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In recent years, we have abstracted physical fractal space from biological structures and movements within living organisms, revealing the profound intrinsic connections between fractional order time and fractional-dimensional space, and providing partial explanations for the sources and orders of fractional order. We have confirmed that the topological invariants of fractal cells, the order of physical components, and the mismatch of spatiotemporal order are important factors determining the fractional order of operators. This paper is a continuation of the previous work. Inspired by bone fractal operators, this article attempts to identify other factors that affect the order of operators. Specifically, the following contents are included: (1) originating from the bone fractal operators, we present the construction process of the “apparent half-order” system; (2) using the Schiessel–Blumen model as the comparative object, we analyze the origin and characteristics of the “γ-order” system; (3) using the continued fraction theory and operatorization thought as the link, we establish the design and control method for general fractional-order systems, and discuss the factors affecting the order of fractional-order operators.

This paper focuses on the invariant properties of fractal operators and aims to achieve the axiomatization of the theory of fractal operators. Building upon the derivative and integral theorems of the Laplace transform, we redefine the time differential operator and demonstrate that the newly defined operator exhibits form invariance under the Laplace transform. This property is further generalized to encompass broader classes of operators, including non-rational and fractional fractal operators. Inspired by Klein’s concept of “invariance under transformation groups”, we propose a postulate asserting the “form invariance of operators under the Laplace transform group”. Based on this postulate, we clarified the algebraic operational rules of fractal operators and constructed a rigorously axiomatized theory of fractal operators.

This paper reports an interesting phenomenon in which fractional-order effects can be induced by the mismatch of the differential orders of space and time; that is, fractional-order effects can be induced by space–time symmetry breakage. Classical mathematical equations can be transformed into differential equations of undetermined operators. We confirmed that the presence of fractional-order operator solutions in operator differential equations is contingent upon the mismatch of differential orders of space and time, which can induce both fractional operators in the time domain and fractional operators in the space domain. The introduction of symmetry breakage and operators of space and time offers novel insights into understanding nonlocal phenomena within the space–time continuum.

In this paper, we present a re-established functional fractal circuit model of arterial blood flow that incorporates the shunt effect of the branch vessels. Under the background of hemodynamics, we abstracted a family of fractal operators and investigate the kernel function and properties thereof. Based on fractal operators, the intrinsic relation between Bessel function and Struve function was revealed, and some new special functions were found. The results provide mathematical tools for biomechanics and automatic control.

This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing on the main part of bone fractal operators: p+α2-type differential operator, discussing the convolutional kernel function image; (3) exploring the fractional-order correlation between the error function and other special functions from the perspective of fractal operators.

In this paper, we explore the self-convolution of the kernel function of the aortic fractal operator. Previous research has established a model named “physical fractal”, and confirmed that the hemodynamics of the aorta can be inscribed by a fractal operator and that the dominant component of the kernel function of the fractal operator is a weighted first-order Bessel function. These studies primarily focus on solving the fractal operator kernel function and examining the overall properties of the physical fractal. As we began to investigate the internal structure of physical fractals, we discovered that studying the powers of fractal operators is a necessary step. In this paper, we introduce the concept of kernel function self-convolution, establish its connection with the power of the fractal operator, and derive a series of invariant properties for the self-convolution of the aortic operator kernel function. These invariant properties, in turn, are deeply and intrinsically related to the invariant properties of the Bessel functions. The research findings of this paper enrich hemodynamics and biomechanics in physical fractal space and extend the scope of using fractal operators to characterize the dynamics of living organisms.

