Applied Mathematics and Mechanics >
Homogenization-based numerical framework of second-phase reinforced alloys integrating strain gradient effects
Received date: 2025-01-09
Revised date: 2025-05-05
Online published: 2025-06-30
Supported by
Project supported by the National Natural Science Foundation of China (Nos. 11922206, 11702089, and 12272132) and the Postgraduate Scientific Research Innovation Project of Hunan Province (No. CX20240388)
Copyright
The acuurate prediction of the time-dependent mechanical behavior and deformation mechanisms of second-phase reinforced alloys under size effects is critical for the development of high-strength ductile metals and alloys for dynamic applications. However, solving their responses using high-fidelity numerical methods is computationally expensive and, in many cases, impractical. To address this issue, a dual-scale incremental variational formulation is proposed that incorporates the influence of plastic gradients on plastic evolution characteristics, integrating a strain-rate-dependent strain gradient plasticity model and including plastic gradients in the inelastic dissipation potential. Subsequently, two minimization problems based on the energy dissipation mechanisms of strain gradient plasticity, corresponding to the macroscopic and microscopic structures, are solved, leading to the development of a homogenization-based dual-scale solution algorithm. Finally, the effectiveness of the variational model and tangent algorithm is validated through a series of numerical simulations. The contributions of this work are as follows: first, it advances the theory of self-consistent computational homogenization modeling based on the energy dissipation mechanisms of plastic strain rates and their gradients, along with the development of a rigorous multi-level finite element method (FE2) solution procedure; second, the proposed algorithm provides an efficient and accurate method for evaluating the time-dependent mechanical behavior of second-phase reinforced alloys under strain gradient effects, exploring how these effects vary with the strain rate, and investigating their potential interactions.
Haidong LIN , Yiqi MAO , Shujuan HOU . Homogenization-based numerical framework of second-phase reinforced alloys integrating strain gradient effects[J]. Applied Mathematics and Mechanics, 2025 , 46(7) : 1273 -1294 . DOI: 10.1007/s10483-025-3268-7
| [1] | LI, L. and LIU, G. Study on a straight dislocation in an icosahedral quasicrystal with piezoelectric effects. Applied Mathematics and Mechanics (English Edition), 39(9), 1259–1266 (2018) https://doi.org/10.1007/s10483-018-2363-9 |
| [2] | LIN, H., MAO, Y., LIU, W., and HOU, S. Concurrent cross-scale and multi-material optimization considering interface strain gradient. Computer Methods in Applied Mechanics and Engineering, 421, 116749 (2024) |
| [3] | LIU, J., ZHANG, Y., and CHU, H. Modeling core-spreading of interface dislocation and its elastic response in anisotropic bimaterial. Applied Mathematics and Mechanics (English Edition), 38(2), 231–242 (2017) https://doi.org/10.1007/s10483-017-2163-9 |
| [4] | ZHOU, J. H., WANG, J., RITCHIE, R. O., WU, Y. C., CHENG, J. W., WANG, L., YIN, X. W., JIANG, Y. F., and REN, J. G. Ductile ultrastrong China low activation martensitic steel with lamellar grain structure. International Journal of Plasticity, 171, 103813 (2023) |
| [5] | LUO, G., LI, L., FANG, Q., LI, J., TIAN, Y., LIU, Y., LIU, B., PENG, J., and LIAW, P. K. Microstructural evolution and mechanical properties of FeCoCrNiCu high entropy alloys: a microstructure-based constitutive model and a molecular dynamics simulation study. Applied Mathematics and Mechanics (English Edition), 42(8), 1109–1122 (2021) https://doi.org/10.1007/s10483-021-2756-9 |
| [6] | LIN, H. and HOU, S. Cross-scale optimization of advanced materials for micro and nano structures based on strain gradient theory. Computer Methods in Applied Mechanics and Engineering, 411, 116010 (2023) |
| [7] | WEI, Y. Particulate size effects in the particle-reinforced metal-matrix composites. Acta Mechanica Sinica, 17(1), 45–58 (2001) |
