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Mullins effect

The Mullins effect is a stress-softening phenomenon observed in rubber-like materials, particularly filled elastomers such as carbon black-reinforced rubbers, where the material exhibits reduced stiffness during subsequent loading cycles after an initial deformation, resulting in lower stress for the same strain level up to the previously reached maximum stretch.^[1] This behavior is accompanied by significant hysteresis in the stress-strain curve, energy dissipation, residual strain, and induced anisotropy.^[2] First systematically investigated by L. Mullins in 1947, the effect challenges accurate mechanical modeling of elastomers due to its dependence on deformation history.^[3] The Mullins effect occurs prominently in filled rubbers and crystallizing unfilled rubbers like natural rubber, but is absent or minimal in non-crystallizing pure gums such as polydimethylsiloxane.^[1] Experimental observations, including uniaxial tension tests, reveal that softening stabilizes after the first cycle and can be partially recovered through annealing at elevated temperatures (e.g., 80°C) or storage in vacuum, indicating a reversible microstructural change rather than permanent damage.^[1] Mullins' seminal work in the Journal of Rubber Research demonstrated this through cyclic stretching experiments on vulcanized rubbers, highlighting how pre-stretching alters subsequent modulus and tensile properties.^[3] Despite over seven decades of research, the precise physical mechanisms remain debated, with leading hypotheses including the rupture of filler-rubber bonds or clusters, slippage of polymer chains, and strain-induced crystallization in unfilled materials.^[2] Phenomenological models, such as those incorporating damage parameters in hyperelastic strain-energy functions, and microstructurally motivated approaches like chain scission theories, have been developed to capture the effect in finite element simulations for applications in tires, seals, and biomedical devices.^[4] These models emphasize the effect's path-dependence and multiaxial nature, essential for predicting long-term performance in engineering contexts.^[1]

Overview

Definition

The Mullins effect refers to the stress-softening phenomenon observed in rubber-like materials, known as elastomers, during the initial cycles of mechanical loading, characterized by a reduction in stress response for a given strain level in subsequent cycles compared to the first loading cycle.^[5] This effect is particularly prominent in filled elastomers, such as those reinforced with carbon black or silica, and manifests as a nonlinear degradation in the material's stiffness that persists after the initial deformation history. The phenomenon was named after researcher Leonard Mullins, who contributed key experimental and theoretical insights into its behavior in rubber vulcanizates.^[5] Core features of the Mullins effect include hysteresis, which represents energy dissipation primarily as heat during loading-unloading cycles, residual strain that remains after unloading, and the induction of anisotropy in the material's subsequent deformation response.^[6]^[7] These characteristics distinguish the Mullins effect from other time-dependent behaviors, as it is inherently nonlinear and predominantly occurs during the first few loading cycles, after which the stress-strain response tends to stabilize without further significant softening at strains below the historical maximum.^[1] In contrast to linear viscoelasticity, which involves recoverable time-dependent relaxation across all strain levels, the Mullins effect involves a permanent alteration in the material's structure that affects only the loading paths up to previously experienced strains.^[8] A representative example of the Mullins effect is observed in uniaxial tension tests of elastomer samples, where the stress-strain curve for the initial loading exhibits higher stresses at each strain level than the curves for reloading cycles, provided the maximum strain does not exceed the prior peak; this results in a characteristic "memory" of the maximum deformation, with reloading following a softened path up to that strain before potentially aligning with the original curve beyond it.^[6]

Key Characteristics

The Mullins effect manifests primarily as a stress-softening phenomenon in rubber-like materials during cyclic loading, characterized by the formation of a hysteresis loop in the stress-strain response. In a typical uniaxial tension test, the initial loading path follows a nonlinear curve typical of hyperelastic behavior, but upon unloading, the stress level drops significantly below the loading path for a given strain, forming a closed loop that indicates dissipative energy loss. This hysteresis is most pronounced in the first cycle and arises from irreversible changes within the material structure.^[9] Representative stress-strain plots illustrate this as a wide area enclosed by the loading and unloading branches, with the unloading curve often exhibiting a gentler slope, reflecting reduced stiffness. The softening exhibits strong strain dependence, becoming more evident and severe at higher strain levels, where the reduction in stress can be significant compared to the virgin loading path. Upon reloading, the stress-strain curve closely retraces the previous unloading path until the strain exceeds the prior maximum, at which point it deviates upward, resuming a path parallel to but below the original loading curve. This behavior underscores the path-dependent nature of the effect, with softening confined to strains below the historical maximum. In visual representations, secondary loading paths in multi-cycle plots appear as a series of nested loops, each progressively closer to the primary path but offset due to accumulated softening.^[9] Over successive cycles, the Mullins effect diminishes rapidly, typically stabilizing after approximately 10 loading-unloading cycles, where the hysteresis loop narrows and the response approaches a repeatable, though softened, hyperelastic curve. Full recovery of the initial stiffness is not immediate; it requires extended rest periods (e.g., days to weeks at room temperature) or thermal annealing, during which partial reversal of the softening occurs. Associated with this is a permanent set, or residual strain, observed upon complete unloading, which can be significant and often reduces over time, alongside a damage-like reduction in the effective modulus that persists until recovery.^[9] In multiaxial loading scenarios, the effect introduces directional softening, leading to induced anisotropy in the material's response, where the degree of softening varies with loading direction relative to prior deformations. Stress-strain curve plots under biaxial or shear conditions reveal asymmetric loops and varying modulus reductions across principal directions, highlighting the phenomenon's sensitivity to deformation history. This is particularly notable in filled elastomers, where the effect amplifies due to interactions between the polymer matrix and reinforcing particles.^[9]

