Accurate simulation for strength-degrading effects of geomaterials via a decoupling approach to treating tension-compression asymmetry

doi:10.1007/s10483-026-3348-6

摘要/Abstract

Abstract:

This study focuses on a new and high-efficiency approach in a unified sense of accurately simulating strength-degrading effects for geomaterials, including non-symmetric hardening-to-softening effects in tension and compression as well as non-symmetric tensile and compressive stiffness-degrading effects during unloading. It is intended to bypass both modeling and numerical complexities involved in existing approaches. To this goal, new elastoplastic equations are established with new numerical techniques. With a decoupling technique of treating tension-compression asymmetry, the foregoing complex effects are automatically incorporated as inherent response features of the new elastoplastic equations, thus bypassing usual modeling complexities. A new numerical technique of renormalizing piecewise spline functions is introduced to resolve the central yet tough issue of obtaining accurate and unified expressions for the tensile and compressive strength functions, thus bypassing usual numerical complexities and uncertainties in treating numerous unknown parameters and multiple ad hoc criteria. As such, the new approach is not only of wide applicability for various geomaterials but also of high computational efficiency with no more than three adjustable parameters. Toward validating the efficacy of the new approach, numerical examples for granite, salt rock, and sandstone-concrete combined body as well as plain concrete, high-performance concrete, and ultrahigh-performance concrete are presented by comparing model predictions with multiple data sets for strength-degrading effects in tension and compression.

Key words: geomaterial, elastoplastic, tension-compression asymmetry, hardening, softening, stiffness degradation, decoupling technique, high-efficiency scheme

中图分类号:

. [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(2): 283-302.

Quanpu LIU, Haonan HE, Siyu WANG, Lin ZHAN, O. BRUHNS, Heng XIAO. Accurate simulation for strength-degrading effects of geomaterials via a decoupling approach to treating tension-compression asymmetry[J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(2): 283-302.

