Fractal Modeling of Nonlinear Flexural Wave Propagation in Functionally Graded Beams: Solitary Wave Solutions and Fractal Dimensional Modulation Effects

Abstract

In this study, a new nonlinear dynamic model was established for functionally graded material (FGM) beams with layered/porous fractal microstructures, aiming to reveal the cross-scale propagation mechanism of flexural waves under large deflection conditions. The characteristics of layered/porous microstructures were equivalently mapped to the fractal dimension index. In the framework of the fractal derivative, a fractal nonlinear wave governing equation integrating geometric nonlinear effects and microstructure characteristics was derived, and the coupling effect of finite deformation and fractal characteristics was clarified. Four groups of deflection gradient traveling wave analytical solutions were obtained by solving the equation through the extended minimal (G′/G) expansion method. Compared with the traditional (G′/G) expansion method, the new method, which is concise and expands the solution space, generates additional csch² soliton solutions and csc² singular-wave solutions. Numerical simulations showed that the spatiotemporal fractal dimension can dynamically modulate the amplitude attenuation, waveform steepness, and phase rotation characteristics of kink solitary waves in beams. At the same time, it was found that the decrease in the spatial fractal dimension will make the deflection curve of the beam more gentle, revealing that the fractal characteristics of the microstructure have an active control effect on the geometric nonlinearity. This model provides theoretical support for the prediction and regulation of the wave behavior of fractal microstructure FGM components, and has application potential in acoustic metamaterial design and engineering vibration control.

1. Introduction

Functional gradient materials composed of two or more heterogeneous materials, with their excellent physical properties and flexible structural adjustability, have important applications in engineering practice in aerospace, new energy, microelectronics, and the chemical and biomedical fields [1,2,3]. Mechanical waves inevitably carry key information such as material composition distribution characteristics and mechanical parameters during the propagation of such materials. These characteristics are revealed by changes in parameters such as the phase characteristics, amplitude attenuation rate, and propagation rate during the wave propagation process, which provide key data support for non-destructive testing and performance evaluation of materials [4,5]. Unfortunately, the actual preparation of functionally graded materials is often accompanied by microscopic pores and layered heterostructures. Such internal inhomogeneity significantly affects the wave propagation characteristics, making the traditional homogeneous medium wave theory difficult to directly apply [6,7,8]. The wave theory of traditional functionally graded materials is mostly based on the assumption of continuous medium, and the material gradient is processed by a layered model or equivalent parameter method. However, this model has difficulty characterizing the microstructural effects induced by preparation defects, leading to systematic deviations between theoretical predictions and experimental observations [9,10]. In recent years, multiscale modeling methods based on fractal geometry have provided a new way to describe the mechanical behavior of such complicated media [11,12,13]. As a mathematical tool to describe the nonlocal effect in hierarchical and porous media [14,15], fractal derivative has shown unique advantages in the fields of porous seepage and anomalous diffusion [16,17,18,19,20,21], but its application in nonlinear wave propagation problems still needs to be further explored.

In research on nonlinear wave propagation, building precise analytical solutions has multiple scientific advantages. Such mathematical solutions can not only visually present complex nonlinear dynamic phenomena such as soliton collision and shock wave formation through three-dimensional visualization technology, but also provide theoretical decoders to reveal the coupling mechanism between material constitutive relations and wave modes. At the same time, it constitutes the cornerstone of derivative research such as dispersion control and energy dissipation analysis [22]. Although the nonlinear wave equation has not yet formed a unified analytical theoretical framework, after half a century of methodological innovation, a variety of effective solutions have been developed, such as the inverse scattering method [23]; the hyperbolic tangent method [24]; the extended hyperbolic function method [25,26,27]; the Jacobi elliptic function expansion method and its extended form [28,29,30]; the F-function expansion method [31]; the exponential function method [32]; the Kudryashov method [33]; the Riccati function expansion method [34]; the Khater method [35]; the sine–cosine method [36]; the (G′/G) function expansion method [37,38]; the (G′/G,1/G) function expansion method [39]; the (1/G) function expansion method [40]; the (1/G′) function expansion method [41]; the (G′/G²) function expansion method [42]; the extended (G′/G) function expansion method [43,44]; and the extended minimal (G′/G) function expansion method [45]. Among these nonlinear fluctuation equation solution methods, the extended minimal (G′/G) method [45] modified by the authors improves the computational efficiency while maintaining the constructive ability of the traditional methods, combining theoretical simplicity and application effectiveness.