Historical Development

Early Observations

The earliest documented observation of stress softening in rubber under cyclic loading, a precursor to the Mullins effect, was reported by French physicists Henri Bouasse and Z. Carrière in 1903. In their study of vulcanized natural rubber, they noted a reduction in stress required for subsequent stretching cycles after an initial deformation, describing the behavior as a form of "fatigue" or material softening without in-depth mechanistic analysis or quantification. This phenomenon was observed in unfilled rubber samples subjected to repeated tensile cycles, marking the first recognition of cyclic-induced stiffness loss in elastomers.^[10] Throughout the early 20th century, similar softening behaviors were sporadically mentioned in French and German scientific literature on rubber mechanics, often in the context of hysteresis during deformation. These reports typically attributed the observed energy dissipation and stress reduction to viscous effects within the polymer network, viewing them as inherent viscoelastic responses rather than a distinct, history-dependent phenomenon. For instance, studies on rubber elasticity in the 1910s and 1920s, such as those exploring tensile properties for industrial applications, described hysteresis loops in loading-unloading curves but lacked separation of viscous damping from permanent softening.^[1] However, these early investigations had significant limitations that prevented a comprehensive understanding of the effect. Research emphasized single-cycle or quasi-static tests, with minimal systematic exploration of multi-cycle loading protocols to isolate repeatable softening. There was no explicit acknowledgment of strain-history dependence, where the material's response varies based on prior maximum deformations, leading to inconsistent interpretations across experiments. Additionally, observations emerged in practical contexts like raw rubber samples used in early tire and conveyor belt testing around the 1910s–1930s, where softening contributed to performance degradation, but these were rarely quantified or formally named, remaining anecdotal within engineering reports.^[11] These preliminary reports provided initial evidence of cyclic softening in rubber, paving the way for more rigorous investigations in the mid-20th century.^[10]

Mullins' Contributions

Leonard Mullins, a British rubber scientist at the British Rubber Producers' Research Association (later the Malaysian Rubber Producers' Research Association and Tun Abdul Razak Research Centre) in Hertford, UK, conducted pioneering studies on the mechanical behavior of filled rubbers from the 1940s through the 1960s. His research focused on the inelastic responses of vulcanized rubbers reinforced with fillers like carbon black, laying the groundwork for understanding stress-softening phenomena in these materials.^[12] In his 1947 publication, Mullins first systematically documented the stress-softening observed in carbon-black filled natural rubber under cyclic loading, demonstrating that the modulus decreases significantly after the initial stretch and depends on the maximum prior strain achieved. This work quantified the cyclic behavior, showing that softening is more pronounced in filled rubbers compared to unfilled ones and is partially reversible, with recovery occurring slowly at room temperature (less than 20%) but accelerated by heat treatment, such as up to 80% recovery after heating to 100°C for two days. Mullins introduced the concept of recoverable "damage" in the rubber structure, attributing partial restoration to annealing effects that reverse chain rearrangements induced by deformation.^[12] Mullins' 1969 review synthesized decades of empirical observations, confirming the effect's presence across deformation modes including uniaxial tension, compression, shear, and biaxial stretching, while emphasizing its strain-dependent nature in filled elastomers. He highlighted that the degree of softening scales with the stiffening ability of the filler, shifting the perception of the phenomenon from simple hysteresis to a distinct form of recoverable inelasticity that influences long-term material performance. This comprehensive analysis solidified the effect's recognition in rubber science.^[13] The phenomenon became known as the Mullins effect in the 1960s, directly honoring his contributions, and his findings profoundly impacted rubber engineering by informing standards for fatigue resistance and preconditioning protocols in applications like tires and seals. Mullins' empirical insights, particularly the proportionality of softening to maximum prior strain and its partial reversibility, remain foundational for subsequent studies on elastomer durability.^[1]

