0
Research Papers

Solution of the Contact Problem of a Rigid Conical Frustum Indenting a Transversely Isotropic Elastic Half-Space

[+] Author and Article Information
X.-L. Gao

Professor
Fellow ASME
Department of Mechanical Engineering,
University of Texas at Dallas,
800 West Campbell Road,
Richardson, TX 75080-3021
e-mail: Xin-Lin.Gao@utdallas.edu

C. L. Mao

Doctoral Student
Department of Architecture,
Texas A&M University,
College Station, TX 77843

1Corresponding author.

Manuscript received May 29, 2013; final manuscript received July 14, 2013; accepted manuscript posted July 29, 2013; published online September 23, 2013. Editor: Yonggang Huang.

J. Appl. Mech 81(4), 041007 (Sep 23, 2013) (12 pages) Paper No: JAM-13-1219; doi: 10.1115/1.4025140 History: Received May 29, 2013; Revised July 14, 2013; Accepted July 29, 2013

The contact problem of a rigid conical frustum indenting a transversely isotropic elastic half-space is analytically solved using a displacement method and a stress method, respectively. The displacement method makes use of two potential functions, while the stress method employs one potential function. In both the methods, Hankel's transforms are applied to construct potential functions, and the associated dual integral equations of Titchmarsh's type are analytically solved. The solution obtained using each method gives analytical expressions of the stress and displacement components on the surface of the half-space. These two sets of expressions are seen to be equivalent, thereby confirming the uniqueness of the elasticity solution. The newly derived solution is reduced to the closed-form solution for the contact problem of a conical punch indenting a transversely isotropic elastic half-space. In addition, the closed-form solution for the problem of a flat-end cylindrical indenter punching a transversely isotropic elastic half-space is obtained as a special case. To illustrate the new solution, numerical results are provided for different half-space materials and punch parameters and are compared to those based on the two specific solutions for the conical and cylindrical indentation problems. It is found that the indentation deformation increases with the decrease of the cone angle of the frustum indenter. Moreover, the largest deformation in the half-space is seen to be induced by a conical indenter, followed by a cylindrical indenter and then by a frustum indenter. In addition, the axial force–indentation depth relation is shown to be linear for the frustum indentation, which is similar to that exhibited by both the conical and cylindrical indentations—two limiting cases of the former.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Spencer, A. J. M., 1972, Deformations of Fibre-Reinforced Materials, Oxford University Press, Oxford, UK.
Dong, X. N., Zhang, X., Huang, Y., and Guo, X. E., 2005, “A Generalized Self-Consistent Estimate for the Effective Elastic Moduli of Fiber-Reinforced Composite Materials With Multiple Transversely Isotropic Inclusions,” Int. J. Mech. Sci., 47, pp. 922–940. [CrossRef]
Ning, X., Zhu, Q., Lanir, Y., and Margulies, S. S., 2006, “A Transversely Isotropic Viscoelastic Constitutive Equation for Brainstem Undergoing Finite Deformation,” ASME J. Biomech. Eng., 128, pp. 925–933. [CrossRef]
Liu, M., and Yang, F. Q., 2012, “Finite Element Analysis of the Spherical Indentation of Transversely Isotropic Piezoelectric Materials,” Modell. Simul. Mater. Sci. Eng., 20, p. 045019. [CrossRef]
Chen, S., Yan, C., and Soh, A., 2008, “Non-Slipping JKR Model for Transversely Isotropic Materials,” Int. J. Solids Struct., 45, pp. 676–687. [CrossRef]
Guo, X., and Jin, F., 2009, “A Generalized JKR-Model for Two-Dimensional Adhesive Contact of Transversely Isotropic Piezoelectric Half-Space,” Int. J. Solids Struct., 46, pp. 3607–3619. [CrossRef]
Elliott, H. A., 1949, “Axial Symmetric Stress Distributions in Aeolotropic Hexagonal Crystals. The Problem of the Plane and Related Problems,” Math. Proc. Cambridge Philos. Soc., 45, pp. 621–630. [CrossRef]
Elliott, H. A., 1948, “Three-Dimensional Stress Distributions in Hexagonal Aeolotropic Crystals,” Math. Proc. Cambridge Philos. Soc., 44, pp. 522–533. [CrossRef]
Shield, R. T., 1951, “Notes on Problems in Hexagonal Aeolotropic Materials,” Math. Proc. Cambridge Philos. Soc., 47, pp. 401–409. [CrossRef]
Green, A. E., and Zerna, W., 1968, Theoretical Elasticity, 2nd ed., Oxford University Press, Oxford, UK.
Lodge, A. S., 1955, “The Transformation to Isotropic Form of the Equilibrium Equations for a Class of Anisotropic Elastic Solids,” Q. J. Mech. Appl. Math., 8, pp. 211–225. [CrossRef]
Okumura, I. A., 1987, “Generalization of Elliott's Solution to Transversely Isotropic Solids and its Application,” Struct. Eng./Earthquake Eng., 4, pp. 185–195.
Wang, M. Z., and Wang, W., 1995, “Completeness and Nonuniqueness of General Solutions of Transversely Isotropic Elasticity,” Int. J. Solids Struct., 32, pp. 501–513. [CrossRef]
