Isotropic Materials Failure Behavior
Isotropic Materials Failure Behavior
Abstract and Keywords
The failure theory for isotropic materials was derived in the previous chapter, with the end result being the mathematical forms of the failure criteria. Now it is important to visualize the failure surfaces or envelopes in threedimensional principal stress space. A gallery of different failure envelopes is presented across the full range of T/C values, from the Mises cylinder at T/C = 1 to the brittle limit form at T/C = 0. The fracture cutoffs from the basic paraboloidal form commence at T/C = 1/2 with only an infinitesimal effect there, but becoming progressively larger as T/C diminishes, and becoming totally dominant at T/C = 0. The division of these failure envelopes into ductile and brittle regions is introduced as a preview to the full treatment in Chapter 8. Because the entire theory is calibrated by only the two properties, T and C, a related question concerns why these two properties are unique in this application. Why cannot any two strength properties suffice for formulating the basic theory? This question is taken up by considering the possible use of the shear strength and a dilatational strength. The particular example of a cracked solid as analyzed by Budiansky and O’Connell is used to reveal the necessity of using the T and C properties.
Keywords: principal stress, Mises, fracture, paraboloid, ductile, brittle, uniaxial strengths
The present chapter and the subsequent two chapters are immediate and necessary continuations of the preceding chapter on the derivation of failure theory and failure criteria for isotropic materials. No matter how rigorous a derivation may be, it is of little value if it is not readily assimilated and readily interpretable and if it does not offer direct facility in evaluation and in applications. Several different twodimensional and threedimensional stress states with associated failure envelopes will be presented and discussed here to broaden the understanding. Other conditions and decision junctures will also be introduced: for example, failure types, such as ductile or brittle failure.
The failure criteria for isotropic materials are given by eqs. (4.18) and (4.19) in Chapter 4. As specialized to principal stresses they are
and if
the following fracture criteria also apply:
The criterion (5.1) is referred to as the polynomialinvariants form, after its method of derivation.
(p.51) 5.1 Failure Behavior in Two Dimensions
As mentioned earlier, the fracture criteria (5.2) intersect the paraboloid (5.1) and produce three flattened surfaces on it. This is most easily illustrated in twodimensional stress space σ _{11} and σ _{22} with the other stresses vanishing, Fig. 5.1, which is for $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ = 1/3. The stresses are nondimensionalized by dividing by C.
Referring to the biaxial stress state,the $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ = 1 case would be the usual Mises ellipse. At $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ = 1/2 the fracture criterion (5.2) is just tangent to the shifted and slightly smaller ellipse (5.1) at the intercepts with the axes. The $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ = 1/3 case, Fig. 5.1, is typical of cast iron and some ceramics, showing a very pronounced fracture cutoff effect. The limiting case $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$→0 reveals that applied stress can be sustained only if the mean normal stress is negative and no component of normal stress is positive. The intuitive division into ductile and brittle regions shown in Fig. 5.1 will be found later to actually have a physical basis.
5.2 Failure in Principal Stress Space
The complete failure surface in threedimensional principal stress space is composed of two parts: the paraboloid (as determined by polynomial invariants), and a competing fracture mode of failure that produces planar “fracture cutoffs” from the paraboloid. Mainly the geometric aspects of the failure surfaces will be shown and described here. The physical and (p.52) mathematical bases of these are the subjects of nearly the entire book, and will not be immediately involved here.
The complete failure criteria are given by eqs. (5.1) and (5.2). These failure criteria are calibrated by two failure properties, the uniaxial tensile and compressive strengths T and C. The paraboloidal part of the overall failure criterion, from the method of polynomial invariants, has a symmetry axis that makes equal angles with the three principal stress axes. The fracture criterion planes are normal to the three principal stress axes.
In the following graphics, stress is nondimensionalized by the uniaxial compressive strength, C. The examples differ only through the variation of the tensile/compressive strength ratio, $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$. The different $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ values correspond to seven examples from widely different classes of materials: namely, ductile metals, ductile polymers, brittle polymers, cast irons, ceramics, glasses, and geological materials.
All of these cases are at the same scale, thus allowing direct comparisons between them. In this sequence of computer graphics the scale and the view angle are selected to best observe the brittle end of the sequence rather than the ductile end, since the latter is quite well understood. The ductile cases will be further illustrated later. A grouping of smallsized images is shown first in Fig. 5.2, followed by larger, individual images in Fig. 5.3a–5.3g. In some cases the stress axes are shown only in the negative coordinate directions.
The two shadings refer to the two different and competing failure criteria. They do not imply a ductileversusbrittle distinction, which will be brought in later.
The unusual scaling for the Mises criterion shown in Fig. 5.3a is necessary for the onetoone comparison capability with the other cases. When displayed by itself the Mises criterion usually is taken in a long, slender cylindrical form, as shown in Fig. 5.4.
