VII. THE DUCTILE/BRITTLE TRANSITION, GAGING DUCTILITY LEVELS

Transcription

1 VII. THE DUCTILE/BRITTLE TRANSITION, GAGING DUCTILITY LEVELS In setting out to theoretically characterize and dierentiate ductile behavior rom brittle behavior it is necessary to irst have a credible theory o ailure as the starting platorm. Historically there was one exceptional mathematical ailure criterion and a long list o ailures. Perhaps the ailures would be better described as mis-guided attempts. In any case, none o them were successul. Only the Mises criterion (and to a much lesser extent the Tresca criterion) or (but only or) ductile metals was an unqualiied and very important contribution. All other materials types such as polymers, ceramics, glasses, brittle metals, and minerals did not yield to successul mathematical ailure criteria description. The cases o non-metallic materials usually have been approached with empirical ailure criteria on a case by case basis. Such empirical orms do not have established limits o validity, and represent little more than curve its over narrow ranges o data. For a general approach to ailure criteria to be successul it must originate rom some uniying concept or some special type o physical identiication. This will be considered under the next heading, ollowed by other closely linked subsections on ductility and brittleness. I the vitally important areas o ductile and brittle behaviors are to be realistically treated they must be closely interwoven with a general treatment o ailure criteria that applies across the ductile and brittle materials classes. All o these matters will be taken up now. Organizing Principle In searching or an organizing principle or ailure criteria, there is an overwhelming example that at least supplies the motivation and inspiration or trying to do so. This is the Periodic Table o the Elements. Its organizing principle is that o the sequence o atomic numbers that then admits urther classiication when grouped in matrix orm with rows and columns.

2 Is there such a uniying and organizing basis or materials ailure? In pursuing this possibility, attention will be ocused upon isotropic materials. In scanning across the various materials types, there is an apparent, even obvious dierentiation, but still with a connecting thread between them. This dierence is that some types o materials have about the same strength in simple tension as in compression, while some others have ar less strength in tension than in compression. Still other groups all in between these two groups. Since the times o antiquity it was always well known and well utilized that gold ell into the irst group. It also was learned how to live with the restrictions imposed on the second group or all the common minerals used in ancient construction. However, in the era o ailure criteria activity, the realization o the tensile versus compressive strength dierences never progressed beyond that point. There only was an awareness o the negative consequences o the possible disparity, no recognition o a possible deeper meaning. The present program o ailure characterization is based upon the hypothesis that general isotropic ailure behavior is completely organized by and determined by the spectrum o T/C values, where T and C are the uniaxial tensile and compressive strengths. The spectrum o T/C values captures the entire spread rom brittle behavior to ductile behavior over the range 0! T C! 1 This brittle to ductile change with the variation o the T/C s is an integral and important part o the organizing structure. Inact, it provides the portal to all urther developments. The T/C ratio at a particular value will be reerred to as the materials type. Thus there is a continuous spectrum o materials types, not just a sequence o discrete groups. The traditional materials groups, polymers, ceramics, etc., have rather speciic bands o T/C s, although there can be and are overlaps. The individual strength properties T and C have very dierent roles and unctions to ulill in synthesizing the related ailure criterion. Both T and C are involved (but unspeciied) at the ductile limit o T/C=1. In contrast, only C is involved (but unspeciied) at the brittle limit, T/C=0, because T!0. Accordingly it is advantageous to nondimensionalize stress

3 with respect to C (using T would not be possible) and then have the ensuing ailure criterion be expressed entirely in terms o the T/C spectrum. Following this course take! ˆ ij =! ij C The organization o ailure theory based upon the T/C spectrum does not automatically determine the ailure criterion, but it does indicate that only two properties suice or the speciication. In Sections II and VI the consequent ailure criteria or isotropic materials are derived and given in orms that can be put into the ollowing equivalent orms involving only T/C. The overall ailure criterion or homogeneous and isotropic materials is actually composed o two separate criteria. The irst part covers the ull range o T/C's while the second part is racture related and covers the brittle part o the range. POLYNOMIAL INVARIANTS FAILURE CRITERION " 1! T % $ ' (( ˆ # C& 1 + ( ˆ 2 + ( ˆ 3 ) [(( ˆ 1! ( ˆ ) 2 + (( ˆ 2 2! ( ˆ ) 2 + (( ˆ 3 3! ( ˆ ) 2 ] ) T 1 C (1) and