Beam structure is the most widely used basic structure in engineering. Most of the existing studies on wave propagation in functionally graded material beams are linear wave problems [46]. However, the linear fluctuation theory is only a simplified theory of the actual fluctuation, and the geometric nonlinear effect in beams with large deflection should not be neglected. If the material nonuniformity caused by the laminar and porous features of functional gradient materials is considered, the wave equation will show a double nonlinear coupling (material inhomogeneity + geometric nonlinearity). The complexity of the dual nonlinear coupling problem has meant that there are a lack of research results on nonlinear wave propagation in beams of functionally gradient materials, and even fewer on nonlinear wave-bending wave propagation [47,48]. However, such research results will help with the non-destructive testing of functionally graded materials, and will also reveal many interesting and important phenomena [49,50,51]. In this study, a fractal nonlinear model based on fractal derivative theory that simultaneously characterizes material nonuniformity and finite deformation was developed to reveal the propagation properties of bending waves in beams of functional gradient materials with layered and porous microstructures. The main contributions are the following three points.

• The layered porous microstructure characteristics of the material were equivalently mapped to the fractal dimension index, and the nonlinear propagation control equation of the bending wave in the functionally graded material beam based on the fractal derivative was derived. The model takes into account both the geometric nonlinear effect caused by large deflection and the non-uniformity of the material, which makes the derivation of the wave equation more complicated.
• Using the extended minimal (G′/G) method modified by the author, four sets of deflection gradient exact traveling wave solutions including various forms were obtained. The comparative study shows that this method generates additional csch² and csc² solutions alongside the traditional (G′/G) method, which is simple and effective while expanding the solution space.
• Numerical simulations find that the spatiotemporal fractal dimension may modify the waveform, amplitude, and rotation of the deflection gradient kink isolated wave in the beam. In addition, the deflection profile of the beam corresponding to the fractal kink isolated wave was derived, and numerical simulation of this deflection profile finds that reducing the spatial fractal dimension index has a suppressive effect on the geometric nonlinearity.

Below, we list the derivation of the model, the solution of the model, the discussion of the solution, the numerical simulation analysis, and the conclusion.

2. Derivation of Fractal Nonlinear Propagation Equations for Bending Waves in Beams

Based on the theoretical framework of the Rayleigh beam, this section integrates the geometric nonlinear characteristics induced by a large deflection effect and the layered–porous composite characteristics of functionally graded materials; introduces the fractal derivative mathematical tool and Lagrange material description method; and constructs the fractal space–time nonlinear wave equation of a bending wave in the beam structure of functionally graded materials with an equal cross-section under the fractal space–time dimension. This paper uses the following form of fractal derivative definition [14,15].

$${\displaystyle \frac{df}{d{x}^{\alpha }}}=\underset{\Delta x=x_A-x_B\to L_0}{\lim }{\displaystyle \frac{f(A)-f(B)}{x_A^{\alpha }-x_B^{\alpha }}}.$$

(1)

In this definition, the fractal medium is regarded as a fractal space. The distance between A and B in the fractal space will no longer be the linear distance $x_A-x_B$ in the continuous medium space, it will be $x_A^{\alpha }-x_B^{\alpha }$ (see Equation (47) of Reference [14]).

The research object is a functionally graded straight four-prism beam with uniform cross-section and porous characteristics. The plane bending vibration analysis adopts the following coordinate system: the x-axis is constructed with the geometric axis of the prismatic beam when it is not loaded, its neutral axis is taken as the y-axis of the coordinate system, and the direction of motion of the deflection curve is determined as the positive direction of the z-axis.

In this mechanical model, the action of external loads along the axial direction (x-axis) of the beam is explicitly excluded, and $u_x$ is used to denote the displacement along the x-axis, $u_y$ is used to denote the displacement along the y-axis, $u_z$ is used to denote the displacement along the z-axis, and W is used to denote the deflection in the direction of the z-axis. The system of fractal dimension parameters is defined as follows: $\alpha$ denotes the temporal fractal dimension and $\beta$ denotes the fractal dimension in the direction of the spatial coordinate system (x, y, z). In the framework of fractal derivative, the following displacement expression is adopted.

$$u_x=-z^{\beta }{\displaystyle \frac{\partial W}{\partial x^{\beta }}},u_y=0,u_z=W,$$

(2)

Based on the geometrically nonlinear ontological relations under large deflection conditions, the positive strain in the longitudinal (x-axis) direction of a functional gradient beam with fractal microstructural features is

$${\epsilon }_x=-z^{\beta }{\displaystyle \frac{{\partial }^{2}W}{\partial x^{2\beta }}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\partial W}{\partial x^{\beta }}}\right)}^2.$$

