Mullins Effect Research Articles

ABSTRACTThe dynamic behavior of seismically isolated structures is governed by the force‐deformation response of the isolation devices. Consequently, significant efforts have been made to accurately simulate the behavior of different types of devices. High damping rubber bearings (HDRBs) are among the most widely manufactured and used isolators in practice. Given the internal structure of these devices and the characteristic behavior of the rubber compound, HDRBs show highly nonlinear behavior with strong coupling between deformation directions, which is challenging to simulate numerically. Capturing these complex multiaxial interactions is essential for reliably predicting device behavior and ensuring the dynamic stability of the isolation system during seismic events, therefore, a holistic multiaxial modeling approach is critical. This study presents a robust and sufficiently accurate numerical model for simulating the multiaxial behavior of HDRBs under large deformations. This elaborate model includes: bidirectional shear response that accounts for stiffness degradation with load‐direction dependency, including scragging (long‐term degradation) and Mullins effect (short‐term degradation), and temporary hardening; coupling between axial and shear response, including axial stiffness softening due to lateral displacement and shear stiffness variability due to axial load variation; axial instability due to large compressive loads; and cavitation under tensile forces. The proposed model is validated using a wide range of load patterns applied to an HDRB, as well as experimental results from the literature. The proposed model demonstrates good agreement with experimental data, accurately simulating HDRB responses across diverse validation tests, including double bidirectional shear tests in rotated directions, cyclic shear response under different axial loads, tensile loads, bidirectional deformation history with an elliptical orbit, extremely large deformations (beyond the design limits), and dynamic analyses. The results show that the model provides reliable predictions of the static and dynamic behavior of HDRBs under different load patterns, including deformations until the onset of failure. The proposed model has been implemented in OpenSees and is openly available at the supplementary repository https://github.com/JAGallardo1992/HDRB_model.

Abstract In recent times, large language models have become popular in materials science for the development of new materials and formulations. An attempt is made in this work to utilize large language models such as the ones used in commercial ChatGPT versions for artificial intelligence applications. In this paper, an assessment is made for the energy absorption in rubber formulations during different cycles of loading and unloading which is intended to stabilize stress softening in rubber-like materials and in each case for the same maximum load. The experimental data obtained by earlier investigators is used for this purpose. Chat GPT4o has been used for the purpose for which the graphical data has been culled from the graphs available for Mullins effect using the commercially available graphical plotting and analysing software. No attempt is made here to go into reasons for behaviour in Mullins effect nor the various aspects of constitutive models for the Mullins effect.

A visco-hyperelastic constitutive model with Mullins effect for polyurea under repeated impact load

Influence of Gel on the Payne Effect and Mullins Effect of Butadiene Rubber

Abstract Stress softening, known as the Mullins effect, has a significant effect on the durability and performance of filler-reinforced rubber, making it a critical issue in designing products for practical applications. While empirical equations are widely used, they fail to capture the intricate and nonlinear behaviors that are characteristic of filler-reinforced rubber. To address this limitation, this study developed a simplified equation to predict the Mullins effect. The model is based on the assumption that the Mullins effect originates from the destruction of particle aggregation structures, and the relationship between the degree of destruction and the stretch ratio is expressed using extreme value statistics. Validation against experimental data revealed that the equation accurately predicts the behavior of rubber reinforced with carbon black (CB) or silica. Additionally, in systems with CB-filled rubber, the equation demonstrated good agreement with the experimental results, even when the CB content was varied. These findings suggest that the proposed model is versatile and effective for predicting the Mullins effect under different conditions, providing a useful tool for understanding and optimizing the performance of filler-reinforced rubber in practical applications.

