Network Alteration Theory Research Articles

Multiscale experimental characterization and physical-based constitutive modeling of silicone adhesive

Silicone adhesive has been widely used in assembly glass curtain walls which leads to its mechanical behaviors under heavy investigation. Due to its complexity, the microstructure of silicone adhesive stays unclear. A detailed description of microstructures of silicone adhesive is the basis to obtain an in-depth understanding of its mechanical behavior and the related mechanisms. In this work, series of multiscale experiments, uniaxial tension, uniaxial compression, simple shear and scanning electron microscopy (SEM) observation, were conducted to characterize mechanical behaviors and microstructure features of silicone adhesive. Stress-strain curves obtained through macroscale experiments were presented. Morphology and dispersion of filler clusters in silicone adhesive were analyzed using Gaussian mixed models. A modification of molecular chain spatial distribution was made to non-affine network model to consider anisotropic damage evolution of polymer matrix. A simplified term was further introduced to account for load-bearing capacity of filler clusters based on fractal theory where fractal dimensions of clusters work as a connection between microscale and macroscale. The modified model was further extended to describe Mullins effect combined with network alteration theory. The results of model validation and comparison demonstrated that due to the consideration of polymer chain anisotropic evolution and filler cluster microscopic features, the proposed model can predict the mechanical behavior of silicone adhesive under different loading conditions as well as the stress softening and permanent set of Mullins effect more accurately.

Experimental study and model on the low‐cycle fatigue behavior of thermoplastic vulcanizates

AbstractThermoplastic vulcanizates (TPVs) is composed of crosslinked rubber and plastic and have recently been used to replace traditional rubber. To investigate their constitutive behavior throughout the fatigue life, a series of tensile fatigue tests were carried out. The experimental results show significant fatigue softening of TPV, which can be regarded as the joint action of fatigue damage and fatigue creep. After many cycles, the viscous flow can also be restored; thus, the fatigue creep was eliminated by the cyclic‐pause‐cyclic experiment. The cyclic curves were obtained when the fatigue damage was the only effect. Based on a visco‐hyperelastic constitutive model, the evolution rate of the constitutive model parameters during fatigue damage was obtained using the network alteration theory. Considering fatigue damage and fatigue creep effect together, a fatigue‐softening constitutive model is proposed. The model can predict the constitutive behavior of TPVs over their entire fatigue life.

A Constitutive Model for Mechanical Behaviors of Novel Double Network Hydrogels with Mechanophores

Double network hydrogels (DN hydrogels) with high stretchability and toughness have attracted broad research interest. Recently, a kind of novel tough DN hydrogels was designed by means of the reactive strand extension strategy, which introduced mechanophores into the first network. When the strands of the first network reach their nominal stretching limit, the mechanophores allow the strands to survive through force-coupled reactions instead of fracture. As a consequence, the novel hydrogels can achieve a better mechanical performance compared with the conventional DN hydrogels. In this work, we aim to develop a constitutive model for the novel DN hydrogels. The model is based on the worm-like single-chain model by introducing bond deformation. The network alteration theory is used to account for the damage behaviors. The theoretical framework is capable of clarifying the difference in mechanical behaviors between conventional and novel DN hydrogels, which demonstrates the importance of bond deformation on the mechanical behaviors of DN hydrogels.

A hyperelastic-damage model based on the strain invariants

We develop a unified framework to describe the hyperelastic and damage behaviors of soft materials. The free energy density consists of both the crosslinked part of Langevin chains and the entanglement part described by a tube model. The micro–macro transition is obtained by using the averaging value over a microsphere. Consequently, the free energy density only depends on the strain invariants. The damage mechanism is then incorporated via the network alteration theory for the crosslinked part, while the deformation can also induce dilation of the tube and subsequent a decrease in the modulus for the entangled part. The hyperelastic model captures the Mooney Rivlin plot with a first decrease in the small deformation regime and then an increase in the large deformation region. The hyperelastic model can also accurately reproduce the stress–strain relationships in the uniaxial and biaxial loading conditions for various elastomer systems. The damage model can describe the stress response of filled rubbers in uniaxial and biaxial loading conditions, which shows a considerable improvement compared with the damage model without the entanglement effect. Since the model only depends on the strain invariants, the model can be easily implemented into the commercial finite element software Abaqus through a UHYPER. The finite element model can accurately reproduce the experimentally measured strain distribution of filled rubbers with complex geometry in the loading process. These results clearly show that the hyperelastic-damage model developed in this work has strong potentials to predict the mechanical responses of soft materials for practical applications.

