1、 - NASA CONTRACT-O_. 88 Exponential Stress Distribution 88 Discussion and Conclusion 89 APPENDIX D - STRESS CONCENTRATIONS IN NON-ADJACENT FIBERS . 92 APPENDIX E - ELASTIC STRAIN ENERGY . 94 Fiber Energy Change 94 Matrix Energy Change . 95 Net Energy Change 96 APPENDIX F - ANALYSIS OF THE CUMULATIVE
2、 GROUP MODE OF FAILURE . 97 REFERENCES . 102 FIGURES 104 V Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TENSILE FAILURE CRITERIA FOR FIBER COMPOSITE MATERIALS By B. Walter Rosen and Carl H. Zweben Materials Sciences Corporation SUMMARY An analytic
3、al model of the tensile strength of fiber com- posite materials has been developed. The analysis provides in- sight into the failure mechanics of these materials and defines criteria which serve as tools for preliminary design material selection and for material reliability assessment. The model inc
4、orporates both dispersed and propagation type failures and includes the influence of material heterogeneity. The important effects of localized matrix damage and post-failure matrix shear stress transfer are included in the treatment. The model is used to evaluate the influence of key parameters on
5、the failure of several commonly used fiber-matrix systems. Analyses of three possible failure modes have been de- veloped. These modes are the fiber break propagation mode, the cumulative group fracture mode, and the weakest link mode. In the former, adjacent fibers fracture sequentially at posi- ti
6、ons which are within a short distance of a planar surface. Eventually the propagation becomes unstable and the plane be- comes the fracture plane. In the cumulative group mode dis- tributed fiber fractures increase in size and number until the damaged regions have weakened one cross-section so that
7、it can no longer carry the applied load. In the weakest link mode, an initial fiber fracture causes an immediate propagation to failure. Application of the new model to composite material systems has indicated several results which require attention in the de- velopment of reliable structural compos
8、ites. Prominent among these are the size effect and the influence of fiber strength variability. 1 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-INTRODUCTION At the present stage of development of composite materials and their applications, there a
9、re many new and improved high performance fiber and matrix materials. At such a time the desire to utilize reliable, high-strength composites makes the need for an understanding of the tensile failure of fiber composite materials self-evident. However, despite widespread attempts to use limited expe
10、rimental data to substantiate simplis- tic concepts of the failure process, it is equally evident that this failure process is extremely complex. The primary factor contributing to the complexity of this problem is the variability of the fiber strength. There are two important consequences of a wide
11、 distribution of in- dividual fiber strengths. First, all fibers will not be stressed to their maximum value at the same time. Thus, the strength of a group of fibers will not equal the sum of the strengths of the individual fibers, nor even their mean strength value. Second, those fibers which brea
12、k earliest will cause perturbations of the stress field resulting in localized high interface shear stresses, and in stress concentrations in adjacent fibers. Thus, progressive damage may well result. In earlier studies, approxi- mate models of different possible failure modes have been formulated.
