Wave propagation in an elastic solid with a line of discontinuity or finite crack

Authors:
G. C. Sih and J. F. Loeber

Journal:
Quart. Appl. Math. **27** (1969), 193-213

DOI:
https://doi.org/10.1090/qam/99830

MathSciNet review:
QAM99830

Full-text PDF Free Access

Abstract | References | Additional Information

Abstract: With the aid of integral transforms, a method is presented for solving the problem of scattering of plane harmonic compression and shear waves by a line of discontinuity or crack of finite width embedded in an elastic medium of infinite extent. When the incoming waves are applied in an arbitrary direction, the scattered-wave field may be determined by separating the crack-surface boundary conditions into functions even and odd with respect to the variable along the line crack. The problem is reduced to the evaluation of a system of coupled Fredholm integral equations with special emphasis placed on finding the *near-field* solution which consists of a knowledge of the detailed structure of the displacements and stresses in a small region around the crack vertex. Dynamic stress-intensity factors, the critical values of which govern the condition of crack propagation, are defined and found to be dependent on the incident wave length and Poisson's ratio of the medium. At certain wave lengths, they are larger than those encountered under static loading. Such information is of particular importance in perdicting the fracture strength of structures subjected to oscillating loads.

**[1]**P. M. Morse and H. Feshbach,*Methods of theoretical physics*. Part II, McGraw-Hill, New York, 1955 MR**0059774****[2]**D. S. Jones,*The theory of electromagnetism*, Macmillan, New York, 1964 MR**0161555****[3]**V. A. Fock,*Electromagnetic diffraction and propagation problems*, Pergamon Press, New York, 1965 MR**0205569****[4]**Y. H. Pao,*Dynamic stress concentration in an elastic plate*, Appl. Mech.**29**, 299-305 (1962)**[5]**Y. H. Pao and C. C. Mow,*Dynamic stress concentration in an elastic plate with rigid circular inclusion*, Proceedings of Fourth U. S. National Congress of Applied Mechanics**1**, 335-345 (1962) MR**0151024****[6]**S. A. Thau and Y. H. Pao,*Stress-intensification near a semi-infinite rigid-smooth strip due to diffraction of elastic waves*, J. Appl. Mech.**34**, 119-126 (1967)**[7]**B. Noble,*Methods based on the Wiener-Hopf technique*, Pergamon Press, New York, 1958 MR**0102719****[8]**G. C. Sih,*Some elastodynamic problems of cracks*, Internat. J. Fracture Mech. 4, 51-68 (1968)**[9]**J. F. Loeber and G. C. Sih,*Diffraction of anti-plane shear waves by a finite crack*, Acoust. Soc. Amer. 44, 90-98 (1968)**[10]**D. Ang and L. Knopoff,*Diffraction of scalar elastic waves by a finite crack*, Proc. Nat. Acad. Sci.**51**, 593-598 (1964) MR**0161568****[11]**R. A. Schmeltzer and M. Lewin,*Function-theoretic solution to a class of dual integral equations and an application to diffraction theory*, Quart. Appl. Math.**21**, 269-283 (1964) MR**0155162****[12]**J. W. Miles,*Homogeneous solutions in elastic wave propagation*, Quart. Appl. Math.**18**, 37 (1960) MR**0111291****[13]**M. Papadopoulos,*Diffraction of plane elastic waves by a crack with application to a problem of brittle fracture*, J. Austral. Math. Soc.**3**, Part 3, 325-339 (1963) MR**0152239****[14]**N. I. Muskhelishvili,*Some basic problems of mathematical theory of elasticity*, P. Noordhoff, Holland, 1953 MR**0058417****[15]**E. T. Whittaker and G. N. Watson,*Modern analysis*, 4th Ed., Cambridge University Press, London, 1962 MR**0178117****[16]**G. N. Watson,*Theory of Bessel functions*, 2nd Ed., Cambridge University Press, London, 1958**[17]**G. C. Sih and H. Liebowitz,*Mathematical theories of brittle fracture, in Mathematical fundamentals of fracture*, Vol. 2, 67-190, Academic Press, New York, 1968

Additional Information

DOI:
https://doi.org/10.1090/qam/99830

Article copyright:
© Copyright 1969
American Mathematical Society