An approach to assess the thermal aging effects on the coupling between inelasticity and network alteration in filled rubbers

The development of a constitutive representation for predicting the durability of filled rubbers upon aggressive environmental conditions is crucial for numerous technical examples but still remains largely an open issue. This paper presents a constitutive model for the prediction of thermal aging effects on the inelastic response of filled rubbers. A network alteration theory is proposed for the thermally driven network degradation by considering a collection of chemical and physical changes occurring inside the rubber-filler material system. The network decomposition considers scission-rebound mechanisms of links (between filler aggregates and between elastically active cross-linked chains) and changes in movements of the free chains superimposed into the (newly born and original) chemically linked network. The material kinetics is designed using multi-step (physical) stress relaxation experiments performed at room temperature on a sulfur-vulcanized styrene–butadiene rubber containing different amounts of carbon-black fillers, and previously submitted to an accelerated chemical stress relaxation procedure, i.e. aging at constant stretch for various temperatures and exposure times. The time–temperature equivalence principle is used to construct master curves of the alteration kinetics for both the cross-linked chains and the superimposed free chains. The amplified effect of the fillers is also considered in our network decomposition to account for the cross-linked chains/fillers interactions and the free chains/fillers interactions. The capabilities of the proposed network alteration based constitutive model to describe and predict the material response evolution in terms of stiffening, hysteresis area increase and physical relaxation increase are shown. The chemically induced “plastic-like” deformation and stress relaxation are predicted using the developed model. The results show the importance of the inelastic effects on the prediction of long-term material response due to thermally activated degradation. From an applicative point of view, the model makes possible to estimate the operational life of pre-strained rubber mechanical components with respect to loss of pressure and flexibility.

Constitutive modelling for the mullins effect with permanent set and induced anisotropy in particle-filled rubbers

Rubber industry plays an important part in various application fields and particle-filled rubbers with reinforced mechanical properties are very essential. The stress softening during cyclic loading, known as the Mullins effect, are evident in filled rubbers. The Mullins softening can induce permanent set and anisotropy. It is crucial to understand these phenomena well. In this paper, we propose a new constitutive model that integrates the micro-sphere model with the network alteration theory. The physical mechanisms of the Mullins effect in particle-filled rubbers are analyzed elaborately and comprehensively to propose new directional damage equations for the network alteration parameters. Both the softening effects in stretched directions and the stiffening effects in contracted directions are taken into account in the modelling. The resulting theoretical model is then applied to predict the stress softening, the residual stretch and the induced anisotropy in vsrious filled rubbers. Moreover, the prediction effect of the current model is compared with those by applying two existing directional macromolecular models. The comparisons with other model predictions and with the experimental data are then analyzed intensively to reveal the underlying mechanisms of the Mullins effect in the different filled rubbers. The results demonstrate the capability of the proposed model to quantitatively predict the Mullins effect with permanent set and induced anisotropy and to clearly reveal the major mechanisms of the Mullins effect for various filled rubbers.

A Constitutive Model for Alginate-Based Double Network Hydrogels Cross-Linked by Mono-, Di-, and Trivalent Cations

In this contribution, a micro-mechanically based constitutive model is proposed to describe the nonlinear inelastic rubber-like features of alginate-based double network hydrogel cross-linked via various counterions. To this end, the lengthening of the polysaccharide polymer chain after a fully stretched state is characterized. A polymer chain is firstly considered behaving entropically up to the fully stretched state. Then, enthalpic behavior is accounted for concerning the following lengthening. To calculate enthalpic behavior, the macroscopic material properties, such as elastic modulus, are integrated into the proposed model. Thus, a new energy concept for a polymer chain is proposed. The model is constituted by the proposed energy concept, the network decomposition model, the Arruda–Boyce eight chain model and the network alteration theory. The model is compared against the cyclic tensile test data of alginate-based double network hydrogels cross-linked via mono-, di-, and trivalent cations. Good agreement between the model and experiments is obtained.

Micromechanical modeling of the multi-axial deformation behavior in double network hydrogels

The Mullins-type damage behaviors in double network hydrogels have attracted a broad research interest in recent years. However, most of current works focus on characterizing and modeling the uniaxial deformation behaviors of these materials. In this work, we combine experimental and theoretical approaches to investigate the anisotropic damage behaviors of double network hydrogels revealed in multi-axial deformation conditions. We demonstrate that an isotropic damage model based on the eight-chain model and the network alteration theory fails to capture the stress response in multi-axial loading tests. An anisotropic damage theory based on the microsphere model has also been developed, while both the affine and non-affine approaches are adopted to obtain the micro-macro mapping. The results show that the affine microsphere model cannot describe the experimental results in pure shear and unequal biaxial tests. Remarkably, the non-affine microsphere model with three parameters captures all the important features of the experimental observations. This is because the non-affine model accurately predicts the directional damage of the primary cross-linked network. The non-affine microsphere model is also able to describe the damage cross-effect in double network hydrogels. The developed theoretical framework can promote the fundamental understanding of the anisotropic damage behaviors in various types of tough gels.

