Diagnóstico de necesidades

The procedure for the Coulomb–Mohr (or Mohr–Coulomb) theory uses the Mohr’s circle construction as shown in Fig. 3.3, Mohr [3.3].

T –C x y τ σ φ

On the normal stress,σ, versus shear stress, τ, axes plot the circles for uniaxial tension and compression to failure, as shown in Fig. 3.3. Then two linear failure envelopes are taken as just being tangent to the two circles. In order to evaluate the safety for any given three-dimensional state of stress, the maximum and minimum principal stresses are used to form a circle on the plot. If this circle is inside the failure envelopes then there is no failure, otherwise there is failure of the material. For example, take a state of simple shear stress. Form a circle with its center at the origin in Fig. 3.3 and just tangent to the failure envelopes. This circle intersects the τ axis at the values of the shear stress at failure, S, given by

S = TC

T + C (3.10)

where T and C are the failure levels in uniaxial tension and compression. The general characteristics of the construction in Fig. 3.3 are given by

x = TC C − T y = 1 2 √ TC and sinφ = C − T C + T (3.11)

The procedure just outlined for failure assessment from Fig. 3.3 can be expressed in analytical form. Let σ1,σ2, andσ3 be the principal stresses, with σ1 being the largest andσ3 being the smallest in an algebraic sense. From relations (3.10) and (3.11) and Fig. 3.3, the failure criterion can be shown to be given by 1 2  1 T − 1 C  (σ1+σ3) + 1 2  1 T + 1 C  (σ1− σ3)≤ 1 (3.12) The first term in (3.12) shows the normal stress effect, and the second term the maximum shear stress effect. Relation (3.12) directly reduces to

σ1

T −

σ3

Coulomb–Mohr Failure Criterion 27

This extremely simple form constitutes the entire two-property Coulomb– Mohr failure criterion.

In the limit of T = C the Coulomb–Mohr form reduces to the max- imum shear stress criterion of Tresca. In principal stress space this is given by the infinite cylinder of hexagonal cross-section. When T < C the Coulomb–Mohr form becomes a six-sided pyramid in principal stress space, and this six-sided pyramid has but three-fold symmetry rather than six-fold symmetry. An analytical form for this criterion in terms of invari- ants has been given by Schajer [3.4], although it is far easier to directly use the forms given here. The Coulomb–Mohr theory has been used in applications as diverse as nano-indentation and large-scale geophysics.

As with the Drucker–Prager example, it is not difficult to find real- istic examples where the Coulomb–Mohr criterion produces blatantly incorrect results. Some examples are as follows. Take a three-dimensional compressive stress state specified by

σ1 =σ2=−σ σ3 =−2σ

(3.14)

Substituting (3.14) into (3.13) gives 

−1 + 2T C



σ ≤ T (3.15)

Thus the Coulomb–Mohr criterion asserts that materials such as brittle polymers and a wide range of other materials can withstand unlimited compressive stresses in the stress state (3.14) when

T

C ≤

1

2 (3.16)

Unfortunately this allows unlimited compressive stresses.

This prediction is completely and egregiously unrealistic. The stress state (3.14) can be thought of as an hydrostatic pressure of magnitude σ with a superimposed uniaxial compressive stress also of magnitude σ. Now, superimposed hydrostatic pressure does have a strengthening effect on any other stress state. From typical data, an hydrostatic pressure equal to twice the uniaxial compressive failure stress would be expected to approximately double the superimposed uniaxial compressive strength.

The failure theory from Chapter 4 applied to the stress state (3.14) with T /C = 1/2 gives the failure solution as σ = 2.23C, in agree- ment with the typical data expectation. The Coulomb–Mohr prediction of infinite strength in this stress state for these materials classes is totally unacceptable.

The Coulomb–Mohr form was very appealing in its simplicity of concept and its presumed generality and ease of use. There was great enthusiasm for it in the early 1900s when it came into general use. Then it was shown by von Karman [3.5] and B¨oker [3.6] to have considerable lim- itations for brittle materials—specifically, natural minerals. Much effort has been expended over the years to “correct” and generalize this most simple and direct form of the Coulomb–Mohr criterion, but to no special advantage or usefulness. It has continued to be used—perhaps because of the perceived lack of a suitable two-property alternative—but never- theless the above examples unequivocally prove that the Coulomb–Mohr theory is fundamentally incorrect and should never be used, even though it is widely disseminated and codified.

3.6

The Bottom Line

Of the four standard and widely used failure criteria for isotropic materials,

Mises Tresca

Drucker-Prager Coulomb-Mohr

only the Mises criterion is of lasting significance. Although applicable only to very ductile metals, it is the classic result in that specific application, and it surely will endure. As shown, the other three criteria fail to establish the credentials necessary for permanency, and in fact can be misleading and patently incorrect in particular materials applications. Worst of all, none of the four can be used as general purpose failure criteria, as they are completely inappropriate to the task. Furthermore, it is probably obvious that adding another parameter or two to the Coulomb–Mohr or Drucker–Prager forms would be exceedingly unlikely to convert what is a completely unacceptable form into a transparently successful form. A totally new and fundamentally different approach is clearly required.

References 29

Although the results of this chapter do not lead to any useful general failure form(s), they do provide a hint of where to look for opportun- ity. In particular, the Coulomb–Mohr criterion is completely linear in the stress components and is unacceptable. Other linear and quasi-linear forms such as the Drucker–Prager criterion are also unacceptable for the same reason. It is probable that a general failure theory would involve some non-linear physical effects in some manner.

Going even further, it is still conceivable that there could be a reas- onably straightforward failure descriptor, but it probably must have a non-linear formalism to support it. And if the Mises criterion gives any indication, this general non-linear formalism could be quadratic in nature. This possible opening will be pursued in Chapter 4.

References

[3.1] Drucker, D. C. and Prager, W. (1952). “Soil Mechanics and Plastic Analysis or Limit Design,” Quart. of Applied Mathematics, 10, 157–65.

[3.2] Christensen, R. M. (2006). “A Comparative Evaluation of Three Isotropic, Two Property Failure Theories,” J. Appl. Mech., 73, 852–9.

[3.3] Mohr. O. (1900). “Welche Umstande Bedingen die Elastizitatsgrenze und den Bruch eins Materials,” Zeitschrift des Vereins Deutscher Ingenieure, 44, 1524–30.

[3.4] Schajer, G. S. (1998). “Mohr–Coulomb Failure Criterion Expressed in Terms of Stress Invariants,” J. Appl. Mech., 65, pp. 1066–8. [3.5] von Karman, T. V. (1912). “Festigkeitversuche unter allseitigem

Druck,” Mitteilungen Forschungsarbeit Gebiete Ingenieurs, 118, 27–68.

[3.6] B¨oker, R. (1915). “Die Mechanik der Bleibenden Formanderung in Kristallinish Aufgebauten Korpern,” Mitteilungen Forschungsarbeit auf dem Gebeite Ingenieurwesens, 24, 1–51.

The Failure Theory for Isotropic