ABSTRACTThis study reports the behavior of a microporous polyacrylate dielectric elastomer under controlled mechanical and electrical loading to assess its suitability for soft actuators in robotics. The investigation focuses on the behavior of elastomers subjected to cyclic mechanical loading with prestrain conditions, evaluating stress‐softening, energy dissipation, and residual strain over multiple cycles to understand the suppression of the Mullins effect. Further, the impact of repeated electrical loading, with and without preload conditions, is analyzed by examining area strain, energy conversion efficiency, and electrical breakdown strength to evaluate the feasibility of the actuator for real‐world applications. The findings reveal that the stresses in the elastomer with reserved strain conditions reduce significantly (about 80%) between the first two cycles and diminish to stabilization with repeated mechanical loading. Also, repeated electrical loading in an actuator made of prestrained elastomer, under preload conditions, is observed to stabilize actuation and improve breakdown strength from 6 to 22 kV. This work demonstrates the importance of considering the preload and prestrain conditions in stabilizing the performance of microporous elastomers, enhancing their reliability for soft actuators and similar electromechanical devices.

Solvent-free polyelectrolyte elastomers, which are resistant to leakage, hold significant promise for large-scale engineering applications of stretchable ionotronic devices. However, the viscoelastic nature of ionized polymer networks introduces complexities in mechanical performance, highlighting the need for a deeper understanding of their visco-hyperelastic properties. In this study, a poly[1-[2-acryloyloxyethyl]-3-butylimidazolium bis(trifluoromethane) sulfonimide-co-methyl acrylate] elastomer is synthesized as the model material, with controlled covalent crosslinker densities and tailored ionic-to-neutral segment ratios to systematically modify its molecular structures. Through experimental mechanical characterizations—including tensile, hysteresis, and relaxation tests—the effects of network structure and strain rates on the material's responses are investigated. The results reveal a significant rate dependence and the Mullins effect. To model these behaviors, the Yeoh hyperelastic model, incorporating the Mullins effect, is employed to describe the nonlinear elastic response, while a nonlinear power law model is introduced to capture the time-dependent viscoelastic deformation. The proposed modeling framework demonstrates excellent agreements with the experimental data, effectively capturing the complex mechanical behaviors in various tests. This study provides valuable insights into the visco-hyperelastic behaviors of polyelectrolyte elastomers by mapping microscopic molecular structures to macroscopic mechanical performance.

The advancement of soft multi-digital grippers with both adaptive grasping and in-hand manipulation capabilities remains a challenging issue for the development of human-like dexterous manipulation. Despite four decades of research, the most advanced grippers remain encumbered by excessive complexity and a lack of robustness. The field of soft robotics presents a promising avenue for reducing the level of complexity and enhancing the safety of grasping and interaction with the environment. This work presents a methodology for the design and control of a soft finger, with the objective of ultimately developing a highly dexterous gripper. To address these challenges, it is essential to master the design, fabrication process, and behavior of the finger’s soft material. The iterative design and fabrication process requires a comprehensive understanding of the theoretical and experimental aspects, as detailed in this paper. Given that the finger is constructed from silicone, the proposed methodology and outcomes demonstrate the importance of accounting for the Mullins effect and conducting finger training prior to controlling the pneumatic soft finger. The proposed hard real-time control architecture guarantees the robustness of the analysis and control of the finger’s behavior, while also offering perspectives for coordinated multi-fingered manipulation.

A physically based constitutive model captures the Mullins effect due to different damage mechanisms

The present study investigated the dynamic behavior of structures made of re-entrant unit cells subjected to cyclic compressive loading limited to the elastic range. The structures were assembled from printed polymer re-entrant cells in six combinations. Through the given compression cycles for three different amplitude values, strain-force relationships, which had the shape of a hysteresis loop, were obtained. Under compression, all unit cells of the structures deformed uniformly, though only for a certain amount of strain, whereas with larger changes, they underwent uncontrolled deformation. Experiments showed that structures composed of more than one unit cell exhibit different mechanical characteristics. It was observed that the width of the hysteresis loop depended on the degree of closing the structure and on the compression amplitude. The obtained hysteresis curves for different amplitudes also testify to the occurrence of the Mullins effect for these polymeric auxetic structures. Taking into account the maximum values of changes in dimensions for a given compression cycle, Poisson's ratio values were determined, which were negative and below unity. The effect of strut thickness on the NPR was confirmed, decreasing its negative value along with the increasing thickness.