(3)

Hooke’s law is used in the x-axis upward direction; that is, ${\sigma }_x=E{\epsilon }_x$, where ${\sigma }_x$ and E denote the stress and modulus of elasticity in the x-axis upward direction, respectively. Thus, the straining energy over a unit distance in a prismatic fraction beam when large deflections are considered can be expressed as follows:

$$\begin{array}{ll}V\hfill & =0.5{\displaystyle {\iint }_{S^{\beta }}{\sigma }_x}{\epsilon }_{x}d{y}^{\beta }d{z}^{\beta }=0.5E{\displaystyle {\iint }_{S^{\beta }}{\left(-z^{\beta }\frac{{\partial }^{2}W}{\partial x^{2\beta }}+{\left(\frac{\partial W}{\partial x^{\beta }}\right)}^2\right)}^2}d{y}^{\beta }d{z}^{\beta }\hfill \\ & =0.5EI{\left({\displaystyle \frac{{\partial }^{2}W}{\partial x^{2\beta }}}\right)}^2+{\displaystyle \frac{1}{8}}ES{\left({\displaystyle \frac{\partial W}{\partial x^{\beta }}}\right)}^4,\hfill \end{array}$$

(4)

After considering the rotational inertia, the dynamic energy of a unit length of a prismatic fractal beam consists of not only the dynamic energy of transverse motion, but also the rotational dynamic energy, which is expressed by the mathematical formula as follows

$$\begin{array}{ll}T\hfill & =0.5\rho {\displaystyle {\iint }_{S^{\beta }}\left[{\left(\frac{{\partial }_{x}D}{\partial t^{\alpha }}\right)}^2+{\left(\frac{{\partial }_{z}D}{\partial t^{\alpha }}\right)}^2\right]}d{y}^{\beta }d{z}^{\beta }=0.5\rho {\displaystyle {\iint }_{S^{\beta }}\left[{\left(-z^{\beta }\frac{{\partial }^{2}W}{\partial x^{\beta }\partial t^{\alpha }}\right)}^2+{\left(\frac{\partial W}{\partial t^{\alpha }}\right)}^2\right]}d{y}^{\beta }d{z}^{\beta }\hfill \\ & =0.5\rho I{\left({\displaystyle \frac{{\partial }^{2}W}{\partial x^{\beta }\partial t^{\alpha }}}\right)}^2+{\displaystyle \frac{1}{2}}\rho S{\left({\displaystyle \frac{\partial W}{\partial t^{\alpha }}}\right)}^2,\hfill \end{array}$$

(5)

where the physical quantity $\rho$ denotes the density of the beam material, the parameter $I={\displaystyle {\iint }_{S^{\beta }}{z}^{2\beta }}d{y}^{\beta }d{z}^{\beta }$ is defined as the rotational inertia taking into account the spatio-temporal fractal modulation, and $S={\displaystyle {\iint }_{S^{\beta }}d{y}^{\beta }d{z}^{\beta }}$ denotes the equivalent cross-sectional area corresponding to the fractal geometric modification. Combining Equations (4) and (5) yields the expression for the total power densities of the fractionally shaped beams for a unit of length in the spatio-temporal fractal dimension as

$$L=T-V=0.5\rho I{\left({\displaystyle \frac{{\partial }^{2}W}{\partial x^{\beta }\partial t^{\alpha }}}\right)}^2+0.5\rho S{\left({\displaystyle \frac{\partial W}{\partial t^{\alpha }}}\right)}^2-0.5EI{\left({\displaystyle \frac{{\partial }^{2}W}{\partial x^{2\beta }}}\right)}^2-0.125ES{\left({\displaystyle \frac{\partial W}{\partial x^{\beta }}}\right)}^4.$$

(6)

According to the Hamilton variational principle, the fractal nonlinear control model of bending wave in functionally graded beams can be obtained by the term-by-term variation of Equation (6) as follows.

$${\displaystyle \frac{{\partial }^{2}W}{\partial t^{2\alpha }}}-r^2{\displaystyle \frac{{\partial }^{4}W}{\partial x^{2\beta }{\partial }^2{t}^{\alpha }}}+r^2{c}_0^2{\displaystyle \frac{{\partial }^{4}W}{\partial x^{4\beta }}}-{\displaystyle \frac{c_0^2}{2}}{\displaystyle \frac{\partial }{\partial x^{\beta }}}{\left({\displaystyle \frac{\partial W}{\partial x^{\beta }}}\right)}^3=0.$$

(7)