Physical Mechanisms

Molecular-Level Explanations

At the molecular level, the Mullins effect in unfilled or lightly filled elastomers arises primarily from chain-scale mechanisms involving polymer dynamics. One key process is the irreversible sliding and disentanglement of polymer chains during the initial stretch, which reduces the entanglement density and subsequently lowers the modulus. This reconfiguration occurs as chains align and slip past one another under strain, leading to a permanent alteration in the network topology that manifests as stress softening upon subsequent loading. Neutron scattering studies have supported this by revealing changes in chain conformations and reduced entanglement densities post-deformation in stretched polymer networks. Molecular dynamics simulations further confirm that such disentanglement alone can account for essential features of the effect, including residual strain and strain-history dependence, without invoking additional microstructural changes. In natural rubber, strain-induced crystallization contributes significantly to the softening observed in the Mullins effect. Under extension, polymer chains align and form crystalline domains that act as physical crosslinks, enhancing stiffness during loading; however, these crystals melt during unloading, resulting in apparent softening in reload cycles. While the crystallization-melting cycle is largely reversible, hysteresis arises from trapped defects or incomplete recrystallization, contributing to energy loss and permanent set. This mechanism is particularly pronounced in unfilled natural rubber due to its stereoregular structure, as evidenced by early X-ray diffraction observations and subsequent studies quantifying crystallinity evolution during cyclic deformation. Temporary alterations in chemical bonds also play a role, particularly in vulcanized rubbers where sulfur crosslinks can undergo scission and reformation. During the first stretch, weaker polysulfidic bonds break preferentially, allowing network reconfiguration and stress relaxation; upon relaxation and heating, some reformation occurs, enabling partial recovery. This bond breakage-reformation process reduces the effective crosslink density temporarily, aligning with the observed softening and distinguishing it from permanent damage. Seminal work proposed this as a primary cause, with later investigations linking sulfidic linkage inhomogeneities to the extent of softening in isoprene-based elastomers. Energy dissipation in these molecular processes stems from internal friction associated with chain uncoiling and sliding, rather than viscous effects. As chains extend and disentangle, frictional interactions between segments generate heat, contributing to the hysteresis loop without altering the overall network irreversibly in subsequent cycles. This dissipation mechanism is distinct from filler-induced losses and is sufficient to explain the progressive softening in pure elastomer systems.

Role of Fillers and Reinforcement

In filled elastomers, particulate fillers such as carbon black and silica play a pivotal role in amplifying the Mullins effect by introducing heterogeneous stress fields and additional energy dissipation pathways at the filler-polymer interfaces. These fillers, typically incorporated at volume fractions of 10-30%, enhance the initial modulus but lead to pronounced softening upon cyclic loading due to the disruption of interfacial bonds and networks. Unlike intrinsic polymer chain mechanisms in unfilled materials, the filler-induced effects dominate in reinforced systems, where the softening manifests at lower strains and with greater hysteresis.^[1] The primary mechanism involves the initial strong adhesion between filler particles and the rubber matrix, which breaks during the first stretch through progressive debonding at the interfaces. This debonding reduces the effective load-bearing capacity of the fillers, thereby diminishing reinforcement and inducing macroscopic softening. Experimental observations indicate that such interfacial failure creates minor voids, comprising only a few percent of volume at stretches up to 300%, further contributing to the loss of stiffness.^[1]^[14] Polymer chains adsorbed onto filler surfaces undergo desorption under applied strain, enabling slippage along the filler-rubber boundary and generating hysteresis through frictional dissipation. This process is exacerbated in systems with high filler volume fractions, where a larger interfacial area promotes more extensive chain mobilization and incomplete re-adsorption upon unloading. The slippage is often reversible under thermal annealing, suggesting a physical rather than chemical origin for the bond alterations.^[1]^[9] At the nanoscale, the aggregated structure of fillers, such as carbon black clusters, is disrupted during loading, leading to the formation of weak zones or voids within the network. This breakdown of the filler percolation network partially recovers through re-aggregation and chain re-adsorption, but residual damage persists, limiting full restoration of initial properties. The effect is particularly evident in non-crystallizing rubbers, where filler dynamics provide the dominant softening pathway.^[1]^[14]^[15] Compared to unfilled elastomers, the Mullins effect is significantly more pronounced in filled rubbers, occurring at much lower strains—for instance, at around 10% extension in styrene-butadiene rubber reinforced with 50 parts per hundred rubber (phr) of carbon black—while unfilled variants typically require higher deformations or crystallization to exhibit comparable softening. This amplification, often by factors of several times, also results in greater residual strain after unloading in filled systems.^[1] Recent insights from network-based models highlight the microscopic pull-out of filler particles from the polymer matrix as a key contributor to the strain-induced anisotropy observed in the Mullins effect. These discrete approaches, building on filler-chain interaction morphologies, demonstrate how directional debonding and network reconfiguration explain the directional dependence of softening in reinforced elastomers.^[16]

