Abstract
Additive Manufacturing involves the interaction of different physical processes. In some printing technologies, an additional chemical reaction takes place. A thermomechanical process with phase change is present if the 3D-print is based on melting. In the case of photopolymerization, an external heat source accelerates the hardening of a liquid due to a chemical reaction. Hence, mechanical, thermal, and chemical fields need to be coupled.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. André, P. Wriggers, Thermo-mechanical behaviour of rubber materials during vulcanization. Int. J. Solids Struct. 42(16–17), 4758–4778 (2005)
H.D. Baehr, S. Kabelac, Thermodynamik, 16th edn. (Springer, 2016)
C. Bonacina, G. Comini, A. Fasano, M. Primicerio, Numerical solution of phase-change problems. Int. J. Heat Mass Trans. 16(10), 1825–1832 (1973)
A.N. Brooks, T.J.R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1–3), 199–259 (1982)
H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids (Oxford Science Publications, 1959)
M. Chiumenti, M. Cervera, A. Salmi, C.A. De Saracibar, N. Dialami, K. Matsui, Finite element modeling of multi-pass welding and shaped metal deposition processes. Comput. Methods Appl. Mech. Eng. 199(37–40), 2343–2359 (2010)
B.D. Coleman, M.E. Gurtin, Thermodynamics with internal state variables. J. Chem. Phys. 47(2), 597–613 (1967)
C.A. de Saracibar, M. Cervera, M. Chiumenti, On the constitutive modeling of coupled thermomechanical phase-change problems. Int. J. Plast. 17(12), 1565–1622 (2001)
A.C. Eringen, C.B. Kafadar, Polar field theories, in Continuum Physics, ed. by A.C. Eringen. Polar and Nonlocal Field Theories, vol. IV (Academic Press, New York, 1976), pp. 1–74
N.R. Eyres, D.R. Hartree, J. Ingham, R.J. Sarjant, J.B. Wagstaff, The calculation of variable heat flow in solids. Phil. Trans. R. Soc. Lond. A 240(813), 1–57 (1946)
M.E. Glicksman, Principles of Solidification: An Introduction to Modern Casting and Crystal Growth Concepts (Springer, New York, 2011)
P. Haupt, Continuum Mechanics and Theory of Materials, 2nd edn. (Springer, Berlin, Heidelberg, Wien, 2002)
G.A. Holzapfel, Nonlinear Solid Mechanics A Continuum Approach for Engineering (Wiley, Chichester, 2000)
T.R.J. Hughes, The Finite Element Method (Prentice Hall, Englewood Cliffs, NJ, 1987)
H. Ji, D. Chopp, J.E. Dolbow, A hybrid extended finite element/level set method for modeling phase transformations. Int. J. Num. Methods Eng. 54(8), 1209–1233 (2002)
A. Krawietz, Materialtheorie (Springer, Berlin, Heidelberg, 1986)
L.D. Landau, E.M. Lifshitz, Fluid mechanics: Volume 6 of Course of Theoretical Physics, 2rd edn. (Pergamon Press, 1987)
R. Landgraf, M. Rudolph, R. Scherzer, J. Ihlemann, Modelling and simulation of adhesive curing processes in bonded piezo metal composites. Comput. Mech. 54(2), 547–565 (2014)
A. Lion, P. Höfer, On the phenomenological representation of curing phenomena in continuum mechanics. Arch. Mech. 59(1), 59–89 (2007)
D. Lüdecke, C. Lüdecke. Thermodynamik: Physikalisch-chemische Grundlagen der thermischen Verfahrenstechnik (Springer, 2013)
L.E. Malvern, Introducation to the Mechanics of a Continuous Medium (Prentice-Hall Inc, Englewood Cliffs, NJ, 1969)
J. Marsden, T.J.R. Hughes, Mathematical Foundations of Elasticity (Prentice-Hall Inc, Englewood Cliffs, 1983)
C. Miehe, Zur numerischen Behandlung thermomechanischer Prozesse. Ph.D. thesis, Universität Hannover, Germany (1988)
W.D. Rolph, K.J. Bathe, An efficient algorithm for analysis of nonlinear heat transfer with phase changes. Int. J. Num. Methods Eng. 18(1), 119–134 (1982)
L.D. Schmidt, The Engineering of Chemical Reactions, vol. 2 (Oxford University Press, USA, 1998)
H.R. Schwarz, Methode Der Finiten Elemente: Eine Einführung Unter Berücksichtigung Der Rechenpraxis (Teubner, Stuttgart, 1991)
M. Silhavy, The Mechanics and Thermodynamics of Continuous Media (Springer, Berlin, Heidelberg, Wien, 1997)
S. Sourour, M.R. Kamal, Differential scanning calorimetry of epoxy cure: isothermal cure kinetics. Thermochimica Acta 14(1–2), 41–59 (1976)
C. Truesdell, W. Noll, The Non-Linear Field Theories of Mechanics, 3rd edn. (Springer, Berlin, Heidelberg, New York, 2004)
C. Truesdell, R. Toupin. The classical field theories, in Principles of Classical Mechanics and Field Theory/Prinzipien der Klassischen Mechanik und Feldtheorie (Springer, 1960), pp. 226–858
V.R. Voller, C. Prakash, A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems. Int. J. Heat Mass Trans. 30(8), 1709–1719 (1987)
S.L. Wang, R.F. Sekerka, A.A. Wheeler, B.T. Murray, S.R. Coriell, R.J. Braun, G.B. McFadden, Thermodynamically-consistent phase-field models for solidification. Physica D: Nonlinear Phenomena 69(1–2), 189–200 (1993)
P. Wriggers, Nonlinear Finite Element Methods (Springer Science & Business Media, 2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Weißenfels, C. (2022). Differential Equations. In: Simulation of Additive Manufacturing using Meshfree Methods. Lecture Notes in Applied and Computational Mechanics, vol 97. Springer, Cham. https://doi.org/10.1007/978-3-030-87337-0_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-87337-0_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-87336-3
Online ISBN: 978-3-030-87337-0
eBook Packages: EngineeringEngineering (R0)