| [8] | ORLOVA, D. and BERINSKII, I. Multiscale analysis of a 3D fibrous collagen tissue. International Journal of Engineering Science, 195, 104003 (2024) |
| [9] | EITELBERGER, J. and HOFSTETTER, K. Prediction of transport properties of wood below the fiber saturation point—a multiscale homogenization approach and its experimental validation, part I: thermal conductivity. Composites Science and Technology, 71(2), 134–144 (2011) |
| [10] | SHRIMALI, B., LEFèVRE, V., and LOPEZ-PAMIES, O. A simple explicit homogenization solution for the macroscopic elastic response of isotropic porous elastomers. Journal of the Mechanics and Physics of Solids, 122, 364–380 (2019) |
| [11] | LUO, Z., TANG, Q., MA, S., WU, X., FENG, Q., SETCHI, R., LI, K., and ZHAO, M. Effect of aspect ratio on mechanical anisotropy of lattice structures. International Journal of Mechanical Sciences, 270, 109111 (2024) |
| [12] | WEI, H., HUANG, X., XIE, W., JIANG, X., ZHAO, G., and ZHANG, W. Multiscale modeling for the impact behavior of 3D angle-interlock woven composites. International Journal of Mechanical Sciences, 276, 109382 (2024) |
| [13] | BRASSART, L. and STAINIER, L. Effective transient behaviour of heterogeneous media in diffusion problems with a large contrast in the phase diffusivities. Journal of the Mechanics and Physics of Solids, 124, 366–391 (2019) |
| [14] | MAO, Y., WANG, C., WU, Y., and CHEN, H. S. Homogenization-based chemomechanical properties of dissipative heterogeneous composites under transient mass diffusion. International Journal of Solids and Structures, 288, 112623 (2024) |
| [15] | LUCCHETTA, A., AUSLENDER, F., BORNERT, M., and KONDO, D. A double incremental variational procedure for elastoplastic composites with combined isotropic and linear kinematic hardening. International Journal of Solids and Structures, 158, 243–267 (2019) |
| [16] | JIANG, Y., LI, L., and HU, Y. A physically-based nonlocal strain gradient theory for crosslinked polymers. International Journal of Mechanical Sciences, 245, 108094 (2023) |
| [17] | COULAIS, C., KETTENIS, C., and VAN HECKE, M. A characteristic length scale causes anomalous size effects and boundary programmability in mechanical metamaterials. Nature Physics, 14(1), 40–44 (2017) |
| [18] | LIU, D. B., HE, Y. M., DUNSTAN, D. J., ZHANG, B., GAN, Z. P., HU, P., and DING, H. M. Toward a further understanding of size effects in the torsion of thin metal wires: an experimental and theoretical assessment. International Journal of Plasticity, 41, 30–52 (2013) |
| [19] | FLECK, N. A., MULLER, G. M., ASHBY, M. F., and HUTCHINSON, J. W. Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia, 42(2), 475–487 (1994) |
| [20] | LIU, D. B., HE, Y. M., TANG, X. T., DING, H. M., HU, P., and CAO, P. Size effects in the torsion of microscale copper wires: experiment and analysis. Scripta Materialia, 66(6), 406–409 (2012) |
| [21] | FENG, G. and NIX, W. D. Indentation size effect in MgO. Scripta Materialia, 51(6), 599–603 (2004) |
| [22] | ASHBY, M. F. Work hardening of dispersion-hardened crystals. Philosophical Magazine, 14(132), 1157–1178 (1966) |
| [23] | TAYLOR, M. B., ZBIB, H. M., and KHALEEL, M. A. Damage and size effect during superplastic deformation. International Journal of Plasticity, 18(3), 415–442 (2002) |
| [24] | ZENG, S., WANG, K., WANG, B., and WU, J. Vibration analysis of piezoelectric sandwich nanobeam with flexoelectricity based on nonlocal strain gradient theory. Applied Mathematics and Mechanics (English Edition), 41(6), 859–880 (2020) https://doi.org/10.1007/s10483-020-2620-8 |
| [25] | YU, Z. and WEI, Y. Cross-scale indentation scaling relationships of strain gradient plastic solids: influence of inclusions near the indenter tip. Acta Mechanica Sinica, 38(1), 221257 (2022) |
| [26] | LAM, D. C. C., YANG, F., CHONG, A. C. M., WANG, J., and TONG, P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51(8), 1477–1508 (2003) |
| [27] | NGUYEN, T. N., SIEGMUND, T., TOMAR, V., and KRUZIC, J. J. Interaction of rate- and size-effect using a dislocation density based strain gradient viscoplasticity model. Journal of the Mechanics and Physics of Solids, 109, 1–21 (2017) |
| [28] | ASHBY, M. F. The deformation of plastically non-homogeneous materials. The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics, 21(170), 399–424 (1970) |
