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

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

A. Sadeghirad, R. M. Brannon, and J. Burghardt

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

MMS stands for “Method of Manufactured Solutions,” which is a rather sleazy sounding name for what is actually a respected and rigorous method of verifying that a finite element (or other) code is correctly solving the governing equations.

A simple introduction to MMS may be found on page 11 of The ASME guide for verification and validation in solid mechanics. The basic idea is to analytically determine forcing functions that would lead to a specific, presumably nontrivial, solution (of your choice) for the dependent variable of a differential equation. Then you would verify a numerical solver for that differential equation by running it using your analytically determined forcing function. The difference between the code’s prediction and your selected manufactured solution provides a quantitative measure of error.