The Mises criterion at T/C = 1 represents one of the limiting cases of failure behavior, the ductile limit. Its cylindrical form is the limiting case of the general paraboloid. The other limiting case of failure is that of the brittle limit, at T/C = 0. This case is similar to the geological materials graphic, being only slightly different when taken to the limit where the apex of the paraboloid coincides with the coordinate origin. Also at this (p.53) limit, each of the three fracture planes contains the coordinate origin along with two of the axes, thereby requiring that no tensile component of threedimensional stress exist anywhere in the domain. The brittle limit failure surface is shown in Fig. 5.5. This form has a triangular pyramid shape near the origin, then transitioning into the paraboloid with corners between the two, as shown.
The two limiting cases of failure are remarkably different—indeed, fundamentally different. The most distinctive features of the ductile limit failure criterion are its independence of hydrostatic stress and its symmetry of a type in tensile and compressive stresses. In contrast, the most distinctive features of the brittle limit failure criterion are a strong dependence upon hydrostatic stress and the disallowance of any tensile components of stress, giving it a special type of asymmetry.
(p.54) At a different scale and view angle, the ductile polymers case at T/C = 3/4 appears as in Fig. 5.6. While not having the capability of the ductile limit, the form in Fig. 5.6 still reveals the very considerable tensile stress capabilities for this class of materials. Rather than examine other specific cases, some general geometric characteristics will now be noted.
(p.55) In the examples and in general, the complete failure surface is axisymmetric for $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ ≥ 1/2, but for $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ 〈 1/2 it has threefold symmetry. It is at and below $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ = 1/2 that brittle behavior becomes much more prevalent with the onset of the fracture criterion. Even though the character of the failure surfaces changes at $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ = 1/2, there is no discontinuity of form for the failure surface at this value of $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$.
As $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ diminishes the apex of the paraboloid moves toward the coordinate origin. For $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ 〈 1/2 the intersection of the fracture planes with the paraboloidal surface generates elliptical forms. These elliptical (p.56) intersections become larger as $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ diminishes from the value of 1/2. Going in the other direction the three elliptical intersections reduce to points at $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ = 1/2, because the fracture planes become tangent to the paraboloid.
For a sufficiently small value of $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ the adjacent elliptical intersections become tangent with each other. For even smaller values of $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ the elliptical intersections become truncated and contiguous, as shown in Fig. 5.3g, the $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ = 1/12 graphic. The point of tangency occurs at
These examples show the intricate and nonintuitive failure surface changes that occur in going across the full spectrum of isotropic materials types. They also show how extremely limited the allowable tensile stress regions become relative to the allowable compressive stress regions as $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ decreases.
Although the fracture cutoffs are nominally brittle, it does not follow that the remaining paraboloidal part of the failure surface is entirely ductile. For any particular value of $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ there is a mean normal stress controlled plane which is normal to the symmetry axis of the paraboloid and which divides the domain into ductile and brittle parts. This ductile/brittle division plane cuts across the intersection ellipses, where they exist.
(p.57) Despite the exceptionally wide range of failure surface forms for isotropic materials, all are determined by and calibrated by only two failure properties: the uniaxial tensile strength and the uniaxial compressive strength.
5.3 DuctileversusBrittle Failure
The distinction between ductile failure and brittle failure is one of the cornerstones of materials science, in addition to being of utmost importance for materials applications. Ductile–brittle transitions as influenced by some control variable, such as temperature, have been very extensively examined. Basic theoretical studies of the transition effect have usually (but not always) been posed in the context of the activation and flow of atomic scale dislocations in facecentered cubic and bodycentered cubic metals, or in more idealized forms. Despite the large amount of research already carried out on the ductile–brittle transition, very little of it has been successfully reduced to standard practice and a uniformity of understanding for the full spectrum of materials types. The specification of ductileversusbrittle failure criteria for all homogeneous materials has continued to be indistinct and elusive.
Some perspective on the problem can be gained as follows. The same as with temperature, pressure can be used to control ductile–brittle effects. Sufficiently large superimposed pressure can convert what is normally thought of as a brittle material into a ductile material. Bridgeman was awarded the Nobel Prize at least partially for illuminating the problem nearly a hundred years ago. However, positive or negative pressures are but a part of the full stress tensor, and because of this they are more complex than scalar temperature variation. This complexity has been something of a barrier to the understanding of ductile–brittle failure effects.
Proceeding further with this ductile–brittle reconnaissance, all stress states can be decomposed into a dilatational part (as with hydrostatic stress) and a distortional part (as with shear stress). Consider the three following stress states: uniaxial tension, eqibiaxial tension, and eqitriaxial tension. The decomposition of these stress states into dilatational, and distortional parts are as shown in Table 5.1.
Table 5.1 Uniaxial tension, eqibiaxial tension, and eqitriaxial tension
Dilatational 
Distortional (shear) 
Total 


