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Scale effects in engineering failures
Engineering FailureAnalysis, Vol 1, No. 3 pp. 201-214, 1994 Copyright~) 1994ElsevierScienceLtd Printedin Great Britain.All rightsreserved 1350-6307/94$7.00 + 0.00
Pergamon
1350-6307(94)00015-8
SCALE EFFECTS
IN ENGINEERING
FAILURES
A. G. ATKINS Department of Engineering, University of Reading, PO Box 225, Whiteknights, Reading RG6 2AY, U.K.
(Received 11 April 1994) Abstract -- A brief review is given of what may be meant by a "catastrophic engineering failure", particularly as it relates to situations involving fracture. Whether "fast fracture" is necessarily the same as "unstable" fracture--and whether it makes any difference anyway--is discussed. Since recoverable elastic energy, and irreversible remote plasticity, are volume-dependent quantities, but fracture work depends only on area, cube/square energy scaling principles are inherent in all mechanics of fracture. These different dependencies translate into the well-known experience that components and structures, made of materials which are appreciably ductile in laboratory-size testpieces, behave in a progressively less ductile fashion the larger they get. Eventually, above some critical size, their behaviour is globally elastic, and fractures are brittle. Conversely, when deformation zones are kept very small in normallybrittle materials, plastic deformation is possible, as shown by limiting sizes in the communution of powders and by the ability to micro-machine glass. Analytical studies and experimental investigations of scaled cracked bodies, covering the whole range from globally elastic in the large, through to quasi-rigid plastic in the small are described and assessed, including consideration of dynamic progressive fracture in frameworks.
1. INTRODUCTION The image of a catastrophic engineering failure is of something (usually large) that fails unexpectedly and dramatically. Examples would be the Ronan Point tower block failing by progressive collapse [1]; metal shell structures failing by buckling, e.g. the Bucharest Exhibition Centre under snow loading [2]; large ships and pressure vessels failing by brittle fracture [3] and so on. Equal damage to buildings, structures and vehicles may be produced in collisions, but that is not unexpected in that there is an obvious cause of the damage. Catastrophic failures on the other hand seem spontaneous and often happen for no apparent reason at the time. Only after an accident investigation is the cause usually, but not always, identified. As with all fabricated articles, if something goes wrong in use, the fault may be with the original design, with the materials used or with the manufacturing: the designer may have got the sums wrong; he may have specified a material having inappropriate properties; even though an appropriate material has been specified, an improper one has actually been used through a mistake or because the manufacturer thought it wouldn't make any difference to use a substitute; manufacturing standards or inspection procedures are not good enough; the component or structure may have been inappropriately modified after installation; and so on. Engineering design theories have developed often as a result of learning from catastrophic failures, e,g. wind-induced loadings on the Tacoma Narrows bridge. Again, the discipline of "fracture mechanics" arose from trying to explain the premature brittle failure of large steel structures, particularly the all-welded ships of the Second World War. (In fact, it may be said that an explanation was already available in the seminal Griffith theory, but it was not understood at the time that Griffith could be applied to any displacement-reversible fractured body, even if made of "ductile" steel and not merely of something obviously brittle such as the glass studied by Griffith.) In this paper we concentrate on failure by fracture, with or without preceding 201
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A . G . ATKINS
and accompanying plastic flow. J. E. Gordon has said that "The worst sin in an engineering material is not lack of strength or lack of stiffness, desirable as these properties are, but lack of toughness, i.e. lack of resistance to the propagation of cracks". He also said that "the history of attempts to prevent cracks spreading, or to evade the consequences, is almost the history of engineering" [4]. It is common experience that different materials fracture in different ways. Brittle fractures are globally elastic and the broken bits (if you can find all of them!) may be fitted back together to reconstruct the original article. Hence the success of archaeologists with broken pottery and our ability to mend furniture. In contrast, ductile fractures involve a great deal of irreversible plasticity both preceding and accompanying the fracture. To refit bits back together would involve much unbending and unbuckling of regions far removed from the fracture path (cf. metal food containers with curled-up lids after opening). There is, of course, a complete spectrum of elastoplastic fracture behaviour from that involving a small amount of remote plasticity (for which modified elastic fracture mechanics analyses may apply) to that involving extensive remote plasticity with hardly any elastic springback, for which the methods of rigid-plastic fracture mechanics apply. It is a pity that we traditionally classify materials as being ductile or brittle based upon tests in the laboratory, for there is a scale effect in the mechanics of fracture which produces unexpected transitions in behaviour. That is, components and structures made from brittle materials can behave in a ductile fashion, and those made from ductile materials can behave in a brittle fashion. It depends on size and comes about because recoverable elastic energy and irreversible remote plasticity are volume-dependent quantities, but fracture work depends only on area. Hence cube/square energy scaling principles are inherent in all mechanics of fracture. It is the transition from ductile behaviour in the small to brittle behaviour in the large which has figured prominently in catastrophic engineering failures. Crack velocities in large components and structures turn out to be much greater than in small laboratory-size models and this has compounded the unexpectedness of sudden violent failures. This can then involve consideration of "dynamic fracture mechanics" which either means the initiation of a stationary crack under rapidly-applied loading, or rapid propagation and possible arrest of a crack under slowly varying load. In turn stress wave effects in the cracked body (reflexion from boundaries etc.) may become important. For a review of these sorts of things, see Chapters 8-10 of reference [5]. Even in the small, and certainly in the large, it has been established that there is the danger in loaded frameworks of domino-like progressive dynamic fracture, for which quasi-static calculations on defective frameworks are inappropriate. Examples are given later.
2. ELASTIC F R A C T U R E Figure 1 shows a variety of load (X), displacement (u) responses in globally-elastic fracture. Beyond the point P at which cracks initiate and start to propagate, the load may decrease, remain constant or increase monotonically with increasing load-point displacement; or drop suddenly to complete fracture; or fluctuate with arrested load drops followed by increases in load up to other bursts of cracking. The general incremental work equation for elastic fracture mechanics is X d u = dA + R d A
(1)
where A is elastic strain energy, R is toughness and A is crack area. For linear elastic fracture, dA = d ( X u / 2 ) and manipulation gives X 2 = 2 R { ( d / d A ) ( u / X ) } -1
(2)
which is the well-known "compliance calibration" equation usually written with Gc (the strain energy released from the cracked body) in place of R (the specific work
Scale effects in engineering failures P
203
P
0
,-I
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Id
(b)
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Displacement (d)
(c) P
U
p~
O
..1
Displacement
u
(e) Flo. 1. Typical load--displacement diagrams for: (a) unstable cracking, (b) stable cracking with decreasing load, (c) stable cracking with constant load, (d) stable cracking with increasing load, and (e) "stick-slip" cracking. Reprinted by permission of the Council of the Institution of Mechanical Engineers from J. Strain Analysis.
required for fracture). Stress intensity factor equations have the same form as Eqn (2), with K 2 = ER and the (d/dA)(u/X) term implicit. For a given crack/body geometry, loci of constant A and R obeying Eqn (1) may be superimposed on load-displacement Cartesian co-ordinates. The most general type of plot for bodies with linear stiffness is shown in Fig. 2 (which relates to a cracked ring loaded in compression), segments of which for other testpieces may be identified in Fig. 1. Insofar as catastrophic fractures may be "unstable" fractures, we remember that energetic stability depends upon external constraints. In particular we identify displacement control (du/u > 0) and load control (dX/X > 0). These conditions concern the vertical and horizontal tangents to the constant toughness loci shown in Fig. 2. Load control is impossible unless crack propagation occurs under increasing load; displacement control is impossible unless cracking occurs under increasing displacement. The stiffness lines passing through the two tangents to the R locus determine the minimum sizes of crack for which stable crack growth may be achieved according to (du/u > 0) or (dX/X > 0). It follows that the conditions shown in Fig. l(a) correspond to instability under both load and displacement control; those in Fig. l(b), (c) and (d) are all stable under displacement-control and Fig. l(d) depicts stability under load-control. In Fig. l(a) there is so much energy available at first cracking that energy could be removed from the system and still leave enough to feed the crack growth. Unless the displacement is reduced in some way at the instant of cracking, instability will result, the system going "bang" with, perhaps, fragments being thrown off. It is possible to
A. G. ATKINS
204
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/
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Displacement: l~m Fxo. 2. Load--deflexion diagram for cracking of a ring under diametrical compression.
