We present in this study a new approach for predicting the plastic and shakedown limits of structures composed of orthotropic materials. In this approach, the Hill yield criterion is introduced to Melan’s theorem. By formulating the problem by means of the finite element method and solving the resulting large-scale nonlinear optimization problem we successfully predict the plastic and shakedown limits of structures having complex geometries made from multi-orthotropic materials. Several numerical examples are elaborated in this study for evaluating the accuracy, general applicability, as well as the efficiency of the established numerical scheme. Overall, the study confirms that the direct method can be extended and adopted as a viable means for design and analysis of structures made of orthotropic materials.
MelanE. Zur plastizität des räumlichen kontinuums. Arch Appl Mech1938; 9: 116–126.
2.
KoiterWT. A new general theorem on shakedown of elastic-plastic structures. Proc Koninkl Ned Akad Wet B1956; 59: 24–34.
3.
WeichertDHachemiASchwabeF. Application of shakedown analysis to the plastic design of composites. Arch Appl Mech1999; 69: 623–633.
4.
HachemiAMouhtamidSWeichertD. Progress in shakedown analysis with applications to composites. Arch Appl Mech2005; 74: 762–772.
5.
WeichertDPonterA. Limit states of materials and structures: direct methods. Dordrecht: Springer Netherlands, 2009.
6.
HachemiAChenMChenG, et al. Limit state of structures made of heterogeneous materials. Int J Plast2014; 63: 124–137.
7.
ChenGBezoldABroeckmannC, et al. On the size of the representative volume element used for the strength prediction: a statistical survey applied to the particulate reinforce metal matrix composites (PRMMCs). In: Advances in direct methods for materials and structures. Cham, Switzerland: Springer, 2018, 51–72.
8.
KarmarkarN. A new polynomial-time algorithm for linear programming. In: Proceedings of the Sixteenth Annual ACM Symposium on Theory of Computing, 1 December 1984, 302–311. New York: ACM.
9.
NguyenADHachemiAWeichertD. Application of the interior-point method to shakedown analysis of pavements. Int J Numer Methods Eng2008; 75: 414–439.
10.
SimonJWWeichertD. Interior-point method for lower bound shakedown analysis of von Mises-type materials. In: Limit state of materials and structures. Dordrecht: Springer, 2013, 103–128.
11.
GarceaGLeonettiL. A unified mathematical programming formulation of strain driven and interior point algorithms for shakedown and limit analysis. Int Numer Methods Eng2011; 88: 1085–1111.
12.
WrightM. The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bull Am Math Soc2005; 42: 39–56.
13.
PastorJTurgemanSBoehlerJP. Solution of anisotropic plasticity problems by using associated isotropic problems. Int J Plast1990; 6: 143–168.
14.
ZhangYGLuMW. Computational limit analysis of anisotropic axisymmetric shells. Int J Press Vessels Pip1994; 58: 283–287.
15.
YuHSSloanSW. Limit analysis of anisotropic soils using finite elements and linear programming. Mech Res Commun1994; 21: 545–554.
16.
YuHSSloanSW. Finite element limit analysis of reinforced soils. Comput Struct1997; 63: 567–577.
17.
CapsoniACorradiLVenaP. Limit analysis of orthotropic structures based on Hill’s yield condition. Int J Solids Struct2001; 38: 3945–3963.
18.
ZhangHTLiuYHXuBY. Lower bound limit and shakedown analysis of orthotropic structures. Eng Mech2006; 23: 11–16.
19.
LiHX. Limit analysis of composite materials with anisotropic microstructures: a homogenization approach. Mech Mater2011; 43: 574–585.
20.
LiHX. Kinematic shakedown analysis of anisotropic heterogeneous materials: a homogenization approach. J Appl Mech2012; 79: 041016.
21.
ZhangHLiuYXuB. Plastic limit analysis of ductile composite structures from micro-to macro-mechanical analysis. Acta Mech Solida Sin2009; 22: 73–84.
22.
ChenMHachemiA. Progress in plastic design of composites. In: Direct methods for limit states in structures and materials. Dordrecht: Springer, 2014, 119–138.
23.
ChenMHachemiAWeichertD. Shakedown and optimization analysis of periodic composites. In: Limit state of materials and structures. Dordrecht: Springer, 2013, 45–69.
24.
ChenGWangHBezoldA, et al. Strengths prediction of particulate reinforced metal matrix composites (PRMMCs) using direct method and artificial neural network. Compos Struct2019; 223: 110951.
25.
ChenGJiangKZhangL, et al. A Bayesian statistics based investigation of binder hardening’s influence on the effective strength of particulate reinforced metal matrix composites (PRMMC). Int J Mech Sci2019; 159: 151–164.
26.
ChenGZhangLBezoldA, et al. Statistical investigation on influence of grain size on effective strengths of particulate reinforced metal matrix composites. Comput Methods Appl Mech Eng2019; 352: 691–707.
27.
SimonJW. Direct evaluation of the limit states of engineering structures exhibiting limited, nonlinear kinematical hardening. Int J Plast2013; 42: 141–167.
ChenMHachemiAMarkertB. Direct evaluation of limit states of heterogeneous materials under cyclic thermo-mechanical loadings. PAMM2014; 14: 363–364.
30.
ChenGSimonJWWeichertD. A direct evaluation of plastic shakedown limit of a piping nozzle structure under superposed multiple loadings. In: ASME 2013 Pressure Vessels and Piping Conference. New York: ASME.
31.
HuangSXuYBezoldA, et al. A direct method-based strength evaluation of the cast aluminum beam used in a high-speed train. Proc IMechE F2019; 233: 896–905.
32.
HuangSXuYChenG, et al. A numerical shakedown analysis method for strength evaluation coupling with kinematical hardening based on two surface model. Eng Fail Anal2019; 103: 275–285.
33.
HillR. A theory of the yielding and plastic flow of anisotropic metals. Proc R Soc Lond A1948; 193: 281–297.
34.
TsaiSWWuEM. A general theory of strength for anisotropic materials. J Compos Mater1971; 5: 58–80.
35.
FrançoisAAbdelkaderHSaidM, et al. Application of lower bound direct method to engineering structures. J Global Optim2007; 37: 609–630.
36.
CarvelliVCenZZLiuY, et al. Shakedown analysis of defective pressure vessels by a kinematic approach. Arch Appl Mech1999; 69: 751–764.
37.
ChenSLiuYCenZ. Lower bound shakedown analysis by using the element free Galerkin method and non-linear programming. Comput Methods Appl Mech Eng2008; 197: 3911–3921.
