##
**Generalized variational principles for thermo-chemo-mechanical coupling systems based on decomposition of internal energy.**
*(English)*
Zbl 1500.74001

Summary: A novel theoretical framework of thermo-chemo-mechanical coupling problems is proposed in this paper based on a decomposition of the internal energy into the free internal energy and the dissipation energy. Fundamental equations (i.e., constitutive equations, divergence and gradient equations) along with corresponding boundary conditions are all included in this framework, on which a pair of dual-complementary Hu-Washizu type generalized variational principles (GVPs) with explicit functionals is established. Therein, two energy functionals complementing to each other are introduced for the first time. The newly developed GVPs treat all variables as independent ones, and some well-known variational principles in the literature can be reduced from them by setting some preconditions, which indicates that these variational principles share the same theoretical basis and shows the universality and adaptability of the present theory.

### MSC:

74A15 | Thermodynamics in solid mechanics |

74H80 | Energy minimization in dynamical problems in solid mechanics |

74F05 | Thermal effects in solid mechanics |

74F25 | Chemical and reactive effects in solid mechanics |

### Keywords:

free internal energy; dissipation energy; dual-complementary Hu-Washizu variational principle; energy functional; Gibbs equation; thermodynamic force
PDF
BibTeX
XML
Cite

\textit{J.-H. Zheng} and \textit{Z. Zhong}, Acta Mech. 233, No. 9, 3725--3745 (2022; Zbl 1500.74001)

Full Text:
DOI

### References:

[1] | Bekas, DG; Tsirka, K.; Baltzis, D.; Paipetis, AS, Self-healing materials: a review of advances in materials, evaluation, characterization and monitoring techniques, Compos. Part. B-Eng., 87, 92-119 (2016) |

[2] | Chester, SA; Anand, L., A coupled theory of fluid permeation and large deformations for elastomeric materials, J. Mech. Phys. Solids., 58, 11, 1879-1906 (2010) · Zbl 1225.74034 |

[3] | Swaminathan, N.; Qu, JM, A mechanical-electrochemical theory of defects in ionic solids, Ceram. Eng. Sci. Proc., 27, 4, 125-136 (2007) |

[4] | Brassart, L.; Suo, ZG, Reactive flow in large-deformation electrodes of lithium-ion batteries, Int. J. Appl. Mech., 4, 3, 1250023 (2012) |

[5] | Huyghe, JM; Janssen, JD, Thermo-chemo-electro-mechanical formulation of saturated charged porous solids, Transport Porous Med., 34, 1-3, 129-141 (1999) |

[6] | Coussy, O., Poromechanics (2004), West Sussex: John Wiley & Sons, West Sussex · Zbl 1120.74447 |

[7] | Coussy, O., Mechanics and physics of porous solids (2010), West Sussex: John Wiley & Sons, West Sussex |

[8] | Tian, F., Zhong, Z.: Modeling of load responses for natural fiber reinforced composites under water absorption. Compos. Part. A: Appl. Sci. Manuf., 105564 (2019) |

[9] | Tian, F.; Zhong, Z.; Pan, Y., Modeling of natural fiber reinforced composites under hygrothermal ageing, Compos. Struct., 200, 144-152 (2018) |

[10] | Pan, YH; Tian, F.; Zhong, Z., A continuum damage-healing model of healing agents based self-healing materials, Int. J. Damage Mech., 27, 5, 754-778 (2018) |

[11] | Peradzynski, Z., Diffusion of calcium in biological tissues and accompanying mechano-chemical effects, Arch. Mech., 62, 6, 423-440 (2010) · Zbl 1269.76137 |

[12] | Yang, CH; Zhou, S.; Shian, S.; Clarke, DR; Suo, ZG, Organic liquid-crystal devices based on ionic conductors, Mater. Horiz., 4, 6, 1102-1109 (2017) |

[13] | Gibbs, J.W.: The scientific papers of J. Willard Gibbs. (1878) · JFM 10.0759.01 |

[14] | Biot, MA, General theory of three-dimensional consolidation, J. Appl. Phys., 12, 2, 155-164 (1941) · JFM 67.0837.01 |

[15] | Rice, JR; Cleary, MP, Some basic stress diffusion solutions for fluid-saturated elastic porous-media with compressible constituents, Rev. Geophys., 14, 2, 227-241 (1976) |

[16] | Rambert, G.; Grandidier, JC; Aifantis, EC, On the direct interactions between heat transfer, mass transport and chemical processes within gradient elasticity, Eur. J. Mech. A-Solids, 26, 1, 68-87 (2007) · Zbl 1111.74012 |

[17] | Rambert, G.; Jugla, G.; Grandidier, JC; Cangemi, L., A modelling of the direct couplings between heat transfer, mass transport, chemical reactions and mechanical behaviour. Numerical implementation to explosive decompression, Compos. Part A-Appl. Sci. Manuf., 37, 4, 571-584 (2006) |

[18] | Loeffel, K.; Anand, L.; Gasem, ZM, On modeling the oxidation of high-temperature alloys, Acta Mater, 61, 2, 399-424 (2013) |

[19] | Loeffel, K.; Anand, L., A chemo-thermo-mechanically coupled theory for elastic-viscoplastic deformation, diffusion, and volumetric swelling due to a chemical reaction, Int. J. Plasticity, 27, 9, 1409-1431 (2011) · Zbl 1426.74137 |

[20] | Zhang, XL; Zhong, Z., Thermo-chemo-elasticity considering solid state reaction and the displacement potential approach to quasi-static chemo-mechanical problems, Int. J. Appl. Mech., 10, 10, 1850112 (2018) |

