Exploring the possibilities of FDM filaments comprising natural fiber-reinforced biocomposites for additive manufacturing

Mahdi Rafiee; Roozbeh Abidnejad; Anton Ranta; Krishna Ojha; Alp Karakoç; Jouni Paltakari; Mahdi Rafiee; Roozbeh Abidnejad; Anton Ranta; Krishna Ojha; Alp Karakoç; Jouni Paltakari

doi:10.3934/matersci.2021032

AIMS Materials Science

2021, Volume 8, Issue 4: 524-537. doi: 10.3934/matersci.2021032

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Research article Special Issues

Exploring the possibilities of FDM filaments comprising natural fiber-reinforced biocomposites for additive manufacturing

1.
Aalto University, Department of Bioproducts and Biosystems, Vuorimiehentie 1, Espoo, Finland
2.
Aalto University, Department of Communications and Networking, Maarintie 8, Espoo, Finland

Received: 15 April 2021 Accepted: 03 June 2021 Published: 24 June 2021

In recent years, with the recent advancements in the field of additive manufacturing, the use of biobased thermoplastic polymers and their natural fiber-reinforced biocomposite filaments have been rapidly emerging. Compared to their oil-based counterparts, they provide several advantages with their low carbon footprints, ease of reusability and recyclability and abundancy, and comparable price ranges. In consideration of their increasing usage, the present study focused on the development and analysis of biocomposite material blends and filaments by merging state-of-the-art manufacturing and material technologies. A thorough suitability study for fused deposition modeling (FDM), which is used to manufacture samples by depositing the melt layer-by-layer, was carried out. The mechanical, thermal, and microstructural characterization of birch fiber reinforced PLA composite granules, in-house extruded filaments, and printed specimens were investigated. The results demonstrated the printability of biocomposite filaments. However, it was also concluded that the parameters still need to be optimized for generic and flawless filament extrusion and printing processes. Thus, minimal labor and end-products with better strength and resolutions can be achieved.

Keywords:

Citation: Mahdi Rafiee, Roozbeh Abidnejad, Anton Ranta, Krishna Ojha, Alp Karakoç, Jouni Paltakari. Exploring the possibilities of FDM filaments comprising natural fiber-reinforced biocomposites for additive manufacturing[J]. AIMS Materials Science, 2021, 8(4): 524-537. doi: 10.3934/matersci.2021032

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Abstract

1. Introduction

Since pioneering works of Pecora and Carroll's ^[1], chaos synchronization and control have turned a hot topic and received much attention in various research areas. A number of literatures shows that chaos synchronization can be widely used in physics, medicine, biology, quantum neuron and engineering science, particularly in secure communication and telecommunications ^[1,2,3]. In order to realize synchronization, experts have proposed lots of methods, including complete synchronization and $Q$ - $S$ synchronization ^[4,5], adaptive synchronization ^[6], lag synchronization^[7,8], phase synchronization ^[9], observer-based synchronization ^[10], impulsive synchronization ^[11], generalized synchronization ^[12,13], lag projective synchronization ^[14,15], cascade synchronization et al ^{[16,17,18,19,20]}. Among them, the cascade synchronization method is a very effective algorithm, which is characterized by reproduction of signals in the original chaotic system to monitor the synchronized motions.

It is know that, because of the complexity of fractional differential equations, synchronization of fractional-order chaotic systems is more difficult but interesting than that of integer-order systems. Experts find that the key space can be enlarged by the regulating parameters in fractional-order chaotic systems, which enables the fractional-order chaotic system to be more suitable for the use of the encryption and control processing. Therefore, synchronization of fractional-order chaotic systems has gained increasing interests in recent decades ^{[21,22,23,24,25,26,27,28,29,30,31]}. It is noticed that most synchronization methods mentioned in ^{[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]} work for integer-order chaotic systems. Here, we shall extend to cascade synchronization for integer-order chaotic systems to a kind of general form, namely function cascade synchronization (FCS), which means that one chaotic system may be synchronized with another by sending a signal from one to the other wherein a scaling function is involved. The FCS is effective both for the fractional order and integer order chaotic systems. It constitutes a general method, which can be considered as a continuation and extension of earlier works of ^[13,16,19]. The nice feature of our method is that we introduce a scaling function for achieving synchronization of fractional-order chaotic systems, which can be chosen as a constant, trigonometric function, power function, logarithmic and exponential function, hyperbolic function and even combinations of them. Hence, our method is more general than some existing methods, such as the complete synchronization approach and anti-phase synchronization approach et al.

To sum up, in this paper, we would like to use the FCS approach proposed to study the synchronization of fractional-order chaotic systems. We begin our theoretical work with the Caputo fractional derivative. Then, we give the FCS of the fractional-order chaotic systems in theory. Subsequently, we take the fractional-order unified chaotic system as a concrete example to test the effectiveness of our method. Finally, we make a short conclusion.

