Abstract
This paper develops a coupled chemo-mechanical model for stress-assisted diffusion in the framework of micropolar elasticity. The two-way interaction of mechanical and chemical driving forces as well as internal microstructure of the polar media is taken into account. The fundamental governing equations of chemo-mechanics with kinetics driven by diffusion and stress are developed, and mathematical expressions for stress and concentration fields are derived. Using Airy stress functions, a simplified formulation for two-dimensional chemo-elasticity problems under chemical equilibrium is presented. Using a perturbation approach to solve the presented system of equations, closed-form expressions for different field parameters can be obtained. As an illustrative case study, expressions for the stress and solute concentration in an infinite plate with circular hole embedded in a chemical medium have been written. The derived equation and proposed solution approach can be applied to various plane micropolar chemo-elasticity problems for studying the interaction between mechanical and chemical driving forces as well as material length-scale parameters.
Similar content being viewed by others
References
Parfitt, D., Kordatos, A., Filippatos, P., Chroneos, A.: Diffusion in energy materials: Governing dynamics from atomistic modelling. Appl. Phys. Rev. 4, 031305 (2017)
Shaw, D.: Atomic Diffusion in Semiconductors. Springer, Berlin (2012)
Bonef, B., Shah, R.D., Mukherjee, K.: Fast diffusion and segregation along threading dislocations in semiconductor heterostructures. Nano Lett. 19, 1428–1436 (2019)
Kazakov, N.F.: Diffusion Bonding of Materials. Elsevier, New York (2013)
Köhler, M., Junker, P., Balzani, D.: Continuum multiscale modeling of absorption processes in micro- and nanocatalysts. Arch. Appl. Mech. 92, 2207–2223 (2022)
Wu, C.H.: The role of Eshelby stress in composition-generated and stress-assisted diffusion. J. Mech. Phys. Solids 49, 1771–1794 (2001)
Yang, F.: Interaction between diffusion and chemical stresses. Mater. Sci. Eng. A 409, 153–159 (2005)
Yamaue, T., Doi, M.: The stress diffusion coupling in the swelling dynamics of cylindrical gels. J. Chem. Phys. 122, 084703 (2005)
Dong, X., Feng, X., Hwang, K.-C.: Stress–diffusion interaction during oxidation at high temperature. Chem. Phys. Lett. 614, 95–98 (2014)
Seo, H.K., Park, J.Y., Chang, J.H., Dae, K.S., Noh, M.-S., Kim, S.-S., et al.: Strong stress-composition coupling in lithium alloy nanoparticles. Nat. Commun. 10, 1–8 (2019)
Wang, X., Li, X., Mei, J., Zhao, K.: Doping kinetics in organic mixed ionic–electronic conductors: Moving front experiments and the stress effect. Extreme Mech. Lett. 54, 101739 (2022)
Peradzyński, Z., Kazmierczak, B.: On mechano-chemical Calcium waves. Arch. Appl. Mech. 74, 827–833 (2005)
Bahramifar, S., Haftbaradaran, H., Mossaiby, F.: Cohesive modeling of crack formation in two-phase planar electrodes subject to diffusion induced stresses using the distributed dislocation method. Int. J. Mech. Sci. 194, 106183 (2021)
Prussin, S.: Generation and distribution of dislocations by solute diffusion. J. Appl. Phys. 32, 1876–1881 (1961)
Podstrigach, Y.S., Pavlina, V.: Differential equations of thermodynamic processes in n-component solid solutions. Soviet Mater. Sci. 1, 259–264 (1966)
Podstrigach, Y.S., Shevchuk, P.: Diffusion phenomena and stress relaxation in the vicinity of a spherical void. Soviet Mater. Sci. 4, 140–145 (1969)
Larché, F., Cahn, J.W.: A linear theory of thermochemical equilibrium of solids under stress. Acta Metall. 21, 1051–1063 (1973)
Larcht’e, F., Cahn, J.: The effect of self-stress on diffusion in solids. Acta Metall. 30, 1835–1845 (1982)
Larche, F., Cahn, J.W.: The interactions of composition and stress in crystalline solids. J. Res. Natl. Bur. Stand. 89, 467 (1984)
Swaminathan, N., Qu, J., Sun, Y.: An electrochemomechanical theory of defects in ionic solids. I. Theory. Phil. Mag. 87, 1705–1721 (2007)
Swaminathan, N., Qu, J.: Interactions between non-stoichiometric stresses and defect transport in a tubular electrolyte. Fuel Cells 7, 453–462 (2007)
Suo, Y., Shen, S.: Coupling diffusion–reaction–mechanics model for oxidation. Acta Mech. 226, 3375–3386 (2015)
Wang, H., Suo, Y., Shen, S.: Reaction–diffusion–stress coupling effect in inelastic oxide scale during oxidation. Oxid. Met. 83, 507–519 (2015)
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, 107–123 (2021)
Chester, S.A., Anand, L.: A thermo-mechanically coupled theory for fluid permeation in elastomeric materials: application to thermally responsive gels. J. Mech. Phys. Solids 59, 1978–2006 (2011)
Zheng, S., Li, Z., Liu, Z.: The fast homogeneous diffusion of hydrogel under different stimuli. Int. J. Mech. Sci. 137, 263–270 (2018)
