Axisymmetric Slipless Indentation of an Infinite, Elastic Cylinder

Yau, W.F.

doi:10.1137/0115018

Cited by 113 publications

(73 citation statements)

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“…This makes the numerical solution of such equations quite difficult. To overcome this difficulty, the Schmidt method [Morse and Feshbach 1958;Yau 1967] is used to solve the dual integral equations (45)- (48). From the nature of the displacement along the crack line, it can be shown that the jumps of the displacements across the crack surface are finite, differentiable, and continuum functions.…”

Section: Solution Of the Dual Integral Equationsmentioning

confidence: 99%

“…So the semi-infinite integral in Equations (55)- (56) can be evaluated numerically by Filon's method. Equations (55)- (56) can now be solved for the coefficients a n and b n by the Schmidt method [Morse and Feshbach 1958;Yau 1967]. Briefly, Equations (55)- (56) can be rewritten as…”

Section: Solution Of the Dual Integral Equationsmentioning

confidence: 99%

“…In the present paper, the effect of the lattice parameter of functionally graded materials on dynamic stress fields near Mode-I crack tips is investigated using nonlocal theory in functionally graded materials with the Schmidt method [Morse and Feshbach 1958;Yau 1967]. When the lattice parameter of materials tends to zero, the present problem will revert to the same problem as in [Zhou et al 2004].…”

Section: Introductionmentioning

confidence: 97%

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The nonlocal theory solution of a mode-I crack in functionally graded materials subjected to harmonic stress waves

Zhou

Liang

2006

JOMMS

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In this paper, the dynamic behavior of a finite crack in functionally graded materials subjected to harmonic stress waves is investigated by means of nonlocal theory. The traditional concepts of nonlocal theory are extended to solve the dynamic fracture problem of functionally graded materials. To overcome mathematical difficulties, a one-dimensional nonlocal kernel is used instead of a twodimensional one for the dynamic problem to obtain the stress fields near the crack tips. To make the analysis tractable, it is assumed that the shear modulus and the material density vary exponentially and vertically with respect to the crack. Using the Fourier transform and defining the jumps of the displacements across the crack surfaces as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Unlike classical elasticity solutions, it is found that no stress singularities are present near crack tips. Numerical examples are provided to show the effects of the crack length, the parameter describing the functionally graded materials, the frequency of the incident waves, the lattice parameter of the materials and the material constants upon the dynamic stress fields near crack tips.

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Section: Solution Of the Dual Integral Equationsmentioning

confidence: 99%

Section: Solution Of the Dual Integral Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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The nonlocal theory solution of a mode-I crack in functionally graded materials subjected to harmonic stress waves

Zhou

Liang

2006

JOMMS

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“…Equation (3.9) can be solved for the coefficients a, b by a modified version of Schmit's method (14). Once the displacement and rotation at the boundary are found, this analysis is considered to be complete.…”

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confidence: 99%

The effect of couple‐stresses on the stress concentration around a moving crack

Itou

1981

International Journal of Mathematics and Mathematical Sciences

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“…Thus Oda, Shibahara and Miyamoto calculated the effects of band loads to determine the influence coefficients for contour displacements in infinite full cylinders [9]. By use of integral transforms Yau [10] solved indentation problems: the mixed boundary situation arises from assigned displacements in a finite band and assigned (zero) stresses out of this region. The loading is represented in series of Chebyshef polynomials, whose Fourier transform leads to Bessel functions.…”

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confidence: 99%

Analytic solutions of axisymmetric problems in elastic cylinders

Cicala

1985

Meccanica

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MECCANICA 20 (198'5), 43- , till recently subjected to numerous studies. The present paper, inspecting the conceptual outline of analytic investigations referred to, aims at facilitating the use of the abundant literature by engineers.Many deem that actual progress in numerical aids and availability of computation programs make these studies obsolete. On the contrary, it could be argued that, as these developments open to access by engineers a wider field of advanced problems, the need for analytic exploration becomes more and more impellent. Moreover, attention is called to the fact that the present diffusion of small computers makes more analytical approaches now practicable.Paper is incomplete: the important question of stress concentrations is not touched on, assuming that the analysis of these singularities is to be based on the study of the prevailing regular region of the stress field.Special attention is devoted to an investigation [3] being carried out in contact with the writer: its relevance in connection with literature is examined here.2. Generalities. The usual cylindric reference is adopted, denoting by Rr, Rz radial and axial coordinates, R being a reference length. Let V 2 be the axisymmetric Laplace operator 3 2 + r -1 a r + ~}z with d r = ~/br, a z = b/az. Material * Department of Structures, Politecnico di Torino. is assumed to be elastic isotropic, with tangential modulus G and Poisson's ratio v, both constant. The displacement field contains only radial and axial components, denoted by Ru r, Ru~. Let 2Ga r be the normal stress in radial direction: similar notation is used for circumferential o 0, axial a z and for the only tangential stress rrz. Non-dimensional variables are used without further notice.Homogeneous axisymmetric solutions are usually constructed by use of the biharmonic Love function • ([4], p. 276). From it, by differentiations, displacements and stresses are derived. Note that the expressions for the stresses can be obtained from the displacements through the elasticity relationships: moreover the stresses thus obtained comply with the equilibrium condition for radial forces. Therefore this formulation obeys the compatibility conditions and one ecluilibrium condition, for any choice of • whereas the equilibrium of axial forces requires that • be biharmonic. Therefore the denomination of stress function for • adopted by some Authors, is misleading as it suggests a false analogy with proper stress functions. These lead to an equilibrium configuration of stresses independently of their properties, which are dictated by compatibility of deformations(i). Distinction between stress and displacement functions is necessary when energy methods are adopted.A straightforward formulation is furnished by the equilibrium equations in terms of displacements. One of these may be written in the form V20 = 0 where 0 is the cubic dilatation, expressed for axisymmetric situations by the relation 0 = ~rUr -1" r'lu r 4" ~zUz.(The other equation writesConsider the solutions expressed by products o...

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Axisymmetric Slipless Indentation of an Infinite, Elastic Cylinder

Cited by 113 publications

References 3 publications

The nonlocal theory solution of a mode-I crack in functionally graded materials subjected to harmonic stress waves

The nonlocal theory solution of a mode-I crack in functionally graded materials subjected to harmonic stress waves

The effect of couple‐stresses on the stress concentration around a moving crack

Analytic solutions of axisymmetric problems in elastic cylinders

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