Here, $c_0=\sqrt{E/\rho }$, $r=\sqrt{I/S}$. In the next section, the extended minimal (G′/G) expansion method [43] modified by the authors is used to obtain the deflection-gradient precise traveling wave solution of Equation (7) with the auxiliary equation used by the method being $G^{″}+hG=0$. The main difference between this method and the method in reference [52] is that the setting of the proposed solution is different. Solving the auxiliary equation and utilizing its generalized solution, under different conditions, the expression of (G′/G) can be obtained as follows:

$$\left({\displaystyle \frac{G^{\prime }}{G}}\right)=\left\{\begin{array}{l}\sqrt{-h}\left(C_1\sinh \left(\sqrt{-h}\xi \right)+C_2\cosh \left(\sqrt{-h}\xi \right)\right)/\left(C_1\cosh \left(\sqrt{-h}\xi \right)+C_2\sinh \left(\sqrt{-h}\xi \right)\right),h<0\\ \sqrt{h}\left(-C_1\sin \left(\sqrt{h}\xi \right)+C_2\cos \left(\sqrt{h}\xi \right)\right)/\left(C_1\cos \left(\sqrt{h}\xi \right)+C_2\sin \left(\sqrt{h}\xi \right)\right),h>0\\ C_2/(C_1+C_2\xi ),h=0\end{array}\right.,$$

(8)

Among them, C₁ and C₂ are arbitrary constants, and after an equivalent deformation, the expression for (G′/G) can also be expressed in the following form.

$$\left({\displaystyle \frac{G^{\prime }}{G}}\right)=\left\{\begin{array}{l}\sqrt{-h}\tanh \left(\sqrt{-h}\xi +{\xi }_0\right),h<0,\tanh ({\xi }_0)=C_2/C_1,\left|C_2/C_1\right|<1\\ \sqrt{-h}\coth \left(\sqrt{-h}\xi +{\xi }_0\right),h<0,\coth ({\xi }_0)=C_2/C_1,\left|C_2/C_1\right|>1\\ \sqrt{h}\cot \left(\sqrt{h}\xi +{\xi }_0\right),h>0,\cot ({\xi }_0)=C_2/C_1\\ -\sqrt{h}\tan \left(\sqrt{h}\xi +{\xi }_0\right),h>0,\tan ({\xi }_0)=-C_2/C_1\\ C_2/(C_1+C_2\xi ),h=0\end{array}\right..$$

(9)

3. Deflection Gradient Traveling Wave Solution of Fractal Nonlinear Propagation Equation of Bending Wave in Beam

The traveling wave transformation $W(x,t)=W(\xi ),\xi =x^{\beta }-{(ct)}^{\alpha }$ is performed on Equation (7), and $w(\xi )=\partial W/\partial \xi =\partial W/\partial x^{\beta }$ is taken to obtain the following equation.

$$c^{2\alpha }{\displaystyle \frac{\partial w}{\partial \xi }}-r^2{c}^{2\alpha }{\displaystyle \frac{{\partial }^{3}w}{{\partial }^3\xi }}+r^2{c}_0^2{\displaystyle \frac{{\partial }^{3}w}{{\partial }^3\xi }}-{\displaystyle \frac{c_0^2}{2}}{\displaystyle \frac{\partial }{\partial \xi }}{\left(w\right)}^3=0.$$

(10)

where $w(\xi )$ denotes the deflection gradient. Taking

$$p_1=-c^{2\alpha }/(r^2(c^{2\alpha }-c_0^2)),p_2=c_0^2/(2r^2(c^{2\alpha }-c_0^2)),$$

(11)

After organizing Equation (10), integrate the equation about $\xi$ once, and taking the integration constant in the integrated equation as $C$, we get

$${\displaystyle \frac{{\partial }^{2}w}{{\partial }^2\xi }}+p_{1}w+p_2{w}^3-C=0.$$

(12)

Using the equilibrium principle to balance the two terms ${\displaystyle \frac{{\partial }^{2}w}{{\partial }^2\xi }}$ and $p_2{w}^3$ in Equation (12) with respect to the power of (G′/G), the proposed solution of Equation (12) can be obtained as

$$w(\xi )=a_0+a_1{G}^{\prime }/G+b_1{\left(G^{\prime }/G\right)}^{-1}.$$

(13)

The coefficients of each degree of power of (G′/G) are extracted by substituting Equation (13) into Equation (12), which is processed by polynomial expansion and combining like terms. Because each power of (G′/G) is a function of variable $\xi$ which can be arbitrarily valued, in order to make the whole equation equal to zero, we can only take the coefficient of each power of (G′/G) in the equation equal to zero, thus generating a closed system composed of seven nonlinear equations.

