Skew Meshes: Effective Width
The foregoing considerations are, of course, applicable even when a crack hand of width hc is embedded in a finite element of size h and is inclined with respect to the size of the element (Fig. 8.6.3a), an issue that has been used repeatedly in various fields. The idea of embedding a band of strain softening in a finite clement was first developed for plastic shear bands (Pietmszczak and Mroz 1981), and the subsequent development of a finite element with an embedded crack band (Wiliam, Bicanic and Sturc 1986; Wiliam, Pramono and Sture 1989) was mathematically analogous. Recently, a general and fully consistent three – dimensional formulation for an embedded strain-softening band in general finite elements was presented by Dvorkin, Cuitino and Gioia (1990).
In the present formulation, the only modification that is necessary is to substitute a proper value for the volume fraction of the crack band within the element /. For square meshes, Baz. ant and Oh (1983a) proposed to use an effective bandwidth for the element hb such that
where 0 is the angle between the band and the base of the element. An approximate generalization of this rule to irregular elements was proposed in Bazant (1985a). In general, however, such extrapolations to irregular elements can hardly be satisfactory, and a more general approach is needed.
The problem, however, is not trivial. The reason is that the volume fraction, when the crack band is inclined with respect to the element side, or the element is irregular, is not well defined: it depends on the precise position of the crack band with respect to the element. This is illustrated in Figs. 8.6.3b and c, for which the volume fractions are in the proportion 2:1, approximately. Therefore, either information on the position of the band within the element must be given —which is not possible if ordinary elements are used— or else, the bandwidth must be defined in an average sense. The average can be obtained in the following manner. Consider a bidimensiona! clement of thickness b, arbitrary size and shape, and arbitrarily oriented with respect to the band, which is drawn horizontal (Fig. 8.6.3d). Let у be the axis normal to the band, with its origin at the lowest point of the clement. If the band is located at distance y, as shown in the figure, the volume of the band is approximately given by V(y) = hcbc(y) where c(y) is the length of the intercept of the center of the band with the element. Let y?(y)dy be the probability that the band lies at a distance between у and у ~ dy. Then the average volume of the band is given by
Ц, — hcb f (f(y)c(y)dy (8.6.11)
where hp is the maximum ordinate of the element, which coincides with the projected element size. If the probability density is uniform, then yz — 1 jhv and we get
but b fglp c(y) dy is the volume of the element, and therefore,
This indicates that for equally probable distributions, the element bandwidth coincides with the projected size of the element (projected on the normal of the crack band). This coincides with the formula proposed by Bazant (1985a) for rectangular meshes, and has been implemented in commercial finite element codes (e. g., SBETA; Cervenka and Pukl 1994). However, it must be clearly understood that it is an average value, which may differ appreciably from the actual value for a particular clement in a particular mesh.
The foregoing calculation is based on the assumption that the strain is uniform within the clement, which is generally not the case because quadratic or higher order elements are used with various possible integration schemes (i. e., distribution of integration points). In such cases, the analysis would have to be redone starting from the virtual work equation (8.6.7), a task that is not straightforward. Rots (1988) used a trial-and-error method to determine the effective bandwidths of the elements for a particular problem, and Oliver (1989) proposed an objective formulation of an integral to define hb. Cervenka et al. (1995) empirically found that using the projected element size gave a still larger dissipation for inclined bands, and proposed a correction factor 7 so that
hb = yhp
Figure 8.6.4 (a) Skew cracking orientation at the integration points, (b) inclined cracking in the shaded element induces stress in the uncracked element (from Rots 1989). (c) Load-CMOD curves for a compact specimen under mixed mode loading (from Rots 1989).
Here 7 varies linearly from 1 for clement sides perpendicular to the band to 7 — 1.5 for sides at 45°; in the case of irregular elements, an average side angle needs to be used. With this correction, they obtained results approximately independent of the orientation of the mesh, for mode I cases (bending and tension).
ft seems that using conventional finite elements with plain smearing, as used in most finite element codes, implies a variable degree of uncertainty in the definition of the element bandwidth for meshes skew to the band. There arc two alternatives to circumvent this problem: (1) use remeshing techniques to achieve a mesh in which the band runs parallel and perpendicular to the sides of the elements it crosses (see Fig. 8.7.3b), or (2) use enriched elements with embedded strain discontinuities similar to those described in Section 7.2.3. In fact, the displacement discontinuity in Oliver’s element depicted in Fig. 7.2.10 is numerically implemented as a thin band with large, but finite, constant strain, very similar to an embedded crack band.