Application of the strain invariant failure theory (SIFT) to metals and fiber–polymer composites

Abstract

The strain invariant failure theory (SIFT) model, developed to predict the onset of irreversible damage of fiber–polymer composite laminates, may be also applied to metals. Indeed, it can be applied to all solid materials. Two initial failure mechanisms are considered – distortion and dilatation. The author's experiences are confined to the structures of transport aircraft; phase changes in metals and self-destruction of laminates during curing are not covered. Doing so would need additional material properties, and probably a different failure theory. SIFT does not cover environmental attack on the interface between fibers and resin; it covers only cohesive failures within the fibers or resin, or within a homogeneous piece of metal. In the SIFT model, each damage mechanism is characterized by its own critical value of a strain invariant. Each mechanism dominates its own portion of the strain domain; there is no interaction between them. Application of SIFT to metals is explained first. Fiber–polymer composites contain two discrete constituents; each material must be characterized independently by its own two invariants. This is why fiber–polymer composites need four invariants whereas metals require only two. There is no such thing as a composite material, only composites of materials. The “composite materials” must not be modeled as homogeneous anisotropic solids because it is then not even possible to differentiate between fiber and matrix failures. The SIFT model uses measured material properties; it does not require that half of them be arbitrarily replaced by unmeasurable properties to fit laminate test data, as so many earlier composite failure criteria have. The biggest difference in using SIFT for metals and fiber-reinforced materials is internal residual thermal and moisture absorption stresses created by the gross dissimilarity in properties between embedded fibers and thermoset resin matrices. These residual stresses consume so much of the strength of unreinforced polymers for typical thermoset resins cured at high temperature, like epoxies, that little strength is available to resist mechanical loads. (Thermoplastic polymers suffer far less in this regard.) The paper explains how SIFT is used via worked examples, which demonstrate the kind of detailed information that SIFT analyses can generate.

Keywords:

• metal
• fiber–polymer composite
• SIFT model
• dilatational and distortional failure mechanism
• strength
• laminate strength
• matrix failure
• fiber failure
• uniaxial load
• biaxial load
• residual thermal stress
• chemical shrinkage
• elastic constant
• failure envelope

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Corrigendum

Acknowledgements

This paper has been greatly improved by the diligent and extensive comments from the referees. They have worked tirelessly to enable the SIFT model to be understood by as wide an audience as possible. The author appreciates this help very much. He also takes this opportunity to thank a colleague, Tom Wu, who prepared the analyses and illustrations in Figures 13, 14, 18, 19, and 21. Those people familiar with the author's decades-long attempts to bring enlightenment to the process of predicting the strength of fiber–polymer composites will be aware of some of the many obstacles he has had to face. What is not so well known is the encouragement and helpful advice from a great many people all around the world who have inspired the author to continue in the face of adversity. There are too many such helpers to name them all personally, but rest assured, your help has been appreciated. The help from Prof. Kelly has been particularly appreciated. Something else that has helped immeasurably, in the sense that no progress could have been made without it, is the great amount of test data, generated over 40 years or so, that has been needed to provide sanity checks on the approaches and clues to orientate the thinking. It is unfortunate that much of the test data is as unreliable as many of the existing failure theories, but even bad data serves a useful role in highlighting a good data. It is also very significant that both of the inventors of the SIFT model worked at aircraft factories, where they received many helpful peripheral inputs from the composite structures built there that would not have been available in a purely academic environment. Finally, it should be noted that, while not the result of a large team effort, SIFT is the result of a collaboration between two inquisitive research engineer-scientists who would have been unlikely to have ever completed the task on their own. The author supplied the fiber-failure criterion and most of the physics, while Gosse provided the matrix-failure criterion, did all of the numerical analyses, the test specimen selection and development – and all of the work since our initial collaboration. Thank you, Jon Gosse, for providing closure to the author's many years of study in this field. And I am honored to be the person who helped you bring your own work to fruition. We both owe a debt to Gosse's colleague, S. Christensen, for the fine experimental work that provided the necessary critical values of the invariants.

Notes

Notes

1. See 11,12 for discussions of some of these shortcomings, as well as arguments in favor of the hypothesis that the fibers were failing by distortion (then called shear).