图/表 6

参考文献 47

[48]	BRUHNS, O. T. Large deformation plasticity: from basic relations to finite deformation. Acta Mechanica Sinica, 36, 472–492 (2020)
[49]	XIAO, H., BRUHNS, O. T., and MEYERS, A. Elastoplasticity beyond small deformations. Acta Mechanica, 182, 31–111 (2006)
[50]	XIAO, H., BRUHNS, O. T., and MEYERS, A. Logarithmic strain, logarithmic spin and logarithmic rate. Acta Mechanica, 124, 89–105 (1997)
[51]	DRUCKER, D. C. and PRAGER, W. Soil mechanics and plastic analysis or limit design. Quarterly of Applied Mathematics, 10, 157–165 (1952)
[52]	BRUHNS, O. T., XIAO, H., and MEYERS, A. Self-consistent eulerian rate type elastoplasticity models based upon the logarithmic stress rate. International Journal of Plasticity, 15, 479–520 (1999)
[53]	XIAO, H., BRUHNS, O. T., and MEYERS, A. Thermodynamic laws and consistent Eulerian formulation of finite elastoplasticity with thermal effects. Journal of the Mechanics and Physics of Solids, 55, 338–365 (2007)
[54]	LODE, W. Versuche über den Einfluß der mittleren Hauptspannung auf die Fließgrenze. Journal of Applied Mathematics and Mechanics, 5, 142–144 (1925)
[55]	FROMM, H. Stoffgesetze des isotropen Kontinuums, inbesondere bei zähplastischem Verhalten. Ingenieur-Archiv, 4, 432–466 (1933)
[56]	NOVOZHILOV, V. V. On relations between stresses and strains in non-linear elastic media (in Russian). Journal of Applied Mathematics and Mechanics, 15, 183–194 (1951)
[57]	KOLUPAEV, A. V., YU, M. H., ALTENBACH, H., and BOLCHOUN, A. Comparison of strength criteria based on the measurements on concrete. Journal of Engineering Mechanics, 144, 04018028 (2018)
[58]	ALTENBACH, H. and KOLUPAEV, A. V. General forms of limit surface: application for isotropic materials. Material Modeling and Structural Mechanics, Springer, Cham, 19–94 (2022)
[59]	BRUHNS, O. T. Some remarks on Lode angle and Lode parameter. Proceedings in Applied Mathematics and Mechanics, 24, e202400004 (2024)
[60]	XIAO, H. An explicit, direct approach to obtaining multiaxial elastic potentials that exactly match data of four benchmark tests for rubbery materials-part 1: incompressible deformations. Acta Mechanica, 223, 2039–2063 (2012)
[61]	LIU, Q. P., WANG, H. Y., WANG, S. Y., XIAO, H., and BRUHNS, O. T. An accurate and unified study on non-symmetric tensile and compressive responses of elastoplastic bars and beams until failure. Acta Mechanica, 234, 6561–6577 (2023)
[62]	VON KÁRMÁN, T. Festigkeitsversuche unter allseitigem Druck. Z. VDI., 55, 1749–1759 (1911)
[63]	BELL, J. F. Volume I: The Experimental Foundations of Solid Mechanics, Springer, Berlin (1984)
[64]	ZHAN, L., WANG, S. Y., BRUHNS, O. T., and XIAO, H. High-efficiency algorithms for metal fatigue failure effects under multi-axial repeated loadings. International Journal for Numerical Methods in Engineering, 123, 1277–1293 (2022)
[65]	XIAO, H., BRUHNS, O. T., and MEYERS, A. Free rate-independent elastoplastic equations. Journal of Applied Mathematics and Mechanics, 94, 461–476 (2014)
[66]	XIAO, H. Thermo-coupled elastoplasticity models with asymptotic loss of the material strength. International Journal of Plasticity, 63, 211–228 (2014)
[67]	XIAO, H., BRUHNS, O. T., and MEYERS, A. Surprises due to new elastoplastic equations of Prandtl-Reuss type. Acta Mechanica Sinica, 41, 425087 (2025)
[68]	WANG, H., CAI, L., and XIAO, H. A unified model for obtaining stress-strain relationship under spherical indenter loading and test application. Applied Mathematics and Mechanics (English Edition), 46(8), 1591–1608 (2025) https://doi.org/10.1007/s10483-025-3283-6
[1]	KARSAN, I. D. and JIRSA, J. O. Behavior of concrete under compressive loadings. Journal of the Structural Division, 95, 2543–2564 (1969)
[2]	OKUBO, S. and FUKUI, K. Complete stress-strain curves for various rock types in uniaxial tension. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 33, 549–556 (1996)
[3]	GRAYBEAL, B. A. Compressive behavior of ultrahigh-performance-fiber-reinforced concrete. ACI Materials Journal, 104, 146–152 (2007)
[4]	THOMAS, J. and RAMASWAMY, A. Mechanical properties of steel fiber-reinforced concrete. Journal of Materials in Civil Engineering, 19, 385–392 (2007)
[5]	GRAYBEAL, B. A. Development of direct tension test method for ultra-high-performance fiberreinforced concrete. ACI Materials Journal, 110, 177–186 (2013)
[6]	SHAH, S. P. Determination of fracture parameters (KIcs and CTOD_c) of plain concrete using three-point bend tests. Materials and Structures, 23, 457–460 (1990)