relate the upper bound triangular work area of Fig. l(a) to the fracture toughness, and Turner's r/-factor (well-known in elastoplastic fracture) was first introduced for unstable elastic fracture [6]. Of course, the material may have a rate-, temperature-, or environment-dependent toughness which leads to the stick-slip behaviour shown in Fig. l(e). Furthermore, the crack may not be sharp, which affects stability. The stiffness of the loading system and the energy it stores will play its part in achieving stability or not. The following analogy of an ice-skater demonstrates the influence of external constraints on stability [5]. Consider: (a) a skater who falls through ice that has cracked because of his weight; (b) a skater who happens to be under an overhanging tree, and who can hang on to a bough to prevent himself dropping through the ice, even though the ice sheet may be breaking; (c) the skater who not only can grab on to a tree as soon as he sees cracks appear but can also haul himself up off the ice sheet. In all cases the strain energy in the ice sheet that helps to feed the cracks is caused by the deflection of the ice under the skater's weight. In the first case, the cracking is catastrophic and unstable because the descent of the skater's body provides more energy to the ice sheet than is required for balanced stable cracking. In the second case, the skater's body is prevented from dropping, and if he grabs on to the tree as soon as he sees cracks initiating he may prevent further cracking if his body is locked at a fixed height and can supply no further energy to the system. In the third case, if he quickly lifts himself off the ice sheet upon sight of the first cracks, he may prevent catastrophic cracking altogether in that he removes energy from the ice sheet by changing the way his weight is supported (i.e. by his feet and hence the ice, or by his arms and hence the tree). All testing machines themselves deflect under load (so, in practice, would the bough of the tree). "Stiff" machines store less energy than "soft"
Scale effects in engineering failures
205
machines under the same load. The behaviour of soft machines corresponds with (i) above; hard with (ii) above. However most loading systems are unlike the skater in (iii) who removes energy from the loading system, so that catastrophic fracture can result in even the stiffest loading devices. In the field, loading of structures, vehicles and so on is far softer. A homely example of stability in fracture is in the design of nutcrackers, the common "squeezed lever" sort of which often leads to crushed kernels owing to an inability to control the displacement once cracking starts. Gurney and Mai [7] provide a comprehensive analysis of energetic stability in elastic fracture and show that there are always geometric factors as well as material-dependent factors to consider in stability. As regards crack velocities, it follows from Eqn (1) that
A/a = X/(2R + u dX/dA).
(3)
LEFM K-formulae enable u dX/dA to be determined for any cracked body (see Chap. 3 of [5]) from which the ratio of crack velocity ~ to load-point velocity ti is given. It is not uncommon to find that stable cracks propagate at one or two orders of magnitude greater than the load-point velocity. Even so, cracking is "stable"--it just happens to be fast, but yet not so fast that a kinetic energy term associated with ~ has to be included in Eqn (1). Upper limiting ranges for ti for quasi-static fracture are some 6 ms -1 in metals and 2 ms -1 in polymers [8]. Hence "fast fractures" are not always "unstable" fractures, but the distinction in a practical case may be unimportant unless there is the possibility of "crack arrest" (where an unstable crack "runs out of steam" and stops within the body). The sharpness of cracks is important here. It is often forgotten that LEFM formulae presume fully-constrained sharp cracks, which is why one gets away with a single parameter (Kc, R, Go) to characterise elastic fracture: in ductile fracture, on the other hand, the specific work or fracture (R, J~) or COD (6c) is hydrostatic-stress-state-dependent, because the cracks are blunt. If cracks are initially blunt in elastic fracture cases, the fracture load goes to higher levels and there is excess energy at initiation. In what would be otherwise stable propagation cases, instability can sometimes result. There may, or may not, be crack arrest of the "sharpened" crack during propagation. The case of truly unstable fast fracture is shown in Fig. 3 where the variation of ~/a is shown with crack length for the cracked ring geometry. Negative values of ~/ti are predicted for the unstable part of the locus, because in that region ~ is negative to satisfy Eqn (1); and in Eqn (3) dX/dA is so big negatively that the denominator becomes negative. At the vertical tangent stability point for du/u > 0, the denominator of Eqn (3) is zero, and therefore (~/a) ~ o¢. The negative velocities predicted for cracks shorter than the critical length will never be seen in normal circumstances and the fracture is catastrophic. Initially-blunt cracks make the situation worse. The cracked ring specimen provides a valuable illustration of the relationship between so-called R-curves and stability. A material with a constant toughness R shows a rapid change from zero to R as soon as propagation commences. Some materials display a transition region of increasing R with crack growth up to a higher constant level during globally elastic fracture, and this can promote stability in otherwise unstable situations. However, instability may eventually occur after only a short period of crack growth. Instead of the loci of constant A and R for stable cracking on Xu Cartesian co-ordinates in Fig. 2, we show loci of constant X and u on GA or RA co-ordinates in Fig. 4. Remember G relates to energy "availability", and R to energy "requirement". These loci have maxima (to understand this consider changes in R loci cut by vertical and horizontal lines in Fig. 2). The mexima occur at the starting crack lengths corresponding with the vertical and horizontal tangents in Fig. 2, either side of which instability or stability occurs. The material RA curve is superimposed for two different starter crack lengths on either side of the maxima. The relative magnitudes of the slopes dR/dA (indicating energy required) and 3G/OA)u o r X (indicating energy available), determine how much of the R curve may be picked up before a catastrophic fracture stops measurements. Clearly, to obtain
206
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-
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S fable Region for ~X>O
I I
600-
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lengfh (ram)
800-
6'0
eb
Sfable Region for A u > 0
&1200 1600-
Fro. 3. Plots of (t~/fi) vs. a for the cracked ring in Fig. 2.
~
d
Ei
a (ram) FIG. 4. G vs. a c u r v e s f o r constant load and constant displacement f o r a s t e e l r i n g fracturing under diametrical compression. R vs. a curve is also s h o w n .
experimentally the complete R curve requires testpieces with different minimum starter crack lengths in order to achieve complete stability. This fact is not always realised by experimenters who quote their last R-value before instability as the critical toughness, whereas their result is really testpiece-dependent.
Scale effectsin engineeringfailures
207
Further consideration of crack stability (including stability in elastoplastic fracture), and experimental tricks to promote crack stability in testing, may be found in Chap. 8
of [51.
3. SCALE EFFECTS When load-deflexion tests are performed on different size bodies of nominally crack-free material it is possible to normalise the different load-extension curves into one stress-strain curve, from which elastic moduli and plastic flow properties are traditionally determined. Variations sometimes do occur depending on the size of the body (e.g. segregation in castings, strengthening in fine wires), but it is customary to consider that elastic moduli and yield stresses are independent of the size of the material, and that it is translated into traditional "strength of materials" design practice. Load-extension curves given by globally elastic cracked bodies are used directly to determine Jc, Gc or Kc, consistent values of which will be obtained from the appropriate formula (to correspond with the specific work R absorbed in the process zone), irrespective of the overall size of the crack body. However, unlike nominally crack-free bodies, the loads at fracture in larger bodies do not change in proportion to the changed cross-sectional area compared with smaller cracked bodies. In fact, as shown later, cracking loads usually increase less rapidly than areas. Thus, at constant fracture toughness the stresses to cause fracture can get smaller as the size of the cracked body increases. This is dangerous, as the cracking stress may be brought below the safety-factored yield stress in sufficiently large structures, and the structure will unexpectedly break in a brittle fashion even though small samples of the same material will be ductile in the laboratory. Fracture in the plastic range also does not follow the laws of geometric scaling. Size-effects in plastic flow itself are small, but are significant for initiation and propagation of cracks in plastic flow fields. The "ductility" (i.e. extension before fracture) of scaled bodies undergoing elastoplastic fracture diminishes with absolute size. The largest bodies may deform merely in the elastic range before cracking, as mentioned earlier. This behaviour goes against the conventional wisdom of using a size-independent safety-factored flow stress in traditional design procedures. In fatigue fracture, any growth-rate law which incorporates the stress intensity factor K (such as the Paris equation) must exhibit a scale effect, whereby (usually) the fatigue life at a given stress is smaller the larger the body [9]. Irwin in 1954 recognised that explanations of size-effects prior to that time neglected the influences of elastic energy in the cracked bodies: he remarked that [10] " . . . disregarding the size effects due to changes in materials properties with size caused by problems of metallurgical processing, previous studies have shown that fracture size effects of significant magnitude exist in solids whether they are brittle or ductile and whether the material is a metal or a plastic. The most elaborate precautions to eliminate differences due to material quality, lack of precise similarity of dimensional ratios, specimen surface preparation, and the experimental loading devices have failed to remove fracture size effects. It is apparent that these are fundamental to the nature of the severing p r o c e s s . . . " The particular symposium, at which he spoke, was concerned with low-temperature effects but he cautioned that reduced ductility with temperature (the usual "metallurgical" explanation of World War II T2 tanker and Liberty ship failures), was not the sole reason for brittle fractures in large structures. There are mechanics reasons for scale-effects which are concerned with predicting the relationships between critical loads, stresses, displacements and crack lengths in small and large components under monotonic elastic and plastic loading, and in fatigue. The well-known Griffith formula shows that in geometrically-similar bodies the
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A . G . ATKINS
fracture stress in the large prototype (p) body is smaller than in the small model (m) body. We have Op = ~/ EpRp/Trap
and
Orn =
~/ EmRm/rra m .
