Taking the advantages of tracking discontinuous material surfaces explicitly within the continuous mechanics framework, the phase field method has been successfully applied to study crack propagation and damage in brittle materials. Considering that the complex micro-structural inclusion-matrix interaction dominates the damage nucleation for heterogeneous PBX, we develop a phase field inclusion model based on the Eshelby's inclusion theory, and carry out the numerical simulation study of damage initiation and evolution. The phase field energy consists of the elastic energy and the inclusion-matrix interaction energy. Combining with the Mori-Tanaka method, the effective elastic moduli of heterogeneous PBXs with different volume fractions are derived. For the proposed phase field inclusion model, the nonlinear debonding effects and damage distribution can be characterized by the phase field order parameter directly. It owns explicity in physical mechanism, and completeness in mathematical theory. We apply this model to compute the high volume fraction heterogeneous PBXs with typical circular and polygonal inclusions, and investigate the influences of loading, inclusion shape, volume fraction and computational parameters on the debonding mechanism. Numerical results indicate that the inclusion-matrix micro-structural evolution promotes interface debonding and the formation of macroscopic material failure, which coincides with experimental observations.

This work extends the idea of the traditional complex-amplitude expanded phase field crystal (APFC) model using the Ginzburg-Landau approach. A fast structural APFC model is proposed as a quick and effective method for describing different crystal structures. Taking square and rectangular phases as examples, we systematically determine the structure-dependent parameters in the fast structure APFC model and validates its effectiveness through numerical simulations. In particular, when dealing with rectangular phases, it is found that this method not only solves the stability problem of the rectangular phase but also describes the structural phase transition between rectangular and orthorhombic layered phases, demonstrating the capability of the model in describing multiple structural phase transitions. Finally, through simulating the classic rotation-shrinking of a circular grain, we confirm the ability of the model for correctly predicting physical laws and reveal the roles of different crystal symmetries on the rotation-shrinking behavior of the grain. The proposed method in this paper can effectively promote the application of APFC models in the simulation research of more and larger material systems.

Memristor plays an important role in modeling nonlinear circuits and systems. Based on the proposed smooth quadratic generic memristor, this paper proposes a memristor-based Lorenz chaotic system. Different from the chaotic system on account of memristor feedback, this system takes a variable of the original Lorenz chaotic system as an inner state variable of memristor, so as to ensure that the system dimension does not increase. Stability analysis shows that the system has the same equilibrium point and stability as the original Lorenz system, namely, one unstable saddle point and two unstable saddle foci. By means of bifurcation diagram, Lyapunov exponent spectra, and phase plot, the dynamics of the proposed memristive system are revealed. The simulated results show that the memristive Lorenz chaotic system possesses coexisting bistable mode and self-similar bifurcation structures. What is more interesting is that the amplitude of the system can be regulated by changing the inner parameters of the generic memristor. Finally, the equivalent circuits of memristor and memristive system are designed and also synthesized by analog components. Simulation results confirm the correctness of the numerical simulations.