Stress-strain curves Considere construction

Rg. 8.4 The Considere construction. Possible forms of true-stress-strain curves (upper curves), the tangents drawn to them from the point e = -1 on the strain axis and the corresponding nominal-stress-strain curves. The curves are schematic for clarity in showing the construction for real polymers the initial part of the curve would generally be much steeper and the maximum, when it occurs, is generally at a much lower value of strain than that shown. See example 8.1 for a more realistic version of the curves in (c). 

This is the well-known Considere (1885) criterion for impending necking for a rigid plastic, rate-insensitive material. It is expressed graphically with a simple construction in Fig. 10.1 showing the intersection of the stress-strain curve with the curve of the decreasing slope of that curve, where the point of intersection is at u, gives the uniform extension strain. In Fig. 10.1, for strains e < u,dr/d > t e)... 

In terms of the Considere construction, the criterion for a material to neck and cold draw is indicated by the ability to draw a second tangent to the true stress-strain curve from the point = — 1 (Fig. 4). Another important aspect of yielding is that, while there is a clear drop in the engineering stress at (or near)... 

Equation (11.3) defines a geometric condition for the true stress-strain curve, corresponding to the simple construction due to Considere shown in Figure 11.3. The ultimate stress is obtained when the tangent to the true stress-strain curve da/(U is given by the line from the point A = 0 on the extension axis. The angle a in Figure 11.3 is defined by... 

The onset of ductile failure in sohds is determined by the Considire construction, in which a maximum in the stress-strain curve causes an instability that manifests itself as a neck. This concept is unhkely to be apphcable to the onset of necking in polymer melts. All constitutive equations, including the Maxwell model. Equation 9.16, predict a maximum in the stress-strain curve for stretching at a constant stretch rate, and this maximum normally occurs prior to the attainment of steady state. Hence, hteral interpretation of the construction as a sufficient condition for failure would imply that uniform uniaxial extensional experiments could never be carried out past the force maximum, which often corresponds to a relatively low strain such an interpretation is clearly contrary to substantial experimental experience in extensional rheometry, and several experimental studies focusing specifically on the Considere construction have shown that it does not predict the experimental onset of necking in melts. 

Equation (14.7) corresponds to the slope of the tangent to the curve cr, vs. e drawn from the point e = -1 or X, = 0. Figure 14.7 shows the true stress versus nominal strain curves for polymer samples A and B. Curves B] and B2 are compatible with curve B of Figure 14.6. The so-called Considere construction, Eq. (14.7), is satisfied with the tangent to the curves drawn from E = — 1. The tangential point corresponds to the maximum observed in the curve vs. and therefore with the maximum load that the specimen can support. In practice, the Considere construction is used as a criterion to decide when a polymer will form an unstable neck or form a neck accompanied by cold drawing. 

Figure 14.7 Considere construction for tensile strain. The maximum and minimum values of a in Figure 14.6 are given by the tangents to the curve of true stress ct, from 8 = -1. Polymer A forms an unstable neck polymers Bj and B2 form stable necks.

Fig. 5.25 Schematic curves of true stress, cr against nominal strain, e for polymers showing the Considere construction, (a) Unstable neck, (b) stable neck.

When the results of a tensile test are plotted on a graph of true stress against linear strain, show that the point at which necking starts is the point of contact of the tangent to the curve that passes through the point on the strain axis where the strain equals —1. This is the construction of Considere. 

---