Abstract
Gradient plasticity theories are of utmost importance for accounting for size effects in metals, especially on the grain scale. Today, there are several methods used to derive the governing equations for the additional degrees of freedom in gradient plasticity theories. Here, the equivalence between an extended principle of virtual power and an extended energy balance is shown. The energy balance of a Boltzmann continuum is supplemented by contributions based on a scalar-valued degree of freedom. It is considered to be invariant with respect to a change of observer. This yields unambiguously the existence of a corresponding micro-stress vector, which is presumed from the outset in the context of an extended principle of virtual power. A thermodynamically consistent nonlocal evolution equation for the additional, scalar-valued degree of freedom is obtained by evaluation of the dissipation inequality in terms of the Clausius–Duhem inequality. Partitioning the nonlocal flow rule yields a partial differential equation, often referred to as micro-force balance. The approach presented is applied to derive a slip gradient crystal plasticity theory regarding single slip. Finally, the distribution of the plastic slip is exemplified with respect to a laminate material consisting of an elastic and an elastoplastic phase.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bardella, L.: Size effects in phenomenological strain gradient plasticity constitutively involving the plastic spin. Int. J. Eng. Sci. 48(5), 550–568 (2010)
Bayerschen, E., Böhlke, T.: Power-law defect energy in a single-crystal gradient plasticity framework: a computational study. Comput. Mech. 58(1), 13–27 (2016)
Bayerschen, E., Stricker, M., Wulfinghoff, S., Weygand, D., Böhlke, T.: Equivalent plastic strain gradient plasticity with grain boundary hardening and comparison to discrete dislocation dynamics. Proc. R. Soc. A 471, 1–19 (2015)
Beegle, B.L., Modell, M., Reid, R.C.: Legendre transforms and their application in thermodynamics. AIChE J. 20(6), 1194–1200 (1974)
Bertram, A.: Elasticity and Plasticity of Large Deformations: An Introduction. Springer, Berlin (2005)
Bertram, A.: Solid Mechanics: Theory, Modeling, and Problems. Springer, Heidelberg (2015)
Bertram, A., Krawietz, A.: On the introduction of thermoplasticity. Acta Mech. 223(10), 2257–2268 (2012)
Capriz, G.: Continua with Microstructure. Springer, New York (1989)
Capriz, G., Podio-Guidugli, P., Williams, W.: On balance equations for materials with affine structure. Meccanica 17(2), 80–84 (1982)
Cermelli, P., Gurtin, M.E.: Geometrically necessary dislocations in viscoplastic single crystals and bicrystals undergoing small deformations. Int. J. Solids Struct. 39(26), 6281–6309 (2002)
Coleman, B.D., Gurtin, M.E.: Thermodynamics with internal state variables. J. Chem. Phys. 47(2), 597–613 (1967)
Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13(1), 167–178 (1963)
Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. Hermann, Paris (1909)
dell’Isola, F., Seppecher, P., Della Corte, A.: The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc. R. Soc. A 471(2183), 1–25 (2015)
dell’Isola, F., Madeo, A., Seppecher, P.: Cauchy tetrahedron argument applied to higher contact interactions. Arch. Ration. Mech. Anal. 219(3), 1305–1341 (2016)
Dunn, J.E., Serrin, J.: On the thermomechanics of interstitial working. In: Dafermos, C.M., Joseph, D.D., Leslie, F.M. (eds.) The Breadth and Depth of Continuum Mechanics, pp. 705–743. Springer, Berlin (1986)
Ericksen, J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5(1), 23–34 (1961)
Eringen, A.C.: Simple microfluids. Int. J. Eng. Sci. 2(2), 205–217 (1964)
Eringen, A.C.: Mechanics of Micromorphic Continua. In: Kröner, E. (ed.) Mechanics of Generalized Continua, pp. 18–35. Springer, Berlin (1968)