[7]	BAŽANT, Z. P. and KAZEMI, M. T. Determination of fracture energy, process zone length and brittleness number from size effect, with application to rock and concrete. International Journal of Fracture, 44, 111–131 (1990)
[8]	KARIHALOO, B. L. and NALLATHAMBI, P. An improved effective crack model for the determination of fracture toughness of concrete. Cement and Concrete Research, 19, 603–610 (1989)
[9]	XU, S. and REINHARDT, H. W. Crack extension resistance and fracture properties of quasi-brittle softening materials like concrete based on the complete process of fracture. International Journal of Fracture, 92, 71–99 (1998)
[10]	KUHN, C., SCHLÜTER, A., and MÜLLER, R. On degradation functions in phase field fracture models. Computational Materials Science, 108, 374–384 (2015)
[11]	SCHRÖDER, J., PISE, M., BRANDS, D., GEBUHR, G., and ANDERS, S. Phase-field modeling of fracture in high performance concrete during low-cycle fatigue: numerical calibration and experimental validation. Computer Methods in Applied Mechanics and Engineering, 398, 115181 (2022)
[12]	SUN, Y., ROUBIN, E., SHAO, J. F., and COLLIAT, J. B. FE modeling of concrete with strong discontinuities for 3D shear fractures and comparison with experimental results. Engineering Fracture Mechanics, 251, 107752 (2021)
[13]	BAŽANT, Z. P. and OH, B. H. Microplane model for progressive fracture of concrete and rock. Journal of Engineering Mechanics, 111, 559–582 (1985)
[14]	CAROL, I. and BAŽANT, Z. P. Damage and plasticity in microplane theory. International Journal of Solids and Structures, 34, 3807–3835 (1997)
[15]	BAŽANT, Z. P. and ZI, G. Microplane constitutive model for porous isotropic rocks. International Journal for Numerical and Analytical Methods in Geomechanics, 27, 25–47 (2003)
[16]	CHEN, X. and BAŽANT, Z. P. Microplane damage model for jointed rock masses. International Journal for Numerical and Analytical Methods in Geomechanics, 38, 1431–1452 (2014)
[17]	SIMO, J. C. and JU, J. W. Strain- and stress-based continuum damage model: I, formulation. International Journal of Solids and Structures, 23, 821–840 (1987)
[18]	LUBLINER, J., OLIVER, J., OLLER, S., and ONATE, E. A plastic-damage model for concrete. International Journal of Solids and Structures, 25, 299–326 (1989)
[19]	CAROL, I., RIZZI, E., and WILLAM, K. A unified therory of elastic degradation and damage based on a loading surface. International Journal of Solids and Structures, 31, 2835–2865 (1994)
[20]	SWOBODA, G., SHEN, X. P., and ROSAS, L. Damage model for jointed rock mass and its application to tunelling. Computers and Geotechnics, 22, 183–203 (1998)
[21]	SALARI, M. R., SAEB, S., WILLAM, K. J., PATCHET, S. J., and CARRASCO, R. C. A coupled elastoplastic damage model for geomaterials. Computer Methods in Applied Mechanics and Engineering, 193, 2625–2643 (2004)
[22]	CICEKLI, U., VOYIADJIS, G. Z., and ABU AL-RUB, R. K. A plasticity and anisotropic damage model for plain concrete. International Journal of Plasticity, 23, 1874–1900 (2007)
[23]	NGUYEN, V. P., STROEVEN, M., and SLUYS, L. J. Multiscale failure modeling of concrete: micromechanical modeling, discontinuous homogenization and parallel computations. Computer Methods in Applied Mechanics and Engineering, 201, 139–156 (2012)
[24]	ZHANG, X. C., LU, Y., WU, X. G., WANG, P. Y., LI, Y., LIU, Y., SHEN, C., ZHANG, H. M., and ZHANG, D. Constitutive model for ultra-high performance concrete (UHPC) considering the size effect under cyclic compressive loading. Construction and Building Materials, 368, 130499 (2023)
[25]	WU, J. Y., LI, J., and FARIA, R. An energy release rate-based plastic-damage model for concrete. International Journal of Solids and Structures, 43, 583–612 (2006)
[26]	ZHANG, J., LI, J., and JU, J. W. 3D elastoplastic damage model for concrete based on novel decomposition of stress. International Journal of Solids and Structures, 94-95, 125–137 (2016)
[27]	ZHANG, J. C. Experimental and modelling investigations of the coupled elastoplastic damage of a quasi-brittle rock. Rock Mechanics and Rock Engineering, 51, 465–478 (2018)
[28]	SARIKAYA, A. and ERKMEN, R. E. A plastic-damage model for concrete under compression. International Journal of Mechanical Sciences, 150, 584–593 (2019)
[29]	MURAKAMI, S. Continuum Damage Mechanics: a Continuum Mechanics Approach to the Analysis of Damage and Fracture, Springer, Berlin (2012)
