Determination of Characteristic Length

The characteristic length is a parameter that controls the spread of the nonlocal weight function. It may be defined as the diameter of an averaging region (line segment, circle, or sphere in one-, two-, or three – dimensions) with a uniform weight function that has the same volume as the actual weight function used. The characteristic length £ cannot be directly measured but must be inferred indirectly from test of suitable types. There arc two types of tests suitable for this purpose: (1) the use of size effect, and (2) the use of elastically restrained tests. Let us examine each of the two possibilities.

(a) Use of size effect. The size effect is the most blatant and most important manifestation of nonlocality. It is necessary to carry out tests of geometrically similar notched specimens of sufficiently different sizes and determine the size effect plot (Chapter 6). Then the characteristic length of the nonlocal model needs to be varied until the finite element calculations match the experimentally determined size effect curve in the optimum way. Generally, it is observed that the transitional size Dq of the size effect plot (intersection of the horizontal and inclined asymptotes) is approximately (but not exactly) proportional to the value of characteristic length £. Therefore, an effective strategy is to assume characteristic length £’, calculate by a nonlocal finite element code the nominal strength of specimens of different sizes, and trace the size effect curve. Optimum lilting of this curve with the size effect law makes it possible lo obtain the horizontal and vertical asymptotes and determine their intersection Dq. Then the best estimate of the corrected characteristic length is

£ — i’D:/D’0 (13.2.1)

The process is then repeated and the value of £ corrected iteratively. Normally no more than two corrections are required for convergence.

(b) Elastically restrained tensile test. Another approximate way of determining £ was proposed by Bazant and Pijaudier-Cabot (1989). Л long prismatic specimen of concrete, with a thickness of only a few aggregate sizes, is cast and many longitudinal thin steel rods are glued to its surfaces by epoxy as shown in Fig. 13.2.7. It is assumed that the glued steel bars are sufficient to force the strain in the specimen to be uniformly distributed, and for this reason the specimen must be as thin as possible. If that is the case, the tensile load-deflection diagram directly yields the stress-strain curve for the fracture process zone of concrete. This is illustrated in Fig. 13.2.7c, where the inclined straight line of slope Iis gives the stress carried by steel bars and epoxy alone, and the shaded zone represents the additional contribution due to concrete. If the slope of the load-deflection curve is always positive, localization should not happen according to uniaxial localization analysis. Thus, plotting the results in terms of the average stress and average strain, the shaded area in Fig. 13.2.7c gives the energy Ws dissipated per unit volume of the fracture process z. onc, on the average. Hence, the average width of the softening zone h should approximately be given by

h = Gf/Ws (13.2.2)

which has the dimension of length because Gy ~ J/m2 and Ws ~ J/m3. The fracture energy Gy is determined by any of the previously discussed methods. A particular nonlocal model is then needed to correlate h and £, although it may be assumed that h?» £ (Bazant and Pijaudier-Cabot 1989).

In practice, however, it turned out that this method gives only a crude estimate of the characteristic length because the specimen with tensile restraining elastic bars does not behave uniaxially. The deformation becomes nonuniform transversely and there is some degree, although not a large degree, of localization, as transpired from a thorough investigation by Berthaud, Ringot and Schmitt (1991). Further development would be required before £ can be accurately determined by this method.