Effect of boundary conditions on shakedown analysis of heterogeneous materials

• Published in: Applied mathematics and mechanics Vol. 45; no. 1; pp. 39 - 68
• Main Authors: Gong, Xiuchen, Nie, Yinghao, Cheng, Gengdong, Li, Kai
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2024; Springer Nature B.V
• Edition: English ed.
• Subjects: Apexes; Applications of Mathematics; Bearing capacity; Boundary conditions; Classical Mechanics; Elastic limit; Elastoplasticity; Fluid- and Aerodynamics; Load bearing elements; Lower bounds; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Shakedown analysis; Stress distribution; Theorems; Ultimate loads; heterogeneous material; 74C99; shakedown analysis; effect of boundary conditions; O344.5; self-equilibrium stress field (SSF)
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-024-3073-9

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.