Residual strength and toughness of damaged composites

A study has been made of the effect of prior damage on the tensile strength and toughness of a number of carbon and glass fibre reinforced plastics materials. The methods of introducing this prior damage were by tension fatigue cycling, by transverse compression, by repeated impact, and by stress corrosion. The effects of the damage were assessed by measuring the un-notched and notched tensile strengths and, in some instances, the work of fracture of notched samples. Acoustic emission monitoring and microstructural studies were carried out in support of the mechanical tests. It appears that under various circumstances, the effects of microstructural damage may result in independent changes in the notched and un-notched strengths of a composite, although the pattern of changes is not simple. The toughness to strength ratio,K _Q/_f, may rise or fall, depending on the material and the damaging conditions. However, the results for all of the tests presented here still fall within the 90% confidence limits for theK _Q/_f relationship previously identified.     

Deformation of a carbon-epoxy composite under hydrostatic pressure

This paper describes the behaviour of a carbon-fibre reinforced epoxy composite when deformed in compression under high hydrostatic confining pressures. The composite consisted of 36% by volume of continuous fibres of Modmur Type II embedded in Epikote 828 epoxy resin. When deformed under pressures of less than 100 MPa the composite failed by longitudinal splitting, but splitting was suppressed at higher pressures (up to 500 MPa) and failure was by kinking. The failure strength of the composite increased rapidly with increasing confining pressure, though the elastic modulus remained constant. This suggests that the pressure effects were introduced by fracture processes. Microscopical examination of the kinked structures showed that the carbon fibres in the kink bands were broken into many fairly uniform short lengths. A model for kinking in the composite is suggested which involves the buckling and fracture of the carbon fibres.List of symbols d diameter of fibre - E _f elastic modulus of fibre - E _m elastic modulus of epoxy - G _m shear modulus of epoxy - k radius of gyration of fibre section - l length of buckle in fibre - P confining pressure (= ₂ = ₃) - R radius of bent fibre - V _f volume fraction of fibres in composite - _t, _c bending strains in fibres - angle between the plane of fracture and ₁ - ₁ principal stress - ₃ confining pressure - _c strength of composite - _f strength of fibre in buckling mode - _n normal stress on a fracture plane - _m strength of epoxy matrix - shear stress - tangent slope of Mohr envelope - slope of pressure versus strength curves in Figs. 3 and 4.     

Compression strength of carbon,glass and Kevlar-49 fibre reinforced polyester resins

The compression behaviour of a series of polyester resins of various compositions and in different states of cure has been investigated. Their mechanical characteristics having been established, the same range of resins was then used as a matrix material for a series of composites reinforced with carbon, glass and aromatic polyamide fibres. The composites were unidirectionally reinforced, having been manufactured by pultrusion, and were compression tested in the fibre direction after a series of experiments to assess the validity of a simple testing procedure. Rule of Mixtures behaviour occurred in glass-polyester composites up to limiting volume fractions (V _f) of 0.31 for strength and 0.46 for elastic modulus, the compression modulus being equal to the tensile modulus, and the apparent fibre strength being in the range 1.3 to 1.6 GPa at this limiting V _f. At a V _f of 0.31 the strengths of reinforced polyesters were proportional to the matrix yield strength, _my, and their moduli were an inverse exponential function of _my. For the same matrix yield strength a composite with an epoxy resin matrix was stronger than polyester based composites. At V _f=0.30, Kevlar fibre composites behaved as though their compression modulus and strength were much smaller than their tensile modulus and strength, while carbon fibre composites were only slightly less stiff and weaker in compression than in tension. The compression strengths of the polyester resins were found to be proportional to their elastic moduli.     

Carbon fibre-reinforced cement

This review describes fabrication processes for aligned fibre and random fibre carbonreinforced cement and links important process parameters with composite theory. The way in which the material fits into the general framework of crack constraint and matrix cracking theories is discussed. A broad survey is made of the mechanical properties, durability and dimensional stability of a variety of carbon-reinforced cement composites, and economic constraints on potential applications are considered.List of symbols b breadth of three-point bend specimen - d depth of three-point bend specimen - E _c composite Young's modulus - E _f fibre Young's modulus - E _m matrix Young's modulus - l fibre length - l _c fibre critical transfer length - l _s specimen span in three-point bend test - m Weibull modulus - r fibre radius - P applied load - V _f fibre volume fraction - V _m matrix volume fraction - x length of fibre needed to transfer load _mu V _m - x _d crack spacing in a composite with short, aligned fibres - _fu fibre ultimate strain - _mu matrix ultimate strain - _fu fibre ultimate strength - _mu matrix ultimate strength - _cu composite ultimate strength - _MOR modulus of rupture - _T tensile strength - interlaminar shear strength - _i interfacial shear strength - _m matrix work of fracture - _F work of fracture     

