Abstract
The paper examines an effect of boundary constraints applied to the enhanced degrees of freedom of partition of unity based discontinuous elements. To highlight the present issue the problem is studied in a one-dimensional setting. In particular, an example of a one-dimensional bar element crossed by a set of discontinuities having a finite elastic stiffness clearly shows a need for proper approximation of the displacement field within a discontinuous element in order to correctly represent the structural response. While the discontinuous elements with boundary constraints applied to the enhanced degrees of freedom display an unrealistic dependence of the global response on the locations of the discontinuities, the discontinuous elements with complete approximation of the discontinuous part of the displacement field provide the expected global response independent of the locations of the discontinuities.
Highlights
The partition of unity concept [2], which allows a local enrichment of the standard finite element basis by special functions has been widely used to model the displacement discontinuity in a number of applications, e.g., the quasi-brittle failure of natural stones such as Massangis limestone [6] or continuous-discontinuous modeling of failure in high performance fiber-reinforced cement composites [7]
Recall the problem of localization of the inelastic deformation in problems free of initial stress concentrators. This problem has been addressed, e.g., in the habilitation thesis of Brocca [5] and recently revisited in [1] using the concept of the partition of unity method, which allows the necessary splitting of the total displacement field into elastic and inelastic displacements associated with the crack opening
The study showed a possible depreciation of the results when an element crossed by a discontinuity containing a boundary node that had to be eliminated by the boundary constraints
Summary
The partition of unity concept [2], which allows a local enrichment of the standard finite element basis by special functions has been widely used to model the displacement discontinuity in a number of applications, e.g., the quasi-brittle failure of natural stones such as Massangis limestone [6] or continuous-discontinuous modeling of failure in high performance fiber-reinforced cement composites [7]. In this framework, the discontinuity in the displacement field is introduced by enriching the standard finite element polynomial basis with the Heaviside function [8]. Application to a one-dimensional problem is discussed in Section 3 and compared to the analytical solutions provided by the conventional chain of spring elements
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