Surface Stress Sensitive Film
Surface Stress Sensitive Films is an optical instrument for measurements of skin friction and pressure. The basis of this measurement is an elastic film that deforms under the action of the applied loads. The reaction of the film is monitored by imaging the surface. The film reaction is then modeled using finite element analysis resulting in a continuous distribution of skin friction and pressure over the filmed surface.
Some insight into the operation of the S3F is gained by considering the simplified response of the film to normal and tangential forces. The surface of the film will undergo a tangential displacement due to an applied tangential load but will not compress or yield. The response of the film may be visualized by considering a series of markers on the surface of the film. The markers will be displaced as the film shears and this displacement is a function of the film properties, the shear modulus and thickness. Upon removal of the load the film will return to its original shape.
Upon application of a normal force, the film will deform but will not compress or yield. The local thickness of the film will be modified by the presence of the load near the point of action. Upon removal of the load, the film will return to its original shape. The stressed film thickness is a function of the applied normal force, the thickness of the film, and the shear modulus. Note that the film responds to gradients in pressure, not changes in static pressure. This can be a significant advantage for several reasons. First, this results in a shear sensor that is insensitive to static pressure. Generally, pressure forces are orders of magnitude larger than skin friction forces and thus the response of the skin friction sensor suffers due to cross-talk between the normal and tangential response. Second, the sensor is a gradient sensor and thus can be tuned for applications that require larger or smaller sensitivity. A more detailed description of the film behavior is included later.
The process of measuring pressure and shear is accomplished in two steps. First, the normal and tangential deformation of the film is optically measured. These deformations are then converted to forces using a physical stress/strain model of the film. In the first generation system, the normal deformation of the film is measured using fluorescence and the tangential deformation is measured using a cross-correlation technique. The experimental setup for this S3F measurement system is shown here.
All three components of deformation can be extracted from a pair images taken by a single hi-resolution CCD camera. The normal component is measured using the fluorescence signal emitted from a dye embedded in the S3F. Two images are acquired, an unloaded or flow-off image and loaded or flow-on image. The ratio of these images is a linear function of film thickness. The tangential displacement measurement is accomplished using the same pair of images. The surface of the film is lightly doped with small particles and the tangential displacement map is obtained by spatially cross-correlating the flow-off and flow-on images.
The film is lightly doped with a fluorescent dye and exposed to an illumination source to excite the dye. An image of the film fluorescence is recorded at an unloaded condition (wind-off) and a second image is recorded at the loaded condition (wind-on). The fluorescence of the film is a linear function of the film thickness so a ratio of the wind-off to wind-on image is an effective measurement of the film thickness. Variation in the film thickness represent the presence of a pressure gradient.
An image of the film is recorded at an unloaded condition (wind-off) and a second image is recorded at the loaded condition (wind-on). The tangential displacement of the film surface is determined using a cross-correlation between the wind-off and wind-on images. By superimposing the tangential displacement (vectors) and the thickness variations (colormap), a visualization of the flow can be produced. Note that this image must be converted to forces using an FEA model.
Finite Element Model
The process of converting film deformations to physical loads accomplished using a simple 2D FEA model of the film. A 1D load applied to the film surface as shown above; in this case the film deformation can be treated in 2D space. Since the S3F is an elastic solid, it is deformed by the applied force, a point in the solid originally at (x,y) is moved to (X,Y) upon application of the load. If the displacement vector is small, Hooke’s law relates the stress tensor inside the solid to the deformation (strain) tensor.
The response of the film can is modeled using the response functions of the film to individual normal and tangential loads. The response of the film to a normal load at the surface includes a normal response function and a tangential response function. Similarly, the response of the film to a tangential load includes both a normal and tangential response. The result is the response matrix g(x). The elastic reaction can be expressed as the convolution of the response matrix and the applied load components. If the response matrix, g(x), can be determined experimentally or by a Finite Element model, the applied loads, L(x), can be determined by de-convolution of Equation.
Finally, the reaction equation is written in a discreet form. An arbitrary set of loads applied at discreet surface locations on the film surface and the reaction of the film is a function of each load times the reaction matrix function. The response of the film is a superposition of the response to the individual forces, and therefore, the response can be written as a system of linear equations. The film reactions (R) are measured experimentally and the response function can be determined experimentally, or by FEA model. This system of linear equations with unknown loads has a diagonally dominant matrix and can be solved by inversion.
Some insight into the behavior of the film can be gained by modeling the response functions using a finite element model. An arbitrary force, normal or tangential, is applied to the film surface and the force is distributed over a specified contact area. The amplitude of the film reaction, normal, tangential, and cross talk, to the force is determined for a range of contact areas. The contact area is then normalized by the film thickness, this ratio of a contact area to film thickness defines a spatial frequency. A force that is acting over an area equal to the film thickness has a spatial frequency of 1. A force acting over a contact area larger than the film thickness has a spatial frequency of less than 1, and a force acting over a contact area smaller than the film thickness has a spatial frequency of greater than 1.
The response functions of the film, normal, tangential, and cross-talk are plotted here at each spatial frequency. Focusing on the spatial frequencies below 0.1, the zone inside the green box, highlights the key result from this analysis. Here, the force is acting on a contact area greater than 5 times the film thickness. In this region, the tangential response to a pure shear has normalized amplitude of 1. Assuming a normal force of equal magnitude is applied over the same contact area, the amplitude of the response is significantly smaller by a factor of ~100. This means that the S3F response to skin friction is substantially magnified compared to the response to pressure. One of the major limitations for measurements of skin friction is that the skin friction forces are substantially smaller than the pressure forces. The response characteristics of the S3F enhance the response of the skin friction forces relative to the pressure forces. This behavior allows the S3F to operate as a skin friction sensor, even in the presence of much stronger pressure forces.
Inspection of amplitude/frequency plot suggests that the response of the S3F to normal forces could be modeled mathematically using the pressure gradient rather than pressure. This model is valid for spatial frequencies below 0.1. The result of this modeling is given here, the normal (Y) and tangential (X) reaction of the film is written in terms of the film thickness (h), shear modulus (m), tangential force (Fx), and normal force (P). Note that the X reaction of the film is a function of the applied tangential force and the pressure gradient. Furthermore, the pressure gradient is weighted by the film thickness. These equations may be used as a design tool for the films. If one is interested in skin friction, a thin film will mitigate the response to pressure gradients relative to shear forces. For example a 100-micron film with a 1-kPa/m pressure gradient and a 1-Pa shear would respond to the shear force with an amplitude that is 20X larger than the amplitude of the response to the pressure force. Conversely, pressure forces would be more significant in the response of a thick film.