2. σ _T is the measured lamina-level transverse stress at failure, while τ is the measured lamina-level in-plane shear stress at failure.

3. This inevitable interaction contradicts a very widespread belief amongst composites theoreticians that the longitudinal strength of the fibers is determined uniquely by the longitudinal stress. Performing tests only on isolated fibers cannot possibly tell whether or not their strength is sensitive to any transverse loads that may be applied when the fiber is embedded in a multi-directional laminate. In the US aerospace industry, the empirical truncated-maximum-strain model has been widely accepted, and applied, for years. The failure envelope for this model has cut-offs in the tension-compression (shear) quadrants. But academia still insists that carbon fibers cannot possibly fail by distortion (called shear by the author in his earlier works), and there is still an unwillingness to accept that the compressive strength of carbon fibers should be the same as the tensile strength, because such an acknowledgement would invalidate much of the test data used to characterize the strength of composite laminas. The measured differences are actually due to problems with the experimental technique and the difficulty of actually making unidirectional laminates with truly straight fibers. The acceptance of superior test fixtures and the backing out of the data from tests on cross-plied laminates is beginning to create awareness that the strengths should be the same, but there is as yet, no universal acceptance of the commonality of fiber failure mechanism.

4. There is some ambiguity about the critical value for J ₁ after yielding, for Material 1, because the cross-section then involves both dilatation failures and distortional yielding. This means that the present interpretation of the ultimate strain-to-failure may be an over-simplification. On p. 368 of 31, Timoshenko cites the following explanation of what is actually happening:

• In studying the phenomena of fracture, L. Prandtl suggested (in 1907) that two types of fracture are identified: (1) cohesive or brittle fracture perpendicular to the tensile force and (2) shear fracture. In testing cylindrical specimens of structural steel, we obtain the so-called cup-and-cone fracture. In the center of this, the surface is perpendicular to the axis of the specimen and is of the brittle type. The outer portion of the fracture forms a conical surface inclined at approximately 45 deg to the direction of tension; this represents shear fracture. (In 1928) P. Ludwik noticed that fracture starts at the center of the minimum cross section of the neck and spreads, as a brittle fracture, over the middle portion of the cross section, while the outer material continues to stretch plastically.

This might mean that Gosse and the author were not the first researchers to develop SIFT, only the first to apply it to fiber–polymer composites. Timoshenko does not indicate whether Prandtl was merely drawing a qualitative distinction between what he clearly recognized were two different failure mechanisms, or if he actually established a quantitative failure criterion based on that understanding. What this description does show, however, is that the governing failure mechanism (as distinct from the deformation process), after yielding, is eventually one of dilatation. Therefore it is possible to locate the entire dilatational failure lines in Figure 3 where they are, from any single reference point on each of them, since the strains in the figure are macroscopic strains, as could be measured with a strain gauge or extensometer, not microscopic strains. Since the material is homogeneous, these would be the same if the stress distribution was uniform, as it can be assumed to be elsewhere on this diagram. It is the microscopic strain at this point that is unknown. The explanation given by Timoshenko makes it unlikely that SIFT could ever contribute to establishing the stress at ultimate failure, but there are existing theories of plasticity that can help in that regard. All that SIFT can do is to identify when to stop the yielding process at the onset of fracture. It is also quite likely that the coarse value of J ₁ developed in this manner would not suffice for the task suggested in the next section of analyzing what happens when cracks first initiate, since the local three-dimensional stress field would differ from that in a rod under tension. On the other hand, because SIFT is a strain-based theory, and is not tied in any way to the actual load on the rod, it might well be possible to establish the correct value of J ₁ in spite of the complicated stress state where the rod is merely yielding and hasn’t yet failed. The author does not know, but suggests that some fracture mechanics experts might be interested in doing parametric studies, based on a range of assumed critical SIFT invariants, to see whether or not SIFT can do what fracture mechanics has not – to predict when a non-pre-existing crack first appears.

