Some Principles of Brittle Material Design

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1 75-GT-112 Copyright 1975 by ASME $3.00 PER COPY $1.00 TO ASME MEMBERS The Society shall not be responsible for statements or opinions advanced in papers or in discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printea only if the paper is published in an ASME journal or Proceedings. Released for general publication upon presentation. Full credit should be given to ASME, the Technical Division, and the author(s). Some Principles of Brittle Material Design W. H. DUKES Assistant Director of Engineering, Bell Aerospace Co., New Orleans, La. This paper presents the basic principles which must be followed in designing structures to utilize completely brittle materials to carry tensile loads or stresses in an efficient and reliable manner. Brittleness introduces a need for highly refined stress analysis methods. Brittleness also requires the statistical treatment of material strength because of sensitivity to microscopic flaws which are distributed statistically in size and shape. It proves, in fact, to be impractical to do this and still achieve the extremely low probabilities of failure expected of a primary structure. The use of statistical methods to describe the material strength properties should also be extended to the statistical treatment of loads applied to the component, because it is the overall probability of failure of the component which is significant. These methods must also be extended to cover such effects as nonuniform stress distributions, stress gradient and fatigue, and methods for doing this are suggested. Finally, the use of statistical methods raises problems in the verification of a design since the successful test of a small number of components is not likely to include material at the lower extremes of the strength distribution curve. The use of the proof test permits this problem to be circumvented by comparing component destruction test results with expected failure loads based on the actual material strength in each individual component. Contributed by the Gas Turbine Division of The American Society of Mechanical Engineers for presentation at the Gas Turbine Conference & Products Show, Houston, Texas, March 2-6, Manuscript received at ASME Headquarters December 13, Copies will be available until December 1, THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS, UNITED ENGINEERING CENTER, 345 EAST 47th STREET, NEW YORK, N.Y

2 Some Principles of Brittle Material Design W. H. DUKES INTRODUCTION Ceramic materials have been used for structural purposes for centuries; every stone building is an example, and certainly the use of concrete has become extensive over the last hundred years. In all of these applications, the general problem of brittleness, which is reflected as an inability of the materials to sustain tension stresses in a reliable and predictable manner, was avoided, rather than solved, by employing the materials in compression. In glass, which is another brittle material which is widely used structurally, the problem has been circumvented in a different manner for applications in which significant structural loads arise. This involves the introduction of residual compressive stresses into the glass surface so that tension stresses, which would be introduced by externally applied loads, are nullified, In recent years, there has been an increas- Ing interest in using brittle ceramic materials in engineering applications where tension stress must be accepted, and where a highly reliable structural element must be achieved, This interest results primarily from the ability of ceramics to sustain much higher temperatures than metals, and high temperatures are the key to efficiency in all energy conversion machines. Also, in many applications where acceptable operating temperatures can be achieved with metallic materials, there is an interest in ceramics for achieving lower cost or longer life components. Thus, there is a requirement to face, rather than circumvent, the problem of designing reliable and efficient structures with brittle materials. This paper is intended to lay down the basic principles which must apparently be followed to achieve this goal. The word apparently is used because the paper is mainly speculative and based on very little experimental evidence, except perhaps the evidence of failures. Fortunately, there is now a major program underway, sponsored by ARPA, to develop the necessary technology, It is hoped that this program will show many of these speculations to be correct; it may show some of them to be unnecessary. The point is that a very large step is needed in brittle material design technology, to get from speculation to a proven technique; this step has been initiated with the ARPA-sponsored Ford and Westinghouse program, and should be continued, DEFINITIONS Before discussing the problems that brittleness introduces into the design process, it is necessary to define, with some precision, what