Development of the network alteration theory for the Mullins softening of double-network hydrogels

Double-network hydrogels (DN gels) with excellent mechanical properties have great application prospects as an innovative soft material. The Mullins effect originally discovered in rubber materials is also observed in this material. The network alteration theories have achieved many successes in capturing this stress-softening behavior of DN gels and exploring the underlying damage mechanisms. In this paper, we develop a network alteration theory to account for the softening behaviors of DN gels under both uniaxial tension and pure shear and simultaneously explore more about the damage origins. The advantage of the use of worm-like chain model for DN gels is illustrated. New network evolutions of the network alteration parameters are further developed on a basis of an existing network alteration theory. The model predictions of the new model for the experimental results of DN gels under both uniaxial tension and pure shear are compared with those of the old one. The results demonstrate that the new model is more physically sound and offers much better predictive capabilities. It is found that the dangling chains, chain disentanglements and the degradation of the interactions between the two networks during deformation play significant roles in the Mullins softening of DN gels. This paper gives new insights into the microdamage origins of DN gels and significantly improve the network alteration theory for DN gels.

A constitutive model for multi network elastomers pre-stretched by swelling

Multi network elastomers (MNEs), composed of a single sacrificial network and other matrix networks, exhibit appealing mechanical properties. In this paper, we develop a constitutive model for MNEs prepared by multiple swellings. Firstly, the swelling process is analyzed. The free energy of a swollen elastomer consists of the strain energy of deformed polymer chains and the energy of mixing. We analyze the mechanical equilibrium and chemical potential equilibrium during the swelling of MNEs. The degree of swelling of MNEs is affected by the preparation conditions, like the component of the solution, as well as the microscopic physical quantities, such as the chain length and density of polymer chains of the networks. Secondly, the free energy of the completed MNE subjected to external loading is composed of the strain energy of each network. The chains of the sacrificial network (first network) can be destroyed gradually when the applied load increases. We adopt the network alteration theory to describe the above progressive damage of the sacrificial network. In addition, the matrix networks are fully elastic. We verify this model using the experimental data including stress–stretch curves of elastomers with double and triple networks, as well as the step cycle curves of triple network elastomers (TNEs). This model predicts well for the swelling-induced pre-stretches of each network under specified preparation conditions. It relates the stress to the stretch of MNEs under external loading, and indicates the stress contribution of each network, and the damage evolution of the sacrificial network. Our model is instructive for designing MNEs with desired mechanical properties.

Strain and filler ratio transitions from chains network to filler network damage in EPDM during single and cyclic loadings

Chains and filler network damage were investigated during single and multiple cycles on a series of vulcanized EPDM containing various filler contents. In both series of experiments, a strain and a filler ratio transitions for damage mechanisms were identified. For low filler content (≤40 phr), damage mechanisms mostly occur in the elastically active rubber network consistent with the chains network alteration theory associated with irreversible chains scission/bond breakage. For high filler content (>40 phr), a strain transition occurs with damage initially located in the elastically active rubber network, but subsequently localizes in the vicinity of the filler-filler network. This is ascribed to filler re-aggregation with strain, improving its load-bearing capacity, that may release the immobilized rubber formed by the chains that are occluded and bonded to fillers. During cyclic experiments, such reversible release involving loss of weak physical bonds and chains slippage yields in a progressive cavity closing with cyclic accumulation that prevents further irreversible damage of the elastically active rubber network.

A constitutive model for hysteresis: the continuum damage approach for filled rubber-like materials

In this contribution, a constitutive model is proposed for filled rubber-like materials to describe hysteresis. The model is based on the network decomposition concept. Rubber matrix is decomposed into purely elastic and deformable networks. The Gent model is considered to model the purely elastic network. The deformable network is simulated by using the freely jointed chain concept, a probability density function of the chain length and the network alteration theory. Damage in a network is assumed as a debonding of polymer chains from filler surface, which evolves with respect to the chain length over the set of available chains. In the consecutive loading cycles, a new network rearrangement is accounted for the characterization of the hysteresis behavior. Debonded chains in the damaged network are considered to rebond up to the point of the permanent set. The model contains seven material parameters and was successfully tested on the sequential cyclic uniaxial tensile test data of filled rubbers having different filler concentrations.

