## A variational model of elasto-plastic behavior of materials.(English)Zbl 07464181

Summary: We present a mesoscale phase field model of the elasto-plastic behavior of materials as an extension of the model of plastic slip put forth by Ambrosio et al. (J. Elast. 110:201-235, 2013). In the proposed model, we consider a new strain energy density decomposition based on the degradation of the shear modulus of the material and a new formulation of the surface plastic energy term that accounts for the energy that is consumed when gliding occurs across slip bands. The resulting constitutive model leads to the introduction of an additional strain contribution that we term the gliding strain tensor. With this formulation, the phase field distribution captures the nature of the activation process of slip planes and is capable of characterizing a wide range of plastic behavior, from very diffuse to highly localized, as evidenced by several two-dimensional numerical results.
Our results also show that this mesoscopic model is able to capture the Hall-Petch effect, wherein the strength of some polycrystalline materials is inversely proportional to the square root of the average grain size. This means that the size effect does not necessarily need to be explained with models at microscopic scale but can be captured by mesoscopic models. The proposed model is also consistent with the classical theory of plasticity in terms of the postulate of incompressibility and the von Mises yield criterion.

### MSC:

 35Qxx Partial differential equations of mathematical physics and other areas of application 35R35 Free boundary problems for PDEs 74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) 74G65 Energy minimization in equilibrium problems in solid mechanics 74R20 Anelastic fracture and damage

FEniCS
Full Text:

### References:

 [1] Alessi, R.; Freddi, F.; Mingazzi, L., Phase-field numerical strategies for deviatoric driven fractures, Comput. Methods Appl. Mech. Eng., 359 (2019) · Zbl 1441.74198 [2] Alnæs, M. S.; Blechta, J.; Hake, J.; Johansson, A.; Kehlet, B.; Logg, A.; Richardson, C.; Ring, J.; Rognes, M. E.; Wells, G. N., The FEniCS project version 1.5, Arch. Numer. Softw., 3, 100, 9-23 (2015) [3] Ambrosio, L.; Lemenant, A.; Royer-Carfagni, G., A variational model for plastic slip and its regularization via $${\Gamma }$$-convergence, J. Elast., 110, 201-235 (2013) · Zbl 1326.74112 [4] ASTM E1820-20b: Standard Test Method for Measurement of Fracture Toughness. Standard, ASTM International, West Conshohocken, PA (2020). www.astm.org [5] Braides, A., $$\Gamma$$-Convergence for Beginners (2002) · Zbl 1198.49001 [6] Conti, S.; Focardi, M.; Iurlano, F., Phase field approximation of cohesive fracture models, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33, 1033-1067 (2015) · Zbl 1345.49012 [7] Cottrell, A. H.; Bilby, B. A., Dislocation theory of yielding and strain ageing of iron, Proc. Phys. Soc., 62, 49-62 (1948) [8] Freddi, F.; Royer-Carfagni, G., Plastic flow as an energy minimization problem. Numerical experiments, J. Elast., 116, 53-74 (2014) · Zbl 1298.74092 [9] Freddi, F.; Royer-Carfagni, G., Phase-field slip-line theory of plasticity, J. Mech. Phys. Solids, 94, 257-272 (2016) [10] Gilman, J. J.; Johnston, W. G., Dislocations in Lithium Fluoride Crystals, 147-222 (1962) [11] Hahn, G., A model for yielding with special reference to the yield-point phenomena of iron and related bcc metals, Acta Metall., 10, 727-738 (1962) [12] Hall, E., The deformation and ageing of mild steel: III discussion of results, Proc. Phys. Soc. B, 64, 747-753 (1951) [13] Hatherly, M.; Malin, A., Shear bands in deformed metals, Scr. Metall., 18, 449-454 (1984) [14] Humphreys, J.; Hatherly, M., Recrystallization and Related Annealing Phenomena (2004), Amsterdam: Elsevier, Amsterdam [15] Jia, D.; Ramesh, K.; Ma, E., Effects of nanocrystalline and ultrafine grain sizes on constitutive behavior and shear bands in iron, Acta Mater., 51, 3495-3509 (2003) [16] Li, Y.; Bushby, A. J.; Dunstan, D. J., The Hall-Petch effect as a manifestation of the general size effect, Proc. R. Soc. A, 472 (2016) [17] Logg, A.; Mardal, K. A.; Wells, G. N., Automated Solution of Differential Equations by the Finite Element Method (2012), Berlin: Springer, Berlin · Zbl 1247.65105 [18] Mumford, D.; Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems, Commun. Pure Appl. Math., 42, 577-685 (1989) · Zbl 0691.49036 [19] Petch, N., The cleavage strength of polycrystals, J. Iron Steel Inst., 174, 25-28 (1953) [20] Sylwestrowicz, W.; Hall, E., The deformation and ageing of mild steel, Proc. Phys. Soc., 64, 495-502 (1951) [21] Verhoosel, C.; de Borst, R., A phase-field model for cohesive fracture, Int. J. Numer. Methods Eng., 96, 43-62 (2013) · Zbl 1352.74029 [22] Zhang, Y. T.; Qiao, J. L.; Ao, T., Strain softening of materials and Lüders-type deformations, Model. Simul. Mater. Sci. Eng., 15, 147-156 (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.