Uniaxial tension 

σ_{11} 
1/3 
2/3 
1 

σ_{22} 
1/3 
–1/3 
0 

σ_{33} 
1/3 
–1/3 
0 

Eqibiaxial tension 

σ_{11} 
2/3 
–2/3 
0 

σ_{22} 
2/3 
1/3 
1 

σ_{33} 
2/3 
1/3 
1 

Eqitriaxial tension 

σ _{11} = σ _{22} = σ _{33} = $\tilde{T}$, no shear 
Each of the two distortional (shear) stress states can be further decomposed into two simple shear stress states, as with
(p.58) For further investigation, consider only those materials that are normally thought of as being very ductile, such as aluminum, copper, and so on. Such isotropic materials have the uniaxial tensile and compressive yield stresses as being identical, T = C, to within experimental discrimination. Also, the same failure values occur for eqibiaxial tension, for these materials. Other materials types have $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C\u30081$, or much less in some cases. For convenience, refer to the distortional part of the stress tensor as shear. Designate by S a measure of the shear part of the stress tensor (the Mises stress for example), and by D designate a measure of the dilatational part, both at yield/failure.
Normally it is only the shear part that controls yield/failure in these T = C materials, but it is both S and D that relate to the ductileversusbrittle failure mode type. For present purposes, take a possible ductility index for these three cases as specified by S/D. In the uniaxial tension case, normalize this S/D ratio to the value of 1. It then follows that the S/D ratio for the eqibiaxial tension case will have the value of 1/2, because it has the same value for S as in uniaxial tension (the sign change shown in the table is irrelevant) but its value for D is twice as large as that in uniaxial tension. Finally, for the eqitriaxial tension case there is no shear, S = 0, so its ductility index is S/D = 0.
In this manner and for these three cases there is a ductility scale that goes from 0 to 1, with the value 0 being that of no ductility (major brittleness) and that of 1 being major ductility. Collecting these three cases for materials with T = C then gives (p.59)
The triaxial case falls into the brittle category, since there are no shear stresses to activate dislocation flow in these T = C materials. That an eqitriaxial tension test is exceedingly difficult to perform is not of concern here, as only the type of failure is being considered. Also, for this simple exercise, the fact that the idealized Mises criterion would say there is no failure under triaxial tension is of no relevance.
The present reasoning and discussion is for background purposes only, and does not represent a specific technical approach. Nevertheless, it does quite clearly indicate that there is a primary linkage between (i) ductileversusbrittle failure behavior and (ii) the type of stress state which is imposed. Any comprehensive failure theory must account for these effects. They can be just as important as the effects due to temperature variation. They comprise the litmus tests (criteria) indicating and differentiating ductile yield behavior from brittle failure behavior.
A general criterion for the ductile–brittle transition is developed in Chapter 8. The theory given there interestingly determines that the state of eqibiaxial tension for materials with T = C does in fact lie on the borderline transition between ductile yield and brittle failure. This same effect is again shown in Fig. 5.7, as a preview of Chapter 8. More importantly, the theory covers all isotropic materials types having T ≤ C, and all stress states.
The failure criteria (5.1) and (5.2) have been examined in considerable detail and developed into a criterion for ductileversusbrittle behavior. (p.60) Thus the modified paraboloid in stress space is subdivided into regions of brittle failure versus regions of ductile failure. The brittle regions are not just the fracture planes produced by (5.2), but also include portions of the paraboloid (5.1). For given values of T and C the specific stress state which is imposed determines whether the failure will be of ductile or of brittle nature. The resultant stress state division into ductile and brittle regions would not actually be expected to have a sharp dividing line between them, but rather to be of a transition zone nature in reality.
The ductile–brittle criterion is discussed and illustrated at length in Chapter 8. This ductile–brittle behavior can also be viewed for a particular stress state as showing the change as a function of the $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ variation. For example, for uniaxial tension it is found that
while for simple shear the result is