$${\left(G^{\prime }/G\right)}^{-3}:b_1^3{p}_2+2h^2{b}_1=0,$$

$${\left(G^{\prime }/G\right)}^{-2}:3p_2{a}_0{b}_1^2=0,$$

$${\left(G^{\prime }/G\right)}^{-1}:3a_0^2{b}_1{p}_2+3a_1{b}_1^2{p}_2+2h{b}_1+b_1{p}_1=0,$$

$${\left(G^{\prime }/G\right)}^0:p_2{a}_0^3+6p_2{a}_0{a}_1{b}_1+p_1{a}_0+C=0,$$

$${\left(G^{\prime }/G\right)}^1:3a_0^2{a}_1{p}_2+3a_1^2{b}_1{p}_2+2h{a}_1+a_1{p}_1=0,$$

$${\left(G^{\prime }/G\right)}^2:3p_2{a}_0{a}_1^2=0,$$

$${\left(G^{\prime }/G\right)}^3:a_1^3{p}_2+2a_1=0.$$

Programming to solve this system of equations consisting of seven coefficients of zero yields four sets of meaningful parametric solutions as shown below.

$$a_0=0,a_1=\pm \sqrt{-2/p_2},b_1=\pm \sqrt{-2h^2/p_2},2h-6\left|h\right|=-p_1,C=0.$$

(14)

$$a_0=0,a_1=\pm \sqrt{-2/p_2},b_1=\mp \sqrt{-2h^2/p_2},2h+6\left|h\right|=-p_1,C=0$$

(15)

$$a_0=0,a_1=\pm \sqrt{-2/p_2},b_1=0,2h=-p_1,C=0$$

(16)

$$a_0=0,a_1=0,b_1=\pm \sqrt{-2h^2/p_2},2h=-p_1,C=0$$

(17)

The values of Equations (14)–(17), combined with Equations (8) or (9), are substituted into Equation (13) to obtain different exact solutions of Equation (10) under different parameter conditions. The exact solution is integrated once about $\xi$, and the exact traveling wave solution of Equation (7) can be obtained by combining the fractal traveling wave transformation.

For Equation (14):

(1)

With h < 0, the expression (G′/G) for the corresponding case is first found in Equations (8) or (9), and then it is replaced in Equation (13) with the parameter expressions in Equations (11) and (14) to get the analytical solution for Equation (10) in the below form.

$$w_{1,2}(x,t)=w_{1,2}(\xi )=\pm {\displaystyle \frac{\sqrt{2}{c}^{\alpha }}{c_0}}\coth \left(\sqrt{{\displaystyle \frac{c^{2\alpha }}{2r^2(c_0^2-c^{2\alpha })}}}(x^{\beta }-{(ct)}^{\alpha })+2{\xi }_0\right).$$

(18)

(2)

With h > 0, the expression (G′/G) for the corresponding case is first found in Equations (8) or (9), and then it is replaced in Equation (13) with the parameter expressions in Equations (11) and (14) to obtain the analytical solution for Equation (10) in the below form.

$$w_{3,4}(x,t)=w_{3,4}(\xi )=\pm {\displaystyle \frac{2c^{\alpha }}{c_0}}\csc \left[\sqrt{{\displaystyle \frac{c^{2\alpha }}{r^2(c_0^2-c^{2\alpha })}}}(x^{\beta }-{(ct)}^{\alpha })+2{\xi }_0\right].$$

(19)

(3)

With h = 0, the computation reveals that b₁ = 0. Similarly, an exact solution of the following form is obtained.

$$w_{5,6}(x,t)=w_{5,6}(\xi )=\pm \sqrt{-2/p_2}{C}_2/\left(C_1+C_2\xi \right).$$

(20)

For Equation (15):

(1)

With h < 0, the expression (G′/G) for the corresponding case is first found in Equations (8) or (9), and then it is replaced in Equation (13) with the parameter expressions in Equations (11) and (15) to obtain the analytical solution for Equation (10) in the below form.

$$w_{7,8}(x,t)=w_{7,8}(\xi )=\pm 2\sqrt{{\displaystyle \frac{2h}{p_2}}}\csc \mathrm{h}\left[2\left(\sqrt{-h}(x^{\beta }-{(ct)}^{\alpha })+{\xi }_0\right)\right].$$

(21)

(2)

With h > 0, the expression (G′/G) for the corresponding case is first found in Equations (8) or (9), and then it is replaced in Equation (13) with the parameter expressions in Equations (11) and (15) to obtain the analytical solution for Equation (10) in the below form.

$$w_{9,10}(x,t)=w_{9,10}(\xi )=\pm 2\sqrt{-{\displaystyle \frac{2h}{p_2}}}\tan \left(2\sqrt{h}(x^{\beta }-{(ct)}^{\alpha })+2{\xi }_0\right).$$

(22)

(3)

With h = 0, the calculation reveals that b₁ = 0, at which point the solution is the same as the solution to Equation (20).

