Unloading and Reloading

For general applications in finite element programs, one must also specify what happens when, after partial or full cracking, the material is unloaded or reloaded. Experimentally observed behavior at unloading and reloading is rather complicated and looks approximately as sketched in Fig. 8.3.4a which is characterized by hysteretie loops of considerable area (Reinhardt and Cornelissen 1984; Hordijk 1991). In most finite element programs, however, it is assumed that unloading and reloading are linear. In the next section, devoted to the uniaxial softening models, we show how these linear unloading-reloading curves are generated within theoretical frameworks that can be easily generalized to the general three-dimensional models.

If the detailed uniaxial unloading-reloading curves need to be reproduced, the expressions developed to generate realistic unloading-reloading curves in cohesive crack models (Section 11.7.4) are easily incorporated into the crack band model through the basic relationship (8.3.1).

A simpler rule, called the secant-tangent rule, was proposed in the frame of crack band models by Bazant and Chem (1985a, b), as illustrated in Fig. 8.3.4b. Given the stress-strain diagram for monotonic

Unloading and Reloading Подпись: (8.3.13)

stretching, a — ір{є), the secant-tangent rule assumes that the unloading has always the same slope as the secant-modulus for virgin loading at the same strain value, i. e.,

Graphically, this means (Fig. 8.3.4b) that segment 23 is parallel to the secant 05, segment 34 is parallel to the secant 06, etc., where points 5, 6, 7 are obtained from points 2, 3,4 by vertical piojections onto the virgin stress-strain curve.

For reloading one may assume either the same path as for unloading, or, better, a straight line reloading up to point 8 on the strain axis and then either a straight line back to point 1 where unloading started or a straight line 89 parallel to the secant 01. The tangent-secant rule underestimates the area of the hysteresis loops, but it has the advantage that it yields, without any additional material parameters, an approximately correct location of point 4 at which the initial elastic slope is resumed. Furthermore, using the rule shown by curve 489, point 9 at which the virgin curve is reached again is approximately correct. The tangcnt-sccant rule was applied in a finite element program for combined smeared cracking, creep, and shrinkage of concrete (Bazant and Chern 1985a, b).