- Email: [email protected]
The difference between real and imaginary interactions in composite failure predictions. A reply to Mr Edge
test, the transverse tension strain in the compression ply at failure is well in excess of that required to fail a 90 ° coupon (typically by a ratio of about 2:1), even when it is assumed that failure occurs at 60% of the uniaxial level. This level of transverse tension strain will mean a reduction of the ply transverse stiffness to about 40% of the uncracked value. This in turn must represent a reduction in the support given to the fibres by the matrix. This mechanism may provide a better explanation of the apparent loss in compressive strength than that put forward by Hart-Smith 3, particularly as not many workers have reported fibre shear failures. There is also the question of how to interpret the uniaxial compression control result in the work of Swanson and Nelson 7, the authors offering a different view to that of Hart-Smith (compare Fig. 6 of Reference 7 with Fig. 22 of Reference 10). This critically affects the interpretation of the whole series of compression-tension results.
CONCL UDING REMA RKS Hart-Smith 3 correctly states that nearly all composite structures have so far been certified by test rather than analysis. This is not surprising, considering that, as he also tells us, less than 1% of such structures are governed by unnotched strength. The necessary models to obtain notched and damage tolerant data from unnotched properties, let alone unidirectional or constituent properties, simply did not exist. The Hart-Smith approach seems to be dominated by the 'realism' of this present empirical certification route. Its best role seems to lie in initial sizing calculations in conjunction with empirical notch and damage tolerance knockdown factors. While the unnotched laminate on an aircraft can and must be designed as far as possible to avoid loading combinations such as those shown in Fig. 1, it is not possible to do this in the critical region round, for example, a notch. The geometry and stress gradients, and above all the matrix-dominated through-thickness stresses induced by the in-plane loading discontinuities, mean that interactions of this type are largely unavoidable.
If we are to find a way out of this costly empirically based impassse, we need tools of greater theoretical depth. This is the basis of the current interest in micromechanical modelling, and why we are seeing renewed interest in phemonena such as interactions, features which HartSmith is so keen to dismiss from consideration.
REFERENCES 1 Hart-Smith, L. J. 'Should fibrous composite failure modes be interacted or superimposed?' Composites 24 (1993) pp 53 56
2 Hart-Smlth, L. J. 'An inherent fallacy in composite interaction failure curves' Composites 24 (1993) pp 523 524 3 Hart-Smith, L. J. 'The role of biaxial stresses in discriminating between meaningful and illusory composite failure theories' Composite Structures 25 (1993) pp 3 20 4 Sanders, R. C. and Grant, P. 'The strength of laminated plates under in-plane loading' BAe Warton Report no SOR(P)I30 (January 1982) 5 Anon'Failure criteria for an individual layer of a fibre reinforced composite laminate under in-plane loading' Engineering Sciences Data Unit (ESDU) Item 83014 (June 1986) 6 Fleck, N. A. Presentation to Workshop on Compressive Failure, Cambridge University Engineering Dept, July 1993
7 Swanson,S. R. and Nelson, M. 'Failure properties of carbon/epoxy laminates under tension-compression biaxial stress' Proc Third Japan US Conf on Composite Materials, Tokyo, 1986
8 Parry,T. V. and Wronski, A. S. 'Kinking and tensile, compressive and interlaminar shear failure in carbon fibre reinforced plastic beams tested in flexure"J Mater Sei 16 (1981) p 439 9 Parry,T. V. and Wronski, A. S. 'Kinking and compressive failure in uniaxially aligned carbon fibre tested under superposed hydrostatic pressure' J Mater Sci 17 (1982) p 3656 10 Hart-Smith, L. 3. 'A scientific approach to composite laminate strength prediction' Douglas Paper 8467. Presented to lOth A S T M Symp on Composite Materials: Testing and Design, San Fransisco, CA, USA. April 1990
AUTHOR Mr Edge is a Specialist in Composites in the Structures Unit (postcode W310C) of British Aerospace Defence, Military Aircraft Division, Warton Aerodrome, Preston PR4 lAX, UK.
The difference between real and imaginary interactions in composite failure predictions. A reply to Mr Edge L. J. HART-SMITH (McDonnell Douglas Corporation, USA) The author is indebted to Mr Edge, of British Aerospace, for starting the open debate he has sought for so long over composite failure theories. He hopes that others will now participate, too. Whether or not the author is 100%
correct in the failure model he has proposed for fibre/ polymer composites, there can be no doubt that examples he has cited have exposed a need for major changes in what is currently taught on this subject.
