《纺织复合材料》课程参考文献（Composite Materials Handbook，Volume 3）Chapter 5 Design and Analysis.pdf_P11-P15

MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis =Um Vm+ULf vf 5.2.2.4(d where uif is the longitudinal conductivity of the fibers.The results of Equations 5.2.2.4(c)and(d)are valid for any fiber distribution and any fiber cross-section. The problem of transverse conductivity is mathematically analogous to the problem of longitudinal shearing (Reference 5.2.2(a)).All results for the effective longitudinal shear modulus Gi can be inter- preted as results for transverse effective conductivity In particular,for the composite cylinder assem- blage model um Vm+ue(1+vf) um(1+vf)+uevm Vf 5.2.2.4(e) =m+ +Vm (f-"m)2'm These results are for isotropic fibers.For carbon and graphite fibers ur should be replaced by the trans- verse conductivity uat of the fibers(Reference 5.2.2.1(e)).As in the elastic case,there is reason to be- lieve that Equation 5.2.2.4(e)accurately represents all cases of circular fibers which are randomly distrib- uted and not in contact.Again the hexagonal array numerical analysis results coincide with the number predicted by Equation 5.2.2.4(e). To interpret the results for the case of moisture diffusivity,the quantity um is interpreted as the diffusiv- ity of the matrix.Since moisture absorption of fibers is negligible,ur is set equal to zero.The results are then =Um Vm Vm 5.2.2.4(⑤ 吃=Hm1+v These equations describe the moisture diffusivity of a composite material. 5.2.3 Fiber composites:strength and failure The mathematical treatment of the relationships between the strength of a composite and the proper- ties of its constituents is considerably less developed than the analysis for the other physical property re- lationships discussed in Section 5.2.2.Failure is likely to initiate in a local region due to the influence of the local values of constituent properties and the geometry in that region.This dependence upon local characteristics of high variability makes the analysis of the composite failure mechanisms much more complex than the analyses of the physical properties previously discussed. Because of the complexity of the failure process,it may be desirable to regard the strength of a unidi- rectional fiber composite subjected to a single principal stress component as a quantity to be measured experimentally,rather than deduced from constituent properties.Such an approach may well be the prac- tical one for fatigue failure of these composites.Indeed,the issue of determining the degree to which het- erogeneity should be considered in the analysis of composite strength and failure is a matter for which there exists a considerable degree of difference of opinion.At the level of unidirectional composites,it is well to examine the effects upon failure of the individual constituents to develop an understanding of the nature of the possible failure mechanisms.This subject is discussed in the following sections.The gen- eral issue of the approach to failure analysis is treated further in laminate strength and failure. The strength of a fiber composite clearly depends upon the orientation of the applied load with re- spect to the direction in which the fibers are oriented as well as upon whether the applied load is tensile or compressive.The following sections present a discussion of failure mechanisms and composite- constituent property relations for each of the principal loading conditions. 5-13

MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-13 1 * µµ µ = m vm + Lf vf 5.2.2.4(d) where µ1f is the longitudinal conductivity of the fibers. The results of Equations 5.2.2.4(c) and (d) are valid for any fiber distribution and any fiber cross-section. The problem of transverse conductivity is mathematically analogous to the problem of longitudinal shearing (Reference 5.2.2(a)). All results for the effective longitudinal shear modulus 1 * G can be interpreted as results for transverse effective conductivity 2 * µ . In particular, for the composite cylinder assemblage model 2 * m m m f f m f f m m f f m m m = v + (1+ v ) (1+ v ) + v = + v 1 (- ) + v 2 µ µ µ µ µ µ µ µµ µ L N M O Q P 5.2.2.4(e) These results are for isotropic fibers. For carbon and graphite fibers µf should be replaced by the transverse conductivity µ2f of the fibers (Reference 5.2.2.1(e)). As in the elastic case, there is reason to believe that Equation 5.2.2.4(e) accurately represents all cases of circular fibers which are randomly distributed and not in contact. Again the hexagonal array numerical analysis results coincide with the number predicted by Equation 5.2.2.4(e). To interpret the results for the case of moisture diffusivity, the quantity µm is interpreted as the diffusivity of the matrix. Since moisture absorption of fibers is negligible, µf is set equal to zero. The results are then 1 * m m 2 * m m f = v = v 1+ v µ µ µ µ 5.2.2.4(f) These equations describe the moisture diffusivity of a composite material. 