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 
This simple problem is taken to be governed by the equations of equilibrium 
Tag Archives: material
Tutorial: Large deformation kinematics
Summarizes the meaning of the deformation gradient tensor, stretches, rotations, etc. Also shows how material line segments, volumes, and area vectors change in response to deformation.
You may download the rest of the document here
Tutorial: multi-linear regression
The straight line is the linear regression of a function that takes scalars (x-values) as input and returns scalars (y-values) as output. (figure from GANFYD)
You’ve probably seen classical equations for linear regression, which is a procedure that finds the straight line that best fits a set of discrete points 
The pink parallelogram is the multi-linear regression of a function that takes vectors (gray dots) as input and returns vectors (blue dots) as output
The bottom of this post provides a link to a tutorial on how to generalize the concept of linear regression to fit a function 
Course offering: ME 7960 (special topics) Computational Constitutive Modeling
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.
Publication: Experimental Assessment of Unvalidated Assumptions in Classical Plasticity Theory
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.
Available Online:
http://www.mech.utah.edu/~brannon/pubs/7-BrannonBurghardtSAND-Report2009-0351.pdf
http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=948711
Publication: Determining a Surrogate Contact Pair in a Hertzian Contact Problem
Sanders, A. P. and R. M. Brannon (2011). “Determining a Surrogate Contact Pair in a Hertzian Contact Problem.” Journal of Tribology 133(2): 024502-024506.
Hertzian substitution concept: An arbitrary contact pair (a) with given principal curvatures and orientation, is substituted with a simpler contact pair (b) consisting of a spheroid and a plane
Laboratory testing of contact phenomena can be prohibitively expensive if the interacting bodies are geometrically complicated. This work demonstrates means to mitigate such problems by exploiting the established observation that two geometrically dissimilar contact pairs may exhibit the same contact mechanics. Specific formulas are derived that allow a complicated Hertzian contact pair to be replaced with an inexpensively manufactured and more easily fixtured surrogate pair, consisting of a plane and a spheroid, which has the same (to second-order accuracy) contact area and pressure distribution as the original complicated geometry. This observation is elucidated by using direct tensor notation to review a key assertion in Hertzian theory; namely, geometrically complicated contacting surfaces can be described to second-order accuracy as contacting ellipsoids. The surrogate spheroid geometry is found via spectral decomposition of the original pair’s combined Hessian tensor. Some numerical examples using free-form surfaces illustrate the theory, and a laboratory test validates the theory under a common scenario of normally compressed convex surfaces. This theory for a Hertzian contact substitution may be useful in simplifying the contact, wear, or impact testing of complicated components or of their constituent materials.
Available Online:
http://www.mech.utah.edu/~brannon/pubs/7-2011-SandersBrannon-IJT.pdf
Publication: A model for statistical variation of fracture properties in a continuum mechanics code
H.W. Meyer Jr. and R.M. Brannon
[This post refers to the original on-line version of the publication. The final (paper) version with page numbers and volume is found at http://dx.doi.org/10.1016/j.ijimpeng.2010.09.007. Some further details and clarifications are in the 2012 posting about this article]
Simulation results for a reference volume of 0.000512 cm^3 ; sf is the size effect factor
Continuum mechanics codes modeling failure of materials historically have considered those materials to be homogeneous, with all elements of a material in the computation having the same failure properties. This is, of course, unrealistic but expedient. But as computer hardware and software has evolved, the time has come to investigate a higher level of complexity in the modeling of failure. The Johnsone-Cook fracture model is widely used in such codes, so it was chosen as the basis for the current work. The CTH finite difference code is widely used to model ballistic impact and penetration, so it also was chosen for the current work. The model proposed here does not consider individual flaws in a material, but rather varies a material’s Johnsone-Cook parameters from element to element to achieve in homogeneity. A Weibull distribution of these parameters is imposed, in such a way as to include a size effect factor in the distribution function. The well-known size effect on the failure of materials must be physically represented in any statistical failure model not only for the representations of bodies in the simulation (e.g., an armor plate), but also for the computational elements, to mitigate element resolution sensitivity of the computations.The statistical failure model was tested in simulations of a Behind Armor Debris (BAD) experiment, and found to do a much better job at predicting the size distribution of fragments than the conventional (homogeneous) failure model. The approach used here to include a size effect in the model proved to be insufficient, and including correlated statistics and/or flaw interactions may improve the model.
