Elsevier

Mechanics of Materials

Volume 161, October 2021, 103992
Mechanics of Materials

Research paper
Modeling the mechanical behaviors of multiple network elastomers

https://doi.org/10.1016/j.mechmat.2021.103992Get rights and content

Highlights

  • A constitutive model is developed for multiple network elastomers.

  • The contribution of bond deformation is incorporated into the single chain model.

  • The model can capture the stress response of multiple network elastomers.

  • The model can describe the Mullins effect in multiple network elastomers.

Abstract

Compared with traditional single network elastomers, multiple network elastomers (MNEs) can achieve high stiffness and toughness. MNEs can be assumed as elastomer composites, where the first network elastomer serves as fillers and the other networks are assumed as matrix. Thus, similar to nanoparticle filled rubbers, MNEs also exhibit the Mullins effect. In addition, the mechanical response of MNEs shows significant dependence on the prestretching ratio of the filler network. In this work, a constitutive model is developed to describe the complex mechanical behaviors of MNEs. For a single chain, the Langevin statistics has been modified to incorporate the contribution of bond deformation. The polydisperse network of the filler network is progressively damaged with deformation, which is the underlying mechanism for the Mullins effect. Through incorporation the effect of prestretch caused by the matrix, the model also fully captures the stress–strain response of MNEs with varied stretching ratios of the filler network, which is beyond the ability of the classic damage models based on the Langevin statistics.

Introduction

Soft materials have been widely applied in the areas of biomedical devices, stretchable electronics and soft robotics (Li et al., 2021, Kim et al., 2020, Xu and Zhao, 2020, Xiao and Huang, 2020, Liu et al., 2021). It is highly desirable to design tough soft materials, which can be achieved through several different strategies. It has been long recognized that mechanical properties of natural rubbers can be significantly improved through adding carbon black nanoparticles, resulting in the wide application of filled rubbers in tires (Tee et al., 2018). Another widely used approach is developed by Gong et al. (2003) to create double network hydrogels. Compared with each single network, the double network hydrogels can achieve a pronounced improvement in stretchability and toughness. Ducrot et al. (2014) has extended this method to synthesize multiple network elastomers (MNEs). The Young’s modulus of the single network of acrylate-based elastomers is less than 1 MPa while the strength is only around 0.5 MPa. These values can be increased to 5 MPa for the modulus and more than 10 MPa for the strength for MNEs (Ducrot et al., 2014, Millereau et al., 2018). In addition, it is found that the modulus and stiffness can be tuned through controlling the prestretching ratio of the filler network (Millereau et al., 2018).

Regardless the filled rubbers, double network hydrogels and MNEs, these tough soft materials all exhibit the stress softening effect, also named as the Mullins effect (Morovati et al., 2020, Xiao et al., 2021a, Xiang et al., 2020). The typical behavior is the stress level in the reloading process is below that of the virgin specimens until the deformation surpasses the previous maximum deformation (Zhang et al., 2021). Modeling the Mullins effect has been an active topic for several decades (Govindjee and Simo, 1991, Diani et al., 2009, Dargazany and Itskov, 2013, Pan and Zhong, 2017, Qi et al., 2018, Wang and Chester, 2018, Diani and Le Tallec, 2019, Zhong et al., 2019, Liao et al., 2020). Though most of these models are developed for filled rubbers, the models can be extended for MNEs. Bacca et al. (2017) developed a damage model for the Mullins effect of MNEs. The model assumes that the filler network has a uniform length distribution and the stretch limit increases with deformation. Lavoie et al. (2019) assumed that the filler is polydisperse and can be damaged progressively with deformation. Vernerey et al. (2018) and Buche and Silberstein (2021) have developed continuum-level models for MNEs based on a statistical approach, which have the ability to predict the stress–strain curves of MNEs in the cyclic loading conditions. Zhong et al. (2020) formulated a theory to analyze the effects of the preparation condition on the swelling behaviors of MNEs. However, these models are either complex or not fully calibrated with a complete set of the experimental results of MNEs. Thus, this work aims to extend the model based on progressive rupture of chains to fully describe the mechanical responses for various MNEs.