4 COORDINATED FRACTURE CRITERION ˆ! 1 " T C ˆ! 2 " T C ˆ! 3 " T C # % % $ % % &% Also applicable i T C " 1 2 (2) where principal stresses are employed. The individual values o T and o C only tell the speciic strength properties or a particular material, but it is the T/C ratio that signiies the materials type that is so pivotal in ormulating and in interpreting this general theory o ailure. A little later the T/C spectrum will also be ound to be o irst importance in ductile/brittle matters. The two coordinated ailure criteria or isotropic materials, (1) and (2), now will be used to develop a ramework or characterizing and quantitatively dierentiating ductile behavior rom brittle behavior. Beore doing so, a inal observation on the historical search or ailure criteria may be in order. Criteria (1) and (2) reduce to the Mises criterion at T/C=1, and they also reduce to a physically meaningul brittle limit at T/C=0. Thus the Mises criterion is a unctional part o this general orm. The maximum normal stress criterion, originally proposed by Lamé and Rankine, has a supericial resemblance to the racture criterion (2). I it, (2), were to be considered as a ree standing ailure criterion, it would be necessary to omit (ignore) the condition or its applicability shown in (2). In some current as well as past sources o inormation it is stated that the Mises criterion covers ductile materials and the maximum normal stress criterion covers brittle materials. The irst hal o this assertion is partially correct but the second hal o it is completely allacious. For example, common iron is conventionally considered to be a brittle material. When one compares the maximum normal stress criterion with typical data or the ailure o iron, the lack o correlation is glaringly obvious. The comparison o the coordinated ailure criteria (1) and (2) with brittle material behavior is quite good, see Section VI.

5 It is indeed somewhat surprising that this association o T/C values with the controlling physical behavior was not probed or at least recognized a great many years ago. But that is just a curiosity, a ootnote to history. Building up a viable theory o behavior is another matter entirely and only the latter counts or anything, anytime. Characterizing Ductility The conventional methods or characterizing ductility and brittleness are as ollows. Brittleness is usually taken to be the sudden, unexpected interruption o the linear elastic stress strain loading orm. Not only is this unannounced ailure occurrence annoying, it oten is dangerous, sometimes extremely dangerous. Some warning o impending ailure would be tremendously helpul. This simple classiication o idealized brittle behavior will suice or initial purposes. Ductility is much more complex. The usual method o identiying ductility is with imposed uniaxial tension. I the material proceeds with a strain hardening state ater exceeding some yield stress, and i it is capable o experiencing large strains in that state then it is said to be ductile. This view is unduly restrictive and it certainly doesn t suice or describing ductility or three dimensional stress states. There is a need or a more inclusive deinition o ductility. For present purposes, relevant examples o brittle and ductile behaviors are given in Fig. 1. Fig. 1 Failure types

6 In these diagrams, " and # are taken to represent any stress strain loading state, one, two, or three dimensional. But the progressive loading is always taken to be o the proportional type. That Case (i) is brittle is obvious. So too is Case (iii) obviously ductile. However, Case (ii) is where the complexity arises. Case (ii) is crucial, and will irst be approached in the manner now shown. The distinguishing characteristic o ductility is here taken to be the capacity to absorb mechanical energy and convert it into irrecoverable heat energy, or involve some other irreversible damage process. Perhaps the best way to characterize ductility is that it involves mechanical energy dissipation, which is completely absent in the case o idealized brittle ailure. Obviously Case (ii) has a smaller degree o ductility than does Case (iii). How important is this? Any ductility is much more than no ductility. Even in this sense, a small degree o ductility is very helpul and it aords an important capacity or damage tolerance. In non-uniorm stress states, e. g. stress concentrations, some ductility allows material mobility to adjust to local conditions without causing macroscopic ailure. With the above considerations in mind, Case (ii) will be taken with the ollowing method o broadly (not precisely) distinguishing ductile rom brittle behavior. Fig. 2 shows Case (ii) with some designations.

