Deformation profile

The layers are heated to 280°C or higher for a few seconds up to several minutes. The coated metal coils are then processed, i.e., deformed, profiled, or stamped. The resulting coated products are broad in scope they are employed in a variety of areas from packaging and the vehicle industry to household items and building materials. 

Figure 2.21. Schematic representation of colloid probe-PDMS droplet interaction during the AFM experiment. Solid line depicts the undeformed profile of the PDMS droplet and the rigid colloid probe. Dashed line shows the deformed profile of the PDMS droplet.

For a one-material case, analytic solutions exist for both the deformation profile of the elastic material as well as the pressure and stress distributions for the indenter (approximating wafer features). Consider a single-layer pad that is thick relative to the vertical deformation and has a deformation force applied over a circular region of radius a. The deformation is given by a set of two equations that represent deformations within and outside the circular radius over which the force is applied. The deformation at any radius r less than a is given by [59] ... 

An important observation regarding the equations representing the deformation is that the shape of the deformation (with only vertical scaling factors) is independent of the material constants and only dependent on the area of force application. The deformation shape is therefore similar for different material properties. The material properties and applied force control only the actual amount of the deformation. This is ideal for our initial goal that was to determine the typical deformation profile of the polish pad. All that remains is the determination of the appropriate length scale to use to represent the deformation. 

The ratio between these relative deformations is Ifn and can be used to define the deformation profile or length scale. Due to the presence of a softer back pad, more deformation is expected for the stacked pad but the shape, which is the main concern, will be approximately similar [45,46], The deformation is relatively small compared to the region of apphcation of the force. Using approximate material properties for the ICIOOO pad (Young s modulus of 2.9 x 10 Pa [41] and approximate Poisson ratio of 1/3) with force applied in a circular region of radius 2 mm, and a local pressure of 7 psi, the maximum deflection is about 6 fira. This deformation is referenced to the origin as illustrated in Fig. 13. It is also important to note that the transition shape is very gradual and this sets the polish limit for the down areas. 

Fig. 14. One-dimensional cross section of an elliptic weighting filter. The characteristic length is defined as the section length when the relative weight has dropped to 2/a. The filter shape corresponds to the deformation profile of an elastic material under distributed load in a circle of radius Z./2.

Figure 11.23 RFM rate of deformation profile (1/s units) between the rolls using a power law viscosity model with a power law index, n, of 0.5.

Methods of objective measurement of cereal foam structures are reviewed, including image analysis, confocal microscopy and x-ray tomography. The analysis of foam structures and their relationship with mechanical and rheological properties is described, and also the relationships between these structures and sensory descriptors such as crispness, crunchiness and texture. The size, shape and anisotropy of bubbles and their cell walls in foams are seen as critical in determining their fracture properties and sensory perception of crispness. Techniques for measuring crispness using acoustic emission and force-deformation profiles are discussed. 

The recoverable deformation of the pellets was quantifled from the linear portion of the load/deformation profile obtained when measuring the strength of pellets by Aulton et al. (88). Alternatively, the elastic properties as a storage modulus" can be measured by the application of dynamic mechanical analysis (DMA) (89). The values obtained by this method for a series of pellet formulations were found to be considerably greater than those obtained by application of the former method (81). which must therefore be considered as an estimate of the real value. It did rank the pellet formulations in the same order as the DMA. 

The boundary problem for Eq. (58) was solved [117] using the Shooting Algorithm implemented with Mathematica 4 [121]. Comparison with analytical solutions in the uniform limit demonstrated agreement to within 2% for deformation profiles and to within 5% for the elastic energy. 

Calculations were carried out for different mesh spacings h (0.5, 1, and 2 A). Regardless of the value of h, the deformation profiles u r) were in excellent agreement (to within 2%) with the results described in Section III for both values of 6. However, the total deformation free energy is very sensitive to the choice of h. 

We have solved the uniform 6 = 1) elastic problem numerically for two inclusions at various separations, d. We chose mq = 3.4 A and varied 5. The deformation profiles along the line connecting the centers of two insertions are presented in Fig. 11 for s = 5min- As d increases, the well in u x) becomes deeper. It reaches its minimal value. 

We presented a new approach allowing reconciliation of the requirement of selfadjustment of the membrane deformation profile (relaxed boundary conditions) with (1) the notion that membrane properties must be modified at short distances from the inser-... 

In the present extrusion of HDPE ribbons, the deformation patterns were examined by the deformation of parallel ink marks preimprinted on the surface of a HDPE (Figure 1-d). At EDR >12, the low and high MW HDPE exhibited a typical shear parabola and a W-shape deformation profile, respectively, with both characteristics enhanced at higher EDR as shown in Figure 2. These characteristics of the deformation patterns are in well agreement with our previous observations (3) and further confirm the previous conclusion that there is no significant effect of cutting a billet into two halves and/or coextrusion of a film with the split billet halves on the deformation flow patterns. 

On the other hand, the displacement is accompanied by a rotation of the local surface tangent by a bending angle il> (Fig. 3.7c). Trivially, tan( ) = dw/dx. The calculation of the actual deformation profile w(x) is a common problem in elasticity, known as the bending of a beam without a distributed load. Mathematically, for weak deformation, which is assumed as usual, w(x) has to satisfy the differential equation ... 

Equation (3) in Harden et which gives the deformation profile of a cell moved... 

Figures 3.8a and 3.8b illustrate that the deformation profile w x) and the surface tangent dw/dx, respectively, are both sensitive to the mechanical boundary conditions at the edges.

Fig. 3.8. Dependence of the substrate deformation on the mechanical boundary conditions. (a) The deformation profile w x). (b) The tangent dwjdx of the surface. The solid lines correspond to case A (no bending at the edges) while the dashed lines are for case B (bending is allowed at the edges).

Taking into account the deformation profile calculated above in Eqs (3.8) and (3.9) we can transform Eq. (3.18) into... 

A definition of these angles is given in Fig, 1, The deformation profile is dependent on the dielectric constants C and Ej., the elastic constants for splay, twist and bend Kn, K22> 33 the total twist (90-23q), the tilt angle at the surface of the substrates ao, and the applied voltage. The optical response depends in addition on the refractive indices ne and viq and the ratio of wavelength to cell thickness. For display applications a finite tilt at the surfaces is required to avoid areas of opposite tilt. Therefore the deformation profiles are calculated for various combinations of K33/K11, Ae/ej and using Berreman s program. All calculations are performed for 10 im cells and a pretilt ao=l°. 

In the case where the membrane is deformed, the deformation profiles can be compared to a variety of theories [16,17,27, 33, 245-247]. Both in coarse-grained [30,234] and atomistic [248] simulations, it was reported that membrane thickness profiles as a function of the distance to the protein are not strictly monotonic, but exhibit a weakly oscillatory behavior. This feature is not compatible with membrane models that predict an exponential decay [16,17,27], but it is nicely captured by the coupled elastic monolayer models discussed earlier [22, 28, 30]. Coarsegrained simulations of the Lenz model showed that the coupled monolayer models describe the profile data at a quantitative level, with almost no fit parameters except the boundary conditions [30, 244]. 

To determine the angle 0(z), the total free energy F which consists of the elastic deformation, optical field, and electric field energy is expressed in terms of z). The symmetry of the problem and the rigid boundary conditions require that the maximum deformation angle 0 =0 z — d/2)y and (z = 0) = (z = c ) = 0. Variation of the total free energy leads to an Euler equation from which the solution to the deformation profile is obtained as... 

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