True stress

True stress-strain curves for plastic flow... 

The apparent difference between the curves for tension and compression is due solely to the geometry of testing. If, instead of plotting load, we plot load divided by the actual area of the specimen, A, at any particular elongation or compression, the two curves become much more like one another. In other words, we simply plot true stress (see Chapter 3) as our vertical co-ordinate (Fig. 8.7). This method of plotting allows for the thinning of the material when pulled in tension, or the fattening of the material when compressed. 

This equation is given in terms of true stress and true strain. As we said in Chapter 8, tensile data are usually given in terms of nominal stress and strain. From Chapter 8 ... 

To see what is going on physically, it is easier to return to our first condition. At low stress, if we make a little neck, the material in the neck will work-harden and will be able to carry the extra stress it has to stand because of its smaller area load will therefore be continuous, and the material will be stable. At high stress, the rate of workhardening is less as the true stress-true strain curve shows i.e. the slope of the o/e curve is less. Eventually, we reach a point at which, when we make a neck, the workhardening is only just enough to stand the extra stress. This is the point of necking, with... 

At still higher true stress, do/de, the rate of work-hardening decreases further, becoming insufficient to maintain stability - the extra stress in the neck can no longer be accommodated by the work-hardening produced by making the neck, and the neck grows faster and faster, until final fracture takes place. 

Assuming that a diamond-pyramid hardness test creates a further nominal strain, on average, of 0.08, and that the hardness value is 3.0 times the true stress with this extra strain, construct the curve of nominal stress against nominal strain, and find ... 

Chapter 8 Nominal and True Stress and Strain, Energy of Deformation... 

Corrosion effect of forming Elongation X gauge length Standard hydropress specimen test True stress-strain curve Uniformity of characteristics... 

Appendix B General Properties and Data on Elastomers and Plastics 175 Table B.IO True Stress at Break of Selected Melt-Mixed Rubber-Plastic Blends ... 

B.IO True stress at break of selected melt-mixed rubber-... 

Generally there is a stiffening effect in compression compared to tension. As a first approximation one could assume that tension and compression behaviour are the same. Thomas has shown that typically for PVC, the compression modulus is about 10% greater than the tensile modulus. However, one needs to be careful when comparing the experimental data because normally no account is taken of the changes in cross-sectional area during testing. In tension, the area will decrease so that the true stress will increase whereas in compression the opposite effect will occur. 

True stress, o, is defined as o = F/A where A is the actual area of cross section of the member corresponding to the load F. 

Figure 1.9 Comparison between nominal and true stress-strain.

The rate of strain hardening, do /ds, at any given value of the true stain is given by the slope of the true stress-true strain plot at that strain and is called the modulus of strain hardening. 

A plot of log o against log s should thus yield a straight line whose slope is n and which makes an intercept equal to log fej on the log o axis (at s = 1). Thus the constant kj represents the true stress at unit true strain and is termed the strength coefficient. The exponent n is known as the strain hardening exponent. 

The true stress for the uniaxial case is obtained by substituting Equation (20) into Equation (15) as... 

Values of stress and strain obtained from Figure 1 and from similar plots of data obtained on the other elastomers yield the plots of Xo vs. (X — 1) in Figure 2, where Xo is the true stress, i.e., the force per unit cross-sectional area of the deformed specimen. The data at strains up to 1.0 (100% elongation) give straight lines whose slopes equal the equilibrium tensile moduli, E values of 1 /3 are given in Table I. 

---