Abstract
A multidomain macro-hybrid mixed variational subpotential approach of models in continuum mechanics is developed. Variational principles of mass conservation, balance momentum and energy, as well as material constitutive relations, are formulated as subpotential subdifferential evolution systems. Under the basis of variational divergence formulae and subdifferential generalized boundary conditions, variational mixed initial/boundary-value inclusion problems are established in reflexive Banach trajectory real functional frameworks. Multidomain motion velocity, mass conservation density, material constitutive stress, deformation porosity as well as balance internal energy evolution coupled variational mechanical systems are achieved. As a distinctive modeling feature, multidomain synchronizing transmission constraints are implemented in terms of dual Lagrangian internal boundary subpotential subdifferential inclusions. Lastly, evolution internal variational approximations, in general, for nonconforming macro-hybrid mixed finite element spatial discretizations as well as semi-implicit time marching fully discrete schemes are briefly discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gurtin, M.: An Introduction to Contimuum Mechanics. Academic Press, San Diego (2003)
Gurtin, M., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continuum. Cambridge University Press, Cambridge (2010)
Alduncin, G.: Composition duality methods for mixed variational inclusions. Appl. Math. Optim. 52, 311–348 (2005)
Alduncin, G.: Composition duality methods for evolution mixed variational inclusions. Nonlinear Anal. Hybrid Syst. 1, 336–363 (2007)
Alduncin, G.: Variational formulations of nonlinear constrained boundary value problems. Nonlinear Anal. 72, 2639–2644 (2010)
Temam, R.: Analyse Numérique. Presses Universitaires de France, Paris (1970)
Ciarlet, G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Fortin, M., Glowinski, R. (eds.): Méthodes de Lagrangien Augmenté: Applications à la Résolution Numérique de Problèmes aux Limites. Dunod-Bordas, Paris (1982)
Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia (1989)
Lions, J.-L.: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod/Gauthier-Villars, Paris (1969)
Alduncin, G.: Subdifferential and variational formulations of boundary value problems. Comput. Methods Appl. Mech. Eng. 72, 173–186 (1989)
Gurtin, M.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, New York (2000)
Akagi, G., Ôtani, M.: Evolution inclusions governed by subdifferentials in reflexive Banach spaces. J. Evol. Equ. 4, 519–541 (2004)
Alduncin, G.: Proximal penalty-duality algorithms for mixed optimality conditions. J. Fixed Point Theory Appl. 19, 1775–1791 (2017)
Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, New York (2010)
Brézis, H.: Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. Math. Studies, vol. 5. North-Holland, Amsterdam/New York (1973)
Halphen, B., NGuyen, Qs.: Sur les matériaux standard généralisés. Journal Mécanique 14, 39–63 (1975)
Bodovillé, G.: The implicit standard material theory for modelling the nonassociative behaviour of metals. Arch. Appl. Mech. 71, 426–435 (2001)
Le Tallec, P.: Numerical Analysis of Viscoelastic Problems. Masson, Paris (1990)
Temam, R.: A generalized Norton–Hoff model and the Prandt–Reuss law of plasticity. Arch. Ration. Mech. Anal. 95, 137–183 (1986)
Perzyna, P.: Fundamental problems in viscoplasticity. Rec. Adv. Appl. Mech. 9, 243–377 (1966)
Prager, W., Hodge, P.: The Theory of Perfectly Plastic Solids. Wiley, New York (1951)
Coulibaly, M., Sabar, H.: Micromechanical modeling of linear viscoelastic behavior of heterogeneous materials. Arch. Appl. Mech. 81, 345–359 (2011)
Dinzart, F., Sabar, H.: Homogenization of the viscoelastic heterogeneous materials with multi-coated reinforcements: an internal variables formulation. Arch. Appl. Mech. 84, 715–730 (2014)
Ekeland, I., Temam, R.: Analyse Convexe et Problèmes Variationnels. Dunod/Gauthier-Villars, Paris (1974)
Adams, A., Fournier, J.F.: Sobolev Spaces. Academic Press, Amsterdam (2008)
Yosida, K.: Functional Analysis. Springer, New York (1974)
Alduncin, G.: Macro-hybrid variational formulations of constrained boundary value problems. Numer. Funct. Anal. Optim. 28, 751–774 (2007)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Alduncin, G. Mixed variational continuum mechanics modeling: a multidomain subpotential approach. Arch Appl Mech 92, 3381–3404 (2022). https://doi.org/10.1007/s00419-022-02242-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-022-02242-x
Keywords
- Set-valued variational formulation
- Multidomain mixed initial boundary-value problem
- Continuum mechanics modeling
- Maximal monotone operator
- Subpotential subdifferential
- Lagrangian internal boundary dual transmission