2. The method of FCS for fractional-order chaotic systems

As for the fractional derivative, there exists a lot of mathematical definitions ^[32,33]. Here, we shall only adopt the Caputo fractional calculus, which allows the traditional initial and boundary condition assumptions. The Caputo fractional calculus is described by

$\begin{eqnarray} \frac{d^q f(t)}{dt^q} = \frac{1}{{\Gamma}(q-n)}\int^t_0 \frac{f^{(n)}(\xi)}{(t-\xi)^{q-n+1}}d\xi, \quad\quad n-1 \lt q \lt n . \end{eqnarray}$

(2.1)

Here, we give the function cascade synchronization method to fractional-order chaotic systems. Take a fractional-order dynamical system:

$\begin{eqnarray} \frac{d^q x}{d t^q} = f(x) = L x+N( x) \end{eqnarray}$

(2.2)

as a drive system. In the above $x = (x_1, x_2, x_3)^T$ is the state vector, $f: R^3 \rightarrow R^3$ is a continuous function, $L x$ and $N(x)$ represent the linear and nonlinear part of $f(x)$ , respectively.

Firstly, on copying any two equations of $(2.2)$ , such as the first two, one will obtain a sub-response system:

$\begin{eqnarray} \frac{d^q y}{d t^q} = L_1 y+N_1(y, x_3)+\tilde{U} \end{eqnarray}$

(2.3)

with $y = (X_1, Z)^T$ . In the above, $x_3$ is a signal provided by (2.2), while $\tilde{U} = (u_1, u_2)^T$ is a controller to be devised.

For the purpose of realizing the synchronization, we now define the error vector function via

$\begin{eqnarray} \tilde{e} = y-\tilde{Q}(\tilde{x})\tilde{x} \end{eqnarray}$

(2.4)

where $\tilde{e} = (e_1, e_2)^T$ , $\tilde{x} = (x_1, x_2)^T$ and $\tilde{Q}(\tilde{x}) = {\rm diag}({Q}_1(x_1), {Q}_2(x_2))$ .

Definition 1. For the drive system (2.2) and response system (2.3), one can say that the synchronization is achieved with a scaling function matrix $\tilde{Q}(\tilde{x})$ if there exists a suitable controller $\tilde{U}$ such that

$\begin{eqnarray} \lim\limits_{t \to \infty }{||\tilde{e}||} = \lim\limits_{t \to \infty }{||y-\tilde{Q}(\tilde{x})\tilde{x}||} = 0. \end{eqnarray}$

(2.5)

Remark 1. We would like to point out that one can have various different choices on the scaling function $\tilde{Q}(\tilde{x})$ , such as constant, power function, trigonometric function, hyperbola function, logarithmic and exponential function, as well as limited quantities of combinations and composite of the above functions. Particularly, when $\tilde{Q}(\tilde{x}) = I$ and $-I$ ( $I$ being a unit matrix), the problem is reducible to the complete synchronization and anti-phase synchronization of fractional-order chaotic systems, respectively. When $\tilde{Q}(\tilde{x}) = \alpha I$ , it becomes to the project synchronization. And when $\tilde{Q}(\tilde{x})$ = ${\rm diag}(\alpha_1, \alpha_2)$ , it turns to the modified projective synchronization. Hence, our method is more general than the existing methods in ^[4,13].

It is noticed from (2.5) that the system (2.3) will synchronize with (2.2) if and only if the error dynamical system (2.5) is stable at zero. For this purpose, an appropriate controller $\tilde{U}$ such that (2.5) is asymptotical convergent to zero is designed, which is described in the following theorem.

Theorem 1. For a scaling function matrix $\tilde{Q}(\tilde{x})$ , the FCS will happen between (2.2) and (2.3) if the conditions:

(i) the controller $\tilde{U}$ is devised by

$\begin{eqnarray} \tilde{U} = \tilde{K}\tilde{e}-N_1(y, x_3)+\tilde{Q}(\tilde{x})N_1(\tilde{x})+\tilde{P}(\tilde{x})\tilde{x} \end{eqnarray}$

(2.6)

(ii) the matrix $\tilde{K}$ is a $2\times 2$ matrix such that

$\begin{eqnarray} L_1+\tilde{K} = -\tilde{C}, \end{eqnarray}$

(2.7)

are satisfied simultaneously. In the above, $\tilde{P}(\tilde{x}) = {\rm {diag}}(\dot{Q}_1(x_1)\frac{ d^q x_1}{d t^q}, \dot{Q}_2(x_2)\frac{ d^q x_2}{d t^q})$ , $\tilde{K}$ is a $2\times 2$ function matrix to be designed. While $\tilde{C} = (\tilde{C}_{ij})$ is a $2\times 2$ function matrix wherein

$\begin{eqnarray} \tilde{C}_{ii} \gt 0 \quad \quad{\rm and} \quad \quad \tilde{C}_{ij} = -\tilde{C}_{ji}, \quad i\neq j. \end{eqnarray}$

(2.8)

Remark 2. It needs to point out that the construction of the suitable controller $\tilde{U}$ plays an important role in realizing the synchronization between (2.2) and (2.3). Theorem 2 provides an effective way to design the controller. It is seen from the theorem that the controller $\tilde{U}$ is closely related to the matrix $\tilde{C}$ . Once the condition $(2.8$ ) is satisfied, one will has many choices on the controller $\tilde{U}$ .