Available Online:
http://www.mech.utah.edu/~brannon/pubs/7-2011MeyerBrannon_IE_1915_final_onlinePublishedVersion.pdf
http://www.sciencedirect.com/science/article/pii/S0734743X10001466
Publication: On the thermodynamic requirement of elastic stiffness anisotropy in isotropic materials
Measure of anisotropy for Zircon, Quartz, Uranium, Titanium, Hornblende, and Copper.
T. Fuller and R.M. Brannon
In general, thermodynamic admissibility requires isotropic materials develop reversible deformation induced anisotropy (RDIA) in their elastic stiffnesses. Taking the elastic potential for an isotropic material to be a function of the strain invariants, isotropy of the elastic stiffness is possible under distortional loading if and only if the bulk modulus is independent of the strain deviator and the shear modulus is constant. Previous investigations of RDIA have been limited to applications in geomechanics where material non-linearityand large deformations are commonly observed. In the current paper, the degree of RDIA in other materials is investigated. It is found that the resultant anisotropy in materials whose strength does not vary appreciably with pressure, such as metals, is negligible, but in materials whose strength does vary with pressure, the degree of RDIA can be significant. Algorithms for incorporating RDIA in a classical elastic–plastic model are provided.
Available Online:
http://www.mech.utah.edu/~brannon/pubs/7-2011FullerBrannonInducedElasticAnisotropy.pdf
http://www.sciencedirect.com/science/article/pii/S0020722511000024
Publication: A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations
A. Sadeghirad, R. M. Brannon, and J. Burghardt
Three snapshots of the model with 248 particles in simulation of the radial expansion of a ring problem using: (a) CPDI method and (b) cpGIMP
A new algorithm is developed to improve the accuracy and efficiency of the material point method for problems involving extremely large tensile deformations and rotations. In the proposed procedure, particle domains are convected with the material motion more accurately than in the generalized interpolation material point method. This feature is crucial to eliminate instability in extension, which is a common shortcoming of most particle methods. Also, a novel alternative set of grid basis functions is proposed for efficiently calculating nodal force and consistent mass integrals on the grid. Specifically, by taking advantage of initially parallelogram-shaped particle domains, and treating the deformation gradient as constant over the particle domain, the convected particle domain is a reshaped parallelogram in the deformed configuration. Accordingly, an alternative grid basis function over the particle domain is constructed by a standard 4-node finite element interpolation on the parallelogram. Effectiveness of the proposed modifications is demonstrated using several large deformation solid mechanics problems.
Available Online:
http://www.mech.utah.edu/~brannon/pubs/7-2011-SadeghiradBrannonBurghardt-NME.pdf
http://onlinelibrary.wiley.com/doi/10.1002/nme.3110/abstract
Publication: On a viscoplastic model for rocks with mechanism-dependent characteristic times
A. F. Fossum and R. M. Brannon
Rate Dependance
This paper summarizes the results of a theoretical and experimental program at Sandia National Laboratories aimed at identifying and modeling key physical features of rocks and rock-like materials at the laboratory scale over a broad range of strain rates. The mathematical development of a constitutive model is discussed and model predictions versus experimental data are given for a suite of laboratory tests. Concurrent pore collapse and cracking at the microscale are seen as competitive micromechanisms that give rise to the well-known macroscale phenomenon of a transition from volumetric compaction to dilatation under quasistatic triaxial compression. For high-rate loading, this competition between pore collapse and microcracking also seems to account for recently identified differences in strain-rate sensitivity between uniaxial-strain ‘‘plate slap’’ data compared to uniaxial-stress Kolsky bar data. A description is given of how this work supports ongoing efforts to develop a predictive capability in simulating deformation and failure of natural geological materials, including those that contain structural features such as joints and other spatial heterogeneities.
Available Online:
http://www.mech.utah.edu/~brannon/pubs/7-2006FossumBrannonMechanismDependentViscoplasticity.pdf
University of Utah CSM Group 