A Hamiltonian graph model for the cooperative toughening of crystalline phases and covalent adaptable networks in semi-crystalline thermoset epoxy

Jing Zhang, Haibao Lu*, Ahmed Elmarakbi, Yongqing Fu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


The existence of bond exchange reactions and covalent adaptable networks (CANs) in thermoset epoxy has facilitated its self-healing and reversible mechanical capabilities. However, the toughening mechanisms and cooperative coupling of these crystal phases and CANs in a semi-crystalline thermoset epoxy have not been well understood. In this study, a Hamiltonian graph model is formulated to examine toughening mechanisms in the semi-crystalline thermoset epoxy based on the vertices and paths, both of which are employed to describe the crystalline phases and CANs, respectively. A free-energy equation is also developed based on the tail and tie free energy functions to investigate the cooperative coupling of crystal phases and CANs. The crystal phases increase the cross-linking density of the CANs, which helps the crystal phases with a homogeneous dispersion. Moreover, an extended Maxwell model is developed along with the Hamiltonian graph to explore the coupling effect of crystal phase and CAN on the mechanical behaviors of semi-crystalline thermoset epoxy. A constitutive stress–strain relationship is then proposed to describe the self-healing and toughening behaviors of semi-crystalline thermoset epoxy. The stress–strain relationship of semi-crystalline polymers, which incorporates crystal phases and CANs, has been thoroughly investigated using the analytical results obtained from the proposed Hamiltonian graph model. Finally, the effectiveness of the proposed model is verified using the finite element analysis method and a set of experimental data.
Original languageEnglish
Article number065023
Number of pages11
JournalSmart Materials and Structures
Issue number6
Publication statusPublished - 23 May 2023

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