All stress states have $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ designated regions of ductile or brittle behavior. Some particular stress states are entirely of the ductile type for all values of $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ while some are entirely brittle.
The limiting case of the yield/failure paraboloid at $T\phantom{\rule{negativethinmathspace}{0ex}}/\phantom{\rule{negativethinmathspace}{0ex}}C$ = 1 is given by the Mises cylinder, Mises [5.1], Fig. 5.7. The ductile–brittle criterion divides the Mises cylinder into ductileversusbrittle failure regions as shown. Most of the common stress states are on the ductile side of the division, but if the mean normal stress part of the stress state is sufficiently great, brittle failure will occur. This might seem to be a surprising result, but on further consideration it is to be expected. First consider the effect of temperature. Temperature variation can determine whether a given material fails in a ductile or a brittle manner. So too can pressure control the ductileversusbrittle failure character. Sufficient pressure can convert a nominally brittle material into a (p.61) ductile material. Conversely, negative pressure can convert a nominally ductile material into a brittle material, as shown in Fig. 5.7.
The development and consequences of this comprehensive yield/failure theory for isotropic materials have been examined in many different ways, and will be taken up throughout this book. For applications where both yield strength and brittle failure are viewed together as generalized failure, the failure criteria (5.1) and (5.2) comprise the entire specification needed for analysis.
5.4 T and C versus S and D
If, as argued here, there exists a comprehensive theory of failure for isotropic materials, then the most favorable situation would be that of two calibrating failure properties in coordination with the two elastic properties of physical/mechanical behavior. The most common sets of elastic properties are either μ and k or E and ν. The present theory of isotropic material failure employs the uniaxial strengths T and C as associated with E and ν. But this suggests that one could as well take the shear strength, S, and the positive dilatational strength, D, as associated with μ and k. This alternative approach is a matter worth examining.
The fact that all stress states can be built up from the superpositions of states of distortion and dilatation could lead one to suspect that S and D may be more fundamental than T and C. Furthermore, this superposition capability suggests (but does not prove) that two failure properties should indeed suffice for describing all isotropic materials failures. Proof can follow only from experimental verification. Note that the symbols S and D used here are a little different from the like symbols used in Section 5.3, but they have the same general implications.
Next, the failure criterion used here and based upon T and C will be recalled. Then it will be expressed in terms of S and D to determine whether the formulation becomes more clarified, less clarified, or completely opaque compared with that using T and C.
The failure criterion (criteria) developed and used here are represented by (5.1) and (5.2). Take S as the failure stress in shear, and D as the failure stress in positive dilatation. Henceforth, D will be referred to as dilation failure. Then (5.1) becomes
The fracture criterion (5.2) becomes
which is expressed in terms of the largest principal stress.
The first observation to be made is that the fracture criterion (5.2), when expressed in terms of S and D, (5.5), is completely devoid of physical interpretation. If the comparison of using T and C versus S and D involved only the polynomialinvariants criterion there would be little to choose between them. But the real problem is with the fracture criterion. The fracture criterion is very important, and not just in applications. It admits the limiting case of perfectly brittle behavior, without which there is no closure to the theory, and no actual assurance of generality.
Consider the relationships of T and C with E and ν, and correspondingly that of S and D with μ and k. From an operational, linear elasticity point of view, knowing the properties μ and k is absolutely equivalent to knowing E and ν. But from a physical point of view and especially with implications for failure, the two sets of elastic properties may or may not be entirely equivalent.
The possibly deeper meaning of E and ν versus μ and k is that the scale of the loadbearing resistance or capability in the one case is concentrated into one property, E, rather than into two properties, μ and k. The Poisson’s ratio ν serves a completely different role, as will be explored in Chapter 14.
The limiting cases for the two elastic property forms are specified by
E and ν

