Localization in an Elastic-Softening Bar

Consider a homogeneous bar of length L (Fig, 8.2.2a) that has a stress-strain curve of the elastic-softening type, as depicted in Fig. 8.2.2b, and is characterized by a linear elastic behavior up to the peak, followed by strain-softening. We can then write the strain on the softening branch as

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Localization in an Elastic-Softening Bar


where E is the elastic modulus and fJ is the inelastic fracturing strain, graphically defined as shown in Fig. 8.2.2b. Unloading from the peak is assumed to be fully elastic. We further assume that the softening branch is unique, i. e., that a unique relationship exists between a and as long as є* increases monotonically:

сг = (/>(є/) (8.2.2)

This function can be extracted from the cr-e curve and plotted independently as shown Fig. 8.2.2c. We can also compute the work – yp required to fully break a unit volume of material (the fracture energy density): it is the area under the cr-e curve, and so the area under the о-є-f curve:

rOC rOO.

7F~ cr = / Ф(є^) (8.2.3)

Jo Jo

Note that Figs. 8.2.2 and 7.1.3 and the foregoing integral arc similar to the definition of Gp in (7.1.8). It might seem that the correspondence is immediate and logical. It is not.

Consider a quasi-static process in which the bar is monotonically stretched. Up to the peak, the strain is uniform, equal to the elastic strain. At peak, just as seen before, a bifurcation can occur so that a portion of the bar, of length h < L, continues stretching, while the rest of the bar unloads elastically (Fig. 8.2.2d). The total elongation of the bar in the softening branch is thus:

AL — ~{L-h)+ h = –L – I – e! h (8.2.4)

where we see that the first term in the last inequality is the elastic elongation. Therefore, we can define the fracturing elongation as

Localization in an Elastic-Softening Bar

Figure 8.2.2 (a) Homogeneous bar. (b) Elastic-softening stress strain curve, (c) Stress-fracturing strain curve, (d) Bar with a softening band of length h.

On the other hand, the total work supply required to break the whole specimen is just the work required to break the softening portion (the remainder is always elastic) so that


Wf — A / a d(he^) — Ah / a def — Ah^p (8.2.6)

Jo Jo

wiiere A is the area of the cross section of the bar.

Up to now il has been arbitrary, but what is its preferred value? To find it, we apply again the maximum second-order work condition {62W* = max) and find immediately that the thennomechanical solution is h — 0, in complete concordance with the previous result N — oo for equally sized elements. It follows from this essential result and from (8.2.5) and (8.2.6) that, according to this model, both the inelastic displacement and the fracture work are zero. This is physically unacceptable and contrary to experiment.