(4)
Reasonably assuming that Ep = E m and Rp = Rm, division of the two relations gives
(5)
Op = Om//~
where 3. = ap/am is the "scaling factor", i.e. the ratio of characteristic lengths between prototype and model. For a scale factor of 100, the elastic fracture stress of a large body is 1/10th of the fracture stress of a small geometrically-similar body. Corresponding relations exist for the ratio of loads (~A3/2) and displacements (~V~) at fracture, see Chap. 9 of [5]. It is the fracture toughness that is size-invariant not the fracture stress, and we see why the fracture strength of brittle materials varies with testpiece geometry since in "Kc = o~/rra. Y(a/w)", Y(a/w) is different for different specimens broken in different ways; Kc is the material property (dependent, of course, on rate, temperature, environment and so on). Most large prototype engineering structures are not, of course, proportionately scaled-up versions of model testpieces. For example, ships' plates can be tested for toughness in full-thickness in the laboratory, but the height and width dimensions of a real hull are impossible to duplicate; again, in pipe steel, annular cross-sections can readily be tested, but the considerble length of pipelines cannot. Clearly geometric similarity between laboratory model and prototype does not hold, yet brittle fracture in such structures made of ductile materials is well known. A generalised approach for scaling in non-proportionately scaled elastic structures does not exist [11]. So, for purposes of illustration, consider centre-cracked panels (CCP) as in Fig. 5 (but where 3. is not the same for all dimensions). We have
hp _ ~.h;
Bp _ ~.B;
Wp _ Zw;
hm
Bm
Wm
ap _ A,.
and
(6)
am
It may be shown that Xp _ ).B~tw
Up -- )L3a/2
17p __
1
(7)
Om V~ " ~a' llm ~"w' Notice that only Za comes into (Tp/Om, which says that all small cracks in large plates gm
behave like Griffith cracks.
Flo. 5. "Model" and "prototype" centre-cracked panels.
Scale effects in engineering failures
209
Burst tests on mild steel pipes by Duffy et al. [12] provide examples of the application of these scaling laws. Using Folias's correction [13] for the effect of curvature in the pipe to enable the flat panel algebra to be used, we obtain (Orh°°p)P
-- ~
(O.oop)m
(8)
Z3/2
Table 1 shows the ratio of hoop stresses at fracture (converted from the burst pressure) in prototypes and models all having the same length pipe and where the product of radius of curvature and pipe thickness is constant (i.e.).BAh = 1). There is a good agreement with a ).a 3/2 relation. This problem is of particular interest since it turns out that hoop stresses at fracture in full-size pipelines are larger than those in laboratory CCP coupons containing the same crack length. These predictions are borne out by the results of Duffy et al. [12] Hence cracking stresses in larger structures are not necessarily always lower than those in small structures as predicted by geometrically similar scaling which gives Crp = trm/V~. When ). varies in different directions Op may even be greater than Ore. Hence some large structures containing flaws may not fail in a brittle fashion and design based on a criterion of plastic yielding and collapse may be adequate. Even so, it is important to realise that a size-effect does exist in fracture, unlike yielding, and that unexpected brittle failures can occur. For a factor of safety f, the strength-of-materials design working stress is Cry/f. Brittle fracture will occur if Op < Oy/f, where Op is the stress to cause cracking without yielding. For plate structures containing some sort of central crack, Crp = am/ V ~ from Eqn (7). Also o m = Kem/~-~m. Thus for brittle fracture, t or J
~--~--~ "
(9)
For structural steels with Kc/oy ~ 0 . 5 V ~ at room temperature, we conclude that brittle fracture occurs when ap (=).aam) ~ 0.75 m for f = 3. Ship's hatches larger than this have been known to initiate brittle fracture. For Kc/oy reduced to 0.05~mm, ap I> 8 mm for brittle fracture. The effect of body size on crack velocities may also be determined. For geometrically-similar linear bodies,
aJa,, = "V~
(10)
for the same load-point velocity in model and prototype (zi,~ = zip). Hence not only do large structures sometimes unexpectedly fail in a brittle fashion, but they do at high crack speeds. This result has been arrived at for E, R and Kc being constant. When these properties are rate dependent, the model and prototype velocities may be closer together or wider apart, depending upon the sign of K~(a). If K~ decreases with increase of crack velocity, the readier cracking of prototype structures over model structures is made even more easy. For non-proportionately scaled, but linear CCP, we h a v e ~lp/tl m = ~av/~aa. For identical flaw sizes in model and prototype, gtp is ).wam so that "fast" fracture in a ship's hull under bending is readily explained. (Note that the crack velocity is increased, even though (79 = (7m if ).a ---- 1 according to Eqn (7)). Even if the prototype has a larger flaw than the model, ~ will not rise as quickly as ).w, so fast fracture still results. In pipelines, ).w is the scale factor for length of pipe, so that enormous TABLE 1. Scaling laws of fracture in pipes O'hoop (MPa) ap (mm)
172 127
(Ohoop)p/(Ohoop)m
1
(ap/ara) -3/2
1
138 152 0.83 0.76
117 178 0.68 0.61
103 203 0.60 0.50
83 229 0.48 0.42
69 254 0.40 0.36
210
A.G. ATKINS
prototype velocities can be attained in full-scale compared with laboratory models of the same thickness, diameter and initial flaw size. Cracks in pipelines therefore run long distances--hundreds of metres and more--before arresting, and fit the "catastrophic" category. A full dynamic analysis of pipeline fracture is very complicated. The increase in velocity is much larger than for geometrically similar scaling, where ap = ~/~am. It must be pointed out, however, that according to Wilkowski [14] actual measurements of brittle fracture speeds have shown a limiting maximum speed, apmax (100 ms -1) in steel pipelines. This is believed to be caused by elastic flexural stress waves in the pipe causing a sinusoidal fracture path. Fracture speeds in laboratory testing of fiat specimens (e.g. fiat plates, DCB etc.) can be higher due to the absence of flexural stress waves (apmax ~ 2800 m s-a). Analysis of elastoplastic fracture is often tackled using the J-integral approach, which is really the mechanics of fully reversible non-linear elastic fracture, the justification for which is the equivalence (providing no unloading occurs) between Hencky total strain plasticity theory with proportional loading and non-linear elasticity. Evidence shows that this is acceptable for the initiation of cracks in elastoplastic flow fields, but is not correct for ensuing propagation, except in the rare cases of cracking under constant or increasing load. Elastoplastic propagation requires new analysis and an exact closed-form solution for beam geometries has recently been presented [15], based upon separation of remote flow from fracture events close to the crack tip. This clarifies much of the confusion about JR curves (the basis of which does not separate these components of work) and their supposed geometryindependence. Scaling laws for events up to crack initiation in elastoplastic fracture described by the J~ approach must coincide with those for non-linear elastic fracture. For a body obeying a power law between load and displacement, viz. X = k ( A ) u n where k ( A ) is the function relating crack starting area to "stiffness", we have for geometrically similar non-linear bodies having invariant R [16] Xp g m
-
~(n+2)/(n+l)
Op _ ,~-n/(n+l),
Up -- ~n/(n+l).
am
um
(11)
We see that stresses to cause cracking in large non-linear structures are even smaller than those in large linear structures. When the structure is highly non-linear (n is large) Op = Crm/Z rather than Op = O m / ~ . Similar procedures may be applied to particular non-proportionately scaled, non-linear bodies, and for crack velocity ratios, see Chap. 9 of [5]. In terms of energy absorbed up to initiation, a 3?-factor applies whatever the value of n because it is only the fracture work which is being supplied in theory. Afterwards however, in the practical elastoplastic case, the remote flow accompanying propagation must scale as ~3 and the nett result of these different laws is that the total energy absorbed from just loading to final part separation scales as )x, where 2 < x < 3 [17]. This is confirmed by many empirical relations in the literature. How to relate x to the "basic" properties of strength and toughness, and to the cracked geometry itself, is explained in [17]. There is a number of early reports of size-effects in the fracture of metal testpieces undergoing plastic flow before rupture [17-20]. Tests on geometrically-similar notched specimens on a variety of metals revealed two principal effects, that (a) the extent of plastic flow before cracking is smaller in larger testpieces; (b) the relative amount of total work done after peak load is smaller in larger testpieces. Figure 6 shows results from Hagiawara [21] for tests on a variety of notched and un-notched scaled I-beams. In this figure, the load co-ordinate is normalised by ~z (where ~. is the scaling factor) to give stress which, in turn, is non-dimensionalised by the different flow stresses ay of the various steels from which the beams were made: again, the displacement is normalised to (engineering) strain by dividing by ~.. The
211
Scale effects in engineering failures
100
~
80
/~
6 0 S
X
60 o
E
20
5
100
I
150
I
200 250 Normatised deflection 61k (mm)
I
300
F1G. 6. Elastoplastic fracture of geometrically-similar notched and un-notched I-beams (after Hagiawara e t al. [21], with permission).
relevance of all this to catastrophic fracture is that a designer may believe that large components and structures will behave in the ductile fashion of the small "model" pieces, whereas instead they fracture at much smaller displacements, and go bang with a more energetic, and faster-velocity, cracking event. The relative displacements at crack initiation depicted in Fig. 6 follow the ~,/~+z) relation given by Eqn (11) with n = 0.21. It is clear that whereas plastic flow dominates small behaviour, quasi-elastic fracture characterises the large, and that there are transitions in mode of behaviour as the size changes. In Fig. 7, EFG represents initial yield and BCD extensive yielding (which corresponds with unlimited plastic flow or collapse in a rigid-perfectly plastic solid). It
II ~ B
/
?
