ABSTRACT
For illustration purposes only one starting stress state 6start is considered-the stress state at the apex, . Trial state increments are normalised with respect to the triaxial tensile strength ftt and will reach both transition and apex stress return zones. The number of iterations for convergence in the transition zone (cutting plane algorithm used) is illustrated on Fig. 3. An increased reduction of the tensile strength is clearly obtained for larger degrees of softening (Fig. 4). Percentage errors in stress defined as
M.PASTOR, P.MIRA and C.RUBIO
Sector Ing. Computational CETA, Centro de Estudios y Exp. de Obras Publicos, Madrid, Spain
M.QUECEDO
1 Introduction Computation of failure conditions in engineering structures is needed to ascertain their safety. The analyst needs not only the limit load (if any), but the overall behaviour, and the failure mechanism, the latter will allow him to reinforce the weaker parts while the overall behaviour will indicate wether the failure can be of catastrophic type or not. Limit load computations are known to be influenced by the type of element [1], and specially how it performs near the incompressibility limit. In addition to this, an eigenvalue analysis can detect non-zero eigenvalues responsible for a spurious numerical hardening [2]. Quite often, remedial measures aiming to improve the behaviour near incompressibility improve other aspects as well, as it happens with selective integration and strain projection methods [3], or viceversa, as shown by Simo and Rifai [4]. These effects have made the analysts abandon elements of poor performance such as triangles in favour of more complex and
robust ones (like the 15N CST, the or the element designed by Ortiz [5]). If the material is of softening type, the problem becomes ill-posed, and the solution depends on the mesh size used. To circumvect this problem, it has to be regularized either by using a rate-dependent material [6], or by enriching the continuum description [7,8,9]. Failure mechanism is known to depend on mesh alignment. Numerical experiments done by Larsson [10] and Bicanic [11] showed how different alignments produced either clearly defined or smeared mechanisms. The situation, is however, worse, as spurious and unrealiastic mechanisms can be predicted under certain circumstances [12]. This paper aims to describe how different elements perform in spreading the shear band and how the predicted solution is affected by particular arrangements of elements, and by the existence of privileged directions in the mesh, proposing a possible remedial strategy based on adaptive remeshing [13,14]
2 Shear band smearing Finite elements tend to diffuse sharp gradients and discontinuities such as shear bands over elements. The result makes it difficult sometimes to identify which is the mechansm causing failure, and, gives an overall "load-displacement" response less fragile than the real, one with a peak load which can be higher. Therefore, the prediction gives over-conservative results. It is important, therefore, to reduce this smearing of the band either by reducing the mesh size or by adaptive remeshing techniques, as it was shown in Ref. [14]. The simplest linear triangle presents severe locking problems which may result in much wider bands, as it can be seen in Fig. 1. There, a test problem consisting on a vertical cut on a Tresca elastoplastic material has been discretized by randomly oriented triangles resulting from subdivision of a regular mesh of quadrilaterals. Material properties are E=100 kPa, and H=-500 Pa. The load was applied through a rigid plate 1000 times stiffer. The contours of equivalent plastic shear strain show an important diffusion of the band over 10 element widths. Analysis of the deformed mesh, not being shown in the picture, shows a diffuse mode of failure similar to the barrelling of laboratory specimens of soft cohesive soils, rather than a clearly defined shear band. If the triangles are now aligned along the direction of the shear band, (Fig. 1.b), the definition increases dramatically, reducing to 3 element sides or 4 nodes. The deformed mesh shows now a clearly defined failure mechanism consisting on the sliding of the upper block along a 45° inclined surface. The same mesh has been reproduced in Fig. 1c with 15N cubic strain triangles, keeping constant the spacing between the nodes. Comparing the definition of the band, it appears that it is the linear triangle which produces the sharper band when properly aligned. If it is not the case, locking will appear and, sometimes, even wrong failure mechanisms will be pre
3 Mesh alignment effects To ascertain the robustness of elements against mesh alignment effects, the same test example has been considered, using now a mesh aligned perpendicularly to the band direction. (Fig. 2). The linear triangle is unable to provide the correct mechanism, and only a shear band originating at the left upper corner develops. Comparing the maximum vertical force applied to the footing in both cases, their maximum values coincide, but in the second case, the softening is prevented by severe locking exhibited by the elements. Taking into account the fact that plastic deformation is isochoric, the difference can be related to the fact that in the first case, movement of the nodes relative to the opposite side is a translation paralel to it, while in the second, the relative movement activated by the shear band mechanism would be normal to it. The behaviour exhibited by the six-node and the 15N CST is less affected by mesh alignment as it can be seen in Fig 2.b for the CST triangle.