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While the comments by Edge I might suggest very different approaches to the subject from what the author has proposed, a little clarification will show that there are far fewer real differences than his critique suggests. There is, however, a vital difference in the focus of the work. The present author's works refer exclusively to the behaviour of plies within a multidirectional structural laminate, while every example cited in Edge's critique refers to unidirectional plies in isolation. This might not seem significant to some, because of the widespread but erroneous belief that there is no difference, but the distinction is vital. Failure predictions for fibredominated mechanisms transfer essentially unchanged from isolated to embedded layers of strong, stiff fibres set in a soft matrix. However, that is only rarely the case for matrix-dominated behaviour. The other fibres alter the residual thermal stresses in the matrix and inhibit the spread of any matrix microcracks in ways that cannot possibly be characterized by testing and analysis of unidirectional plies in isolation. It would appear that Edge may have misunderstood the author's criticisms of meaningless interaction curves between unrelated phenoment~. Every example Edge cites refers to an interaction between different stress components affecting the same mode of failure. Every example the author criticized concerned interactions between physically unrelated failure modes. Nothing in the author's works precludes the kinds of interaction Edge advocates. Indeed, the author's failure envelope depicted in Fig. 3 of Reference 2 shows clearly that, in the tension-compression (shear) quadrants, the strength of the fibres is governed by an interaction between the longitudinal and transverse stresses. There is no such interaction in the popular maximum-strain failure model, in which failure is predicted to be governed by the axial fibre stress alone or, sometimes, by a transverse matrix stress alone. The author has no trouble accepting that certain failure modes are affected by more than one stress and that their characterization must include an interaction between each of the relevant stress components. The author's concern is with abstract mathematical failure theories in which no attempt is made to select only those stresses that have an influence on a given failure mode, but to include them all, in physically oversimplified but mathematically unnecessarily complicated universal failure models. This shotgun approach leads to such nonsensical interactions as the predicted enhancement of biaxial tension strengths by a decrease in the unidirectional compressive strength, for example. The most telling condemnation concerns the prediction that, according to theories like that of Tsai and Wu 3, the biaxial compressive strength should be increased by decreasing the transverse tension strength, as illustrated in Reference 4. Edge's specific comments do not address the kinds of interaction the author has objected to. Turning now to the specific points raised by Edge in his riposte 1, the first concerns, the effect of in-plane shear stress on the uniaxial compressive strength of a unidirectional lamina. The author has no quarrel with the data presented, but makes two observations. First, the theory on which Edge's Equation (1) is based requires an assumed level of fibre waviness to match the 162
COMPOSITES. NUMBER 2. 1994
data. The model does not work for perfectly collimated unidirectional fibres. (An earlier micromechanical model, based on the absence of sufficient matrix shear stiffness to stabilize perfectly aligned fibres, predicted compressive strengths far higher than observed by test.) Edge does not cite the level of waviness implied by his Fig. 1, which is the only way of assessing how good these data are. Second, even if the phenomenon characterized by the data in his Fig. 1 is true for unidirectional laminates, no evidence is provided that it still occurs, at the same level of severity, in the presence of strong, stiff + 45 ° cross-plies that would inevitably react most of any applied shear loading. The resistance to out-of-plane buckling will be improved by thorough interspersion of the cross-plies, which precludes the kind of fibre wash known to be severe for thick filament-wound tubes. Using Edge's own logic, if the shear in the matrix reduces the support offered to the fibres and this in turn reduces the compressive loads the fibres could carry, stiffening the matrix by fibres at a different orientation in adjacent plies would be expected to suppress the mechanism described by Edge and possibly increase the compressive strength. One can make theoretical projections about the strengths of multidirectional laminates, on the assumption that the loss of strength depicted in Fig. 1 of Reference 1 would remain, for each 0° ply. But that is all they are - - predictions that can be validated only by equivalent tests of multidirectional laminates. Does any reader have such data? Is the reason for the author's plea in Reference 5 for biaxial testing at the laminate level as the only basis for assessing and rejecting composite failure models any clearer now? Any curve-fit model, no matter what it is meant to represent, can be validated only by checking against data that were not used to establish the coefficients. The data Edge provided in his Fig. 1 might