~C
E
t/3
/ CracVkelocity
I
/
/
~
I
........
/
,F ....
~
i
H
P
D Extensive yleldtng or plashc collapse
._5 Inihal yteldmg
I I
0
5tze or 5cahng Factor [ X)
third transition
Upper frans#lon
Lower fransthon
Fie. 7. Size-effect in fracture. Cracking stress falls with increasing size; crack velocity increases with increasing size. Yield stress remains invariant.
212
A . G . ATKINS
is not obvious how to set the level of BCD in a workhardening solid. One point of view is to say BCD corresponds with the least stress level at which the plastic zones reach an external boundary of the cracked body. The fracture stress curve M . . . N will be given by the appropriate algebra for the fracture stress at all sizes, i.e. which expresses elastic fracture, elastoplastic fracture and plastic fracture. The lowering of the fracture stress as the size increases is the cause of premature catastrophic failures (at least "premature" if the behaviour shown in Fig. 7 is not understood by the designer). The vertical distances before or between the yielding and yielding/fracture lines are some indication of the body displacement before fracture: on the left of the illustration there would be gross deformation before fracture but on the right not very much. Furthermore, Eqn (10) says that the crack velocity ratio increases as V)~ on geometrically-similar structures, so that the unexpected failures do really go off with a bang!
4. PROGRESSIVE DYNAMIC FRACTURE Our previous discussions have, perhaps, emphasised fracture in large monolithic bodies, where there is lots of shear connexion between elements. In quasi-static fracture, it is the lack of shear connexion that can make bodies notch-insensitive and where, in the case of the extensible biological materials for example, it is the shape of the stress-strain curve that makes them difficult to tear, n o t that they have a high specific work of fracture [22]. What of bodies composed of discrete elements, such as frameworks? Should one element of such a framework (in a bridge, offshore structure, airframe etc.) fail prematurely we posed [23] the question as to whether: (a) other members are unaffected by the fracture; (b) a few members break in sequence but the progression arrests before the structure as a whole fails; (c) a series of fractures is triggered with elements failing one after the other, leading to complete collapse of the structure. Central to the philosophy of fail-safe, or damage tolerant, design is that the damaged structure should still safely carry the design loads. This aim is achieved by employing structures with many redundancies where, even with load shedding from a number of damaged or broken elements, the damaged structure can still bear the design loads. Generally, failure or damage to individual components may be caused by plastic hinging, fracture, buckling and so on, or by combinations of different mechanisms. In investigations of the load-carrying capacity of a structure with one or more failed members, it is customary to perform static analyses of the structure with elements removed. Clearly it is possible to get a quasi-static cascade of failures once one member has failed when the remaining damaged structure cannot sustain the applied loads statically. But an equilibrium analysis which predicts that a damaged structure should be safe will not be conservative and may not provide a correct picture of events. For, when a premature failure of an element takes place with the structure under load, energy stored in the structure is released and this inaugurates a state of transient vibration about the new equilibrium position. The members of the structure will therefore experience transient loads and displacements greater than the values given by static analysis, in the manner of a mass vibrating on a spring. Clearly, there is the possibility that these dynamic stresses and strains may produce failure in another member of the structural system even though the new steady-state stresses and strains in the damaged system may be below failure levels. Failure of a second member will excite further vibration and either more failures will occur or the process may arrest. There are remarkably few reports of analytical and experimental investigations of dynamic progressive failure, whether produced by collapse, buckling, or fracture.
Scale effects in engineering failures
213
Pretlove [24] considered dynamic effects in the progressive fracture of a simple uniaxial tension structure. He demonstrated that there are structures which appear to be safe from progressive fracture on the basis of static calculations but which are unsafe when transient motions are taken into account. We extended that work to a 12-member radially-tied structure with two degrees of freedom. This simplified problem was suggested by the reported failure of a multiply-guyed mooring post associated with an offshore oil installation. In this case, with the mooring ropes under tension a failure in one is said to have led to progressive failure in several others even though, according to static calculations, the system was supposedly fail-safe. This problem is unlikely to be complicated by damping, plasticity or buckling, and, owing to its relative simplicity, statistical variations in the strength of the members were included in the analysis. The nature of cascaded failures was predicted by means of a Monte-Carlo type computer simulation and was compared with actual experiment. The analysis is relevant to guyed masts on land and, indirectly, to cable-net structures. Figure 8(a) shows the geometry of the experiment using a stiff steel outer ring, with tensioned spokes connected to a central mass which is free of restraint. Figure 8(b) shows the safety boundaries given by experiment and predicted by theory: a~ = a/af, where of is the "normal" failure stress of a spoke and a is the (lower) stress when premature failure of the member occurs. Premature failure may arise through locally embrittled material, pre-existing flaws, or be produced by external causes (battle damage, accidents etc.). We may conclude from Fig. 8(b) that, if a partly broken structure is incapable of bearing the applied loading statically, transient overloading of members simply aids failures in other members and spreads the structural damage.
(a)
~/~rFiinXgd
outer
(b) 1.0
!
~q~ Safe
~
I
\
®
statically\
-
I Unsafe ~ ~ ! dynamically k
0.5
Unsafe e~ e~
o a_
I
0.6
0.7
*+\ , ~
M
,\ t
0.8
L
0
0.9
1.0
FIG. 8. (a) Schematic of the spoke wheel structure. (b) Safety boundaries for the spokedwheel structure taking into account the statistics of member failure load. (®) Computer stimulation: probability of static survival. ( ~ ) Computer simulation: probability of dynamic survival. (x) Experimental: probability of static survival. (+) Experimental: probability of dynamic survival. (---) Theoretical boundaries for zero spread of member failure load. (--) Theoretical boundaries for 6.1% S.D. of member failure load.
214
A . G . ATKINS
However, and this was the significant finding of [23, 24], the transient overloads may induce progressive fracture even though static calculations predict safety.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
A. Bignell, B. Peters and C. Pym, in Catastrophic Failures, p. 107, Open University Press (1977). N. Constantin, Private Communication. G. M. Boyd (ed.), Brittle Fracture in Steel Structures, Butterworths, London (1970). J. E. Gordon, The New Science of Strong Materials, Penguin Books, London (1968). A. G. Atkins and Y-W Mai, Elastic and Plastic Fracture, Ellis Horwood/Wiley, Chichester (1985/88). C. E. Turner, Mater. Sci. Engr. 11,275 (1973). C. Gurney and Y-W Mai, Engr. Fract. Mech. 4, 853 (1972). C. Gurney and K. M. Ngan, Proc. Roy. Soc. A325, 207 (1971). K. E. Puttick and A. G. Atkins, Int. J. Fract. 23, R51 (1983). G. R. Irwin, Syrup. on Effects of Temperature on Brittle Behaviour. Philadelphia, ASTM, 176 (1954). Y-W Mai and A. G. Atkins, Int. J. Mech. Sci. 20,437 (1978). A. R. Duffy, G. G. McClure, R. J. Eiber and W. A. Maxey, in Fracture, Vol. 5 (ed. Liebowitz), Academic Press, New York (1969). E. S. Folias, Int. J. Fract. 1, 104 (1965). G. M. Wilkowski, Private Communication. A. G. Atkins and Chert Zhong (To be published). Y-W Mai and A. G. Atkins, Int. J Mech. Sci. 17,673 (1975). A. G. Atkins, Int. J. Mech. Sci. 30, 173 (1988). T. E. Stanton and R. G. C. Batson, Proc. Inst. Civil Engrs 211, 67 (1921). J. G. Docherty, Engineering 133,645 (1932). J. G. Docherty, Engineering 139, 211 (1935). K. Hagiawara, H. Takenabe and H. Kawano, Int. J. Impact. Engr. 1,257 (1983). Y-W Mai and A. G. Atkins, J. Phys. D: Appl. Phys. 22, 48 (1989). A. J. Pretlove, M. Ramsden and A. G. Atkins, Int. J. Impact Engr. 11,539 (1991). A. J. Pretlove, in Proc. Int. Conf. Steel Struct., Budva, Yugoslavia, 749 (1986).
