# Estoplastic shell analysis in DYNA3D: a finite element code developed at Lawrence Livermore National Laboratory analyzes thin shells under transient, dynamic loads.

Elastoplastic Shell Analysis In DYNA3DA finite element code developed at Lawrence Livermore National Laboratory analyzes thin shells under transient, dynamic loads.

Computer simulation of the elastoplastic behavior of thin shell structures under transient dynamic loads plays an important role in many programs at Lawrence Livermore National Laboratory (LLNL) in Livermore, Calif. Often the loads are severe and the structure undergoes plastic (or permanent) deformation. These simulations are effectively performed using DYNA3D, an explicit nonlinear finite element code developed at LLNL for simulating and analyzing the large-deformation dynamic response of solids and structures. It is generally applicable to problems where the loading and response are of short duration and contain significant high-frequency components. Typical problems of this type include the contact of two impacting bodies and the resulting elastoplastic structural behavior.

Several of LLNL's programs rely heavily on the ability to accurately model the large-deformation elastoplastic behavior of thin shell structures. Three programs that routinely use DYNA3D for this purpose include Nuclear Systems Safety, Nuclear Design, and Enhanced Safety.

The Nuclear Systems Safety Program is concerned with safe transportation of radioactive waste. DYNA3D analyzes waste-canister damage under projectile impact and drop loads.

In the Nuclear Design Program, DYNA3D analyzes shipping containers under simulated drop conditions to verify structural integrity. To model the shipping containers, the simulations rely heavily on shell element formulations capable of representing elastoplastic behavior.

In the Enhanced Safety Program, shell elements with elastoplastic behavior are used in DYNA3D to simulate an aircraft crashing into a stationary U.S. Air Force B1-B bomber. The objective of the simulation is to determine the loads that such an event would impart to the weapons carried by the B1-B. Figure 1a is a cutaway view of the finite element model used for this analysis. To simulate the impact, the model is given an initial velocity toward the crash barrier on the right in the figure. Figure 1b shows the post-impact geometry.

The objective of this investigation was to examine and improve upon the elastoplastic shell modeling capability in DYNA3D. This included a basic review of both the shell element formulations and the computational methods used in the elastoplastic constitutive algorithms. Special attention was to be paid to accuracy/speed trade-offs between the various algorithms. The outcome of these studies provides guidance for the analyst in choosing the most cost-effective set of approximations to solve a particular problem to a given level of accuracy.

This article summarizes the development of a new four-node quadrilateral finite element shell formulation, the YASE shell, and compares two basic methods (the stress-resultant and the thickness-resultant methods) employed in elastoplastic constitutive algorithms for shell structure modeling.

In the thickness-integration method, plane-stress constitutive algorithms are applied at a number of integration points through the thickness of the shell. We evaluated seven algorithms, each with a different combination of plane-stress constitutive integration algorithm and number of thickness-integration points.

Shell Element Formulation

The finite element shell formulation plays a key role in the elastoplastic modeling of thin shell structures. Accuracy, speed, and robustness are heavily influenced by the assumptions embedded in the element formulation. Accuracy requires a formulation that can represent large rigid-body motions. Speed in an explicit analysis context requires one-point shell element integration. Robustness requires an effective stabilization procedure to suppress zero-energy, or hourglass, modes, introduced by the one-point element integration procedure.

We have developed a new shell element formulation that satisfies these requirements and is more accurate and robust than earlier formulations, while retaining comparable speed. In conjunction with the new element formulation, we have derived a new consistent stabilization procedure.

The newly developed YASE shell element is a four-node quadrilateral based on a three-field weak form of the governing equations. The three-field form of the governing equations allows independent interpolation of displacement, strain, and stress fields within a finite element. The associated Euler equations are equilibrium, strain-displacement, and the constitutive relations. In-plane accuracy is attained through a three-field adaptation of the stress fields.

All one-point shell element procedures require stabilization to suppress spurious zero-energy, or hourglass, modes. Failure of the stabilization algorithm to control these zero-energy modes results in a nonphysical solution, as shown in Figure 2a. This example problem is a corner-supported plate subjected to a point load at the center. The presence of hourglass modes is manifested by the characteristic checkerboard pattern of displacements, wherein adjacent nodes are displaced in opposite directions. The correct solution to this simple problem is shown in Figure 2b. Note the smooth curves in the correct solution. Stabilization procedures that require the user to select values for hourglass control parameters frequently have accuracy characteristics that depend heavily on the choice of values. In practice, it is difficult to choose appropriate values for hourglass stabilization parameters that yield an accurate solution.