(i) $\nu =1/2$

(ii) $\nu =1$
(p.63) no restriction on E (positive values)
μ and k

(i) $k/\mu \to \mathrm{\infty}\phantom{\rule{1em}{0ex}}\mathrm{o}\mathrm{r}\text{}\text{}\mu /k\to 0$

(ii) $k/\mu \to 0\text{}\phantom{\rule{2pt}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{o}\mathrm{r}\text{}\text{}\mu /k\to \mathrm{\infty}$
Some formulations with μ and k involve indeterminacies and discontinuities, whereas those with E and ν usually do not. A nontrivial example will be given that illustrates this.
An evaluation example is needed that not only involves stress and strain under elasticity conditions, but also the example should have implications or significance for failure behavior. Since isotropy is being considered here it must comply with that as well. The ideal example turns out to be that for the effective behavior of an elastic medium having a randomly oriented and distributed array of cracks.
In a major work, Budiansky and O’Connell [5.2] presented the solution for the effective properties of an isotropic, elastic material containing a damage pattern of randomly oriented cracks. This problem and solution is ideal for present purposes. It brings in elastic properties, damage, and failure significance. The only potential problem with using this work is that some have questioned the meaning of its prediction of material collapse at a value of ɛ = 9/16, where ɛ is the crackdensity variable of circular cracks.
This result for material collapse is sometimes criticized as being arbitrary, or arbitrarily small. So, before using the results of this work, its validity will be reinforced by showing the logic and consistency of its prediction of the loss of material integrity at ɛ = 9/16.
For circular cracks the crack density is defined as
where a is the radius of the randomly distributed cracks, with $\u3008{a}^{3}\u3009$ being an average, and n the number of cracks existing within the volume V. For extension to more general cracks of convex shape, the crack density is taken as
(p.64) where A is the crack area and ρ its perimeter.
To verify the prediction of material collapse at ɛ = 9/16, consider different possible crack patterns at material collapse. The faces of some of the platonic solids will be taken to represent the fully developed crack patterns at collapse. The only platonic solids or combination of platonic solids that pack perfectly in 3space are the cube and a 2:1 combination of tetrahedra and octahedra.
Dodecahedra do not pack in 3space, although they come fairly close to doing so. The dihedral angle for a dodecahedron is θ = 116.°57. If this angle were 120° there would be perfect packing. As an estimate for that obtained from the dodecahedron, take the number of sides of the single cell as twelve, recognizing that if there were perfect packing the appropriate number would be six, since each face would be shared with the neighboring cell. This then recognizes the imperfect packing, leaving a small amount of material as being left over to form a “skeleton” of remaining connected material at or near collapse. In solving for ɛ for the case of the dodecahedron the volume will be taken as that of the dodecahedron itself. Thus this estimate for ɛ can be viewed as an upper bound.
The respective values of ɛ for these three cellular cases of fully developed crack patterns are determined from (5.7) to be as shown in Fig. 5.8.
In terms of the crack pattern symmetries, the first two cases have cubic symmetry, while the third is isotropic. Based on these results it is completely reasonable and rational to expect that ɛ = 9/16 gives the minimum crack density value for material collapse due to an isotropic pattern of cracks, and correspondingly for a random distribution of cracks.
As a consequence of the above comparison, the results of Budiansky and O’Connell will be taken as the rigorous solution of the problem. These results will now be used for present purposes. As an aside, the (p.65) generalized selfconsistent method is the preferred effective properties approach for twophase media, while the selfconsistent method is the preferred approach for singlephase media containing cracks or grain boundaries. It is the latter method that was used by Budiansky and O’Connell.
The notation of Budiansky and O’Connell takes E_{c}, ν_{c}, μ_{c}, and k_{c} as the effective elastic properties of the medium containing the cracks. The same symbols without the subscripts are the properties of the uncracked material.