Remark 3. Based on the fact that the fractional orders themselves are varying parameters and can be applied as secret keys when the synchronization algorithm is adopted in secure communications, it is believed that our method will be more suitable for some engineering applications, such as chaos-based encryption and secure communication.

Proof: Let's turn back to the error function given in (2.4). Differentiating this equation with respect to $t$ and on use of the first two equations of (2.2) and (2.3), one will obtain the following dynamical system

$\begin{eqnarray} \frac{d^q \tilde{e}}{d t^q}& = &\frac{d^q y}{d t^q}-\tilde{Q}(\tilde{x})\frac{d^q x}{d t^q}-\tilde{P}(\tilde{x})\tilde{x}\\ & = &L_1 y+N_1(y, x_3)+\tilde{U}-\tilde{Q}(\tilde{x})[L_1 \tilde{x}+N_1(\tilde{x})]-\tilde{P}(\tilde{x})\tilde{x}\\ & = &L_1\tilde{e}+N_1(y, x_3)-\tilde{Q}(\tilde{x})N_1(\tilde{x})-\tilde{P}(\tilde{x})\tilde{x}+\tilde{K}\tilde{e}-N_1(y, x_3)+\tilde{Q}(\tilde{x})N_1(\tilde{x})+\tilde{P}(\tilde{x})\tilde{x}\\ & = &(L_1+\tilde{K})\tilde{e}. \end{eqnarray}$

(2.9)

Assuming that $\lambda$ is an arbitrary eigenvalue of matrix $L_1+\tilde{K}$ and its eigenvector is recorded as $\eta$ , i.e.

$\begin{eqnarray} ( L_1+\tilde{K} )\eta = \lambda \eta, \quad \eta\neq 0. \end{eqnarray}$

(2.10)

On multiplying (2.10) by $\eta^{\mathbf {H}}$ on the left, we obtain that

$\begin{eqnarray} \eta^{\mathbf {H}}( L_1+\tilde{K} )\eta = \lambda \eta^{\mathbf {H}}\eta \end{eqnarray}$

(2.11)

where ${\mathbf {H}}$ denotes conjugate transpose. Since $\bar{\lambda}$ is also an eigenvalue of $L_1+\tilde{K}$ , we have that

$\begin{eqnarray} \eta^{\mathbf {H}}( L_1+\tilde{K} )^{\mathbf {H}} = \bar{\lambda} \eta^{\mathbf {H}}. \end{eqnarray}$

(2.12)

On multiplying (2.12) by $\eta$ on the right, we derive that

$\begin{eqnarray} \eta^{\mathbf {H}}( L_1+\tilde{K} )^{\mathbf {H}}\eta = \bar{\lambda} \eta^{\mathbf {H}}\eta \end{eqnarray}$

(2.13)

From (2.11) and (2.13), one can easily get that

$\begin{eqnarray} \lambda+\bar{\lambda}& = &\eta^{\mathbf {H}}[( L_1+\tilde{K} )^{\mathbf {H}}+( L_1+\tilde{K} )]\eta/\eta^{\mathbf {H}}\eta\\[0.06in] & = &-\eta^{\mathbf {H}}(\tilde{C}+\tilde{C}^{\mathbf {H}})\eta/\eta^{\mathbf {H}}\eta\\[0.06in] & = &-\eta^{\mathbf {H}}\Lambda\eta/\eta^{\mathbf {H}}\eta \end{eqnarray}$

(2.14)

with $\Lambda = \tilde{C}+\tilde{C}^{\mathbf {H}}$ . Since $\tilde{C}$ satisfy the condition (2.8), one can know that $\Lambda$ denotes a real positive diagonal matrix. Thus we have $\eta^{\mathbf {H}}\Lambda\eta > 0$ . Accordingly, we can get

$\begin{eqnarray} \lambda+\bar{\lambda} = 2\mathbf {Re}(\lambda) = -\eta^{\mathbf {H}}\Lambda\eta/\eta^{\mathbf {H}}\eta \lt 0, \end{eqnarray}$

(2.15)

which shows

$\begin{eqnarray} |\mathbf{arg}\lambda| \gt \frac{\pi}{2} \gt \frac{q\pi}{2}. \end{eqnarray}$

(2.16)

According to the stability theorem in Ref. ^[34], the error dynamical system (2.9) is asymptotically stable, i.e.