For Equation (16):

(1)

With h < 0, the expression (G′/G) for the corresponding case is first found in Equations (8) or (9), and then it is replaced in Equation (13) with the parameter expressions in Equations (11) and (16) to obtain the analytical solution of Equation (10) in the below form.

$$w_{11,12}(\xi )=\pm \sqrt{{\displaystyle \frac{2h}{p_2}}}\left({\displaystyle \frac{C_1\sinh \left(\sqrt{-h}\xi \right)+C_2\cosh \left(\sqrt{-h}\xi \right)}{C_1\cosh \left(\sqrt{-h}\xi \right)+C_2\sinh \left(\sqrt{-h}\xi \right)}}\right).$$

(23)

(2)

With h > 0, the expression (G′/G) for the corresponding case is first found in Equations (8) or (9), and then it is replaced in Equation (13) with the parameter expressions in Equations (11) and (16) to obtain the analytical solution for Equation (10) in the below form.

$$w_{11,12}(\xi )=\pm \sqrt{{\displaystyle \frac{2h}{p_2}}}\left({\displaystyle \frac{C_1\sinh \left(\sqrt{-h}\xi \right)+C_2\cosh \left(\sqrt{-h}\xi \right)}{C_1\cosh \left(\sqrt{-h}\xi \right)+C_2\sinh \left(\sqrt{-h}\xi \right)}}\right).$$

(24)

(3)

With h = 0, the calculation reveals that b₁ = 0, at which point the solution is the same as the solution to Equation (20).

For Equation (17):

(1)

With h < 0, the expression (G′/G) for the corresponding case is first found in Equations (8) or (9), and then it is replaced in Equation (13) with the parameter expressions in Equations (11) and (17) to obtain the analytical solution for Equation (10) in the below form.

$$w_{15,16}(\xi )=\pm \sqrt{{\displaystyle \frac{6h}{p_2}}}{\left({\displaystyle \frac{C_1\sinh \left(\sqrt{-h}\xi \right)+C_2\cosh \left(\sqrt{-h}\xi \right)}{C_1\cosh \left(\sqrt{-h}\xi \right)+C_2\sinh \left(\sqrt{-h}\xi \right)}}\right)}^{-1}.$$

(25)

(2)

With h > 0, the expression (G′/G) for the corresponding case is first found in Equations (8) or (9), and then it is replaced in Equation (13) with the parameter expressions in Equations (11) and (17) to obtain the analytical solution of Equation (10) in the below form.

$$w_{17,18}(\xi )=\pm \sqrt{-{\displaystyle \frac{6h_2}{p_2}}}{\left({\displaystyle \frac{-C_1\sin \left(\sqrt{h_2}\xi \right)+C_2\cos \left(\sqrt{h_2}\xi \right)}{C_1\cos \left(\sqrt{h_2}\xi \right)+C_2\sin \left(\sqrt{h_2}\xi \right)}}\right)}^{-1}.$$

(26)

(3)

With h = 0, at which point b₁ = 0, the solution at this point is not discussed, since it is constant at this point.

4. Discussion of Analytical Solution and Numerical Simulation Analysis

In this section, firstly, the similarities and differences between the extended minimal (G′/G) method and the (G′/G) method and the (G/G′) method are expounded by means of a comparison of the groups. Next, the fractal kink solitary wave is searched from a large number of accurate solutions for the deflection gradient, and some of the parameters are calculated. Then, the two-dimensional and three-dimensional numerical simulation of fractal kink solitary waves is carried out. Finally, the deflection curve of the beam corresponding to the fractal kink solitary wave is given and the numerical simulation is carried out. Through an analysis of the numerical simulation results, it can be seen that the space–time fractal dimension can modulate the amplitude, waveform, and rotation of the deflection gradient solitary wave in the functionally graded beam. When the spatial fractal dimension index decreases, the geometric nonlinearity is suppressed.