13、These include an assessment of the failure resulting from fracture of the weakest link; of the fiber break propaga- tion resulting from internal stress concentrations; and of the failure resulting from the cumulative weakening effect of dis- tributed fiber fractures. The present study utilizes stati
14、stical analyses to assess the effects of the occurrence of damage at scattered locations within the material followed by an increase in the size and number of these damaged regions as the stress level is increased. The results of this study provide an integrated approach to the definition of the mod
15、e and level of tensile failure for fiber composite materials. The new failure model includes the limiting 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-effects of matrix or interface strength and thereby enhances the understanding of crack arrest
16、 mechanisms within a composite. The results are not only of value for assessing the relative merits of different constituent properties, but also provide a basis for evaluating material reliability and assessi g 7 damage tolerance for fiber composite materials. In an attempt to present clearly the m
17、ajor concepts introduced in this paper, all details of the analyses have been relegated to a series of six appendices. Thus, following a brief outline of the background to the present problem, the body of the paper is composed of three descriptive sections. The first, the development of failure mode
18、ls and failure criteria; the second, the results of the application of the new analysis to both real and idealized composite systems; and the final, the implications of the results of this study. The approach taken in this paper is consistent with the new materials engineering concepts. Thus, one ma
19、y expect that materials will be tailored to suit the requirements of their application. Choice of constituents is a new freedom which will be exploited by the designer in time to come. Thus, the analytical understanding of material behavior must be adequate to assess a priori the relative merits of
20、various potential combinations of constituents. The required analyses should be viewed as preliminary design tools for this selection process. Final determination of material properties for the actual design will be obtained experimentally after this analytical screening process. The present definit
21、ion of criteria for tensile failure of composites is consistent with this philosophy. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Af Ef = Fb) Gm I J L L g Ln M N P PI (0) LIST OF SYMBOLS Cross-sectional area of an Fiber extensional modulus indivi
22、dual fiber Fiber strength distribution Matrix shear modulus Number of adjacent broken fibers Fiber index denoting position of fiber relative to last broken fiber Specimen length Fiber gage length in strength test Influence coefficient definint force in fiber n due to a unit displacement of fiber 0.
23、Number of axial layers or links = L/8 Number of fibers in a typical cross-section Applied load on a fiber at infinity = cr,Af Probability of having a crack of size I in a composite (see Eq. A.14) Applied load when matrix failure occurs Transitional probability (see Eq. 2.4 for example) Probability o
24、f failure of a group of I fibers (see Eq. A.16) u()JJ1 A.2 = Displacements of core of broken fibers, intact fiber, and average material, respectively used in approximate model (see Appendix B) a = Half length of inelastic zone dld2 = Effective fiber spacing parameters used in 3D model for load conce
25、ntrations (see Fig. B.5) = f Subscript indicating fiber g(I) = Number of intact fibers surrounding I broken fibers Y = Surface energy kE,k; = Effective load concentration factors associated with exponential and linear stress variations, respectively (see Appendix C) 4 Provided by IHSNot for ResaleNo
26、 reproduction or networking permitted without license from IHS-,-,-kI = m = = m n = nlpn2 = P,(U) = qq2 = rI(s) = r ,r = a b t = uo,u1,u2 = Vf X AV AvF cl,P LIST OF SYMBOLS CONTINUED Load concentration factor associated with I broken fibers Number of fibers in approximate model of Ref. B.2 Subscript
27、 indicating matrix Number of broken fibers in core Parameters used in calculating Q, (see Eq. 2.5) Probability that a crack will initiate in a given layer and grow to size I (see Eq. A12) Probability of failure of overstressed fibers (see Eqs. A.4 and A.51 Probability that one d fibers will break Ra
28、dii used in 3D model for load concentrations (see Fig. B.5) (rbdl) / (rad2) Nondimensional axial displacements of core, intact fiber, and average material used in approximate model (see Appendix B) Fiber volume fraction Coordinate parallel to fiber axis Avf + AVm Energy required to open a crack that
29、 will extend to next fiber Elastic energy released when an isolated fiber breaks (Eq. El) Elastic energy released when matrix fractures (Eq.EG) Nondimensional half length of inelastic zone Weibull distribution parameter Weibull parameters used in Cumulative Group Mode of Failure Analysis 5 Provided
30、by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-LIST OF SYMBOLS CONTINUED a -l/B Ineffective length Elastic ineffective length defined in Eq. 1.4 Ineffective length associated with I adjacent broken fibers Ineffective length associated with a group of g broke
31、n fibers Representative ineffective length used in calculating pI (see Appendix A) Post-failure shear stress parameter Fraction of undisturbed fiber stress oo, parameter = 2ir/n in Appendix B, exponent parameter in AppFl:dix C 2n/g Nominal fiber stress = Statistical mode of cumulative weakening fail