[21] | Zhang, XL; Zhong, Z., A coupled theory for chemically active and deformable solids with mass diffusion and heat conduction, J. Mech. Phys. Solids, 107, 49-75 (2017) |

[22] | Zhang, XL; Zhu, PP; Zhong, Z., A chemo-mechanically coupled continuum damage-healing model for chemical reaction-based self-healing materials, Int. J. Solids Struct., 236-237, 1 (2022) |

[23] | Qin, B.; Zhong, Z., A theoretical model for thermo-chemo-mechanically coupled problems considering plastic flow at large deformation and its application to metal oxidation, Int. J. Solids Struct., 212, 102-123 (2021) |

[24] | Luo, E.; Kuang, JS; Huang, WJ; Luo, ZG, Unconventional Hamilton-type variational principles for nonlinear coupled thermoelastodynamics, Sci. China Ser. A Math. Phys. Astron, 45, 6, 783-794 (2002) · Zbl 1145.74313 |

[25] | Luo, E.; Zhu, HJ; Yuan, L., Unconventional Hamilton-type variational principles for electromagnetic elastodynamics, Sci. China-Phys. Mech. Astron., 49, 1, 119-128 (2006) · Zbl 1148.74301 |

[26] | Yang, QS; Qin, QH; Ma, LH; Lu, XZ; Cui, CQ, A theoretical model and finite element formulation for coupled thermo-electro-chemo-mechanical media, Mech. Mater., 42, 2, 148-156 (2010) |

[27] | Hu, SL; Shen, SP, Non-equilibrium thermodynamics and variational principles for fully coupled thermal-mechanical-chemical processes, Acta Mech., 224, 12, 2895-2910 (2013) · Zbl 1401.82029 |

[28] | Fernández-Cara, E.; Münch, A., Strong convergent approximations of null controls for the 1D heat equation, SéMA J., 61, 1, 49-78 (2013) · Zbl 1263.35121 |

[29] | Bouklas, N.; Landis, CM; Huang, R., A nonlinear, transient finite element method for coupled solvent diffusion and large deformation of hydrogels, J. Mech. Phys. Solids., 79, 21-43 (2015) · Zbl 1349.74130 |

[30] | Chen, J.; Wang, H.; Yu, P.; Shen, S., A finite element implementation of a fully coupled mechanical-chemical theory, Int. J. Appl. Mech., 9, 3, 1750040 (2017) |

[31] | Yu, PF; Wang, HL; Chen, JY; Shen, SP, Conservation laws and path-independent integrals in mechanical-diffusion-electrochemical reaction coupling system, J. Mech. Phys. Solids., 104, 57-70 (2017) · Zbl 1442.74011 |

[32] | Xue, S-L; Li, B.; Feng, X-Q; Gao, H., Biochemomechanical poroelastic theory of avascular tumor growth, J. Mech. Phys. Solids., 94, 409-432 (2016) |

[33] | Xue, S-L; Li, B.; Feng, X-Q; Gao, H., A non-equilibrium thermodynamic model for tumor extracellular matrix with enzymatic degradation, J. Mech. Phys. Solids, 104, 32-56 (2017) |

[34] | Edelen, DGB, On the existence of symmetry relations and dissipation potentials, Arch. Ration. Mech. Anal., 51, 3, 218-227 (1973) · Zbl 0269.73003 |

[35] | Kuang, ZB, Variational principles for generalized dynamical theory of thermopiezoelectricity, Acta Mech., 203, 1-2, 1-11 (2009) · Zbl 1161.74018 |

[36] | Yu, PF; Shen, SP, A fully coupled theory and variational principle for thermal-electrical-chemical-mechanical processes, J. Appl. Mech.-T ASME, 81, 11, 111005 (2014) |

[37] | Kuang, ZB, Energy and entropy equations in coupled nonequilibrium thermal mechanical diffusive chemical heterogeneous system, Sci. Bull., 60, 10, 952-957 (2015) |

[38] | Kuang, ZB, Variational principles for generalized thermodiffusion theory in pyroelectricity, Acta Mech., 214, 3-4, 275-289 (2010) · Zbl 1261.74013 |

[39] | Santillán, M., Chemical kinetics, stochastic processes, and irreversible thermodynamics (2014), Heidelberg: Springer, Heidelberg · Zbl 1295.92003 |

[40] | Finlayson, BA, The method of weighted residuals and variational principles (1972), New York: Academic Press, New York · Zbl 0319.49020 |

[41] | Demirel, Y., Modeling of thermodynamically coupled reaction-transport systems, Chem. Eng. J., 139, 1, 106-117 (2008) |

[42] | Hu, HC, Variational principles of theory of elasticity with applications (1984), Beijing: Science Press, Beijing |

[43] | Cai, M., Hydration ageing of plant fiber reinforced composites (2018), Shanghai: Tongji University, Shanghai |

[44] | Simo, JC, On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects, Comput. Methods Appl. Mech. Eng., 60, 2, 153-173 (1987) · Zbl 0588.73082 |

[45] | Yu, Y.; Wu, H., Significant differences in the hydrolysis behavior of amorphous and crystalline portions within microcrystalline cellulose in hot-compressed water, Ind. Eng. Chem. Res., 49, 8, 3902-3909 (2010) |

[46] | Zheng, JH; Jiang, CY; Zhong, Z., Continuum mechanics for thermo-chemo-mechanical coupling system based on decomposition of internal energy, Sci. Sin. Techs., 49, 10, 1168-1176 (2019) |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.