Observation and prediction of the electrical and mechanical properties of nanocomposites under dynamic deformation conditions are critical for wearable devices and soft electronics. Despite extensive research, a comprehensive understanding of the mechanical characteristics of composites subjected to various repetitive deformations remains limited. The intrinsic mechanical properties of a composite undergo significant changes after cyclic deformation, and these changes are strongly influenced by the magnitude of deformation, type and content of fillers, and other variables. This study identified softening and unexpected stiffening effects in carbon nanotube-based composites after repeated tensile deformation. The Mullins effect was evident during cyclic stretching within the pre-strain region; however, a stiffening effect occurred beyond this region. To understand this behavior, we quantitatively evaluated three key factors—filler aspect ratio, pre-strain level, and number of cycles—to determine the mechanical properties of the composite under cyclic deformation. This was achieved using systematic experiments and molecular dynamics simulations. Existing theoretical models that predict the mechanical properties of composites fail to account for the property changes under dynamic deformation. To address this limitation, we developed a formula using symbolic regression to predict the tensile strength of the composites after cyclic deformation, demonstrating its robustness and broad applicability.

A complementary energy-based constitutive model for the Mullins effect

Silicone elastomers filled with silica exhibit a softening effect under cyclic loading, also known as the Mullins effect. There are various theories as to the origin of this effect. One theory cites the breaking of secondary bonds as the reason, while another theory cites molecular slipping and the loosening of entanglements as the reason for the softening. In contrast to previous studies, foamed silicone rubbers are investigated here. In addition to the silica, the samples investigated here contain thermally expandable microspheres, resulting in two types of filler (silica and the microspheres as thermoplastic particles). Two different kinds of these microspheres with different particle sizes and expansion temperatures are used. To study the effect, hysteresis tensile tests (Mullins tests) as well as FTIR analyses, density measurements and microscopy images were carried out. The Mullins tests were evaluated for their curve shape, their hysteresis area of the first cycle and the residual strain. The residual strain shows a correlation with the porosity, as the larger microspheres have a higher porosity and a higher residual strain. The area of the first hysteresis correlates with the number of microspheres, but not with the porosity. The larger microspheres have a smaller hysteresis area but a higher porosity. The FTIR analysis showed that the stabilizer used to produce the microspheres is based on silicon oxide. As a result, there are secondary bonds between the microspheres and the silicone elastomer, which are broken at the beginning.

The modulus and toughness of rubbers can be improved by incorporating fillers into the rubber matrix. Filled rubbers exhibit a strain‐induced softening phenomenon known as the Mullins effect. Upon deformation, filled rubbers can dissipate strain energy, which results in enhanced resistance to crack growth. An in‐depth understanding of the energy dissipation accompanying crack growth is important for the predictive modeling of rubber fracture. Herein, an experimental study on the fracture toughness of filled and unfilled rubbers with a focus on characterizing the energy dissipation caused by the Mullins effect during crack growth is presented. A particle tracking method is used to measure the nonlinear deformation fields in notched rubber specimens when subjected to monotonic tensile loading. Using the measured deformation fields and an independently calibrated constitutive model, the evolution of stress fields during crack growth is quantitatively evaluated, based on which the energy dissipation and intrinsic toughness are determined with the assumption of steady‐state crack growth. It is found that the filled rubber exhibits more energy dissipation during crack growth than the unfilled rubber. More importantly, the intrinsic toughness of the filled rubber (5000–8000 J m−2) is also much higher than that of the unfilled rubber (150–300 J m−2).

The incorporation of reversible sacrificial bonds is an important strategy for enhancing the mechanical properties of elastomers. However, the research on the viscoelasticity of vulcanized rubber with a reversible sacrificial bond network lags seriously. In this paper, the effects of metal coordination bonds on the mechanical properties of butadiene-styrene-vinylpyridine rubber vulcanizates (VPR) were systematically investigated. The experimental results showed that lower temperature and higher strain rate under smaller strain were advantageous in reflecting the significant contribution of sacrificial bonding. Compared with the conventional rubber nanocomposites, the sacrificial bond enhanced the energy dissipation while also improving reversible hysteresis energy and its proportion, revealing the origin of better self-healing and damping properties. The high-temperature rearrangement of sacrificial bonds promoted not only the recovery of mechanical properties, but also the recovery of covalent networks. The evolution mechanism of the network structure and viscoelasticity research under different thermal-mechanical coupling conditions could help to control the rubber network structure, providing a promising method for the regulation of the nonlinear behavior of rubber composites.