well provide a useful interaction for a slightly different set of combined stresses acting on a laminate. Neither the author's theory nor any of those he has criticized accounts for the effect of transverse shear applied normal to the surface of the laminate, rather than in-plane, on the in-plane compressive strength. This is one situation the designer tries to avoid because he knows intuitively that there would be an appreciable loss of strength but, until now, he has been unable to characterize it. (It should be noted that micromechanical research on this topic is proceeding in the UK, the USA and in France. The longterm outcome that the author expects from this work is improved materials that are not subject to such limitations in compressive strength, not the continued use of more complex analyses of inferior materials.) Edge then turns to the enhancement of the axial compression strength of unidirectional fibres by radially symmetric transverse compression. As Edge acknowledges, the data he provided can also be explained by the author's theory for shear failure of the fibres. Indeed, the author's explanation may even be the more plausible. It should first be noted, however, that the data refer to triaxial compression only, not to the more usual case of biaxial in-plane compression, essentially in the absence of significant normal loads. The explanation of the data given here would imply that the enhancement of strength shown by Edge would not occur for only biaxial compression. The author's shear-failure model
Hart-Smith's analysis for triaxial compression k = + oh °max °c ET/E L
from as obvious, or even convincing, as he implies. First, why should the softer matrix, after it has yielded, provide better support for the fibres than it did when it was stiffer, before it yielded? See his Fig. 3 and the present Fig. 1, which show far more increase in fibre axial strength for the higher applied transverse stresses than for small transverse stresses. Second, given that his Fig. 1 implies that the fibre strength in compression is governed by the sum of the shear stresses in the matrix from the applied shear load and that caused by fibre waviness, how can triaxial compression (which applies no shear) increase the strength - - without contradicting the hypothesis on which his Fig. 1 was based? Even the older compression-failure models based on an interaction between perfectly straight fibres and the stiffness of the matrix would not seem to be affected by triaxial compression. If the present author's explanation of the phenomenon in Fig. 1 is incorrect, what is the right one?
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Work by Collings 6 on transversely compressed unidirectional carbon/epoxy laminates may shed some light on this issue. He recorded that, when matrix failures were prevented by external constraints in the test fixture, the fibres failed by shear--just as this author's theory would predict.
15 - O 1 Linear
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behaviour
I
I
I
I
I
I
50
100
150
200
250
300
Hydrostatic pressure (MN m-2} Fig 1. Effect of hydrostatic stress on axial compressive strength of unidirectional composite laminae
would predict failure when the following differences developed between the longitudinal, transverse and normal strains. (eL - eT) = (EL -- EN) = (1 + VLT)e0= constant Edge only speculates that 'when reliable biaxial compression~compression test results are available, the strain at failure m a y p r o v e [italics by the present author] to be significantly greater than the uniaxial value'. He admits not having any such data today, which undermines his implication that the author's theory is inadequate in this regard. Using the triaxial compression data supplied by Edge, the author's theory would predict that the axial compressive strength of the fibres should be enhanced because the radial transverse compression decreases the difference between the longitudinal and transverse stresses in the fibres, permitting an increase in axial stress to regenerate the same difference as occurred when there was no transverse stress acting in any direction. The axial strength would be raised by an increment proportional to (but not equal to) the transverse stress, because of the orthotropy of the fibres. The strain-hardening effect shown in Edge's Fig. 3 may well be the result of the non-linear softening of the matrix once it has been loaded past its proportional limit. The author's theory would predict this, as shown in Fig. 1 here. The coefficient k is essentially a constant (the value 0.5 was used in preparing Fig. 1), defined by various Poisson's ratios and EL and Ex, the longitudinal and transverse moduli, respectively. Edge's assertion that 'the physical explanation seems intuitively obvious, as the increased transverse stress can be expected to inhibit fibres bowing and buckling' is far