is meant in this discussion by the term. Brittleness, for design purposes, is considered as implying two specific characteristics: (a) a linear stress-strain curve from zero strain to failure, and (b) a very low value of K I, relative to metals of similar strength, The low C KI C implies that ceramics are very sensitive to cracks, flaws, defects, etc,, and it is convenient to consider this as an individual characteristic, but it is undoubtedly related directly to the lack of any ductility or yielding capability, which, in turn, results from the first characteristic. High concentrations of stress at the tip of a crack cannot be relieved by yielding, and when these very localized stresses reach the ultimate strength of the material, failure occurs. Consequently, the low toughness may be directly the result of the linear stress-strain properties. Conversely, there may be a limited amount of ductility present, but it is all used in local yielding at crack tips resulting in a straight line stress-strain curve to failure when the material is tested in bulk. For the designer, the distinction is not important; whatever the mechanism, ceramic materials, as a class, exhibit the characteristics defined in the foregoing, Ceramic materials frequently have other troublesome characteristics; for instance, differential expansion of different phases in a complex material, and phase changes at certain temperatures, involving volumetric changes and thermal stresses. It is for the materials development specialist to minimize these problems, and when they have been minimized, they may still require special treatment by the designer, but

3 they should not be confused with the characteristic by stress analysts or treated as a "fatigue" of "brittleness," as contrasted with the "ductility"problem. Much of the subject of fatigue is a which is present in all structural metallic device to avoid precise stress analysis, since materials in some degree, material fatigue and stress concentrations are invariably lumped together, Furthermore, the REQUIREMENTS FOR BRITTLE MATERIAL DESIGN fatigue problem becomes more and more acute as higher strength materials, generally of lower Reduced to its most simple form, brittleness ductility, are pursued in search of structural introduces two new aspects into the structural efficiency, As ductility decreases, the practice design problem: of using approximate stress analysis methods and neglecting local stress peaks becomes less and 1 The need for highly refined stress less acceptable, analysis methods Similarly, material strength is usually 2 The need to treat material strength treated in a deterministic manner because it is in a statistical manner, far more convenient than statistical treatment, The need for highly refined stress analysis methods is a consequence of the inability of the material to yield, so that failure begins at the point of maximum stress, however localized this stress might be. For instance, if a bolted attachment were to be made with brittle materials, not only must the stress concentration due to the hole be considered, but tolerances on bolt and hole diameters, and surface irregularities on the bolt and in the hole, will tend to create point contacts with very high local stresses, The statistical treatment of material strength results from the sensitivity of the material to flaws, and since these flaws are statistically distributed in size and shape, then strength becomes a statistical function, It might be argued that careful process control would ensure "reproducible" flaws, from sample to sample, and hence reproducible strength properties, Improvements in both the level and the variability of mechanical properties have, in fact, been achieved by better process control, but, in general, the relationship between the numerous processing parameters and the variability in mechanical properties is not understood, and strength variability in ceramic materials is far greater than in metallic materials. It must be appreciated that the two new aspects of design, introduced by the use of brittle materials, do not make brittle material design a special case, In fact, brittle material design, in so far as it involves refined stress analysis methods and the statistical treatment of material strength, is the more general case, while the conventional design practice with ductile material is the special case, an approximation permitted by the forgiving characteristic of ductility, Local stress concentrations at holes, fillets, changes in cross section, applied loads, and in very short beams, where simple bending theory does not apply, have been neglected and tha variphilit,r in r3netila matallir matarials is so small and the distribution curve so sharply peaked, that this approximation is acceptalbe. Again, as metallic materials were improved in strength, at a cost in ductility and toughness, the limitations of these methods began to show in catastrophic failures leading to the development of fracture mechanics. Fracture