On the depth of cylindrical indentation of an elastic half-space for two types of displacement constraints

For cylindrical indentation of elastic half-space the relationship between the depth of indentation delta and the applied force F is nonlinear, in contrast to the linear relationship between the height of the contact zone delta_0 and the force F. While the latter is independent of the boundary conditions used to specify the rigid-body translation, the former depends on a selected datum for vertical displacement. The depth of the indentation is determined for any permissible value of the length b, which specifies the points of the free surface where the vertical displacement is required to be zero, w(b)=0. From the condition that the work of the indentation force is equal to the work of the contact pressure, it follows that the indentation is geometrically and physically possible under imposed boundary conditions w(b)=0 provided that b>=b_min. The numerical value of b_min is found to be about 10 times greater than the semi-width of the contact zone a, based on the numerical precision in fulfilling the work condition W_F=W_p. If a datum is taken to be at a point at some distance h below the load, there is an alternative closed-form expression for delta in terms of F, which involves the Poisson ratio nu. For nu=1/3, it is found that h_min is about 21a. A simple expression relating the permissible values of h and b is derived, which is linear for large values of h and b

Image force on a straight dislocation emitted from a cylindrical void

AbstractThe image force exerted by the free surface of a cylindrical circular void on a nearby straight dislocation depends on whether the dislocation has arrived at its location by the emission from the surface of the void, or by the glide from infinity. In the context of elasticity theory, in the first case, the dislocation has been created by imposing the displacement discontinuity along the cut from the free surface of the void to the center of the dislocation, and, in the second case, from the center of the dislocation to infinity. The explicit expressions for the two corresponding image forces are derived and compared. It is shown that the attraction from the free surface of the void is stronger in the first case, particularly for smaller voids. Furthermore, in the case of dislocation emitted from the surface of the void, the interaction energy depends on the cut used to impose the displacement discontinuity, but not in the case of a dislocation approaching the void from infinity. The relevance of the obtained results for the materials science problems is discussed

ON THE MECHANICAL RESPONSE OF A PRESSURIZED FUNCTIONALLY-GRADED CYLINDER

A pressurized functionally-graded cylinder is considered made of the material whose elastic moduli vary with the radial distance according to the power-law relation. Some peculiar features of the mechanical response are noted for an incompressible functionally-graded material with the power of radial inhomogeneity equal to two. In particular, it is shown that the maximum shear stress is constant throughout the cylinder, while the displacement changes proportional to 1/r along the radial distance. No displacement takes place at all under equal pressures applied at both boundaries.A pressurized functionally-graded cylinder is considered made of the material whose elastic moduli vary with the radial distance according to the power-law relation. Some peculiar features of the mechanical response are noted for an incompressible functionally-graded material with the power of radial inhomogeneity equal to two. In particular, it is shown that the maximum shear stress is constant throughout the cylinder, while the displacement changes proportional to 1/r along the radial distance. No displacement takes place at all under equal pressures applied at both boundaries

Determination of the belt force before the gross slip

The mechanics of belt friction before the state of gross slip is considered. The variation of the belt force within the contact region is evaluated based on the assumption of gradual growth of slip from the pull-end to the hold-end of the belt, as the pull force increases towards its Euler's value. The local pressure and friction forces exerted by the belt on the cylinder are also determined. Both flat and V-shaped belts are considered. The total pressure and friction forces are evaluated at an arbitrary stage of slip growth. They are neither proportional nor orthogonal to each other, unless the state of gross slip is reached throughout the contact range

CHAPTER 05 : FINITE STRAIN ELASTICITY

Elastic deformation does not cause irreversible rearrangement of internal structure, and the corresponding Helmholtz free energy is a function of stress and temperature only. Restricting consideration to isothermal elastic deformation (θ = 0), Eqs. (4.3.7) and (4.3.9) givė ψ = ∂ψ ∂E (n

Thermodynamic analysis based on the second-order variations of thermodynamic potentials

An analysis of the Gibbs conditions of stable thermodynamic equilibrium based on the constrained minimization of the four fundamental thermodynamic potentials, is presented with a particular attention given to the previously unexplored connections between the second-order variations of thermodynamic potentials. These connections are used to establish the convexity properties of all potentials in relation to each other, which systematically deliver thermodynamic relationships between the specific heats, and the isentropic and isothermal bulk moduli and compressibilities. The comparison with the classical derivation is then given