Surface strain tensor

A model that pictures the interface as a thin layer with high viscosity (the Bous-sinesq model [2]) has gained widespread acceptance. For such a layer, one has to write down a special phenomenological correlation that connects the surface stress tensor with the rate of surface strain tensor [25). Paper [26] uses this model to examine the problem of the influence of surfactants on the dynamics of a free interface. 

The second contribution to the surface energy is new because it is based on an extension of the atomic distances in the surface area by a force called surface stress. The symbol used for the surface stress in this book is T as in the review of Linford. The surface stress is a tensor with the tensor components T, T, T, and T. The tensor is only independent in its direction for an isotropic sohd. The surface stress causes an elastic deformation of the surface described by the surface strain tensor e. ... 

Elastic surface strain tensor Plastic surface strain tensor... 

Total surface strain tensor Phase shift Activity coefficient Electrosorption valency Mean activity coefficient Activity coefficient of an ion... 

The local change in area from the undeformed configuration to the strained configuration represented by efj is the trace of the surface strain tensor e j,. It follows that the surface energy per unit area in the deformed configuration is C s(l + e j.) = Ug. In terms of this measure of surface energy density, (1.6) becomes... 

In what follows the Kirchhoff-Love model of the shell is used. We identify the mid-surface with the domain in R . However, the curvatures of the shell are assumed to be small but nonzero. For such a configuration, following (Vol mir, 1972), we introduce the components of the strain tensor for the mid-surface,... 

Surface stress — The surface area A of a solid electrode can be varied in two ways In a plastic deformation, such as cleavage, the number of surface atoms is changed, while in an elastic deformation, such as stretching, the number of surface atoms is constant. Therefore, the differential dUs of the internal surface energy, at constant entropy and composition, is given by dUs = ydAp + A m g m denm, where y is the interfacial tension, dAp is the change in area due to a plastic deformation, gnm is the surface stress, and enm the surface strain caused by an elastic deformation. Surface stress and strain are tensors, and the indices denote the directions of space. From this follows the generalized Lippmann equation for a solid electrode ... 

At least for weak strain and weak curvature, the influences of these strain and curvature phenomena on the flame structure can be characterized in terms of a single quantity, an effective curvature of the flame with respect to the flow or the total stretch of the flame surface produced by the flow with respect to the moving, curved flame. By evaluating b from the rate-of-strain tensor in the products just behind the flame, the last of these quantities may be expressed nondimensionally as... 

Here, etj is the ij component of the rate-of-strain tensor. Because the leading-order approximation of the shape is a sphere, these conditions with /0 = 0 are just the exact interface boundary conditions applied at the surface of a spherical drop. The only possible confusion with these conditions is that all terms appear to be 0(1) except for the Ca l term in (7-210). It should be noted, however, that this is just the dimensionless form of the capillary pressure jump for a spherical drop, i.e.,... 

Kinematics of Mixing Spencer and Wiley [1957] have found that the deformation of an interface, subject to large unidirectional shear, is proportional to the imposed shear, and that the proportionality factor depends on the orientation of the surface prior to deformation. Erwin [1978] developed an expression, which described the stretch of area under deformation. The stretch ratio (i.e., deformed area to initial area) is a function of the principal values of the strain tensor and the orientation of the fluid. Deformation of a plane in a fluid is a transient phenomenon. So, the Eulerian frame of deformation that is traditionally used in fluid mechanical analysis is not suitable for the general analysis of deformation of a plane, and a local Lagrangian frame is more convenient [Chella, 1994]. 

Finally, we introduce the coneept of intrinsic surface stress [52Her, 1876] which is equal to the work required to deform a surface. It is direetly related to the surfaee energy and its derivative with respect to strain. For crystalhne surfaces, strain is a tensor, and thus the surface stress, Xy, is also a tensor of second... 

Equation (4.11) incorporates the bulk (Fv) and surface (Fs) parts of Helmholtz free energy F depending on the order parameter t and strain tensor components Uij. 

G. Generalization to Nondiagonal Surface Stress and Strain Tensors... 

Network stress is then related to the nite strain tensor with an elastic material law, and normal stress is set to zero at the gel surface. This closes the mathematical problem for the four independent variables, ip, Vj and T. 

The strain at a point of the surface S is understood to be consistent with the limiting value of bulk strain Cjj as the observation point approaches the surface from within R. The surface strain expressed as a three-dimensional tensor field over the surface with outward unit normal n, is... 

The forces applied to the material per unit surface in the plane z = h and in the plane z = 0 are given by [7.4]. We consider a case where stresses are kept constant in time, and we analyze the situation where the material s response is also independent from time. In the case of a solid, it wiU take a short while for the strain tensor to stabihze. In the case of a fluid, it is the strain rate tensor that will become independent with time after a short period. 

P. Curie and J. Curie discovered the piezoelectric effect in 1880. It was found that, when a compressive or a tensile force was applied on some crystals along some special directions (for example, a quartz) electrical charges could be created on the corresponding surfaces of the crystal and the size of the created charge was proportional to the strength of the apphed force. This phenomenon is called the piezoelectric effect . All ferroelectric crystals show a piezoelectric effect. The piezoelectric effect can be described by piezoelectric equations. On the basis of thermodynamic principles, piezoelectric equations can be derived (e.g., see Xu, 1991). These equations describe linear relationships between the four variables stress tensor [T], strain tensor [S], electrical field vector E and electric displacement vector D. The piezoelectric equations can be expressed as four kinds of equations, depended on the variables. Selecting E and Tas variables, we have ... 

---