Modeling Approaches

Phenomenological Models

Phenomenological models for the Mullins effect adopt an empirical, continuum mechanics framework to capture stress softening in filled elastomers by introducing a scalar damage parameter into standard hyperelastic constitutive equations. These models modify the strain energy density function of isotropic hyperelastic materials, such as the neo-Hookean or Mooney-Rivlin forms, by scaling the response with a damage variable η (where 0 < η ≤ 1) to represent the reduction in stiffness after initial loading. The effective Cauchy stress is then given by σ_eff = η σ_nom, where σ_nom is the nominal hyperelastic stress derived from the undamaged strain energy density W_0. A common formulation for the evolution of the damage parameter relies on the maximum strain energy density W_max achieved during prior loading history, expressed as

\eta = 1 - c \left(1 - e^{-b W_{\max}}\right),

where c (0 < c ≤ 1) controls the maximum damage level and b governs the rate of softening onset. This equation is fitted to uniaxial or biaxial cyclic test data by minimizing the error between predicted and experimental stress-strain curves, typically using least-squares optimization on unloading paths up to several strain levels. The parameters c and b are material-specific, determined empirically for each elastomer composition and filler content. To account for cycle dependence, these models distinguish between loading and unloading phases: during primary loading, η = 1 (no damage), while on unloading and subsequent reloading up to the prior W_max, the strain energy is scaled by η, and an unloading factor (often 1 - γ, with 0 < γ < 1) further reduces the response to approximate hysteresis loop closure. The damage "ratchets" forward with each new maximum strain, ensuring path-dependent softening without recovery on reloading within the same cycle. This approach qualitatively reproduces the S-shaped stress-strain curves observed in cyclic tests. The primary advantages of phenomenological models lie in their simplicity, requiring only a few additional parameters beyond the base hyperelastic model, which facilitates straightforward implementation in finite element simulations for component-level analysis of rubber structures. They effectively capture qualitative features like hysteresis and residual strain under quasi-static cyclic loading. However, these models lack a physical basis, relying on black-box fitting that necessitates recalibration for different materials or loading conditions, and they often exhibit poor predictive accuracy under multiaxial deformation paths due to the scalar nature of η. An illustrative example is the extension of the Ogden hyperelastic model with a Mullins damage term, which was applied in simulations of filled rubber components starting in the late 1990s to predict softening in automotive bushings and seals.

Micromechanical Models

Micromechanical models of the Mullins effect aim to capture stress softening by incorporating microstructural physics, such as polymer chain networks or filler-matrix interactions, within representative volume elements (RVEs) or statistical frameworks. These approaches derive constitutive relations from molecular-scale mechanisms, contrasting with purely empirical formulations by linking macroscopic response to microscale alterations like chain scission or interfacial debonding. Seminal contributions emphasize affine or phantom network theories and inclusion-based homogenizations to predict anisotropic and multiaxial behaviors. Network models represent rubber as a statistical ensemble of polymer chains in affine or phantom configurations, where softening arises from chain breakage and reformation during loading cycles. In these frameworks, the material is modeled as a cross-linked network where initial stretching induces irreversible scission of weaker links (e.g., van der Waals or partial covalent bonds), reducing chain density and increasing average chain length, thereby simulating the observed stress drop upon reloading. A prominent example is the modification of the eight-chain model, originally proposed by Arruda and Boyce for hyperelasticity, extended to account for network alteration. The strain energy density in the base eight-chain model is given by