Edge's counter-proposal is that the transverse compression inhibits fibre bowing and buckling. However, the author's understanding of buckling equations would suggest that only the transverse stiffness could influence the axial buckling stress, not the transverse stress, p e r se. To be fair, it is true that a micromechanical analysis might suggest that the fibres could be destabilized by Poisson-induced gaps opening up between the fibres and the surrounding matrix, gaps that could close under radial compression. This would create a situation in which the axial strength was enhanced by a transverse stress rather than stiffness. Conversely, accounting for the radial shrinkage of the matrix around the fibres during cooldown after curing at elevated temperature might well ensure that such gaps would never exist, except at elevated temperature. It might take a whole series of tests to correctly explain the phenomena reported in Fig. 3 of Reference 1. The final issue addressed in Edge's article is the combination of axial compression of fibres with transverse tension loads. His suggestion--that a superior explanation of the cut-offs the author has proposed might again be the destabilizing of the fibres because of matrix cracking rather than that the strength of the fibres could be influenced by biaxial stresses--had already been proposed to the author by American composite experts at the MIL-HDBK-17 meeting in Portland, Oregon, in June 1993. A separate article has been prepared in response to this suggestion but, briefly, following this alternative hypothesis through to its unavoidable conclusion would put an end to the use of advanced composites in aircraft structures. As the author has stated 7, once real matrix failures actually do dominate the behaviour of composite laminates, neither their strength nor their life can be predicted. A reassessment of this position in the light of the suggestion by Edge and others makes it clear that the author had then grossly underestimated the magnitude of the problem. Far from enhancing the failure envelope as the matrix failed progressively, as is normally assumed to be the case,
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under this hypothesis the failure envelope would shrink with every foray into the transverse tension domain, restricting the use of composites to only a single application of load(s). Worse, it calls into question the very definition of a failure envelope, which should be the demarcation between undamaged and broken material (or structures). If structurally significant damage is permitted within an alleged failure envelope, predicting the failure of a structure subject to multiple loads in varying sequences and environments is not possible without knowing the complete load/environment history of the structure. Merely changing the sequence of application the same loads could then change the strength of the structure. Even if some designer were sufficiently misguided to use such an inferior composite material on his product, no chief of stress would ever endorse the use of a material for which the stress analysis could not be completed within the life span of the analyst. In conclusion, the author would like to reassure Mr Edge that he is very much in favour of including in composite strength predictions any and all physically real interactions between stress components. What he has objected to is the m a n y abstract mathematical interactions in published theories that can easily be shown to be false because they yield absurd predictions. Also, the theme of Reference 4 is that each distinct failure mode must be characterized independently and the results superimposed, not interacted. Verbal feedback received about the first of the author's Designer's Corner articles has indicated widespread understanding of and support for this position. Finally, the author feels obliged to point out his concern that, by emphasizing interactions at the unidirectional lamina level, Edge may be overlooking the very different interactions that exist when the same lamina is part of a multidirectional laminate. To imply that the
164
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characterization of matrix cracking for unidirectional laminae in isolation is also applicable in the presence of adjacent layers of fibres in different directions is equivalent to suggesting that crack arrestors have no effect on the residual strength of metallic skin-stringer aircraft structures.
REFERENCES 1 2 3 4 5
Edge, E. C. 'Does transverse and shear loading affect the compression strength of unidirectional CFC? A reply to Dr HartSmith' Composites 25 (1994) pp 159-161 Hart-Smith, L. J. 'The truncated maximum-strain composite failure model' Composites 24 (1993) pp 587-591 Tsai, S. W. and Wu, E. M. 'A general theory of strength for anisotropic materials' J Composite Mater 5 (1974) pp 58-80 Hart-Smith, L. J. 'Should fibrous composite failure modes be interacted or superimposed?' Composites 24 (1993) pp 53-55 Hart-Smith,L. J. 'The role of biaxial stresses in discriminating between meaningful and illusory composite failure theories' McDonnell Douglas Paper MDC 91K-0077. Presented to 9th DOD/NASA/FAA Conf on Fibrous Composites in Structural Design, Lake Tahoe, NV, 4 7 November 1991. Published in Composite Structures 25 (1993) pp 3-20
6 7
Coilings, T. A. 'Transverse compression behaviour of unidirectional carbon fibre reinforced plastics' Composites 5 (May 1974) pp 108-116 Hart-Smith,L. J. 'Some observations on the analysis of in-plane matrix failures in fibrous composite laminates' Douglas Paper 8558 Presented to 4th ASTM Syrup on Composite Materials. Fatigue and Fracture, Indianapolis, IN, ~ 7 May 1991. In Proceedings, ASTM STP 1156 edited by W. W. Stinchcomb and N. E. Ashbaugh (American Society for Testing and Materials, Philadelphia, 1993) pp 363-380
AUTHOR Dr Hart-Smith is a McDonnell Douglas Corporation Fellow with McDonnell Douglas Aerospace in Long Beach, CA, USA.
