Elastic half space

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]... 

Similarly, the assumption that the contact area is small enough that the particle can be represented by an elastic half space allows the radii of the two contacting particles to be combined into a single effective radius that represents how the contacting shapes interact. 

There are two major sources of the deformation in contact-mode SFM the elasticity of the cantilever and the adhesion between the tip and sample surface. For purely elastic deformation, a variety of models have been developed to calculate the contact area and sample indentation. The lower limit for the contact diameter and sample indentation can be determined based on the Hertz model without taking into account the surface interactions [79]. For two bodies, i.e. a spherical tip and an elastic half-space, pressed together by an external force F the contact radius a and the indentation depth 8 are given by the following equations ... 

There is a minimum length scale for loading of the subpad. Consider a point asperity contacting the surface of the abrasive layer. As the related stress transfers through the abrasive layer, it spreads out, so that the subpad sees the point load on the surface of the abrasive as a pressure distribution. Consider two point loads on an elastic half space as shown in Figure 2. 

The crack, semi-infinite in length, is assumed to propagate along the interface of two linearly elastic half spaces with a steady state velocity of a under small-scale yielding conditions, which implies that the region of pullout is small compared with typical specimen dimension. The interfaces are reinforced by chains which obey the pullout laws stated above. The steady state condition implies that all quantities are independent of time with respect to an observer moving with the crack tip. 

Van Landigham et al. reviewed nanoindentation of polymers, [40, 41] including a summary of the most common analyses of load-indentation data. Chief among these methods is an analysis of indentation load-penetration curves according to the Oliver-Pharr method. [42] This method is based on relationships developed by Sneddon for the penetration of a flat elastic half space by different probes with particular axisymmetric shapes (e.g., aflat-ended cylindrical punch, a paraboloid of revolution, or a cone) [43], More recently, Withers and Aston discussed indentation in the context of plasticity and viscoelasticity [44]. 

For this aim, we use a numerical model which is on the one hand to some degree physical, and on the other hand simple enough that it allows to perform long simulations. The basic version of the model consists of a segmented two-dimensional strike-slip fault in a three-dimensional elastic half space and is inherently discrete, because it does not arise from discretizing a continuous model. 

Kalpna R.C. 2000. Green s function based stress diffusion solutions in the porous elastic half space for time varying finite reservoir loads. Phys, Earth Planet. Int. 120 pp93-101. 

The effective value of must scale with the contact radius a, for an elastic half space, since this is the only length scale in the problem. An approximate equivalence between and a gives the correct scaling as in Eq. (4). This stiffness can be compared, with the contact stiffness of the elastomer itself [see Eq. 

FIGURE 39.3 Schematic illustration of three types of film response (a)elastic half-space behavior, dominated by bulk compression of the film, (b) plate behavior, dominated by bending deformation of the film, and (c) membrane behavior, dominated by axial stretching of the film. 

This section considers plate and membrane behavior, as opposed to scenarios where the film is thick enough to be considered an elastic half-space (as in Figure 39.2a). Thus, the applications in Section 39.4.2 thus correspond directly to the theory described in Section 39.3.2. The analysis of two relatively thick films (compared to crack length) requires numerical solutions not discussed in Section 39.3.2 and hence is discussed in Section 39.4.3.1 in conjunction with pressure-deflection relationships. 

One can naturally derive similar design guidelines for chamber dimensions comparable to the film thickness (i.e., the elastic half-space regime) or valve displacements greater than the film thickness (i.e., the membrane regime). 

There are several major techniques that are used to extract the mechanical properties of ceUs. The models and experiments are interconnected the experiments provide parameters for the models and, in turn, the models are the basis for the interpretation of the experiments. One common technique is micropipette aspiration, where a pipette is sealed on the surface of a cell, negative pressure is appUed inside the pipette, and a portion of the ceU is aspirated into the pipette. The height of the aspirated portion is considered as an inverse measure of the ceU stiffness. The same technique is used to observe the time response of the ceU to the appUcation of pressure, and in this case, the corresponding relaxation time is a measure of the cell s viscoelastic properties. The experiment with the micropipette aspiration of a red blood ceU was interpreted by considering the ceU membrane (including the cytoskeleton) as a nonlinear elastic half-space... 