Eringen, A.C., Suhubi, E.S.: Nonlinear theory of simple micro-elastic solids—I. Int. J. Eng. Sci. 2(2), 189–203 (1964)
Eugster, S.R., dell’Isola, F.: Exegesis of the Introduction and Sect. I from Fundamentals of the Mechanics of Continua by E. Hellinger. Z. Angew. Math. Mech. 97(4), 477–506 (2017)
Eugster, S.R., dell’Isola, F.: Exegesis of Sect. II and III.A from “Fundamentals of the Mechanics of Continua” by E. Hellinger. Z. Angew. Math. Mech. 98(1), 31–68 (2018)
Eugster, S.R., dell’Isola, F.: Exegesis of Sect. III.B from “Fundamentals of the Mechanics of Continua” by E. Hellinger. Z. Angew. Math. Mech. 98(1), 69–105 (2018)
Forest, S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135(3), 117–131 (2009)
Forest, S.: Questioning size effects as predicted by strain gradient plasticity. J. Mech. Behav. Mater. 22(3–4), 101–110 (2013)
Forest, S., Guéninchault, N.: Inspection of free energy functions in gradient crystal plasticity. Acta Mech. Sin. 29(6), 763–772 (2013)
Fox, N.: A continuum theory of dislocations for polar elastic materials. J. Mech. Appl. Math. 19(3), 343–355 (1966)
Fox, N.: On the continuum theories of dislocations and plasticity. J. Mech. Appl. Math. 21(1), 67–75 (1968)
Germain, N., Besson, J., Feyel, F.: Simulation of laminate composites degradation using mesoscopic non-local damage model and non-local layered shell element. Model. Simul. Mater. Sci. Eng. 15(4), 425–434 (2007)
Germain, P.: The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J. Appl. Math. 25(3), 556–575 (1973)
Giorgio, I.: Numerical identification procedure between a micro-Cauchy model and a macro-second gradient model for planar pantographic structures. Z. Angew. Math. Phys. 67(4), 1–17 (2016)
Green, A., Naghdi, P., Rivlin, R.: Directors and multipolar displacements in continuum mechanics. Int. J. Eng. Sci. 2(6), 611–620 (1965)
Green, A.E., Rivlin, R.S.: Simple force and stress multipoles. Arch. Ration. Mech. Anal. 16(5), 325–353 (1964)
Green, A.E., Rivlin, R.S.: On Cauchy’s equations of motion. Z. Angew. Math. Phys. 15(3), 290–292 (1964)
Green, A.E., Rivlin, R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17(2), 113–147 (1964)
Gurtin, M.E., Anand, L., Lele, S.P.: Gradient single-crystal plasticity with free energy dependent on dislocation densities. J. Mech. Phys. Solids 55(9), 1853–1878 (2007)
Hellinger, E.: Die allgemeinen Ansätze der Mechanik der Kontinua. Encyclopädie der Mathematischen Wissenschaften 4(4), 601–694 (1913)
Krawietz, A.: Materialtheorie. Springer, Berlin (1986)
Krawietz, A.: Classical mechanics recast with Mach’s principle. Technol. Mech. 35(1), 49–59 (2015)
Landau, L., Lifshitz, E.: Mechanics. Pergamon Press, Oxford (1969)
Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28(4), 265–283 (1968)
Mariano, P.M.: Trends and challenges in the mechanics of complex materials: a view. Philos. Trans. R. Soc. A 374(2066), 1–31 (2016)
Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, New York (1994)
Maugin, G.A.: The method of virtual power in continuum mechanics: application to coupled fields. Acta Mech. 35(1), 1–70 (1980)
Maugin, G.A.: The saga of internal variables of state in continuum thermo-mechanics (1893–2013). Mech. Res. Commun. 69, 79–86 (2015)
Maugin, G.A.: Non-Classical Continuum Mechanics: A Dictionary. Springer, Singapore (2017)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)
Misra, A., Placidi, L., Scerrato, D.: A review of presentations and discussions of the workshop “Computational Mechanics of Generalized Continua and Applications to Materials with Microstructure” that was held in Catania 29–31 October 2015. Math. Mech. Solid 22(9), 1891–1904 (2017)
Müller, I.: Thermodynamics. Pitman, Boston (1985)
Neff, P., Ghiba, I.D., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: the relaxed linear micromorphic continuum. Continu. Mech. Thermodyn. 26(5), 639–681 (2014)