$\begin{eqnarray} \lim\limits_{t \to \infty }{||\tilde{e}||} = \lim\limits_{t \to \infty }{||y-\tilde{Q}(\tilde{x})\tilde{x}||} = 0, \end{eqnarray}$

(2.17)

which implies that synchronization can be achieved between (2.2) and (2.3). The proof is completed.

Next, on copying the last two equations of (2.2), one will get another sub-response system:

$\begin{eqnarray} \frac{d^q z}{d t^q} = L_2 z+N_2(z, X_1)+\bar{U} \end{eqnarray}$

(2.18)

where $X_1$ is a synchronized variable in (2.3), $z = (X_2, X_3)^T$ and $\bar{U} = (u_3, u_4)^T$ is the controller being designed.

Here, we make analysis analogous to the above. Now we define the error $\bar{e}$ via

$\begin{eqnarray} \tilde{e} = z-\bar{Q}(\bar{x})\bar{x} \end{eqnarray}$

(2.19)

where $\bar{e} = (e_3, e_4)^T$ , $\bar{x} = (x_2, x_3)^T$ and $\bar{Q}(\bar{x}) = {\rm diag}({Q}_3(x_2), {Q}_4(x_3))$ . If devising the the controller $\bar{U}$ as

$\begin{eqnarray} \bar{U} = \bar{K}\bar{e}-N_2(z, X_1)+\bar{Q}(\bar{x})N_2(\bar{x})+\bar{P}(\bar{x})\bar{x} \end{eqnarray}$

(2.20)

and $L_2+\bar{K}$ satisfying

$\begin{eqnarray} L_2+\bar{K} = -\bar{C} \end{eqnarray}$

(2.21)

where $\bar{P}(\bar{x}) = {\rm {diag}}(\dot{Q}_3(x_2)\frac{ d^q x_2}{d t^q}, \dot{Q}_4(x_3)\frac{ d^q x_3}{d t^q})$ , $\bar{C} = (\bar{C}_{ij})$ denotes a $2\times 2$ function matrix satisfying

$\begin{eqnarray} \bar{C}_{ii} \gt 0 \quad \quad{\rm and} \quad \quad \bar{C}_{ij} = -\bar{C}_{ji}, \quad i\neq j, \end{eqnarray}$

(2.22)

then the error dynamical system (2.19) satisfies

$\begin{eqnarray} \lim\limits_{t \to \infty }{||\bar{e}||} = \lim\limits_{t \to \infty }{||z-\bar{Q}(\bar{x})\bar{x}||} = 0. \end{eqnarray}$

(2.23)

Therefore, one achieve the synchronization between the system (2.2) and (2.18). Accordingly, from (2.5) and (2.23), one can obtain that

$\begin{eqnarray} \left\{\begin{array}{l} \lim\limits_{t \to \infty }{||X_1-Q_1(x_1)x_1||} = 0, \\[0.08in] \lim\limits_{t \to \infty }{||X_2-Q_3(x_2)x_2||} = 0, \\[0.08in] \lim\limits_{t \to \infty }{||X_3-Q_4(x_3)x_3||} = 0. \end{array}\right. \end{eqnarray}$

(2.24)

which indicates the FCS is achieved for the fractional order chaotic systems.

3. Applications of FCS method to fractional-order unified chaotic system

In the sequel, we shall extend the applications of FCS approach to the fractional-order unified chaotic system to test the effectiveness.

The fractional-order unified chaotic system is described by:

$\begin{eqnarray} \left\{\begin{array}{l} \\ \frac{d^{q} x_{1}}{d t^q} = (25a+10)(x_2-x_1), \\[0.1in] \\ \frac{d^q x_{2}}{d t^q} = (28-35a)x_1-x_1x_3+(29a-1)x_2, \\[0.1in] \\ \frac{d^q x_{3}}{d t^q } = x_1x_2-\frac{a+8}{3}x_3, \end{array}\right. \end{eqnarray}$

(3.1)

where $x_i, (i = 1, 2, 3)$ are the state parameters and $a\in[0, 1]$ is the control parameter. It is know that when $0\leq a < 0.8$ , the system (3.1) corresponds to the fractional-order Lorenz system ^[35]; when $a = 0.8$ , it is the Lü system ^[36]; while when $0.8 < a < 1$ , it turns to the Chen system ^[37].

