Abstract
Metal forming processes require wear-resistant tool surfaces in order to ensure a long life cycle of the expensive tools together with a constant high quality of the produced components. Thermal spraying is a relatively widely applied coating technique for the deposit of wear protection coatings. During these coating processes, heterogeneous coatings are deployed at high temperatures followed by quenching where residual stresses occur which strongly influence the performance of the coated tools. The objective of this article is to discuss and apply a thermo-mechanically coupled simulation framework which captures the heterogeneity of the deposited coating material. Therefore, a two-scale finite element framework for the solution of nonlinear thermo-mechanically coupled problems is elaborated and applied to the simulation of thermoviscoplastic material behaviour including nonlinear thermal softening in a geometrically linearised setting. The finite element framework and material model is demonstrated by means of numerical examples.
Similar content being viewed by others
References
Berthelsen R, Tomath D, Denzer R, Menzel A (2016) Finite element simulation of coating-induced heat transfer: application to thermal spraying processes. Meccanica 51(2):291–307
Berthelsen R, Wiederkehr T, Denzer R, Menzel A, Müller H (2014) Efficient simulation of nonlinear heat transfer during thermal spraying of complex workpieces. World J Mech 04(09):289–301
Bertram A, Krawietz A (2012) On the introduction of thermoplasticity. Acta Mech 223(10):2257–2268
Bosco E, Kouznetsova VG, Geers MGD (2015) Multi-scale computational homogenization-localization for propagating discontinuities using X-FEM. Int J Numer Methods Eng 102(3–4):496–527. doi:10.1002/nme.4838
Chatzigeorgiou G, Charalambakis N, Chemisky Y, Meraghni F (2016) Periodic homogenization for fully coupled thermomechanical modeling of dissipative generalized standard materials. Int J Plast 81:18–39
Chatzigeorgiou G, Chemisky Y, Meraghni F (2015) Computational micro to macro transitions for shape memory alloy composites using periodic homogenization. Smart Mater Struct 24(3):035,009
Feyel F, Chaboche JL (2000) FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Comput Methods Appl Mech Eng 183(3–4):309–330
Fleischhauer R, Božić M, Kaliske M (2016) A novel approach to computational homogenization and its application to fully coupled two-scale thermomechanics. Comput Mech 58(5):769–796
Francfort GA (1983) Homogenization and linear thermoelasticity. SIAM J Math Anal 14(4):696–708
Hill R (1972) On constitutive macro-variables for heterogeneous solids at finite strain. Proc R Soc Lond A 326(1565):131–147
Hohenemser K, Prager W (1932) Über die Ansätze der Mechanik isotroper Kontinua. ZAMM 12(4):216–226
Javili A, Saeb S, Steinmann P (2017) Aspects of implementing constant traction boundary conditions in computational homogenization via semi-Dirichlet boundary conditions. Comput Mech 59(1):21–35
Klusemann B, Denzer R, Svendsen B (2012) Microstructure-based modeling of residual stresses in WC-12Co-sprayed coatings. J Therm Spray Tech 21(1):96–107
Kouznetsova V, Brekelmans WAM, Baaijens FPT (2001) An approach to micro-macro modeling of heterogeneous materials. Comput Mech 27(1):37–48
Larsson F, Runesson K, Saroukhani S, Vafadari R (2011) Computational homogenization based on a weak format of micro-periodicity for RVE-problems. Comput Methods Appl Mech Eng 200(1–4):11–26
Liu IS (2002) Continuum mechanics. Springer, Berlin
Luenberger DG, Ye Y (2008) Linear and nonlinear programming. International series in operations research and management science, 3rd edn. Springer, New York
Mercer BS, Mandadapu KK, Papadopoulos P (2015) Novel formulations of microscopic boundary-value problems in continuous multiscale finite element methods. Comput Methods Appl Mech Eng 286:268–292
Miehe C, Schröder J, Schotte J (1999) Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials. Comput Methods Appl Mech Eng 171(3–4):387–418
Oñate E, Owen R (2007) Computational plasticity. No. 7 in Computational methods in applied sciences. Springer, Dordrecht
Özdemir I, Brekelmans WAM, Geers MGD (2008) Computational homogenization for heat conduction in heterogeneous solids. Int J Numer Methods Eng 73(2):185–204
Özdemir I, Brekelmans WAM, Geers MGD (2008) FE\(^2\) computational homogenization for the thermo-mechanical analysis of heterogeneous solids. Comput Methods Appl Mech Eng 198(3–4):602–613
Pamin J, Wcisło B, Kowalczyk-Gajewska K (2017) Gradient-enhanced large strain thermoplasticity with automatic linearization and localization simulations. J Mech Mater Struct 12(1):123–146
Pina JC, Kouznetsova VG, Geers MGD (2015) Thermo-mechanical analyses of heterogeneous materials with a strongly anisotropic phase: The case of cast iron. Int J Solids Struct 63:153–166
