Illustrated below is the solution to an idealized problem of a linear elastic annulus (blue) subjected to twisting motion caused by rotating the T-bar an angle . The motion is presumed to be applied slowly enough that equilibrium is satisfied.
This simple problem is taken to be governed by the equations of equilibrium , along with the plane strain version of Hooke’s law in which Cauchy stress is taken to be linear with respect to the small strain tensor (symmetric part of the displacement gradient). If this system of governing equations is implemented in a code, the code will give you an answer, but it is up to you to decide if that answer is a reasonable approximation to reality. This observation helps to illustrate the distinction between verification (i.e., evidence that the equations are solved correctly) and validation (evidence that physically applicable and physically appropriate equations are being solved). The governing equations always have a correct answer (verification), but that answer might not be very predictive of reality (validation).
Stills from YouTube video of buried roadside explosive
As one of four institutions collaborating with the University of Colorado — Boulder, the CSM group in the Department of Mechanical Engineering at the University of Utah, will be developing constitutive models for soils, as well as full-scale simulation capabilities in Uintah to predict blast and ejecta from shallow buried explosives (such as roadside improvised explosive devices). The $1.1M slated for CSM work presumes the project will last 5 years. For more information, see the University of Colorado’s press release.
Constitutive modeling refers to the development of equations describing the way that materials respond to various stimuli. In classical deformable body mechanics, a simple constitutive model might predict the stress required to induce a given strain; the canonical example is Hooke’s law of isotropic linear elasticity. More broadly, a constitutive model predicts increments in some macroscale state variables of interest (such as stress, entropy, polarization, etc.) that arise from changes in other macroscale state variables (strain, temperature, electric field, etc.).
Constitutive equations are ultimately implemented into a finite element code to close the set of equations required to solve problems of practical interest. This course describes a few common constitutive equations, explaining what features you would see in experimental data or structural behavior that would prompt you to select one constitutive model over another, how to use them in a code, how to test your understanding of the model, how to check if the code is applying the model as advertised in its user’s manual, and how to quantitatively assess the mathematical and physical believability of the solution.
Sanders, A., I. Tibbitts, D. Kakarla, S. Siskey, J. Ochoa, K. Ong, and R. Brannon. (2011). “Contact mechanics of impacting slender rods: measurement and analysis.” Society for Experimental Mechanics Annual Meeting. Uncasville, CT, June 13-16.
Images of a typical contact patch
To validate models of contact mechanics in low speed structural impact, slender rods with curved tips were impacted in a drop tower, and measurements of the contact and vibration were compared to analytical and finite element (FE) models. The contact area was recorded using a thin-film transfer technique, and the contact duration was measured using electrical continuity. Strain gages recorded the vibratory strain in one rod, and a laser Doppler vibrometer measured velocity. The experiment was modeled analytically using a quasi-static Hertzian contact law and a system of delay differential equations. The FE model used axisymmetric elements, a penalty contact algorithm, and explicit time integration. A small submodel taken from the initial global model economically refined the analysis in the small contact region. Measured contact areas were within 6% of both models’ predictions, peak speeds within 2%, cyclic strains within 12 microstrain (RMS value), and contact durations within 2 µs. The accuracy of the predictions for this simple test, as well as the versatility of the diagnostic tools, validates the theoretical and computational models, corroborates instrument calibration, and establishes confidence thatthe same methods may be used in an experimental and computational study of the impact mechanics of artificial hip joint.
Global model results comparison with analytical and experimental results for speed at the midpoint of one of the rods
R. Brannon, J.A. Burghardt, D. Bronowski, and S. Bauer
Common isotropic yield surfaces. Von Mises and Drucker-Prager models are often used for metals. Gurson’s function, and others like it, are used for porous media. Tresca and Mohr-Coulomb models approximate the yield threshold for brittle media. Fossum’s model, and others like it, combine these features to model realistic geological media.
This report investigates the validity of several key assumptions in classical plasticity theory regarding material response to changes in the loading direction. Three metals, two rock types, and one ceramic were subjected to non-standard loading directions, and the resulting strain response increments were displayed in Gudehus diagrams to illustrate the approximation error of classical plasticity theories. A rigorous mathematical framework for ﬁtting classical theories to the data,thus quantifying the error, is provided. Further data analysis techniques are presented that allow testing for the effect of changes in loading direction without having to use a new sample and for inferring the yield normal and ﬂow directions without having to measure the yield surface. Though the data are inconclusive, there is indication that classical, incrementally linear, plasticity theory may be inadequate over a certain range of loading directions. This range of loading directions also coincides with loading directions that are known to produce a physically inadmissible instability for any nonassociative plasticity model.
A plot of the frequency-dependent wave propagation velocity for the case study problem with an overlocal plasticity model, with the elastic and local hardening wave speeds shown for reference (left). Stress histories using an overlocal plasticity model with a nonlocal length scale of 1m and a mesh resolution of 0.125m (right)
The following series of three articles (with common authors J. Burghardt and R. Brannon of the University of Utah) describes a state of insufficient experimental validation of conventional formulations of nonassociative plasticity (AKA nonassociated and non-normality). This work provides a confirmation that such models theoretically admit negative net work in closed strain cycles, but this simple prediction has never been validated or disproved in the laboratory!