5.2.3 Fiber composites: strength and failure The mathematical treatment of the relationships between the strength of a composite and the properties of its constituents is considerably less developed than the analysis for the other physical property relationships discussed in Section 5.2.2. Failure is likely to initiate in a local region due to the influence of the local values of constituent properties and the geometry in that region. This dependence upon local characteristics of high variability makes the analysis of the composite failure mechanisms much more complex than the analyses of the physical properties previously discussed. Because of the complexity of the failure process, it may be desirable to regard the strength of a unidirectional fiber composite subjected to a single principal stress component as a quantity to be measured experimentally, rather than deduced from constituent properties. Such an approach may well be the practical one for fatigue failure of these composites. Indeed, the issue of determining the degree to which heterogeneity should be considered in the analysis of composite strength and failure is a matter for which there exists a considerable degree of difference of opinion. At the level of unidirectional composites, it is well to examine the effects upon failure of the individual constituents to develop an understanding of the nature of the possible failure mechanisms. This subject is discussed in the following sections. The general issue of the approach to failure analysis is treated further in laminate strength and failure. The strength of a fiber composite clearly depends upon the orientation of the applied load with respect to the direction in which the fibers are oriented as well as upon whether the applied load is tensile or compressive. The following sections present a discussion of failure mechanisms and compositeconstituent property relations for each of the principal loading conditions

MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis 5.2.3.1 Axial tensile strength One of the most attractive properties of advanced fiber composites is high tensile strength.The sim- plest model for the tensile failure of a unidirectional fiber composite subjected to a tensile load in the fiber direction is based upon the elasticity solution of uniform axial strain throughout the composite.Generally. the fibers have a lower strain to failure than the matrix,and composite fracture occurs at the failure strain of the fibers alone.This results in a composite tensile strength,Fu,given by: Fu=VfF+vmom 5.2.3.1 where Fu -the fiber tensile strength o the stress in the matrix at a strain equal to the fiber failure strain The problem with this approach is the variability of the fiber strength.Non-uniform strength is charac- teristic of most current high-strength fibers.There are two important consequences of a wide distribution of individual fiber strengths.First,all fibers will not be stressed to their maximum value simultaneously. Secondly,those fibers which break earliest during the loading process will cause perturbations of the stress field near the break,resulting in localized high fiber-matrix interface shear stresses.These shear stresses transfer the load across the interface and also introduce stress concentrations into adjacent un- broken fibers. The stress distribution at each local fiber break may cause several possible failure events to occur. The shear stresses may cause a crack to progress along the interface.If the interface is weak,such propagation can be extensive.In this case,the strength of the composite material may differ only slightly from that of a bundle of unbonded fibers.This undesirable mode of failure can be prevented by a strong fiber-matrix interface or by a soft ductile matrix which permits the redistribution of the high shear stresses. When the bond strength is high enough to prevent interface failure,the local stress concentrations may cause the fiber break to propagate through the matrix,to and through adjacent fibers.Alternatively,the stress concentration in adjacent fibers may cause one or more of such fibers to break before failure of the intermediate matrix.If such a crack or such fiber breaks continue to propagate,the strength of the com- posite may be no greater than that of the weakest fiber.This failure 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 increases will produce new fiber failures at other locations in the material.An accumulation of dispersed internal damage results. It can be expected that all of these effects will occur before material failure.That is,local fractures will propagate for some distance along the fibers and normal to the fibers.These fractures will initiate and grow at various points within the composite.Increasing the load will produce a statistical accumulation of dispersed damage regions until a sufficient number of such regions interact to provide a weak surface, resulting in composite tensile failure. 5.2.3.1.1 Weakest link failure The weakest link failure model assumes that a catastrophic mode of failure is produced with the oc- currence of one,or a small number of,isolated fiber breaks.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 (e.g.,Reference 5.2.3.1.1(a))have been applied by Zweben (Reference 5.2.3.1.1(b))to determine the expected stress at which the first fiber will break.For practical materials in realistic structures,the calculated weakest link failure stress is quite small and,in general, failure cannot be expected in this mode. 5.2.3.1.2 Cumulative weakening failure If the weakest link failure mode does not occur,it is possible to continue loading the composite.With increasing stress,fibers will continue to break randomly throughout the material.When a fiber breaks, there is a redistribution of stress near the fracture site.The treatment of a fiber as a chain of links is ap- 5-14

MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-14 5.2.3.1 Axial tensile strength One of the most attractive properties of advanced fiber composites is high tensile strength. The simplest model for the tensile failure of a unidirectional fiber composite subjected to a tensile load in the fiber direction is based upon the elasticity solution of uniform axial strain throughout the composite. Generally, the fibers have a lower strain to failure than the matrix, and composite fracture occurs at the failure strain of the fibers alone. This results in a composite tensile strength, 1 tu F , given by: 1 tu f f tu m mt F = v F + v σ 5.2.3.1 where f tu F - the fiber tensile strength m t σ - the stress in the matrix at a strain equal to the fiber failure strain The problem with this approach is the variability of the fiber strength. Non-uniform strength is characteristic of most current high-strength fibers. There are two important consequences of a wide distribution of individual fiber strengths. First, all fibers will not be stressed to their maximum value simultaneously. Secondly, those fibers which break earliest during the loading process will cause perturbations of the stress field near the break, resulting in localized high fiber-matrix interface shear stresses. These shear stresses transfer the load across the interface and also introduce stress concentrations into adjacent unbroken fibers. The stress distribution at each local fiber break may cause several possible failure events to occur. The shear stresses may cause a crack to progress along the interface. If the interface is weak, such propagation can be extensive. In this case, the strength of the composite material may differ only slightly from that of a bundle of unbonded fibers. This undesirable mode of failure can be prevented by a strong fiber-matrix interface or by a soft ductile matrix which permits the redistribution of the high shear stresses. When the bond strength is high enough to prevent interface failure, the local stress concentrations may cause the fiber break to propagate through the matrix, to and through adjacent fibers. Alternatively, the stress concentration in adjacent fibers may cause one or more of such fibers to break before failure of the intermediate matrix. If such a crack or such fiber breaks continue to propagate, the strength of the composite may be no greater than that of the weakest fiber. This failure 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 increases will produce new fiber failures at other locations in the material. An accumulation of dispersed internal damage results. It can be expected that all of these effects will occur before material failure. That is, local fractures will propagate for some distance along the fibers and normal to the fibers. These fractures will initiate and grow at various points within the composite. Increasing the load will produce a statistical accumulation of dispersed damage regions until a sufficient number of such regions interact to provide a weak surface, resulting in composite tensile failure. 5.2.3.1.1 Weakest link failure The weakest link failure model assumes that a catastrophic mode of failure is produced with the occurrence of one, or a small number of, isolated fiber breaks. 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 (e.g., Reference 5.2.3.1.1(a)) have been applied by Zweben (Reference 5.2.3.1.1(b)) to determine the expected stress at which the first fiber will break. For practical materials in realistic structures, the calculated weakest link failure stress is quite small and, in general, failure cannot be expected in this mode. 5.2.3.1.2 Cumulative weakening failure If the weakest link failure mode does not occur, it is possible to continue loading the composite. With increasing stress, fibers will continue to break randomly throughout the material. When a fiber breaks, there is a redistribution of stress near the fracture site. The treatment of a fiber as a chain of links is ap-

MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis propriate to the hypothesis that fracture is due to local imperfections.The links may be considered to have a statistical strength distribution which is equivalent to the statistical flaw distribution along the fibers. Additional details for this model are given in References 5.2.3.1.1(a)and 5.2.3.1.2.The cumulative weakening model does not consider the overstress on adjacent fibers or the effect of adjacent laminae. 5.2.3.1.3 Fiber break propagation failure The effects of stress perturbations on fibers adjacent to broken fibers are significant.The load con- centration in the fibers adjacent to a broken fiber increases the probability that a second fiber will break. Such an event will increase the probability of additional fiber breaks,and so on.The fiber break propaga- tion mode of failure was studied by Zweben (Reference 5.2.3.1.1(b)).The occurrence of the first fracture of an overstressed fiber was proposed as a measure of the tendency for fiber breaks to propagate,and, hence,as a failure criterion for this mode.Although the first multiple break criterion may provide good correlations with experimental data for small volumes of material,it gives very low failure stress predic- tions for large volumes of material.Additional work in this area can be found in References 5.2.3.1.3(a) and (b). 5.2.3.1.4 Cumulative group mode failure As multiple broken fiber groups grow,the magnitude of the local axial shear stress increases and ax- ial cracking can occur.The cumulative group mode failure model (Reference 5.2.3.1.4)includes the ef- fects of the variability of fiber strength,load concentrations in fibers adjacent to broken fibers,and matrix shear failure or interfacial debonding which will serve to arrest the propagating cracks.As the stress level increases from that at which fiber breaks are initiated to that at which the composite fails,the material will have distributed groups of broken fibers.This situation may be considered as a generalization of the cu- mulative weakening model.In practical terms,the complexity of this model limits its use. Each of these models has severe limitations for the quantitative prediction of tensile strength.How- ever,the models show the importance of variability of fiber strength and matrix stress-strain characteris- tics upon composite tensile strength. 