Summary: Necessity of Localization Limiters
The foregoing simple analysis corresponds to static loading and shows that the simple stress-strain model with strain softening leads to unacceptable behavior both physically and computationally: (1) the softening zone has a z. ero width and volume; (2) the inelastic strain and fracture work are z. ero; and (3) the computational results are mesh-unobjective.
Further analyses indicate that similar conclusions apply to dynamic situations. For example, Bazant and Belytschko (1985) analyzed the problem of two converging elastic waves propagating from the ends of a bar towards its center, where they add up to exceed the tensile strength. The results show that failure is instantaneous and occurs again over a zone of zero width, and with zero energy dissipation. Belytschko, Bazant et ai. (1986) reached similar conclusions for converging elastic waves in a sphere or a cylinder; although the fracture pattern was chaotic, with fracture occurring at many locations, the results still had zero measure and were mesh unobjective (see also Bazant and Cedolin 1991, Sec. 13.1).
The conclusion is that these models are not suitable at all because they allow localization in a region of zero volume. Therefore, if a continuum formulation based on stress-strain curves with strain softening is to be used, it is necessary to complement it with some conditions that prevent the strain from localizing into a region of measure zero. Such conditions are generically called localization limiters (Bazant and Belytschko 1985).
The model with the simplest localization limiter is the crack band model that we introduce next. Models with more general limiters are the nonlocal models that are presented in Chapter 13.
8.4 Consider a bar with a triangular stress-strain curve defined as Ее = о for Еє < and Ее — (1 + m)f’t ■- mo < for fi < Ее < (1 + m)f, where E = 30 GPa, f = 3 MPa, E = 30 GPa, and m = 21; the stress is z. ero for Ее > (1 + rn)ft. Determine the load-elongation curve and the energy supplied to break the bar if its length is 0.5 m and the softening localizes in a zone of width (a) h — 25 cm, (b) h — 10 cni, (c) h = 3 cm, (d) h — 1 cm.
8.5 In the previous exercise, determine the width of the softening zone for whicli the stress drops vertically to zero right after the peak.
8.6 Consider a bar that has an exponential stress-strain curve defined as
Fe= a f0r Ee ~ f>t 18 2 71
I a + mfl ln(/(‘/cr) for Ее < ft ( ■ • )
in which E = 27 GPa, ft = 3.1 MPa, m = 12. Determine the load-elongation curve and the energy supplied to break the bar if its length is 0.5 m and the softening localizes in a zone of width (a) h —– 25 cm, (b) h – 10 cm, (c) h = 3 cm, (d) h = 1 cm.
8.7 In the previous exercise, determine the width of the softening zone for whicli the tangent to the stress – elongation curve right after the peak becomes vertical.