According to the FCS method in section 2, we take (3.1) as the drive system. On copying the first two equation, we get a sub-response system of (3.1):

$\begin{eqnarray} \left\{\begin{array}{l} \\ \frac{d^q X_{1}}{d t^q} = (25a+10)(Z-X_1)+u_1, \\[0.12in] \\ \frac{d^q Z}{d t^q} = (28-35a)X_1-Z x_3+(29a-1)Z+u_2, \end{array}\right. \end{eqnarray}$

(3.2)

where $\tilde{U} = (u_1, u_2)^T$ is a controller to be determined. In the following, we need to devise the desired controller $\tilde{U}$ such that (3.1) can be synchronized with (3.2). For this purpose, we set the error function $\tilde{e} = (e_1, e_2)$ via :

$\begin{eqnarray} \tilde{e} = (e_1, e_2) = (X_1-x_1(x_1^2+\alpha_1), Z-x_2\tanh{x_2}). \end{eqnarray}$

(3.3)

On devising the controller $\tilde{U}$ as (2.6), one can get that the error dynamical system is

$\begin{eqnarray} \frac{d^q \tilde{e}}{d t^q} = (L_1+\tilde{K})\tilde{e}, \end{eqnarray}$

(3.4)

where

$\begin{equation} \begin{split} & L_1 = \left( \begin{array}{cc} -10-25a & -10-25a\\ 28-35a & 29a-1 \\ \end{array} \right), \end{split} \quad\quad\quad \begin{split} & N_1(y, x_3) = \left( \begin{array}{c} 0 \\ -X_1x_3 \\ \end{array} \right){.} \end{split} \end{equation}$

(3.5)

If choosing, for example, the matrix $\tilde{K}$ as

$\begin{equation} \begin{split} & \tilde{K} = \left( \begin{array}{cc} -\lambda_1+25a+10 & x_1+x_1x_2-25a\\ -x_1-x_1x_2+35a-38 & -\lambda_2-29a+1 \\ \end{array} \right), \end{split} \end{equation}$

(3.6)

where $\lambda_1 > 0$ and $\lambda_2 > 0$ , then one can obtain that

$\begin{equation} \begin{split} & \tilde{C} = \left( \begin{array}{cc} -\lambda_1 & x_1+x_1x_2+10\\ -x_1-x_1x_2-10 & -\lambda_2 \\ \end{array} \right). \end{split} \end{equation}$

(3.7)

Therefore the dynamical system (3.4) becomes

$\begin{eqnarray} \frac{d^q \tilde{e}}{d t^q} = \begin{split} &\left( \begin{array}{cc} -\lambda_1 & x_1+x_1x_2\\ -x_1-x_1x_2 & -\lambda_2 \\ \end{array} \right)\tilde{e}. \end{split} \end{eqnarray}$

(3.8)

According to Theorem 2, the synchronization is realized in the system (3.1) and (3.2).

Subsequently, on copying the last two equations of (3.1), we get another sub-response system:

$\begin{eqnarray} \left\{\begin{array}{l} \\ \frac{\partial^q X_{2}}{\partial t^q} = (28-35a)X_1-X_1 X_3+(29a-1)X_2+u_3, \\[0.15in] \\ \frac{\partial^q X_{3}}{\partial t^q} = X_1X_2-\frac{a+8}{3}X_3+u_4, \end{array}\right. \end{eqnarray}$

(3.9)

where $\bar{U} = ({u_3, \ u_4})^T$ is the controller needed. When choosing the error function $\bar{e} = (e_3, e_4)$ as:

$\begin{eqnarray} \bar{e} = (e_3, e_4) = (X_2-\alpha_2 x_2, X_3-x_3(\alpha_3+e^{-x_3})), \end{eqnarray}$

(3.10)

and the controller $\bar{U}$ as (2.20), where

$\begin{equation} \begin{split} & L_2 = \left( \begin{array}{cc} 29a-1 & 0\\ 0 & -\frac{a+8}{3} \\ \end{array} \right), \end{split} \quad\quad\quad \begin{split} & N_2(z, X_1) = \left( \begin{array}{c} -X_1X_3 \\ X_1X_2 \\ \end{array} \right), \end{split} \end{equation}$

(3.11)

and the matrix $\bar{K}$ is chosen by

$\begin{equation} \begin{split} & \bar{K} = \left( \begin{array}{cc} -\lambda_3-29a+1 & 1+x_2x_3+e^{-x_3}\\ -1-x_2x_3-e^{-x_3} & -\lambda_4-\frac{a+8}{3} \\ \end{array} \right), \end{split} \end{equation}$

(3.12)

where $\lambda_3 > 0$ and $\lambda_4 > 0$ . Calculations show that the error dynamical system (2.19) becomes

$\begin{eqnarray} \frac{d^q \bar{e}}{d t^q} = \begin{split} &\left( \begin{array}{cc} -\lambda_3 & 1+x_2x_3+e^{-x_3}\\ -1-x_2x_3-e^{-x_3} & -\lambda_4 \\ \end{array} \right)\bar{e}. \end{split} \end{eqnarray}$

(3.13)

which, according to the stability theorem, indicates that $\bar{e}$ will approach to zero with time evolutions. Therefore, the FCS is realized for the fractional-order unified chaotic system.