CONCL UDING REMA RKS Hart-Smith 3 correctly states that nearly all composite structures have so far been certified by test rather than analysis. This is not surprising, considering that, as he also tells us, less than 1% of such structures are governed by unnotched strength. The necessary models to obtain notched and damage tolerant data from unnotched properties, let alone unidirectional or constituent properties, simply did not exist. The Hart-Smith approach seems to be dominated by the 'realism' of this present empirical certification route. Its best role seems to lie in initial sizing calculations in conjunction with empirical notch and damage tolerance knockdown factors. While the unnotched laminate on an aircraft can and must be designed as far as possible to avoid loading combinations such as those shown in Fig. 1, it is not possible to do this in the critical region round, for example, a notch. The geometry and stress gradients, and above all the matrix-dominated through-thickness stresses induced by the in-plane loading discontinuities, mean that interactions of this type are largely unavoidable.
If we are to find a way out of this costly empirically based impassse, we need tools of greater theoretical depth. This is the basis of the current interest in micromechanical modelling, and why we are seeing renewed interest in phemonena such as interactions, features which HartSmith is so keen to dismiss from consideration.
REFERENCES 1 Hart-Smith, L. J. 'Should fibrous composite failure modes be interacted or superimposed?' Composites 24 (1993) pp 53 56
2 Hart-Smlth, L. J. 'An inherent fallacy in composite interaction failure curves' Composites 24 (1993) pp 523 524 3 Hart-Smith, L. J. 'The role of biaxial stresses in discriminating between meaningful and illusory composite failure theories' Composite Structures 25 (1993) pp 3 20 4 Sanders, R. C. and Grant, P. 'The strength of laminated plates under in-plane loading' BAe Warton Report no SOR(P)I30 (January 1982) 5 Anon'Failure criteria for an individual layer of a fibre reinforced composite laminate under in-plane loading' Engineering Sciences Data Unit (ESDU) Item 83014 (June 1986) 6 Fleck, N. A. Presentation to Workshop on Compressive Failure, Cambridge University Engineering Dept, July 1993
7 Swanson,S. R. and Nelson, M. 'Failure properties of carbon/epoxy laminates under tension-compression biaxial stress' Proc Third Japan US Conf on Composite Materials, Tokyo, 1986
8 Parry,T. V. and Wronski, A. S. 'Kinking and tensile, compressive and interlaminar shear failure in carbon fibre reinforced plastic beams tested in flexure"J Mater Sei 16 (1981) p 439 9 Parry,T. V. and Wronski, A. S. 'Kinking and compressive failure in uniaxially aligned carbon fibre tested under superposed hydrostatic pressure' J Mater Sci 17 (1982) p 3656 10 Hart-Smith, L. 3. 'A scientific approach to composite laminate strength prediction' Douglas Paper 8467. Presented to lOth A S T M Symp on Composite Materials: Testing and Design, San Fransisco, CA, USA. April 1990
AUTHOR Mr Edge is a Specialist in Composites in the Structures Unit (postcode W310C) of British Aerospace Defence, Military Aircraft Division, Warton Aerodrome, Preston PR4 lAX, UK.
The difference between real and imaginary interactions in composite failure predictions. A reply to Mr Edge L. J. HART-SMITH (McDonnell Douglas Corporation, USA) The author is indebted to Mr Edge, of British Aerospace, for starting the open debate he has sought for so long over composite failure theories. He hopes that others will now participate, too. Whether or not the author is 100%
correct in the failure model he has proposed for fibre/ polymer composites, there can be no doubt that examples he has cited have exposed a need for major changes in what is currently taught on this subject.
