Applied Mathematics and Mechanics >
Effect of boundary conditions on shakedown analysis of heterogeneous materials
Received date: 2023-08-31
Online published: 2023-12-26
Supported by
the National Natural Science Foundation of China(52075070);the National Natural Science Foundation of China(12302254);the Dalian City Supports Innovation and Entrepreneurship Projects for High-Level Talents(2021RD16);the Liaoning Revitalization Talents Program(XLYC2002108);Project supported by the National Natural Science Foundation of China (Nos. 52075070 and 12302254), the Dalian City Supports Innovation and Entrepreneurship Projects for High-Level Talents (No. 2021RD16), and the Liaoning Revitalization Talents Program (No. XLYC2002108)
Copyright
The determination of the ultimate load-bearing capacity of structures made of elastoplastic heterogeneous materials under varying loads is of great importance for engineering analysis and design. Therefore, it is necessary to accurately predict the shakedown domains of these materials. The static shakedown theorem, also known as Melan's theorem, is a fundamental method used to predict the shakedown domains of structures and materials. Within this method, a key aspect lies in the construction and application of an appropriate self-equilibrium stress field (SSF). In the structural shakedown analysis, the SSF is typically constructed by governing equations that satisfy no external force (NEF) boundary conditions. However, we discover that directly applying these governing equations is not suitable for the shakedown analysis of heterogeneous materials. Researchers must consider the requirements imposed by the Hill-Mandel condition for boundary conditions and the physical significance of representative volume elements (RVEs). This paper addresses this issue and demonstrates that the sizes of SSFs vary under different boundary conditions, such as uniform displacement boundary conditions (DBCs), uniform traction boundary conditions (TBCs), and periodic boundary conditions (PBCs). As a result, significant discrepancies arise in the predicted shakedown domain sizes of heterogeneous materials. Built on the demonstrated relationship between SSFs under different boundary conditions, this study explores the conservative relationships among different shakedown domains, and provides proof of the relationship between the elastic limit (EL) factors and the shakedown loading factors under the loading domain of two load vertices. By utilizing numerical examples, we highlight the conservatism present in certain results reported in the existing literature. Among the investigated boundary conditions, the obtained shakedown domain is the most conservative under TBCs. Conversely, utilizing PBCs to construct an SSF for the shakedown analysis leads to less conservative lower bounds, indicating that PBCs should be employed as the preferred boundary conditions for the shakedown analysis of heterogeneous materials.
Xiuchen GONG, Yinghao NIE, Gengdong CHENG, Kai LI . Effect of boundary conditions on shakedown analysis of heterogeneous materials[J]. Applied Mathematics and Mechanics, 2024 , 45(1) : 39 -68 . DOI: 10.1007/s10483-024-3073-9
| 1 | KÖNIG, J. Shakedown of Elastic-Plastic Structures, Elsevier, Amsterdam (2012) |
| 2 | PENG, H., LIU, Y., CHEN, H., and ZHANG, Z. Shakedown analysis of bounded kinematic hardening engineering structures under complex cyclic loads: theoretical aspects and a direct approach. Engineering Structures, 256, 114034 (2022) |
| 3 | DO, H., and NGUYEN-XUAN, H. Limit and shakedown isogeometric analysis of structures based on Bézier extraction. European Journal of Mechanics A-Solids, 63, 149- 164 (2017) |
| 4 | LI, K., CHENG, G., WANG, Y., and LIANG, Y. A novel primal-dual eigenstress-driven method for shakedown analysis of structures. International Journal for Numerical Methods in Engineering, 122 (11), 2770- 2801 (2021) |
