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Failure criteria of the type considered here are not needed for all mater-ials. For example, elastomers and “soft” biological tissues probably have no need for explicit three-dimensional failure criteria. On the other hand, for engineering materials with essential stiffness characteristics, there are limits to this performance and failure criteria are of vital importance in applications. Mainly those materials that possess a range of linear elastic behavior (or nearly so) are those that require the characterization of the limits of that performance. These include metals, the glassy (but not rubbery) polymers, some wood products, ceramics, glasses, materi-als of construction, many geological materimateri-als, fiber composite materimateri-als, and many others as well. Most of these materials are isotropic or nearly isotropic.

The requisite characteristic is that of an effectively linear elastic beha-vior up to, but not beyond, some load level. Failure and linear elasticity share a special type of inverse and complementary relationship. The ideal linear elastic behavior is terminated by the failure, and all the non-linearity is subsumed into the failure criterion. Historically, linear elasticity theory was the progenitor of all classical field theories, but the related failure theory is required to complete the characterization.

A failure criterion takes the form of a surface in stress space. The stress space can either be the 3-space of the principal stresses, or the larger space of the full stress tensor, or smaller subspaces. The comparison with geometry is obvious. A geometric surface in 3-space is described by a scalar function of the coordinates f (x_i) = 1. In the stress context, the problem is to find the scalar equivalent for the failure criterion problem, F (σij) = 1, giving the failure surface in stress space.

So the problem distills to finding the proper form for the scalar function—here called the scalar potential of failure. The failure poten-tial is not to be confused with the plastic potenpoten-tial, which has a different meaning and use.

Theoretical Assessment 73 The failure criterion derivation of Section 4.4 will be briefly summar-ized here in a slightly different form, so that it can be compared with another failure criterion in the overall theoretical evaluation process. The form of the failure potential will be taken from the same mathematical representation as that used with the elastic energy potential, since it too is a scalar and in some broad sense related. However, significant and format-ive differences between the two will arise immediately thereafter. As with elastic energy, take the failure potential as a polynomial expansion in the invariants of the stress tensor, as appropriate to isotropy. This then is the method of polynomial invariants, giving

φ = a0+ a₁I₁+ a₂I₁²+ a₃J₂+− − − (6.1) I₁ is the first invariant of the stress tensor, and J₂ the second invariant of the deviatoric stress tensor, as

I₁ =σ1+σ2+σ3

J₂= 1 2

(σ1− σ2)²+ (σ2− σ3)²+ (σ3− σ1)² (6.2)

where principal stresses are used. See Chapter 4 for the more general case not involving principal stresses. The expansion is terminated at terms of second degree—the same as with the elastic energy representa-tion. Parameter a₀ merely establishes the datum and is not needed here.

Sometimes a coefficient different from 1/2 is used in the J₂ expression, which is acceptable only so long as it is used consistently.

In the case of the elastic energy potential the resultant form must be positive definite and therefore a₁ = 0, but for the failure potential there is no such requirement and so a₁ = 0. However, for a physical condition that the homogeneous and isotropic material not fail under hydrostatic compression, it is required that a₂ = 0, leaving the failure potential as

φ = a1I₁+ a₃J₂ (6.3) Then the failure criterion becomes

a₁I₁+ a₃J₂ ≤ 1 (6.4)

Finally, evaluating a₁ and a₃ in (6.4) in terms of the uniaxial tensile and compressive strengths, T and C , gives

1

As the overall failure criterion, (6.5) is necessary but not sufficient.

Sufficiency requires that a competitive fracture criterion in the more brittle range of behavior also be satisfied. From Section 4.5 this is given by

σ1 ≤ T

There is both geometric and physical significance to the commencement of the fracture criterion at T/C = 1/2. It is at this value of T/C that the three fracture planes (6.6) in principal stress space are just tangent to the paraboloid (6.5) of the failure potential. The “if” condition in (6.6) can be written either as the inequality shown or as an inequality with equality because of the tangency condition. The fracture criteria (6.6) then introduces the occurrence of “corners” into the yield/failure surface characterization for T/C < 1/2.

The forms (6.5) and (6.6) comprise the complete, two-property failure criterion for homogeneous and isotropic materials. In effect, the failure criterion and the elastic energy (from which the stress–strain relations follow) are joint constitutive relations. They are independent, but one without the other is incomplete. The elastic energy has two properties (E and ν) and one condition (positive definiteness), while the failure potential also has two properties (T and C ) and one condition (no failure under hydrostatic compression).

Placing the above results aside until experimental evaluation in the next section, a contemporary example of a postulated failure criterion will now be given as a case study in contrasts. A research group has

Theoretical Assessment 75 for several years advocated that the material failure criterion (for the isotropic matrix phase in composites) be written directly in uncoupled, two-parameter form as

I₁ ≤ α

J₂≤ β (6.7)

The first of these is said to associate with a fracture behavior, and the second with yielding and flow. Taking both α and β to be calibrated by both T and C then gives

I₁≤ T

J₂ ≤ C² (6.8)

It is not clear what the physical basis of (6.7) and (6.8) may be. The equations merely reflect that J₂ applies to ductile metals, and there are also (erroneous) claims that I₁ completely covers brittle materials, so the proponents take both of them as independently and simultaneously required for all glassy polymers (and by extension for all materials). This is typical of the many schemes that have been tried. Certainly all obvious ways of combining the invariants (including I₃ or J₃) have been proposed and in some cases claimed to be the long-sought form for failure criteria.

Most of these attempts do not receive a critical comparison with data on well-characterized homogeneous and isotropic materials. Rather, they usually involve comparisons with a window of data from non-uniform stress states in complicated forms or structures where many other factors are also in effect, but not acknowledged nor even recognized.

With regard to the possibility of forms (6.8) being a useful, well-posed failure criterion, consider the following. One of the basic tenets of mater-ials science is that superimposed pressure increases the strength of the material. This happens in all cases except the limiting case of a Mises material. The failure criteria derived here, (6.5) and (6.6), embody this effect in an interactive manner. In contrast, the forms (6.8) predict that pressure has no effect at all on the uniaxial compressive strength nor on the shear strength, while they provide an unusual prediction for the effect of pressure on the uniaxial tensile strength.

Even more objectionable is the prediction of (6.8) in the case of the ductile limit at T = C . For two-dimensional biaxial stresses, (6.8) gives only a portion of the usual Mises ellipse because the first of (6.8) truncates

it by a straight-line cutoff in the first quadrant. In the range of polymer behavior, (6.8) still exhibits the same characteristic. At T = C the derived failure criteria (6.5) and (6.6) directly recover the full Mises criterion.

All of these difficulties with forms (6.7) and (6.8) still occur when the invariants are interpreted in terms of strains rather than stresses. This is easily shown by substituting the isotropic stress–strain relations into (6.7), whichever way they are specified, to directly obtain the other form.

Thus (6.7), whether in their stress or strain forms, fail to pass stand-ard tests of physical consistency. All tentative failure criteria should be subjected to conceptual tests of the types just discussed. The cloud of ill-conceived failure criteria over the years has done much to obscure and impede progress.

There also have been a few very clever constructs, such as the Coulomb–Mohr form, but unfortunately they do not coordinate at all well with the physics of failure. This will be further shown through com-parisons with data in the next section, since the Coulomb–Mohr criterion is by far the most substantial and prominent of all historical forms.