Series Coupling of Two Equal Strain Softening Elements: Imperfection Approach
Consider two nominally identical elements 1 and 2 coupled in series as shown in Fig. 8.1.1a. Assume that each element has a load-elongation (P-AL) curve displaying softening as sketched in Fig. 8.1.1b for element 1. In this plot, the full line is for monotonic extension, and the dashed line corresponds to unloading (shortening) right at the peak. The question is: What is the load-elongation response of the series coupling of the two elements?
A quick answer, extrapolated from the more usual cases of hardening structures would be: Just multiply the elongation by a factor of two. Wrong! Softening breaks down the usual rules. To clarify this, we lake first the imperfection approach to bifurcation. In this approach, one realizes that no two elements can be really identical. One of them must have a strength (peak load) slightly smaller than the other one. Assume that such is the case for element 1. So the element 2, whose curve is depicted in Fig. 8.1,1c, has a strength only slightly larger than clement 1. The difference is so slight that it cannot be discerned at the scale of the drawing.
As the series coupling is extended, both elements 1 and 2 load up until the peak A’ of element 1 is reached. Upon further extension, element 1 must begin to soften, following path A’-S’ with decreasing load. Since the load on both elements is identical, the load on element 2 must decrease, too. But since element 2 has not yet reached the peak, it is not going to soften. It is going to unload following the path A”-U".
Therefore, as soon as one element reaches the peak, further straining leads to softening of this clement and to unloading of the other. We say that strain localizes into one element due to softening. Fig. 8.1.Id shows the resulting P-AL curve as a full line. The dotted line represents the (wrong) result obtained by assuming that both elements go into the softening regime (we call it homogeneous deformation, same extension in each clement). Note that the rising portion of the curve (the hardening part) displays a displacement that is twice the displacement for a single element, the classical result. The difference lies only in the softening portion of the curve.
The foregoing result (see also Bazant andCedolin 1991, See. 13.2) is based on the idea that the strength of the two elements cannot be identical. Note that the amount by which they differ is immaterial. The same will happen if the difference were only one part in 1012, which is much less than what can be experimentally delected.
We have assumed that element 1 was the weaker element. In practice, wc cannot know a priori which of the two elements is going to break. We can only state that, if the loading system is perfectly symmetric, the probabilities of failure through one or other element must be equal, so that 50% of tests will show failure of element 1 and 50% failure of element 2.
The foregoing discussion makes use of inhoniogeneities or imperfections to get a general conclusion. However, (his result may be also obtained on the basis of thermodynamics. To do so, we consider a series coupling of two identical elements, and consider the possibility of bifurcation at the peak load. The two possible resulting paths are depicted in Fig. 8.1.2a. Path Л-Н (dotted line) corresponds to a homogeneous deformation, while path A L (full line) corresponds to softening that localizes into one of the elements, while the other unloads. Which is the preferred path? Following Baz. ant and Cedolin (1991, Sec. 10.2), for the correct path, the second-order work <52W = ~SPSu for imposed displacement increment <5u must be minimum, or, alternatively, the second-order complementary work 52W* – 6P 6u — <S2W for imposed load increment SP must be maximum.
Fig. 8.1.2b shows the graphical representation of the second-order work and second-order complementary work for a softening incremental process. Note that the values of the second-order areas are negative because SP < 0. Therefore, the foregoing principles may be restated by expressing that the second-order area below the P-и curve must be maximum at fixed 6//,, and that the second-order area over the P-u curve must be minimum at fixed SP.
Figs. 8.1.2c-d show the application of the foregoing principles to our case, ft is obvious that the correct path is that for which the localization occurs (see also Baz. ant and Cedolin 1991, Sec 13.2).