Hertz model elastic deformation

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. 

The elasticity was quantitatively determined by analyzing the recorded force curves with the help of the Hertz model. The Hertz model describes the elastic deformation of two spherical surfaces touching imder the load, which was calculated theoretically in 1882 by Hertz. Other effects, such as adhesion or plastic deformation, were not included in this model. Sneddon extended the calculation to other geometries. For a cone pushing onto a flat sample, the relation between the indentation 5 and the loading force F can be expressed as ... 

Compressive measurements provide a means to determine specimen stiffness, Young s modulus of elasticity, strength at failure, stress at yield, and strain at yield. These measurements can be performed on samples such as soy milk gels (Kampf and Nussi-novitch, 1997) and apples (Lurie and Nussi-novitch, 1996). In the case of convex bodies, where Poisson s ratio is known, the Hertz model should be applied to the data in order to determine Young s modulus of elasticity (Mohsenin, 1970). It should also be noted that for biological materials, Young s modulus or the apparent elastic modulus is dependent on the rate at which a specimen is deformed. 

Elastic deformation. For small loads we can use the Hertz model as a simple approximation. The microcontacts are thereby assumed to be spherical. Hertz theory predicts an actual contact surface for an individual sphere on a plane (see Eq. 6.64) ... 

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 ... 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

Mathematical modelling of the compression of single particles 2.5.2.7 Hertz model. The mechanics of a sphere made of a linear elastic material compressed between two flat rigid surfaces have been modelled for the case of small deformations, normally less then 10% strain (Hertz, 1882). Hertz theory provides a relationship between the force F and displacement hp as follows ... 

AFM can also be used to probe local mechanical properties of thin films of food biopolymers, which are difficult to measure using traditional rheological methods. Several mechanical models have been developed to analyze the Young s modulus of food systems. One of the simplest models, the Hertz model, assumes that only the elastic deformation exists in a surface with spherical contacts, and the adhesion force can be neglected (Hugel and Seitz 2001). Equation (8.2) describes the relationship between the loading force, F and the penetration depth, d, where a is the radius of contact area, R the curvature of the tip radius, Vi and the Poisson s ratios of the two contact materials that have Young s modulus, Ei and E2. ... 

The resolution of the contact mode depends on a contact area at the tip apex. The contact diameter (a) can be estimated with the Hertz model in which the contact area increases with applied force (F) on the tip due to elastic deformation between tip and sample. 

In this context, Mazur has corrunented [32b] that for films dried at T > MFT, fire extent of deformation for an elastic particle should be predicted by the Hertz model, not the JKR model. Thus the reduced neck diameter (x/a) should be proportional to fl (Equation 14.3), and not rfirectly dependent on particle size... 

Figure 31 shows the schematic of a particle of diameter d attached to a flat surface. Here, P is the external force exerted on the particle, a is the contact radius, and Fad is the adhesion force. The classical Hertz contact theory provides for the elastic deformation of bodies in contact, but neglects the adhesion force. Several models for particle adhesion to flat surfaces were developed in the past that improves the Hertz model by including the effect of adhesion (van der Waals) force. 

Figure 17.1b shows the profile of the deformed elastic plane predicted by the JKR model. The profile is depicted under the condition of the same indentation S as in the Hertzian model in Figure 17.1a. Owing to the existence of adhesion, the JKR model requires more contact region and less loading force than the Hertz model.

Near the contact, the vertical arrows at the dashed contour schematically represent the surface forces which cause an additional deformation of the elastic sphere thus increasing the contact radius from aH (Hertz) to aJKR (JKR). The contact radius for the JKR model is a function of the external load, the work of adhesion, the radius of the contacting sphere (or the reduced radii of the contacting spheres, if two spheres are in contact) and the elastic constant K (a combination of the Young s moduli and the Poisson s ratios of the contacting materials), defined as... 

Modelling of the true contaet area between surfaees requires consideration of the deformation that occurs at the peaks of asperities as they come into contact with mating surfaces. Purely elastic contact between two solids was first described by H Hertz [7], The Hertzian contact area (yl between a sphere of radius r and a flat surface... 

Hertzian Model. Hertz described in a general form a contact phenomenon between two spheres of radii R andi , respectively, where the deformation mechanism is totally elastic. R oo, if one of the objects in contact is a plane. Based on his analysis and for symmetry reasons, the area of contact between the sphere and the plane is a circle and its area is... 

When two solids are in contact, deformation takes place in the local contact zone resulting in a contact force. This suggests that the contact force is directly related to the amount of local deformation or indentation of the two solids. The best-known force model for the contact between two spheres of isotropic material was developed by Hertz based on the theory of elasticity [9]. With radii Rj and R of the two spheres i and j, and masses mj and mj, the contact force f follows the relation... 

Based on the presented theory, it is intended to construct the contact force model between the two spheres. The initial indentation velocity between the two spheres is = 0.3 m/s. The speed of deformation waves is 2.6xlO m/s, which provides a limiting value of 0.(126 m/s for the impact to be considered elastic. Hence, the Hertz contact force model with permanent indentation is a valid one. The generalized parameter K is calculated from equation (2), with v = 0.33, to be equal to 5.50x10 N/m -. The equivalent mass of the two spheres is obtained from equation (7) as m = 0.046 kg. From equations (12), (13), and (14), the unknown parameters in the contact force model are evaluated as... 

During uniaxial loading of a comparatively soft spherical elastic pellet with a smooth stiff wall (flat surface), the contact area of the particle deforms as a circle. The model of Hertz (Hertz, 1882) describes the force and internal pressure distribution as well as the contact radius depending on the particle radius and moduh of elasticity behavior of the two contacting materials. 

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