Effect of magnesium content on the stress exponent and effective stress in the steady state creep of Al-Mg alloys

The stress exponent of steady state creep,n, and the internal ( _i) and effective stresses ( _e) have been determined using the strain transient dip test for a series of polycrystalline Al-Mg alloys creep tested at 300° C and compared with previously published data. The internal or dislocation back stress, _i, varied with applied stress,, but was insensitive to magnesium content of the alloy, being represented by the empirical equation _i=1.084 ^1.802. Such an applied stress dependence of _i can be explained by using an equation for _i of the form _i (dislocation density)^1/2 and published values for the stress dependence of dislocation density. Values of the friction stress, _f, derived using the equation _e/=(1–c) (1– _f/), indicate that _f is not dependent on the magnesium content. A constant value of _f can best be rationalized by postulating that the creep dislocation structure is relatively insensitive to the magnesium content of the alloy.On leave from Engineering Materials Department, University of Windsor, Windsor, Ontario N9B 3P4, Canada.     

Fatigue behaviour of borosilicate glass-ceramic matrix,nicalon (silicon carbide) fibre composites

Uniaxial fatigue damage analyses were performed on borosilicate glass-ceramic matrix, Nicalon (silicon carbide) fibre reinforced unidirectional composites. The fibre volume fraction varied from about 0.25 to 0.60. Load-controlled tension-tension fatigue tests (R ratio = 0.1) were conducted at room temperature and 540°C (1000°F). The fatigue life was found to decrease with increasing cyclic stress level and a power-law relationship of the form _app = _uts(2N _f)^b was established where _app is the applied maximum stress, _uts the monotonic tensile strength, N _f is the number of cycles to failure and b is the fatigue strength exponent. The fatigue damage evolution manifested itself as a decrease in stiffness of the composite with fatigue cycles. This stiffness drop was associated with matrix cracking followed by fibre-matrix debonding and fibre sliding breakage/pull-out, and final failure, respectively at 540°C. The damage evolution at room temperature was associated with degradation of the matrix followed by steady breakage of fibres with no debonding/pull-out, leading to eventual failure of the net section of the composite. In general, quantitative microscopic observations of debonded and pulled-out fibres showed a good correlation with the observed reduction in stiffness. A predictive model to interpret the drop in stiffness is presented and validated using experimental results from the current study.     

The longitudinal electrical conductivity of metal matrix composites at cryogenic temperature in the presence of a longitudinal magnetic field

In this paper we use the results of Part I to derive two integral expressions for the electrical conductivity of metal matrix composite materials when a magnetic field, B, is added to a small electric field also parallel to fibres. One expression applies to strong magnetic fields meaningR ₀/a < 1, whereR ₀ =m ^* v _F/eB when the Fermi velocity is perpendicular to the magnetic field. WhenB , the integral expression reduces to the well known conductivity value = ₀(1 -V _f), where ₀ is the bulk matrix conductivity andV _f is the fibre volume fraction. For weak magnetic fields,R ₀/a > 1, then the electrical conductivity is expressed by the sum of two integrals. When B 0, the electrical conductivity reduces to the integral expression obtained in our earlier results when there is only a longitudinal electric field. In this paper we correct an incorrect derivation of the composite conductivity in the absence of a magnetic field published earlier [J. Mater. Sci.21 (1986) 2409].     

Interfacial debonding and fibre pull-out stresses

Based on a theoretical model developed previously by the authors in Part II of this series for a single fibre pull-out test, a methodology for the evaluation of interfacial properties of fibre-matrix composites is presented to determine the interfacial fracture toughness G _c, the friction coefficient , the radial residual clamping stress q _o and the critical bonded fibre length z _max. An important parameter, the stress drop , which is defined as the difference between the maximum debond stress _d ^* and the initial frictional pull-out stress _fr, is introduced to characterize the interfacial debonding and fibre pull-out behaviour. The maximum logarithmic stress drop, In(), is obtained when the embedded fibre length L is equal to the critical bonded fibre length z _max. The slope of the In()-L curve for L bigger than z _max is found to be a constant that is related to the interfacial friction coefficient . The effect of fibre anisotropy on fibre debonding and fibre pull-out is also included in this analysis. Published experimental data for several fibre-matrix composites are chosen to evaluate their interfacial properties by using the present methodology.On leave at the Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.     