C O M P O S I T E S . N U M B E R 2 . 1994
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While the comments by Edge I might suggest very different approaches to the subject from what the author has proposed, a little clarification will show that there are far fewer real differences than his critique suggests. There is, however, a vital difference in the focus of the work. The present author's works refer exclusively to the behaviour of plies within a multidirectional structural laminate, while every example cited in Edge's critique refers to unidirectional plies in isolation. This might not seem significant to some, because of the widespread but erroneous belief that there is no difference, but the distinction is vital. Failure predictions for fibredominated mechanisms transfer essentially unchanged from isolated to embedded layers of strong, stiff fibres set in a soft matrix. However, that is only rarely the case for matrix-dominated behaviour. The other fibres alter the residual thermal stresses in the matrix and inhibit the spread of any matrix microcracks in ways that cannot possibly be characterized by testing and analysis of unidirectional plies in isolation. It would appear that Edge may have misunderstood the author's criticisms of meaningless interaction curves between unrelated phenoment~. Every example Edge cites refers to an interaction between different stress components affecting the same mode of failure. Every example the author criticized concerned interactions between physically unrelated failure modes. Nothing in the author's works precludes the kinds of interaction Edge advocates. Indeed, the author's failure envelope depicted in Fig. 3 of Reference 2 shows clearly that, in the tension-compression (shear) quadrants, the strength of the fibres is governed by an interaction between the longitudinal and transverse stresses. There is no such interaction in the popular maximum-strain failure model, in which failure is predicted to be governed by the axial fibre stress alone or, sometimes, by a transverse matrix stress alone. The author has no trouble accepting that certain failure modes are affected by more than one stress and that their characterization must include an interaction between each of the relevant stress components. The author's concern is with abstract mathematical failure theories in which no attempt is made to select only those stresses that have an influence on a given failure mode, but to include them all, in physically oversimplified but mathematically unnecessarily complicated universal failure models. This shotgun approach leads to such nonsensical interactions as the predicted enhancement of biaxial tension strengths by a decrease in the unidirectional compressive strength, for example. The most telling condemnation concerns the prediction that, according to theories like that of Tsai and Wu 3, the biaxial compressive strength should be increased by decreasing the transverse tension strength, as illustrated in Reference 4. Edge's specific comments do not address the kinds of interaction the author has objected to. Turning now to the specific points raised by Edge in his riposte 1, the first concerns, the effect of in-plane shear stress on the uniaxial compressive strength of a unidirectional lamina. The author has no quarrel with the data presented, but makes two observations. First, the theory on which Edge's Equation (1) is based requires an assumed level of fibre waviness to match the 162
COMPOSITES. NUMBER 2. 1994
data. The model does not work for perfectly collimated unidirectional fibres. (An earlier micromechanical model, based on the absence of sufficient matrix shear stiffness to stabilize perfectly aligned fibres, predicted compressive strengths far higher than observed by test.) Edge does not cite the level of waviness implied by his Fig. 1, which is the only way of assessing how good these data are. Second, even if the phenomenon characterized by the data in his Fig. 1 is true for unidirectional laminates, no evidence is provided that it still occurs, at the same level of severity, in the presence of strong, stiff + 45 ° cross-plies that would inevitably react most of any applied shear loading. The resistance to out-of-plane buckling will be improved by thorough interspersion of the cross-plies, which precludes the kind of fibre wash known to be severe for thick filament-wound tubes. Using Edge's own logic, if the shear in the matrix reduces the support offered to the fibres and this in turn reduces the compressive loads the fibres could carry, stiffening the matrix by fibres at a different orientation in adjacent plies would be expected to suppress the mechanism described by Edge and possibly increase the compressive strength. One can make theoretical projections about the strengths of multidirectional laminates, on the assumption that the loss of strength depicted in Fig. 1 of Reference 1 would remain, for each 0° ply. But that is all they are - - predictions that can be validated only by equivalent tests of multidirectional laminates. Does any reader have such data? Is the reason for the author's plea in Reference 5 for biaxial testing at the laminate level as the only basis for assessing and rejecting composite failure models any clearer now? Any curve-fit model, no matter what it is meant to represent, can be validated only by checking against data that were not used to establish the coefficients. The data Edge provided in his Fig. 1 might well provide a useful interaction for a slightly different set of