Chemical recycling of polymers is critical for improving the circular economy of plastics and environmental sustainability. Traditional thermoset polymers have generally been considered permanently crosslinked materials that are difficult or impossible to recycle. Herein, we demonstrate that by activating ‘dormant’ covalent bonds, traditional polycyanurate thermosets can be recycled into the original monomers, which can be circularly reused for their original purpose. Through retrosynthetic analysis, we redirected the synthetic route from forming conventional C–N bonds via irreversible cyanate trimerization to forming the C–O bonds through reversible nucleophilic aromatic substitution of alkoxy-substituted triazine derivatives by alcohol nucleophiles. The new reversible synthetic route enabled the synthesis of previously inaccessible alkyl-polycyanurate thermosets, which exhibit excellent film properties with high chemical resistance, closed-loop recyclability and reprocessing capability. These results show that ‘apparently dormant’ dynamic linkages can be activated and utilized to construct fully recyclable thermoset polymers with a broader monomer scope and increased sustainability. Alkyl and aryl polycyanurate networks have now been prepared through polymerization of diols and substituted triazines via a dynamic S N Ar reaction. When treated with excess mono alcohol or phenol, the polycyanurate networks can be depolymerized into the starting monomers, which can be separated and reused, thus achieving closed-loop recycling.

Polymers have become an integral component of modern life, and their applications are essentially everywhere. After the world's first synthetic polymer was invented in 1909, various polymers, such as nylon, polyethylene, Teflon, and silicone, have been developed and used in a wide range of applications. Today, polymers are intended to be designed with different properties in addition to conventional load-bearing to extend their functions and potential applications. A type of polymer, called an active polymer, is at the forefront of polymer research, which can sense environmental stimuli (e.g., heat, light, moisture) and change their properties or configurations as a direct response. Such polymers have been demonstrated to be used in a wide range of applications, such as actuators, sensors, energy harvesters, etc. Before these polymers can be widely used to benefit society, a detailed study of their structure-processing-property relationship is in high demand. This thesis focuses on two representative active polymers with i) dynamic bonding among polymer chains and ii) the dynamic bonding on the chain backbone. For the first one, the recently emerged covalent adaptable networks (CANs) with exchangeable bonds on the chain backbone are selected as the material platform. Comprehensive chemomechanics constitutive models were developed to study the effects of bond exchange reactions (BERs) and relaxation of polymer chains on the thermomechanical behaviors of CANs. Then the model was extended to investigate the moisture induced malleability of polyimine CANs, as well as the BER-induced material stiffening of two-stage UV-curable resins during the heat processing. The chemomechanics theory is also incorporated with the reaction kinetics of exchangeable bonds and small solvent molecules to study the depolymerization and recycling kinetics of epoxy composites. For the second type of active polymers, the main-chain liquid crystal elastomers (LCEs) are selected as the material platform for investigation. In combination with the digital light processing (DLP) 3D printing technique, innovative LCE lattice foams are fabricated as enhanced energy dissipative materials. Their energy dissipation abilities were studied with different lattice porosities, connectivity, temperatures, and loading rates were studied. A new optical characterization technique is developed to examine the kinetics of mesogen alignment and reorientation of LCEs upon deformation. It is shown that the polarized optical measurement enables in situ measurements of mesogen alignment, providing opportunities to link the microscale network structure with the mechanical responses of LCEs. Finally, the deformation mechanisms of 3D printed LCE lattice structures are examined using finite element simulations, which serves as a significant first-step investigation for the following optimization design of lattice structures. Overall, the research in this thesis reveals the material-process-property relationship of active polymers using an integrated experimental-theoretical approach. It provides valuable guidance of the design, processing, and applications of active polymers.

In recent years, our research on biomechanical and biophysical problems has involved a series of symmetry issues. We found that the fundamental laws of the aforementioned problems can all be characterized by fractal operators, and each type of operator possesses rich invariant properties. Based on the invariant properties of fractal operators, we discovered that the symmetry evolution laws of functional fractal trees in the physical fractal space can reveal the intrinsic correlations between special functions. This article explores the fractional-order correlation between special functions inspired by bone fractal operators. Specifically, the following contents are included: (1) showing the intrinsic expression in the convolutional kernel function of bone fractal operators and its correlation with special functions; (2) proving the following proposition: the convolutional kernel function of bone fractal operators is still related to the special functions under different input signals (external load, external stimulus); (3) using the bone fractal operators as the background and error function as the core, deriving the fractional-order correlation between different special functions.