Ideal strength

As we showed in Chapter 6 (on the modulus), the slope of the interatomic force-distance curve at the equilibrium separation is proportional to Young s modulus E. Interatomic forces typically drop off to negligible values at a distance of separaHon of the atom centres of 2rg. The maximum in the force-distance curve is typically reached at 1.25ro separation, and if the stress applied to the material is sufficient to exceed this maximum force per bond, fracture is bound to occur. We will denote the stress at which this bond rupture takes place by d, the ideal strength a material cannot be stronger than this. From Fig. 9.1... 

Let us now see whether materials really show this strength. The bar-chart (Fig. 9.2) shows values of Oy/E for materials. The heavy broken line at the top is drawn at the level it/E = 1/15. Glasses, and some ceramics, lie close to this line - they exhibit their ideal strength, and we could not expect them to be stronger than this. Most polymers, too, lie near the line - although they have low yield strengths, these are low because the moduli are low. 

All metals, on the other hand, have yield strengths far below the levels predicted by our calculation - as much as a factor of 10 smaller. Even ceramics, many of them, yield at stresses which are as much as a factor of 10 below their ideal strength. Why is this ... 

Explain what is meant by the ideal strength of a material. Show how dislocations can allow metals and alloys to deform plastically at stresses that are much less than the ideal strength. Indicate, giving specific examples, the ways in which metals and alloys may be made harder. 

To further illustrate the concepts of strength and consolidation pressure, consider an idealized strength test, as shown in Figure 9. In this idealized test, the cohesive strength of the bulk solid is measured in two distinct steps ... 

An instructive two-dimensional calculation that reveals the stress magnifying effects of flaws is that of an elliptical hole in an elastic solid as depicted in fig. 2.12. The crucial idea is that, despite the fact that the specimen is remotely loaded with a stress uq which may be lower than the ideal strength needed to break bonds in a homogeneous fashion, locally (i.e. in the vicinity of the crack-like defect) the stresses at the termination of the major axis of the hole can be enhanced to values well in excess of the remote load. The exact solution to this problem can be found in any of the standard references on fracture and we will content ourselves with examining its key qualitative features. 

For a typical metal, the shear modulus is measured in gigapascals, implying an ideal strength of the same order. This value is to be contrasted with typical... 

A stress-strain curve for a Cu whisker is shown in fig. 8.8. It is seen that the stress scale associated with whiskers is in the gigapascal range in support of our hypothesis that in whiskers the ideal strength may be more approximately realized. It is not surprising that the ideal strength estimate is not reproduced exactly since our argument assumed a contrived sinusoidal dependence to the energy associated with uniform slip. 

An interesting estimate of the significance of the core parameter can be constructed by harking back to our discussion of the ideal strength concept. There, we noted that when shear stresses reach a value on the order of /r/27r, the forces... 

For the special case in which we make the simplifying assumption that the interplanar potential takes the simple cosine form given in eqn (8.2) and advocated earlier in the context of the ideal strength model, this problem allows for direct analytic solution. In particular, the solution is... 

One can thus see that the ratio between real and ideal strengths of solid is determined by the ratio between the size of molecules (interatomic distance), b, and the size of a defect. 

Idealized strength profile in homogeneous clay, (a) NC strength profile (b) UC strength profile (c) OC strength profile. 

In some cases crack advance in a polymer could occur by close-to-ideal de-cohesion or, alternatively, by ideal cavitation when the local crack-tip stress reaches either the ideal de-cohesion strength or, alternatively, the cavitation strength estimated from the universal... 

Metals exhibit the maximum stress only in whisker form because they permit dislocation glide at low stresses, and whiskers are almost free of dislocations, in bcc metals, improved potential models will lead to a better understanding of the ideal strength than has so far been gained from either the Orowan-Polanyi approach or the use of the Morse potential predictions, for example, of the fracture stress for a-Fe whiskers in the (111) direction using the Orowan-Polanyi equation are 46 GPa, where the maximum tensile stress obtained by Brenner was 13.1 GPa (Brenner, 1956), at an elongation close to 0.05 (see paper by Kiinzi, this volume). 

We may, however, inquire whether a small fraction of this ideal strength can be exploited. The theory that will be developed will show that, indeed, a small fraction of the ideal strength can be achieved in practice. But this fraction is strongly dependent on the rate of the separation, on geometry, and on the presence or absence of imperfections in the interface and in the matter that adjoins it, as well as being dependent on the rheological properties of the adhesive and the substrate. 

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