W = N k \Theta \left[ \beta \lambda_{\text{chain}} - \ln \left( \frac{\sinh \beta}{\beta} \right) \right],

where N is the chain density, k \Theta is the thermal energy, \lambda_{\text{chain}} = \sqrt{I_1 / 3} is the chain stretch (with I_1 the first invariant of the right Cauchy-Green tensor), and \beta = \mathcal{L}^{-1}(\lambda_{\text{chain}} / \sqrt{n}) involves the inverse Langevin function \mathcal{L} and n the number of Kuhn segments per chain. To incorporate the Mullins effect, Marckmann et al. introduced evolving parameters: chain density N = N_0 \exp(-\kappa \bar{\lambda}) decreases due to breakage, while n = n_0 \exp(\mu \bar{\lambda}) increases as longer chains form, with \bar{\lambda} the maximum historical chain stretch and \kappa, \mu > 0 material constants; this yields pseudo-elastic reloading curves matching cyclic data.^[17] Such models simulate recovery through limited chain reformation, often with time-dependent kinetics in phantom network variants where junctions fluctuate independently of affine deformation. Filler-based micromechanical models treat filled rubbers as composites with rigid particles embedded in a hyperelastic matrix, using RVEs to homogenize the response where rubber chain absorption onto filler particles causes softening. These approaches model the filler-rubber interface interactions with nonlinear effects under strain, leading to altered load transfer and effective stiffness. A key formulation uses a double-inclusion method within the RVE to predict effective stress-strain relations, employing the incremental Mori-Tanaka scheme to integrate stresses over particle orientations for multiaxial loading, predicting anisotropy from aligned fillers post-processing; the full derivation involves solving Eshelby inclusion problems for each orientation angle \theta, yielding \sigma_{\theta} = \int_0^{2\pi} \mathbf{T}(\theta) : \boldsymbol{\epsilon} \, d\theta / 2\pi, where \mathbf{T} is the Eshelby tensor and \boldsymbol{\epsilon} the macroscopic strain. This captures the stress drop from filler interactions, validated against uniaxial and shear tests on carbon black-filled rubbers.^[18] Advanced micromechanical models extend these ideas using Eshelby tensors to represent fillers as ellipsoidal inclusions in a viscoelastic network, incorporating recovery through reattachment of breakable chains. In the 2006 model by De Tommasi, Puglisi, and Saccomandi, rigid ellipsoids are embedded in an incompressible rubber matrix, with softening due to a fraction of chains that break and reattach viscoelastically; the homogenized stress is derived via mean-field approximation, \boldsymbol{\sigma} = (1 - V_f) \boldsymbol{\sigma}_m + V_f \mathbf{A} : \boldsymbol{\sigma}_i, where \mathbf{A} is the strain concentration tensor from Eshelby solutions, and recovery is modeled by a relaxation time for chain rebonding. This framework predicts three-dimensional isotropic softening and cyclic recovery, outperforming phenomenological models in multiaxial validation against experiments on silica-filled rubbers, where it better captures strain-history dependence and azimuthal shear responses without ad hoc parameters.^[19] These micromechanical approaches have been validated to match experimental data in filled silica rubbers more accurately than phenomenological models for multiaxial loading, as they inherently link softening to microstructural evolution like chain scission and debonding. Recent advancements, such as 2024 work by Abdusalamov et al., employ deep symbolic regression on cyclic test data to rediscover network parameters (e.g., evolving N and n) in eight-chain formulations, yielding interpretable equations that recover Mullins softening without prior assumptions, enhancing model discovery from data in complex filler systems.^[20]

Experimental Observations

Measurement Methods

The Mullins effect is typically quantified through cyclic mechanical testing of elastomer specimens, with uniaxial tension or compression serving as the standard protocol. Samples are subjected to repeated loading-unloading cycles at controlled strain rates ranging from 0.01 to 1 s⁻¹, extending to maximum strains of 100-300%, and encompassing 3-10 cycles to capture the initial softening and subsequent stabilization. These tests reveal the characteristic stress softening, where the reloading curve falls below the initial loading path up to the prior maximum strain.^[21] Key metrics for assessing the effect include the damage ratio D = 1 - \frac{S_{\text{reload}}}{S_{\text{virgin}}}, where S denotes the secant modulus calculated from the stress-strain curve at a fixed strain level, quantifying the extent of softening by comparing reloaded and virgin material responses. Additional measures encompass the hysteresis loop area, which represents energy dissipation per cycle, and residual strain \varepsilon_{\text{res}}, the permanent deformation retained after unloading.^[21] These parameters provide a comprehensive evaluation of the material's degradation without requiring complex post-processing. Testing employs universal testing machines, such as Instron systems, equipped with extensometers for precise displacement measurement; for anisotropic assessments, optical strain mapping techniques, like digital image correlation, are integrated to track local deformations.^[22] ^[23] Specimens are typically prepared as dogbone shapes to ensure uniform strain in the gauge section and prevent premature necking or slippage at grips. Protocols emphasize sample preconditioning through low-strain cycles to stabilize initial viscoelastic responses, followed by primary cyclic loading. For recovery investigations, annealing at elevated temperatures (e.g., 80°C) allows observation of partial or full restoration of stiffness, often conducted in a stress-free state.^[23] Multiaxial extensions utilize biaxial stretchers or bulge test rigs to impose equibiaxial tension, enabling evaluation of direction-dependent softening. ^[23] Challenges in measurement include rate dependence, as higher strain rates amplify viscoelastic contributions that mask pure Mullins softening, necessitating low-rate conditions for isolation. Ensuring grip adhesion and avoiding specimen slippage or necking in dogbone geometries is critical, particularly at large strains, to maintain test validity. To facilitate cross-material comparisons, data are normalized by plotting reduced stress—nominal stress divided by the elasticity factor (\lambda - \lambda^{-2})—against the first strain invariant I_1 = \lambda^2 + 2\lambda^{-1}, highlighting invariant-based softening trends independent of specific loading paths.^[21]