A physically-based damage model for soft elastomeric materials with anisotropic Mullins effect

In this paper, we develop a new physically-based damage model to describe the anisotropic Mullins effect of elastomeric materials. The damage model is based on our previous hyperelastic model (Xiang et al., 2018), which has a strain energy density function composed of the crosslinked part and the entangled part. We describe the damage of the crosslinked network using a network alteration theory. Chains along various directions suffer from different degrees of damage and thus cause the anisotropy. The irreversible degradation on constraints for entanglement leads to the reduction of the entangled modulus with the increasing of historical maximal stretch. This model can well capture the stress-softening, anisotropy and residual deformation (permanent set) with only five parameters that have definite physical meanings. It also predicts the damage behaviors under various states of deformation, for both unfilled and filled elastomers. We use the model to fit the experiment data of VHB 4905, silicon rubber, carbon-black filled styrene butadiene rubber (SBR) and carbon-black filled nature rubber (NR). The damage induced anisotropy, which is validated by directional loading experiments of silicon rubber and carbon-black filled NR, can be characterized. We compare this model with other existing similar ones, and show its competitive advantages in characterizing the comprehensive Mullins effect with concise and explicit mathematical expression and physical meaning.

Changes in functional connectivity dynamics with aging: A dynamical phase synchronization approach

The dynamics of the human brain network has attracted broad attention, in recognition of the concept that functional connectivity is not static, but changes its pattern over time, even in the resting state. We hypothesized that analysis of continuously captured time-varying instantaneous phase synchronization between signals from different brain regions might add another dimension to already identified network dynamics. To validate this hypothesis as an aid to elucidating the physiological mechanisms of aging, we examined time-series of instantaneous phase synchronization events in resting-state EEG activity across the brain, in healthy younger and healthy older subjects. We then characterized the temporal dynamics of phase synchronization using multiscale entropy, which quantifies the complexity of brain signal dynamics over multiple time scales. The results of surrogate analyses confirmed that the temporal dynamics of phase synchronization arise from deterministic processes in the neural network system. Group comparison showed region-specific enhanced complexity of temporal dynamics of phase synchronization in older subjects in alpha band predominantly in frontal brain regions, which was not identified by a comparative phase synchronization approach such as phase lag index. Enhanced complexity of temporal dynamics of functional connectivity in older subjects might reflect a general network alteration theory in aging. This is a first report describing the importance of capturing the dynamics of instantaneous phase synchronization and characterizing its temporal organization. Applying this method to neurophysiologic data may provide a novel understanding of dynamical neural network processes in both healthy and pathological conditions.

Mullins thresholds in context of the network alteration theories

Rubbers undergo a softening phenomenon in first cycles of loading where it is known this softening is more pronounced in filled rubbers. Existence of a clear physical interpretation for this phenomenon called Mullins effect is of great importance in development of constitutive relations and numerical simulations. In this paper, a recently proposed network alteration theory is extended to obtain a general form of evolution laws based on the physical description of the material molecular network. The resulting evolution laws are defined based on different measures to evaluate efficiency and accuracy of each measure in predicting the Mullins softening in context of the network alteration theories. Two experimental data sets on natural rubber are thoroughly investigated to determine a reliable threshold for this effect. It is concluded that, the maximum principal stretch and the Euler‒Almansi strain are not good thresholds for Mullins effect at all. Other investigated measures are relatively efficient at small stretches but they mostly become less and less accurate at large stretches. However, the stored strain energy preserves its accuracy at the whole stretch range.

Development of a network alteration theory for the Mullins - softening of filled elastomers based on the morphology of filler–chain interactions

• A new 2-step method is proposed for accurate determination of the network parameters. • Active monomers are not necessarily decreasing with deformation but can even increase. • It is concluded, physical filler–chain bonds are main broken links during deformation. • Inspired by the Langmuir's theory, evolution of material network is formulated. • A physically-motivated network alteration theory is proposed for Mullins-softening. The modified network alteration theory of Marckmann is thoroughly discussed. It is shown that, this theory gives inconsistent results when different hyperelastic models are used to obtain evolutions of the network parameters during deformation. A new general 2-step method is proposed for more accurate determination of the network parameters. Using this method, it is shown that the number of active monomers in material molecular network is not necessarily a decreasing function of the applied deformation but it can even increase. To find out the physical mechanism responsible for this phenomenon and also to see how it does not contravene the principle of mass conservation, the nature of broken links is thoroughly investigated with focus on the role of filler particles. It is concluded that, for filled elastomers, the weak physical filler–chain interactions are the main broken links during deformation; hence, their morphology is precisely noticed to propose a physically-motivated network alteration theory for the Mullins effect by determining the evolution of the network parameters. Inspired by the Langmuir's theory, the evolution laws are formulated and the developed relations are applied to experimental data and good correlations are obtained. It is concluded that, the developed relations well estimate the alterations of the material network and also provide a useful tool for modeling the Mullins-softening.