combined stresses acting on a laminate. Neither the author's theory nor any of those he has criticized accounts for the effect of transverse shear applied normal to the surface of the laminate, rather than in-plane, on the in-plane compressive strength. This is one situation the designer tries to avoid because he knows intuitively that there would be an appreciable loss of strength but, until now, he has been unable to characterize it. (It should be noted that micromechanical research on this topic is proceeding in the UK, the USA and in France. The longterm outcome that the author expects from this work is improved materials that are not subject to such limitations in compressive strength, not the continued use of more complex analyses of inferior materials.) Edge then turns to the enhancement of the axial compression strength of unidirectional fibres by radially symmetric transverse compression. As Edge acknowledges, the data he provided can also be explained by the author's theory for shear failure of the fibres. Indeed, the author's explanation may even be the more plausible. It should first be noted, however, that the data refer to triaxial compression only, not to the more usual case of biaxial in-plane compression, essentially in the absence of significant normal loads. The explanation of the data given here would imply that the enhancement of strength shown by Edge would not occur for only biaxial compression. The author's shear-failure model
Hart-Smith's analysis for triaxial compression k = + oh °max °c ET/E L
from as obvious, or even convincing, as he implies. First, why should the softer matrix, after it has yielded, provide better support for the fibres than it did when it was stiffer, before it yielded? See his Fig. 3 and the present Fig. 1, which show far more increase in fibre axial strength for the higher applied transverse stresses than for small transverse stresses. Second, given that his Fig. 1 implies that the fibre strength in compression is governed by the sum of the shear stresses in the matrix from the applied shear load and that caused by fibre waviness, how can triaxial compression (which applies no shear) increase the strength - - without contradicting the hypothesis on which his Fig. 1 was based? Even the older compression-failure models based on an interaction between perfectly straight fibres and the stiffness of the matrix would not seem to be affected by triaxial compression. If the present author's explanation of the phenomenon in Fig. 1 is incorrect, what is the right one?
~,t/~/ II
20
c~
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19 •
,
.
I
Expertmental ana theoretical work of Wronski and Parry, cited by Edge ~ . . . x . x ~ / '
18
Q.
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,~/"\~'v /~,x,
/"~/
¢)
//%\
t" /
.
~.~
17
J
i/~
d
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~~
J I~3
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Work by Collings 6 on transversely compressed unidirectional carbon/epoxy laminates may shed some light on this issue. He recorded that, when matrix failures were prevented by external constraints in the test fixture, the fibres failed by shear--just as this author's theory would predict.
15 - O 1 Linear
>
0
behaviour
I
I
I
I
I
I
50
100
150
200
250
300
Hydrostatic pressure (MN m-2} Fig 1. Effect of hydrostatic stress on axial compressive strength of unidirectional composite laminae
would predict failure when the following differences developed between the longitudinal, transverse and normal strains. (eL - eT) = (EL -- EN) = (1 + VLT)e0= constant Edge only speculates that 'when reliable biaxial compression~compression test results are available, the strain at failure m a y p r o v e [italics by the present author] to be significantly greater than the uniaxial value'. He admits not having any such data today, which undermines his implication that the author's theory is inadequate in this regard. Using the triaxial compression data supplied by Edge, the author's theory would predict that the axial compressive strength of the fibres should be enhanced because the radial transverse compression decreases the difference between the longitudinal and transverse stresses in the fibres, permitting an increase in axial stress to regenerate the same difference as occurred when there was no transverse stress acting in any direction. The axial strength would be raised by an increment proportional to (but not equal to) the transverse stress, because of the orthotropy of the fibres. The strain-hardening effect shown in Edge's Fig. 3 may well be the result of the non-linear softening of the matrix once it has been loaded past its proportional limit. The author's theory would predict this, as shown in Fig. 1 here. The coefficient k is essentially a constant (the value 0.5 was used in preparing Fig. 1), defined by various Poisson's ratios and EL and Ex, the longitudinal and transverse moduli, respectively. Edge's assertion that 'the physical explanation seems intuitively obvious, as the increased transverse stress can be expected to inhibit fibres bowing and buckling' is far