Material-Specific Behaviors

In unfilled natural rubber, the Mullins effect results in moderate stress softening, with high recoverability attributed to strain-induced crystallization that facilitates chain reorganization upon unloading.^[24] This softening becomes particularly prominent at engineering strains exceeding 200%, where the material's inherent ability to crystallize under tension contributes to reduced hysteresis in subsequent cycles compared to filled variants.^[25] In filled synthetic rubbers, such as styrene-butadiene rubber (SBR) reinforced with carbon black, the Mullins effect is more severe, accompanied by persistent residual strains that reflect irreversible chain detachment from filler surfaces.^[26] The type of filler significantly influences hysteresis; carbon black induces greater energy dissipation and slower recovery than silica fillers due to stronger polymer-filler interactions, resulting in amplified softening under repeated deformation.^[27] Silicone elastomers exhibit a minimal Mullins effect, primarily due to their weak intermolecular chain interactions and low tendency for network rearrangement, leading to predominantly viscous dissipation with little permanent set in unfilled formulations.^[28] In these materials, cyclic loading produces smaller hysteresis loops and faster stress recovery compared to carbon-chain-based elastomers, as the effect is largely confined to surface slippage in any minor filler inclusions rather than bulk chain disentanglement.^[29] An analogous softening phenomenon occurs in biological tissues like arterial walls, where the Mullins effect arises from the progressive recruitment and uncoiling of collagen fibers during initial stretching, followed by strain-stiffening in subsequent cycles as aligned fibers bear load more efficiently.^[30] This fiber recruitment mechanism results in anisotropic damage that stabilizes after preconditioning, mimicking the hysteresis observed in synthetic elastomers but driven by microstructural realignment rather than filler interactions.^[31] In swollen or highly crosslinked elastomer variants, the Mullins effect is reduced in hydrogels due to increased chain mobility and dilution of network constraints in the swollen state, which diminishes stress softening and hysteresis during cyclic deformation.^[32] Conversely, in variants with high crosslink density, the effect is enhanced as dense networks resist disentanglement, promoting greater bond scission and pronounced softening under load, particularly at strains around 300%.^[33]

Applications

In Rubber Engineering

In rubber engineering, the Mullins effect significantly influences the design and performance of tire components, where initial stress softening reduces handling stiffness during the first loading cycles, potentially compromising vehicle stability and ride quality.^[34] To mitigate this, manufacturers incorporate preconditioning cycles during production, subjecting tires to controlled deformation sequences that stabilize the material response and minimize further softening under operational loads.^[35] This approach ensures consistent mechanical properties from the outset, as the effect largely stabilizes after 3-5 cycles, preventing excessive hysteresis that could lead to uneven wear or reduced traction.^[35] For seals and mounts, the Mullins effect can alter sealing pressure and damping characteristics. Engineers address this by incorporating safety factors in designs to account for modulus loss post-initial loading, ensuring long-term functionality under cyclic service conditions. In automotive bushings, for instance, this softening directly impacts noise, vibration, and harshness (NVH) performance by reducing isolation efficiency and altering frequency response.^[36] Mitigation strategies focus on material formulation and processing to limit the extent of softening. Filler optimization, such as using lower aspect ratio particles, reduces strain amplification and network disruption in filled rubbers, thereby attenuating the effect.^[29] Additives like silane coupling agents enhance filler-rubber interface stability, promoting better dispersion and minimizing chain detachment during deformation.^[37] Thermal preconditioning, involving controlled heating to promote recovery of internal structures, further stabilizes the material before assembly.^[38] The Mullins effect precedes fatigue cracking in rubber components, with initial softening cycles depleting energy reserves and accelerating damage accumulation. Models incorporating this phenomenon predict endurance of a limited number of cycles before significant degradation in filled systems, guiding component lifespan assessments.^[39] Industry standards like ASTM D5992 provide protocols for cyclic testing of rubber products, enabling evaluation of dynamic properties and Mullins-induced changes under vibratory conditions to ensure compliance in engineering applications.