mechanics treats flaws individually because the critical flaw size for even the least tough metallic material, is quite large. If toughness and critical flaw size decrease, the statistical methods will become necessary as they presently are for ceramics. The first of the new aspects imposed by brittleness, the need for refined stress analysis, can, fortunately, now be satisfied, Finiteelement methods of stress analysis have been developed to a high degree and have been well correlated with experiment, and these will produce any level of stress analysis refinement required. The only limitation is the cost of making the analysis. Usually, this can be minimized by applying finite-element methods to a large complex structure on a relatively coarse, macroscopic scale, then isolating sections on the assumption the boundary conditions are known from the more general analysis and carrying out a more refined and localized analysis. This process can be carried out from a complete hull or airframe analysis at one extreme, to the analysis of the tip of a crack at the other. Thus, this aspect of brittle material design is solved, except perhaps for continuing work at cost reduction, permitting more and more elements, or degrees of freedom, to be considered in a specific problem with a specific budgetary limitation. The second of the new aspects, the statistical treatment of material strength, is a far more difficult problem. We consider, first, the case of a simple tension member of constant stress and constant cross section. If a number of such members, of a particular ceramic material, are 3

4 I- 8 0 ^ 3 f- F- U. U. Q S O O K K Q Q LL LL 2 H. I'LLCU JIKCJJ PROOF STRESS Fig. 1 Typical strength variability curve for a ceramic RATIO: UNIFORM IENSILE PROOF STRESS MAXIMUM FENDING STRESS loaded to failure and the results plotted as failure stress versus probability of failure, the result will be a typical "S" curve, shown in Fig. 1. While this curve has been treated extensively by mathematicians and statisticians, when it is associated with strength of a practical structure, the facts are as follows. The curve is plotted against a vertical ordinate, denoting failure probability, that extends from zero to 1.0. Even if 20 to 30 identical test specimens are used, the test points will spread between about 0.1 and 0.9, whereas the failure probability of interest for establishing allowable stresses is of the order of l0-7 or This is down in the tail of the curve and extrapolation from the test points, whether it be done by mathematical curve fitting or by eye or any other means, is almost meaningless. It is equally meaningless to anticipate testing 106 or 10 7 specimens, which would be necessary to obtain one or two result of interest. Furthermore, the problem is grossly compounded when other important parameters, such as temperature, stress gradient, strain rate, stress state, and stress sequence are introduced. The concept of a proof test is now introduced. With this concept, each specimen is first stressed to a predetermined stress level, and those which fail are rejected before the remainder are subsequently tested to failure. The curve which results is truncated, as shown by the broken line in Fig. 1, the fifficult tail is eliminated, and the stress level corresponding to zero failure probability, which defines the left-hand end curve, is completely known. Fitting a curve to the test data is then a matter of interpolation, rather than extrapolation, a far more accurate 4 Fig. 2 Effect of a uniform tensile proof test on failure probability of a bar in bending procedure. In principal, this concept of proof testing seems to be an essential requirement to the use of ceramic materials. All components must be proof tested, to eliminate those with flaws sufficiently large that some predetermined strength level cannot be attained. Obviously, the selection of this strength level now effects not only the weight of the structure, since it determines the working stress, but it also has an important economic effect since the higher the stress level selected, the larger is the percentage of the parts made, which will be rejected. Considering again, the truncated material strength curve, it will also be evident that if the proof stress represents a zero probability of failure stress level and the design conditions are such that the probability of failure, under ultimate loads is to be 10-7 or 10-8, then stress to be associated with ultimate load is essentially equal to the proof test stress. Evidently, the whole question of statistics is avoided, For a simple tensile member, this would be so, but most real situations involve a structural element designed for some comples stress situation which it is difficult to reproduce in a proof test. Furthermore, there may be numerous design conditions, each critical for a different part of the component, and it may be impossible to find a single proof test which will produce the maximum stresses at all points in the component, at the same time.