Pergamon
1350-6307(94)00015-8
SCALE EFFECTS
IN ENGINEERING
FAILURES
A. G. ATKINS Department of Engineering, University of Reading, PO Box 225, Whiteknights, Reading RG6 2AY, U.K.
(Received 11 April 1994) Abstract -- A brief review is given of what may be meant by a "catastrophic engineering failure", particularly as it relates to situations involving fracture. Whether "fast fracture" is necessarily the same as "unstable" fracture--and whether it makes any difference anyway--is discussed. Since recoverable elastic energy, and irreversible remote plasticity, are volume-dependent quantities, but fracture work depends only on area, cube/square energy scaling principles are inherent in all mechanics of fracture. These different dependencies translate into the well-known experience that components and structures, made of materials which are appreciably ductile in laboratory-size testpieces, behave in a progressively less ductile fashion the larger they get. Eventually, above some critical size, their behaviour is globally elastic, and fractures are brittle. Conversely, when deformation zones are kept very small in normallybrittle materials, plastic deformation is possible, as shown by limiting sizes in the communution of powders and by the ability to micro-machine glass. Analytical studies and experimental investigations of scaled cracked bodies, covering the whole range from globally elastic in the large, through to quasi-rigid plastic in the small are described and assessed, including consideration of dynamic progressive fracture in frameworks.
1. INTRODUCTION The image of a catastrophic engineering failure is of something (usually large) that fails unexpectedly and dramatically. Examples would be the Ronan Point tower block failing by progressive collapse [1]; metal shell structures failing by buckling, e.g. the Bucharest Exhibition Centre under snow loading [2]; large ships and pressure vessels failing by brittle fracture [3] and so on. Equal damage to buildings, structures and vehicles may be produced in collisions, but that is not unexpected in that there is an obvious cause of the damage. Catastrophic failures on the other hand seem spontaneous and often happen for no apparent reason at the time. Only after an accident investigation is the cause usually, but not always, identified. As with all fabricated articles, if something goes wrong in use, the fault may be with the original design, with the materials used or with the manufacturing: the designer may have got the sums wrong; he may have specified a material having inappropriate properties; even though an appropriate material has been specified, an improper one has actually been used through a mistake or because the manufacturer thought it wouldn't make any difference to use a substitute; manufacturing standards or inspection procedures are not good enough; the component or structure may have been inappropriately modified after installation; and so on. Engineering design theories have developed often as a result of learning from catastrophic failures, e,g. wind-induced loadings on the Tacoma Narrows bridge. Again, the discipline of "fracture mechanics" arose from trying to explain the premature brittle failure of large steel structures, particularly the all-welded ships of the Second World War. (In fact, it may be said that an explanation was already available in the seminal Griffith theory, but it was not understood at the time that Griffith could be applied to any displacement-reversible fractured body, even if made of "ductile" steel and not merely of something obviously brittle such as the glass studied by Griffith.) In this paper we concentrate on failure by fracture, with or without preceding 201
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A . G . ATKINS
and accompanying plastic flow. J. E. Gordon has said that "The worst sin in an engineering material is not lack of strength or lack of stiffness, desirable as these properties are, but lack of toughness, i.e. lack of resistance to the propagation of cracks". He also said that "the history of attempts to prevent cracks spreading, or to evade the consequences, is almost the history of engineering" [4]. It is common experience that different materials fracture in different ways. Brittle fractures are globally elastic and the broken bits (if you can find all of them!) may be fitted back together to reconstruct the original article. Hence the success of archaeologists with broken pottery and our ability to mend furniture. In contrast, ductile fractures involve a great deal of irreversible plasticity both preceding and accompanying the fracture. To refit bits back together would involve much unbending and unbuckling of regions far removed from the fracture path (cf. metal food containers with curled-up lids after opening). There is, of course, a complete spectrum of elastoplastic fracture behaviour from that involving a small amount of remote plasticity (for which modified elastic fracture mechanics analyses may apply) to that involving extensive remote plasticity with hardly any elastic springback, for which the methods of rigid-plastic fracture mechanics apply. It is a pity that we traditionally classify materials as being ductile or brittle based upon tests in the laboratory, for there is a scale effect in the mechanics of fracture which produces unexpected transitions in behaviour. That is, components and structures made from brittle materials can behave in a ductile fashion, and those made from ductile materials can behave in a brittle fashion. It depends on size and comes about because recoverable elastic energy and irreversible remote plasticity are volume-dependent quantities, but fracture work depends only on area. Hence cube/square energy scaling principles are inherent in all mechanics of fracture. It is the transition from ductile behaviour in the small to brittle behaviour in the large which has figured prominently in catastrophic engineering failures. Crack velocities in large components and structures turn out to be much greater than in small laboratory-size models and this has compounded the unexpectedness of sudden violent failures. This can then involve consideration of "dynamic fracture mechanics" which either means the initiation of a stationary crack under rapidly-applied loading, or rapid propagation and possible arrest of a crack under slowly varying load. In turn stress wave effects in the cracked body (reflexion from boundaries etc.) may become important. For a review of these sorts of things, see Chapters 8-10 of reference [5]. Even in the small, and certainly in the large, it has been established that there is the danger in loaded frameworks of domino-like progressive dynamic fracture, for which quasi-static calculations on defective frameworks are inappropriate. Examples are given later.
2. ELASTIC F R A C T U R E Figure 1 shows a variety of load (X), displacement (u) responses in globally-elastic fracture. Beyond the point P at which cracks initiate and start to propagate, the load may decrease, remain constant or increase monotonically with increasing load-point displacement; or drop suddenly to complete fracture; or fluctuate with arrested load drops followed by increases in load up to other bursts of cracking. The general incremental work equation for elastic fracture mechanics is X d u = dA + R d A
(1)
where A is elastic strain energy, R is toughness and A is crack area. For linear elastic fracture, dA = d ( X u / 2 ) and manipulation gives X 2 = 2 R { ( d / d A ) ( u / X ) } -1
(2)
which is the well-known "compliance calibration" equation usually written with Gc (the strain energy released from the cracked body) in place of R (the specific work
Scale effects in engineering failures P
203
P
0
,-I
O ..d
Displacement
Q
0
U
(a)
/
"O O ,.d
0
P
Displacement
Id
(b)
O O
,-1
Displacement
Displacement (d)
(c) P
U
p~
O
..1
Displacement
u
(e) Flo. 1. Typical load--displacement diagrams for: (a) unstable cracking, (b) stable cracking with decreasing load, (c) stable cracking with constant load, (d) stable cracking with increasing load, and (e) "stick-slip" cracking. Reprinted by permission of the Council of the Institution of Mechanical Engineers from J. Strain Analysis.
required for fracture). Stress intensity factor equations have the same form as Eqn (2), with K 2 = ER and the (d/dA)(u/X) term implicit. For a given crack/body geometry, loci of constant A and R obeying Eqn (1) may be superimposed on load-displacement Cartesian co-ordinates. The most general type of plot for bodies with linear stiffness is shown in Fig. 2 (which relates to a cracked ring loaded in compression), segments of which for other testpieces may be identified in Fig. 1. Insofar as catastrophic fractures may be "unstable" fractures, we remember that energetic stability depends upon external constraints. In particular we identify displacement control (du/u > 0) and load control (dX/X > 0). These conditions concern the vertical and horizontal tangents to the constant toughness loci shown in Fig. 2. Load control is impossible unless crack propagation occurs under increasing load; displacement control is impossible unless cracking occurs under increasing displacement. The stiffness lines passing through the two tangents to the R locus determine the minimum sizes of crack for which stable crack growth may be achieved according to (du/u > 0) or (dX/X > 0). It follows that the conditions shown in Fig. l(a) correspond to instability under both load and displacement control; those in Fig. l(b), (c) and (d) are all stable under displacement-control and Fig. l(d) depicts stability under load-control. In Fig. l(a) there is so much energy available at first cracking that energy could be removed from the system and still leave enough to feed the crack growth. Unless the displacement is reduced in some way at the instant of cracking, instability will result, the system going "bang" with, perhaps, fragments being thrown off. It is possible to
A. G. ATKINS
204
V /
I
/
I
d
I
Load, D/sp/acement /
/
I
I
'A '~ncreasmg 200 160
120 C3
-~
80
i
;s
:o
i
2s
3'o
Ys
Displacement: l~m Fxo. 2. Load--deflexion diagram for cracking of a ring under diametrical compression.
