International Journal of Applied Mechanics

High-damping rubber materials utilized in high-damping rubber isolation bearings are frequently subjected to multiple deformations during the occurrence of earthquakes. Typically, large combined compression–shear deformations of the material could potentially cause compressive shear damage to rubber bearings. During this process, visco-hyperelastic properties of rubber materials will greatly change, which would significantly impact the seismic performance of rubber bearings. Thus, to give out a deep insight into their variations, it is necessary and urgent to develop a high-performance numerical method to investigate this process. This paper proposed a visco-hyperelastic constitutive modeling approach for high-damping rubber materials based on the experimental assessment of combined quasi-static compression–cyclic shear deformation process. Within the thermodynamic framework, the Clausius–Duhem inequality associated with the intrinsic dissipation of the material was firstly derived in accordance with the Lagrangian formulism. Then, stress–strain relations were obtained upon considering the occurrence of entropy production due to viscous dissipation. In the model, Stumpf–Marczak strain energy density function, which satisfies the Baker–Ericksen (B–E) inequality, was harnessed to describe the hyperelasticity of the material. By introducing higher orders of strain and strain rates and taking their couplings into account, a generalized viscous dissipation potential was proposed to capture nonlinear strain and strain rate-sensitivity effects of the material. To identify constitutive parameters, the deformation gradient was particularized for the combined quasi-static compression–cyclic shear deformation process. And, an inverse identification procedure was carried out at different levels of compression stress. The prediction results revealed that the proposed model exhibits remarkable prediction ability and adaptivity for different rubber materials during this process. Several new insights were highlighted on the variations of visco-hyperelastic characteristics of high-damping rubber materials with respect to the compression stress. The accuracy of the model was further validated by design parameters including initial shear modulus, secant shear modulus and equivalent viscous damping factor. This work could provide a fundamental guideline for the optimization and reliability analysis of high-damping rubber isolation bearings used in the field of seismic engineering.

Hydrogels can change their size upon swelling. The swelling ratio is the same for all directions in the stress-free state. Dielectric elastomers can reduce their thickness and expand the area upon an electric field. Similarly, the expansion ratio in the plane is also the same for different directions. This isotropic shape change effect limits the function of these soft materials in certain circumstances. To address this issue, recent works have shown that the incorporation of fibers into the polymer matrix can induce an anisotropic response upon external stimulus. In this work, we develop multi-field coupling models for both fiber-reinforced hydrogels and dielectric elastomers. For the former, the change in free energy is caused by the stretching of polymer chains and fibers and the mixing of solvents and polymer networks. The Fickian-type law is adopted for the solvent diffusion. The free energy density for the latter consists of a mechanical part, considering the deformation of both polymer matrix and fibers, and an electric polarization component. Gauss’s law is adopted to obtain the distribution of the electric field. The multi-field models are then implemented for finite element analysis. We consider the stimulus-responsiveness of bilayer strips with an active layer and a passive layer. Without fibers, the strips bend upon the external stimulus. In contrast, the shape changes to helix shapes, which can be further tuned by changing the distribution of fibers. The work provides an efficient design tool for self-folding structures based on stimulus-responsive polymers.

Covalent adaptable network (CAN) in thermoset epoxy shows good recyclability due to its bond exchange reactions (BERs), which result in low mechanical strength and toughness. Many experimental studies on using the semi-crystalline structures and ions with the CAN have shown to improve the mechanical strength and toughness of these thermoset epoxies. However, their toughening mechanisms and coupling effects have not been well understood. Herein, we propose an extended Maxwell model to explore toughening mechanisms and coupling effects of the semi-crystallization process and formation of ion phases on CANs, which enables the thermoset epoxy with hybrid chemomechanical yielding. A constitutive stress–strain relationship is formulated to characterize the mechanical toughening and hybrid chemomechanical yielding behavior, where the first stage of yielding is originated from the effect of ions on amorphous phases, and the second stage of yielding is due to the coupling effect of semi-crystallization and formation of ions. The proposed models have been verified using the experimental results, and provide a fundamental approach to understanding the toughening mechanisms and coupling effects of semi-crystallization and ions in thermoset epoxy.

In this paper, the formulation of the mechanical behavior of visco-hyperelastic materials is developed in a thermo-dynamically compatible framework. In this viewpoint, the stress power is assumed to be equal to the sum of the reversible (elastic) energy rate stored and the irreversible energy rate dissipated. Based on this assumption and using the second law of thermodynamics, the constitutive modeling framework is obtained for the rate-dependent materials. Inspired by the constitutive law of a Newtonian fluid and also, models of density strain energy of an elastic material, a framework for the dissipative energy rate related to the irreversible part is proposed for viscoelastic materials. In this framework, the reversible energy part is considered a function of the right Cauchy–Green deformation tensor, and the irreversible energy part is considered a function of the same tensor’s rate. To evaluate the proposed model for nonlinear viscoelastic behavior modeling, the results of tests performed on non-compressible materials at different rates are used, and the proposed model provided acceptable results. Finally, the proposed model is implemented to find a closed-form analytical solution for mechanical behavior modeling of the thick-walled cylindrical shells made of visco-hyperelastic materials, also, stress and stability analyses are assessed for these types of cylinders. To achieve this goal, the effect of pre-stretch on the stability of cylinders has been substantially investigated. Furthermore, based on the proposed model, the effect of the parameters such as the wall thickness and behavior of the material used in the cylinder are examined on the stability of open-ended and closed-ended cylinders.

