Evaluating the Applicability Limits of the Beam Approximation in the Analysis of Problems on the Compression of Bodies Along Cracks

Abstract

In the investigation of the problems on the compression of a material along interacting parallel cracks, there appears a natural question of the necessity for considering two “limit” alternatives of the mutual location of defects. Those are the situations when the cracks are far away from each other (the separation distance between the cracks or between a near-surface crack and the boundary surface of the body is considerably larger than the linear dimensions of the crack), and when, conversely, the cracks are located very closely to each other (or the crack is very close to the boundary surface of the body). The first, simpler, limit case has been partially discussed in Chaps. 3 and 4 when investigating plane and spatial problems on the compression of bodies along the arrays of interacting cracks. The result of that discussion has been intuitively expected, and its conclusions have correlated with physical considerations: by making the distance between the cracks (or the distance between the near-surface crack and the body boundary) increasingly larger, we arrive at the situation when cracks interaction between themselves (or the interaction of a near-surface crack with the body surface) produces an ever weaker effect on the values of critical loads. In the extreme case when the distance between the cracks (or between the near-surface crack and the body boundary) tends to infinity, we do obtain the anticipated result—critical compressive loads asymptotically approach the loads for a single isolated crack in an unbounded body (they, in turn, coincide with the critical compressive loads of the surface instability of the examined material and with the first in the loading history loss of the continuity of integral equation kernels in terms of the derived eigenvalue problem). In the other limit case, when the distance between the cracks is becoming not larger but, on the contrary, closer, the situation is more intricate. It is evident that when the cracks are becoming closer, critical compressive loads will tend to zero; however, the determination of what exactly the asymptotics of such behavior will be does require a special investigation in the framework of the precise approach considered here. The approximate approach, which is widely used in the investigations of problems on the compression of bodies along cracks, is referred to in literature as “beam approximation” (see Chap. 1). In it, the material containing cracks is substituted with a thin-walled member separated by the cracks, and then its instability in the Eulerian sense is investigated for a chosen a priori fastening on the edges. Evidently, it is oriented towards the situation of the closest distance between the cracks, therefore, it is a fortiori inapplicable for the cracks that are rather distant from one another (the assessment of the applicability of this approach in terms of the “thin-wall” parameter is required). On the other hand, the abovementioned beam approximation may result in significant (up to hundreds percent) errors even for very closely spaced cracks due to an inadequate choice of boundary conditions on the edges of the examined thin-walled member (the assessment of the propriety of the fastening type chosen for a particular geometry of cracks location). In this chapter, based on the results of investigating the problems on the compression of bodies along the interacting cracks in the rigorous linearized formulations, the limits of the applicability of the beam approximation are assessed. We use the approach that provides the values of critical compression parameters and the theoretical strength limit for the compression of unbounded bodies along flat cracks in closely located parallel planes and for the compression of semi-bounded bodies along near-surface cracks located closely to the boundary surface. This approach permits the passages to the limit when the separation distance between the parallel coaxial cracks or between a near-surface crack and the body boundary tends to zero. Due to it, the applicability limits of beam approximation to such problems are determined. Besides, for various design diagrams within beam approach we determine boundary conditions on the edges of the respective idealized two-dimensional objects that are as if separated by the cracks. They, when specified, produce the results that are the closest to those obtained in the framework of the rigorous three-dimensional approach. It should be noted that this chapter has been co-authored by M. V. Dovzhik and includes the results obtained both by him personally and by the authors of this book. Those were presented, in particular, in the following publications [13, 14, 15, 16, 17, 18, 19, 27].