5 im APPLIED STRESS 6 Fig. 3 Weibull curves from repeated stress testing Fig. 4 Method for generating Weibull curves for repeated stress conditions It can be shown that by reverting to the statistical approach, any convenient proof test can be used, regardless of the stress distributions of operating conditions, The benefits of the proof test can ba obtained even if the maximum stresses occurring during proof do not coincide in location with the maximum stresses during operation. For this purpose, the truncated strength distribution curve for the material is again required but not only is this an interpolation through test points, rather than an extrapolation, but the accuracy with which the si pe of the curve is predicted, is not critical. This is because the component failure probability is related not to the stress level at each point, but to the difference between design stress level and proof stress level, at each point. These differences can be made as small as desired, depending on how closely the proof stress distribution matches the operational stress distribution. This is illustrated in a twofold manner. First by the expression which defines failure probability: where: S =^ k I M ( l M I Ull 0) S = probability of failure a = applied stress in element of volume V c'p = proof stress in same element O'o = material constant Second by a simple example as in Fig. 2. The expression, which is an approximation to the Weibull relationship, based on the assumption that the failure probability is extremely small, shows directly that any distribution of proof stress will reduce failure probability. The example shows how failure probability is improved for a simple rectangular beam loaded at the center, if the specimens are first proof tested with a simple uniaxial tension stress, LOAD STATISTICS The next step in this problem is to extend the concept of failure probability to include the probability of attaining the applied loads used for design, In a typical airframe component, for instance, while the limit loads have a probability of occurrence of approximately 0.01; normal 1-g steady flight loads are 1.0; ultimate loads are probably The point is that it is overall probability of failure which should be defined, and this probability is the product of the probability of load occurrence times the probability of material failure at the stress produced by that load. Thus, depending on the shapes of the material strength probability curve and the load probability curve, the critical probability of failure may or may not occur at ' I ultimate" load. With this concept, the usual approach of deterministic loads and factors of safety loses its meaning. Even such an apparently deterministic case as a turbine wheel, in brittle materials, should be designed on the basis of curves defining probability of occurrence for such conditions as overspeed levels and temperature gradients during start-up or shutdown. None of the design conditions are truly deterministic; though the difficulty of obtaining probabilistic data in most design situations,is certainly recognized. This concept calls for a complete change in the approach to design criteria with a specification of acceptable failure probability, 5

6 rather than safety factors. ENVIRONMENTAL CONDITIONS In addition to the foregoing considerations, the basic problems of brittle material design must be extended further. There are problems of the effect of stress state and stress gradient on material failure probability and problems of how to accommodate material damage due to repeated loads and load and environmental spectra. Speculatively, on the assumption that failure is precipitated by the combination of a high stress state and a macroscopic flaw, it would not be expected that stress state would affect failure, but that maximum principle tension stress is the significant parameter. Similarly, stress gradient would not be expected to be significant, although the stress distribution is important. The total failure probability of a component is a summation of the failure probability of each element which, in turn, is dependent on the stress level in that element. Thus, a non-uniform stress distribution has a definite effect on failure probability. Again speculating, fatigue failure with brittle materials might be handled in two ways: 1 Using the conventional approach with experimentally determined S-n data, applied load spectra and a cumulative damage law 2 Based on fracture mechanics principles. If the first method is applied to brittle materials, it will be necessary, for each material and each temperature, to generate Weibull curves for material subjected to various cycles at a specified stress level. What is required is shown in Fig. 3, where a point such as "A" shows the probability that the material will fail when subjected to 10,000 cycles of the stress, Q4. Actually, it is not possible to select a stress level such that failure occurs precisely at the required number of cycles particularly since the level will vary from sample to sample in a statistical manner. A way around this is shown in Fig. 4. The tests must be conducted at preselected stress levels, the number of cycles to failure observed, and extrapolation used to obtain the stress level at which the particular piece of material would have failed for a particular number of cycles. These extrapolations, in turn, need an average S-n curve, drawn through all test data. This approach to fatigue is probably not practical. To obtain adequate fatigue data with conventional metallic materials is difficult and extremely costly. To add to this problem additional parameters such as