Our new stabilization procedure does not require the user to supply values for the control parameters. In the case of elastic material behavior, the new procedure with one-point shell element integration yields results identical to the fully integrated element. This represents a substantial improvement in accuracy over previously existing one-point shell element formulations.

We have devised a stabilization procedure for elastoplastic shell analysis that accounts for the effects of plasticity on the higher-order modes of the element. This procedure properly treats plasticity due to membrane deformations and both in- and out-of-plane bending deformations.

A simple example illustrates the improved behavior of the new explicit YASE shell element. An undeformed mesh of an arch clamped at one end was subjected to in-plane bending by a point load at the tip. Final displacements, at the middle of the tip node, under the load (where tip displacement is given as a factor of the arch width or thickness) are: the new explicit YASE shell with one-point integration, 0.12; the Belytschko-Tsay shell (currently the default shell element in DYNA3D), 0.33; and a fully integrated version of the YASE shell that serves as the reference standard, 0.105.

The result using the YASE shell with one-point integration is within 15 percent of the reference (fully integrated) result, while the result using the Belytschko-Tsay element exceeds the reference value by more than a factor of three. While achieving this increase in accuracy, the explicit YASE element (with one-point integration) required only 10 percent more computer time than the Belytschko-Tsay element. Thus, the slight increase in cost appears well justified.

Constitutive Algorithms

Thickness-Integration Method. Most methods for elastoplastic shell analysis use continuum-based material models (i.e., stress is related to strain). One of the basic assumptions in the formulations of many shell elements, including the YASE shell, is that the "through-the-thickness" normal stress is negligibly small. As far as numerical constitutive evaluations are concerned, each shell element is composed of a number of lamina, with each lamina satisfying the postulates of plane stress. (Plane stress theory assumes that the out-of-plane normal stress is negligibly small.) These lamina stresses are used to evaluate the integrals of stress through the element thickness appearing in the computation of the element moment and force resultants. The advantage of this thickness-integration method is accurate representation of both partial yielding of a section and complex loading states, but at the expense of speed. The extent of approximation and level of cost can be partially controlled through the choice of the number of numerical integration points used in the thickness integration.

Stress-Resultant Method. An alternative method uses material models formulated directly in terms of shell stress resultants (i.e., forces and moments are directly related to displacements and curvatures). This stress-resultant method offers potentially greater speed since no thickness integration is performed and the material model is evaluated only once per shell element, but at the price of accuracy. Inaccuracies in the stress-resultant method occur from the assumption of full-section yielding, which is basic to most stress-resultant methods. Physically, yielding begins at a point in the cross section and spreads until the entire section has yielded. This causes a gradual reduction in the shell's stiffness. Thus, the assumption of full section yield replaces this gradual stiffness reduction with an abrupt change from the elastic stiffness to the post-yield stiffness.

Inaccuracies also result from the approximation method used to integrate the constitutive equation. Stress-resultant plasticity theory involves multiple yield surfaces that intersect in a nonsmooth fashion. This leads to considerable complexity in the associated constitutive integration algorithm. Speed requirements in an explicit-analysis environment require simplifications to be made in these sophisticated procedures and the modifications introduce errors into the integration.

Evaluation of Modeling Methods. Current efforts in constitutive modeling focus on evaluating the performance/cost (i.e., accuracy/speed) trade-offs between the thickness-integration plasticity and the stress-resultant plasticity methods for constitutive modeling in shells.

In order to assess the accuracy/speed trade-offs between the various methods for constitutive modeling, a model problem was chosen for study. The selected problem is a cantilevered plate with a point load applied at one of the free corners. This load induces bending, shear, and normal forces in the plate. A comparison of the out-of-plane displacement and required computer time was made for: the stress-resultant plasticity method; seven different thickness-integration shell plasticity methods, each with a different combination of plane stress plasticity algorithms and number of thickness-integration points. Results of the comparison are given in the table.

