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Large Strains of Shells in Statics & Dynamics

There are many analyses of shells where large strain conditions need be simulated, for example, in the analyses of crush or crash conditions of motor cars. Additional large problem areas are the analyses of metal forming processes and rubber seals.

In ADINA 8.7, new 3D-shell elements are available for such simulations, a 3-node element and a 4-node element, referred to, respectively, as MITC3 and MITC4 3D-shell elements.

These elements build upon the conventional MITC elements, with 5 or 6 degrees of freedom at each node, but have these additional features, when invoked by degrees of freedom:

  • 2 additional through-the-thickness strain degrees of freedom to allow large strains through the element thickness

  • warping degrees of freedom to allow the transverse fibers to warp, that is, these fibers originally straight, when allowed, do not remain straight during the deformations

The 2 through-the-thickness degrees of freedom allow the element to model a constant and a linear strain distribution through the shell thickness. The 2 extra warping degrees of freedom (3 warping dofs are used in case there are 3 usual rotational dofs) allow the transverse fibers to warp in a quadratic displacement, see Figure 1. Thus, from a displacement interpolation point of view, the elements can be thought of as 3D solid elements when the additional dofs are invoked.



(a) Usual 5 dofs



(b) Additional dofs for 3D-shell elements


Figure 1 3- and 4-node shell elements, degrees of freedom



However, in the 3D-shell elements, the additional dofs are only and simply invoked when needed in addition to the usual shell dofs. The pre- and post-processing is not changed and ill-conditioning due to a small shell thickness does not exist (as it does when 3D solid elements are used). Of course, as for the usual MITC elements, the mixed interpolations of strains are employed to avoid shear locking, but in addition, to also avoid volumetric locking in large strain plasticity and rubber analyses, the u/p interpolation approach with pressure degrees of freedom is used.

The elements thus do not use the usual assumption of zero stress through the shell thickness, and are employed with 3D material models.

The elements are used in static analyses, and in implicit and explicit integration dynamic solutions.

An important point is that the elements can be employed in static analyses and implicit dynamic solutions since no reduced integration with hourglass control is used and there are no artificial stability factors in the formulation. Hence, artificial explicit solutions with mass scaling need not be used when the physics asks for a static or slow-motion dynamic solution. Instead, the physical situation is properly modeled.

The 3D-shell elements can be used for usual shell analyses, but the power of the 3D-shell elements lies in that — hierarchically — additional deformation effects are included in the analysis, namely 3D effects that are specifically needed in large strain shell conditions.

Figure 2 shows the analysis of a rubber block undergoing large strains in bending. The block is modeled with equally spaced 3D-shell elements with nodes on the midsurface. Of course, initially the midsurface is equidistant from the top and bottom surfaces.

As the block is subjected to a prescribed bending rotation, the material in compression thickens and the material in tension thins. Therefore the midsurface shifts to be closer to the tension side of the block, as shown in the movie below.

The 3D-shell model gives a very good response prediction as compared to the analytical solution for this problem.



(a) Schematic of problem




(b) Movie showing deformation of the rubber block




(c) Comparison of numerical and analytical results


Figure 2 Cantilever in large strains



Figure 3 shows a crash tube subjected to the impact of a heavy weight. A large strain plasticity material model is employed. This problem is solved using, both, implicit and explicit time integration, and very good comparison between the two results is observed.

The area under the curves in Figure 3(c) give the energy absorbed by the crash tube. The absorbed energies, for both the implicit and explicit curves, are within 2% of the initial kinetic energy of the heavy mass.



(a) Schematic of problem




(b) Movie showing the crash tube in cross-section




(c) Comparison of results using implicit and explicit time integration


Figure 3 Crash analysis of a tube



The new 3D-shell elements are very powerful and strengthen the offering in ADINA for shell analyses in a very significant way.


Keywords:
Shells, large strains, large deformations, MITC, 3D solid, u/p formulation, crashworthiness, metal forming, solid-shell, 3D-shell, axial crush test, crash tube


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