5. Interfacial failures should be included as an additional possible failure mechanism, if necessary, that would not be covered by a SIFT-type analysis, but by some other means that addressed environmental degradation. However, the effect of the first local interfacial failures is to alleviate the local stress concentrations that could cause them, permitting higher strains prior to failure in both the fibers and the resin matrix. This effect is automatically included in measured reference lamina strengths. Also, it is less prevalent than is commonly believed, in aerospace structures, at least, because of the intense residual radial compressive stresses developed around the fibers, in thermoset resins that are cured at elevated temperatures. These internal stresses create significant resistance to interfacial failures by friction along the length and a benign pre-stress interfacial stress under transverse loads. That would not be the case with room-temperature-cured resins, but chemical shrinkage of the resin would also result in the resin being shrunk around the fibers.

6. Drying out a laminate on the surface, while the interior is still moist can create tensile residual stresses near the surfaces. The absorbed moisture, otherwise, tends to alleviate some portion of the residual thermal stresses.

7. If this is known not to be the case, one can integrate this contraction using the variable coefficient of thermal expansion. For typical aerospace engineering applications, this refinement is not normally needed.

8. StressCheck™ is a finite-element analysis software product developed and supported by ESRD, Inc. of St. Louis, Missouri. It is one of the first commercially available FEA products to utilize the p-version of the finite-element method.

9. Henceforth, the word ‘composites’ will be used here in place of the ambiguous more common reference to ‘composite materials’.

10. There is a necessary caveat to the extent that the in situ measured critical invariant for distortional failure of the fibers might be modified by the choice of matrix or the sizing on the fiber. This would need to be checked.

11. The appropriate level of conservatism for designs is then introduced by rational knock-down and safety factors with respect to accurate material properties instead of allowing the scatter between inferior test results to achieve some uncontrolled reduction in design allowables, which may or may not be the same.

12. While outside the scope of the issues discussed here, it should be noted that the test results are extremely sensitive to the load-introduction tabs at each end of the coupon. This was discussed at length by the MIL-HDBK-17 committee because of its importance. The author's contributions on this issue are recorded in 19, where references to other such contributions can be found. The critical values established for the SIFT strain invariants can be no better than the quality of the tests used to establish them.

13. The author learned of an intriguing alternative coupon to eliminate fiber wash at a presentation by an engineer from Dassault Aviation at a MIL-HDBK-17 meeting. Instead of using 90° plies to stabilize the 0° fibers, they added skewed fibers oriented at ±θ°, where θ was selected to create a laminate having the same Poisson's ratio as the unidirectional laminas. Both techniques achieved far greater strengths than were measured on all-0°, coupons and both gave the same answer, The use of 90°, cross plies is more compatible with establishing the critical strain invariants when the basic composite layers are woven fabrics rather than unidirectional tapes. But even then, the test data must be converted analytically to equivalent unidirectional lamina data before it is possible to extract the critical invariant values for fiber and resin constituents.

14. The author's counterparts in other US aerospace companies had also recognized that there was a need to truncate the lamina-level failure envelope in the tension-compression quadrants. The truncated maximum-strain failure model was actually developed empirically, simultaneously and independently, at several US aerospace companies, as was revealed by discussions at the MIL-HDBK-17 committee meetings. Some such truncations were sometimes used for entirely different reasons associated with matrix failures since only Douglas Aircraft initially had credible test data with which to quantify the magnitude of the cut-off for fiber-dominated failures.

15. It was another Boeing colleague, Jeff Wollschlager, in St. Louis, who first used StressCheck™ and the unit cell concept for the micromechanical amplification factors needed to apply the SIFT model. But it was Gosse who had already established the sensitivities to fiber volume fraction and identified the location of the critical sites between the fibers, using his own method of finite-element analysis, with multiple fibers to allow a distinction to be made between conditions in the interior of laminates and at their free edges. These established that the interior was the more critical because the stresses had to all drop to zero on free edges. This was what allowed Wollschlager to use a greatly simplified finite-element model.

16. The sample solutions shown here are a selection from a very thorough later cataloguing of all of the standard cases by another colleague at Boeing, Tom Wu, at Huntington Beach.

17. This last requirement of flatness is relaxed on any truly free surface of a structure. It is required only for the interior. However, the stress-free condition applies in both cases, until the thermal strains have been combined with whatever mechanical strains were needed to make the unit cell match the size specified by the lamina-level analysis.

18. Gosse and others of his colleagues have been analyzing the spread of damage in composites, but the work that has been published shows that it contains a lot more than is needed to explain the onset of irreversible damage using the SIFT model as it was originally conceived.