Edge's counter-proposal is that the transverse compression inhibits fibre bowing and buckling. However, the author's understanding of buckling equations would suggest that only the transverse stiffness could influence the axial buckling stress, not the transverse stress, p e r se. To be fair, it is true that a micromechanical analysis might suggest that the fibres could be destabilized by Poisson-induced gaps opening up between the fibres and the surrounding matrix, gaps that could close under radial compression. This would create a situation in which the axial strength was enhanced by a transverse stress rather than stiffness. Conversely, accounting for the radial shrinkage of the matrix around the fibres during cooldown after curing at elevated temperature might well ensure that such gaps would never exist, except at elevated temperature. It might take a whole series of tests to correctly explain the phenomena reported in Fig. 3 of Reference 1. The final issue addressed in Edge's article is the combination of axial compression of fibres with transverse tension loads. His suggestion--that a superior explanation of the cut-offs the author has proposed might again be the destabilizing of the fibres because of matrix cracking rather than that the strength of the fibres could be influenced by biaxial stresses--had already been proposed to the author by American composite experts at the MIL-HDBK-17 meeting in Portland, Oregon, in June 1993. A separate article has been prepared in response to this suggestion but, briefly, following this alternative hypothesis through to its unavoidable conclusion would put an end to the use of advanced composites in aircraft structures. As the author has stated 7, once real matrix failures actually do dominate the behaviour of composite laminates, neither their strength nor their life can be predicted. A reassessment of this position in the light of the suggestion by Edge and others makes it clear that the author had then grossly underestimated the magnitude of the problem. Far from enhancing the failure envelope as the matrix failed progressively, as is normally assumed to be the case,
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under this hypothesis the failure envelope would shrink with every foray into the transverse tension domain, restricting the use of composites to only a single application of load(s). Worse, it calls into question the very definition of a failure envelope, which should be the demarcation between undamaged and broken material (or structures). If structurally significant damage is permitted within an alleged failure envelope, predicting the failure of a structure subject to multiple loads in varying sequences and environments is not possible without knowing the complete load/environment history of the structure. Merely changing the sequence of application the same loads could then change the strength of the structure. Even if some designer were sufficiently misguided to use such an inferior composite material on his product, no chief of stress would ever endorse the use of a material for which the stress analysis could not be completed within the life span of the analyst. In conclusion, the author would like to reassure Mr Edge that he is very much in favour of including in composite strength predictions any and all physically real interactions between stress components. What he has objected to is the m a n y abstract mathematical interactions in published theories that can easily be shown to be false because they yield absurd predictions. Also, the theme of Reference 4 is that each distinct failure mode must be characterized independently and the results superimposed, not interacted. Verbal feedback received about the first of the author's Designer's Corner articles has indicated widespread understanding of and support for this position. Finally, the author feels obliged to point out his concern that, by emphasizing interactions at the unidirectional lamina level, Edge may be overlooking the very different interactions that exist when the same lamina is part of a multidirectional laminate. To imply that the
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characterization of matrix cracking for unidirectional laminae in isolation is also applicable in the presence of adjacent layers of fibres in different directions is equivalent to suggesting that crack arrestors have no effect on the residual strength of metallic skin-stringer aircraft structures.
REFERENCES 1 2 3 4 5
Edge, E. C. 'Does transverse and shear loading affect the compression strength of unidirectional CFC? A reply to Dr HartSmith' Composites 25 (1994) pp 159-161 Hart-Smith, L. J. 'The truncated maximum-strain composite failure model' Composites 24 (1993) pp 587-591 Tsai, S. W. and Wu, E. M. 'A general theory of strength for anisotropic materials' J Composite Mater 5 (1974) pp 58-80 Hart-Smith, L. J. 'Should fibrous composite failure modes be interacted or superimposed?' Composites 24 (1993) pp 53-55 Hart-Smith,L. J. 'The role of biaxial stresses in discriminating between meaningful and illusory composite failure theories' McDonnell Douglas Paper MDC 91K-0077. Presented to 9th DOD/NASA/FAA Conf on Fibrous Composites in Structural Design, Lake Tahoe, NV, 4 7 November 1991. Published in Composite Structures 25 (1993) pp 3-20
6 7
Coilings, T. A. 'Transverse compression behaviour of unidirectional carbon fibre reinforced plastics' Composites 5 (May 1974) pp 108-116 Hart-Smith,L. J. 'Some observations on the analysis of in-plane matrix failures in fibrous composite laminates' Douglas Paper 8558 Presented to 4th ASTM Syrup on Composite Materials. Fatigue and Fracture, Indianapolis, IN, ~ 7 May 1991. In Proceedings, ASTM STP 1156 edited by W. W. Stinchcomb and N. E. Ashbaugh (American Society for Testing and Materials, Philadelphia, 1993) pp 363-380
AUTHOR Dr Hart-Smith is a McDonnell Douglas Corporation Fellow with McDonnell Douglas Aerospace in Long Beach, CA, USA.