In the above, we have revealed that the FCS is achieved for the fractional-order unified chaotic system in theory. In the sequel, we shall show that the FCS is also effective in the numerical algorithm.

For illustration, we set the fractional order $q = 0.98$ and the parameters $\lambda_i (i = 1, \cdots, 4)$ as $(\lambda_1, \lambda_2, \lambda_3, \lambda_4) = (2, 3, 0.5, 0.3)$ . It is noticed that when the value of $a\in[0, 1]$ is given, the system (3.1) will be reduced to a concrete system. For example, when $a = 0.2$ , it corresponds to the fractional-order Lorenz system. The chaotic attractors are depicted in Figure 1. Time responses of states variables and synchronization errors of the Lorenz system are showed in and , respectively. When $a = 0.8$ , it is the fractional-order Lü system. The chaotic attractors, time responses of state variables and synchronization errors are exhibited in –, respectively. When $a = 0.95$ , it turns to the fractional-order Chen system. Numerical simulation results are depicted in Figures 7–9. From the chaotic attractors pictures marked by Figures 1, 4 and 5, one can easily see that the trajectories of the response system (colored red) display certain consistency to that of the drive system (colored black) because of the special scaling functions chosen. Meanwhile, one can also see the synchronization is realized from Figures 3, 6 and 9. Therefore, we conclude that the FCS is a very effective algorithm for achieving the synchronization of the fractional-order unified chaotic system.

Figure 1. FCS of the fractional-order Lorenz system. Here we choose

$(\alpha_1, \alpha_2, \alpha_3) = (0.2, 2, -1.5)$ , initial values

$(x_1, x_2, x_3) = (-1, -0.5, -0.2)$ and

$(X_1, X_2, X_3) = (0.2, 0.3, 0.1)$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. Time responses of state variables

$x_i$ and

$X_i (i = 1, 2, 3)$ for the fractional-order Lorenz system.

DownLoad: Full-Size Img PowerPoint

Figure 3. Synchronization errors of the Lorenz system.

DownLoad: Full-Size Img PowerPoint

Figure 4. FCS of the fractional-order Lü system with

$a = 0.8$ . Here we choose

$(\alpha_1, \alpha_2, \alpha_3) = (-0.5, 2.5)$ , initial values

$(x_1, x_2, x_3) = (0.5, -0.5, 0.2)$ and

$(X_1, X_2, X_3) = (0.15, -0.1, -0.1)$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. Time responses of state variables

$x_i$ and

$X_i (i = 1, 2, 3)$ for the Lü system.

DownLoad: Full-Size Img PowerPoint

Figure 6. Synchronization errors of the Lü system.

DownLoad: Full-Size Img PowerPoint

Figure 7. FCS of the fractional-order Chen system with

$a = 0.95$ . Here we choose

$(\alpha_1, \alpha_2, \alpha_3) = (0.5, 1.5, 3)$ , initial values

$(x_1, x_2, x_3) = (1.5, 0.02, -0.01)$ and

$(X_1, X_2, X_3) = (-2, 0.01, -0.05)$ .

DownLoad: Full-Size Img PowerPoint

Figure 8. Time evolutions of state variables

$x_i$ and

$X_i$

$(i = 1, 2, 3)$ for the Chen system.

DownLoad: Full-Size Img PowerPoint

Figure 9. Synchronization errors of the Chen system.

DownLoad: Full-Size Img PowerPoint

4. Conclusions

Chaos synchronization, because of the potential applications in telecommunications, control theory, secure communication et al, has attracted great attentions from various research fields. In the present work, via the stability theorem, we successfully extend the cascade synchronization of integer-order chaotic systems to a kind of general function cascade synchronization algorithm for fractional-order chaotic systems. Meanwhile, we apply the method to the fractional-order unified chaotic system for an illustrative test. Corresponding numerical simulations fully reveal that our method is not only accuracy, but also effective.

It is worthy of pointing out that the scaling function introduced makes the method more general than the complete synchronization, anti-phase synchronization, modified projective synchronization et al. Therefore, in this sense, our method is applicable and representative. However, the present work just study the fractional-order chaotic system without time-delay. It is known that in many cases the time delay is inevitably in the real engineering applications. Lag synchronization seems to be more practical and reasonable. Hence, it will be of importance and interest to study whether the FCS method can be used to realize the synchronization of fractional-order chaotic systems with time-delay. We shall considered it in our future work.

Acknowledgments

The authors would like to express their sincere thanks to the referees for their kind comments and valuable suggestions. This work is supported by the National Natural Science Foundation of China under grant No.11775116 and No.11301269.

Conflict of interest

We declare that we have no conflict of interests.

