Top

Acta Mechanica

26-04-2024 | Original Paper

New expanding cavity model for conical indentation and its application to determine an intrinsic length scale of polymeric materials

Author: Georgiy M. Sevastyanov

Published in: Acta Mechanica

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

New expanding spherical cavity model (ECM) for conical indentation is proposed. For polymeric materials description, the model incorporates isotropic non-monotonic strain hardening. For capturing the indentation size effect (ISE), the model incorporates the strain gradient dependence in yield strength based on lower-order strain gradient plasticity assumptions. Specifically, the forward gradient of the equivalent (accumulated) plastic strain is utilized as a non-local part of the yield strength. To predict the indentation depth-dependent hardness based on the proposed model, it is sufficient to numerically integrate one nonlinear ODE of the first order, and then calculate the definite integral. For the local perfect plasticity model, the hardness is obtained as an analytical expression that differs from known ECMs. The hardness estimate obtained numerically using the proposed model is compared with the experimental ISE data for polycarbonate (PC) and polymethyl methacrylate (PMMA). For the local perfect plasticity model, the formula obtained in the study is compared with the experimental data on the hardness of preliminary work-hardened materials. In both cases, the model shows good agreement with the experimental data. Fitting the experimental data on ISE, we found that intrinsic length scale of PMMA should be near 3 microns and near 9 microns for PC.

Available only for authorised users

Most of the data are given for metallic materials; however, the leftmost experimental points correspond to polymeric materials. For them, it is not clear what is meant by the state of full-hardening, which is indicated in [46] (see, for example, stress–strain curve for PMMA on Fig. 8). In [34] it is indicated that the pre-strain of the samples was in limits 1.1 and 1.5.

In contrast to viscoplastic models, in which the plastic strain rate vanishes at the elastic–plastic boundary and increases rapidly with deepening into the plastic region up to values comparable to those predicted by rate-independent plasticity.

Aifantis, E.C.: On the microstructural origin of certain inelastic models. J. Eng. Mater. T ASME 106(4), 326–330 (1984). https://doi.org/10.1115/1.3225725CrossRef

Aifantis, E.C.: The physics of plastic deformation. Int. J. Plast. 3(3), 211–247 (1987). https://doi.org/10.1016/0749-6419(87)90021-0CrossRef

Aifantis, E.C.: Update on a class of gradient theories. Mech. Mater. 35(3–6), 259–280 (2003). https://doi.org/10.1016/S0167-6636(02)00278-8CrossRef

Alisafaei, F., Han, C.-S.: Indentation depth dependent mechanical behavior in polymers. Adv. Cond. Matter Phys. 2015, 391579 (2015). https://doi.org/10.1155/2015/391579CrossRef

Ames, N.M., Srivastava, V., Chester, S.A., Anand, L.: A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part II: Applications. Int. J. Plast. 25(8), 1495–1539 (2009). https://doi.org/10.1016/j.ijplas.2008.11.005CrossRef

Anand, L.: On H. Hencky’s approximate strain-energy function for moderate deformations. J. Appl. Mech. 46(1), 78–82 (1979). https://doi.org/10.1115/1.3424532MathSciNetCrossRef

Atkins, A.G., Tabor, D.: Plastic indentation in metals with cones. J. Mech. Phys. Solids 13(3), 149–164 (1965). https://doi.org/10.1016/0022-5096(65)90018-9CrossRef

Balta Calleja, F.J., Flores, A., Michler, G.H.: Microindentation studies at the near surface of glassy polymers: Influence of molecular weight. J. Appl. Polym. Sci. 93, 1951–1956 (2004). https://doi.org/10.1002/app.20665CrossRef

Bhattacharya, A.K., Nix, W.D.: Finite element analysis of cone indentation. Int. J. Solids Struct. 27(8), 1047–1058 (1991). https://doi.org/10.1016/0020-7683(91)90100-TCrossRef

10.

Bilby, B.A., Lardner, L.R.T., Stroh, A.N.: Continuous distributions of dislocations and the theory of plasticity. In: Actes du IXe congres international de mecanique appliquee (Bruxelles, 1956), V. 8, pp. 35–44 (1957)

11.

Bishop, R.F., Hill, R., Mott, N.F.: The theory of indentation and hardness tests. Proc. Phys. Soc. 57(321), 147–159 (1945). https://doi.org/10.1088/0959-5309/57/3/301CrossRef

12.

Briscoe, B.J., Fiori, L., Pelillo, E.: Nano-indentation of polymeric surfaces. J. Phys. D Appl. Phys. 31, 2395–2405 (1998). https://doi.org/10.1088/0022-3727/31/19/006CrossRef

13.

