Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T11:29:50.978Z Has data issue: false hasContentIssue false
  • English
  • Français

Response of power-law-viscoelastic and time-dependent materials to rate jumps

Published online by Cambridge University Press:  31 January 2011

A.H.W. Ngan  and
B. Tang
A.H.W. Ngan*
Affiliation:
Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, People’s Republic of China
B. Tang
Affiliation:
Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, People’s Republic of China
*
a) Address all correspondence to this author. e-mail: [email protected]
Get access

Abstract

Nonlinear viscoelastic problems are in general not analytically solvable. However, it is shown here that, for any viscoelastic materials describable by a constitutive law with linear elastic and (in general) nonlinear viscous elements arranged in any network fashion, such as the Maxwell or standard linear solid arrangements, it is always possible to eliminate the viscous terms by replacing the displacement, strain, and stress fields of the problem by the jumps in rates of these fields. After the viscous terms are eliminated, the problem is reduced to a linear elastic problem defined on the same spatial domain and with the same elastic constant as in the original viscoelastic problem. Such a reduced elastic problem is analytically solvable in many practical cases, and the solution yields a relation between jumps in the load rate and the displacement rate, pertinent to the boundary conditions in the original problem. Such a relation can often be used as the basis for an experimental scheme to measure the elastic constants of materials. The material can be time- or strain-dependent, and the value of the elastic constant measured corresponds to the time instant or the strain value when the jump in load or displacement rate is implemented.

Keywords

Elastic propertiesNanoindentationViscoelasticity
Type
Articles
Information
Journal of Materials Research , Volume 24 , Issue 3 , March 2009 , pp. 853 - 862
Copyright
Copyright © Materials Research Society 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Johnson, K.L.: Contact Mechanics (Cambridge University Press, Cambridge, UK, 1985), p. 93.CrossRefGoogle Scholar
2.Radok, J.R.M.: Viscoelastic stress analysis. Q. Appl. Math. 15, 198 (1957).CrossRefGoogle Scholar
3.Findley, W.N., Lai, J.S., and Onaran, K.: Creep and Relaxation of Nonlinear Viscoelastic Materials (Dover Publications Inc., NY, 1976).Google Scholar
4.Odqvist, F.K.G.: Theory of creep under combined stresses with application to machinery. Roy. Swedish Acad. Eng. Res. Proc. 141, 31 (1936).Google Scholar
5.Feng, G. and Ngan, A.H.W.: Effects of creep and thermal drift on modulus measurement using depth-sensing indentation. J. Mater. Res. 17, 660 (2002).CrossRefGoogle Scholar
6.Ngan, A.H.W., Wang, H.T., Tang, B., and Sze, K.Y.: Correcting power-law viscoelastic effects in elastic modulus measurement using depth-sensing indentation. Int. J. Solids Strut. 42, 1831 (2005).CrossRefGoogle Scholar
7.Tang, B., Ngan, A.H.W., and Lu, W.W.: Viscoelastic effects during depth-sensing indentation of cortical bone tissues. Philos. Mag. 86, 5653 (2006).CrossRefGoogle Scholar
8.Sneddon, I.N.: The relation between load and penetration in the axisymmetris Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3, 47 (1965).CrossRefGoogle Scholar
9.Pharr, G.M., Oliver, W.C., and Brotzen, F.R.: On the generality of the relationship between contact stiffness, contact area, and elastic modulus during indentation. J. Mater. Res. 7, 613 (1992).CrossRefGoogle Scholar
10.Oliver, W.C. and Pharr, G.M.: An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564 (1992).CrossRefGoogle Scholar
11.Tang, B. and Ngan, A.H.W.: Accurate measurement of tip-sample contact size during nanoindentation of viscoelastic materials. J. Mater. Res. 18, 1141 (2003).CrossRefGoogle Scholar
12.Tang, B., Ngan, A.H.W., and Lu, W.W.: An improved method for the measurement of mechanical properties of bone by nanoindentation. J. Mater. Sci.-Mater. Med. 18, 1875 (2007).CrossRefGoogle ScholarPubMed
13.Oliver, W.C. and Pethica, J.B.: Method for continuous determination of the elastic stiffness of contact between two bodies. U.S. Patent No. 4 848 141, July 18, 1989.Google Scholar
14.Gu, F., Amsden, B., and Neufeld, R.: Sustained delivery of vascular endothelial growth factor using alginate beads. J. Controlled Release 96, 463 (2004).CrossRefGoogle ScholarPubMed
15.de Vos, P., Faas, M.M., Strand, B., and Calafiore, R.: Alginatebased microcapsules for immunoisolation of pancreatic islets. Biomaterials 27, 5603 (2006).CrossRefGoogle ScholarPubMed
16.Payne, R.G., Yaszemski, M.J., Yasko, A.W., and Mikos, A.G.: Development of an injectable, in situ crosslinkable, degradable polymeric carrier for osteogenic cell populations. Part 1. Encapsulation of marrow stromal osteoblasts in surface cross-linked gelatin microparticles. Biomaterials 23, 4359 (2002).CrossRefGoogle Scholar
17.Perka, C., Arnold, U., Spitzer, R.S., and Lindenhayn, K.: The use of fibrin beads for tissue engineering and subsequential transplantation. Tissue Eng. 7, 359 (2001).CrossRefGoogle ScholarPubMed
18.Gorodetsky, R., Vexler, A., Levdansky, L., and Marx, G.: Fibrin microbeads (FMB) as biodegradable carriers for culturing cells and for accelerating wound healing. Methods Mol. Biol. 238, 11 (2004).Google ScholarPubMed
19.Chan, B.P., Li, C.H., Au-yeung, K.L., Sze, K.Y., and Ngan, A.H.W.: A microplate compression method for elastic modulus measurement of soft and viscoelastic collagen microspheres. Ann. Biomed. Eng. 36, 1254 (2008).CrossRefGoogle ScholarPubMed
20.Ngan, A.H.W. and Tang, B.: Viscoelastic effects during unloading in depth-sensing indentation. J. Mater. Res. 17, 2604 (2002).CrossRefGoogle Scholar
21.Tang, B. and Ngan, A.H.W.: Nanoindentation measurement of mechanical properties of soft solid covered by a thin liquid film. Soft Materials 5, 169 (2007).CrossRefGoogle Scholar
22.Oyen, M.L.: Spherical indentation creep following ramp loading. J. Mater. Res. 20, 2094 (2005).CrossRefGoogle Scholar
23.Oyen, M.L.: Sensitivity of polymer nanoindentation creep measurements to experimental variables. Acta Mater. 55, 3633 (2007).CrossRefGoogle Scholar