7 Fig. 2 Failure designations The conditions o brittle versus ductile behavior are taken to satisy! " # E,! >> # E, Brittle Ductile These orms give a general and conventional threshold or ductile behavior. In the second o these, the strain at ailure may be as little as only 30 or 40 or 50% greater than the elastic projection, or there to be the inception o signiicant ductility. Otherwise, the mechanical ailure behavior is taken to be brittle as in Case (i) or close to it. So brittle behavior allows a moderate generalization beyond that shown in Case (i). Still, there is a grey area, an uncertain area, in between the behaviors o totally brittle and ideally ductile types. Also, this operational deinition o and distinction between ductile and brittle behaviors is stress state dependent, and it is taken to apply to all stress states.

8 Again, it should be emphasized that even a small degree o ductility can be extremely advantageous, especially as compared with idealized brittle behavior, Case (i). These designations will suice or having a physical understanding o the possible and likely dierences between ductile and brittle behaviors. The clear message is that ductile versus brittle ailure behavior is not just an on/o switch. There is a graded and graduated change rom one to the other. To say these are diicult matters would be an understatement. They pose long standing, classically diicult, unresolved issues. The undamental task is that o quantiying ductility levels. None o this so ar accomplishes that purpose. To make progress on theoretically gaging ductility levels requires a drastic change in direction and approach rom the vague and rather loose traditional discussions o ductility at the macroscopic level. Now a new direction will be sought. In so doing, it should be recognized that any real improvement in characterizing ductility (or lack o) or homogeneous materials must be applicable to all stress states, not just careully selected, particular ones. The Ductile/Brittle Transition, the Failure Number or Gaging Ductility Levels A given material is generally taken to be ductile or brittle based upon experience and exposure. In this heuristic but eminently practical approach, ceramics are said to be brittle, some metals are brittle and some are ductile, and polymers are all over the place. There is nothing inherently wrong with intuitive and practical approaches but it is likely that there could be a more organized approach to this problem. First there should be some measurable but nondimensional physical characteristic(s) that determines the materials type as it relates to its expected ailure type, ductile or brittle. Not surprisingly this missing identiier will here be taken to be the strength ratio T/C. But i one tries to use only the T/C tag to make general assertions about ductile versus brittle ailure, one must also ask i this could be true under all conditions. It is quickly seen that there must be qualiiers, especially that o the stress state under which the ailure characteristic is to be determined. Generally tensile type stress states are ar more likely to promote brittle behavior, or a given material, than are generally compressive type stress states. The objective here is to ind a rational

9 method or classiying and quantiying expected ailure behavior, ductile or brittle with appropriate gradations in between, based upon the materials type, speciied by T/C, and the speciic imposed state o stress at ailure. In the papers containing archive reerences given in Section II and discussed in Section VI, a criterion or ductile versus brittle ailure is given as I 1 < 3T! C, I 1 > 3T! C, Ductile Brittle (3) where I 1 =! 11 +! 22 +! 33 is the irst invariant in terms o the ailure stresses. These relations, or speciied values o T and C then designate which stress states are expected to represent brittle ailure and which ones have ductile ailure. The irst invariant is at its value on the ailure envelopes (1) or (2) and thus expressed in terms o T and C or the imposed stress state. here, Now express relations (3) in non-dimensional orm or urther use! ˆ ii < 3 T C "1, Ductile (4)! ˆ ii > 3 T C "1, Brittle where! ˆ ii is the nondimensionalized irst invariant o the ailure stress state rom (1) or (2). The relations in (4) actually only give the transition rom ductile to brittle conditions when the inequalities are replaced by an equality. Thus, or a particular stress state the value o T/C can be ound at which the stress state is at the borderline between ductile and brittle ailures but it tells nothing about the relative scales o ductility or brittleness in general ailure conditions. It will be helpul to recall here the outline o the derivation o (4). Take the two dimensional state o biaxial ailure stresses as shown in Fig. 3.

10 Fig. 3 Ductile/Brittle Transition, 2-D example Relations (1) and (2) produce the intersection o ailure modes shown in Fig. 3. The domain is divided into ductile and brittle regions at the intersection o the ailure modes, as shown. I the ductile brittle division were taken through the other intersection corners nearer the origin in Fig. 3 it would quickly be ound to lead to unacceptable results. This process is repeated, taking T/C as a continuous variable to ind the unctional relationship between T/C and the division into ductile and brittle regions. The unctional relationship is then taken to continue into the range where T/C$1/2. The end result is the ductile/brittle criteria, (4). The act that this ductile/brittle division is at the intersection o ailure modes (1) and (2) requires that this be done in two dimensional stress space as a subspace o three dimensional stress space. It cannot be done

