
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12544-3
10.1016/j.heliyon.2024.e36513
e36513
Research Article
Bending of bidirectional functionally graded nonlocal stress-driven beam
Indronil D. indronil.devnath@gmail.com

Department of Civil and Environmental Engineering, North South University, Bangladesh
16 8 2024
15 9 2024
16 8 2024
10 17 e365131 7 2024
16 8 2024
16 8 2024
© 2024 The Author
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
This paper provides a comprehensive analysis, using nonlocal stress-driven integral theory, of the static behavior of a nanoscale beam of bidirectionally graded materials. After a brief explanation of the mathematical formulation of BDFGMs, the work done and strain energy expressions derived from the displacement field are discussed. Variational formulations and Hamilton's principle are used to develop the equilibrium equation. An analytical development of the nonlocal kernel for stress-driven integral theory and formulated governing equation which was nondimensionalized later. Explicit equations for displacement and moment are obtained by solving this equation using the Laplace transformation. Three different boundary conditions are examined, and differences in the maximum displacement with respect to the nonlocal parameter and the two material FGM parameters are displayed both visually and in table form. The results exhibit excellent agreement and provide a standard for further research when they are closely compared to the existing numerical data. This work contributes to the knowledge of BDFGMs under nonlocal effects generated by stress-driven integral theory and offers solutions that have been confirmed for further investigation.

Highlights

Highlights for the research study “Bending of Bidirectional Functionally Graded Nonlocal Stress-Driven Beam” are.• Innovatively derived biexponential kernel through analytical assumptions, advancing structural mechanics.

• Pioneer application of integral forms in nonlocal stress-driven models for rigorous analysis.

• Validated predictive accuracy using Laplace transformation, enhancing modeling precision.

• First complete research of bidirectional functionally graded materials in stress-driven models.

• Insights into optimizing structural designs under diverse loading and support condition, setting new standards.

Keywords

Stress driven integral model
Bidirectional material gradation
Laplace transform
Beam bending displacements
==== Body
pmc1 Introduction

The strain-driven nonlocal elastic model of Eringen [1] offered two paradoxes. First thing is the model in differential form was not fully equivalent to its integral form [2], and the next are some bending solutions of nonlocal elastic beams were identical to the classical (local) solution, e.g., a nanosized cantilever beam with a point load anywhere in the domain [3]. Scientists came up with various modifications of the nonlocal elasticity, for example, the couple stress-based strain gradient theory [4] and the nonlocal strain gradient theory [5].

Romano and Barretta came up with a stress-driven, nonlocal elasticity model. The bending displacement field is the input variable, and the elastic curvature field is the output of an integral convolution law between the bending interaction field and an averaging kernel [6]. So, a stress-driven integral constitutive law was the natural way to solve well-posed nonlocal elastic problems for nanostructures, and the new model was shown to be useful for designing the structures of nanodevices [7]. Romano, Barretta, and Diaco investigated nonlocal integral constitutive laws in a general setting, discussed the effectiveness of the stress-driven nonlocal elastic model, and described a general solution procedure for nonlocal elastic beams [8]. Apuzzo et al. reported free vibrations of Euler-Bernoulli nano-beams by the stress-driven nonlocal integral model [9]. The model is extended toward Timoshenko beams [10] by Barretta with other researchers. Adopting a special Helmholtz averaging kernel in the convolution, Barretta et al. proposed a consistent stress-driven nonlocal integral model for nonisothermal structural analysis of elastic nano- and microbeams [11]. Also, Barretta et al. reported closed-form solutions for the bending of functionally graded nanobeams, with the conclusion that local-nonlocal mixtures model based on the stress-driven theory are mathematically and mechanically appropriate for nano systems [12]. Barretta, Faghidian, and Luciano reported how the ill-posedness of the strain-driven theory is overcome by the stress-driven model and the preferred Helmholtz kernel to solve for natural frequencies and mode shapes of nano-rods using an effective analytical solution strategy [13]. Barretta et al. investigated size-dependent buckling of compressed Bernoulli-Euler nano-beams by stress-driven nonlocal continuum mechanics and numerically calculated buckling loads of compressed nanobeams [14]. Barretta, Faghidian, and Marotti de Sciarra extended the stress-driven nonlocal integral model of elasticity for one directional nanostructures to Kirchhoff axisymmetric nanoplates [15]. Barretta et al. established that the stress-driven nonlocal model for FG Timoshenko nano-beams provides an effective tool for dynamical analyses of stubby composite parts of nano-electro-mechanical systems [16]. Pinnola et al. [17] modeled the axial and flexural dynamic behaviors of elastic nanobeams by nonlocal strain and stress gradient approaches after generalizing the variational static formulation contributed by Barretta and Marotto de Sciarra [18]. Several researchers have utilized the stress-driven nonlocal integral model in recent years to overcome the inconsistencies of nanobeam bending, buckling, and vibration observed in the strain-driven method. In the next three paragraphs, a literature review of stress-driven nanobeam analysis on buckling, bending, and vibration is presented, respectively.

Zhang, Qing, and Gao provided an exact bending analysis of microbeams under varying loads and boundary conditions and demonstrated a consistent toughening effect with the nonlocal parameter for Euler-Bernoulli and Timoshenko beams [19]. In another effort, they have conducted a static bending study of functionally graded (FG) curved nanobeams based on the Timoshenko beam theory and discovered that the shear-deformable effect gets more pronounced as the nonlocal parameter increases [20]. Farajpour, Howard, and Robertson presented a two-dimensional stress-driven nonlocal integral model for the bending and transverse vibration of rectangular nanoplates and obtained numerical solutions utilizing two differential quadrature methods [21]. Ouakad et al. presented the effects of material properties, nonlocal parameters, Lorentz, and electric forces on maximum static deflections and natural frequencies of actuated hybrid carbon/boron-nitride nanotubes (CBNNT) subjected to thermal loads [22]. Pinnola et al. developed an effective and consistent computational methodology based on nonlocal two-noded finite elements to analyze the bending behavior of straight elastic beams [23]. Limkatanyu et al. proposed a flexibility-based nonlocal frame element for size-dependent bending analyses of nanosized frame-like structures where the material's small-scale effect is consistently represented by the stress-driven nonlocal integral model, with a demonstration that the material nonlocality yields a stiffer system response [24]. Critical buckling loads for Euler-Bernoulli and Timoshenko beams were provided by Darban et al. in their investigation of the instability of nanobeams supported by a two-parameter elastic foundation [25,26]. Stress-driven nonlocal Bernoulli-Euler and Timoshenko integral models were developed by Chi et al. for the size-dependent buckling analysis of FG nanobeams [27,28].

