ABSTRACT
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
II. Interface in Problems of Continuum Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
III. Basic Notion of Geometry and Kinematics of Surface . . . . . . . . . . . . . . . . . . . . . . 237
A. The Geometry of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
B. Kinematics of Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
1. Orthogonal Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
IV. Material Displacement Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
A. Componental Formulation and Consideration . . . . . . . . . . . . . . . . . . . . . . . . . 241
B. Material Displacement Derivative of Basic Surface Quantities . . . . . . . . . . . . 245
1. The List of Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
2. General Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
a. Material Displacement Derivative of Higher Order . . . . . . . . . . . . . . 251
3. Decomposition of a General Tensor Field . . . . . . . . . . . . . . . . . . . . . . . . 251
4. The Decomposition of Displacement (Material) Derivative . . . . . . . . . . . . 254
V. The Theory of Singular Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
A. Singular Surfaces Associated with a Motion . . . . . . . . . . . . . . . . . . . . . . . . . 259
B. Conditions of Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
1. Geometrical Conditions of Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . 261
2. Kinematical Conditions of Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . 261
3. Dynamical Conditions of Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . 263
VI. Balance Laws of Bulk Material and Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
A. Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
1. Transport Theorem for a Volume which Contains
Discontinuity Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
2. Transport Theorem for a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
B. Balance Laws for a Single Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
C. Nonmaterial Interface: Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 271
1. Material Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
2. Impermeable Material Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
D. Balance Laws for a Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
VII. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
The objective of this chapter is to present perspective of physics of interfaces from standpoint of
continuum physics. This approach develops the foundations more carefully than the traditional
approach where there is a tendency to hurry on to the applications, and moreover, provides a back-
ground for advanced study in modern nonlinear continuum physics. Our ultimate intension was to
present the subject in a sound manner as clear as possible. We hope that the text provides enough
insights for understanding of terminology used in scientific state-of-the-art papers and to find the
“right and straightforward path” in the scientific world of material surface phenomena. In what
follows a few words of general meaning are worthwhile.
