Transient disturbances in a half-space during the first stage of frictionless indentation of a smooth rigid die of arbitrary shape

Authors:
A. R. Robinson and J. C. Thompson

Journal:
Quart. Appl. Math. **33** (1975), 215-223

DOI:
https://doi.org/10.1090/qam/99664

MathSciNet review:
QAM99664

Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: A solution is obtained by the method of self-similar potentials for the time-spatial distribution of the contact stress between a homogeneous, isotropic, linearly elastic half-space and a smooth rigid die having an arbitrary indenting velocity and shape. The solution holds as long as the outward speed of the contact zone does not fall below the speed of the dilatational wave in the elastic medium. A proof is given that the instantaneous value of the force required to indent the die during this stage of contact is directly proportional to the product of the area of contact and the velocity of indentation at that instant.

**[1]**Ph. Frank and R. von Mises,*Differential and integral equations of mathematical physics*, Chapter XII, by S. L. Sobolev (Russian edition only), ONTI, Moscow-Leningrad, 1937**[2]**V. I. Smirnov and S. L. Sobolev,*Sur une méthode nouvelle dans le problème plan des vibrations elastiques*, Trud. Inst. Seism. Akad. Nauk SSR**20**(1932)**[3]**V. I. Smirnov and S. L. Sobolev,*On the application of a new method of investigation of the elastic vibrations in the space with axial symmetry*, Trud. Inst. Seism. Akad. Nauk SSSR**29**(1933)**[4]**B. V. Kostrov,*Self-similar dynamic problems of pressing of a rigid die into an elastic half space*(in Russian), Mekhanika i Mashionostroenie**4**, 54 (1964)**[5]**A. R. Robinson and J. C. Thompson,*Exact solutions of some dynamic problems of indentation and transient loadings of an elastic half space*, Technical Report, Structural Research Series No. SRS-350, University of Illinois, Urbana, Illinois (1969)**[6]**A. R. Robinson and J. C. Thompson,*Transient stress states in an elastic half space resulting from the frictionless indentation of a rigid wedge-shaped die*, ZAMM**54**(1974)**[7]**A. R. Robinson and J. C. Thompson,*Transient stresses in an elastic half space resulting from the frictionless indentation of a rigid axially symmetric conical Die*, Cambridge Philosophical Society**74**(1974)**[8]**J. R. Willis,*Self-similar problems in elastodynamics*, Philos. Trans. Roy. Soc. London Ser. A**274**(1973), 435–491. MR**0337105**, https://doi.org/10.1098/rsta.1973.0073**[9]**Frederick Reyes Norwood,*Similarity solutions in plane elastodynamics*, Internat. J. Solids and Structures**9**(1973), 789–803 (English, with Russian summary). MR**0329395**

Additional Information

DOI:
https://doi.org/10.1090/qam/99664

Article copyright:
© Copyright 1975
American Mathematical Society