relate the upper bound triangular work area of Fig. l(a) to the fracture toughness, and Turner's r/-factor (well-known in elastoplastic fracture) was first introduced for unstable elastic fracture [6]. Of course, the material may have a rate-, temperature-, or environment-dependent toughness which leads to the stick-slip behaviour shown in Fig. l(e). Furthermore, the crack may not be sharp, which affects stability. The stiffness of the loading system and the energy it stores will play its part in achieving stability or not. The following analogy of an ice-skater demonstrates the influence of external constraints on stability [5]. Consider: (a) a skater who falls through ice that has cracked because of his weight; (b) a skater who happens to be under an overhanging tree, and who can hang on to a bough to prevent himself dropping through the ice, even though the ice sheet may be breaking; (c) the skater who not only can grab on to a tree as soon as he sees cracks appear but can also haul himself up off the ice sheet. In all cases the strain energy in the ice sheet that helps to feed the cracks is caused by the deflection of the ice under the skater's weight. In the first case, the cracking is catastrophic and unstable because the descent of the skater's body provides more energy to the ice sheet than is required for balanced stable cracking. In the second case, the skater's body is prevented from dropping, and if he grabs on to the tree as soon as he sees cracks initiating he may prevent further cracking if his body is locked at a fixed height and can supply no further energy to the system. In the third case, if he quickly lifts himself off the ice sheet upon sight of the first cracks, he may prevent catastrophic cracking altogether in that he removes energy from the ice sheet by changing the way his weight is supported (i.e. by his feet and hence the ice, or by his arms and hence the tree). All testing machines themselves deflect under load (so, in practice, would the bough of the tree). "Stiff" machines store less energy than "soft"
Scale effects in engineering failures
205
machines under the same load. The behaviour of soft machines corresponds with (i) above; hard with (ii) above. However most loading systems are unlike the skater in (iii) who removes energy from the loading system, so that catastrophic fracture can result in even the stiffest loading devices. In the field, loading of structures, vehicles and so on is far softer. A homely example of stability in fracture is in the design of nutcrackers, the common "squeezed lever" sort of which often leads to crushed kernels owing to an inability to control the displacement once cracking starts. Gurney and Mai [7] provide a comprehensive analysis of energetic stability in elastic fracture and show that there are always geometric factors as well as material-dependent factors to consider in stability. As regards crack velocities, it follows from Eqn (1) that
A/a = X/(2R + u dX/dA).
(3)
LEFM K-formulae enable u dX/dA to be determined for any cracked body (see Chap. 3 of [5]) from which the ratio of crack velocity ~ to load-point velocity ti is given. It is not uncommon to find that stable cracks propagate at one or two orders of magnitude greater than the load-point velocity. Even so, cracking is "stable"--it just happens to be fast, but yet not so fast that a kinetic energy term associated with ~ has to be included in Eqn (1). Upper limiting ranges for ti for quasi-static fracture are some 6 ms -1 in metals and 2 ms -1 in polymers [8]. Hence "fast fractures" are not always "unstable" fractures, but the distinction in a practical case may be unimportant unless there is the possibility of "crack arrest" (where an unstable crack "runs out of steam" and stops within the body). The sharpness of cracks is important here. It is often forgotten that LEFM formulae presume fully-constrained sharp cracks, which is why one gets away with a single parameter (Kc, R, Go) to characterise elastic fracture: in ductile fracture, on the other hand, the specific work or fracture (R, J~) or COD (6c) is hydrostatic-stress-state-dependent, because the cracks are blunt. If cracks are initially blunt in elastic fracture cases, the fracture load goes to higher levels and there is excess energy at initiation. In what would be otherwise stable propagation cases, instability can sometimes result. There may, or may not, be crack arrest of the "sharpened" crack during propagation. The case of truly unstable fast fracture is shown in Fig. 3 where the variation of ~/a is shown with crack length for the cracked ring geometry. Negative values of ~/ti are predicted for the unstable part of the locus, because in that region ~ is negative to satisfy Eqn (1); and in Eqn (3) dX/dA is so big negatively that the denominator becomes negative. At the vertical tangent stability point for du/u > 0, the denominator of Eqn (3) is zero, and therefore (~/a) ~ o¢. The negative velocities predicted for cracks shorter than the critical length will never be seen in normal circumstances and the fracture is catastrophic. Initially-blunt cracks make the situation worse. The cracked ring specimen provides a valuable illustration of the relationship between so-called R-curves and stability. A material with a constant toughness R shows a rapid change from zero to R as soon as propagation commences. Some materials display a transition region of increasing R with crack growth up to a higher constant level during globally elastic fracture, and this can promote stability in otherwise unstable situations. However, instability may eventually occur after only a short period of crack growth. Instead of the loci of constant A and R for stable cracking on Xu Cartesian co-ordinates in Fig. 2, we show loci of constant X and u on GA or RA co-ordinates in Fig. 4. Remember G relates to energy "availability", and R to energy "requirement". These loci have maxima (to understand this consider changes in R loci cut by vertical and horizontal lines in Fig. 2). The mexima occur at the starting crack lengths corresponding with the vertical and horizontal tangents in Fig. 2, either side of which instability or stability occurs. The material RA curve is superimposed for two different starter crack lengths on either side of the maxima. The relative magnitudes of the slopes dR/dA (indicating energy required) and 3G/OA)u o r X (indicating energy available), determine how much of the R curve may be picked up before a catastrophic fracture stops measurements. Clearly, to obtain
206
A . G . ATK1NS
22oo~ 7moJ c~lz, w Q
-
tO00
S fable Region for ~X>O
I I
600-
I 'o
lengfh (ram)
800-
6'0
eb
Sfable Region for A u > 0
&1200 1600-
Fro. 3. Plots of (t~/fi) vs. a for the cracked ring in Fig. 2.
~
d
Ei
a (ram) FIG. 4. G vs. a c u r v e s f o r constant load and constant displacement f o r a s t e e l r i n g fracturing under diametrical compression. R vs. a curve is also s h o w n .
experimentally the complete R curve requires testpieces with different minimum starter crack lengths in order to achieve complete stability. This fact is not always realised by experimenters who quote their last R-value before instability as the critical toughness, whereas their result is really testpiece-dependent.
Scale effectsin engineeringfailures
207
Further consideration of crack stability (including stability in elastoplastic fracture), and experimental tricks to promote crack stability in testing, may be found in Chap. 8
of [51.
3. SCALE EFFECTS When load-deflexion tests are performed on different size bodies of nominally crack-free material it is possible to normalise the different load-extension curves into one stress-strain curve, from which elastic moduli and plastic flow properties are traditionally determined. Variations sometimes do occur depending on the size of the body (e.g. segregation in castings, strengthening in fine wires), but it is customary to consider that elastic moduli and yield stresses are independent of the size of the material, and that it is translated into traditional "strength of materials" design practice. Load-extension curves given by globally elastic cracked bodies are used directly to determine Jc, Gc or Kc, consistent values of which will be obtained from the appropriate formula (to correspond with the specific work R absorbed in the process zone), irrespective of the overall size of the crack body. However, unlike nominally crack-free bodies, the loads at fracture in larger bodies do not change in proportion to the changed cross-sectional area compared with smaller cracked bodies. In fact, as shown later, cracking loads usually increase less rapidly than areas. Thus, at constant fracture toughness the stresses to cause fracture can get smaller as the size of the cracked body increases. This is dangerous, as the cracking stress may be brought below the safety-factored yield stress in sufficiently large structures, and the structure will unexpectedly break in a brittle fashion even though small samples of the same material will be ductile in the laboratory. Fracture in the plastic range also does not follow the laws of geometric scaling. Size-effects in plastic flow itself are small, but are significant for initiation and propagation of cracks in plastic flow fields. The "ductility" (i.e. extension before fracture) of scaled bodies undergoing elastoplastic fracture diminishes with absolute size. The largest bodies may deform merely in the elastic range before cracking, as mentioned earlier. This behaviour goes against the conventional wisdom of using a size-independent safety-factored flow stress in traditional design procedures. In fatigue fracture, any growth-rate law which incorporates the stress intensity factor K (such as the Paris equation) must exhibit a scale effect, whereby (usually) the fatigue life at a given stress is smaller the larger the body [9]. Irwin in 1954 recognised that explanations of size-effects prior to that time neglected the influences of elastic energy in the cracked bodies: he remarked that [10] " . . . disregarding the size effects due to changes in materials properties with size caused by problems of metallurgical processing, previous studies have shown that fracture size effects of significant magnitude exist in solids whether they are brittle or ductile and whether the material is a metal or a plastic. The most elaborate precautions to eliminate differences due to material quality, lack of precise similarity of dimensional ratios, specimen surface preparation, and the experimental loading devices have failed to remove fracture size effects. It is apparent that these are fundamental to the nature of the severing p r o c e s s . . . " The particular symposium, at which he spoke, was concerned with low-temperature effects but he cautioned that reduced ductility with temperature (the usual "metallurgical" explanation of World War II T2 tanker and Liberty ship failures), was not the sole reason for brittle fractures in large structures. There are mechanics reasons for scale-effects which are concerned with predicting the relationships between critical loads, stresses, displacements and crack lengths in small and large components under monotonic elastic and plastic loading, and in fatigue. The well-known Griffith formula shows that in geometrically-similar bodies the
208
A . G . ATKINS
fracture stress in the large prototype (p) body is smaller than in the small model (m) body. We have Op = ~/ EpRp/Trap
and
Orn =
~/ EmRm/rra m .
