Elasticity solution surface

During the processing of composite materials in a hot press or an autoclave, the laminate is usually kept flat until cure is complete. If the platen surfaces are assumed frictionless, the effect of the constraints is to require that the curvatures K] and k2 be zero throughout cure. To develop the elastic solution under these constrained conditions, the laminated plate theory may be used with conditions of N = 0 and jc, = 0. The resulting midplane strains are given by... 

The oscillating bubble method proves to be very convenient and precise for the evaluation of the non-equilibrium elasticity of surfaces in a wide range of frequencies of external disturbances and surface coverage (adsorption of surfactant) [103-105]. It is based on registration of the sinusoidal variation of bubble volume. The bubble is situated in a capillary containing surfactant solution in which oscillations of different frequencies and amplitudes are created. The treatment of the U = f(ft)) curves (where U is the tension needed to initiate oscillations of constant amplitude) allows the determination of Marangoni elasticities [105]. 

Solution parameters such as viscosity of solution, polymer concentration, molecular weight of polymer, electrical conductivity, elasticity and surface tension, those are which attribute to polymer and its solution charaeteristies has important effect on morphology [18]. 

An elastic network can be formed both in bulk, induced by a heating cycle causing denaturation of the protein, and at an interface, induced by adsorption, which likely caused changes in the conformation. The network formed at the air-protein solution interface can be considered as a kind of two dimensional gel. At low pH, a gel is more readily formed in bulk (lower heating temperature required)30,31 as well as at the air-protein solution surface. Likely, both phenomena are related to a lower stability of the molecule against conformational changes caused by external factors. 

In many ceramics, pore shapes can be complex and although elastic solutions are available for ellipsoidal pores, these solutions are complex. For a spherical pore, the maximum tensile stress occurs at the pore surface for = tt/2 and is given by... 

The behavior of PAA solutions in response to agitation and aeration was different from the behavior in CMC and XTN solutions. In PAA solutions, small spherical, as well as inverted tear drop bubbles, were observed. This latter shape is the result of the interaction of elastic and surface tension forces, and has been reported to occur in stagnant [26,27] as well as in mildly stirred solutions [28]. It is known that in free climb motion, PAA bubbles have lower terminal rise velocities than... 

Systematic studies of the foam rheology [932-939] show that the power-law index varies between 0.25 and 0.5 depending on the elasticity of the individual air-solution surfaces. If the elasticity is lower than 10 mN/m, then n is close to 0.25, whereas for large surface elasticity (>100 mN/m) n increases to 0.5. 

This displacement estimate must also be obtained from the elasticity solution for the half space problem it is exactly twice the normal displacement of the surface at the center of the contact area relative to the normal displacement at a remote point on the half space surface in the limit as r/a —> oo. 

The radial and hoop stresses are plotted versus position in Fig. 9.3, where it is easily seen that the boundary condition of a (r = a) = -p0 is met on the inner boundary and the limit case of uniform stresses at long time for incompressible behavior is also apparent. The elastic solution is included, which overlays the viscoelastic response at t=0. While the internal pressure applied is compressive leading to compressive radial stresses at aU positions and all times, the hoop stress at the inner surface is tensile due to the expansion of the cylinder. While the hoop stress remains tensile for all time for an elastic cylinder, this tensile stress relaxes in the viscoelastic cylinder, ultimately becoming compressive. 

It should be noted that the preceding representations are merely mathematical models enabling the analyses to extend beyond the earlier elastic solutions. The actual adhesive bonds do not yield in the classical ductile metal sense. What actually happens is that the adhesive fails under the tensile component of the applied combination of shear and peel stresses, as explained by Gosse [8]. The failure mode for the peel-dominated case is a simple single fracture surface parallel to the adherends which, once started, will not arrest. Under dominant in-plane shear loads, however, the failure mode is a series of hackles inclined at... 

Before discussing the results for the crack near the interface one should observe that, because of the crack surface interference near the crack tips implied by (25), the results given by the linear elasticity solution (22)-(28) are physically inadmissible. One may attempt to remove this objectionable feature of the solution within the confines of continuum solid mechanics in various ways. 

One would be in an ad-hoc fashion to assume that, because of the tendency toward interpenetration, near the crack tips the crack surfaces would come in smooth contact and form a cusp, and the resulting contact region would consist of a single uninterrupted zone rather than the sum of a series of discrete zones as implied by the oscillatory nature of the elastic solution (see Comninou [ll], Atkinson [l2]). Another way is to assume that near the crack tip the linear theory is not valid and to use a large deformation nonlinear theory. An asymptotic analysis using such a theory was provided by Knowles and Sternberg [l3] for the plane stress interface crack problem in two bonded dissimilar incompressible Neo-Hookean materials which shows no oscillatory behavior for stresses or... 

In the equivalent circular base approach the impedance function of a foundation is obtained from the elastic solution of a rigid massless circular base resting on the surface of the soil for each degree of freedom independently. The impedance function for each degree of freedom is a frequency dependent complex expression, where its real part represents the elastic stiffness (spring constant) of the soil-foundation system and its imaginary part represents the damping in the soil-foundation system. The impedance function is expressed as ... 

The magnitude and distributions of stress in the knee are different from the hip. In the hip, the spherical contacting surfaces are highly conforming, and the effective (von Mises) stress levels are below yield, and, thus, below the onset of irrecoverable plastic deformation. Consequently, for hip components, UHMWPE can reasonably be considered to behave as an elastic material at the continuum level. Elasticity solutions have been developed to calculate... 

It is well known that many parameters, such as viscosity, elasticity, conductivity, surface tension and distance between tip and collection screen, can influence the transformation of polymer solutions into nanofibers through electrospinning. Solution viscosity is one of the most important factors. Since both polymers have significantly different molecular weights, four different solutions of N6,6 with various concentrations were prepared. We used a mixed solvent consisting of formic acid and chloroform with the ratio of 75/25 (v/v). Table 4.1 shows the solution viscosities for each concentration of low and high molecular weight N6,6. 

Hertz [27] solved the problem of the contact between two elastic elliptical bodies by modeling each body as an infinite half plane which is loaded over a contact area that is small in comparison to the body itself. The requirement of small areas of contact further allowed Hertz to use a parabola to represent the shape of the profile of the ellipses. In essence. Hertz modeled the interaction of elliptical asperities in contact. Fundamental in his solution is the assumption that, when two elliptical objects are compressed against one another, the shape of the deformed mating surface lies between the shape of the two undeformed surfaces but more closely resembles the shape of the surface with the higher elastic modulus. This means the deformed shape after two spheres are pressed against one another is a spherical shape. 

Boussinesq and Cerruti made use of potential theory for the solution of contact problems at the surface of an elastic half space. One of the most important results is the solution to the displacement associated with a concentrated normal point load P applied to the surface of an elastic half space. As presented in Johnson [49]... 

---