C. I. Kim

The Effects of Surface Elasticity on an Elastic Solid With Mode-III Crack: Complete Solution

We examined the effects of surface elasticity in a classical mode-III crack problem arising in the antiplane shear deformations of a linearly elastic solid. The surface mechanics are incorporated using the continuum based surface/interface model of Gurtin and Murdoch. Complex variable methods are used to obtain an exact solution valid everywhere in the domain of interest (including at the crack tip) by reducing the problem to a Cauchy singular integro-differential equation of the first order. Finally, we adapt classical collocation methods to obtain numerical solutions, which demonstrate several interesting phenomena in the case when the solid incorporates a traction-free crack face and is subjected to uniform remote loading. In particular, we note that, in contrast to the classical result from linear elastic fracture mechanics, the stresses at the (sharp) crack tip remain finite.

Effect of surface elasticity on an interface crack in plane deformations

We consider the effect of surface elasticity on an interface crack between two dissimilar linearly elastic isotropic homogeneous materials undergoing plane deformations. The bi-material is subjected to either remote tension (mode-I) or in-plane shear (mode-II) with the faces of the (interface) crack assumed to be traction-free. We incorporate surface mechanics into the model of deformation by employing a version of the continuum-based surface/interface theory of Gurtin & Murdoch. Using complex variable methods, we obtain a semi-analytical solution valid throughout the entire domain of interest (including at the crack tips) by reducing the problem to a system of coupled Cauchy singular integro-differential equations, which is solved numerically using Chebychev polynomials and a collocation method. It is shown that, among other interesting phenomena, our model predicts finite stress at the (sharp) crack tips and the corresponding stress field to be size-dependent. In particular, we note that, in contrast to the results from linear elastic fracture mechanics, when the bi-material is subjected to uniform far-field stresses (either tension or in-plane shear), the incorporation of surface effects effectively eliminates the oscillatory behaviour of the solution so that the resulting stress fields no longer suffer from oscillatory singularities at the crack tips.

Bud formation of lipid membranes in response to the surface diffusion of transmembrane proteins and line tension

We study the formation of membrane budding in model lipid bilayers with the budding assumed to be driven by means of diffusion of trans-membrane proteins over a composite membrane surface. The theoretical model for the lipid membrane incorporates a modified Helfrich-type formulation as a special case. In addition, a spontaneous curvature is introduced into the model in order to accommodate the effect of the non-uniformly distributed proteins in the bending response of the membrane. Furthermore, we discuss the effects of line tension on the budding of the membrane, and the necessary adjustments to the boundary conditions. The resulting shape equation is solved numerically for the parametric representation of the surface, which has one to one correspondence to the membrane surface in consideration. Our numerical results successfully predict the vesicle formation phenomenon on a flat lipid membrane surface, since the present analysis is not restricted to the conventional Monge representation often adopted to the problems of this kind for the obvious computational simplicity, despite its limited capability to describe the deformed configuration of membranes. In addition, we show that line tension at the interface of the protein-concentrated domain makes a significant contribution to the bud formation of membranes.

An Elliptic Inhomogeneity Subjected to a General Class of Nonuniform Remote Loadings in Finite Elasticity

We consider a composite material containing an elastic elliptic inhomogeneity which is assumed to be perfectly bonded to a surrounding matrix of similar elastic material. In fact, both the inhomogeneity and the matrix belong to the same class of compressible hyperelastic materials of harmonic-type but each has its own distinct material properties. We consider finite plane deformations of the inhomogeneity-matrix system and obtain the complete solution when the system is subjected to classes of nonuniform remote (Piola) stress characterized by stress functions described by general polynomials of order n ≥ 1 in the corresponding complex variable z used to describe the matrix.