(4)
Reasonably assuming that Ep = E m and Rp = Rm, division of the two relations gives
(5)
Op = Om//~
where 3. = ap/am is the "scaling factor", i.e. the ratio of characteristic lengths between prototype and model. For a scale factor of 100, the elastic fracture stress of a large body is 1/10th of the fracture stress of a small geometrically-similar body. Corresponding relations exist for the ratio of loads (~A3/2) and displacements (~V~) at fracture, see Chap. 9 of [5]. It is the fracture toughness that is size-invariant not the fracture stress, and we see why the fracture strength of brittle materials varies with testpiece geometry since in "Kc = o~/rra. Y(a/w)", Y(a/w) is different for different specimens broken in different ways; Kc is the material property (dependent, of course, on rate, temperature, environment and so on). Most large prototype engineering structures are not, of course, proportionately scaled-up versions of model testpieces. For example, ships' plates can be tested for toughness in full-thickness in the laboratory, but the height and width dimensions of a real hull are impossible to duplicate; again, in pipe steel, annular cross-sections can readily be tested, but the considerble length of pipelines cannot. Clearly geometric similarity between laboratory model and prototype does not hold, yet brittle fracture in such structures made of ductile materials is well known. A generalised approach for scaling in non-proportionately scaled elastic structures does not exist [11]. So, for purposes of illustration, consider centre-cracked panels (CCP) as in Fig. 5 (but where 3. is not the same for all dimensions). We have
hp _ ~.h;
Bp _ ~.B;
Wp _ Zw;
hm
Bm
Wm
ap _ A,.
and
(6)
am
It may be shown that Xp _ ).B~tw
Up -- )L3a/2
17p __
1
(7)
Om V~ " ~a' llm ~"w' Notice that only Za comes into (Tp/Om, which says that all small cracks in large plates gm
behave like Griffith cracks.
Flo. 5. "Model" and "prototype" centre-cracked panels.
Scale effects in engineering failures
209
Burst tests on mild steel pipes by Duffy et al. [12] provide examples of the application of these scaling laws. Using Folias's correction [13] for the effect of curvature in the pipe to enable the flat panel algebra to be used, we obtain (Orh°°p)P
-- ~
(O.oop)m
(8)
Z3/2
Table 1 shows the ratio of hoop stresses at fracture (converted from the burst pressure) in prototypes and models all having the same length pipe and where the product of radius of curvature and pipe thickness is constant (i.e.).BAh = 1). There is a good agreement with a ).a 3/2 relation. This problem is of particular interest since it turns out that hoop stresses at fracture in full-size pipelines are larger than those in laboratory CCP coupons containing the same crack length. These predictions are borne out by the results of Duffy et al. [12] Hence cracking stresses in larger structures are not necessarily always lower than those in small structures as predicted by geometrically similar scaling which gives Crp = trm/V~. When ). varies in different directions Op may even be greater than Ore. Hence some large structures containing flaws may not fail in a brittle fashion and design based on a criterion of plastic yielding and collapse may be adequate. Even so, it is important to realise that a size-effect does exist in fracture, unlike yielding, and that unexpected brittle failures can occur. For a factor of safety f, the strength-of-materials design working stress is Cry/f. Brittle fracture will occur if Op < Oy/f, where Op is the stress to cause cracking without yielding. For plate structures containing some sort of central crack, Crp = am/ V ~ from Eqn (7). Also o m = Kem/~-~m. Thus for brittle fracture, t or J
~--~--~ "
(9)
For structural steels with Kc/oy ~ 0 . 5 V ~ at room temperature, we conclude that brittle fracture occurs when ap (=).aam) ~ 0.75 m for f = 3. Ship's hatches larger than this have been known to initiate brittle fracture. For Kc/oy reduced to 0.05~mm, ap I> 8 mm for brittle fracture. The effect of body size on crack velocities may also be determined. For geometrically-similar linear bodies,
aJa,, = "V~
(10)
for the same load-point velocity in model and prototype (zi,~ = zip). Hence not only do large structures sometimes unexpectedly fail in a brittle fashion, but they do at high crack speeds. This result has been arrived at for E, R and Kc being constant. When these properties are rate dependent, the model and prototype velocities may be closer together or wider apart, depending upon the sign of K~(a). If K~ decreases with increase of crack velocity, the readier cracking of prototype structures over model structures is made even more easy. For non-proportionately scaled, but linear CCP, we h a v e ~lp/tl m = ~av/~aa. For identical flaw sizes in model and prototype, gtp is ).wam so that "fast" fracture in a ship's hull under bending is readily explained. (Note that the crack velocity is increased, even though (79 = (7m if ).a ---- 1 according to Eqn (7)). Even if the prototype has a larger flaw than the model, ~ will not rise as quickly as ).w, so fast fracture still results. In pipelines, ).w is the scale factor for length of pipe, so that enormous TABLE 1. Scaling laws of fracture in pipes O'hoop (MPa) ap (mm)
172 127
(Ohoop)p/(Ohoop)m
1
(ap/ara) -3/2
1
138 152 0.83 0.76
117 178 0.68 0.61
103 203 0.60 0.50
83 229 0.48 0.42
69 254 0.40 0.36
210
A.G. ATKINS
prototype velocities can be attained in full-scale compared with laboratory models of the same thickness, diameter and initial flaw size. Cracks in pipelines therefore run long distances--hundreds of metres and more--before arresting, and fit the "catastrophic" category. A full dynamic analysis of pipeline fracture is very complicated. The increase in velocity is much larger than for geometrically similar scaling, where ap = ~/~am. It must be pointed out, however, that according to Wilkowski [14] actual measurements of brittle fracture speeds have shown a limiting maximum speed, apmax (100 ms -1) in steel pipelines. This is believed to be caused by elastic flexural stress waves in the pipe causing a sinusoidal fracture path. Fracture speeds in laboratory testing of fiat specimens (e.g. fiat plates, DCB etc.) can be higher due to the absence of flexural stress waves (apmax ~ 2800 m s-a). Analysis of elastoplastic fracture is often tackled using the J-integral approach, which is really the mechanics of fully reversible non-linear elastic fracture, the justification for which is the equivalence (providing no unloading occurs) between Hencky total strain plasticity theory with proportional loading and non-linear elasticity. Evidence shows that this is acceptable for the initiation of cracks in elastoplastic flow fields, but is not correct for ensuing propagation, except in the rare cases of cracking under constant or increasing load. Elastoplastic propagation requires new analysis and an exact closed-form solution for beam geometries has recently been presented [15], based upon separation of remote flow from fracture events close to the crack tip. This clarifies much of the confusion about JR curves (the basis of which does not separate these components of work) and their supposed geometryindependence. Scaling laws for events up to crack initiation in elastoplastic fracture described by the J~ approach must coincide with those for non-linear elastic fracture. For a body obeying a power law between load and displacement, viz. X = k ( A ) u n where k ( A ) is the function relating crack starting area to "stiffness", we have for geometrically similar non-linear bodies having invariant R [16] Xp g m
-
~(n+2)/(n+l)
Op _ ,~-n/(n+l),
Up -- ~n/(n+l).
am
um
(11)
We see that stresses to cause cracking in large non-linear structures are even smaller than those in large linear structures. When the structure is highly non-linear (n is large) Op = Crm/Z rather than Op = O m / ~ . Similar procedures may be applied to particular non-proportionately scaled, non-linear bodies, and for crack velocity ratios, see Chap. 9 of [5]. In terms of energy absorbed up to initiation, a 3?-factor applies whatever the value of n because it is only the fracture work which is being supplied in theory. Afterwards however, in the practical elastoplastic case, the remote flow accompanying propagation must scale as ~3 and the nett result of these different laws is that the total energy absorbed from just loading to final part separation scales as )x, where 2 < x < 3 [17]. This is confirmed by many empirical relations in the literature. How to relate x to the "basic" properties of strength and toughness, and to the cracked geometry itself, is explained in [17]. There is a number of early reports of size-effects in the fracture of metal testpieces undergoing plastic flow before rupture [17-20]. Tests on geometrically-similar notched specimens on a variety of metals revealed two principal effects, that (a) the extent of plastic flow before cracking is smaller in larger testpieces; (b) the relative amount of total work done after peak load is smaller in larger testpieces. Figure 6 shows results from Hagiawara [21] for tests on a variety of notched and un-notched scaled I-beams. In this figure, the load co-ordinate is normalised by ~z (where ~. is the scaling factor) to give stress which, in turn, is non-dimensionalised by the different flow stresses ay of the various steels from which the beams were made: again, the displacement is normalised to (engineering) strain by dividing by ~.. The
211
Scale effects in engineering failures
100
~
80
/~
6 0 S
X
60 o
E
20
5
100
I
150
I
200 250 Normatised deflection 61k (mm)
I
300
F1G. 6. Elastoplastic fracture of geometrically-similar notched and un-notched I-beams (after Hagiawara e t al. [21], with permission).
relevance of all this to catastrophic fracture is that a designer may believe that large components and structures will behave in the ductile fashion of the small "model" pieces, whereas instead they fracture at much smaller displacements, and go bang with a more energetic, and faster-velocity, cracking event. The relative displacements at crack initiation depicted in Fig. 6 follow the ~,/~+z) relation given by Eqn (11) with n = 0.21. It is clear that whereas plastic flow dominates small behaviour, quasi-elastic fracture characterises the large, and that there are transitions in mode of behaviour as the size changes. In Fig. 7, EFG represents initial yield and BCD extensive yielding (which corresponds with unlimited plastic flow or collapse in a rigid-perfectly plastic solid). It
II ~ B
/
?