| 5 | CHEN, G., BEZOLD, A., and BROECKMANN, C. Influence of the size and boundary conditions on the predicted effective strengths of particulate reinforced metal matrix composites (PRMMCs). Composite Structures, 189, 330- 339 (2018) |
| 6 | CHEN, G., XIN, S., ZHANG, L., and BROECKMANN, C. Statistical analyses of the strengths of particulate reinforced metal matrix composites (PRMMCs) subjected to multiple tensile and shear stresses. Chinese Journal of Mechanical Engineering, 34 (1), 1- 12 (2021) |
| 7 | HACHEMI, A., CHEN, M., CHEN, G., and WEICHERT, D. Limit state of structures made of heterogeneous materials. International Journal of Plasticity, 63, 124- 137 (2014) |
| 8 | LE, C., NGUYEN, P., ASKES, H., and PHAM, D. A computational homogenization approach for limit analysis of heterogeneous materials. International Journal for Numerical Methods in Engineering, 112 (10), 1381- 1401 (2017) |
| 9 | LI, H., and YU, H. A non-linear programming approach to kinematic shakedown analysis of composite materials. International Journal for Numerical Methods in Engineering, 66 (1), 117- 146 (2006) |
| 10 | NGUYEN, P., and LE, C. Failure analysis of anisotropic materials using computational homogenised limit analysis. Computers Structures, 256, 106646 (2021) |
| 11 | MAGOARIEC, H., BOURGEOIS, S., and DÉBORDES, O. Elastic plastic shakedown of 3D periodic heterogeneous media: a direct numerical approach. International Journal of Plasticity, 20 (8-9), 1655- 1675 (2004) |
| 12 | GARCEA, G., and LEONETTI, L. A unified mathematical programming formulation of strain driven and interior point algorithms for shakedown and limit analysis. International Journal for Numerical Methods in Engineering, 88 (11), 1085- 1111 (2011) |
| 13 | MELAN, E. Zur Plastizität des räumlichen Kontinuums. Ingenieur-Archiv, 9 (2), 116- 126 (1938) |
| 14 | KOITER, W. General theorems for elastic plastic solids. Progress in Solid Mechanics, 1, 165- 221 (1960) |
| 15 | WEICHERT, D. On the influence of geometrical nonlinearities on the shakedown of elastic-plastic structures. International Journal of Plasticity, 2 (2), 135- 148 (1986) |
| 16 | FRANÇOIS, A., ABDELKADER, H., HOAI, AN L., SAID, M., and TAO, P. Application of lower bound direct method to engineering structures. Journal of Global Optimization, 37 (4), 609- 630 (2007) |
| 17 | RI, J., and HONG, H. A basis reduction method using proper orthogonal decomposition for shakedown analysis of kinematic hardening material. Computational Mechanics, 64 (1), 1- 13 (2019) |
| 18 | CHEN, M. and HACHEMI, A. Progress in plastic design of composites. Direct Methods for Limit States in Structures and Materials(eds. SPILIOPOULOS, K. and WEICHERT, D.), Springer, Dordrecht, 119-138 (2014) |
| 19 | KLEBANOV, J., and BOYLE, J. Shakedown of creeping structures. International Journal of Solids Structures, 35 (23), 3121- 3133 (1998) |
| 20 | YAN, J., CHENG, G., LIU, S., and LIU, L. Comparison of prediction on effective elastic property and shape optimization of truss material with periodic microstructure. International Journal of Mechanical Sciences, 48 (4), 400- 413 (2006) |
| 21 | YAN, J., CHENG, G., LIU, S., and LIU, L. Prediction of equivalent elastic properties of truss materials with periodic microstructure and the scale effects (in Chinese). Chinese Journal of Solid Mechanics, 26 (4), 421- 428 (2005) |
| 22 | HEITZER, M., POP, G., and STAAT, M. Basis reduction for the shakedown problem for bounded kinematic hardening material. Journal of Global Optimization, 17 (1), 185- 200 (2000) |
| 23 | PENG, H., LIU, Y., and CHEN, H. A numerical formulation and algorithm for limit and shakedown analysis of large-scale elastoplastic structures. Computational Mechanics, 63, 1- 22 (2019) |
| 24 | PENG, H., and LIU, Y. Stress compensation method for structural shakedown analysis. Key Engineering Materials, 794, 169- 181 (2019) |
| 25 | TARN, J., DVORAK, G., and RAO, M. Shakedown of unidirectional composites. International Journal of Solids Structures, 11 (6), 751- 764 (1975) |
