Dynamic Response Analysis for a Terminal Guided Projectile with a Trajectory Correction Fuze

Rupeng Li, Dongguang Li, Jieru Fan
2019 IEEE Access  
Shear and membrane locking phenomena are fundamental issues of shell finite element models. A family of refined shell elements for laminated structures has been developed in the framework of Carrera Unified Formulation, including hierarchical elements based on higher-order Legendre polynomial expansions. These hierarchical elements were reported to be relatively less prone to locking phenomena, yet an exhaustive evaluation of them regarding the mitigation of shear and membrane locking on
more » ... e locking on laminated shells is still essential. In the present article, numerically efficient integration schemes for hierarchical elements, including also reduced and selective integration procedures, are discussed and evaluated through single-element p-version finite element models. Both shear and membrane locking are assessed quantitatively through the estimation of strain energy components. The numerical results show that the fully integrated hierarchical shell elements can overcome the shear and membrane locking effectively when a sufficiently high polynomial degree is reached. Reduced and selective integration schemes can help with the mitigation of locking on lower-order hierarchical shell elements. which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 0123456789().,-: volV Li et al. Adv. Model. and Simul. in Eng. Sci. (2019) 6:8 Page 2 of 24 mitigation of shear and membrane locking on refined multi-layered shell finite elements remains to be quantitatively assessed. The locking phenomena are caused by the greatly overestimated stiffness of thin structures and will lead to a loss of convergence rate of the numerical solution. If no treatment is introduced, the meshes of the shell FE models have to be immensely refined, which makes the analysis numerically prohibitive. Shear locking, caused by the so-called "parasitic shear" in the bending of a thin shell, is a typical locking phenomenon [13] . Due to the incompetence of the shell elements in capturing the bending deformation appropriately, the strain energy is absorbed by the shear mode erroneously. As the shell structures become thinner, the transverse shear energy approaches zero, physically. On the other hand, membrane locking can be observed on shell elements when bending deformation is incorrectly accompanied by the stretching of the mid-surface, and the membrane energy overshadows the bending part [14, 15] . Pioneering simple remedies to the locking phenomena are the reduced integration and selectively reduced integration techniques [13, 16] . Zienkiewicz et al. [13] pointed out that by reducing the order of numerical integration, the stiffness of displacement-based finite elements can be decreased. The main idea of selective integration is to reduce the shear stiffness, and the reduced quadrature is selectively used on the stiffness component related to transverse shear. This method is reported to be useful in bending problems yet was found less effective compared with uniformly reduced integration on all the stiffness components for general shell problems [13] . This reduced integration approach brings significant improvement to the convergence rate. The equivalence of the reduced integration procedure with mixed formulation was demonstrated by Malkus and Hughes [16] and Zienkiewicz and Nakazawa [17] . A drawback of the reduced integration technique is the introduction of "spurious modes" due to the erroneously evaluated stiffness matrix. A typical example is the "hour-glass" mode of four-node bi-linear shell element with reduced integration. Zienkiewicz and Taylor [18] commented that for general applications mixed elements are preferred than reduced integration procedures. This numerical singularity problem can also be avoided by using alternative techniques such as the Mixed Interpolation of Tensorial Components (MITC) proposed by . In the MITC formulation, the shear locking can be overcome by the additional independent interpolation functions for the transverse shear strains. This approach is also referred to as the "assumed shear strain field" method [23] . The link between MITC formulation and the Hellinger-Reissner mixed variation principle was demonstrated by Bathe et al. [22] . The mathematical justification of MITC formulation was established through the Babuska-Brezzi conditions [24] . In the framework of CUF, MITC has been successfully applied to build locking-free refined elements with variable kinematics for multi-layered plates by and for shell structures by Cinefra and Valvano [11] and Carrera et al. [26] . An extension of MITC technique to beam elements was also addressed by Carrera et al. [28] . Very recent developments of four-node MITC elements were presented by Ko et al. [29, 30] . Indeed, shear locking effects are more pronounced on low-order elements [31] . The loss of convergence can be alleviated by adopting higher order elements [32, 33] , such as higherorder hierarchical elements [34, 35] . A combination of hierarchical elements and mixed interpolation method was proposed and applied to isotropic plates based on Reissner-Mindlin assumption by Scapolla and Della Croce [36, 37] . An application of hierarchi-
doi:10.1109/access.2019.2928718 fatcat:nzsfcezktre4pipnnki6z7c5m4