This paper examines the variability of predicted responses when multiple stress-strain curves (reflecting variability from replicate material tests) are propagated through a finite element model of a ductile steel can being slowly crushed. Over 140 response quantities of interest (including displacements, stresses, strains, and calculated measures of material damage) are tracked in the simulations. Each response quantity’s behavior varies according to the particular stress-strain curves used for the materials in the model. We desire to estimate response variability when only a few stress-strain curve samples are available from material testing. Propagation of just a few samples will usually result in significantly underestimated response uncertainty relative to propagation of a much larger population that adequately samples the presiding random-function source. A simple classical statistical method, Tolerance Intervals, is tested for effectively treating sparse stress-strain curve data. The method is found to perform well on the highly nonlinear input-to-output response mappings and non-standard response distributions in the can-crush problem. The results and discussion in this paper support a proposition that the method will apply similarly well for other sparsely sampled random variable or function data, whether from experiments or models. Finally, the simple Tolerance Interval method is also demonstrated to be very economical.
This work examines the variability of predicted responses when multiple stress-strain curves (reflecting variability from replicate material tests) are propagated through a transient dynamics finite element model of a ductile steel can being slowly crushed. An elastic-plastic constitutive model is employed in the large-deformation simulations. The present work assigns the same material to all the can parts: lids, walls, and weld. Time histories of 18 response quantities of interest (including displacements, stresses, strains, and calculated measures of material damage) at several locations on the can and various points in time are monitored in the simulations. Each response quantity's behavior varies according to the particular stressstrain curves used for the materials in the model. We estimate response variability due to variability of the input material curves. When only a few stress-strain curves are available from material testing, response variance will usually be significantly underestimated. This is undesirable for many engineering purposes. This paper describes the can-crush model and simulations used to evaluate a simple classical statistical method, Tolerance Intervals (TIs), for effectively compensating for sparse stress-strain curve data in the can-crush problem. Using the simulation results presented here, the accuracy and reliability of the TI method are being evaluated on the highly nonlinear inputto- output response mappings and non-standard response distributions in the can-crush UQ problem.
This document compares the finite element shell formulations in the Sierra Solid Mechanics code. These are finite elements either currently in the Sierra simulation codes Presto and Adagio, or expected to be added to them in time. The list of elements are divided into traditional two-dimensional, plane stress shell finite elements, and three-dimensional solid finite elements that contain either modifications or additional terms designed to represent the bending stiffness expected to be found in shell formulations. These particular finite elements are formulated for finite deformation and inelastic material response, and, as such, are not based on some of the elegant formulations that can be found in an elastic, infinitesimal finite element setting. Each shell element is subjected to a series of 12 verification and validation test problems. The underlying purpose of the tests here is to identify the quality of both the spatially discrete finite element gradient operator and the spatially discrete finite element divergence operator. If the derivation of the finite element is proper, the discrete divergence operator is the transpose of the discrete gradient operator. An overall summary is provided from which one can rank, at least in an average sense, how well the individual formulations can be expected to perform in applications encountered year in and year out. A letter grade has been assigned albeit sometimes subjectively for each shell element and each test problem result. The number of A's, B's, C's, et cetera assigned have been totaled, and a grade point average (GPA) has been computed, based on a 4.0-system. These grades, combined with a comparison between the test problems and the application problem, can be used to guide an analyst to select the element with the best shell formulation.
Many problems of practical importance involve ductile materials that undergo very large strains, in many cases to the point of failure. Examples include structures subjected to impact or blast loads, energy absorbing devices subjected to significant crushing, cold-forming manufacturing processes and others. One of the most fundamental pieces of data that is required in the analysis of this kind of problems is the fit of the uniaxial stress-strain curve of the material. A series of experiments where mild steel plates were punctured with a conical indenter provided a motivation to characterize the true stress-strain curve until the point of failure of this material, which displayed significant ductility. The hardening curve was obtained using a finite element model of the tensile specimens that included a geometric imperfection in the form of a small reduction in the specimen width to initiate necking. An automated procedure iteratively adjusted the true stress-strain curve fit used as input until the predicted engineering stress-strain curve matched experimental measurements. Whereas the fitting is relatively trivial prior to reaching the ultimate engineering stress, the fit of the softening part of the engineering stress-stain curve is highly dependent on the finite element parameters such as element formulation and initial geometry. Results by two hexahedral elements are compared. The first is a standard, under-integrated, uniform-strain element with hourglass control. The second is a modified selectively-reduced-integration element. In addition, the effects of element size, aspect ratio and hourglass control characteristics are investigated. The effect of adaptively refining the mesh based on the aspect ratio of the deformed elements is also considered. The results of the study indicate that for the plate puncture problem, characterizing the material with the same element formulation and size as used in the plate models is beneficial. On the other hand, using different element formulations, sizes or initial aspect ratios can lead to unreliable results.
In this paper, the advantages and disadvantages of inelastic analysis are discussed. Example calculations and designs showing the implications and significance of factors affecting inelastic analysis are given. From the results described in this paper it can be seen that inelastic analysis provides an improved method for the design of casks. It can also be seen that additional code and standards work is needed to give designers guidance in the use of inelastic analysis. Development of these codes and standards is an area where there is a definite need for additional work. The authors hope that this paper will help to define the areas where that need is most acute.