Mindlin model

The force-displacement relations in contact mechanics are often nonlinear. A prominent example is the transition from stick to slip. Even for nonlinear interactions, there is a strictly quantitative relationship between the shifts of frequency and bandwidth. A/ and AT, on the one hand, and the force acting on the crystal, T(t), on the other. A/ and Ar are proportional to the in-phase and the out-of-phase component of F(t), respectively. Evidently, F(t) cannot be explicitly derived from A/ and Ar. Still, any contact-mechanical model (like the Mindlin model of partial slip) can be tested by comparing the predicted... 

Keywords Contact mechanics Contact stiffness Fretting wear Mindlin model Nonlinear mechanics Quartz crystal resonator Quartz crystal microbalance ... 

Fig. 4 Force-displacement relation as predicted by the Mindlin model. Since the central area, where the contact sticks, decreases with increasing tangential load, the force increases sub-linearly with displacement. The area under the hysteresis loop is the energy dissipated per cycle, AVF...

If roughness plays a role, the Mindlin model does not apply. Still, the Mindlin model is a good example of a broader class of models of partial slip. Bureau et al. have proposed a quantitative extension of the microslip model accounting for multi-contact interfaces [34]. Partial slip also occurs in multiasperity contacts because the microcontacts located at the rim are expected to rupture first. These contacts experience the largest lateral stress and the low-... 

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... 

FIGURE 7.8 Comparison of the linear-spring, Mindlin s no-slip, and Mrndlin and Deresiewicz s models with the experimental data for oblique impact of a particle at different impact angles. (Reprinted from Chem. Eng. Sci., 59, Di Renzo, A. and Di Maio, RR, Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes, 525-541, Copyright 2004, with permission from Elsevier.)... 

Di Renzo and Di Maio [20] simulated oblique impact of a particle to a flat wall at different impact angles using the linear-spring, Mindlin s no-slip, and Mindlin and Deresiewicz s models. The comparison of the results to the experimental findings of Kharaz et al. [21] is shown in Figure 7.8. 

Figure 7.8 shows a very good agreement of the three models with the experimental results, except for shallow impact angles. The linear model surprisingly produced better results compared to Mindlin s no-slip model. 

In the Globals subsection, the contact models for particle-to-particle and particle-to-geometry interactions are defined (see C in Figure 7.15). EDEM has a number of built-in contact models such as Hertz-Mindlin no-slip model (i.e., Hertz model is used for normal contact force calculations [see Section 7.1.4.1.2] and Mindlin no-slip model is used for tangential contact force calculations [Section 7.1.4.1.4]), linear-spring model (see Section 7.1.4.1.1), and JKR adhesive model (see Section 7.1.4.2.1). 

For estimating the contact force acting on a ball, the Hertz-Mindlin contact model was used. The contact forces of normal and tangential directions, Fn and Ft, were estimated using the following equations ... 

In the case of a collision between two particles, the contact forces are calculated according to a contact model based on the theory developed by Hertz (1882) for the normal impact and a non-slip approximation of the model by Mindlin and Dere-siewicz (1953) for the tangential part of the contact force, as proposed by Tsuji et al. (1992). The normal contact force is... 

The sugar pellets SUGLETS produced by Colorcon (UK) are processed in the simulated Wurster coater. These particles are studied in the compression, shear, and impact tests to estimate the material parameters for the Hertz-Mindlin contact model of particles used in DEM on the microscale of the simulation. 

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