Abstract Elastic‐porous metal rubber, commonly employed in dynamic environments, has received special attention nowadays. Utilization of equivalent metal rubber models in the finite element analysis becomes imperative because of the complexity of the spatial network structure and the extensive numerical calculations involved in it. In this work, the nonlinear characteristics of metal rubber are represented by hyperelastic constitutive models, and the Mullins effect is discussed. The relative deviations of the displacement‐load curve among the test results and three dynamic models are analyzed: the hysteretic Bergström–Boyce model, the viscoelastic generalized Maxwell model, and the nonlinear spring‐viscous damping element model. The results indicate that the damage value of the Mullins effect is positively correlated with the magnitude of the strain during unloading, and the performance degradation of metal rubber is weakened after multiple cycles. The time domain characteristics of the generalized Maxwell model demonstrate that the dynamic analysis of metal rubber at different frequencies have obvious deviations. The rate correlation of metal rubber affects the dynamic stiffness under different amplitudes This is reason that the nonlinear spring‐viscous damping element model is deviated. Regardless of the frequency or amplitude, the Bergström–Boyce model has been found to exhibit a lower relative deviation and match well with the dynamic experiment.

Computational modeling of vascular tissue damage for the development of safe interventional devices

Silica (SiO₂) serves as a reinforcing filler, while titanium dioxide (TiO₂) functions as a functional filler, imparting exceptional mechanical strength and UV resistance to silicone rubber. However, compatibility issues may arise when both fillers are incorporated into silicone rubber. This study explores potential solutions through processing methods, specifically by comparing the effects of graded filling and mixed filling on the properties of silicone rubber. Field emission scanning electron microscopy reveals that the mixed filling method promotes a more uniform dispersion of fillers, leading to enhanced performance. Curing tests indicate that the effect of the filling method on the curing processing performance of silicone rubber is negligible. Tensile test results reveal that, compared to the graded filling system, the mixed filling system improves the tensile strength and elongation at break of the silicone rubber composites by 13% and 13.4%, respectively, while simultaneously reducing the Mullins effect, thereby enhancing material stability. Thermogravimetric analysis (TGA) and differential thermal analysis (DTG) results indicate that the uniform dispersion achieved through mixed filling optimizes thermal transfer pathways, resulting in superior thermal stability compared to graded filling. Ultraviolet (UV) spectroscopy results demonstrate that composites containing added TiO₂ exhibit almost zero transmittance in the UV range of 200-400 nm, underscoring their excellent UV resistance; the mixed filling method provides superior UV protection compared to the graded fill method. In summary, for the dual-filler system of SiO₂ and TiO₂, the mixed filling approach enhances filler compatibility by improving dispersion, thereby elevating material performance and broadening potential applications.

Time-dependent constitutive behaviors of a dynamically crosslinked glycerogel governed by bond kinetics and chain diffusion

This paper presents a finite element analysis of steady-state crack propagation in viscoelastic soft solids exhibiting Mullins softening. A cohesive-zone model is employed to simulate the localized processes at the tip of a Mode I crack in materials governed by viscoelastic behavior and damage-induced Mullins effects. The study numerically evaluates the intrinsic dissipation characteristics of typical rubber-like materials, focusing on the influence of key factors such as Mullins damage, relaxation modulus, and relaxation time. The impact of these factors on material toughening is examined, with particular emphasis on their role in crack propagation. The results reveal that crack propagation velocity is highly sensitive to the interplay between energy dissipation mechanisms. Specifically, Mullins damage parameters are shown to increase fracture toughness by raising the local energy release rate threshold at the crack tip. Additionally, the relaxation modulus enhances viscous dissipation, further elevating this threshold and subsequently reducing crack propagation velocity. Interestingly, an inverse relationship between relaxation time and crack propagation velocity is observed. The study provides a detailed analysis of the dissipation mechanisms at the crack tip, offering valuable insights for improving material toughness.