In the context of material science functional gradation technology in nanostructure signifies a substantial progress particularly with bidirectional material gradation. Advanced techniques like additive manufacturing with precise control over material composition and gradient distribution are used in fabrication of these materials. They can be used directly in situations where optimized shape and improved mechanical properties are essential [29,30]. Bidirectional FGMs provide improved strength, toughness, flexibility, and ductility as a result of adapted composition gradients that minimize stress concentrations and improve fracture resistance [31,32]. Furthermore, bidirectional FGMs have better thermal stability, allowing for rigidity at high temperatures, which is necessary for aircraft equipment and renewable energy technological innovations. These materials have real-world applications in aerospace engineering, biomedical implants, and renewable energy systems. Their exceptional mechanical qualities and biocompatibility lead to breakthroughs in lightweight constructions, medical implants, and sustainable energy solutions [33,34]. The bidirectional gradient design Functionally graded materials provide productivity and durability in a wide range of industrial applications by enhancing mechanical efficacy and enabling their usage in composite designs and thermal management systems [35]. Ongoing research and advancements in fabrication methods and application domains are expected to substantially enhance the production and deployment of these advanced materials, eventually shaping the discipline of material engineering.

D. Indronil and I.M. Nazmul investigate some mechanical properties of bidirectional functionally graded nanobeams, applying comprehensive analytical techniques. In a study of bending analysis [36] they applied the Laplace transformation to find exact solution of deflections and stresses in these intricate arrangements. Their research extends to vibration analysis [37], where the similar analytical unique approach enables exact computations of dynamic responses under varying support condition. In a separate study of stability analysis [38], Mellin transformation has been employed to analyze critical buckling loads while accounting for bidirectional material gradation and nonlocal behavior.

Fakher, Behdad, and Hosseini-Hashemi developed FEM and GDQM for damping vibration analysis of viscoelastic, axially functionally graded nanobeams. They provided benchmark findings to identify the impacts of nonlocal parameters, the FG index, and the damping factor [39]. Luciano et al. explored the free flexural vibrations of Euler-Bernoulli nanobeams with non-rigid edge supports and derived the spatial differential equation's closed-form solution [40]. Bian, Qing, and Gao applied the stress-driven nonlocal integral model with a bi-Helmholtz kernel to investigate the elastostatic tensile and free vibration analysis of microbars [41]. Russillo et al. estimated natural frequencies and modes of vibration for a two-noded stress-driven nonlocal beam element [42]. Darban, Luciano, and Basista evaluated the effects of crack location, crack length, and nonlocality on the free transverse vibrations of nanobeams containing multiple cracks [43]. Shishesaz, Shariati, and Hosseini studied the vibrational response of FG annular nanoplates in conjunction with classical plate theory [44], and Shariati et al. did a similar study on axisymmetric FG plates and found that the natural frequencies increase with an increase in the size-effect parameter with stiffening effects [45].

This research introduces an analytical approach to the study of bidirectional functionally graded materials (BDFGM) within a nonlocal stress-driven model rarely explored in previous literature. By assuming the multiplicativity of a continuous function, the study derives a kernel function as a solution of the analysis to the stress driven convolution constitutive equation. Through the application of Laplace transformation, the study provides numerical and graphical analyses, validating the effectiveness of these models and opening avenues for further exploration in functionally graded materials under varying loading conditions. The paper's subsequent sections are organized as follows: Section 2 discusses the approach, including the derivation of equilibrium equations and the assumptions for integral convolution constitutive equations. This part is divided into two important sections: the first investigates the nonlocal stress-driven model and generates the associated governing equations, while the second presents the exact solution found by analytical approaches. Section 3 discusses the obtained insights and their broader impacts with the purpose of providing a thorough framework for using advanced analytical methods in structural mechanics.

2 Methodology

In this section, we present the rigorous mathematical formulation that underlies the bending analysis of bidirectionally functionally graded nonlocal nanobeams. The investigation begins with the description of key equations that governs the behavior of these nanoscale structures. The mechanical properties of nanobeams in our study are described by the modulus of elasticity, denoted as E(x,z), which exhibits bidirectional functional graded. This signifies that E(x,z) varies as an unrestricted function in both the axial (x) and thickness (z) directions. The correspondence for longitudinal direction is formulated with a material gradation f(x), and on the other hand g(z) stands for the thickness directional gradation. Mathematically, this variation is represented as [[36], [37], [38]].(1) E(x,z)=f(x)g(z)=eβxL[(Ec−Em)(zh+12)k+Em]

The modulus of elasticity Ec stands for the ceramic and Em stand for the metallic behavior. Fig. 1 demonstrates the material distribution of along the beam thickness and length with different FG parameter. The analysis of the deflection of an isotropic beam, regardless of its size, when subjected to transverse stresses can be described by the assumed displacement field, as outlined in Eq. (2), based on the Euler-Bernoulli beam theory.(2) u1=u0x−z∂w∂x,u2=0,u3=wx

Fig. 1 Material distribution in the beam domain.

Fig. 1

The beam displacement u1, u2 and u3 denoted to be aligned with the Cartesian coordinate x, y and z axes respectively. According to the classical beam theory, only nonzero strain component remains is the axial strain εxx determined using Eq. (3).(3) εx(x,z)=∂u1∂x=du0dx−z∂2w∂x2

The stress-resultants, is determined by integrating the distribution of stress components over the cross-section of the beam, as described in Eq. (4).(4) {Nx,Mx}(x)=∫A{1,z}σx(x,z)dA

The strain energy of the beam is mathematically expressed in Eq. (5) as the integral of the stress and strain components over the volume. The expression is further reformed by substituting Eqs. (3), (4) into Eq. (5).(5) δU=∫σxδεxdV=∬σxδ∂u0∂x−zσxδ∂2w∂x2dxdA=−∫0L∂Nx∂xδu0+∂2Mx∂x2δwdx+Nxδu00L+∂Mx∂xδw0L−Mxδ∂w∂x0L

The calculation of the external work due to the transverse load exerted on the beam includes the multiplication of that load with the transverse displacement, as stated in Eq. (6).(6) δW=∫q(x)δwdx

According to Hamilton's principle, it is hypothesized that the Lagrangian, which is defined as the difference between the strain energy and the external work, should be minimized as indicated by Eq. (7).(7) δΠ=δU−δW=0