We combine experimental analysis and modeling to investigate the complex mechanical behaviors of copolymer elastomers. Experimental studies are conducted to analyze the impact of monomer ratio, strain rate, and loading mode on the response of copolymer elastomers, which reveal the distinct hyperelastic, viscoelastic, damage and self-recovery behaviors of the materials. Based on the experimental observations, we develop a finite deformation hyperelastic-viscoelastic constitutive model, which incorporates multiple discrete relaxation processes. For the hyperelastic process, an extended eight-chain model is employed. Different from the classic viscoelastic model, the strain hardening and damage/recovery mechanisms are also included in the relaxation processes. The model successfully describes the complex mechanical responses observed in elastomers, including quasi-static loading, strain rate effects, and stress relaxation phenomena. Additionally, the model captures the behavior of elastomers during loading-unloading cycles in a reasonable manner. The work offers an efficient theoretical tool for the design and practical application of elastomers.

In this paper, we suggest a design strategy for multilayer hydrogel beams with controllable swelling deformation and bending, which can be used in soft devices. We adopt a constitutive model that considers the entanglement chains of the hydrogel and an analytical method for solving multilayer hydrogel beams is developed. First, the reliability of the design strategy is provided by comparing analytical solutions and numerical results. Then, guided by the above design strategy, we quantitatively investigate hydrogel beams with gradient distribution and study the effects of material and distribution on the deformation and bending behavior of the structures. The results show that under non-gradient distribution, all structures with an odd number of total layers only undergo in-plane expansion, while all those with an even number of total layers undergo expansion and bending. However, this law will be broken at the gradient distribution. The introduction of entanglement chains and gradient distribution strategy increase the range of adjustment for deformation and bending. This work is expected to provide new insights into the design of multilayer hydrogel beams for material–structure–function integration.

Viscoelastic mechanical characteristics of the liver play a crucial role in fibrosis and cancer progression. Despite the evolution of liver fibrosis, as a pivotal step toward cirrhosis and hepatoma, originates from molecular dysfunctions and further extends to cellular and tissue levels, current clinical assessment is still largely based on tissue level stiffness. This study introduces a novel multiscale viscoelastic signature-based machine learning model for liver fibrosis evaluation. Utilizing a hybrid hierarchical theory-microrheology approach, we unveil a universal two-stage power-law rheology capturing dynamic mechanical variations in diverse liver conditions. Our analysis of multiscale viscoelastic disparities through a self-similar hierarchical theoretical framework enhances our understanding of fibrosis evolution. Distinct mechanical signatures observed across liver states provide valuable insights for assessing fibrotic individuals and treatment responses at different spatial scales. Furthermore, we propose a series of threshold values for each marker in the diagnosis of liver fibrosis. Notably, based on these new viscoelastic signatures, we eventually propose a Light Gradient Boosting Machine (LightGBM) diagnostic model that outperforms conventional stiffness-based classification, offering superior diagnostic precision for fibrotic treatment. This research contributes to the growing knowledge of viscoelastic characteristics in soft tissues and holds promise for innovative diagnostic strategies in various diseases and cancers.

This work develops a chain scission-induced anisotropic damage constitutive model for double network (DN) hydrogels while considering the damage cross effect. A free energy of a single polymer chain comprising the configuration entropy of the polymer chain and the internal energy of persistent segments is proposed to avoid the force singularity caused by chain stretch limit. Then, a distribution function of the initial stretch ratio of polymer chains in DN hydrogels is derived to consider the randomness of chain length. Based on the micro-sphere full-network model and an energy-form fracture criterion of polymer chains, the damage evolution model of polymer chains in a certain orientation is formulated by the critical initial stretch ratio of the polymer chain, showing that only polymer chains with the initial stretch ratio lower than the critical initial stretch ratio survive in a certain chain stretch. The critical initial stretch ratio is determined by the maximum chain stretch and the maximum first invariant of the deformation state by considering the damage cross effect. Using the developed damage constitutive model, we describe the stress softening, strain hardening behavior and the nonmonotonic stress–strain curve of DN hydrogels. It also reveals that the residual strain of DN hydrogels in the loading cycles is caused by the anisotropic damage of polymer chains in DN hydrogels.