Wang, W., and Shi, M. X., 1998, “On the General Solutions of Transversely Isotropic Elasticity,” Int. J. Solids Struct., 35, pp. 3283–3297. [CrossRef]
Eubanks, R. A., and Sternberg, E., 1954, “On the Axisymmetric Problem of Elasticity Theory for a Medium with Transverse Isotropy,” J. Rat. Mech. Anal., 3, pp. 89–101.
Wang, M. Z., Xu, B. X., and Gao, C. F., 2008, “Recent General Solutions in Linear Elasticity and Their Applications,” ASME Appl. Mech. Rev., 61, p. 030803. [CrossRef]
Lekhnitskii, S. G., 1940, “Symmetrical Deformation and Torsion of a Body of Revolution With Anisotropy of a Special Form,” Prikl. Mat. Mekh., 4, pp. 43–60.
Lekhnitskii, S. G., 1981, Theory of Elasticity of an Anisotropic Body, Mir, Moscow.
Hu, H.-C., 1953, “On the Three-Dimensional Problems of the Theory of Elasticity of a Transversely Isotropic Body,” Acta Sci. Sin., 2(2), pp. 145–151.
Nowacki, W., 1954, “The Stress Function in Three-Dimensional Problems Concerning an Elastic Body Characterized by Transverse Isotropy,” Bull. Acad. Pol. Sci., 2(1), pp. 21–25.
Ding, H.-J., Chen, W. Q., and Zhang, L., 2006, Elasticity of Transversely Isotropic Materials, Springer, Dordrecht, The Netherlands.
Hong, J. M., Ozkeskin, F. M., and Zou, J., 2008, “A Micromachined Elastomeric Tip Array for Contact Printing with Variable Dot Size and Density,” J. Micromech. Microeng.18, p. 015003. [CrossRef]
Zhou, S.-S., Gao, X.-L., and He, Q.-C., 2011, “A Unified Treatment of Axisymmetric Adhesive Contact Problems Using the Harmonic Potential Function Method,” J. Mech. Phys. Solids, 59, pp. 145–159. [CrossRef]
Ejike, U. B. C. O., 1981, “The Stress on an Elastic Half-Space Due to Sectionally Smooth-Ended Punch,” J. Elast., 11, pp. 395–402. [CrossRef]
Lai, W. M., Rubin, D., and Krempl, E., 2010, Introduction to Continuum Mechanics, 4th ed., Elsevier, Burlington, MA.
Zhou, S.-S., and Gao, X.-L., 2013, “Solutions of Half-Space and Half-Plane Contact Problems Based on Surface Elasticity,” Z. Angew. Math. Phys., 64, pp. 145–166. [CrossRef]
Titchmarsh, E. C., 1937, An Introduction to the Theory of Fourier Integrals, Oxford University Press, Oxford, UK.
Busbridge, I. W., 1938, “Dual Integral Equations,” Proc. Lond. Math. Soc., 44, pp. 115–130. [CrossRef]
Harding, J. W., and Sneddon, I. N., 1945, “The Elastic Stresses Produced by the Indentation of the Plane Surface of a Semi-Infinite Elastic Solid by a Rigid Punch,” Math. Proc. Cambridge Philos. Soc., 41, pp. 16–26. [CrossRef]
Ding, H.-J., and Xu, B.-H., 1988, “General Solutions of Axisymmetric Problems in Transversely Isotropic Body,” Appl. Math. Mech., 9(2), pp. 143–151. [CrossRef]
Bodunov, N. M., and Druzhinin, G. V., 2009, “One Solution of an Axisymmetric Problem of the Elasticity Theory for a Transversely Isotropic Material,” J. Appl. Mech. Technol. Phys., 50, pp. 982–988. [CrossRef]
Gao, X.-L., and Zhou, S.-S., 2013, “Strain Gradient Solutions of Half-Space and Half-Plane Contact Problems,” Z. Angew. Math. Phys., 64, pp. 1363–1386. [CrossRef]
Sneddon, I. N., 1965, “The Relation Between Load and Penetration in the Axisymmetric Boussinesq Problem for a Punch of Arbitrary Profile,” Int. J. Eng. Sci., 3, pp. 47–57. [CrossRef]
Hanson, M. T., 1992, “The Elastic Field for Conical Indentation Including Sliding Friction for Transverse Isotropy,” ASME J. Appl. Mech., 59, pp. S123–S130. [CrossRef]
Liu, Y., He, Y., Chu, F., Mitchell, T. E., and Wadley, H. N. G., 1997, “Elastic Properties of Laminated Calcium Aluminosilicate/Silicon Carbide Composites Determined by Resonant Ultrasound Spectroscopy,” J. Am. Ceram. Soc., 80, pp. 142–148. [CrossRef]
Behrens, E., 1971, “Elasic Constants of Fiber-Reinforced Composites With Transversely Isotropic Constituents,” ASME J. Appl. Mech., 38, pp. 1062–1065. [CrossRef]
Danyluk, H. T., Singh, B. M., and Vrbik, J., 1991, “Ductile Penny-Shaped Crack in a Transversely Isotropic Cylinder,” Int. J. Fract., 51, pp. 331–342. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Contact of a conical frustum punch with a transversely isotropic elastic half-space

Grahic Jump Location
Fig. 2

Axial force acting on the punch varying with α (with a/ab = 1.2) for the three different half-space materials

Grahic Jump Location
Fig. 3

Axial force acting on the punch varying with a/ab (with α = 15 deg) for the three different half-space materials

Grahic Jump Location
Fig. 4

Axial force-indentation depth relations for the CAS–SiC half-space indented by a frustum punch (with a/ab = 1.2), a conical punch and a cylindrical punch

Grahic Jump Location
Fig. 5

Axial force-indentation depth relations for the graphite-epoxy half-space indented by a frustum punch (with a/ab = 1.2), a conical punch and a cylindrical punch

Grahic Jump Location
Fig. 6

Axial force-indentation depth relations for the E-glass half-space indented by a frustum punch (with a/ab = 1.2), a conical punch and a cylindrical punch

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In