We can have the complete expression utilizing of the calculus of variations considering the energy equations from (5), (6) as(8) −∫0L[∂Nx∂xδu0+{∂2Mx∂x2+q(x)}δw]dx−[Nxδu0]0L+[∂Mx∂xδw]0L−[Mxδ∂w∂x]0L=0

The equilibrium equation can be derived from the (8) as(9) ∂Nx∂x=0and∂2Mx∂x2+q(x)=0

The classical boundary condition from Eq. (8) can be stated for arbitrary position r in the beam domain as(10) [u0(r)=0orNx(r)=0w(r)=0ordMdx(r)=0M(r)=0ordwdx(r)=0

2.1 Nonlocal stress-driven model

In this study, we will examine a nonlocal continuum where the stress components tij at a particular spot are connected to the strain components εij at that point using graded elasticity E(x,z), as described in Eq. (11). Nevertheless, as per the stress-driven nonlocal model, the nonlocal stress components σij are affected by all strain components within the neighboring region [6], as indicated by Eq. (11).(11) tij(x)=E(x,z)εij(x,z)=∫0Lψ(|x−x‾|,η)σij(x‾,z)dx‾

By considering the single strain component of the Euler-Bernoulli beam theory and substituting Eq. (3) into Eq. (11), Eq. (12) may be obtained as a Fredholm integral equation [6,8].(12) E(x,z)εx(x,z)=∫0Lψ(|x−x‾|,η)σxx(x‾,z)dx‾

In the context of the intermediate connection of corresponding molecule/grain/particle distances, it is pertinent to highlight that the kernel magnitude is universally nonnegative. Mathematically, this can be expressed as the absolute value of the difference between the near-end molecule position denoted as x and the far-end molecule position as x‾ in a given connection, formulated as |x−x‾|. This preparation elucidates the spatial relationship between molecules within the connection, where the absolute value signifies the distance effect on the observatory molecule/particle, providing a quantitative basis for understanding the interaction dynamics. The kernel function ψ(|x−x‾|,η) in Eq. (12) regulates the reduction of strain's impact on nonlocal stress within the nonlocal region, typically exhibiting a decrease in effect as the metric |x−x‾| increases. The function's behavior is influenced by the nonlocal parameter η, resulting in an expansion of the function across wider regions with an increasing η. This kernel function needs to satisfy symmetry, positivity and impulsivity in Ref. [8] as(13) ψ(x−x‾,η)=ψ(x‾−x,η)⩾0

(14) limη→0ψ(x,η)=δ(x)

Here equation (14) Can easily be achieved to(15) limη→0∫−∞∞ψ(x−x‾,η)σ(x‾,z)dx‾=∫−∞∞limη→0ψ(x−x‾,η)σ(x‾,z)dx‾=∫−∞∞σ(x‾,z)δ(x)dx‾=σ(x,z)

Applying a limit when nonlocal parameter η close enough to zero. Hence the elasticity equation can be sliced in to two parts based the kernel properties (13) and it can be adopted as an absolute modulus.(16) E(x,z)εx(x,z)=∫0xψ(x−x‾,η)σ(x‾,z)dx‾+∫xLψ(x‾−x,η)σ(x‾,z)dx‾

This integration can be operated in such a manner and reformulate to cultivate as the Volterra integral equation.(17) E(x,z)εx(x,z)=∫0x(ψ(x−x‾,η)−ψ(x‾−x,η))σ(x‾,z)dx‾+∫0Lψ(x‾−x,η)σ(x‾,z)dx‾

Now for the multiplicativity of an arbitrary continuous function some expression can be assumed to have a further insight based on Eq. (14) can be achieved from [46].(18) ψ(x−x‾,η)=ϕ(x−x‾,η)λ(η)=ϕ(x,η)ϕ(−x‾,η)λ(η)

The equation describes how the behavior of a nanostructure changes when shifted by a distance x‾. The kernel function at the shifted position involves the product of the original position's kernel function and its counterpart at the symmetrically shifted location, scaled by the function λ(η). This equation suggests a dependence on both spatial positioning and the nonlocal function λ(η), providing insights into how the nanostructure's characteristics evolve with spatial translations and specific frequency components.

Now we can describe the kernel function in eq. (18) for different specific position of x‾ where this position x‾={0,x,−x,}.(19) 1.ψ(0,η)=λ(η)

(20) 2.ϕ(x,η)ϕ(−x,η)=1

(21) 3.ψ(x,η)=ϕ(x,η)λ(η)

(22) 4.ϕ(2x,η)=ϕ(x,η)2

So, for a physical system eq. (19) act as an initial condition where the kernel only dependents on the nonlocal function λ(η) at the initial point of x. An additional assumption was made through the analysis as the initial condition of ϕ(0,η)=1 to satisfy eq. (20) which reveals a symmetry property in the nanostructure under study. It suggests that the product of the kernel functions at a position x and its symmetric counterpart (−x) is linked to the value of a unit at the central position (x=0). This reflection symmetry visible in Fig. 2 implies a coordinated behavior in the nonlocal structure, where the characteristics at opposite positions are interrelated.Fig. 2 Distribution of the kernel function when (a) c1=1, (b) c1=−1.

Fig. 2

2.1.1 Lemma 1

For a multiplicative function the total kernel expression under the integration in Eq. (17) isψ(x−x‾,η)−ψ(x‾−x,η)=ϕ(x,η)ϕ(−x‾,η)λ(η)−ϕ(−x,η)ϕ(x‾,η)λ(η)

=λ(η)ϕ(x,η)[ϕ(−x‾,η)−ϕ(−x,η)ϕ(x‾,η)ϕ(x,η)]

Based on Eqs. (20), (22) and Eq. (18) one can writeψ(x−x‾,η)−ψ(x‾−x,η)=λ(η)ϕ(x,η)[ϕ(−x‾,η)−ϕ(−x,η)ϕ(−x,η)ϕ(x‾,η)]

=λ(η)ϕ(x,η)[ϕ(−x‾,η)−ϕ(−2x,η)ϕ(x‾,η)]

So now the constitutive equation can be written by separating the variable functions based on Lemma 1 as(23) E(x,z)εx(x,z)λ(η)=ϕ(x,η)∫0x(ϕ(−x‾,η)−ϕ(−2x,η)ϕ(x‾,η))σ(x‾,z)dx‾+ϕ(−x,η)p0

Assuming an integral constant, p0, in this case, the integral convolution is normalized into a definite one.(24) p0=∫0Lϕ(x‾,η)σ(x‾,z)dx‾

