Bartels, Sören; Roubíček, Tomáš Thermoviscoplasticity at small strains. (English) Zbl 1153.74011 ZAMM, Z. Angew. Math. Mech. 88, No. 9, 735-754 (2008). Summary: We investigate a viscoelastic solid in Kelvin-Voigt rheology involving plasticity coupled with a heat transfer equation through a temperature-dependent yield stress. No hardening is studied, but the evolution of plastic strain is considered to be rate-dependent. A numerical scheme which is semi-implicit in time and employs lowest-order finite elements on weakly acute triangulations in space is devised and its convergence is proved by careful subsequent limit passage. Computational studies underline robustness and efficiency of the method and illustrate physical effects such as the softening of material due to dissipated energy that causes a rise in temperature and a local decrease in the yield stress. Cited in 12 Documents MSC: 74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity) 74F05 Thermal effects in solid mechanics 74S05 Finite element methods applied to problems in solid mechanics 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs Keywords:Kelvin-Voigt rheology; semi-implicit discretization; convergence; temperature-dependent yield stress; lowest-order finite elements PDFBibTeX XMLCite \textit{S. Bartels} and \textit{T. Roubíček}, ZAMM, Z. Angew. Math. Mech. 88, No. 9, 735--754 (2008; Zbl 1153.74011) Full Text: DOI References: [1] H.-D. Alber, Materials with Memory, Lecture Notes in Mathematics 1682 (Springer, Berlin, 1998). [2] Agelet de Saracibar, Int. J. Plast. (UK) 15 pp 1– (1999) [3] Alberty, Numer. Algorithms 20 pp 117– (1999) [4] Boccardo, J. Funct. Anal. 87 pp 149– (1989) [5] Boccardo, J. Funct. Anal. 147 pp 237– (1997) [6] Canadija, Int. J. Plast. (UK) 20 pp 1851– (2004) · Zbl 1066.74515 · doi:10.1016/j.ijplas.2003.11.016 [7] Carstensen, J. Numer. Math. 10 pp 157– (2002) · Zbl 1099.74544 · doi:10.1515/JNMA.2002.157 [8] P.G. Ciarlet, The Finite Element Method for Elliptic Problems (SIAM, Philadelphia, PA, 2002). · Zbl 0999.65129 [9] Colli, Commun. P. D. E. 15 pp 737– (1990) [10] Dal Maso, Arch. Ration. Mech. Anal. (Germany) 180 pp 237– (2006) [11] Faragó, Appl. Numer. Math. 53 pp 249– (2005) [12] Gröger, Z. Angew. Math. Mech. 60 pp 169– (1980) [13] Hakansson, Int. J. Plast. (UK) 21 pp 1435– (2005) [14] W. Han and D.B. Reddy, Plasticity. Mathematical theory and numerical analysis (Springer, New York, 1999). · Zbl 0926.74001 [15] Han, Numer. Math. 87 pp 283– (2000) [16] Korotov, SIAM J. Numer. Anal. 39 pp 724– (2001) [17] Kratochvíl, J. Appl. Phys. 40 pp 3207– (1969) [18] Krejčí, J. Math. Anal. Appl. 209 pp 25– [19] Krejčí, Appl. Math. 43 pp 173– (1998) [20] Miehe, Arch. Appl. Mech. (Germany) 66 pp 45– (1995) [21] A. Mielke, Evolution of rate-independent systems, in: Handbook of Differential Equations: Evolutionary Differential Equations, edited by C. Dafermos and E. Feireisl (Elsevier, Amsterdam, 2005), pp. 461-559. · Zbl 1120.47062 [22] A. Mielke and T. Roubíček, Numerical Approaches to Rate-independent Processes and Applications in Inelasticity, Preprint No. 1169, WIAS, Berlin, submitted. [23] Nicholson, Acta Mech. 142 pp 207– (2000) [24] Nochetto, Math. Comput. 71 pp 1405– (2002) · Zbl 1001.41011 · doi:10.1090/S0025-5718-01-01369-2 [25] Rittel, Int. J. Fract. 99 pp 199– (1999) [26] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations (Springer, Berlin, 1994). · Zbl 0803.65088 [27] T. Roubíček, Nonlinear partial differential equations with applications (Birkhäuser, Basel, 2005). [28] Roubíček, Q. Appl. Math. (USA) [29] Roubíček, Z. Angew. Math. Phys. 55 pp 159– (2004) [30] Srikanth, Int. J. Numer. Methods Eng. (UK) 45 pp 1569– (1999) [31] V. Thomée, Galerkin finite element methods for parabolic problems (Springer, Berlin, 2006). · Zbl 1105.65102 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.