The “Myth of No-Locking” in Nonlinear Analysis of Shells
Much research has been focused on the problem of locking of traditional displacement-based elements. It is well accepted that displacement-based elements should not be used for the incompressible analysis of solids. For linear and nonlinear analyses of incompressible media (specifically, rubber-like materials, large strain inelastic materials) mixed elements based on displacement and pressure interpolations (the u/p elements) are much more effective, see for example Ref. . These elements are now widely employed.
A precise definition of locking is given in Refs. [1-3], for an illustrative Example see page 281 of Ref. . From a practical point of view, when elements or meshes that lock are used, the results of the analysis are much less accurate than they should be. The elements store spurious stresses, deform much less than expected, that is, they "lock".
The problem of shear and membrane locking in the analysis of shells is also well known and much research has also been focused on the development of powerful shell elements, see Refs. [1, 2]. Displacement-based elements lock as well, although the locking is much alleviated by using high-order elements (cubic and higher). But such elements are computationally expensive and in nonlinear analysis not efficient when contact and nonlinear material conditions are present. In current practice, by far, most often low-order 4-node quadrilateral shell elements are used. Only sometimes, the 8- or 9-node quadrilateral or 6-node triangular elements are employed, although these elements (in mixed form) can be much more effective (and hence the practice may change over the years to come).
While the problem of locking is well accepted in the linear analysis of shells, there seems to be the myth, or misunderstanding, that this problem does not exist in the nonlinear analysis of shell structures. There is no basis for that misunderstanding: of course — an element which locks in linear analysis will also lock in nonlinear analysis and the consequences are just as severe. We demonstrate this fact in the solutions of two problems below.
In the first problem, we consider the nonlinear large-displacement analysis of a curved beam structure. The arch is modeled using Timoshenko beam elements. The pure displacement-based elements lock while the mixed elements do not lock, see Refs. [1, 2] where mathematical analyses and numerical results are presented. However, numerical results on locking are usually given for linear analysis, Refs. [1-3], and perhaps this is the reason for the misunderstanding.
Figure 1 shows schematically the problem solved, and the deformed geometry for a non-locking solution. Figure 2 gives the relative error in the energy norm obtained for different beam thicknesses. Four meshes were used to obtain these results, with 4, 8, 16 and 32 Timoshenko theory based beam elements. It is clearly seen that, as expected, the displacement-based elements lock also in the nonlinear analysis. For example, for the case t/L = 0.01 , even with 32 2-node elements, the energy and tip displacement are only 28% of the accurate non-locking values.
In the second problem, we model the cantilever plate loaded by a moment shown in Figure 3 using two meshes A and B of triangular elements, and a uniform mesh C of square elements. Two types of elements are used: the MITC6 triangular shell element for meshes A and B, and the 9-node displacement-based shell element for mesh C. This displacement-based element is not recommended for general use, but we employ it here for illustrative purposes.
Figure 4 gives the results obtained for the three meshes and element types. Good solutions are obtained with the MITC6 element, and even when the mesh is quite distorted as shown, the results are still reasonable. The solution obtained using the displacement-based shell element is good for the first step (almost linear analysis results are obtained), however is very inaccurate due to locking when the load increases. It is interesting to note that while locking occurs the mesh curves accurately as it should, but the moment required for that curvature is much too high.
This example shows that while an element may seem to work quite well in linear analysis, in nonlinear analysis it may lock severely! Actually, a closer look would show that the 9-node displacement-based element also locks in linear analysis, by the definition of locking, see Refs. [1-3].
The above solutions are given to help dispel the misconception that locking of elements and meshes does not exist in nonlinear analysis of shell structures. While the mathematical analyses pertain to linear solutions, see Refs. [1-3] and the references therein, the problem is of course just as severe in nonlinear analysis. Hence it is important to always use general, reliable, and effective shell elements. The ADINA shell elements represent in that regard the state of the art.