On the basis of Eqs. (20), (21) it can be written for input variable x=γ/η with an arbitrary variable γ as(25) ψ(γη,η)ψ(−γη,η)=λ(η)2

Eq. (21) can be broken down for the function ψ(x,η) into separated variable function and can be written as ϕ‾(x/η) based on eqs. (25), (14) as(26) ψx,η=ϕ‾xηληwhere,ϕ0=1

It indicates that the value of ψ at a specific position x and nonlocal factor η is determined by the function ϕ at a position scaled by x/η as a variable, multiplied by the function λ associated with the nonlocal factor η. Equation (22) can be written as(27) ϕ‾(2xη)=ϕ‾(xη)2

This equation suggests a certain symmetry or scaling behavior in the function ϕ, with potential implications for understanding how it responds to changes in the input variable x relative to the nonlocal scaling factor η. This Recurrence relation can be solved as(28) ϕ‾(xη)=(exη)c1

Here c1 is a constant. This functional form suggests a dependence on a potential exponential growth or decay pattern in the behavior of ϕ‾.

The nonlocal elasticity illustrates an interesting scenario in which components attract one another with varying intensities based on their proximity. This behavior may be well modeled with equation (28) where x is the spatial distance between components. The constant c1 affects the rate of growth or decay. As components approach closer together, the function in Fig. 2(a) exhibits exponential expansion with positive values of c1, suggesting a greater attraction. However, as Fig. 2(b) illustrates, exponential decline happens when c1 is negative, indicating that attraction diminishes as component distance increases. This straightforward yet effective formula provides a flexible means of capturing and comprehending the dynamic interactions between nano-elements stacked in various spatial configurations.

For λ(η)=1/2η, this equation recreates the biexponential kernel [47](29) ψ(x,η)=(2ηexη)−1

The definition of strain, as presented in Eq. (3), is subsequently replaced into Eq. (23). The definition of functionally graded modulus of elasticity E(x,z) are in the equation to derive the moment displacement governing equation. Following this substitution, Eq. (23) is multiplied by the variable z and integrated throughout the cross-sectional area of the beam, resulting in the derivation of Eq. (30).(30) −f(x)Izλ(η)∂2w∂x2=ϕ(−x,η)[∫0x{ϕ(x−x‾,η)2−1ϕ(−x‾,η)}Mx(x‾)dx‾+c0]

The equation for bending moment, as defined in Eq. (4), is substituted into Eq. (30), and the equation for moment of inertia of area, as defined with an area integral of the beam cross section along with the thickness directional material gradation g(z) stated in Eq. (31).(31) Iz(n)=∬z2g(z)dydz=3(k2+k+2)Y+k[k2+3k+8](k+1)(k+2)(k+3)Embh312

Here, Y is the material ratio of the beam material gradation denoted as Y=Ec/Em. Eq. (28), can be substituted into Eq. (30) to provide Eq. (32)(32) −f(x)Izλ(η)∂2w∂x2=exη[∫0x2ⅇ−xηsinh(x‾−xη)Mx(x‾)dx‾+c0]

By defining the nondimensional parameter ξ=x/L and w‾=w/L, Eqs. (32), (9) can be normalized with some trigonometric operation as(33) 2κeβξ∂2w‾∂ξ2=−2∫0ξsinh(ξ‾−ξκ)M‾(ξ‾)dξ‾−ⅇξκc0

(34) ∂2M‾∂ξ2+q‾(ξ)=0

The nondimensional form of the other parameters are defined as(35) κ=ηL,q‾ξ=qxL4Iz,M‾ξ=MxL2Iz

Now Eq. (24) can be nondimensionalized with cross sectional area integration over the stress component to evaluate it more decently.(36) c0=∫01eξ‾κM‾(ξ‾)dξ‾

2.2 Analytical solution by laplace transform

For the rest of the study we are assuming a distributed load all over the beam q‾(ξ)=q. Applying Laplace transformation on (33) and (34) to have(37) M‾(s)−(κ2s2−1)(s−β)2W(s−β)−c02(κs+1)+(κ2s2−1){(s−β)w‾(0)+w‾′(0)}=0

(38) s3M‾(s)−sM‾′(0)−M‾(0)s2+q=0

Here two theorems are used to transform those integrodifferential equation. First one can be written for the derivatives as(39) Ls[dnw‾dξn]=snW(s)−∑i=0n−1w‾(i)(0)si+1−n

and another one is the transformation for convolution of two continues function.(40) Ls[v1(x)*v2(x)]=Ls[V1(x)].Ls[V2(x)]

Obviously v1 and v2 are some arbitrary dummy functions and V1 and V2 are the transformed form of those dummy functions. Solving for W(s−β) and M(s).(41) W(s−β)=s[s{s(κ2s2−1)(c3(s−β)+c4)+c1}+c2]−qs3(s−β)2(κ2s2−1)−c02(s−β)2(κs−1)

(42) M‾(s)=c1s+c2s2−qs3

Here, ci where i∈N are used for initial condition of displacement and moment and their derivatives to satisfy the integration purpose. With the help of the above relation, eqns. (36), (42) can be written as(43) M‾(ξ)=c1+c2ξ−qξ22

(44) c0=p1+c1p2+c2p3

Here the constant used in eq. (44) are listed asp1=κq2eκ(2κ+1−2(eκ−1)κ2),

p2=(eκ−1)κeκ,

(45) p3=κeκ((eκ−1)κ−1)

By taking the inverse Laplace Transformation on (41) to solve w(ξ) can be simplified as(46) w‾(ξ)=q[2a1a2κ2+{βξ(βξ+4)+8}]2β4ⅇβξ−βc1+c2(βξ+2)β3ⅇβξ+c1−κ(κq+c2)2a12ⅇa1ξ+κ{κ(c2−κq)+c1}−c02a22κⅇa2ξ+c4ξ+c3

Other constants in the above equation can be written as(47) a1=β+1κ,a2=β−1κ

3 Results and discussions

3.1 Simply supported nanobeams

The boundary conditions for simply supported nanobeams are stated by Eq. (48), which stipulates that the displacements and bending moments must be zero at both ends.(48) w‾0=0,M‾0=0,w‾1=0,M‾1=0