5.2.3.2 Axial compressive strength Both strength and stability failures must be considered for compressive loads applied parallel to the fibers of a unidirectional composite.Microbuckling is one proposed failure mechanism for axial compres- sion(Reference 5.2.3.2(a)).Small wave-length micro-instability of the fibers occurs in a manner analo- gous to the buckling of a beam on an elastic foundation.It can be demonstrated that this instability can occur even for a brittle material such as glass.Analyses of this instability were performed independently in References 5.2.3.2(b)and(c).The energy method for evaluation of the buckling stress has been used for these modes.This procedure considers the composite as stressed to the buckling load.The strain energy in this compressed but straight pattern(extension mode)is then compared to an assumed buck- ling deformation pattern(shear mode)under the same load.The change in strain energy in the fiber and the matrix can be compared to the change in potential energy associated with the shortening of the dis- tance between the applied loads at the ends of the fiber.The condition for instability is given by equating the strain energy change to the work done by the external loads during buckling. The results for the compressive strength,Fu,for the extension mode is given by Feu =2 vf Vf EmEf 5.2.3.2(a 3(1-vf) The result for the shear mode is F=】 Gm 5.2.3.2(b) 1-vf 5-15

MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-15 propriate to the hypothesis that fracture is due to local imperfections. The links may be considered to have a statistical strength distribution which is equivalent to the statistical flaw distribution along the fibers. Additional details for this model are given in References 5.2.3.1.1(a) and 5.2.3.1.2. The cumulative weakening model does not consider the overstress on adjacent fibers or the effect of adjacent laminae. 5.2.3.1.3 Fiber break propagation failure The effects of stress perturbations on fibers adjacent to broken fibers are significant. The load concentration in the fibers adjacent to a broken fiber increases the probability that a second fiber will break. Such an event will increase the probability of additional fiber breaks, and so on. The fiber break propagation mode of failure was studied by Zweben (Reference 5.2.3.1.1(b)). The occurrence of the first fracture of an overstressed fiber was proposed as a measure of the tendency for fiber breaks to propagate, and, hence, as a failure criterion for this mode. Although the first multiple break criterion may provide good correlations with experimental data for small volumes of material, it gives very low failure stress predictions for large volumes of material. Additional work in this area can be found in References 5.2.3.1.3(a) and (b). 5.2.3.1.4 Cumulative group mode failure As multiple broken fiber groups grow, the magnitude of the local axial shear stress increases and axial cracking can occur. The cumulative group mode failure model (Reference 5.2.3.1.4) includes the effects of the variability of fiber strength, load concentrations in fibers adjacent to broken fibers, and matrix shear failure or interfacial debonding which will serve to arrest the propagating cracks. As the stress level increases from that at which fiber breaks are initiated to that at which the composite fails, the material will have distributed groups of broken fibers. This situation may be considered as a generalization of the cumulative weakening model. In practical terms, the complexity of this model limits its use. Each of these models has severe limitations for the quantitative prediction of tensile strength. However, the models show the importance of variability of fiber strength and matrix stress-strain characteristics upon composite tensile strength. 5.2.3.2 Axial compressive strength Both strength and stability failures must be considered for compressive loads applied parallel to the fibers of a unidirectional composite. Microbuckling is one proposed failure mechanism for axial compression (Reference 5.2.3.2(a)). Small wave-length micro-instability of the fibers occurs in a manner analogous to the buckling of a beam on an elastic foundation. It can be demonstrated that this instability can occur even for a brittle material such as glass. Analyses of this instability were performed independently in References 5.2.3.2(b) and (c). The energy method for evaluation of the buckling stress has been used for these modes. This procedure considers the composite as stressed to the buckling load. The strain energy in this compressed but straight pattern (extension mode) is then compared to an assumed buckling deformation pattern (shear mode) under the same load. The change in strain energy in the fiber and the matrix can be compared to the change in potential energy associated with the shortening of the distance between the applied loads at the ends of the fiber. The condition for instability is given by equating the strain energy change to the work done by the external loads during buckling. The results for the compressive strength, 1 cu F , for the extension mode is given by 1 cu f f mf f F = 2 v v E E 3(1- v ) 5.2.3.2(a) The result for the shear mode is 1 cu m f F = G 1- v 5.2.3.2(b)