References

[1]	Parandoush P, Lin D (2017) A review on additive manufacturing of polymer-fiber composites. Compos Struct 182: 36-53. doi: 10.1016/j.compstruct.2017.08.088
[2]	Gopinathan J, Noh I (2018) Recent trends in bioinks for 3D printing. Biomater Res 22: 1-15. doi: 10.1186/s40824-018-0122-1
[3]	Rahim TNAT, Abdullah AM, Md Akil H (2019) Recent developments in fused deposition modeling-based 3D printing of polymers and their composites. Polym Rev 59: 589-624. doi: 10.1080/15583724.2019.1597883
[4]	Karakoç A (2017) A brief review on sustainability criteria for building materials. Juniper Online J Mater Sci 2: 1-3.
[5]	Shahrubudin N, Lee TC, Ramlan R (2019) An overview on 3D printing technology: Technological, materials, and applications. Procedia Manuf 35: 1286-1296. doi: 10.1016/j.promfg.2019.06.089
[6]	Campbell T, Williams C, Ivanova O, et al. (2011) Could 3D printing change the world?: Technologies, potential, and implications of additive manufacturing. Washington, DC: Atlantic Council, 3.
[7]	Dou J, Karakoç A, Johansson LS, et al. (2021) Mild alkaline separation of fiber bundles from eucalyptus bark and their composites with cellulose acetate butyrate. Ind Crop Prod 165: 113436. doi: 10.1016/j.indcrop.2021.113436
[8]	Faruk O, Bledzki AK, Fink HP, et al. (2012) Biocomposites reinforced with natural fibers: 2000-2010. Prog Polym Sci 37: 1552-1596. doi: 10.1016/j.progpolymsci.2012.04.003
[9]	Karakoc A, Bulota M, Hummel M, et al. (2021) Effect of single-fiber properties and fiber volume fraction on the mechanical properties of Ioncell fiber composites. J Reinf Plast Compos 0: 1-8.
[10]	Sixta H, Michud A, Hauru L, et al. (2015) Ioncell-F: A high-strength regenerated cellulose fibre. Nord Pulp Pap Res J 30: 43-57. doi: 10.3183/npprj-2015-30-01-p043-057
[11]	Li X, Tabil LG, Panigrahi S (2007) Chemical treatments of natural fiber for use in natural fiber-reinforced composites: A review. J Polym Environ 15: 25-33. doi: 10.1007/s10924-006-0042-3
[12]	Wei L, McDonald AG (2016) A review on grafting of biofibers for biocomposites. Materials 9: 303. doi: 10.3390/ma9040303
[13]	Kalia S, Kaith BS, Kaur I (2009) Pretreatments of natural fibers and their application as reinforcing material in polymer composites-a review. Polym Eng Sci 49: 1253-1272. doi: 10.1002/pen.21328
[14]	Hayward MR, Johnston JH, Dougherty T, et al. (2019) Interfacial adhesion: improving the mechanical properties of silicon nitride fibre-epoxy polymer composites. Compos Interface 26: 263-273. doi: 10.1080/09276440.2018.1499328
[15]	Nogueira CL, De Paiva JMF, Rezende MC (2005) Effect of the interfacial adhesion on the tensile and impact properties of carbon fiber reinforced polypropylene matrices. Mater Res 8: 81-89. doi: 10.1590/S1516-14392005000100015
[16]	Wong KH, Syed Mohammed D, Pickering SJ, et al. (2012) Effect of coupling agents on reinforcing potential of recycled carbon fibre for polypropylene composite. Compos Sci Technol 72: 835-844. doi: 10.1016/j.compscitech.2012.02.013
[17]	Sudesh K, Iwata T (2008) Sustainability of biobased and biodegradable plastics. Clean-Soil Air Water 36: 433-442 doi: 10.1002/clen.200700183
[18]	Wohlers T (2017) Additive manufacturing and composites: An update. Compos World 3: 6.
[19]	Vroman I, Tighzert L (2009) Biodegradable polymers. Materials 2: 307-344. doi: 10.3390/ma2020307
[20]	Pandey JK, Pratheep Kumar A, Misra M, et al. (2005) Recent advances in biodegradable nanocomposites. J Nanosci Nanotechno 5: 497-526. doi: 10.1166/jnn.2005.111
[21]	Iwata T (2015) Biodegradable and biobased polymers: Future prospects of eco-friendly plastics. Angew Chemie Int Edit 54: 3210-3215. doi: 10.1002/anie.201410770
[22]	Farah S, Anderson DG, Langer R (2016) Physical and mechanical properties of PLA, and their functions in widespread applications—A comprehensive review. Adv Drug Deliver Rev 107: 367-392. doi: 10.1016/j.addr.2016.06.012
[23]	Kakuta M, Hirata M, Kimura Y (2009) Stereoblock polylactides as high-performance biobased polymers. J Macromol Sci 49: 107-140.
[24]	Garrison TF, Murawski A, Quirino RL (2016) Biobased polymers with potential for biodegradability. Polymers 8: 262. doi: 10.3390/polym8070262