In Computational Modeling

The integration of Mullins effect models into finite element analysis (FEA) software enables the simulation of stress softening in elastomers under cyclic loading, facilitating predictions of material degradation without extensive physical testing. In ABAQUS/Standard, the Mullins effect is implemented through the *HYPERELASTIC, MULLINS option, which incorporates damage evolution via user-defined subroutines to capture the path-dependent softening behavior in filled rubbers. This approach allows for quasi-static cyclic simulations where the maximum historical strain invariant governs the damage parameter, ensuring compatibility with hyperelastic base models like Mooney-Rivlin or Ogden.^[40]^[41] Key simulation challenges arise from the inherent path-dependence of the Mullins effect, which complicates convergence in nonlinear solvers due to the need to track maximum strain history across loading cycles. Multiaxial generalizations often rely on strain invariants to extend uniaxial observations, but this introduces numerical instabilities, particularly in large-deformation analyses where iterative updates of the damage variable must balance accuracy and computational efficiency. Phenomenological models implemented via multiplicative decomposition of the deformation gradient help mitigate these issues by separating elastic and damage responses, though they require careful calibration to avoid over-softening artifacts in complex geometries.^[42]^[43] In practical applications, FEA with Mullins effect modeling supports virtual prototyping of tires subjected to cyclic loads, where softening predicts altered contact stresses and fatigue life during rolling simulations. For instance, incorporating the effect in tire models reveals increased strain intensities under repeated deformation, aiding design optimization for durability. Similarly, simulations forecast softening in O-rings used in seals, quantifying hysteresis losses that influence sealing performance over service cycles. Advanced implementations couple Mullins effect with viscoelasticity in ANSYS, using hybrid frameworks to model time-dependent dissipation alongside damage evolution in filled elastomers. Hybrid models also extend to 3D-printed elastomers, where layer-dependent softening is simulated to evaluate print-induced anisotropy under cyclic strain.^[44]^[34]^[45] Validation of these FEA models typically involves correlating simulated hysteresis loops with experimental cyclic tests, often yielding prediction errors of 15-20% in energy dissipation for carbon black-filled rubbers under uniaxial loading. Such discrepancies highlight the limitations of invariant-based generalizations but confirm the models' utility for qualitative trend capture in engineering design. Looking ahead, machine learning-enhanced approaches promise real-time parameter identification, using deep symbolic regression to derive Mullins damage functions from experimental data and accelerate inverse calibration in simulation workflows.^[43]^[46]^[20]