~C
E
t/3
/ CracVkelocity
I
/
/
~
I
........
/
,F ....
~
i
H
P
D Extensive yleldtng or plashc collapse
._5 Inihal yteldmg
I I
0
5tze or 5cahng Factor [ X)
third transition
Upper frans#lon
Lower fransthon
Fie. 7. Size-effect in fracture. Cracking stress falls with increasing size; crack velocity increases with increasing size. Yield stress remains invariant.
212
A . G . ATKINS
is not obvious how to set the level of BCD in a workhardening solid. One point of view is to say BCD corresponds with the least stress level at which the plastic zones reach an external boundary of the cracked body. The fracture stress curve M . . . N will be given by the appropriate algebra for the fracture stress at all sizes, i.e. which expresses elastic fracture, elastoplastic fracture and plastic fracture. The lowering of the fracture stress as the size increases is the cause of premature catastrophic failures (at least "premature" if the behaviour shown in Fig. 7 is not understood by the designer). The vertical distances before or between the yielding and yielding/fracture lines are some indication of the body displacement before fracture: on the left of the illustration there would be gross deformation before fracture but on the right not very much. Furthermore, Eqn (10) says that the crack velocity ratio increases as V)~ on geometrically-similar structures, so that the unexpected failures do really go off with a bang!
4. PROGRESSIVE DYNAMIC FRACTURE Our previous discussions have, perhaps, emphasised fracture in large monolithic bodies, where there is lots of shear connexion between elements. In quasi-static fracture, it is the lack of shear connexion that can make bodies notch-insensitive and where, in the case of the extensible biological materials for example, it is the shape of the stress-strain curve that makes them difficult to tear, n o t that they have a high specific work of fracture [22]. What of bodies composed of discrete elements, such as frameworks? Should one element of such a framework (in a bridge, offshore structure, airframe etc.) fail prematurely we posed [23] the question as to whether: (a) other members are unaffected by the fracture; (b) a few members break in sequence but the progression arrests before the structure as a whole fails; (c) a series of fractures is triggered with elements failing one after the other, leading to complete collapse of the structure. Central to the philosophy of fail-safe, or damage tolerant, design is that the damaged structure should still safely carry the design loads. This aim is achieved by employing structures with many redundancies where, even with load shedding from a number of damaged or broken elements, the damaged structure can still bear the design loads. Generally, failure or damage to individual components may be caused by plastic hinging, fracture, buckling and so on, or by combinations of different mechanisms. In investigations of the load-carrying capacity of a structure with one or more failed members, it is customary to perform static analyses of the structure with elements removed. Clearly it is possible to get a quasi-static cascade of failures once one member has failed when the remaining damaged structure cannot sustain the applied loads statically. But an equilibrium analysis which predicts that a damaged structure should be safe will not be conservative and may not provide a correct picture of events. For, when a premature failure of an element takes place with the structure under load, energy stored in the structure is released and this inaugurates a state of transient vibration about the new equilibrium position. The members of the structure will therefore experience transient loads and displacements greater than the values given by static analysis, in the manner of a mass vibrating on a spring. Clearly, there is the possibility that these dynamic stresses and strains may produce failure in another member of the structural system even though the new steady-state stresses and strains in the damaged system may be below failure levels. Failure of a second member will excite further vibration and either more failures will occur or the process may arrest. There are remarkably few reports of analytical and experimental investigations of dynamic progressive failure, whether produced by collapse, buckling, or fracture.
Scale effects in engineering failures
213
Pretlove [24] considered dynamic effects in the progressive fracture of a simple uniaxial tension structure. He demonstrated that there are structures which appear to be safe from progressive fracture on the basis of static calculations but which are unsafe when transient motions are taken into account. We extended that work to a 12-member radially-tied structure with two degrees of freedom. This simplified problem was suggested by the reported failure of a multiply-guyed mooring post associated with an offshore oil installation. In this case, with the mooring ropes under tension a failure in one is said to have led to progressive failure in several others even though, according to static calculations, the system was supposedly fail-safe. This problem is unlikely to be complicated by damping, plasticity or buckling, and, owing to its relative simplicity, statistical variations in the strength of the members were included in the analysis. The nature of cascaded failures was predicted by means of a Monte-Carlo type computer simulation and was compared with actual experiment. The analysis is relevant to guyed masts on land and, indirectly, to cable-net structures. Figure 8(a) shows the geometry of the experiment using a stiff steel outer ring, with tensioned spokes connected to a central mass which is free of restraint. Figure 8(b) shows the safety boundaries given by experiment and predicted by theory: a~ = a/af, where of is the "normal" failure stress of a spoke and a is the (lower) stress when premature failure of the member occurs. Premature failure may arise through locally embrittled material, pre-existing flaws, or be produced by external causes (battle damage, accidents etc.). We may conclude from Fig. 8(b) that, if a partly broken structure is incapable of bearing the applied loading statically, transient overloading of members simply aids failures in other members and spreads the structural damage.
(a)
~/~rFiinXgd
outer
(b) 1.0
!
~q~ Safe
~
I
\
®
statically\
-
I Unsafe ~ ~ ! dynamically k
0.5
Unsafe e~ e~
o a_
I
0.6
0.7
*+\ , ~
M
,\ t
0.8
L
0
0.9
1.0
FIG. 8. (a) Schematic of the spoke wheel structure. (b) Safety boundaries for the spokedwheel structure taking into account the statistics of member failure load. (®) Computer stimulation: probability of static survival. ( ~ ) Computer simulation: probability of dynamic survival. (x) Experimental: probability of static survival. (+) Experimental: probability of dynamic survival. (---) Theoretical boundaries for zero spread of member failure load. (--) Theoretical boundaries for 6.1% S.D. of member failure load.
214
A . G . ATKINS
However, and this was the significant finding of [23, 24], the transient overloads may induce progressive fracture even though static calculations predict safety.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
A. Bignell, B. Peters and C. Pym, in Catastrophic Failures, p. 107, Open University Press (1977). N. Constantin, Private Communication. G. M. Boyd (ed.), Brittle Fracture in Steel Structures, Butterworths, London (1970). J. E. Gordon, The New Science of Strong Materials, Penguin Books, London (1968). A. G. Atkins and Y-W Mai, Elastic and Plastic Fracture, Ellis Horwood/Wiley, Chichester (1985/88). C. E. Turner, Mater. Sci. Engr. 11,275 (1973). C. Gurney and Y-W Mai, Engr. Fract. Mech. 4, 853 (1972). C. Gurney and K. M. Ngan, Proc. Roy. Soc. A325, 207 (1971). K. E. Puttick and A. G. Atkins, Int. J. Fract. 23, R51 (1983). G. R. Irwin, Syrup. on Effects of Temperature on Brittle Behaviour. Philadelphia, ASTM, 176 (1954). Y-W Mai and A. G. Atkins, Int. J. Mech. Sci. 20,437 (1978). A. R. Duffy, G. G. McClure, R. J. Eiber and W. A. Maxey, in Fracture, Vol. 5 (ed. Liebowitz), Academic Press, New York (1969). E. S. Folias, Int. J. Fract. 1, 104 (1965). G. M. Wilkowski, Private Communication. A. G. Atkins and Chert Zhong (To be published). Y-W Mai and A. G. Atkins, Int. J Mech. Sci. 17,673 (1975). A. G. Atkins, Int. J. Mech. Sci. 30, 173 (1988). T. E. Stanton and R. G. C. Batson, Proc. Inst. Civil Engrs 211, 67 (1921). J. G. Docherty, Engineering 133,645 (1932). J. G. Docherty, Engineering 139, 211 (1935). K. Hagiawara, H. Takenabe and H. Kawano, Int. J. Impact. Engr. 1,257 (1983). Y-W Mai and A. G. Atkins, J. Phys. D: Appl. Phys. 22, 48 (1989). A. J. Pretlove, M. Ramsden and A. G. Atkins, Int. J. Impact Engr. 11,539 (1991). A. J. Pretlove, in Proc. Int. Conf. Steel Struct., Budva, Yugoslavia, 749 (1986).