Modeling and Analysis of the Buckling Phenomena in the Homogeneous and Heterogeneous Biomembranes

Abstract

In this project, nonlinear behavior of biomembrane are modeled as heterogeneous elastic biological systems. In addition to the static behavior of the membranes, their dynamic behavior are modeled to be able to investigate time-dependency of the variables of the systems. Some of the available models are used and some new ones are developed to study static and dynamic analysis of monolayer and bilayer membranes as well as circular axisymmetric biomembranes. The presented models are developed based on the Euler-Bernoulli constitutive law and employed to investigate buckling phenomena in the membranes as one of the most important physical phenomena in biological environment. Static and dynamic behavior of Buckling phenomenon in biological membranes are modeled. The static model results in nonlinear ordinary di erential equation for one-dimensional approximation. In order to extend the model for circular membranes, the criteria of constant length in one-dimensional membranes is changed to constant surface. Moreover, tension-compression and bending springs are added to the model and employed to study buckling of biomembranes. Similar to the procedure of obtaining the equations of static large deformation of the membrane, the equations of motion of the membrane is obtained using free body diagram of an in finitesimal element of the membrane and employing Euler- Bernoulli constitutive law. Hence, nonlinear integro partial di erential equations are obtained t model the dynamic behavior of the membrane. All of the equations, including static and dynamic ones, are changed to the dimensionless forms so that the results can be considered general and can be employed to analyze diff erent systems with diff erent properties. The nondimensional equations of each part of the project are solved using di erent iterative and time-dependent schemes. The schemes are used to obtain the discretized forms of the equations. The discretized equations of all nodes of the domain, with due attention to the considered boundary conditions, are gathered in a matrix and the matrix solved to obtain the solution of the variables at each node and time stage. The solutions obtained for diff erent problems investigated in this project are employed to illustrate variations of diff erent dependent variables of the models with respect to the independent variables and parameters of the problems. As the important step to analyze the problems, diff erent results of the problems investigated in the project are verifi ed using the available information in literature. Membrane pro le are obtained for di erent parameter values and external forces in the stationary condition. In addition, variation of maximum deflection and slope are studied with respect to the variation of diff erent dimensionless parameters of the system. As a verifi cation of the solution, the incompressibility of bilayer membrane is shown as well. Growth of di fferent variables is shown with respect to time employing the solution of dynamic modeling of the membrane. As one of the important parts of this project, e ects of heterogeneity on dynamic behavior of the membrane under buckling is investigated. The heterogeneous region is considered to have di fferent material properties and it position is changed to also study the geometrical e ffects.