Li, Y., Zhang, K., Zheng, B., Yang, F.: Effect of local deformation on the coupling between diffusion and stress in lithium-ion battery. Int. J. Solids Struct. 87, 81–89 (2016)
Qin, B., Zhong, Z.: A diffusion–reaction–deformation coupling model for lithiation of silicon electrodes considering plastic flow at large deformation. Arch. Appl. Mech. 91, 2713–2733 (2021)
Haftbaradaran, H., Qu, J.: Two-dimensional chemo-elasticity under chemical equilibrium. Int. J. Solids Struct. 56, 126–135 (2015)
Chen, Z., Zhu, Y., Wang, Q., Liu, W., Cui, Y., Tao, X., et al.: Fibrous phosphorus: a promising candidate as anode for lithium-ion batteries. Electrochim. Acta 295, 230–236 (2019)
Lee, J.: Surface-engineered flexible fibrous supercapacitor electrode for improved electrochemical performance. Appl. Surf. Sci. 539, 148290 (2021)
Ricoeur, A., Lange, S.: Constitutive modeling of polycrystalline multiconstituent and multiphase ferroic materials based on a condensed approach. Arch. Appl. Mech. 89, 973–994 (2019)
Jamkhande, P.G., Ghule, N.W., Bamer, A.H., Kalaskar, M.G.: Metal nanoparticles synthesis: an overview on methods of preparation, advantages and disadvantages, and applications. J. Drug Deliv. Sci. Technol. 53, 101174 (2019)
Liang, Y., Lai, W.H., Miao, Z., Chou, S.L.: Nanocomposite materials for the sodium–ion battery: a review. Small 14, 1702514 (2018)
Akinwande, D., Brennan, C.J., Bunch, J.S., Egberts, P., Felts, J.R., Gao, H., et al.: A review on mechanics and mechanical properties of 2D materials—graphene and beyond. Extreme Mech. Lett. 13, 42–77 (2017)
Sun, L., Wang, X., Wang, Y., Zhang, Q.: Roles of carbon nanotubes in novel energy storage devices. Carbon 122, 462–474 (2017)
Hui, F., Shi, Y., Ji, Y., Lanza, M., Duan, H.: Mechanical properties of locally oxidized graphene electrodes. Arch. Appl. Mech. 85, 339–345 (2015)
Ghavanloo, E., Fazelzadeh, S.A., Trovalusci, P.: Mechanics of size-dependent materials. Arch. Appl. Mech. 93, 1–3 (2023)
Cosserat, E., Cosserat, F.: Theory of Deformable Bodies, Scientific Library. a Hermann and Sons, Paris (1909)
Eringen, A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 909–923 (1966)
Eringen, A.C.: Mechanics of micromorphic continua. In: Mechanics of Generalized Continua: Proceedings of the IUTAM-Symposium on the Generalized Cosserat Continuum and the Continuum Theory of Dislocations with Applications, Freudenstadt and Stuttgart (Germany) , pp 18–35 (1967, 1968)
Rizzi, G., Hütter, G., Madeo, A., Neff, P.: Analytical solutions of the simple shear problem for micromorphic models and other generalized continua. Arch. Appl. Mech. 91, 2237–2254 (2021)
Mindlin, R., Tiersten, H.: Effects of Couple-Stresses in Linear Elasticity. Columbia Univ, New York (1962)
Toupin, R.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)
Garg, N., Han, C.-S.: Axisymmetric couple stress elasticity and its finite element formulation with penalty terms. Arch. Appl. Mech. 85, 587–600 (2015)
Aouadi, M.: The coupled theory of micropolar thermoelastic diffusion. Acta Mech. 208, 181–203 (2009)
Liu, W., Shen, S.: Coupled chemomechanical theory with strain gradient and surface effects. Acta Mech. 229, 133–147 (2018)
Tsagrakis, I., Aifantis, E.C.: Thermodynamic coupling between gradient elasticity and a Cahn–Hilliard type of diffusion: size-dependent spinodal gaps. Contin. Mech. Thermodyn. 29, 1181–1194 (2017)
Tsagrakis, I., Aifantis, E.C.: Gradient elasticity effects on the two-phase lithiation of LIB anodes. General. Models Non-Class. Approac. Complex Mater. 2, 221–235 (2018)
Frolova, K., Vilchevskaya, E., Bessonov, N., Müller, W., Polyanskiy, V., Yakovlev, Y.: Application of micropolar theory to the description of the skin effect due to hydrogen saturation. Math. Mech. Solids 27, 1092–1110 (2022)
Dyszlewicz, J.: Micropolar Theory of Elasticity, vol. 15. Springer, Berlin (2012)
Flory, P.J.: Thermodynamics of high polymer solutions. J. Chem. Phys. 10, 51–61 (1942)
Huggins, M.L.: Theory of solutions of high polymers1. J. Am. Chem. Soc. 64, 1712–1719 (1942)
Cheng, Z.-Q., He, L.-H.: Micropolar elastic fields due to a spherical inclusion. Int. J. Eng. Sci. 33, 389–397 (1995)
Nowacki, W.: The micropolar thermoelasticity. In” Micropolar Elasticity: Symposium Organized by the Department of Mechanics of Solids, June 1972, pp 105–168 (1974)
Funding
This research has received no funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no conflict of interest.
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 (e.g. a society or other partner) 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
Malaeke, H., Asghari, M. A mathematical formulation for analysis of diffusion-induced stresses in micropolar elastic solids. Arch Appl Mech 93, 3093–3111 (2023). https://doi.org/10.1007/s00419-023-02427-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-023-02427-y