Keywords: Trefftz method, Galerkin technique, point-collocation technique, auxiliary function, mode III fracture.

Trefftz approach can be referred to the boundary-type solution procedure employing the regular T-complete function satisfying the governing equation. Since Trefftz [1] in 1926 applied his approach that was forced to satisfy the boundary condition to achieve the outcomes, numerous papers, concerning fundamentals, applications and analysis, have emerged in the literature. For the Trefftz boundary element method, Cheung et al [2] developed a direct formulation for solving two-dimensional potential problem. Kita et al [3] studied the same problem by the direct formulation and domain decomposition approach. Portela and Charafi [4] applied Trefftz Boundary element formulation to potential problems with thin internal or edge cavities. Sladek et al [5] presented a global and local Trefftz bpundary integral approach to solve Helmholtz equation. Domingues et al [6] extended the Trefftz boundary element approach to the analysis of linear elastic fracture mechanics. Most of the developments in the field can also be found in [7].

This paper presents two indirect Trefftz boundary approaches: Galerkin techniques and the collocation point techniques, which can be applied to mode III fracture problems. First, original formulations and the solution of the mode III crack in elastic bodies are deduced based the Trefftz functions satisfying the Laplace governing equation. Then the stiffness matrix and equivalent nodal flow vector are formed from the approximate solution that satisfies the boundary condition by means of Galerkin method and the collocation point method. To improve the accuracy of the numerical results, especially those near crack tips, an auxiliary function method is introduced and its advantages are explained in detail. This aspect is considered as a new feature of this paper. In addition, a general expression of stress intensity factors is obtained based on special Trefftz functions presented that simplify the calculation. A numerical example is considered to show the application of the proposed approach, and the effect of some important parameters on the order of numerical accuracy is also discussed. This paper presents an approximate approach, namely auxiliary function approach in which the displacements satisfy singular properties near crack tips. In this approach, the displacements and stresses along the left micro size is expressed in terms of some analytical functions, which can simulate the characters along the right micro size. Further, two more advantages of using the auxiliary functions can be seen clearly, one is that the accuracy of results can be promoted near crack tips when the computation involves the region far from crack tips; another is that the close results can be achieved using different auxiliary functions, despite of showing different convergence characteristics for different functions. This feature can be applied to check the correctness of the outcomes. Generally, stress intensity factors (SIF) are evaluated by analysing stress and displacement fields near crack-tips obtained from various numerical methods such as conventional FEM and boundary element method. These procedures are usually complicated and time-consuming. In the light of the special purpose function for crack-tip element, local field distribution such as the stress fields and displacement fields in crack problem can easily be obtained. Hence, high efficiency in solving singular problem by HTBE approach is an attractive option of evaluating SIF from associated with the singular factors in special-purpose elements.

From the numerical example provided, it has been shown that the ratio of decreases gradually when the number of crack elements (or -functions or collocation points) increases and converges at a certain value after the number of crack element is large enough. It is also found that SIF can converge to the same value even using different auxiliary functions, which provides a means for us to check the correctness of the results.

The numerical results are compared with those obtained by conventional finite element model or by other approaches and it demonstrates that the proposed Trefftz boundary element approach is ideally suited for the analysis of fracture problem.

1

Trefftz E, "Ein Gegenstück zum Ritzschen Verfahren", in Proceedings 2nd International Congress of Applied Mechanics, Zurich, pp.131-137, 1926

2

Cheung YK, Jin WG, Zienkiewicz OC, "Solution procedure for harmonic Problems: non-singular, Trefftz fun." Com. App. Num. Meth., 5, 159-69, 1989. doi:10.1002/cnm.1630050304

3

Kita E, Kamiya N, Iio T, "Application of a direct Trefftz method with domain decomposition to 2D potential problems", Eng. Analysis B E., 23, 539-48, 1999. doi:10.1016/S0955-7997(99)00010-7

4

Portela A, Charafi A, "Trefftz boundary element methods for domains with slits". Eng. Analysis Bound. Elem., 20, 299-304, 1997. doi:10.1016/S0955-7997(97)00047-7

5

Sladek J, Sladek V, Keer R Van, "Global and local Trefftz boundary integral formulations", Adv. Eng. Software, 33, 469-476, 2002. doi:10.1016/S0965-9978(02)00050-9

6

Domingues JS, Portela A, Castro PMST, "Trefftz boundary element method applied to fracture mechanics". Eng. Frac. Mech. 64, 67-86, 1999. doi:10.1016/S0013-7944(99)00062-4

7

Qin QH, "Fracture Mechanics of Piezoelectric Materials", WIT Press, Southampton, 2001.

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