Mixed-mode flaws modelling at nanoscale in cell membrane system by time-domain hypersingular integral equation method

Cell membrane system has the protein-lipid-protein sandwich type structure. The membrane nanocomposites (PLP-MNs) typically exhibit pronounced nonlinear viscoplastic response under electro-thermo-elastic coupled incremental loads conditions. These extended coupled incremental loads can cause high extended incremental stresses and may induce flaws initiation, growth and finally lead to fracture or failure of nanocomposites.  This work presents a time-domain hypersingular integral equation (TD-HYE) method for modelling three-dimensional mixed-mode flaws growth in PLP-MNs under extended coupled incremental loads rate through intricate theoretical analysis and numerical simulations.  First, the flaws growth problems are reduced to solving a set of TD-HYEs coupled with boundary integral equations, in which the unknown functions are the extended incremental dislocation discontinuities gradient.  Then, the behaviour of the extended incremental dislocation discontinuities gradient around the flaw front terminating at the interface is analyzed by the time-domain main-part analysis method of TD-HYE. Analytical solutions of the extended singular incremental stresses gradient and extended incremental integral near the flaw front in PLP-MNs are provided.  Also, a numerical method of the TD-HYE for a three-dimensional mixed-mode flaw subjected to extended incremental loads rate is put forward with the extended incremental dislocation discontinuities gradient approximated by the product of basic density functions and polynomials. In addition, typical simulations are presented and relationship between the PLP-MN structures and the superior strength are obtained. The conclusion makes it possible to obtain optimum strength and maximum tolerance of flaws through changing the PLP-MN structure.