11 unambiguously in one dimensional or directly in three dimensional spaces. Deriving the ductile/brittle criteria in this way in two dimensional space leaves the one dimensional and three dimensional behaviors to be critically examined or veriication o predicted behavior using test results. This has been done in Section VI, although much more testing could be done along these lines. As a simple example o relations (4) consider the case o the ailure stress in simple shear. Take S as this ailure stress. Using relations (1) and (2) and (4), along with S/T=(S/C)(C/T) gives the result in Fig. 4. Fig. 4 Shear stress ailure types The change in ailure mode type as well as the type o ailure (ductile or brittle) occurs at T/C=1/3 or a state o simple shear. It is important to visualize the ductile/brittle criteria (4) in three dimensional, principal stress space. The ailure criterion (1) takes the orm o a paraboloid in principal stress space. The racture criterion, or T/C%1/2, cuts o slices rom the paraboloid, leaving three lattened suraces on it. The ductile/brittle criteria (4) at the transition represents a plane which is

12 normal to the axis o symmetry o the paraboloid. The ductile/brittle transition plane rom (4) is given by! ˆ ii = 3 T C "1, D/ B Transition (5) This plane divides the ailure surace into ductile and brittle regions as shown in Fig. 5 or the example T/C=0, the brittle limit. Fig. 5 Ductile/Brittle Transition, 3-D example

13 Even in this limit case, T/C=0, there still are separate ductile and brittle regions. This will be urther discussed a little later. At any T/C value, the ductile/brittle plane cuts across the three elliptical lattened suraces (when they exist, T/C % 1/2) at their 1/4 points nearest the apex o the paraboloid. It would not be expected that the ductile/brittle transition at any T/C value is o a sharp nature. This is not a irst order thermodynamic transition. The signiicance o this division boundary is that the proclivities toward ductile versus brittle ailure are in balance, while on either side o the division the weighting gradually shits to a dominate eect, one way or the other. Planes o equal but unspeciied ductility would be parallel to the ductile/brittle transition plane deined by (5). It would be advantageous to have the means to characterize the degree o ductility (or lack o it) or particular ailure states on either side o the D/B transition plane. To this end, rewrite (5) in an alternative orm. Take & to be speciied by! = 3 T C " ˆ # ii "1 (6) When &=0 (6) just reverts back to the criterion (5) or the transition. But when &'0, (6) has an interpretation as a measure o the degree o ductility or brittleness, as will be shown. This is not surprising since it is already known that &=0 is one point on the ductility/brittleness scale. Positive values o & relect ductility, while negative values indicate brittleness. For a ixed value o T/C, & represents a measure o distance rom the ductile/brittle transition plane to the parallel plane containing the stress state o interest. Examine the case o uniaxial tension. For this stress state ˆ! ii =T/C and so (6) gives! = 2 T C "1 For the range on T/C as 0 % T/C % 1 then!1 " # " 1 Also, it should be noted that or uniaxial tension the ductile/brittle transition is at T/C=1/2. The stress state o uniaxial tension is the only one that has

14 these matching symmetry orms on T/C and on &. Uniaxial tension will be taken as the baseline stress state, and all other stress states will be calibrated relative to it. It comprises the basic template or characterizing ductility. The value &=1 corresponds to ull, perect ductility, while &= -1 corresponds to total brittleness Now consider the state o simple shear stress. In this case ˆ! ii =0 and (6) becomes! = 3 T C "1 For uniaxial tension the ductile /brittle transition value o T/C=1/2 gives &=0. But or simple shear the same T/C=1/2 value gives &=1/2. This means that at this value o T/C, simple shear ailure is more ductile than is uniaxial tensile ailure. And at T/C=2/3, then or simple shear &=1, the ully ductile situation. Thus or the case o simple shear, values o T/C larger that 2/3 simply remain as ully, perectly ductile at &=1. Any stress state that at a particular value o T/C gives a value o & larger than 1 will be assigned as ully ductile, at &=1. Correspondingly any stress state with a value o & less than -1 will be taken as totally brittle at &= -1. In eect, the planes o perect ductility and total brittleness are located at ixed distances rom the D/B plane or all T/C s. It is convenient to rescale the quantity &. For -1 % & % 1, change the variable so that the new scale goes rom 0 to 1. To this end, take Fn = 1 ( 2! +1 ) or using (6) Fn = 1 # % 2 3 T C! ˆ $ " ii & ( (7) ' Consistent with taking & at limits o -1 and 1, as totally brittle or perectly ductile respectively, the new quantity Fn will revert to the limits o 0 or 1 when its value rom (7) is outside these limits. So always 0! Fn! 1

