as evidenced by the average volume fractions shown in Figure 13. The primary index ($m$) has a more pronounced effect on the composition in the FGB, as observable from Figure 13a,c,e, while the secondary index ($n$) exerts a subtler influence, as indicated in Figure 13b,d,f.

This study represents the initial effort to develop gradation functions directly linked to structural performance. While the results are encouraging, substantial work remains to refine these functions further. Several areas for future research are outlined below. Firstly, this study has solely considered SED induced by bending moments. This study has not included SED resulting from other loading modes, such as shear and torsion, which also significantly influence the efficacy of the gradation functions in enhancing structural performance. Secondly, the voxel-based finite element analysis of FGBs is extremely time-intensive. The necessity of using larger voxel sizes for computational feasibility may contribute to the oscillations observed in Figure 7, Figure 8 and Figure 9. Therefore, a more efficient analysis method is needed for FGBs designed using the voxel-based approach. Lastly, while voxel-based finite element analysis is generally regarded as reliable and accurate, experimental validation remains crucial. This is particularly true given that 3D-printed FGBs may contain various defects that could impact their actual performance. Regrettably, access to the requisite voxel-based 3D printer, which is quite costly, is beyond the scope of our current resources.

5. Conclusions

This study has pioneered the development of FGB gradation functions grounded in the strain energy density (SED) induced by bending moments, marking a significant step forward in the link between material gradation and structural performance. These SED-derived functions are designed to autonomously adapt the distribution of phase materials, catering to a variety of loading and boundary conditions. Our findings reveal that these new functions, particularly when the primary and secondary gradation indices are judiciously combined, surpass traditional gradation methods in minimizing beam deflection, showcasing the potential of our approach in optimizing structural performance.

This investigation highlighted the primary gradation index’s pivotal role, which aligns with the beam’s axis, in dictating the composition and longitudinal distribution of phase materials. In contrast, the secondary index primarily influences the material distribution across the beam’s cross-section. This nuanced understanding of the indices’ roles underscores the sophistication and effectiveness of our gradation strategy in tailoring material properties to specific structural demands.

Looking ahead, the methodology developed for constructing gradation functions holds promise for broader applications. It can be extended to encompass various forms of strain energy density, such as those stemming from shear and torsion stresses, paving the way for its application beyond beams to other structural forms like plates and shells. The adaptability of this approach suggests a wide-reaching potential for advancing the design and analysis of functionally graded structural components in diverse engineering applications.