Network alteration theory on Mullins effect in semicrystalline polymers

Three semicrystalline polymers with different molecular structure and crystallinity were investigated to analyze the Mullins effect therein. The polymers exhibited time-dependent stress resistance and stress softening. Recoverable network alteration in both crystalline and amorphous domains was proposed to explain the cyclic loading deformation and relaxation. The crystallites and the entanglements acted as the joints in the network where stress was transferred. The maximum stress first rapidly decreased and the balance stress was reached after ca 50 cycles. The balance stress was higher than the quasi-static stress obtained by normal stress relaxation. However, the balance stress could be eliminated by the following stress relaxation and the residual stress was very close to the quasi-static stress. The different network strength between the strain before and after yielding is also discussed by comparing the balance stress and the quasi-static stress. The stress resistance of the network before yielding was stronger than that after yielding mainly due to crystallite slip. © 2014 Society of Chemical Industry

A visco-hyperelastic damage model for cyclic stress-softening, hysteresis and permanent set in rubber using the network alteration theory

The large deformation time-dependent mechanical response of rubber-like materials under cyclic loading is characterized by stress-softening, hysteresis and permanent set. To describe this set of first-order phenomena a constitutive model integrating the physics of polymer chains and their alteration under cyclic loading is proposed. The time-dependency is considered using a Zener-type framework in which the chain extensibility limit is described with both physical and phenomenological approaches. The efficiency of the proposed constitutive model is illustrated by comparisons with experimental data obtained on a styrene–butadiene rubber submitted to cyclic tension loading up to failure both in constant and variable amplitudes.

A theory for large deformation and damage of interpenetrating polymer networks

Elastomers and gels can be formed by interpenetrating two polymer networks on a molecular scale. This paper develops a theory to characterize the large deformation and damage of interpenetrating polymer networks. The theory integrates an interpenetrating network model with the network alteration theory. The interpenetration of one network stretches polymer chains in the other network and reduces its chain density, significantly affecting the initial modulus, stiffening and damage properties of the resultant elastomers and gels. Double-network hydrogels, a special type of interpenetrating polymer network, have demonstrated intriguing mechanical properties including high fracture toughness, Mullins effects, and necking instability. These properties have been qualitatively attributed to the damage of polymer networks. Using the theory, we quantitatively illustrate how the interplay between polymer-chain stiffening and damage-induced softening can cause the Mullins effect and necking instability. The theory is further implemented into a finite-element model to simulate the initiation and propagation of necking instability in double-network hydrogels. The theoretical and numerical results are compared with experimental data from multiple cyclic compressive and tensile tests.

Modeling the low-cycle fatigue behavior of visco-hyperelastic elastomeric materials using a new network alteration theory: Application to styrene-butadiene rubber

Although several theories were more or less recently proposed to describe the Mullins effect, i.e. the stress-softening after the first load, the nonlinear equilibrium and non-equilibrium material response as well as the continuous stress-softening during fatigue loading need to be included in the analysis to propose a reliable design of rubber structures. This contribution presents for the first time a network alteration theory, based on physical interpretations of the stress-softening phenomenon, to capture the time-dependent mechanical response of elastomeric materials under fatigue loading, and this until failure. A successful physically based visco-hyperelastic model is revisited by introducing an evolution law for the physical material parameters affected by the network alteration. The general form of the model can be basically represented by two parallel networks: a nonlinear equilibrium response and a time-dependent deviation from equilibrium, in which the network parameters become functions of the damage rate (defined as the ratio of the applied cycle over the applied cycle to failure). The mechanical behavior of styrene-butadiene rubber was experimentally investigated, and the main features of the constitutive response under fatigue loading are highlighted. The experimental results demonstrate that the evolution of the normalized maximum stress only depends on the damage rate endured by the material during the fatigue loading history. The average chain length and the average chain density are then taken as functions of the damage rate in the proposed network alteration theory. The new model is found to adequately capture the important features of the observed stress–strain curves under loading–unloading for a large spectrum of strain and damage levels. The model capabilities to predict variable amplitude tests are critically discussed by comparisons with experiments.