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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
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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

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References

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