Substituting the boundary conditions into Eqs. (46), (43), a system of four equation is derived involving those four constants denoted as ci,i∈N. This system of equations is later solved to determine the values of these constants.c1=0,

c2=−λ4s,

c3=r4sλ4s−λ1s,

(49) c4=λ1s−λ3s+(r3s−r4s)λ4s

To better represent the solution this study, introduce new parameter defined as followsr1s=12(1a12−2β2+κ−p2κa22),r2s=12(1a12ⅇa1−2β2ⅇβ+(κ−p2)κa22ⅇa2),

r3s=12κ2−p3κa22ⅇa2−22+ββ3ⅇβ−κa12ⅇa1,r4s=12κ2−p3a22κ−4β3−κa12,

λ1s=4qβ4−qκ22a12+qκ2a1a2β4−qκ3+p12κa22,λ2s=0

(50) λ3s=12(q(8+β(4+β))β4ⅇβ−qκ2a12ⅇa1+2qκ2a1a2β4ⅇβ−(qκ3+p1)κa22ⅇa2),λ4s=−q2

Substituting the constants stated in Eq. (49) into Eq. (46) provides the explicit displacement profile for nonlocal stress driven simply supported beam under a uniformly distributed load across the beam domain presented in (51)(51) wS−S(ξ)=κ(λ4s−qκ)2a12ⅇξa1+q(8+βξ(4+βξ)+2κ2a1a2)2β4ⅇβξ+(2+βξ)λ4sβ3ⅇβξ−(κ2(qκ+λ4s)+c0)2κa22ⅇξa2+r4sλ4s−λ1s+ξ(λ1s−λ3s+(r3s−r4s)λ4s)

The maximum displacement of a simply supported nanobeam typically occurs at the midspan of the beam domain. However, this position can vary depending on the β values. Higher β values shift the maximum displacement location to the left clamped support. According to the nonlocal stress driven beam theory when β approaches zero, the maximum displacement is described by eq. (52)(52) limβ→0wmaxSS=(e2κ−1)2κ3(2κ+1)4eκ−κ28+5384

Fig. 3 shows how the nonlocal parameter κ and the axial FGM parameter β affect the displacement profile of a simple supported beam. In Fig. 3(a), the displacement along the beam's longitudinal axis is illustrated for various values of κ when β diminishes. Higher κ values result in a stronger beam structure, as maximum displacement reduces. Fig. 3(b) shows the displacement profile for several β values. As with κ, increasing the parameter β reduces maximum displacement, indicating that higher β values improve beam stiffness. These figures demonstrate the importance of raising either the nonlocal or axial FGM parameters in minimizing beam deflection and thereby enhancing stiffness.Fig. 3 Effect of the (a) nonlocal parameter and (b) axial FGM parameter on displacement profile of Simply Supported beam.

Fig. 3

Fig. 4 examines how κ and β affect the maximum displacement of a simply supported nanobeam. Fig. 4(a) shows the maximum displacement plotted against κ for several values of the material inhomogeneity constant β. The findings demonstrate that increasing κ leads to a reduction in maximum displacement for all β values, demonstrating improved beam stiffness. Increasing β reduces maximum displacement, indicating that both parameters work together to enhance beam stiffness. Fig. 4(b) shows the greatest displacement against β for κ values. As β grows, maximum displacement falls for all κ values, indicating more stiffness. These findings underscore the importance of both the nonlocal and axial FGM parameters in defining the structural behavior of simply supported nanobeams, with both factors playing critical roles in lowering maximum displacement and increasing stiffness.Fig. 4 Effect of the (a) Nonlocal parameter and (b) Axial FGM parameter on maximum displacement of Simply Supported nanobeam.

Fig. 4

The validity of the analytical expression presented can be confirmed by conducting a comparison between the maximum displacement of different support conditions if limited number existing numerical data shown in Table 1.Table 1 Comparison of maximum displacements for different support condition ×10−3.

Table 1κ2	Clamped – Clamped	Simple-Simple	Clamped -Free	
Barretta [10]	This Study	Barretta [10]	This Study	Barretta [10]	This Study	
0.00	0.26074	0.26042	1.30241	1.30208	12.5013	12.5000	
0.05	0.17968	0.17940	1.24993	1.24966	10.8584	10.8572	
0.10	0.14857	0.14832	1.20693	1.20668	10.2512	10.2499	
0.15	0.12793	0.12770	1.17008	1.16985	9.81417	9.81303	
0.20	0.11287	0.11265	1.13780	1.13758	9.46516	9.46404	
0.25	0.10126	0.10105	1.10908	1.10888	9.17213	9.17103	
0.30	0.09198	0.09179	1.08322	1.08303	8.91862	8.91754	
0.35	0.08438	0.08420	1.05971	1.05953	8.69481	8.69375	
0.40	0.07802	0.07784	1.03817	1.03800	8.49430	8.49326	
0.45	0.07260	0.07244	1.01832	1.01815	8.31264	8.31161	
0.50	0.06793	0.06777	0.99990	0.99975	8.14656	8.14555	

The largest displacement for a simply supported beam, when the transverse FGM parameter k=0, is thoroughly examined in Table 2. Further study in this field may be based on this knowledge, which is essential for developing a greater understanding of the behavior of such beams. The data in Table 2 highlights how the absence of the transverse FGM parameter impacts the structural response. On the other hand, Table 3 illustrates the variation in the transverse inhomogeneity parameter k when different nonlocal parameters are applied, with a fixed value of β=0.Table 2 Effect of axial inhomogeneity on maximum displacements of Simply Supported beam ×10−3.

Table 2κ2	Axial Material inhomogeneity parameter β	
0.00	0.25	0.50	0.75	1.00	1.25	1.50	1.75	2.00	
0.00	1.30208	1.15046	1.01891	0.90455	0.80493	0.71797	0.64190	0.57521	0.51663	
0.01	1.20668	1.06619	0.94437	0.83851	0.74632	0.66587	0.59552	0.53386	0.47971	
0.02	1.13758	1.00518	0.89042	0.79075	0.70399	0.62831	0.56214	0.50417	0.45327	
0.03	1.08303	0.95700	0.84782	0.75302	0.67054	0.59861	0.53574	0.48068	0.43233	
0.04	1.03800	0.91724	0.81265	0.72188	0.64291	0.57407	0.51391	0.46122	0.41498	
0.05	0.99975	0.88345	0.78276	0.69540	0.61941	0.55318	0.49532	0.44465	0.40018	

Table 3 Effect of transverse gradation on maximum displacements of Simply Supported beam ×10−3.