Cheng, L., Guo, T.F.: Void interaction and coalescence in polymeric materials. Int. J. Solids Struct. 44(6), 1787–1808 (2007). https://doi.org/10.1016/j.ijsolstr.2006.08.007CrossRef

14.

Chong, A.C.M., Lam, D.C.C.: Strain gradient plasticity effect in indentation hardness of polymers. J. Mater. Res. 14(10), 4103–4110 (1999). https://doi.org/10.1557/JMR.1999.0554CrossRef

15.

Dorogoy, A., Rittel, D., Brill, A.: Experimentation and modeling of inclined ballistic impact in thick polycarbonate plates. Int. J. Impact Eng 38(10), 804–814 (2011). https://doi.org/10.1016/j.ijimpeng.2011.05.001CrossRef

16.

Dugdale, D.S.: Cone indentation experiments. J. Mech. Phys. Solids 2(4), 265–277 (1954). https://doi.org/10.1016/0022-5096(54)90017-4CrossRef

17.

Durban, D., Baruch, M.: On the problem of a spherical cavity in an infinite elasto-plastic medium. J. Appl. Mech. 43(4), 633–638 (1976). https://doi.org/10.1115/1.3423946CrossRef

18.

Durban, D., Fleck, N.A.: Singular plastic fields in steady penetration of a rigid cone. J. Appl. Mech. 59(4), 706–710 (1992). https://doi.org/10.1115/1.2894032CrossRef

19.

Durban, D., Masri, R.: Dynamic spherical cavity expansion in a pressure sensitive elastoplastic medium. Int. J. Solids Struct. 41(20), 5697–5716 (2004). https://doi.org/10.1016/j.ijsolstr.2004.03.009CrossRef

20.

Durban, D., Masri, R.: Conical indentation of strain-hardening solids. Eur. J. Mech. A Solid 27(2), 210–221 (2008). https://doi.org/10.1016/j.euromechsol.2007.05.007CrossRef

21.

Evans, P.D.: The hardness and abrasion of polymers. PhD dissertation. Department of Chemical Engineering and Chemical Technology. Imperial College London (1987)

22.

Felder, E., Ramond-Angelelis, C.: Mechanical analysis of indentation experiments with a conical indenter. Philos. Mag. 86(33–55), 5219–5230 (2006). https://doi.org/10.1080/14786430600589071CrossRef

23.

Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42(2), 475–487 (1994). https://doi.org/10.1016/0956-7151(94)90502-9CrossRef

24.

Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. Adv. Appl. Mech. 33, 295–361 (1997). https://doi.org/10.1016/S0065-2156(08)70388-0CrossRef

25.

Flores, A., Balta Calleja, F.J., Attenburrow, G.E., Bassett, D.C.: Microhardness studies of chain-extended PE: III. Correlation with yield stress and elastic modulus. Polymer 41(14), 5431–5435 (2000). https://doi.org/10.1016/S0032-3861(99)00755-7CrossRef

26.

Gao, X.L.: New expanding cavity model for indentation hardness including strain-hardening and indentation size effects. J. Mater. Res. 21(5), 1317–1326 (2006). https://doi.org/10.1557/jmr.2006.0158CrossRef

27.

Gao, X.-L.: An expanding cavity model incorporating strain-hardening and indentation size effects. Int. J. Solids Struct. 43(21), 6615–6629 (2006). https://doi.org/10.1016/j.ijsolstr.2006.01.008CrossRef

28.

Gao, X.-L., Jing, X.N., Subhash, G.: Two new expanding cavity models for indentation deformations of elastic strain-hardening materials. Int. J. Solids Struct. 43(7–8), 2193–2208 (2006). https://doi.org/10.1016/j.ijsolstr.2005.03.062CrossRef

29.

Green, A.E., Shield, R.T.: Finite elastic deformation of incompressible isotropic bodies. Proc. R. Soc. Lond. A 202, 407–419 (1950). https://doi.org/10.1098/rspa.1950.0109MathSciNetCrossRef

30.

Han, C.S., Nikolov, S.: Indentation size effects in polymers and related rotation gradients. J. Mater. Res. 22, 1662–1672 (2007). https://doi.org/10.1557/JMR.2007.0197CrossRef

31.

Haward, R.N., Thackray, G.: The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics. Proc. R. Soc. Lond. A 302(1471), 453–472 (1967). https://doi.org/10.1098/rspa.1968.0029CrossRef

32.

Hernot, X., Bartier, O.: An expanding cavity model incorporating pile-up and sink-in effects. J. Mater. Res. 27(1), 132–140 (2012). https://doi.org/10.1557/jmr.2011.394CrossRef

33.