| 26 | WEICHERT, D., HACHEMI, A., and SCHWABE, F. Application of shakedown analysis to the plastic design of composites. Archive of Applied Mechanics, 69 (9), 623- 633 (1999) |
| 27 | WEICHERT, D., HACHEMI, A., and SCHWABE, F. Shakedown analysis of composites. Mechanics Research Communications, 26, 309- 318 (1999) |
| 28 | CHEN, M., HACHEMI, A., and WEICHERT, D. Shakedown and optimization analysis of periodic composites. Limit State of Materials and Structures(eds. DE SAXCÉ, G., OUESLATI, A., CHARKALUK, E., and TRITSCH, J.), Springer, London, 45-69 (2013) |
| 29 | CHEN, M., ZHANG, L., WEICHERT, D., and TANG, W. Shakedown and limit analysis of periodic composites. PAMM: Proceedings in Applied Mathematics and Mechanics, 9 (1), 415- 416 (2009) |
| 30 | RI, J., and HONG, H. A basis reduction method using proper orthogonal decomposition for lower bound shakedown analysis of composite material. Archive of Applied Mechanics, 88 (10), 1843- 1857 (2018) |
| 31 | RI, J., RI, U., HONG, H., and KWAK, C. Eigenstress-based shakedown analysis for estimation of effective strength of composites under variable load. Composite Structures, 280, 114851 (2022) |
| 32 | XIA, Z., ZHOU, C., YONG, Q., and WANG, X. On selection of repeated unit cell model and application of unified periodic boundary conditions in micro-mechanical analysis of composites. International Journal of Solids Structures, 43 (2), 266- 278 (2006) |
| 33 | MURA, T Micromechanics of Defects in Solids, Springer Science & Business Media, Berlin (2013) |
| 34 | MACKENZIE, D., SHI, J., and BOYLE, J. Finite element modelling for limit analysis by the elastic compensation method. Computers Structures, 51, 403- 410 (1994) |
| 35 | CHEN, H. Lower and upper bound shakedown analysis of structures with temperature-dependent yield stress. Journal of Pressure Vessel Technology, 132 (1), 011202 (2010) |
| 36 | BORINO, G., and POLIZZOTTO, C. Dynamic shakedown of structures with variable appended masses and subjected to repeated excitations. International Journal of Plasticity, 12, 215- 228 (1996) |
| 37 | CHRISTIANSEN, E., and ANDERSEN, K. Computation of collapse states with von Mises type yield condition. International Journal for Numerical Methods in Engineering, 46, 1185- 1202 (1998) |
| 38 | HACHEMI, A., and WEICHERT, D. Numerical shakedown analysis of damaged structures. Computer Methods in Applied Mechanics and Engineering, 160, 57- 70 (1998) |
| 39 | SIMON, J. Limit states of structures in n-dimensional loading spaces with limited kinematical hardening. Computers Structures, 147, 4- 13 (2015) |
| 40 | Gurobi Optimization, Inc. Gurobi Optimizer Reference Manual(2023) http://www.gurobi.com |
| 41 | GRANT, M. and BOYD, S. CVX: Matlab Software for Disciplined Convex Programming, version 2.1 (2014) http://cvxr.com/cvx |
| 42 | HORI, M., and NEMAT-NASSER, S. On two micromechanics theories for determining micro-macro relations in heterogeneous solids. Mechanics of Materials, 31 (10), 667- 682 (1999) |
| 43 | NIE, Y., LI, Z., and CHENG, G. Efficient prediction of the effective nonlinear properties of porous material by FEM-cluster based analysis (FCA). Computer Methods in Applied Mechanics Engineering, 383, 113921 (2021) |
| 44 | ANDERSEN, M., POULSEN, P., and OLESEN, J. Partially mixed lower bound constant stress tetrahedral element for finite element limit analysis. Computers Structures, 258, 106672 (2022) |
| 45 | ZHANG, H., LIU, Y., and XU, B. Plastic limit analysis of ductile composite structures from micro- to macro-mechanical analysis. Acta Mechanica Solida Sinica, 22 (1), 73- 84 (2009) |
| 46 | NIE, Y., LI, Z., GONG, X., and CHENG, G. Fast construction of cluster interaction matrix for data-driven cluster-based reduced-order model and prediction of elastoplastic stress-strain curves and yield surface. Computer Methods in Applied Mechanics Engineering, 418, 116480 (2024) |
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