Table 3κ2	Transverse Material inhomogeneity parameter k	
0.0	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	
0.000	1.30208	0.90526	0.76790	0.70176	0.66106	0.63146	0.60764	0.58734	0.56948	0.55347	
0.005	1.24966	0.86881	0.73698	0.67350	0.63444	0.60603	0.58317	0.56369	0.54655	0.53119	
0.010	1.20668	0.83893	0.71163	0.65034	0.61262	0.58519	0.56312	0.54431	0.52776	0.51292	
0.015	1.16985	0.81332	0.68991	0.63049	0.59392	0.56733	0.54593	0.52769	0.51165	0.49726	
0.020	1.13758	0.79089	0.67088	0.61310	0.57754	0.55168	0.53087	0.51314	0.49754	0.48355	
0.025	1.10888	0.77093	0.65395	0.59763	0.56297	0.53776	0.51748	0.50019	0.48498	0.47135	
0.030	1.08303	0.75296	0.63871	0.58370	0.54984	0.52522	0.50541	0.48853	0.47368	0.46036	
0.035	1.05953	0.73662	0.62485	0.57103	0.53791	0.51383	0.49445	0.47793	0.46340	0.45037	
0.040	1.03800	0.72166	0.61216	0.55943	0.52699	0.50339	0.48440	0.46822	0.45398	0.44122	
0.045	1.01815	0.70786	0.60045	0.54873	0.51691	0.49376	0.47514	0.45927	0.44530	0.43278	
0.050	0.99975	0.69506	0.58959	0.53881	0.50756	0.48484	0.46655	0.45096	0.43725	0.42496	

3.2 Clamped nanobeams

Eq. (53) specifies the boundary conditions for clamped nanobeams, which call for displacements and their first derivatives to reach zero at both ends.(53) w‾0=0,w‾'0=0,w‾L=0,w‾'L=0

The boundary conditions given in Eq. (53) are replaced into Eq. (46), provides four equations and a system with four unknown constants, ci,i∈N. The set of equations is then solved to determine those unknown coefficients for clamped beam displayed in Eq. (54).c1=R−1{λ2f(r5f−r7f)+(λ1f−λ3f)r8f−(λ1f+λ2f−λ3f)r6f−λ4f(r5f−r7f−r8f)}

c2=R−1{r3f(λ1f+λ2f−λ3f)+(r1f−r2f)(λ2f−λ4f)−r4f(λ1f−λ3f+λ4f)}

c3=R−1[r2f(r8fλ1f−r6fλ1f−r7fλ2f)+r1fλ2f(r5f−r6f)+r3f{(r5f−r8f)λ1f+r7f(λ2f−λ3f)}+r1fλ3f(r6f−r8f)+r4f{(r6f−r5f)λ1f+r7f(λ3f−λ4f)}+(r2fr7f+r1f(r8f−r5f))λ4f]

c4=R−1[r3f{(r7f−r5f)λ2f+r8f(λ3f−λ1f)}+r4f{r6f(λ1f−λ3f)+(r5f−r7f)λ4f}−(r1f−r2f)(r6fλ2f−r8fλ4f)]

(54) R=r4f(r6f+r7f−r5f)+(r1f−r2f)(r6f−r8f)+r3f(r5f−r7f−r8f)

Following a similar procedure of the previous section, additional terms are included for well representation of those constants in Eq. (54)r1f=12(κ−p2a22κ+1a12−2β2)r2f=12((κ−p2)a22κⅇa2+1a12ⅇa1−2β2ⅇβ)

r3f=12((p2−κ)a2κⅇa2−1a1ⅇa1+2βⅇβ)r4f=12(p2−κa2κ−1a1+2β)

r5f=12((κ2−p3)a22κⅇa2−2(β+2)β3ⅇβ−κa12ⅇa1)r6f=12(κa1ⅇa1+(p3−κ2)a2κⅇa2+2(β+1)β2ⅇβ),

(55) r7f=12(κ2−p3a22κ−4β3−κa12)r8f=12(κa1+p3−κ2a2κ+2β2)

And also, those λi terms areλ1f=4qβ4(a1a2κ2+4)−12(κ2qa12+p1+κ3qa22κ)

λ2f=(a1(p1+κ3q)+a2κ3q)2a1a2κ−q2β3(2a1a2κ2+4)

λ3f=q2β4ⅇβ(2a1a2κ2+(β(β+4)+8))−12(κ2qa12ⅇa1+(p1+κ3q)a22κⅇa2),

(56) λ4f=12((p1+κ3q)a2κⅇa2+κ2qa1ⅇa1)−q2β3ⅇβ((β(β+2)+4)+2a1a2κ2)

Afterward, the constants listed in Eq. (54) are substituted into Eq. (46), yielding the explicit expression for the displacement of clamped nanobeams under a uniformly distributed load according to the stress-driven nonlocal theory. The maximum displacement of this boundary condition is observed at the midspan similar to the simply supported beam though the position of the maximum displacement may vary due to the effect of axial FGM parameter β as shown in Fig. 5(b). The maximum displacement of a clamped nanobeam under these loading conditions, as predicted by the stress-driven nonlocal theory, is specified in Eq. (57) when a limit with β=0 is applied.(57) limβ→0w‾maxC−C=κ(6κ+1)[8e2κκ(κ−1)−8κ2+4κ+1]96(κ−1)[eκ(κ−1)−κ]−96κ3−8κ2−5κ+1384(κ−1)

Fig. 5 Effect of the (a) nonlocal parameter and (b) axial FGM parameter on displacement profile of Clamped beam.

Fig. 5

The consequence of applying nonlocal parameter κ and β material inhomogeneity parameter on the displacement profile of clamped nanobeam system given in Fig. 5. The first plots Fig. 5(a) demonstrates the displacement along the beam domain for different values of κ. This plot reveals that κ increment is responsible for displacement reduction, signifying that higher κ values stand for higher stiffness as we can witness the phenomena in the simply supported case. Furthermore, Fig. 5(b) demonstrates the displacement profile for several β parameter. Similar to the simply supported case increments of β leads to a reduced displacement profile and enhanced beam stiffness. So, it can be established the fact that both the parameter β and κ are accountable for beam stiffness and the relation is proportional.

Fig. 6 shows the effect of κ and β on the maximum displacement of a beam clamped at both sides. Fig. 6(a) visualize the association between maximum displacement and nonlocal parameter κ for different values of β. When κ increases in the domain maximum displacement reduces for those β values which indicates higher stiffness of the clamped beam. Additionally, increments of β follow further reductions in displacement. This particular phenomenon describes the combined effect of both parameter on the beam stiffness. Fig. 6(b) shows relation between maximum displacement with the material inhomogeneity parameter β for various values of nonlocal parameter κ. The graph suggests a consistent lower displacement for higher β value in all the cases of κ values. This conforms that higher β values correspond to higher structural rigidity. So, this finding explains that increasing both the parameter β and κ reduces maximum displacement in other word increases stiffness of the clamped nanobeam.Fig. 6 Effect of the (a) Nonlocal parameter and (b) Axial FGM parameter on maximum displacement of Clamped nanobeam.