Hill, R.: The Mathematical Theory of Plasticity. Clarendon Press, Oxford (1950)

34.

Hirst, W., Howse, M.G.J.W.: The indentation of materials by wedges. Proc. R. Soc. Lond. A 311, 429–444 (1969). https://doi.org/10.1098/rspa.1969.0126CrossRef

35.

Huang, Y., Qu, S., Hwang, K.C., Li, M., Gao, H.: A conventional theory of mechanism-based strain gradient plasticity. Int. J. Plast. 20(4–5), 753–782 (2004). https://doi.org/10.1016/j.ijplas.2003.08.002CrossRef

36.

Johnson, K.L.: The correlation of indentation experiments. J. Mech. Phys. Solids 18(2), 115–126 (1970). https://doi.org/10.1016/0022-5096(70)90029-3CrossRef

37.

Johnson, K.L.: Normal contact of inelastic solids. In: Contact Mechanics, Cambridge University Press, Cambridge, pp. 153–201 (1985). https://doi.org/10.1017/CBO9781139171731.007

38.

Kröner, E.: Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Arch. Ration. Mech. An. 4(1), 273–334 (1959). https://doi.org/10.1007/BF00281393MathSciNetCrossRef

39.

Lam, D.C.C., Chong, A.C.M.: Indentation model and strain gradient plasticity law for glassy polymers. J. Mater. Res. 14(9), 3784–3788 (1999). https://doi.org/10.1557/JMR.1999.0512CrossRef

40.

Lee, E., Liu, D.: Finite-strain elastic-plastic theory with application to plane-wave analysis. J. Appl. Phys. 38(1), 19–27 (1967). https://doi.org/10.1063/1.1708953CrossRef

41.

Lee, E.H.: Elastic-plastic deformation at finite strains. J. Appl. Mech. 36(1), 1–6 (1969). https://doi.org/10.1115/1.3564580MathSciNetCrossRef

42.

Levitas, V.I.: Large Deformation of Materials with Complex Rheological Properties at Normal and High Pressure. Nova Science Publishers, New York (1996)

43.

Liu, D., Dunstan, D.J.: Material length scale of strain gradient plasticity: a physical interpretation. Int. J. Plast. 98, 156–174 (2017). https://doi.org/10.1016/j.ijplas.2017.07.007CrossRef

44.

Lockett, F.J.: Indentation of a rigid/plastic material by a conical indenter. J. Mech. Phys. Solids 11(5), 345–355 (1963). https://doi.org/10.1016/0022-5096(63)90035-8CrossRef

45.

Ma, Q., Clarke, D.R.: Size dependent hardness of silver single crystals. J. Mater. Res. 10(4), 853–863 (1995). https://doi.org/10.1557/JMR.1995.0853CrossRef

46.

Marsh, D.M.: Plastic flow in glass. Proc. R. Soc. Lond. A 279, 420–435 (1964). https://doi.org/10.1098/rspa.1964.0114CrossRef

47.

Mata, M., Casals, O., Alcala, J.: The plastic zone size in indentation experiments: the analogy with the expansion of a spherical cavity. Int. J. Solids Struct. 43(20), 5994–6013 (2006). https://doi.org/10.1016/j.ijsolstr.2005.07.002CrossRef

48.

Meijer, H.E.H., Govaert, L.E.: Mechanical performance of polymer systems: the relation between structure and properties. Prog. Polym. Sci. 30, 915–938 (2005). https://doi.org/10.1016/j.progpolymsci.2005.06.009CrossRef

49.

Naghdi, P.M.: Stresses and displacements in an elastic-plastic wedge. J. Appl. Mech. 24(1), 98–104 (1957). https://doi.org/10.1115/1.4011452MathSciNetCrossRef

50.

Prager, W., Hodge, P.G.: The Theory of Perfectly Plastic Solids. Wiley, New York (1951)

51.

Richeton, J., Ahzi, S., Vecchio, K.S., Jiang, F.C., Adharapurapu, R.R.: Influence of temperature and strain rate on the mechanical behavior of three amorphous polymers: characterization and modeling of the compressive yield stress. Int. J. Solids Struct. 43(7–8), 2318–2335 (2006). https://doi.org/10.1016/j.ijsolstr.2005.06.040CrossRef

52.

Sakharova, N.A., Fernandes, J.V., Antunes, J.M., Oliveira, M.C.: Comparison between Berkovich, Vickers and conical indentation tests: a three-dimensional numerical simulation study. Int. J. Solids Struct. 46(5), 1095–1104 (2009). https://doi.org/10.1016/j.ijsolstr.2008.10.032CrossRef

53.

