臺灣博碩士論文加值系統

We study the indentation problems of film-substrate composites. Axisymmetric indentation problem involving elastic film-substrate composites considered by Yu, et al. [1] is revisited. The results are extended to viscoelastic thin film on elastic substrate with flat-ended indenter. Instead of using surface displacement as one of the boundary conditions, the stress distribution is relative to that for a homogeneous half-space. First, the displacement and stress components are considered based on Navier's equation and Hankel transform. We solved linear algebraic equation with two different boundary conditions, perfectly bonded and freely sliding. Secondly, viscoelastic models are introduced, and elastic solutions are replased by two viscoelastic models, standard linear solid model and Burgers model, which come from inverse of Laplace transform. Finally, we find displacement and stress with different positions, thickness and material constants.
The method is simpler and easier to use and also describe the load-penetration relationship. Compared with the previous method, our solutions match the elastic solutions at t = 0. We can change boundary condition using the same equations. Moreover, the method will be extended to n-layer thin films rested on substrate straightforwardly. For fitting experimental data, it will be a new and convenient solution to understand unknown material constants.

Acknowledgment
Abstract
Contents
Figure Captions
Chapter 1 Introduction
Chapter 2 Theory
2.1 Stress and displacement
2.2 Boundary conditions
2.3 Indenter
2.4 Viscoelastic model
Chapter 3 Results and discussion
3.1 Compare with elastic solution and boundary conditions
3.2 Effect with different parameters
3.3 Other applications
3.4 Uniform normal displacement
Chapter 4 Conclusions
References
Appendix:
Abstract
Figure Captions
Chapter 1 Introduction
Chapter 2 Evaporation and condensation model
Chapter 3 Experiment
Chapter 4 Results and discussion
Summary
Acknowledgment
References
Table 1 & Table 2

