Summary
Making use of the Eringen-Kroener form of the nonlocal constitutive equations and the exponential Fourier transformation, a system of two coupled differential equations of the second order describing the equilibrium of the body is derived. By appeal to the Helmholtz representation, the system is reduced to a single differential equation of the fourth order for the Love function, reminding a Bessel type transform of the biharmonic equation. A solution of this equation is found, and inverse transforms of the stress components using the convolution theorem established. A recourse to the formula of de la Vallée Poussin shows that, in contrast to the classical result, the stress singularity at the point of application of a concentrated force fails to appear, though the stress concentration at that point is extremely high.
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References
Krumhansl, J. A.: Generalized continuum field representations for lattive vibrations. In: Lattice dynamics (Wallis, R. F., ed.) Oxford: Pergamon 1965.
Rogula, D.: Influence of spatial acoustic dispersion on dynamical properties of dislocations, I. Bull. Acad. Pol. Sci., Sci. Techn.,13, 337 (1965).
Kroener, E.: Continuum mechanics and range of atomic cohesion forces. In: Proceed. Int. Conf. Fract., Vol. 1, Jap. Soc. Strength Fract. Mat., Sandai, 1966.
Kunin, I. A.: Model of elastic medium with simple structure and space dispersion (in Russian). Prikl. Mat. Mech.30, 642–652 (1966).
Cochran, W.: Interpretation of phonon dispersion curves. In: Lattice dynamics (Wallis, R. F., ed.). Oxford: Pergamon 1965.
Eringen, A. C.: Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci.10, 425–435 (1972).
Eringen, A. C., Edelen, D. G. B.: On nonlocal elasticity. Int. J. Eng. Sci.10, 233–248 (1972).
Eringen, A. C.: Nonlocal polar field theories. In: Continum physics4 (Eringen, A. C., ed.), pp. 205–267, 1976.
Eringen, A. C.: On Rayleigh surface waves with small wave length. Lett. App. Eng. Sci.1, 11–17 (1973).
Nowinski, J. L.: On the nonlocal aspects of the propagation of Love waves. Int. J. Eng. Sci.22, 383–392 (1984).
Eringen, A. C., Kim, B. S.: Stress concentration at the tip of crack. Mech. Res. Commun.1, 233–237 (1974).
Eringen, A. C.: Screw dislocation in nonlocal elasticity.. Phys. D.: Appl. Phys.10, 671 (1977).
Wang, R.: Ph. D. thesis, Beijing Graduate School, China University of Mining and Technology, Beijing, 1988.
Nowinski, J. L.: On the transmission across the interface of two elastic half-spaces with nonlocal cohesion forces. Acta. Mech.,78, 209–218 (1989).
Fung, Y. C.: Foundations of solid mechanics. Englewood Cliffs: Prentice-Hall 1965.
Scarborough, J. B.: Numerical mathematical analysis. Johns Hopkins Press 1958.
Friedman, B.: Principles and techniques of applied mathematics. New York: Wiley 1956.
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Nowinski, J.L. On a three-dimensional Kelvin problem for an elastic nonlocal medium. Acta Mechanica 84, 77–87 (1990). https://doi.org/10.1007/BF01176089
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DOI: https://doi.org/10.1007/BF01176089