4.1. Comparison Results and Discussion of Solutions

After checking, it is found that the parameter solution group obtained by using the (G′/G) method is the same as the parameter solution group represented by the Formula (16), and the parameter solution group obtained by using the (G/G′) method is the same as the parameter solution group represented by the Formula (17). Due to the arbitrariness of C₁ and C₂, compared with the solutions obtained by Equations (16) and (17), we can see that the exact traveling wave solutions obtained by Equations (16) and (17) are the same. An example is given to illustrate that the extended minimal (G′/G) method relieves the solutions of the (G′/G) method and (G/G′) method, and there are new forms of solutions $w_{3,4}(x,t)$ and $w_{7,8}(x,t)$. This reveals that the extended minimal (G′/G) method is different from the (G′/G) method and the (G/G′) method. The algorithm maps the solution space of the nonlinear wave equation to the solution set of two associated Riccati equations, and in this process, a new waveform solution with hyperbolic function characteristics (csch²/csc²) is generated.

4.2. Parameter Calculation and Fractal Kink Solitary Wave

In order to carry out numerical simulation, a functionally graded material beam made of metal and ceramic is considered. The width of the beam is $b=0.22\mathrm{m}$, the height is $h=0.27\mathrm{m}$, the elastic modulus is $E=70\mathrm{Gpa}$, and the density is $\rho =2.7\times {10}^3{\mathrm{kg}/\mathrm{m}}^3$. The following results can be obtained by calculating the material parameters.

$$\begin{array}{l}{c}_0=\sqrt{E/\rho }\approx 5091,I={\displaystyle {\iint }_{S^{\beta }}{z}^{2\beta }}d{y}^{\beta }d{z}^{\beta }={\displaystyle \frac{4^{1-2\beta }}{3}}{b}^{\beta }{h}^{3\beta },\\ S={\displaystyle {\iint }_{S^{\beta }}d{y}^{\beta }d{z}^{\beta }{=4}^{1-\beta }}{(hb)}^{\beta },r=\sqrt{I/S}=\sqrt{{\displaystyle \frac{h^{2\beta }}{3\cdot 4^{\beta }}}}={\displaystyle \frac{\sqrt{3}}{3}}{\left({\displaystyle \frac{h}{2}}\right)}^{\beta }.\end{array}$$

(27)

In Equation (23), letting C₂ = 0, we will obtain the kinked isolated wave solution, which is organized and rewritten as follows

$$w_{11,12}(x,t)=\pm {\displaystyle \frac{\sqrt{2}{c}^{\alpha }}{c_0}}\tanh \left(\sqrt{{\displaystyle \frac{c^{2\alpha }}{2r^2(c_0^2-c^{2\alpha })}}}(x^{\beta }-{(ct)}^{\alpha })\right).$$

(28)

4.3. Two-Dimensional Numerical Simulation Analysis of Fractal Kink Solitary Wave

The amplitude is recorded as $\mathrm{A}=\sqrt{2}{c}^{\alpha }/c_0$. Next, we examine the influence of the spatio-temporal fractal dimension index on the kink waveform. Take $c=4000$ m/s and draw a two-dimensional plot of Equation (28) with positive coefficients of $(\alpha ,\beta )$ taking $(1,1),(0.99,0.6),(0.98,0.4)$ in turn, as shown in Figure 1. It can be noticed from Figure 1 that as the space–time dimension indexes $(\alpha ,\beta )$ become smaller, the amplitude of the fractal kink solitary wave represented by the induced Equation (28) becomes smaller and the kink width becomes larger, and waveform smoothing is realized, indicating that the space–time fractal dimension index can be used to study the amplitude attenuation law and waveform modulation of the deflection gradient solitary wave in the functionally graded beam.

4.4. Three-Dimensional Numerical Simulation Analysis of Fractal Kink Solitary Wave

In this subsection, three-dimensional numerical simulations are given for each of the three variations of the spatio-temporal dimension index $(\alpha ,\beta )$ (each of the spatio-temporal dimension indexes becomes smaller individually and jointly) for the fractal kink isolated waveforms to further investigate the effect of the spatio-temporal fractal dimension on the kink waveforms.

Figure 2 is a three-dimensional graph of Formula (28) with the change in the spatial fractal dimension $\beta$ when $c=4000$ m/s, $\alpha =1$. It can be noticed from Figure 2 that as $\beta$ gradually decreases, the three-dimensional plot of the kink wave gradually rotates clockwise on the t-x plane.

Taking $c=4000$ m/s, $\beta =1$, the three-dimensional diagram of Equation (28) with time fractal dimension $\alpha$ is drawn, as illustrated in Figure 3. It can be noticed from Figure 3 that with the gradual decrease of $\alpha$, the three-dimensional plot of the kink wave gradually rotates counterclockwise on the t-x plane and the amplitude decreases, and the waveform is smoother. At this time, the amplitude decreases, and the smoother results are in agreement with the results of two-dimensional numerical simulation. The reason for this result is that as $\alpha$ gradually decreases, the nonlinear wave velocity $c^{\alpha }$ decreases, resulting in a weakening of the nonlinear self-steepening effect and a more uniform distribution of energy in the spatial domain.

