Mathematical modeling of nanodispersed hardening of FCC materials

doi:10.1007/s40195-018-0754-0

Abstract

Abstract:

The plastic deformation of the pipe made of Cu-based alloy hardened by incoherent nanoparticles and subjected to the uniform internal pressure was investigated. The limits of elastic and plastic resistance are determined. The insignificant excess in the limit of the elastic resistance enables the plastic deformation in the most part of the pipe wall. The densities of shear-forming dislocations and prismatic dislocation loops are higher in alloys strengthened with coarse particles than in alloys strengthened with fine particles. At small distances between the strengthening particles, this effect is the most pronounced.

Key words: Deformation, Dislocation, Dispersion-strengthening, Nanoparticles, Hardening, Stress, Strain, Elastic-plastic deformation, Pipe

Oleg Matvienko, Olga Daneyko, Tatyana Kovalevskaya. Mathematical modeling of nanodispersed hardening of FCC materials[J]. Acta Metallurgica Sinica (English Letters), 2018, 31(12): 1297-1304.

Figures/Tables 8

Fig. 1 Strain hardening curves (a-c) and the dependence of the density of shear-forming dislocations (d-f), prismatic loops (g-i), dipoles (j-l), vacancy concentration (m-o) on the degree of deformation for a copper-based alloy at different deformation temperatures. Distance between particles: solid line is 40 nm; dashed line is 100 nm. Diameter of particles (nm): 1, 4-1; 2, 5-5; 3, 6-10. The strain rate is 10-2 s-1

Table 1 Parameters of the constitutive relations

Parameters	\(\tau_{0}\) (MPa)	\(\tau_{1}\) (MPa)	\(a_{1}\)
\(\varLambda_{\text{p}} = 40\,\,\,{\text{nm}}\) \(\delta = 1 \,\,{\text{nm}}\)	341.62	163.65	0.027
\(\varLambda_{\text{p}} = 40\,\,{\text{nm}}\) \(\delta = 5\,\,{\text{nm}}\)	380.93	362.12	0.024
\(\varLambda_{\text{p}} = 40\,\,{\text{nm}}\) \(\delta = 10\,\,{\text{nm}}\)	443.88	511.39	0.023
\(\varLambda_{\text{p}} = 100\,\,{\text{nm}}\) \(\delta = 1\,\,{\text{nm}}\)	138.97	66.54	0.034
\(\varLambda_{\text{p}} = 100\,\,{\text{nm}}\) \(\delta = 5\,\,{\text{nm}}\)	145.20	147.26	0.028
\(\varLambda_{\text{p}} = 100\,\,{\text{nm}}\) \(\delta = 10\,\,{\text{nm}}\)	153.13	206.91	0.026

Fig. 2 Radial distribution of normal radial stress \(\sigma_{rr} :\,\delta = 1\,\,{\text{nm, }}\Lambda_{\text{p}} = 40\,\,{\text{nm}};\)1—p?=?15.88 MPa (Rpl = ?0.1 m), 2—16.73 (0.10125), 3—17.50 (0.1025); 4—18.18 (0.10375), 5—18.76(0.105)

Fig. 3 Radial distribution of tangential radial stress \(\sigma_{\varphi \varphi } :\,\delta = 1\,\,{\text{nm, }}\Lambda_{\text{p}} = 40\,\,{\text{nm}};\) 1—p?=?15.88 MPa (Rpl? =? 0.1 m), 2—16.73 (0.10125), 3—17.50 (0.1025), 4—18.18 (0.10375), 5—18.76(0.105)

Fig. 4 Radial distribution of deformation intensity \(a:\,\delta = 1\,\,{\text{nm, }}\Lambda_{\text{p}} = 40\,\,{\text{nm}};\) 1—p?=?15.88 MPa (Rpl=?0.1 m), 2—16.73 (0.10125), 3—17.50 (0.1025), 4—18.18 (0.10375), 5—18.76 (0.105)

Fig. 5 Radial distribution of shear-forming dislocations density: a 1—p = 19.98 MPa, 2—20.03, 3—20.49, 4—21.62; b 1—23.73, 2—24.76, 3—25.84, 4—26.98; c 1—7.45, 2—7.47, 3—7.50, 4—7.55; d 1—7.91, 2—7.99, 3—8.01, 4—8.20

Fig. 6 Radial distribution of prismatic dislocation loops density: a 1—p?=?19.98 MPa, 2—20.03, 3—20.49, 4—21.62; b 1—23.73, 2—24.76, 3—25.84, 4—26.98; c 1—7.45, 2 7.47, 3—7.50, 4—7.55; d 1—7.91, 2—7.99, 3—8.01, 4—8.20

