Crack Band Width

From the foregoing analysis it transpires that, in a finite element formulation with a free element size, the strain-softening curve must be adjusted according to the element size so that the calculations would yield macroscopically consistent results whatever the element size. This is close to saying that the crack band width hc is arbitrary since it is replaced by h,^ without a noticeable effect (as long as the element size
is kept small). This means that hc cannot be determined from fracture tests in which a single crack (or crack band) is formed.

The value of hc, however, does have an effect in those situations where cracking does not localize but remains distributed over large zones. This may happen as a consequence of a dense reinforcement grid or in problems such as shrinkage, where the mass of concrete in front of the drying zone restrains the cracking zone and may (but need not) force the cracking to remain distributed. Thus, the value of hc can be identified only by comparing the results of fracture tests with the results of tests in which the cracking is forced to be distributed. The problem is the same as that in determining the characteristic length for nonlocal models, and we will discuss it in more detail in Chapter 13.

In a crude manner, the value of hc can be approximately identified from fracture tests for specimens of various geometries, in which the cracking is localized to a different extent. This has been done in Bazant and Oh (1983a), with the conclusion that the crack band width hc — 3da where da = maximum aggregate size, is approximately optimal. However, the optimum was weak, and crack band width anywhere between 2d. a and 5da would have given almost equally good results.

A better test for determining hc was conceived by Bazant and Pijaudier-Cabot (1989). Localization was prevented by gluing parallel thin rods on the surface of a uniaxially tensioned prism. However, a uniform field of strain-softening was still not achieved. For details, see Section 13.2.4.

Exercises

8.8 Give a detailed proof of Eq. (8.3.6).

8.9 Determine the uniaxial stress-strain curves for a concrete which, according to experimental measurements, has an elastic modulus of 25 GPa, a tensile strength of 2.8 MPa, and a fracture energy of 95 N/m, and is assumed to display an elastic-softening behavior with triangular softening and a crack band width of 50 mm (approximately equal to 3da with da = 16 mm).

8.10 Determine the uniaxial stress-strain curves to be used for the same material as defined in the previous exercise if the numerical analysis is to be performed using finite elements 20 cm in size.

8.11 Determine the maximum size of the finite elements to be used in a numerical analysis with the same material in order for the stress-strain curve to be stable.

8.12 Determine the uniaxial stress-strain curves for a concrete which, according to experimental measure­ments, has an elastic modulus of 25 GPa, a tensile strength of 2.8 MPa, and a fracture energy of 95 N/m, and is assumed to display an elastic-softening behavior with exponential softening and a crack band width of 50 mm (approximately equal to 3da with da — 16 mm).

8.13 Determine the uniaxial stress-strain curves to be used for the material defined in the previous exercise if the numerical analysis is to be performed using finite elements 20 cm in size.

8.14 Determine the maximum size of the finite elements to be used in a numerical analysis with the foregoing exponential material in order for the stress-strain curve to be stable (not to exhibit snapback).

8.15 For the material defined by Eq. (8.3.9), determine the fracturing strain Єи and stress f’t at which the peak occurs as a function of the constants Ci. p,6, and q. Show that the equation can be rearranged to read

= ЛЄ ехр[-р(£’1 – 1 )/q], in which £ = Є je{.

8.16 For the material defined by Kq. (8.3.9), show that the fracture energy density 7f can be written as

„ = г(р±1

фірі-о/я1 q J

where Г(n) is the Etilerian Gamma function defined as Г(n) = J(j°° xn~le~xdx.

8.17 Consider a material with a stress-strain curve given by a —■ Еєе~Ьє. Show that b’ is indeed the elastic modulus. Determine b in terms of 15 and /,’. Determine, as a function of E and /(‘, (a) the total energy density absorbed by a material element that follows the softening branch; (b) the energy density absorbed by a material element loaded up to the peak and then unloaded, if it is assumed that the unloading is linear with the same elastic modulus as for the initial loading; (c) the density of fracture energy 7f; and (d) show that for this model the ratio h. c/!ch is constant and determine its value. [Answers: (a) 7.39/t’2/E, (b) 1.45/(‘2/b’, (c) 5.94fl2/Е, (d) hc/Lh ss 0.17.]

Figure 8.4.1 Types of stress-strain curves: (a) stiffness degradation; (b) yield limit degradation; (c) mixed behavior; (d) more realistic behavior with nonlinear unloading.