Santos, T., Srivastava, A., Rodriguez-Martinez, J.A.: The combined effect of size, inertia and porosity on the indentation response of ductile materials. Mech. Mater. 153, 103674 (2021). https://doi.org/10.1016/j.mechmat.2020.103674CrossRef

54.

Shapiro, G.S.: Elastic-plastic equilibrium of a wedge and discontinuous solutions in the theory of plasticity. J. Appl. Math. Mech. 16(1), 101–106 (1952). (in Russian)MathSciNet

55.

Sevastyanov, G.M.: Adiabatic heating effect in elastic-plastic contraction / expansion of spherical cavity in isotropic incompressible material. Eur. J. Mech. A Solid 87, 104223 (2021). https://doi.org/10.1016/j.euromechsol.2021.104223MathSciNetCrossRef

56.

Sneddon, I.N.: The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3(1), 47–57 (1965). https://doi.org/10.1016/0020-7225(65)90019-4MathSciNetCrossRef

57.

Sokolovsky, V.V.: Theory of Plasticity. Vischaya shkola, Moscow (1969). (in Russian)

58.

Stolken, J.S., Evans, A.G.: A microbend test method for measuring the plasticity length scale. Acta Mater. 46(14), 5109–5115 (1998). https://doi.org/10.1016/S1359-6454(98)00153-0CrossRef

59.

Studman, C.J., Moore, M.A., Jones, S.E.: On the correlation of indentation experiments. J. Phys. D Appl. Phys. 10(6), 949–956 (1977). https://doi.org/10.1088/0022-3727/10/6/019CrossRef

60.

Tabor, D.: Hardness of Metals. Clarendon Press, Oxford (1951)

61.

Turnbull, A., White, D.: Nanoindentation and microindentation of weathered unplasticised poly-vinyl chloride (UPVC). J. Mater. Sci. 31, 4189–4198 (1996). https://doi.org/10.1007/BF00356438CrossRef

62.

Tvergaard, V., Needleman, A.: Polymer indentation: Numerical analysis and comparison with a spherical cavity model. J. Mech. Phys. Solids 59(9), 1669–1684 (2011). https://doi.org/10.1016/j.jmps.2011.06.006CrossRef

63.

Voyiadjis, G.Z., Song, Y.: Strain gradient continuum plasticity theories: theoretical, numerical and experimental investigations. Int J Plasticity 121, 21–75 (2019). https://doi.org/10.1016/j.ijplas.2019.03.002CrossRef

64.

Weiss A., Durban D.: Cavitation theory applied to polycarbonate ballistic response. In: 28th International Symposium on Ballistics (2014)

65.

Wright, S.C., Huang, Y., Fleck, N.A.: Deep penetration of polycarbonate by a cylindrical punch. Mech. Mater. 13(4), 277–284 (1992). https://doi.org/10.1016/0167-6636(92)90020-ECrossRef

66.

Wu, P.D., van der Giessen, E.: On improved network models for rubber elasticity and their applications to orientation hardening in glassy polymers. J. Mech. Phys. Solids 41, 427–456 (1993). https://doi.org/10.1016/0022-5096(93)90043-FCrossRef

67.

Yuan, H., Chen, J.: Analysis of size effects based on a symmetric lower-order gradient plasticity model. Comp. Mater. Sci. 19(1–4), 143–157 (2000). https://doi.org/10.1016/S0927-0256(00)00149-XCrossRef

68.

Yun, G., Qin, J., Huang, Y., Hwang, K.C.: A study of lower-order strain gradient plasticity theories by the method of characteristics. Eur. J. Mech. A Solid 23(3), 387–394 (2004). https://doi.org/10.1016/j.euromechsol.2004.02.003CrossRef

69.

Zhang, P., Li, S.X., Zhang, Z.F.: General relationship between strength and hardness. Mat. Sci. Eng. A Struct. 529, 62–73 (2011). https://doi.org/10.1016/j.msea.2011.08.061CrossRef

Title: New expanding cavity model for conical indentation and its application to determine an intrinsic length scale of polymeric materials
Author: Georgiy M. Sevastyanov
Publication date: 26-04-2024
Publisher: Springer Vienna
Published in: Acta Mechanica
Print ISSN: 0001-5970
Electronic ISSN: 1619-6937
DOI: https://doi.org/10.1007/s00707-024-03921-2

Springer Professional

New expanding cavity model for conical indentation and its application to determine an intrinsic length scale of polymeric materials

Abstract

Premium Partners

Springer Professional

Abstract

Please log in to get access to your license.

Premium Partners