temperature, proof stress level, and possibly strain rate, etc., and then to further compound the magnitude of the task with significant material variability is almost inconceivable. Furthermore, such a large program could not be justified until the material and its processing were thoroughly established, to ensure the most reproducible material. A possible alternate, which would conceivably eliminate the costly statistical aspects of the foregoing, is the use of fracture mechanics principles. In extending fracture mechanics principles to the prediction of structural performance under repeated loads, it is necessary to determine experimentally, crack growth data as a function of stress level and temperature. With this information, predictions can be made of the growth of an initial flaw as a result of the various stresses applied during the life of the structure, and the requirement is to determine whether the critical size flaw is reached for the maximum stress level during this life. In metallic structures, the question of crack initiation also arises, and experimental data is generally needed on the combination of stress and numbers of cycles required to initiate a crack in an otherwise homogeneous piece of material. This complication, however, is not likely to arise with ceramics since these materials are assumed to contain large numbers of flaws initially. If we assume that critical stress-intensity factors and crack growth rates can be measured for brittle materials, it is reasonable to assume that these characteristics are not statistical in nature. The cause of the variability in the mechanical properties of ceramics is assumed to be the flaws and defects which are the result of processing, together with the sensitivity of the material to the stress concentrations produced by the flaws. The variability is not, in other words, inherent in the material itself. If we consider the growth of a flaw where the crack is propagating through homogeneous material, this crack growth should be reproducible, as a function of the applied stresses, from any material sample. Since a typical ceramic material is likely to contain large numbers of flaws, crack growth characteristics might be affected by the propagation of the crack from one flaw to the next, and in this respect, the size and shape distribution of flaws may influence crack growth characteristics. However, the volume of material affected by a growing crack is very small, 6

7 particularly since critical flaw sizes in ceramics at the stress levels of interest will be only a few thousandths of an inch. In using the fracture mechanics principles with ceramic materials, we begin with the assumption that failure is produced by propagation of a flaw of critical size and that the variability in strength properties is the result of variability in the size and distribution of flaws. Thus, the Weibull strength curve also gives the probability of a flaw of a certain size in any individual sample. Conversely, if a probability level is given, the corresponding maximum flaw size can be determined, The two relationships necessary to do this are / M M., S = I - (^ ' ^1 V 0 \ a 0 where: A =[JF ^^ K 2 M, QU, and QC are Weibull constraints 0= specimen failure stress P = proof test stress level V = specimen volume S = specimen failure probability associated with stress a A = maximum flow size in specimen that fails under stress a KI C = stress-intensity factor W = width of tensile specimen A = geometric factor W With the Weibull parameters determined by strength tests, the first equation can be used to determine the failure stress for any given probability level and the second equation will determine the maximum flaw size in the test piece that fails at the stress level, assuming that the critical stress-intensity factor, KID, has been measured in separate tests. Note that the effect of proof testing the material can be easily included. The proof test eliminates material containing flaws greater than the critical size associated with the proof stress, but in so doing, it decreases the probability value associated with a flaw of any size. In order to determine flaw size information from the Weibull curve, it is necessary to know the value of the geometric factor A/W for the configuration of the test specimens used to conduct the strength tests. This will usually consist of either a square cross-section bend specimen or a round bar tensile specimen. Either case requires the geometric factor for an essentially infinite volume of material with a small crack in the surface. Geometric factors for this crack configuration are available in literature. From the foregoing procedure, if the allowable material failure probability under repeated loads is specified, a maximum "allowable" flaw size can be determined. The flaw size will depend on the proof stress level so that in a complete structural component, the allowable flaw size will vary throughout the component as the stress produced in the proof test varies. Having determined allowable flaw sizes throughout the component, the flaw growth at each point is calculated from flaw growth data, which will be given as a function of stress level and temperature, and from the spectrum of stresses andtemperatures expected throughout the component life. When the maximum size flaw at the end of the structural life has been determined for each element of the component in this manner, a check can be made to determine whether the critical flaw size has been reached at any point for the maximum stress to be experienced at each point. If the critical size is just reached in some location, the probability value used