15 Calibrating to the uniaxial tension stress state is the only one that then gives Fn=1/2 at the T/C values or the ductile/brittle transitions o all the stress states. A slightly dierent orm o (7) is that o Fn = 3 # T % 2 C! ˆ $ " m & ( (7a) ' where! ˆ m is the nondimensionalized mean normal stress at ailure. The quantity Fn will be called the ailure number. It represents a measure o ductility, with Fn=0 being no ductility, total brittleness, and Fn=1 being ull, perect ductility. Rather than expressing Fn within the 0 to 1 interval, it will always be stated as the corresponding percentage, rom 0 to 100%. For Fn=50% the material is at the ductile/brittle transition. This is the case as regards ductility versus brittleness. For Fn below 50% brittleness begins to dominate, while or Fn above 50% ductility begins to dominate. The procedure or using the ailure number is as ollows. The sum o the three normal stresses at ailure,! ˆ ii, are ound rom the ailure theory (1) and (2) or they may be taken directly rom experimental data. These are then put into the ailure number ormula, (7). The two simplest examples are uniaxial tension and compression. The ormer has! ˆ ii =T/C and the latter has! ˆ ii =-1. Putting these into (7) gives the corresponding ailure numbers as unctions o T/C. These can then be evaluated at speciic values o T/C to ascertain the related ductility level. The table below shows the values o T/C or the ductile/brittle transition and the ailure number ormulas or some basic stress states.

16 Stress State D/B Transition T/C Failure Number Fn Uniaxial Tension 1/2 T C Uniaxial Comp. 0 1! # 2 3 T " C +1 $ & % Simple Shear 1/3 3 2! T $ # & " C % Eqi-Biaxial Ten T 2C! 1! T C + " T % $ ' # C & 2 Eqi-Biaxial Comp. Always Ductile 1 + T 2C + 1! T C + " T % $ ' # C & 2 Eqi-Triaxial Ten. Always Brittle T ( C 1! 3 " T % + * $ ' ) 2 # C & -, 1! T C Table 1 Failure Numbers, Fn There are no ailures in eqi-triaxial compression in this ailure theory. Next the ailure numbers or various stress states are arranged in descending order, thus going rom the most ductile stress state to the least ductile, most brittle stress state. Cases over the ull range o T/C s are given.

17 Table 2 gives a comprehensive view o the ductility levels o the ailure mechanisms as a unction o the stress state and the materials type. It is the state o uniaxial tension, not shear, that anchors the results. The state o uniaxial tension is o undamental and singular importance in calibrating the ductile/brittle behavior or all isotropic materials.

18 It is seen that insoar as ductile versus brittle ailure behavior is concerned the sensitivity to the stress state type is just as great as the sensitivity to the materials type. It must be recognized that there are very large dierences in ailure behavior or materials below the 50% ductility level rom those above it. For example, or the uniaxial tensile stress state, the ductility level o 25% would be in the range o T/C s or ceramics, while the 75% level would be representative o T/C s or very ductile polymers. The large majority o cases in Table 2 are at or below the ductility level o 50% indicating problems with expected brittle behavior. It also is apparent in Table 2 that cases below and to the let are dominated by brittle behavior, while those above and to the right are ductile. It is thus possible to identiy ductile versus brittle ailure mode groupings as a unction o materials type and stress state. The key to this division are the cases at the 50% level, the ductile/brittle transition. All o the 50% values are exactly at Fn=1/2. In the our cases in Table 2 at T/C=0 with no ductility, total brittleness, it should be remembered that the corresponding stress levels allowed by the ailure criteria are at zero, so the ductility levels are irrelevant. However this does show that exceedingly small T/C s would have virtually total brittleness. The one case at T/C=0 with a speciied ductility level relects the act that even at T/C=0 a compressive mean normal stress component does stabilize the material and give an allowable ailure stress level with a corresponding ductility level. It is generally true that or a given stress state, the ductility increases with increasing values o T/C. However, such simple rules become invalid when comparing dierent stress states. For example, rom Table 2 it is seen that a T/C=1/3 material in simple shear is more ductile than a T/C=2/3 material in eqi-biaxial tension. Another set o examples involves more complicated stress states than those given in the above tables. Consider two possible stress states where the principal stresses at ailure are in the proportions and 1 : 2 3 :! : 2 3 : 1 3