Taking $c=4000$ m/s, the three-dimensional graph of Equation (28) with the simultaneous change in the space–time fractal dimension $(\alpha ,\beta )$ is drawn as illustrated in Figure 4. It can be observed from Figure 4 that as the synchronization of $(\alpha ,\beta )$ gradually becomes smaller, the amplitude of the kink wave decreases, the waveform is smoother, and the wave does not rotate on the t-x plane.

The results of three-dimensional numerical simulation show that the contribution of spatio-temporal fractal dimension to the rotation angle is the same in magnitude and opposite in direction. It can not only modulate the amplitude and waveform of the deflection gradient kink solitary wave in the beam, but can also modulate the rotation of the wave.

4.5. Numerical Simulation Analysis of the Deflection Curve of the Beam Corresponding to the Fractal Kink Solitary Wave

For the integral of Equation (28) with respect to $\xi$, let the integral constant be 0, and the deflection curve equation of the functionally graded material beam is obtained as follows.

$$W_{11,12}(\xi )=\pm {\displaystyle \frac{2r\sqrt{c_0^2-c^{2\alpha }}}{c_0}}\ln \left[\cosh \left(\sqrt{{\displaystyle \frac{c^{2\alpha }}{2r^2(c_0^2-c^{2\alpha })}}}\xi \right)\right].$$

(29)

where $\xi =x^{\beta }-{(ct)}^{\alpha }$, take $c=4000$ m/s and draw a two-dimensional diagram of Formula (29) with positive coefficients of $(\alpha ,\beta )$ taking $(0.8,0.8),(0.8,0.6),(0.8,0.5)$ in turn, as shown in Figure 5.

It can be seen from Figure 5 that the deflection curve of the beam becomes flatter with the reduction in the spatial fractal dimension $\beta$, indicating that the reduction in the spatial fractal dimension index has an inhibitory effect on the geometric nonlinearity.

5. Conclusions

Based on the framework of fractal derivative theory, this study establishes a fractal dynamic model of nonlinear bending wave propagation in functionally graded material beams and reveals the regulation mechanism of space–time fractal dimension on wave propagation characteristics. The primary conclusions are as below.

• By combining the geometric nonlinear effect and the multi-scale heterogeneity of materials, the bending wave control equation of double nonlinear coupling is derived, which provides a theoretical framework with microstructure characterization ability for the wave analysis of functionally graded materials.
• The deflection gradient traveling wave solutions of the equation in the form of hyperbolic function, trigonometric function and rational function are obtained by using the extended minimal (G′/G) expansion method. By comparing the solution sets of the equations, it is revealed that the extended minimal (G′/G) expansion method can obtain the exact solutions of two new types of csch² and csc² in addition to the traditional (G′/G) expansion method, which confirms the simplicity and universality of the method for solving fractal nonlinear systems.
• The two-dimensional numerical simulation shows that the synergistic reduction of the spatiotemporal fractal dimension can induce the amplitude attenuation and characteristic width expansion of the fractal kink solitary wave and realize waveform smoothing, which is derived from the equivalent characterization of the layered and porous microstructure of the material by the fractal spatiotemporal transformation. The three-dimensional numerical simulation shows that the decrease in the spatial dimension alone causes the clockwise rotation of the fractal kink isolated wave, and the reduction in the time dimension drives the counterclockwise rotation and the amplitude decreases, the waveform is smoother, and the rotation effect is offset when the two cooperate. These reveal the controllable modulation characteristics of the spatiotemporal fractal parameters on the amplitude, waveform and propagation direction of the wave.
• The analytical expression of the deflection corresponding to the fractal kink solitary wave is obtained, and the numerical simulation shows that the deflection curve becomes smoother with the reduction in the spatial fractal dimension, indicating that the reduction in the spatial fractal dimension index has an inhibitory effect on the geometric nonlinearity. This phenomenon shows that reducing the spatial non-locality index may inhibit the strain energy accumulation caused by large deflection and provide theoretical support for the geometric nonlinear design of functionally graded beams.

In this work, a cross-scale wave analysis framework is constructed by fractal derivatives. The regulation of spatio-temporal fractal dimension revealed in this work opens up a new way for nonlinear wave control in complex structures. Follow-up research may further explore the quantitative mapping relationship between fractal parameters and material microstructural parameters.