Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T23:26:05.481Z Has data issue: false hasContentIssue false

4 - Trefftz Trial Functions for Computational Grains for Planar and 3D Problems

Published online by Cambridge University Press:  05 October 2023

Leiting Dong  and
Satya N. Atluri
Leiting Dong
Affiliation:
Beihang University, China
Satya N. Atluri
Affiliation:
University of California, Irvine
Get access

Summary

In this chapter, Trefftz trial functions which satisfy identically all the governing equations of linear elasticity in 2D and 3D problems are summarized. These Trefftz functions are later used in conjunction with boundary variational principles (since all the field equations are satisfied identically inside the Voronoi cell elements), to construct planar and 3D Computational Grains to directly model statistically equivalent representative volume elements (SERVEs) of heterogeneous materials at the microscale. In as much as the Trefftz functions are used as trial solutions, this modeling captures the correct and accurate stress solutions in the matrix, inclusions, and at their interfaces. The presented Trefftz solutions include: (1) Muskhelishvili’s complex functions for 2D problems,(2) Papkovich-Neubar solutions for 3 D problems,and (3) Harmonic functions in spherical coordinates, cylindrical coordinates, and ellipsoidal coordinates.

Keywords

computational grainsTrefftz trial functionsMuskhelishvili’s complex functionsPapkovich-Neubar solutionsspherical harmonicscylindrical harmonicsellipsoidal harmonics
Type
Chapter
Information
Computational Grains
Micromechanical Genome for Heterogeneous Materials
 , pp. 45 - 71
Publisher: Cambridge University Press
Print publication year: 2023

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity. Berlin: Springer Dordrecht, 1953.Google Scholar
Weissmüller, J. and Cahn, J. W., “Mean Stresses in Microstructures Due to Interface Stresses: A Generalization of a Capillary Equation for Solids,” Acta Materialia, vol. 45, no. 5, pp. 1899–1906, 1997, https://doi.org/10.1016/S1359-6454(96)00314-X.Google Scholar
Lurie, A. I. and Belyaev, A., “Three-dimensional problems in the theory of elasticity,” in Theory of Elasticity, Lurie, A. I. and Belyaev, A., Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005, pp. 243–407.Google Scholar
Hobson, E. W., The Theory of Spherical and Ellipsoidal Harmonics. Cambridge: Cambridge University Press,1955.Google Scholar
Liu, C.-S., “A Modified Trefftz Method for Two-Dimensional Laplace Equation Considering the Domain’s Characteristic Length,” Computer Modeling in Engineering \& Sciences, vol. 21, no. 1, 2007, https://doi.org/10.3970/cmes.2007.021.053.Google Scholar
Liu, C.-S., “An Effectively Modified Direct Trefftz Method for 2D Potential Problems Considering the Domain’s Characteristic Length,” Engineering Analysis with Boundary Elements, vol. 31, no. 12, pp. 983–993, 2007, https://doi.org/10.1016/j.enganabound.2007.04.006.CrossRefGoogle Scholar
Romain, G. and Jean-Pierre, B., “Ellipsoidal Harmonic Expansions of the Gravitational Potential: Theory and Application,” Celestial Mechanics and Dynamical Astronomy, vol. 79, no. 4, pp. 235–275, 2001, https://doi.org/10.1023/A: 1017555515763.Google Scholar
Hu, X., A comparison of Ellipsoidal and Spherical Harmonics for Gravitational Field Modeling of Non-spherical Bodies. Department of Geodetic Science: The Ohio State University Columbus, 2012.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×