Fig. 6

According to the analytical solution of Indronil et al. [36,48], the nonlocal parameter does not have any influence on a clamped-clamped nanobeam subjected to a uniformly distributed load as per the strain-driven nonlocal model and confirmed by other researchers. This implies that the maximum displacement of a clamped-clamped nanobeam remains unaffected by the nonlocal parameter. However, the present findings oppose this conclusion.

Table 4, Table 5 shows the maximum displacement of clamped nanobeam under UDL with values of κ2 under different values of β and k respectively. The variation of each parameter is considered with the other parameter set to zero.Table 4 Effect of axial inhomogeneity on maximum displacements of Clamped beam ×10−3.

Table 4κ2	Axial Material inhomogeneity parameter β	
0.00	0.25	0.50	0.75	1.00	1.25	1.50	1.75	2.00	
0.00	0.26042	0.22986	0.20296	0.17927	0.15841	0.14002	0.12380	0.10950	0.09688	
0.01	0.14832	0.13092	0.1156	0.10211	0.09023	0.07976	0.07053	0.06239	0.05520	
0.02	0.11265	0.09944	0.08782	0.07759	0.06859	0.06066	0.05368	0.04752	0.04209	
0.03	0.09179	0.08103	0.07157	0.06326	0.05594	0.04950	0.04383	0.03883	0.03442	
0.04	0.07784	0.06872	0.06071	0.05367	0.04748	0.04203	0.03724	0.03301	0.02929	
0.05	0.06777	0.05983	0.05287	0.04675	0.04137	0.03664	0.03247	0.02881	0.02558	

Table 5 Effect of transverse gradation on maximum displacements of Clamped beam ×10−3.

Table 5κ2	Transverse Material inhomogeneity parameter k	
0.0	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	
0.000	0.26042	0.18105	0.15358	0.14035	0.13221	0.12629	0.12153	0.11747	0.11390	0.11069	
0.005	0.17940	0.12473	0.10580	0.09669	0.09108	0.08700	0.08372	0.08092	0.07846	0.07626	
0.010	0.14832	0.10312	0.08747	0.07994	0.07530	0.07193	0.06922	0.06690	0.06487	0.06305	
0.015	0.12770	0.08878	0.07531	0.06883	0.06483	0.06193	0.05960	0.05760	0.05585	0.05428	
0.020	0.11265	0.07832	0.06644	0.06071	0.05719	0.05463	0.05257	0.05081	0.04927	0.04788	
0.025	0.10105	0.07026	0.05960	0.05446	0.05130	0.04901	0.04716	0.04558	0.04420	0.04295	
0.030	0.09179	0.06382	0.05413	0.04947	0.04660	0.04452	0.04284	0.04141	0.04015	0.03902	
0.035	0.08420	0.05854	0.04966	0.04538	0.04275	0.04083	0.03929	0.03798	0.03683	0.03579	
0.040	0.07784	0.05412	0.04591	0.04195	0.03952	0.03775	0.03633	0.03511	0.03405	0.03309	
0.045	0.07244	0.05036	0.04272	0.03904	0.03678	0.03513	0.03380	0.03268	0.03168	0.03079	
0.050	0.06777	0.04712	0.03997	0.03653	0.03441	0.03287	0.03163	0.03057	0.02964	0.02881	

3.3 Nanocantilevers

The boundary conditions for nanocantilever fixed at the near end and free at the far end are specified at Eq. (58).(58) w0=0,w'0=0,M1=0,M'1=0;

By substituting the conditions outlined in Eq. (58) into Eqs. (46), (43), a system of four equations for the nano cantilevered beam is formulated. Using a methodology similar to that of the preceding sections, the system is solved to determine the values of ci,i∈N as elucidated in Eq. (59).c1=λ4c−λ3c,

c2=−λ4c,

c3=(λ3c−λ4c)r1c+λ4cr3c−λ1c,

(59) c4=(λ3c−λ4c)r2c+λ4cr4c−λ2c

The necessary terms for Eq. (59) are listed below.r1c=121a12+κ−p2κa22−2β2,r2c=122β−1a1+p2−κa2κ,

(60) r3c=12κ2−p3a22κ−4β3−κa12,r4c=122β2+κa1+p3−κ2a2κ,

The λicf terms are given as followsλ1c=q(4+κ2a1a2)β4−qκ3(a12+a22)+a12p12κa22

λ2c=qκ3(a2+a1)+a1p12κa1a2−2q(2+κ2a1a2)β3,

(61) λ3c=−q2,λ4c=−q

Substituting the four constants from Eq. (59) into Eq. (46) yields the explicit expression for the nanocantilever displacement under a uniformly distributed load based on the stress-driven nonlocal theory, shown in Eq. (62).(62) wC−F(ξ)=(λ4c(κ−κ2−p2+p3)+λ3c(p2−κ)−p1−κ3q)2a22κⅇa2ξ+((κ+1)λ4c−λ3c−κ2q)2a12ⅇa1ξ+(2a1a2κ2q+q(βξ(βξ+4)+8))2β4ⅇβξ+(βλ3c+λ4c(β(ξ−1)+2))β3ⅇβξ−λ1c+(λ3c−λ4c)r1c+λ4cr3c+ξ((λ3c−λ4c)r2c+λ4cr4c−λ2c)

The maximum displacement of a nanocantilever occurs at the free end x=1, as presented in Eq. (63).(63) wmaxC−F=a1a2κ2+4β4ⅇβ+(β+4)2β3ⅇβ+(βλ3c+2λ4c)β3ⅇβ+(κ+1)λ4c−λ3c−κ22a12ⅇa1+(λ3c−λ4c)r1c−λ1c+λ4cr3c+λ4c(κ−κ2−p2+p3)+λ3c(p2−κ)−κ3−p12a22κⅇa2+{(λ3c−λ4c)r2c+λ4cr4c−λ2c}

For non FGM beam the displacement can be given as(64) limβ→0wmaxC−F=25κ[(κ−1)eκ−κeκ]+252

Fig. 7 illustrates how κ and β affect the displacement profile of a cantilevered beam. Fig. 7(a) shows displacement along the beam length with κ values. Higher κ values suggest a stronger cantilevered beam with little displacement. Fig. 7(b) shows the displacement curve with β values ranging. Higher β values increase beam stiffness, reducing displacement. These results show that increasing either the nonlocal or axial FGM parameters reduces displacement while enhancing the structural stiffness of the cantilevered beam.Fig. 7 Effect of the (a) nonlocal parameter and (b) axial FGM parameter on displacement profile of Cantilevered beam.