1. H. Y. Yu and S. C. Sanday , The effect of substrate on the elastic properties of films determined by the indentation test -- axisymmetric boussinesq problem, Journal of the Mechanics and Physics of Solids 38(6), 745-764 (1990).
2. M. J. Boussinesq, Applications des Potentiels, Gauthier-Villars, Paris, 1885.
3. J. W. Harding and I. N. Sneddon, The elastic stresses produced by the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch, Proceedings of the Cambridge Philosophical Society 41(1), 16-26 (1945).
4. I. N. Sneddon, Boussinesq problem for a flat-ended cylinder, Proceedings of the Cambridge Philosophical Society 42(1), 29-39 (1946).
5. I. N. Sneddon, Boussinesqs problem for a rigid cone, Proceedings of the Cambridge Philosophical Society 44(4), 492-507 (1948).
6. I. N. Sneddon, The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile, International Journal of Engineering Science 3(1), 47-57 (1965).
7. R. S. Dhaliwal and I. S. Rau, The axisymmetric boussinesq problem for a thick elastic layer under a punch of arbitrary profile, International Journal of Engineering Science 8(10), 843-856 (1970).
8. H. Gao, Elastic contact versus indentation modeling of multi-layered materials, International journal of solids and structures 29(20), 2471-2492 (1992).
9. X. Chen, Numerical study on the measurement of thin film mechanical properties by means of nanoindentation, Journal of Materials Research 16(10), 2974-2982 (2001).
10. C. H. Hsueh, Master curves for Hertzian indentation on coating/substrate systems, Journal of Materials Research 19(1), 94-100 (2004).
11. M. Sakai and J. Zhang, Elastic Deformation of Coating/substrate Composites in Axisymmetric Indentation, Journal of Materials Research 20(08), 2173-2183 (2005).
12. N. Schwarzer, The elastic field in a coated half-space under Hertzian pressure distribution, Surface & Coatings Technology 114(2-3), 292-303 (1999).
13. M. Sakai, Indentation rheometry for glass-forming materials, Journal of Non-crystalline Solids 282(2-3), 236-247 (2001).
14. L. Cheng, Flat-punch indentation of viscoelastic material, Journal of Polymer Science. Part B, Polymer Physics 38(1), 10-22 (2000).
15. L. Cheng, Spherical-tip indentation of viscoelastic material, Mechanics of Materials 37(1), 213-226 (2005).
16. M. Sakai, Time-dependent viscoelastic relation between load and penetration for an axisymmetric indenter, Philosophical Magazine. A, Physics of Condensed Matter, Defects and Mechanical Properties 82(10), 1841-1849 (2002).
17. H. Lu, Measurement of creep compliance of solid polymers by nanoindentation, Mechanics of Time-dependant Materials 7(3/4), 189-207 (2003).
18. M. V. Kumar, Analysis of spherical indentation of linear viscoelastic materials, Current Science 87(8), 1088-1095 (2004).
19. M. R. VanLandingham, Viscoelastic characterization of polymers using instrumented indentation. I. Quasi-static testing, Journal of Polymer Science. Part B, Polymer Physics 43(14), 1794-1811 (2005).
20. W. N. Findley, Creep and relaxation of nonlinear viscoelastic materials: with an introduction to linear viscoelasticity, North-Holland Publisher co., Kidlington, Oxford, UK, 1989.
21. M. E. Gurtin, On the linear theory of viscoelasticity, Archive for Rational Mechanics and Analysis 11(1), 291-356 (1962).
22. R. S. Lakes, On Poisson’s ratio in linearly viscoelastic solids, Journal of Elasticity 85(1), 45-63 (2006).
23. A. Jäger, Identification of viscoelastic properties by means of nanoindentation taking the real tip geometry into account, Meccanica 42(3), 293-306 (2007).
24. S. T. Choi, Flat indentation of a viscoelastic polymer film on a rigid substrate, Acta Materialia 56(19), 5377-5387 (2008).
25. I. F. Kozhevnikov, A new algorithm for computing the indentation of a rigid body of arbitrary shape on a viscoelastic half-space, International Journal of Mechanical Sciences 50(7), 1194-1202 (2008).
26. H. S. Lee, Determination of Viscoelastic Poisson’s Ratio and Creep Compliance from the Indirect Tension Test, Journal of Materials in Civil Engineering 21(8), 416-425 (2009).
27. C. Y. Zhang, Extracting the mechanical properties of a viscoelastic polymeric film on a hard elastic substrate. Journal of Materials Research 19(10), 3053-3061 (2004).
28. S. T. Choi, Measurement of time-dependent adhesion between a polymer film and a flat indenter tip, Journal of Physics. D, Applied Physics 41(7), 074023 (2008).
29. K. L. Johnson, Contact Mechanics. Cambridge: Cambridge University Press, Cambridge, UK, 1985.
30. I. F. Kozhevnikov, A new algorithm for solving the multi-indentation problem of rigid bodies of arbitrary shapes on a viscoelastic half-space, International Journal of Mechanical Sciences 52(3), 399-409 (2010).
31. Y. P. Cao, Geometry independence of the normalized relaxation functions of viscoelastic materials in indentation, Philosophical Magazine 90(12), 1639-1655 (2010).
32. H. Li, New methods of analyzing indentation experiments on very thin films, Journal of MaterialsResearch 25(4), 728-734 (2010).
33. M.Vandamme, Viscoelastic solutions for conical indentation, International Journal of Solids and Structures 43(10), 3142-3165 (2006).
34. Y.C Pan and T. W. Chou, Green's function solutions for semi-infinite transversely isotropic materials. International Journal of Engineering Science 17(5), 545-551 (1979).
35. A. Falade, Elastic fields of two-phase transversely isotropic materials, Philosophical Magazine A 45(5), 791-802 (1982).
36. H. Ding, Green's functions for two-phase transversely isotropic magneto-electro-elastic media, Engineering Analysis with Boundary Elements 29(6), 551-561 (2005).
37. G. A. C. Graham, The contact problem in the linear theory of viscoelasticity, International Journal of Engineering Science 3(1), 27-46 (1965).
38. N. N. Lebedev, Axisymmetric contact problem for an elastic layer, Journal of Applied Mathematics and Mechanics 22(3), 442-450 (1958).
39. A. Y. Alexandrov, (1968). Solution of three-dimensional problems of the theory of elasticity for solids of revolution by means of analytical functions. International Journal of Solids and Structures 4(7): 701-721.
40. J. L. Klemm and R. W. Little, The Semi-infinite elastic cylinder under self-equilibrated end loading, SIAM Journal on Applied Mathematics 19(4), 712-729 (1970).
41. K. Adeerogba, On eigenstresses in dissimilar media, Philosophical Magazine 35(2), 281-292 (1977).
42. M. Rahman, Boussinesq type solution for a transversely isotropic half-space coated with a thin film, International Journal of Engineering Science 38(7), 807-822 (2000).
43. M. Eskandari, Green's functions of an exponentially graded transversely isotropic half-space, International Journal of Solids and Structures 47(11-12), 1537-1545 (2010).
44. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th ed., General Publishing Co., Toronto, Ontario, Canada, 1906.
45. G. E. Dieter, Mechanical Metallurgy, McGraw-Hill Book Co., London, 1961.
46. K. L. Johnson, Contact mechanics, Cambridge University Press, UK, 1985.