Once a specific event in the load-displacement curve has been associated with a particular phase transition, the pressure at which it occurs can be estimated by considering the elastoplastic behavior of the material under the indenter. For the point-force contact, the Sneddon s solution [39] to the problem of the penetration of an axisymmetric punch into an elastic half space predicts the following relation between the applied load P and the indenter displacement A ... 

Fig. 56. Plain and side views of contours of greatest principal stress in elastic half-space in contact with sliding sphere (marked with arrows) for the friction coefficients (a) / = 0.1 and (b) / = 0.5. Motion from left to right, unit of stress pQ. Dashed lines denote the trajectories of the lesser principal stresses, starting from place of maximum tensile stress in the specimen. After Reference [261].

At the feasibility stage, foimdation movements are often estimated with available elastic solutions. The vertical (Pv), horizontal (Ph), and rotational (a) movements of a rigid circular foundation with radius (R) and supported on elastic half-space can be computed from the following equations (Yoimg et al., 1975) ... 

Carrier, W. D. Ill, and Christian, J. T. (1973), Rigid circular plate resting on a non-homogeneous elastic half-space. Geotechnique, 23, No. 1, pp. 67-84. 

The elastic problem for an elastic half-space contacted by a normal point force P was solved by Boussinesq in 1885. The stress field is axisymmetric around the force direction and has the general form, in spherical coordinates. 

Lee, D., et al. Surface instability of an elastic half space with material properties varying with depth. J. Mech. Phys. Solids 56(3), 858-868 (2008)... 

The reason for this strengthening of the joint is the cracking mechanism. Although there does not seem to be a crack at the edge of the wire, there is a virtual crack because the rigid material can be replaced by an elastic half-space as shown in Fig. 13.4(b). The stress in the elastic material rises to infinity at the edge according to the Boussinesq analysis, because of the (1 — pressure... 

Biot [25] showed that the surface of an elastic half-space will become unstable at critical values of strain ratios X3 set up in two perpendicular directions in the surface. The critical condition is... 

Adherence of spheres and flat punches was studied by Johnson et al. (3), Kendall (5), and Maugis and Barquins (2,6). For an axisymmetric rigid punch contacting an elastic half-space, under a load P, over an area of radius a, it can be shown (7,4) that... 

M. Barquins and D. Maugis, "Adhesive Contact of Axisymmetric Punches on an Elastic Half-Space The Modified Hertz-Huber s Stress Tensor For Contacting Spheres," J. Mech. Theor. Appl., 1, 331 (1982). 

D. Maugis and M. Barquins, "Adhesive Contact of a Conical Punch on an Elastic Half-Space," J. Physique, lettres, 42, 295 (1981). 

Glaser SD, Weiss GG, Johnson LR (1998) Body waves recorded inside an elastic half-space by an embedded, wideband velocity sensor. L. Ac. Soc. Am. 104 (3),Pt. 1 1404-1412... 

Consider a solid with an edge dislocation in the middle as shown in Figure 8.1 the solid containing this dislocation is represented by two elastic half-spaces joined by atomic-level forces across their common interface, known as the glide plane (—). The goal of the P-N model is to determine the slip distribution on the glide plane, which minimizes the total energy. The dislocation is characterized by the slip (relative displacement) distribution... 

The same result is obtained for a pressurized debond, ie, a blister, of radius r at the interface between an elastic half-space and a rigid plane (159) because a tensile stress ub applied at infinity is equivalent to a pressure Ob applied to the inner surface of the debonded region if the material is incompressible in bulk, as is assumed here. 

The soil region is modeled as homogeneous linear elastic half space with a horizontal free surface. The ballast and subbase are not considered in this model however, they can be accommodated in the proposed method in a straightforward manner. Hence, ties are assumed to rest directly on the free surface of the halfspace with which they... 

The load P corresponding to the limit of stability will be called the adherence force of the two elastic bodies, and thus will generally depend on the stiffness of the measuring apparatus. In some geometries (unstable geometries), equilibrium is always unstable, and thus the criterion for adherence force is simply G = w. li is the case for the double cantilever beam at fixed load, or for a flat punch on an elastic half-space. It is also the classical Griffith case of a crack in an infinite solid. 

The case of an axisymmetric flat punch of radius a on an elastic half-space was solved by KendalK7) by evaluating the elastic energy = iPb and the potential energy Up = —from the elastic displacement under the load P ... 

D. Maugis and M. Barquins, Adhesive contact of a conical punch on an elastic half-space, J. Phys. (Paris), Lett. 42, L95-L97 (1981). 

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