Fig. 7

The impact of κ and β on the maximum displacement of a cantilevered nano sized beam are plotted in Fig. 8. The relationship between κ and maximum end displacement are demonstrated in Fig. 8(a) for different values of axial FGM parameter β. The effect indicated that as κ increases for a uniformly distributed load, end displacement always decreases covering all alternate values of β, and highlights that higher κ value enhances beam stiffness. Moreover, higher values of β further reduces the displacement which is a proof of the combined stiffness effect of both parameters. However, this statement also true for the relation between maximum end displacement and material inhomogeneity parameter β for arbitrary values of κ visible in Fig. 8(b). Finally, these findings of those two graphs focus on substantial impact of both parameter on the stiffness of the cantilevered nanobeam. Structural rigidity always enhanced for higher values of each parameter result in reduced maximum displacement.Fig. 8 Effect of the (a) Nonlocal parameter and (b) Axial FGM parameter on maximum displacement of Cantilevered nanobeam.

Fig. 8

The effect of the nonlocal parameter is characterized by its less prominent influence. This present stress driven nonlocal model does not show any contradictory results in comparison with strain driven nonlocal beam model studied by Nazmul and Indronil [36]. Their study establishes an opposite effect on the displacement profile for specific values of nonlocal parameter. This kind of paradoxical phenomenon was not witnessed within the analysis of stress driven nonlocal integral model.

The variations of material inhomogeneity parameter β and k with respect to κ2 displayed in Table 6, Table 7 respectively. These variations of each parameter are examined while the other parameter vanishes.Table 6 Effect of axial inhomogeneity on maximum displacements of Cantilever beam ×10−3.

Table 6κ2	Axial Material inhomogeneity parameter β	
0.00	0.25	0.50	0.75	1.00	1.25	1.50	1.75	2.00	
0.00	12.5000	11.9001	11.3472	10.8364	10.3638	9.92571	9.51882	9.14029	8.78754	
0.01	10.2500	9.64931	9.09881	8.59330	8.12822	7.69954	7.30369	6.93752	6.59821	
0.02	9.46404	8.86604	8.32047	7.82171	7.36484	6.94553	6.55997	6.20479	5.87700	
0.03	8.91754	8.32324	7.78310	7.29115	6.84216	6.43155	6.05530	5.70987	5.39214	
0.04	8.49326	7.90304	7.36842	6.88310	6.44158	6.03905	5.67131	5.33468	5.02592	
0.05	8.14555	7.55955	7.03041	6.55150	6.11705	5.72208	5.36221	5.03365	4.73305	

Table 7 Effect of transverse gradation on maximum displacements of Cantilever beam ×10−3.

Table 7κ2	Transverse Material inhomogeneity parameter k	
0.0	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	
0.000	12.5000	8.69048	7.37179	6.73687	6.34615	6.06199	5.83333	5.63849	5.46703	5.31334	
0.005	10.8572	7.54836	6.40298	5.85151	5.51213	5.26532	5.06671	4.89747	4.74855	4.61505	
0.010	10.2500	7.12618	6.04487	5.52423	5.20384	4.97083	4.78333	4.62355	4.48296	4.35693	
0.015	9.81303	6.82239	5.78717	5.28873	4.98200	4.75892	4.57941	4.42645	4.29185	4.17120	
0.020	9.46404	6.57976	5.58136	5.10064	4.80482	4.58968	4.41655	4.26903	4.13922	4.02285	
0.025	9.17103	6.37605	5.40856	4.94273	4.65606	4.44758	4.27982	4.13686	4.01107	3.89831	
0.030	8.91754	6.19981	5.25906	4.80611	4.52737	4.32465	4.16152	4.02251	3.90020	3.79055	
0.035	8.69375	6.04423	5.12709	4.68550	4.41375	4.21612	4.05709	3.92157	3.80232	3.69543	
0.040	8.49326	5.90484	5.00885	4.57744	4.31196	4.11889	3.96352	3.83113	3.71464	3.61021	
0.045	8.31161	5.77855	4.90172	4.47954	4.21974	4.03079	3.87875	3.74919	3.63519	3.53299	
0.050	8.14555	5.66310	4.80379	4.39004	4.13543	3.95026	3.80126	3.67429	3.56256	3.46241	

4 Conclusion

The primary aim of this study was to investigate the impact of nonlocal influences and the material inhomogeneity constants β and k on the static bending characteristics of Euler-Bernoulli beams, using analytically derived solutions based on the stress-driven nonlocal integral model. Hamilton's principle is used to formulate the governing equations and corresponding boundary conditions. The analytical solution to these equations was obtained through the Laplace transformation method. In this study of nonlocal classical beams, various boundary conditions, including simply supported, clamped-clamped, and cantilever, were considered. A thorough comparison between our analytical outcomes and pre-existing analytical and numerical solutions was conducted to verify their precision. The results demonstrate that the stress-driven nonlocal integral model induces varying degrees of stiffening effects on the bending behavior of Euler-Bernoulli nanobeams, depending on the boundary conditions.

Specifically, the stiffness of simply supported and cantilever beams reduces mildly with higher nonlocal parameters. In contrast, for clamped-clamped beams, an increase in the nonlocal parameter results in a rapid reduction in stiffness. The material inhomogeneity constants β and k also play crucial roles in these behaviors. β, which influences the gradation of material properties across the beam's axis, affects how stress distributions respond to nonlocal effects. Higher values of β can lead to more pronounced variations in stiffness, highlighting the importance of material gradation in designing nanobeams. Similarly, the transverse inhomogeneity parameter k impacts the beam's response to bending, with different values of n altering the degree to which nonlocal effects manifest.

The explicit analytical solutions outlined in this study are valuable assets for researchers exploring nano-devices and structures featuring nanobeams. The insights garnered from this research have the potential to significantly influence the design and application of nanostructures. By understanding the interplay between nonlocal parameters and material inhomogeneity constants β and k, engineers can better predict and tailor the mechanical behavior of nanobeams, paving the way for innovative applications in nanotechnology and materials science.

Data availability

Data will be made available on request.

CRediT authorship contribution statement

D. Indronil: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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