References

[1]	E. Orowan, in Proceedings of Symposium on Internal Stresses in Metals and Alloys (Institute of Metals, 1948), pp. 451-454
[2]	V. Arnhold, K. Hummert, in New Materials by Mechanical Alloying Techniques,ed. by E.Arzt, L. Schultz(DGM Informationsgeselischaft Verlag, Oberursel, 1989), p. 263
[3]	J. R?sler, R. Joos, E. Arzt, Metall. Trans. A 23, 5 (1992)
[4]	J.H. Weber, R.D. Schelleng, in Dispersion Strengthened Aluminum Alloys, ed. by Y.-W. Kim, W.M. Griffith(TMS, Warrendale, 1988), pp. 467-482
[5]	F. Dobes, K. Kucharova, A. Orlova, K. Milicka, J. Cadek, Acta Metall.Mater. 42, 1447(1994)
[6]	M.F. Ashby, Philos. Mag. 14, 132(1966)
[7]	M.F. Ashby, K. Johnson, Materials and Design, the Art and Science of Materials Selection in Product Design (Butterworth Heinemann, Oxford, 2002)
[8]	R. Ebeling, M.F. Ashby, Philos. Mag. 13, 124(1966)
[9]	P.B. Hirsch, F.J. Humphreys, in Physics of Strength and Plasticity, ed. by A.S. Argon (MIT Press, Cambridge, 1969), p. 189
[10]	P.M. Hazzledine, P.B. Hirsch, Philos. Mag. 30, 6(1974)
[11]	F.J. Humphreys, P.B. Hirsch, Philos. Mag. 34, 4(1976)
[12]	F.J. Humphreys, A.T. Stewart, Surf. Sci. 31, (1972), pp. 389-421
[13]	F.J. Hymphreys, J.W. Martin, Philos. Mag. 16, 143(1967)
[14]	F.J. Hymphreys, P.B. Hirsch, Proc. R. Soc. Lond. A 318, 1532 (1970)
[15]	J.D. Embury, Metall. Trans. A 16, 12 (1985)
[16]	T. Mori, T. Mura, Mater. Sci. Eng. 26, 1(1976)
[17]	I. Kr?pfl, O. V?hringer, E. Macherauch, Mech. Time Depend. Mater. 3, 1(1999)
[18]	P. Stobrawa, Z.M. Rdzawski, W. G?uchowski, J. Achieve. Mater.Manuf. Eng. 20, 1(2007)
[19]	R.A. Espinoza, R.H. Palma, A.O. Sepúlveda, V. Fuenzalida, G. Solórzano, A. Craievich, D. Smith, T. Fujita, M. López, Mater. Sci. Eng. A 498(1-2), 397 (2007)
[20]	A.A. Zaitsev, V.V. Kurbatkina, E.A. Levashov, Rus. J.Non Ferrous Met. 49, 2(2008)
[21]	N.V. Novikov, G.P. Bogatyreva, R.K. Bogdanov, G.D. Il’nitskaya, M. Isonkin, M.A. Marinich, V.N. Tkach, M.A. Tsysar’, N. Zaitseva, J. Superhard Mater. 33, 4(2011)
[22]	L.I. Tuchinskii, Kompozitsionnye materialy, poluchaemye metodom propitki (Composition Materials Obtained by Impregnation Method) (Metallurgiya, Moscow, 1986)
[23]	K.I. Portnoi, B.N. Babich, Dispersno-uprochnennye materialy (Age-Hardened Materials) (Metallurgiya, Moscow, 1974)
[24]	G. Cao, X. Chen, J.W. Kysar, D. Lee, Y.X. Gan, Mech. Res. Commun. 34, 3(2007)
[25]	N.A. Kulaeva, O.I. Daneyko, T.A. Kovalevskaya, V.A. Starenchenko, Vest. Tambov. Univers. Ser. Estestvennye i tekhnicheskie nauki (Bulletin of Tambov University. A series of natural and technical sciences), 21, 3(2016)
[26]	O.I. Daneyko, T.A. Kovalevskaya, S.N. Kolupaeva, N.A. Kulaeva, M.E. Semenov, Rus. Phys. J. 54, 9(2012)
[27]	T.A. Kovalevskaya, O.I. Daneyko, S.N. Kolupaeva, V.A. Starenchenko, Bull. Rus. Acad. Sci. Phys. 67, 6(2003)
[28]	T.A. Kovalevskaya, O.I. Daneyko, N.A. Kulaeva, S.N. Kolupaeva, Rus. Phys. J. 58, 3(2015)
[29]	T.A. Kovalevskaya, O.I. Daneyko, S.N. Kolupaeva, Deformatsiya i Razrushenie Materialov (J. of Deformation and Destruction of Materials). 2, 1(2006), pp. 29-35
[30]	O.V. Matvienko, O.I. Daneyko, T.A. Kovalevskaya, Rus. Phys. J. 60, 7(2017)
[31]	O.V. Matvienko, O.I. Daneyko, T.A. Kovalevskaya, Rus. Phys. J. 60, 2(2017)
[32]	O.V. Matvienko, O.I. Daneyko, T.A. Kovalevskaya, Rus. Phys. J. 60, 4(2017)
[33]	L.E. Popov, V.S. Kobytev, T.A. Kovalevskaya, Rus. Phys. J. 25, 6(1982)
[34]	J.R. Barber, Elasticity, 3rd edn. (Springer, Dordrecht, 2010)
[35]	S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, 3rd edn. (McGraw Hill, New York, 2010)
[36]	J. Lubliner, Plasticity Theory (Dover Publications Inc., Mineola, 2008)