Reid ACE, Langer SA, Lua RC, Coffman VR, Haan SI, García RE (2008) Image-based finite element mesh construction for material microstructures. Comput Mater Sci 43(4):989–999
Rocha IBCM, van der Meer FP, Nijssen RPL, Sluys LJ (2017) A multiscale and multiphysics numerical framework for modelling of hygrothermal ageing in laminated composites. Int J Numer Methods Eng. doi:10.1002/nme.5542
Rosakis P, Rosakis AJ, Ravichandran G, Hodowany J (2000) A thermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals. J Mech Phys Solids 48(3):581–607
Schindler S, Mergheim J, Zimmermann M, Aurich JC, Steinmann P (2017) Numerical homogenization of elastic and thermal material properties for metal matrix composites (MMC). Continuum Mech Therm 29(1):51–75
Sengupta A, Papadopoulos P, Taylor RL (2012) A multiscale finite element method for modeling fully coupled thermomechanical problems in solids: Multiscale modeling of coupled thermomechanical response. Int J Numer Methods Eng 91(13):1386–1405
Simo JC (1988) A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part I. Continuum formulation. Comput Methods Appl Mech Eng 66(2):199–219
Simo JC (1998) Numerical analysis and simulation of plasticity. Handb Numer Anal 6:183–499
Simo JC, Hughes TJR (1998) Computational inelasticity. No. v. 7 in Interdisciplinary applied mathematics. Springer, New York
Simo JC, Miehe C (1992) Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Comput Methods Appl Mech Eng 98(1):41–104
Steinmann P, Häsner O (2005) On material interfaces in thermomechanical solids. Arch Appl Mech 75(1):31–41. doi:10.1007/s00419-005-0383-8
Temizer I (2012) On the asymptotic expansion treatment of two-scale finite thermoelasticity. Int J Eng Sci 53:74–84
Temizer I, Wriggers P (2011) Homogenization in finite thermoelasticity. J Mech Phys Solids 59(2):344–372
Thorpe ML, Richter HJ (1992) A pragmatic analysis and comparison of HVOF processes. J Therm Spray Tech 1(2):161–170
Wcisło B, Pamin J (2017) Local and non-local thermomechanical modeling of elastic-plastic materials undergoing large strains: Local and non-local thermomechanical modeling of elastic-plastic materials undergoing large strains. Int J Numer Methods Eng 109(1):102–124
Author information
Authors and Affiliations
Corresponding author
A Consistent linearisation of the thermoviscoplastic model
A Consistent linearisation of the thermoviscoplastic model
In order to obtain the consistently linearised Jacobian of the finite element framework, (25), the total derivative of the stress \({\varvec{{\sigma }}}\), (52), and the mechanical heat production \(\hat{r}\), (58), with respect to strain \({\varvec{{\varepsilon }}}\) at constant temperature \(\theta \) must be determined
where \(\bullet \) represents a scalar-valued contraction. Analogously, the total derivatives of the stress and the mechanical heat production with respect to the temperature at constant strain are defined as
In order to find a relation between \(\text {d}\tilde{\mathbf {z}}\) and \(\text {d}{\varvec{{\varepsilon }}}\) and, respectively, between \(\text {d}\tilde{\mathbf {z}}\) and \(\text {d}\theta \), the linearisation of the local residual equation (68) at constant temperature as well as at constant strain must be carried out
In the above equations, the expression (69) is inserted. With this at hand, Eqs. (A.1)–(A.4) can be rewritten as
such that, subsequently, merely the arising partial derivatives are required to evaluate the given expressions. The partial derivative of Eq. (52) with respect to the strain is
with
wherein \({\varvec{{\mathsf {I}}}}^\mathrm{dev}\) denotes the deviatoric and symmetric fourth order identity tensor, it follows
Furthermore, the partial derivative of Eq. (52) with respect to the temperature is given by
Finally, the partial derivative of Eq. (52) with respect to \(\tilde{\mathbf {z}}\) is
The partial derivative of the mechanical heat production, which is given by Eq. (58), with respect to the strain is
The partial derivative of the mechanical heat production with respect to the temperature is
wherein the abbreviation
is introduced. The partial derivative of the plastic mechanical heat production with respect to \(\tilde{\mathbf {z}}\) is
The partial derivatives of the local residual (68) with respect to the strain and as well to the temperature are
With the above partial derivatives, the tangent moduli (A.7)-(A.10) finally read
Furthermore, the total derivative of the heat flux (59) with respect to the temperature gradient \(\nabla _{{\varvec{{x}}}}\theta \) reads
Details regarding the total derivative of the surface heat flux contribution, cf. Eq. (20)\(_4\), can be found in [1].
Rights and permissions
About this article
Cite this article
Berthelsen, R., Denzer, R., Oppermann, P. et al. Computational homogenisation for thermoviscoplasticity: application to thermally sprayed coatings. Comput Mech 60, 739–766 (2017). https://doi.org/10.1007/s00466-017-1436-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-017-1436-x