| [29] | GUAN, X. R., CHEN, Q., QU, S. J., CAO, G. J., WANG, H., RAN, X. D., FENG, A. H., and CHEN, D. L. High-strain-rate deformation: stress-induced phase transformation and nanostructures in a titanium alloy. International Journal of Plasticity, 169, 103707 (2023) |
| [30] | JIANG, K., LI, J., GAN, B., YE, T., CHEN, L., and SUO, T. Dynamically compressive behaviors and plastic mechanisms of a CrCoNi medium entropy alloy at various temperatures. Acta Mechanica Sinica, 38(5), 421550 (2022) |
| [31] | SUN, J., ZHAO, W., YAN, P., ZHAI, B., XIA, X., ZHAO, Y., JIAO, L., and WANG, X. Dynamic shear behavior of forged CrMnFeCoNi high entropy alloy using hat-shaped specimens over a wide range of temperatures and strain rates. International Journal of Plasticity, 170, 103761 (2023) |
| [32] | NGUYEN, T., FENSIN, S. J., and LUSCHER, D. J. Dynamic crystal plasticity modeling of single crystal tantalum and validation using Taylor cylinder impact tests. International Journal of Plasticity, 139, 102940 (2021) |
| [33] | DONG, J. L., LI, F. C., GU, Z. P., JIANG, M. Q., LIU, Y. H., WANG, G. J., and WU, X. Q. Impact resistance and energy dissipation mechanism of nanocrystalline CoCrNi medium entropy alloy nanofilm under supersonic micro-ballistic impact. International Journal of Plasticity, 171, 103801 (2023) |
| [34] | WANG, Y. Y., WANG, J. D., LEI, M. Q., and YAO, Y. A crystal plasticity coupled damage constitutive model of high entropy alloys at high temperature. Acta Mechanica Sinica, 38(11), 122116 (2022) |
| [35] | ZHU, Z. W., ZHANG, G. H., FENG, C., XIAO, S., and ZHU, T. Dynamic impact constitutive model of 6008 aluminum alloy based on evolution dislocation density. Acta Mechanica Sinica, 39(7), 122419 (2023) |
| [36] | JIANG, K., ZHANG, Q., LI, J. G., LI, X., ZHAO, F., HOU, B., and SUO, T. Abnormal hardening and amorphization in an FCC high entropy alloy under extreme uniaxial tension. International Journal of Plasticity, 159, 103463 (2022) |
| [37] | ZHANG, G., ZHANG, K. S., and FENG, L. On plastic anisotropy of constitutive model for rate-dependent single crystal. Applied Mathematics and Mechanics (English Edition), 26(1), 121–130 (2005) https://doi.org/10.1007/BF02438373 |
| [38] | KOUZNETSOVA, V. G., GEERS, M. G. D., and BREKELMANS, W. A. M. Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Computer methods in Applied Mechanics and Engineering, 193(48-51), 5525–5550 (2004) |
| [39] | KOUZNETSOVA, V. G., GEERS, M. G. D., and BREKELMANS, W. A. M. Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. International Journal for Numerical Methods in Engineering, 54(8), 1235–1260 (2002) |
| [40] | AYAD, M., KARATHANASOPOULOS, N., REDA, H., GANGHOFFER, J. F., and LAKISS, H. On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects. International Journal of Solids and Structures, 190, 58–75 (2020) |
| [41] | AUFFRAY, N., BOUCHET, R., and BRéCHET, Y. Strain gradient elastic homogenization of bidimensional cellular media. International Journal of Solids and Structures, 47(13), 1698–1710 (2010) |
| [42] | KACZMARCZYK, ?., PEARCE, C. J., and BI?ANI?, N. Scale transition and enforcement of RVE boundary conditions in second-order computational homogenization. International Journal for Numerical Methods in Engineering, 74(3), 506–522 (2008) |
| [43] | LI, J., ROMERO, I., and SEGURADO, J. Development of a thermo-mechanically coupled crystal plasticity modeling framework: application to polycrystalline homogenization. International Journal of Plasticity, 119, 313–330 (2019) |
| [44] | CHEN, Y. P. and CAI, Y. Y. Simulation of plastic deformation induced texture evolution using the crystallographic homogenization finite element method. Acta Mechanica Solida Sinica, 30(1), 51–63 (2017) |
| [45] | DURAND, B., LEBé, A., SEPPECHER, P., and SAB, K. Predictive strain-gradient homogenization of a pantographic material with compliant junctions. Journal of the Mechanics and Physics of Solids, 160, 104773 (2022) |
| [46] | SHI, L., LI, J., LIU, P., ZHU, Y., and KANG, Z. A multi-step relay implementation of the successive iteration of analysis and design method for large-scale natural frequency-related topology optimization. Computational Mechanics, 73(2), 403–418 (2024) |