[30]	GROSS, D. and SEELIG, T. Fracture Mechanics: With an Introduction to Micromechanics, Springer, Berlin (2006)
[31]	VOYIADJIS, G. Z. Handbook of Damage Mechanics: Nano to Macro Scale for Materials and Structures, Springer, New York (2022)
[32]	WAN, R. and GUO, P. A simple constitutive model for granular soils: modified stress-dilatancy approach. Computers and Geotechnics, 22, 109–133 (1998)
[33]	GAJO, A. and WOOD, D. Severn-trent sand: a kinematic hardening constitutive model: the q-p formulation. Géotechnique, 49, 595–614 (1999)
[34]	MALEKI, M., CAMBOU, B., and DUBUJET, P. Development in modeling cyclic loading of sands based on kinematic hardening. International Journal for Numerical and Analytical Methods in Geomechanics, 33, 1641–1658 (2009)
[35]	QIAN, J. G., YANG, J., and HUANG, M. S. Three-dimensional noncoaxial plasticity modeling of shear band formation in geomaterials. Journal of Engineering Mechanics, 134, 322–329 (2008)
[36]	DI BENEDETTO, H., BLANC, M., TIOUAJNI, S., and EZAOUI, A. Elastoplastic model with loading memory surfaces (LMS) for monotonic and cyclic behaviour of geomaterials. International Journal for Numerical and Analytical Methods in Geomechanics, 38, 1477–1502 (2014)
[37]	HASHIGUCHI, K. Foundation of Elastoplasticity: Subloading Surface Model, Springer, New York (2017)
[38]	CHENG, H. Y., THORNTON, A. R., LUDING, S., HAZEL, A. L., and WEINHART, T. Concurrent multi-scale modeling of granular materials: role of coarse-graining in FEM-DEM coupling. Computer Methods in Applied Mechanics and Engineering, 403, 115651 (2023)
[39]	LV, Y., ZHANG, R., ZHANG, A. L., ZHANG, Z. T., REN, L., XIE, J., and ZHA, E. S. Mechanical and degradation properties of rock under triaxial loading-unloading stresses: comparative insights into coal and sandstone. International Journal of Rock Mechanics and Mining Sciences, 190, 106102 (2025)
[40]	GU, Q. X., ZHANG, Q., LI, Y. P., WU, P. N., and HAN, G. L. An improved elastoplastic model for rocks and application to cyclic loading and unloading triaxial compression tests. Acta Mechanica Sinica, 40, 424053 (2024)
[41]	WANG, H. Y., ZHANG, Q., LIU, R. C., LI, T., QUAN, X. W., and JIANG, Y. J. A cohesive-frictional elastoplastic constitutive model for rock joint. Journal of Rock Mechanics and Geotechnical Engineering, 17, 1068–1084 (2025)
[42]	WANG, T. M., YE, J. H., WU, F. Q., ZHANG, Y. S., and PENG, J. B. Elastoplastic model of statistical mechanics of rock masses (SMRM): theory and verification. Engineering Geology, 346, 107897 (2025)
[43]	YU, H. S. Plasticity and Geotechnics, Springer, Boston (2006)
[44]	JING, L. A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering. International Journal of Rock Mechanics and Mining Sciences, 40, 283–353 (2003)
[45]	JING, L. R. and STEPHANSSON, O. Constitutive models of rock fractures and rock masses-the basics. Developments in Geotechnical Engineering, 85, 47–109 (2007)
[46]	WU, F. Q., WU, J., BAO, H., LI, B., SHAN, Z. G., and KONG, D. H. Advances in statistical mechanics of rock masses and its engineering applications. Journal of Rock Mechanics and Geotechnical Engineering, 13, 22–45 (2021)
[69]	BELLO, I., GONZALEZ-FONTEBOA, B., WARDEH, G., and MARTÍNEZ-ABELLA, F. Characterization of concrete behavior under cyclic loading using 2D digital image correlation. Journal of Building Engineering, 78, 107709 (2023)
[70]	WANG, S. Y., ZHAN, L., XI, H. F., BRUHNS, O. T., and XIAO, H. Unified simulation of hardening and softening effects for metals up to failure. Applied Mathematics and Mechanics (English Edition), 42(12), 1685–1702 (2021) https://doi.org/10.1007/s10483-021-2793-6
[71]	YUAN, S. S., DU, B., and SHEN, M. X. Experimental and numerical investigation of the mechanical properties and energy evolution of sandstone-concrete combined body. Scientific Reports, 14, 5214 (2024)
[72]	LIU, J., WANG, C., WU, Z., WANG, L., XU, H., PEI, J., and DENG, C. A new approach of rock tension-compression cyclic measurement. Géotechnique Letters, 9, 89–93 (2019)
[73]	MENG, Q. B., LIU, J. F., PU, H., HUANG, B. X., ZHANG, Z. Z., and WU, J. Y. Effects of cyclic loading and unloading rates on the energy evolution of rocks with different lithology. Geomechanics for Energy and the Environment, 34, 100455 (2023)
[74]	TAYLOR, R. L. FEAP — A Finite Element Analysis Program, Installation Manual (1992)
[47]	BRUHNS, O. T. The Prandtl-Reuss equations revisited. Journal of Applied Mathematics and Mechanics, 94, 187–202 (2014)