Of the four properties E_{c}, ν_{c}, μ_{c}, and k_{c} it is only ν_{c} as a function of ɛ that can be determined as an uncoupled equation apart from any other of the properties. Once ν_{c} is found, then the other three properties can be determined as functions of ɛ. In general, ν_{c} as a function of ɛ and ν is a cubic equation, but its analytical solution can be found in the special case of ν = 1/2. This solution for circular cracks is given by
Eq. (5.8) is indeterminate at ɛ = 0, but it can be easily evaluated and gives the proper result of ν_{c} = 1/2 at ɛ = 0. Knowing ν_{c}, the other properties can be found, such as for E_{c}, from
Similar forms are found for μ_{c} and k_{c}.
The general solutions for E_{c}, ν_{c}, μ_{c}, and k_{c} have the following characteristics. At $\nu $ = 0,
all are the same linear function of ɛ going from the value of 1 at ɛ = 0 to the value of 0 at ɛ = 9/16. They gradually deviate from linearity as ν is increased from zero, going to maximum deviations at ν = 1/2. Thus it suffices to examine the cases at ν = 1/2 to see the maximum deviations from linearity.
(p.66) The properties behaviors at ν = 1/2 are as shown in Fig. 5.9. The slight deviation of E_{c}/E from linearity will be specified below, and k_{c}/k is discontinuous with ɛ. The variation of ν_{c}/ν in a plot of the type of Fig. 5.9 is a little above that for E_{c}/E, necessarily being concave down with the same end points, and that for μ_{c}/μ is considerably above that for ν_{c}/ν, also concave down. The deviations from linearity in plots of the type in Fig. 5.9 can be found as follows.
The midpoints of the variations, in Fig. 5.9, are at ɛ = 9/32. At this midvalue of ɛ and for this worst case of ν = 1/2, the properties values are found to be
Thus the maximum deviations from linearity are found to be as given in Table 5.2.
Table 5.2 Property deviations from linearity
$\frac{{\mathrm{E}}_{c}}{E}$ 
1.4% deviation from linearity 
$\frac{{\nu}_{c}}{\nu}$ 
7.2% deviation from linearity 
$\frac{{\mu}_{c}}{\mu}$ 
20% deviation from linearity 
$\frac{{k}_{c}}{k}$ 
Discontinuous 
(p.67) The result that k_{c} be discontinuous when ν = 1/2 can be easily reasoned as being physically required. Recognizing the intimate relationship of the effective properties to fracture mechanics, it is reasonable to expect that the dilation failure property, D, would also suffer a discontinuous or nearly discontinuous behavior with variations of ɛ.
It is seen that E_{c} and ν_{c} have a special behavior not shared by μ_{c} and k_{c}. In other sections it has been discussed that failure criteria can be interpreted as the tensorial description of the termination of the elastic range of behavior. In this sense there is an association of elastic properties and failure properties. Plastic flow behavior does not contradict this; it just describes a more complex transition from elastic behavior to failure. In relation to this, the behaviors of E_{c} and ν_{c}, as found by Budiansky and O’Connell [5.2] over the full range of crack densities, show vastly more regularity and smoothness than do μ_{c} and k_{c}. Furthermore, the difference in behavior in this problem between E_{c} and ν_{c} is minor, while the disparity between μ_{c} and k_{c} behaviors is huge.
The special behavior of the Poisson’s ratio ν is quite interesting. Although T and C relate most directly to E itself, ν also may have some meaning for failure behavior. Most but not all engineering materials have Poisson’s ratios that fall within 1/5 ≤ ν ≤ 1/2. Nothing of this type can be said for μ and k except by inference from ν. Furthermore, Poisson’s ratios give some indication of a materials failure type. Within the range stated above, the lower Poisson’s ratios relate to brittle failure, while the midrange and some higher ones usually relate to ductile failure. A much deeper look at the significance of Poisson’s ratio will transpire in Section 14.4.
(p.68) In conclusion of these results and observations, the elastic properties E and ν and their failure conjugates T and C are much more likely to be the fundamental continuum mechanics pairings than are the elastic properties μ and k and their failure conjugates S and D. It may also be noted that as a practical expedient, the uniaxial strengths, T and C, are enormously more easy to experimentally determine with reliability than are the shear and dilation strengths, S and D. Agreeably then, the fundamental forms and the practical approach for failure properties determination are in harmony, not in conflict.
Problem Areas for Study