Noether, E.: Invariant variation problems. Transp. Theory Stat. Phys. 1(3), 186–207 (1971)
Ortiz, M., Repetto, E.: Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47(2), 397–462 (1999)
Peerlings, R., Massart, T., Geers, M.: A thermodynamically motivated implicit gradient damage framework and its application to brick masonry cracking. Comput. Methods Appl. Mech. Eng. 193(30), 3403–3417 (2004)
Placidi, L.: A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model. Continu. Mech. Thermodyn. 28(1), 119–137 (2016)
Placidi, L., Barchiesi, E.: Energy approach to brittle fracture in strain-gradient modelling. Proc. R. Soc. A 474(2210), 1–19 (2018)
Placidi, L., Giorgio, I., Della Corte, A., Scerrato, D.: Euromech 563 Cisterna di Latina 17–21 March 2014 Generalized continua and their applications to the design of composites and metamaterials: a review of presentations and discussions. Math. Mech. Solid 22(2), 144–157 (2017)
Placidi, L., Barchiesi, E., Misra, A.: A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results. Math. Mech. Complex Syst. 6(2), 77–100 (2018)
Placidi, L., Misra, A., Barchiesi, E.: Two-dimensional strain gradient damage modeling: a variational approach. Z. Angew. Math. Phys. 69(3), 1–19 (2018)
Planck, M.: A Survey of Physical Theory. Dover, New York (1960)
Rahali, Y., Giorgio, I., Ganghoffer, J., dell’Isola, F.: Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci. 97, 148–172 (2015)
Seppecher, P., Alibert, J.J., dell’Isola, F.: Linear elastic trusses leading to continua with exotic mechanical interactions. J. Phys. Conf. Ser. 319, 1–13 (2011)
Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin (1997)
Spring, K.W.: Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: a review. Mech. Mach. Theory 21(5), 365–373 (1986)
Svendsen, B.: On the thermodynamics of thermoelastic materials with additional scalar degrees of freedom. Continu. Mech. Thermodyn. 11(4), 247–262 (1999)
Svendsen, B.: Formulation of balance relations and configurational fields for continua with microstructure and moving point defects via invariance. Int. J. Solids Struct. 38(6), 1183–1200 (2001)
Svendsen, B., Bertram, A.: On frame-indifference and form-invariance in constitutive theory. Acta Mech. 132(1), 195–207 (1999)
Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(1), 385–414 (1962)
Toupin, R.A.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17(2), 85–112 (1964)
Truesdell, C., Toupin, R.: The classical field theories. In: Flügge, S. (ed.) Encyclopedia of Physics, pp. 226–793. Springer, Berlin (1960)
Ubachs, R., Schreurs, P., Geers, M.: A nonlocal diffuse interface model for microstructure evolution of tin-lead solder. J. Mech. Phys. Solids 52(8), 1763–1792 (2004)
Vardoulakis, I.: Cosserat Continuum Mechanics: With Applications to Granular Media. Springer, Cham (2019)
Wulfinghoff, S., Bayerschen, E., Böhlke, T.: A gradient plasticity grain boundary yield theory. Int. J. Plast. 51, 33–46 (2013)
Wulfinghoff, S., Forest, S., Böhlke, T.: Strain gradient plasticity modeling of the cyclic behavior of laminate microstructures. J. Mech. Phys. Solids 79, 1–20 (2015)
Yavari, A., Marsden, J.E.: Covariant balance laws in continua with microstructure. Rep. Math. Phys. 63(1), 1–42 (2009)
Acknowledgements
The support of the German Research Foundation (DFG) in the project ‘Dislocation based Gradient Plasticity Theory’ of the DFG Research Group 1650 ‘Dislocation based Plasticity’ under Grant BO1466/5 is gratefully acknowledged. In addition, discussions with Matti Schneider on the topic are gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Francesco dell’Isola.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Prahs, A., Böhlke, T. On invariance properties of an extended energy balance. Continuum Mech. Thermodyn. 32, 843–859 (2020). https://doi.org/10.1007/s00161-019-00763-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-019-00763-5