| Summary: | 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.
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