32、ure mode stress = Non constant stress in the intact fibers adjacent to I broken fibers = Undisturbed fiber stresstat a large distance from site of a fiber break) = Fiber stress = Expected fiber stress level for first fiber break = Matrix shear stress = Nondimensional matrix shear failure stress (see
33、Eq. B.3b) = Matrix shear failure stress = T Y = Matrix shear failure stress = Angle between layer fiber axis and laminate axis = Nondimensional coordinate along fiber axis 6 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-P- - I BACKGROUND The major
34、factor motivating the present study is the non-uniform strength of most current high-strength filaments. This statistical fiber strength distribution is generally attributed to a distribution of imperfections along the length of these brittle fibers. In a composite, one can always ex- pect some fibe
35、r breaks at relatively low stresses. The problem of composite tensile strength is the problem of de- termining effects subsequent to these initial internal breaks. Because the relative importance of the multiplicity of pos- sible modes of subsequent internal damage depends upon local details of the
36、stress field, the problem of composite tensile strength is extremely complex. At each local fiber break, several possible events may occur. In the vicinity of the fiber break the local stresses are highly non-uniform (fig. 1.1) This may result in a crack propagating along the fiber interface or acro
37、ss the composite. In the former case the fibers may separate from the composite after breaking and the composite material may be no stronger than a dry bundle of fibers. In the second case, the composite may fail due to a propagating normal crack or due to a fiber break propagation and the strength
38、of the composite may be no greater than that of the weakest fiber. This latter mode is defined as a “weakest link failure. If the matrix and interface properties are of sufficient strength and toughness to prevent or arrest these failure mechanisms, then continued load increase will produce new fibe
39、r failures at other locations in the material, resulting in a statistical accumulation of internal damage. In actuality, it is to be expected that all these effects will generally occur prior to material failure. That is, frac- tures will propagate along and normal to the fibers and these fractures
40、will occur at various points within the composite. 7 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Previous treatments of these various failure modes will be reviewed briefly in this section. The imperfection sensitivity of contemporary filaments a
41、ffects fiber tensile strength in two important ways. First of all, at a constant gage length there is a significant amount of dispersion in fiber strength. Thus some fibers fail at low stress levels and the average stress at failure of a bundle of fibers will be less than the average strength of the
42、 fibers. Second, because the probability of finding an imperfection of given severity increases with gage length average fiber strength increases with decreasing gage length. Thus the question of average fiber strength can be resolved only by determination of the important characteristic length in t
43、he composite. Fig. 1.2 (Ref.l.1) shows the strength variation of single fibers. Because of this important variability it is not possible to define a unique quantity called “fiber strength“, despite the fact that this term is often found in the literature. Generally, what is meant by the term “fiber
44、strength“ is mean fiber strength at a certain test gage length. Because fibers are generally much stiffer than matrix materials, they carry the bulk of the axial load if the fiber volume fraction, v f is not very small. Therefore the study of the tensile strength of composite materials centers on th
45、e behavior of the fibers and what happens when they break at various locations as a composite is loaded. In this report, attention is directed to the axial load carried by the fibers. (Composite strength is expressed in terms of the average fiber stress at composite failure.) There can be little dou
46、bt of the validity of this assumption for resin-matrix composites. In the case of metal matrix composites it is necessary to superpose a contribution of the matrix to axial load-carrying capacity. This will not affect the results of the present study. Weakest Link Failure When a unidirectional compo
47、site is loaded in axial tension, 8 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-scattered fiber breaks occur through the material at various stress levels. It is possible that one of these fiber breaks may trigger a stress wave or initiate a crack
48、 in the matrix resulting in localized stress concentrations which cause the fracture of one or more adjacent fibers. In turn, the failure of these fibers may result in additional stress waves or matrix cracks, leading to overall failure. This produces a catastropic mode of failure associated with th
49、e occurrence of one, or a small number of, isolated fiber breaks. This is referred to as the “weakest link“ mode of failure. The lowest stress at which this type of failure can occur is the stress at which the first fiber will break. The expressions for the expected value of the weakest element in a statistical population (see e.