1. Do failure criteria have a place and a use for materials such as elastomers that have no effectively linear elastic range of behavior?

2. The existence of significant linear elastic ranges of behavior in engineering materials is an extremely fortunate but rather poorly understood phenomenon. For example, in ductile metals, dislocation emission occurs upon the first small increment of load application, but at some point the rate of emissions multiplies hugely and cascades into macroscopic yield/failure. Many other materials types show similarly sudden changes in behavior with load application. What are the underlying physical mechanisms that cause such abrupt changes in constitutive behavior? Can some or most cases be explained by localization, or is every different materials type explained by a different mechanism?

3. Typical failure criteria approaches focus upon a specific class of materials, such as ductile metals, brittle metals, polymers, and so on. They then fit parameters to data for specific stress states. Can such empirical approaches be taken to apply for other stress states if they do not apply across the various materials classes and thereby certify generality in that way?

4. There are many molecular level theories for amorphous or crosslinked, glassy polymers. Generally they involve many parameters to reflect a variety of features in molecular scale architecture and bonding. How can these be related usefully to a twoproperty macroscopic level failure criterion?

5. The twoproperty failure theory developed here for isotropy has its calibration from T and C, the two uniaxial strengths. Can a viable twoproperty theory be developed in terms of S and D, the shear strength (p.69) and the positive dilatational strength? It should be noted that the present T and C theory cannot be converted to an S and D form, because S and D forms cannot produce the fracture criterion present in the T and C theory. See Section 5.4. Does this preclude an S and D theory?
References
Bibliography references:
[5.1] Mohr, O. (1900). “Welche Unstande Bedingen die Elastizitasgrenze und den Bruch eines Materials,” Zeitschrift des Vereines Deutscher Ingenieure, 44, 1524–30.
[5.2] Budiansky, B. and O’Connell, R. J. (1976). “Elastic Moduli of a Cracked Solid,” Int. J. Solids Structures, 12, 81–97.