[25]	Senatov FS, Niaza KV, Zadorozhnyy MY, et al. (2016) Mechanical properties and shape memory effect of 3D-printed PLA-based porous scaffolds. J Mech Behav Biomed 57: 139-148. doi: 10.1016/j.jmbbm.2015.11.036
[26]	Bose S, Vahabzadeh S, Bandyopadhyay A (2013) Bone tissue engineering using 3D printing. Mater Today 16: 496-504. doi: 10.1016/j.mattod.2013.11.017
[27]	Hutmacher DW, Schantz JT, Lam CXF, et al. (2007) State of the art and future directions of scaffold-based bone engineering from a biomaterials perspective. J Tissue Eng Regen M 1: 245-260. doi: 10.1002/term.24
[28]	Cicala G, Latteri A, Curto B Del, et al. (2017) Engineering thermoplastics for additive manufacturing: A critical perspective with experimental evidence to support functional applications. J Appl Biomater Func 15: 10-18.
[29]	Picard M, Mohanty AK, Misra M (2020) Recent advances in additive manufacturing of engineering thermoplastics: Challenges and opportunities. RSC Adv 10: 36058-36089. doi: 10.1039/D0RA04857G
[30]	Keleş Ö, Anderson EH, Huynh J, et al. (2018) Stochastic fracture of additively manufactured porous composites. Sci Rep 8: 1-12. doi: 10.1038/s41598-018-33863-4
[31]	Blok LG, Longana ML, Yu H, et al. (2018) An investigation into 3D printing of fibre reinforced thermoplastic composites. Addit Manuf 22: 176-186.
[32]	Hausman KK, Horne R (2014) 3D Printing For Dummies, John Wiley & Sons.
[33]	Wang X, Jiang M, Zhou Z, et al. (2017) 3D printing of polymer matrix composites: A review and prospective. Compos Part B-Eng 110: 442-458. doi: 10.1016/j.compositesb.2016.11.034
[34]	El Moumen A, Tarfaoui M, Lafdi K (2019) Modelling of the temperature and residual stress fields during 3D printing of polymer composites. Int J Adv Manuf Tech 104: 1661-1676. doi: 10.1007/s00170-019-03965-y
[35]	Sajaniemi V, Karakoç A, Paltakari J (2019) Mechanical and thermal behavior of natural fiber-polymer composites without compatibilizers. RESM 6: 62-73.
[36]	Karakoç A, Rastogi VK, Isoaho T, et al. (2020) Comparative screening of the structural and thermomechanical properties of FDM filaments comprising thermoplastics loaded with cellulose, carbon and glass fibers. Materials 13: 422. doi: 10.3390/ma13020422
[37]	Kaynak C, Varsavas SD (2019) Performance comparison of the 3D-printed and injection-molded PLA and its elastomer blend and fiber composites. J Thermoplast Compos 32: 501-520. doi: 10.1177/0892705718772867
[38]	Zhuang Y, Song W, Ning G, et al. (2017) 3D-printing of materials with anisotropic heat distribution using conductive polylactic acid composites. Mater Design 126: 135-140. doi: 10.1016/j.matdes.2017.04.047
[39]	Li D, Jiang Y, Lv S, et al. (2018) Preparation of plasticized poly(lactic acid) and its influence on the properties of composite materials. PLOS One 13: e0193520. doi: 10.1371/journal.pone.0193520
[40]	Lafranche E, Oliveira VM, Martins CI, et al. (2015) Prediction of injection-moulded flax fibre reinforced polypropylene tensile properties through a micro-morphology analysis. J Compos Mater 49: 113-128. doi: 10.1177/0021998313514875
[41]	International Organization for Standardization (2012) Plastics—Determination of tensile properties—Part 1: General principles, ISO 527-1: 2012.
[42]	Hua J, Zhao ZM, Yu W, et al. (2011) Mechanical properties and hygroscopicity of polylactic acid/wood-flour composite. J Funct Mater 42: 1762-1764+1767.
[43]	Masirek R, Kulinski Z, Chionna D, et al. (2007) Composites of poly(L-lactide) with hemp fibers: Morphology and thermal and mechanical properties. J Appl Polym Sci 105: 255-268. doi: 10.1002/app.26090
[44]	Yang S, Madbouly SA, Schrader JA, et al. (2015) Characterization and biodegradation behavior of biobased poly(lactic acid) and soy protein blends for sustainable horticultural applications. Green Chem 17: 380-393. doi: 10.1039/C4GC01482K
[45]	Johnson D (2020) Filament extrusion using recycled materials: Experimental investigations on recycled Polylactic Acid (PLA) materials[Master's thesis]. Halmstad University: Sweden.
[46]	Durán Redondo D (2019) Circular economy through plastic recycling process into 3D printed products: A frugal solution for schools[Master's thesis]. Universitat Politècnica de Catalunya: Spain.

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