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Investigation of Combined Aging and Mullins Stress Softening of ...
Nov 11, 2024 · The present study investigated the effects of thermal aging, ultraviolet radiation (UV), and stress softening on the performance properties of rubber.
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https://doi.org/10.1016/j.mechmat.2009.05.002
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[PDF] Molecular insight into the Mullins effect: Irreversible disentanglement ...
Molecular insight into the Mullins effect: Irreversible disentanglement of polymer chains revealed by molecular dynamics simulation.
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Mechanical Properties and Mullins Effect in Natural Rubber ...
Aug 1, 2019 · 11 Mullins L., Effect of stretching on the properties of rubber, Rubber Chemistry and Technology. ... Download PDF. back. Additional links ...
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Physical interpretation of the Mullins softening in a carbon-black ...
Sep 15, 2014 · When comparing the stress-stretch responses of filled and unfilled rubbers, filled rubbers evidence substantial softening, upon first stretch, ...Missing: strength | Show results with:strength
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Study of the Mullins Effect in Carbon Black-Filled Styrene–Butadiene ...
Jul 11, 2022 · This work investigates the microscopic mechanism of the Mullins effect in carbon black (CB)-filled styrene–butadiene rubber (SBR) after application of cyclic ...
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[PDF] Importance of Mullins effect in commercial silicone elastomer ...
After stretching the sample to 20% strain, strain is again increased following the “strain rise path”. The term “2nd test” will be added to the sample name and ...
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Control of Mullins stress softening in silicone elastomer composites ...
Sep 29, 2021 · This study provides fundamental understanding on the relation between filler-polymer interactions, filler morphology and Mullins stress softeningMissing: explanations | Show results with:explanations
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On the Mullins effect and hysteresis of fibered biological materials
This paper presents and compare continuous and discontinuous damage approaches to model softening effects in fibered materials such as soft biological tissues.
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Detection and Characterization of Molecular-Level Collagen ...
Utilizing a recently developed collagen hybridizing peptide (CHP), we demonstrate that a single mechanical overstretch of an artery produces molecular-level ...
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[PDF] Modeling the Mullins effect in swollen rubber - HAL
Jan 22, 2024 · More precisely, the Mullins effect appears to decrease with the increase of swelling degree. For the sake of simplicity, the parameters r ...
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Chemical Bond Scission and Physical Slippage in the Mullins Effect ...
May 24, 2019 · We find that the Mullins effect results from chain extension and chain slippage at a low cross-linking density, whereas at a high cross-linking density, it is ...
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[PDF] MANIFESTATIONS OF THE MULLINS EFFECT IN FILLED ...
... elastomers, principally by the manifestation of the softening phenomenon called the Mullins effect. ... Mullins Effect, and is defined as permanent or semi ...
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Influence of the Mullins effect on the stress–strain state of design at ...
Mullins effect increases the strain intensity value in the tyre. Introduction. Mechanical properties of elastomers can be significantly improved by introduction ...
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[PDF] Mullins effect in the calculation of the stress-strain state of a car tire
In this paper, the stress-strain state calculation takes into account the Mullins effect. Experimental data are shown that demonstrate the substantial ...
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[PDF] Mullins effect in swollen rubber: Experimental investigation ... - HAL
Aug 18, 2021 · The Mullins effect or the stress-softening effect disappears after several cycles of loading, i.e. five cycles for the materials used in the ...Missing: mounts | Show results with:mounts
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(PDF) Investigation on the full Mullins effect using time-dependent ...
Sep 3, 2020 · In this paper, we propose a new method for antivibration applications. The full Mullins effect in the modified hyperelastic model, with stress ...
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[PDF] Fatigue Life Prediction of Natural Rubber in Engine Mounts
This is known as the Mullins effect, which creates a hysteresis loop during cyclic stress strain loading. The effect is shown in Figure 2.1, where the reduction ...
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A review of bushing modelling approaches for MultiBody simulations
A stress-softening phenomenon, also known as Mullins effect ... Experiment-based modeling of cylindrical rubber bushings for the simulation of wheel suspension ...
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[PDF] 1 Influence of Silane Coupling Agents on Filler Network Structure ...
Adding filler dramatically improves the strength of rubber by reinforcement and tailoring the type of filler and the chemistry of the interface between filler ...
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Thermal and Volumetric Signatures of the Mullins Effect in Carbon ...
The Mullins effect, first observed in the 1940s, refers to the stress-softening phenomenon exhibited by filled elastomers during cyclic loading [4]. This ...2. Materials And Experiments · 2.2. Tensile Tests Combining... · 3. Results And Discussion<|control11|><|separator|>
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[PDF] Unified model for Mullins effect and high cycle fatigue life ... - HAL
ABSTRACT: The study describes the basic principles of a general damage model (GDMF) for Mullins effect and high cycle fatigue loadings of rubber materials and ...
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4.7.1 Mullins effect - ABAQUS Theory Manual (v6.5-1)
The Mullins effect material model provided in ABAQUS is intended for modeling the phenomenon of stress softening, commonly observed in filled rubber elastomers.<|control11|><|separator|>
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Development of new constitutive equations for the Mullins effect in ...
These physical considerations are introduced in the 8-chain model because the corresponding strain energy density only depends on two material parameters which ...
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Simulation of Mullins effect and permanent set in filled elastomers ...
Stress softening, also known as Mullins effect, can be observed in filled-elastomers, after subjecting a virgin specimen to an initial load.
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Constitutive modelling of rubbers: Mullins effect, residual strain, time ...
Nov 15, 2021 · In this paper, a hyper-visco-pseudo-elastic constitutive equation is introduced and applied to model the complex behaviour of filled rubbers.
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Advances in numerical modelling of tyre fatigue performance: a review
Aug 19, 2025 · Hyperelastic material properties were applied in the material modelling, while Mullin's effect was considered to simulate softening behaviour ...
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MULLINS EFFECT - ABAQUS Example Problems Manual (v6.6)
This example illustrates the use of the Mullins effect option to model the static and steady-state rolling interaction between a solid rubber disc and a rigid ...Missing: virtual prototyping
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A Constitutive Model for Soft Materials Incorporating Viscoelasticity ...
Aug 6, 2025 · In this work, we develop a coupled viscoelastic and Mullins-effect model to characterize the deformation behavior of the tough gel. We modify ...