- An early (mostly failed) attempt at experimental investigation of unvalidated plasticity assumptions (click to view),
- A simple case study confirming that nonassociativity can cause non-unique and unstable solutions to wave motion problems (click to view),
- An extensive study showing that features like rate dependence, hardening, etc. do not eliminate the instability and also showing that it is NOT related to conventional localization (click to view).
Swan, S. and R. Brannon (2009)
Illustration of stair-stepping typical of finite sampling from a Weibull distribution
Current simulations of material deformation are a balance between computational effort and accuracy of the simulation. To increase the accuracy of the simulated material response, the simulation becomes more computationally intensive with finer meshes and shorter timesteps, increasing the time and resource requirements needed to perform the simulation. One method for improving predictions of brittle failure while minimizing computational overhead is to implement statistical variability for the material properties being simulated. This method has low computational overhead and requires a relatively small increase in resource requirements while significantly increasing the precision of simulation results. Currently, most simulation frameworks inaccurately describe brittle and heterogeneous materials as uniform bodies of equal strength and consistency. This over-simplification underscores the need to implement statistical variability to help better predict material response and failure modes for materials that contain intermittent abnormalities such as changes in hardness, strength, and grain size throughout the specimen. Uintah, the computational framework developed by the University of Utah’s C-SAFE program, has a simplistic native Gaussian distribution function that was hard-coded into select material models. The goal of this research is to create an easily duplicable method for enabling dynamic global variability according to a Weibull distribution in constitutive models in Uintah and to implement said ability into the constitutive model Kayenta. The main application of Kayenta is to simulate geological response to penetration and perforation. For the purpose of simulating failure in brittle geological samples, the Weibull distribution produces realistic statistical scatter in constituent properties that correlates well to flaws and irregularities observed in laboratory tests.
K. Kamojjala, R. M. Brannon (2011)
Snapshot of the deformation in time
The principle of material frame indifference require spatial stresses to rotate with the material, whereas reference stresses must be insensitive to rotation. Testing of a classical uniaxial strain problem with superimposed rotation reveals that a very common approach to strong incremental objectivity taken in finite element codes to satisfy frame indifference(namely working in an approximate un-rotated frame) fails this simplistic test. A more complicated verification example is constructed based on the method of manufactured solutions (MMS) which involves the same character of loading at all points, providing a means to test any nonlinear-elastic arbitrarily anisotropic constitutive model.
T.J. Fuller, R.M. Brannon, O.E. Strack, J.E. Bishop
Displacement profile for Thermo-Kayenta at the end of the simulation. the red dots represent the experimental profiles
A persistent challenge in simulating damage of natural geological materials, as well as rock-like engineered materials, is the development of efficient and accurate constitutive models.The common feature for these brittle and quasi-brittle materials are the presence of flaws such as porosity and network of microcracks. The desired models need to be able to predict the material responses over a wide range of porosities and strain rate. Kayenta  (formerly called the Sandia GeoModel) is a unifi ed general-purpose constitutive model that strikes a balance between rst-principles micromechanics and phenomenological or semi-empirical modeling strategies. However, despite its sophistication and ability to reduce to several classical plasticity theories, Kayenta is incapable of modeling deformation of ductile materials in which deformation is dominated by dislocation generation and movement which can lead to signi cant heating. This stems from Kayenta’s roots as a geological model, where heating due to inelastic deformation is often neglected or presumed to be incorporated implicitly through the elastic moduli.The sophistication of Kayenta and its large set of extensive features, however, make Kayenta an attractive candidate model to which thermal eff ects can be added. This report outlines the initial work in doing just that, extending the capabilities of Kayenta to include deformation of ductile materials, for which thermal e ffects cannot be neglected. Thermal e ffects are included based on an assumption of adiabatic loading by computing the bulk and thermal responses of the material with the Kerley Mie-Gruneisen equation of state and adjusting the yield surface according to the updated thermal state. This new version of Kayenta, referred to as Thermo-Kayenta throughout this report, is capable of reducing to classical Johnson-Cook plasticity in special case single element simulations and has been used to obtain reasonable results in more complicated Taylor impact simulations in LS-Dyna. Despite these successes, however, Thermo-Kayenta requires additional re nement for it to be consistent in the thermodynamic sense and for it to be considered superior to other, more mature thermoplastic models. The initial thermal development, results, and required refinements are all detailed in the following report.
J. Burghardt, B. Leavy, J. Guilkey, Z. Xue, R. Brannon
The capability of the generalized interpolation material point (GIMP) method in simulation of penetration events is investigated. A series of experiments was performed wherein a shaped charge jet penetrates into a stack of aluminum plates. Electronic switches were used to measure the penetration time history. Flash x-ray techniques were used to measure the density,length, radius and velocity of the shaped charge jet. Simulations of the penetration event were performed using the Uintah MPM/GIMP code with several different models of the shaped charge jet being used. The predicted penetration time history for each jet model is compared with the experimentally observed penetration history. It was found that the characteristics of the predicted penetration were dependent on the way that the jet data are translated to a discrete description. The discrete jet descriptions were modified such that the predicted penetration histories fell very close to the range of the experimental data. In comparing the various discrete jet descriptions it was found that the cumulative kinetic energy flux curve represents an important way of characterizing the penetration characteristics of the jet. The GIMP method was found to be well suited for simulation of high rate penetration events.