19 It is o interest to know which one is the more ductile o the two stress states, and by how much. To proceed urther it is necessary to speciy the material o interest. Take the material as a typical, airly ductile epoxy resin with T C = 3 5 = 0.6 As the irst step, note rom (7) that the ailure number or this epoxy in uniaxial tension is Fn = 60%, Uniaxial tension, epoxy It will be o relevance to not only compare the ailure numbers or the two stress states with each other, but also with that or the material in uniaxial tension, which comprises the base line. The answers to these questions o relative ductility levels are certainly not obvious. All three stress states will be shown in the comparison. Using the Fn ormula (7) and the ailure criteria (1) and (2) it is ound that Case! 1 :! 2 :! 3! ˆ 1,! ˆ 2,! ˆ 3 Fn Uniaxial Ten. 1 : 0 : 0 3/5, 0, 0 60% I 1 : 2/3 : -2/3 3/7, 2/7, -2/7 69% II 1 : 2/3 : 1/3 3/5, 2/5, 1/5 30% Table 3 Failure numbers or 3-D stress states at T/C=0.6 It is seen rom this table that Case I is on the ductile side o the scale and Case II is on the brittle side. Judging rom the values o the Fn s the dierences in the degrees o ductility are huge. Furthermore, Case I, the ductile case, is even more ductile than the epoxy its sel is in uniaxial tension. All o these results would be purely guesswork without the rationale o the ailure number, Fn.

20 The previous examples use a rather ductile epoxy as the material type. Now an example will be given in the brittle range. Take a glass type material with T C = 1 9 Consider states o simple shear at ailure as superimposed upon states o hydrostatic pressure. Again using (1), (2) and (7) the nondimensional shear stress versus the nondimensional pressure is ound to be as in Fig. 6. Fig. 6 Shear stress superimposed upon pressure The ailure number values are shown at a sequence o pressures. With no pressure the glass material is very brittle in shear. As the pressure increases, the material transorms into an essentially ductile material, becoming very ductile at the larger pressures. Without a reliable theory o ductile and brittle ailure it would be very diicult to predict how much pressure would be required to give a speciied degree o ductility improvement or a

21 particular material. It also ollows that a state o superimposed hydrostatic tension can convert a ductile material into an essentially brittle one. It is also interesting to note that the change o ailure mode and the ductile/brittle transition are not coincident, Fig. 6. Although such coincidence does occur in two dimensional stress states, as explained earlier it does not occur with three dimensional stress states. Thus there is a large region o ductile racture evident in Fig. 6. The ailure number, as speciied by (7) and by ailure criteria (1) and (2) does give a scale or ranking or the degree o ductility o dierent stress states or any particular isotropic material speciied by its T/C value. This is somewhat like the role and unction o the Reynolds number or luids. Both the Reynolds number, Re, and the ailure number, Fn, provide gages upon the expected behavior due to control/domination by an inherent nonlinear physical eect, inertial in the ormer case and ailure in the latter. The dierence is that the Reynolds number is or broad classes o low types while the ailure number is explicit and speciic to any particular ailure mode. It would not be expected that the ailure number would have the same widespread appeal as the Reynolds number does, it is a little too complicated or that. But Fn does reveal that there is an underlying logical basis or expecting and predicting degrees o ductility in ailure as a unction o the materials type and the imposed state o stress. No less than that should be expected rom any legitimate and comprehensive ailure criterion. Furthermore, the ailure criteria (1) and (2) and the ailure number (ductility level), (7), are very amenable to use with inite element codes in design. Although the main thrust o this website was outlined in the introductory section, Section I, it is worth briely repeating it here. This book explores, delineates, and codiies the main aspects o three dimensional ailure criteria or materials. It does not take the place o peer reviewed archive papers, rather it builds upon such contributions. As such, this work involves interpretations o the subtleties o materials ailure behavior. Nowhere are the diiculties more complex or more subtle than with characterizing ductility and its deleterious complement, brittleness.

22 Richard M. Christensen June 27th 2010 Copyright 2010 Richard M. Christensen