Recently, vitrimers with a reversible N$\to$B coordination dynamic structure have garnered interest not only for their self-healing and reprocessable properties but also for their hydrolytic stability. Reported vitrimers with N$\to$B internal coordination dynamic structure were excessively rigid, limiting their stretchability and toughness. This study synthesized a cross-linking agent (DNCB) with a cyclic nitrogen-boron coordination structure. This agent was then used in copolymerization with n-butyl acrylate (BA) and 4-vinylphenylboronic acid (BOH) to create soft vitrimers. By systematically tuning the composition, soft vitrimers exhibit water resistance along with high viscoelasticity, stretchability, toughness, and adhesive properties against various substrates. The observed mechanical behavior is attributed to the rapid relaxation of N$\to$B coordination bonds and slow relaxation of B-O bonds in boroxine within the polymer network. It is anticipated that these soft vitrimers will be applied in adhesives, flexible electronics, and soft robotics due to their favorable properties.

When in an alternating magnetic field, magneto-thermal sensitive hydrogels can absorb solvent molecules, causing the gel network to swell several times to its original size. By conducting experiments with magnetic PNIPAm hydrogels (M-PNIPAm), the parameters of the theoretical model for magneto-thermal sensitive hydrogels were experimentally determined. Subsequently, the mechanical behavior of M-PNIPAm hydrogel was studied through finite element analysis and tensile experiments. The numerical calculations based on the thermodynamic model with specific parameters showed consistency with experimental results. The comparison indicated that the proposed theory for magneto-thermal sensitive hydrogels can accurately predict the mechanical property of M-PNIPAm hydrogel, providing theoretical support for investigating large deformation of magneto-thermal sensitive hydrogels. Furthermore, we discovered that the variation of magnetic particle content significantly influence the mechanics properties of hydrogel. Our findings provide valuable insights for the research and application of magneto-thermal sensitive hydrogels.

This contribution aims to derive a nonlinear FE formulation for large elastic deformation of hyperelastic thin rubber-like elastomeric materials. The formulation accounts for isotropic as well as transversely isotropic rubber-like materials. Besides, with the developed formulation, deformation of incompressible as well as compressible materials can be investigated. The formulation is applicable for in-plane and out-of-plane deformation of thin membranes. Several problems are inspected including uniaxial and equibiaxial extension of rubber, axisymmetric deformation of annular sheet, stretching a perforated rubber, extension of an anisotropic rubber, and inflation of an initially-flat circular and a square membrane. The results show very good agreement compared with those reported in the literature. Moreover, this model is able to model the inflation of initially flat membranes which is not possible to be done by commercial software.

Shape memory polymers (SMPs) are commonly activated through external heating or rigid embedded heaters, which lack precise temperature control and restrict programmable shape transformations. This study demonstrates integrating thin films of hard electronic materials patterned in fractal designs into SMPs yields novel thermomechanical responses, enabling flexible implementation of stretchable electronic functionality. Space-filling Hilbert, Moore and Peano fractal curves generate distributed electronic circuitry patterns within the SMP matrices. To investigate the coupled electro-thermal-mechanical behavior, a multi-physics modeling approach is adopted. A nonlinear thermo-visco-hyperelastic constitutive model captures the SMP’s shape memory characteristics. Governing equations for the multi-physics problem are formulated. Experimental data validates the model’s predictions of the SMP’s nonlinear material response under uniaxial loading. This validated framework analyzes SMP composites with integrated fractal circuits. Findings reveal composites with higher-order fractal circuits’ exhibit faster, more uniform resistive heating and smaller electrical resistance changes during stretching, enabling consistent and predictable shape recovery under various deformations. Notably, these advantageous heating and resistance characteristics enhance the force recovery rate during the shape memory cycle. Furthermore, the composites demonstrate excellent shape recovery under bending. Remarkably, uniaxial stretching can induce controlled out-of-plane shape transformations, enabling novel actuation modalities. This work provides insights into leveraging fractal circuit integration to enhance SMP performance and expand capabilities, enabling multi-functional actuators, sensors and programmable soft grippers.

Tunable three-dimensional (3D) electromagnetic (EM) metasurfaces are critical for dynamic modulation of EM responses but their construction and tuning mechanism are still complex. Here, we report a simple yet effective 3D reconfigurable EM metasurface, which was obtained from a planar kirigami polyimide substrate printed with periodically arranged copper split-ring resonator. Under mechanical stretch, the two-dimensional (2D) planar metasurface can be uniformly deformed into a 3D state, which is effective for tuning its EM transmission characteristic. By combining mechanics and EM simulations as well as experimental measurements, we revealed the deformation mode and active EM transmission modulation capability of the metasurface. It is shown that at the initial state, the planar kirigami metasurface exhibits ideal frequency selective transmission to transverse electric (TE) wave but allows for complete transmission for transverse magnetic (TM) wave. As the applied strain increases from 0% to 20%, the transmission was adjusted from −17.74dB to −9.74dB for TE wave but merely from 0dB to −3.25dB for TM wave. Meanwhile, the resonant frequency experienced a visible shift for both TE and TM waves. Finally, the equivalent circuit analysis and simulated surface current density were conducted to reveal the tuning mechanism of the proposed metasurface.