to determine the initial flaw sizes is the probability of failure of the component. If a critical flaw size is not reached, either a margin of safety is indicated or the stress levels throughout the component life can be raised by reducing material thicknesses for example. This method of determining the effect of repeated brittle material structure offers the promise that no additional statistical data other than that required to develop the Weibull curve is needed, It does, however, involve numerous assumptions which, with the presentlimited knowledge and experience, still require verification'. DESIGN VERIFICATION Before ceramic materials can be used reliably in structure's, a complete design method must be evolved and verified. The evolution of such a method requires a considerable program to learn how ceramics behave under many different and typical design situations, and this, in turn, must be carried out with numerous materials or classes of materials since behavior probably varies, 7

8 Even before this can be done, however, it may be described in the foregoing is not, in itself a necessary to evolve better behaved materials than sufficient verification for it does not check that are typically available, and certainly it is the correct material strength distribution curve necessary to evolve very closely controlled has been used. And, so far as is known, there processing, is no practical way to do this. With a proof After a design method is evolved, its test, it is far less important, and it can be verification presents a fundamental problem. If verified by conducting a relatively small number ;~ a component is designed to a certain procedure, of tests, and then fabricated and tested successfully, its success does not by any means imply a verification SUMMARY AND CONCLUSIONS of the design process, and a confirmation that ceramics can be used for structural applications. Based on the foregoing discussion, the If one, or a small number of components are elements of brittle material design are summarized fabricated, it is most probable that the material as follows: is average in its mechanical properties, whereas the operational stresses will have been selected 1 The use of finite-element methods of in anticipation of material of the lowest strength stress analysis to carry out refined acceptable by y the proof test, Aecordin Accordingly, 1 an analyses individual component may be operating at a rela- 2 The use of the proof test to truncate the tively high factor of safety, material distribution curve There is, in fact, a danger in conducting 3 The use of statistical methods to relate tests on a small number of components and the failure probability of a component finding that these components work successfully. If it exposed to a spectrum of loads, to the is assumed, from this success, that it is possible proof test to use ceramics in structural applications and 4 The verification of the design method by conducting if, as a consequence, ceramic components are put tests on a small number of specimens but carrying to failure and into operational use, then eventually the minimum interpreting results based on actual strength component will appear, and if a premature material stress. failure results, ceramics will again be condemned, In order to avoid this situation, it is This approach adds a very substantial degree important that any program to verify a design method, for ceramic materials, involves testing of sophistication to structural design. To to destruction, and the results of the tests should achieve this capability will, as mentioned earlier, require a substantial effort which fortunately, be compared with predictions based on the actual has begun. mechanical properties of the components tested. It is certain that successful brittle ceramic This procedure must be extended to include not structures can be made if the rules are followed, only simple static testing, but a simulation of It is equally certain that short cuts will lead the entire history of the component, with load to failure and that such failures could set back spectra and environmental spectra imposed. It is probably necessary to subject the part to a number brittle material technology another 15 years. On the other hand, success could open the of lifetimes, depending on the safety factor door to significant improvements in the efficiency applied to life, with load levels increased in of many energy conversion devices presently accordance with the strength g properties of the actual material of the test component. A success- constrained by material temperature limitations, In addition, however, sophistication in ful test of the appropriate number of lifetimes design technology is certain to have benefits in should then be followed by a static test to failure, metallic structures. The technology of fatigue for one of the most critical loading conditions, and fracture mechanics were forced upon structural with the success of this test again g judged on the basis of the strength properties of the particular designers by the limitations of higher strength metals, and a well-developed brittle material component, The foregoing method, which is only sketched design technology, based on the correct principles, in principle and requires considerable detail should also permit reliable structures to be designed in any metallic materials and should development, will check a design technique, and improve the reliability presently attainable with the proper use of the proof test will avoid the relatively ductile metallics. problems of material strength distribution. If the proof test is not used, the test method 8