All computation times are based on a Cray Y/MP computer, although similar results would be expected on any vector-processing machine. This test problem tends to highlight the influence of the constitutive algorithm on the overall execution time, since the problem does not contain other expensive numerical procedures such as those necessary to detect and treat the contact of two initially separate bodies. "Contact" algorithms contain search algorithms that are inherently scalar, and this causes these procedures to be expensive on vector-processing computers such as the Cray. As a result, the effect of the constitutive model on the execution time of most practical calculations is somewhat less dramatic than indicated in this example.

As shown in the table, the stress-resultant method is the fastest and least accurate, introducing a displacement error of 45 percent.

Next in speed is thickness integration with two points through the thickness. If the stress scaling algorithm (SS-2) is used, then the error is 31 percent, but if the fixed three-iteration algorithm (FTI-2) default in DYNA3D is used, the displacement error is down to 18 percent with only a slight increase in computation time.

The results using thickness integration with three points and either the fixed three-iteration algorithm (FTI-3) or the plane stress subspace algorithm (PSS-3) show a slight increase in error over the comparable results using two thickness integration points. This is somewhat unusual, since using more integration points typically yields better accuracy.

However, the expected trend in improved accuracy is again seen in the results using five thickness integration points. While somewhat slower, thickness integration with five points and the fixed three-iteration algorithm (FTI-5) yields an accurate result, with only 6 percent error.

The two calculations using the (scalar) iterative radial return algorithm (IRR-5 and IRR-10) are included for comparison purposes only; they are too inefficient to be used in practice.

In order to compare their effects on a large-scale analysis, we used four different elastoplastic constitutive algorithms to simulate the axial crush of a box beam. Each algorithm employed one of the following four computational methods: the stress-resultant plasticity method or the thickness-integration shell plasticity method using the plane-stress fixed three-iteration algorithm with two (FTI-2), three (FTI-3), and five (FTI-5) points through the thickness.

The computed deformed geometry for each of the methods is shown in Figure 3 along with their respective computation times. As seen in the figure, the simulation using the stress-resultant method predicts a different deformed shape than the thickness-integration method, regardless of the number of thickness integration points. The stress-resultant calculation produces a fold with the first hinge inward, while all of the thickness-integration calculations predict the first hinge will fold outward. Since the thickness-integration method more accurately represents the physics of the problem, more confidence is placed in the deformed geometry predicted by that method.

This simulation illustrates the dramatic effect that constitutive-model errors can have on the evolution of the solution to a nonlinear problem. In addition, Figure 3 shows that the CPU time saving with the stress-resultant plasticity method in this large analysis is less significant than in the simple problem summarized in the table. The thickness-integration shell plasticity method with two points in the thickness integration uses only 2 percent more computer time than the stress-resultant method.

This investigation indicates that the thickness-integration shell plasticity method is generally to be preferred over the stress-resultant plasticity method. The number of thickness integration points may be reduced to two for an approximate solution at minimum cost; this is generally more accurate than the stress-resultant method. In addition, the thickness-integration method generalizes more easily to complex constitutive models that do not lend themselves to representation in a stress-resultant form.

Thickness integration of plane stress constitutive equations has proven to be the method of choice for elastoplastic shell modeling. Due to the complicated structure of the yield surfaces, the stress-resultant plasticity method offers little speed increase over the thickness-integration shell plasticity method with two points through the thickness and it suffers serious inaccuracies. The thickness-integration method using five points through the shell thickness offers an accurate solution to most elastoplastic shell problems.

Based on these observations, it is recommended that thickness integration with two thickness points be used for preliminary calculations and thickness integration with five thickness points be used for final calculations to assure accurate results.

Robert G. Whirley Lawrence Livermore National Laboratory Livermore, Calif. [Figures 1 to 3 Omitted] [Tabular Data Omitted]

PHOTO : Shell-analysis breakthrough. This sequence of a rigid ball going through a steel plate was simulated with the DYNA3D nonlinear finite element program to illustrate a phenomenon known as petalling, in which fractures propagate radially.

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Author: | Whirley, Robert G. |
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Publication: | Mechanical Engineering-CIME |

Date: | Jan 1, 1991 |

Words: | 2310 |

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