The paper is arranged as follows. Section 2 provides the detailed information of the theoretical model of MNEs. The constitutive relationship of the single chain is first provided, which is further extended for the filler network through incorporation a distribution of chain lengths. The matrix is assumed to exhibit a hyperelastic response without damage. The followed section shows the procedures to determine the model parameters. Section 4 applies the model to simulate the experimental data of MNEs in the literature. The last section provides a concluding remark.

Section snippets

Theory

In this section, a micromechanical model is developed for MNEs. The model for a single chain is first presented, followed by defining the constitutive relation of each network. Finally, the constitutive theory for MNEs is proposed.

Parameter determinations

The experimental data in Millereau (2017) is adopted to calibrate the model. The prestretching ratio of each network is directly related with the synthesized procedures. Two methods were adopted by Millereau (2017) to evaluate the prestretching ratio based on measuring the change in weight and dimension respectively, which yield a close value except for the quadruple network elastomers. Thus, for the quadruple network elastomers, only the prestretching ratio of the filler network λs1=3.42 is

Results and discussion

In order to obtain accurate numerical results, the evaluation of the inverse Langevin function is critical. This information has been provided in Appendix. The model is then applied to simulate the experimental data of MNEs in Millereau (2017) and Millereau et al. (2018). The monomer of filler network is ethyl acrylate, and the cross-linker agent is butanediol bis(acrylate). Ethyl acetate was used as solvent. The as-synthesized sample was then dried before immersion in the matrix solvent, which

Conclusions

In this work, a constitutive model for MNEs is developed. The filler network is assumed to have a broad distribution of chain lengths. The free energy density of each single chain has contribution from chain stretch and bond deformation. The bond deformation can be further obtained through a minimization of the free energy density. The singularity of using the Langevin statistics is then avoided. It also allows a large fraction of chains to sustain the force. When a critical stretch ratio is

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 12022204, 91748209), the Fundamental Research Funds for the Central Universities, China (Grant No. 2021FZZX001-16) and the 111 Project, China (No. B21034).

References (46)

  • QiY. et al.

    Fracture toughness of soft materials with rate-independent hysteresis

    J. Mech. Phys. Solids

    (2018)
  • TalaminiB. et al.

    Progressive damage and rupture in polymers

    J. Mech. Phys. Solids

    (2018)
  • TeeY.L. et al.

    Recent advances on fatigue of rubber after the literature survey by mars and fatemi in 2002 and 2004

    Int. J. Fatigue

    (2018)
  • WangS. et al.

    Experimental characterization and continuum modeling of inelasticity in filled rubber-like materials

    Int. J. Solids Struct.

    (2018)
  • XiangY. et al.

    A general constitutive model of soft elastomers

    J. Mech. Phys. Solids

    (2018)
  • XiaoR. et al.

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

    Int. J. Plast.

    (2021)
  • XuF. et al.

    Thermal wrinkling of liquid crystal polymer shell/core spheres

    Extreme Mech. Lett.

    (2020)
  • ZhangC. et al.

    Hysteresis in glass microsphere filled elastomers under cyclic loading

    Polym. Test.

    (2021)
  • ZhongD. et al.

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

    Extreme Mech. Lett.

    (2020)
  • ZhongD. et al.

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

    Int. J. Solids Struct.

    (2019)
  • BaccaM. et al.

    A model for the Mullins effect in multinetwork elastomers

    J. Appl. Mech.

    (2017)
  • BucheM.R. et al.

    Chain breaking in the statistical mechanical constitutive theory of polymer networks

    (2021)
  • ChuH. et al.

    A network evolution model for recovery of the Mullins effect in filled rubbers

    Int. J. Appl. Mech.

    (2020)
  • Cited by (21)

    • A general continuum damage model for soft composites

      2023, Journal of the Mechanics and Physics of Solids
    • Constrained shape-memory behaviors of multiple network elastomers

      2023, Materials Letters
      Citation Excerpt :

      As shown in Millereau et al. [12], the mechanical behaviors of MNEs strongly depend on the pre-stretch ratio of the first network, and the nominal stress can reach as large as 10 MPa with a maximum stretch ratio of around 2.5. So far, most of the works on MNEs focus on characterizing their mechanical responses in the rubbery state [9,13]. However, with changing the temperature, the glass transition can also occur in MNEs [9].

    View all citing articles on Scopus
    View full text