| [47] | YAN, J., HU, W., and DUAN, Z. Structure/material concurrent optimization of lattice materials based on extended multiscale finite element method. International Journal for Multiscale Computational Engineering, 13(1), 73–90 (2015) |
| [48] | ZHU, Y., LUO, J., GUO, X., XIANG, Y., and CHAPMAN, S. J. Role of grain boundaries under long-time radiation. Physical Review Letters, 120(22), 222501 (2018) |
| [49] | CHENG, G. D., CAI, Y. W., and XU, L. Novel implementation of homogenization method to predict effective properties of periodic materials. Acta Mechanica Sinica, 29(4), 550–556 (2013) |
| [50] | GEERS, M. G. D., KOUZNETSOVA, V. G., and BREKELMANS, W. A. M. Multi-scale computational homogenization: trends and challenges. Journal of Computational and Applied Mathematics, 234(7), 2175–2182 (2010) |
| [51] | KOCKS, U. F. and MECKING, H. Physics and phenomenology of strain hardening: the FCC case. Progress in Materials Science, 48(3), 171–273 (2003) |
| [52] | ZHAO, J., LU, X., YUAN, F., KAN, Q., QU, S., KANG, G., and ZHANG, X. Multiple mechanism based constitutive modeling of gradient nanograined material. International Journal of Plasticity, 125, 314–330 (2020) |
| [53] | LI, L., PENG, J., TANG, S., FANG, Q., and WEI, Y. Micromechanism of strength and damage trade-off in second-phase reinforced alloy by strain gradient plasticity theory. International Journal of Plasticity, 176, 103970 (2024) |
| [54] | HARIHARAN, K. and BARLAT, F. Modified Kocks-Mecking-Estrin model to account nonlinear strain hardening. Metallurgical and Materials Transactions A, 50, 513–517 (2019) |
| [55] | GAO, P. F., FEI, M. Y., ZHAN, M., and FU, M. W. Microstructure- and damage-nucleation-based crystal plasticity finite element modeling for the nucleation of multi-type voids during plastic deformation of Al alloys. International Journal of Plasticity, 165, 103609 (2023) |
| [56] | ZENKOUR, A. M. and ABBAS, I. A. Nonlinear transient thermal stress analysis of temperature-dependent hollow cylinders using a finite element model. International Journal of Structural Stability and Dynamics, 14(7), 1450025 (2014) |
| [57] | AGORAS, M., AVAZMOHAMMADI, R., and PONTE CASTA?EDA, P. Incremental variational procedure for elasto-viscoplastic composites and application to polymer- and metal-matrix composites reinforced by spheroidal elastic particles. International Journal of Solids and Structures, 97, 668–686 (2016) |
| [58] | MIEHE, C., MAUTHE, S., and TEICHTMEISTER, S. Minimization principles for the coupled problem of Darcy-Biot-type fluid transport in porous media linked to phase field modeling of fracture. Journal of the Mechanics and Physics of Solids, 82, 186–217 (2015) |
| [59] | MIEHE, C. Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. International Journal for Numerical Methods in Engineering, 55(11), 1285–1322 (2002) |
| [60] | ANAHID, M., SAMAL, M. K., and GHOSH, S. Dwell fatigue crack nucleation model based on crystal plasticity finite element simulations of polycrystalline titanium alloys. Journal of the Mechanics and Physics of Solids, 59(10), 2157–2176 (2011) |
| [61] | JIANG, J., YANG, J., ZHANG, T., ZOU, J., WANG, Y., DUNNE, F. P. E., and BRITTON, T. B. Microstructurally sensitive crack nucleation around inclusions in powder metallurgy nickel-based superalloys. Acta Materialia, 117, 333–344 (2016) |
| [62] | ZHAN, J., YAO, X., HAN, F., and ZHANG, X. A rate-dependent peridynamic model for predicting the dynamic response of particle reinforced metal matrix composites. Composite Structures, 263, 113673 (2021) |
| [63] | LIU, J. L., HUANG, X. Y., ZHAO, K., ZHU, Z. W., ZHU, X. X., and AN, L. N. Effect of reinforcement particle size on quasistatic and dynamic mechanical properties of Al-Al2O3 composites. Journal of Alloys and Compounds, 797, 1367–1371 (2019) |
| [64] | JIA, B., RUSINEK, A., XIAO, X., and WOOD, P. Simple shear behavior of 2024-T351 aluminum alloy over a wide range of strain rates and temperatures: experiments and constitutive modeling. International Journal of Impact Engineering, 156, 103972 (2021) |
/
| 〈 |
|
〉 |
Email Alert
RSS