Mindlin

The treatment of transverse shear stress effects in plates made of isotropic materials stems from the classical papers by Reissner [6-26] and Mindlin [6-27. Extension of Reissner s theory to plates made of orthotropic materials is due to Girkmann and Beer [6-28], Ambartsumyan [6-29] treated symmetrically laminated plates with orthotropic laminae having their principal material directions aligned with the plate axes. Whitney [6-30] extended Ambartsumyan s analysis to symmetrically laminated plates with orthotropic laminae of arbitrary orientation. 

R. D. Mindlin, Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates, Journal of Applied Mechanics, March 1951, pp. 31-38. 

S. S. Mindlin and N. N. Samosatsky, Production of Polyethylene Products by the Extrusion Method, Goschimizdat Publishers (1959). 

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... 

During a collision, the colliding solids undergo both elastic and inelastic (or plastic) deformations. These deformations are caused by the changes of stresses and strains, which depend on the material properties of the solids and the applied external forces. Theories on the elastic deformations of two elastic bodies in contact are introduced in the literature utilizing Hertzian theory for frictionless contact and Mindlin s approach for frictional contact. As for inelastic deformations, few theories have been developed and the available ones are usually based on elastic contact theories. Hence, an introduction to the theories on elastic contact of solids is essential. 

In this section, an introduction of the general relations of stresses in equilibrium in an infinitely large solid medium is presented, followed by a special application where a concentrated force is acting on a point inside the solid. Also presented is the case of forces on the boundary of a semiinfinite solid medium, which is of importance to the contact of two solid objects. As consequences of the boundary compression, displacements due to the changes of stresses and strains in the region of contact can be linked to the contact force by the Hertzian theory for frictionless contacts and by Mindlin s theory for frictional contacts. For more details on the Hertzian theory for contact, interested readers may refer to books on elasticity [Goldsmith, 1960 Timoshenko and Goodier, 1970 Landau and Lifshitz, 1970]. 

Mindlin, R. D. (1949). Compliance of Elastic Bodies in Contact. Trans. ASME, J. ofAppl. Mech., 16, 259. 

Fig. 9 Changes in the crack initiation times and crack depths in an epoxy resin as a function of the amplitude of the imposed cyclic displacement, a Number of cycles to the initiation of the primary cracks at the edge of the contact zone, b Measured depths of the primary cracks at various number of cycles and displacement amplitudes. Circles 103 cycles, solid diamonds 5 x 103 cycles, squares 5 x 104 cycles, c Calculated values of the maximum tensile stress at the edge of the contact using Hamilton (gross slip condition) or Mindlin—Cattaneo (partial slip condition) theories. The two curves correspond to calculations using the initial (/x = 1.0) and the steady-state (/x = 1.5) values of the coefficient of friction. PSR Partial slip regime, MR mixed regime, GSR gross slip regime...

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-... 

McKenzie et al, Artestb. Analg. 60, 783 (1981) as pre-surgical sedative G. Wallace, L. J, Mindlin, ibid, 63, 571 (1984)-Toxicity data E, I. Goldenthal. Toxicol. Appi Pharmacol 18 185 (1971)- Comprehensive description J, Tsau, N. 

H Mindlin, BF Gilp, LS Elliot, M Chamberlain, Corrosion in DOD Systems Data Collection and Analysis (Phase I) MIAC Report 8, Metals Information Analysis Center, Aug 1995. 

When a tangential force, T is applied, elastic distortion causes the centers of the spheres to be displaced perpendicular to their axis (Mindlin and Deresiewicz, 1953 Dobry et al., 1982) by an amount given in the equation below ... 

Based on Mindlin s equations, Poulos and Davis (1980) have developed a design chart for rigid, cylindrical piles of diameter d and length L, embedded in a xmiform elastic layer of thickness h that rests on a rigid base. The settlement (p) can then be calculated from the below equation ... 

Mindlin, R.D., and Deresiewicz, H. 1953. Elastic spheres in contact under varying oblique forces. Journal of Applied Mechanics 20, ASME 327-344. 

Mindlin, R. D. (1936), Force at a point in the interior of a semi-infinite solid. Journal of Applied Physics, 7, No.5, pp. 195-202. 

D. E. Pettit, C. E. Feddersen, and H. Mindlin, Flaw Growth Behavior of Inconel 718 at Room and Cryogenic Temperatures, NASA Rept. CR-101942, Battelle-Columbus Laboratories, Columbus, Ohio (1969). 

Element type Stiffiiess Stiffness Kirchhoff Mindlin Strain variation... 

A few solutions exist for 3-D PZT bodies. Most well-known solutions for finite PZT plates were obtained from approximated two-dimensional (2-D) equations of extended Mindlin s solutions (Herrmann 1974). But, these solutions are not directly applicable to the analysis of AE sensors commercially available. In order to clarify the frequency response of AE sensor (function W(f) in eq. 3.5) and to optimize the design of PZT elements, resonance characteristics of PZT element were analyzed by using the finite element method (FEM) (Ohtsu Ono 1983). 

Herrmann G (1974) R.D. Mindlin and applied mechanics. Pergamon Press, New York... 

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