Compressive strength of coated rigid-rod polymer fibres

One limitation to the use of high-strength/high-modulus rigid-rod polymer fibres like poly-(p-phenylene benzobisthiazole) (PBZT) and poly-(p-phenylene benzobisoxazole) (PBZO) in composite structures is their low compressive strength. Various theories have been developed to predict compressive strength of rigid-rod fibres. In this study the critical buckling stress for rigid-rod fibres with stiff external coatings has been theoretically modelled assuming that the failure mode in compression is the microbuckling of the fibrils in shear. Our model predicts that significant improvement in fibre compressive strength will occur only when relatively thick coatings, with thickness to diameter (t/D) ratios in excess of > 0.05, are used. Experimentally measured compressive strength of aluminium coated PBZT fibres shows values in good agreement to the theory at t/D ratios of 0.006 and below. Factors related to the selection of suitable coating materials and problems associated with establishing coating performance are identified.Nomenclature P axial compressive load - P _f axial compressive load on the fibre - P _c axial compressive load on the coating - P _cr ⁱ critical buckling load in the ith case - _cr critical buckling stress - _co compressive strength of the uncoated fibre - _c compressive strength of the coated fibre - v(x) lateral deflection of a buckled fibril or coating - V _m amplitude of the lateral deflection in the mth mode - m number of half-sine waves in the deflection mode - x coordinate distance along axial direction - y coordinate distance along radial direction - coordinate distance along circumferential direction - l length of the buckling unit - N number of fibrils in the fibre - D fibre diameter - d fibril diameter - t coating thickness - I _f moment of inertia of the fibril - A _f cross-sectional area of the fibril - E _f tensile modulus of the fibre - E _c tensile modulus of the coating material - E tensile modulus of the coated fibre - G torsional shear modulus of the fibre - v_c Poisson's ratio of the coating material - _f density of the fibre - _c density of the coating material - density of the coated fibre - U _f strain-energy change in the fibre - U _c strain-energy change in the coating - T _f external work done on the fibre - T _c external work done on the coating - d/D - t/D     

Interfacial debonding and fibre pull-out stresses

Two current theories [11, 17] of interfacial debonding and fibre pull-out, which have been developed on the basis of fracture mechanics and shear strength criteria, respectively, are critically compared with experimental results of several composite systems. From the plots of partial debond stress, _d ^p , as a function of debond length, three different cases of the interfacial debond process can be identified, i.e. totally unstable, partially stable and totally stable. The stability of the debond process is governed not only by elastic constants, relative volume of fibre and matrix but more importantly by the nature of bonding at the interface and embedded fibre length,L. It is found that for the epoxy-based matrix composite systems, Gaoet al.'s model [17] predicts the trend of maximum debond stress, _d ^* , very well for longL, but it always overestimates _d ^* for very shortL. In contrast, Hsueh's model [11] has the capability to predict _d ^* for shortL, but it often needs significant adjustment to the bond shear strength for a better fit of the experimental results for longL. For a ceramic-based matrix composite, _d ^* predicted by the two models agree exceptionally well with experiment over almost the whole range ofL, a reflection that the assumed stable debond process in theory is actually achieved in practice. With respect to the initial frictional pull-out stress, _f, the agreement between the two theories and experiments is excellent for all range ofL and all composite systems, suggesting that the solutions for _f proposed by the two models are essentially identical. Although Gaoet al.'s model has the advantage to determine accurately the important interfacial properties such as residual clamping stress,q _o, and coefficient of friction, , it needs some modifications if accurate predictions of _d ^* are sought for very shortL. These include varying interfacial fracture toughness,G _ic with debond crack growth, unstable debonding for very shortL and inclusion of shear deformation in the matrix for the evaluation ofG _ic and fibre stress distribution. Hsueh's model may also be improved to obtain a better solution by including the effect of matrix axial stress existing at the debonded region on the frictionless debond stress, _o.     

Examining the temperature dependence of the mechanical properties for materials in large-diameter tubes

Measurements have been made on the mechanical properties of largediameter tubes made of 09G2S, 14G2SAF, and 17G1S steels and on welded joints in them at strain rates of 200 sec^–1 and 113–293 K. Fivefold microspecimens 1.2 mm in diameter have been employed. The parameters have been determined in the temperature-rate dependence of the Yaroshevich yield point, lower brittleness temperature, and fracture resistance as affected by the ferrite diameter. A method is proposed for predicting the brittleness range from the dependence of the notch bottom narrowing B on the generalized parameter = S_f/_y — _t/_i for a Menaget specimen.Translated from Problemy Prochnosti, No. 4, pp. 43–50, April, 1994.     

On internal stress and activation volume in polymers

The two-site model is developed for the analysis of stress relaxation data. It is shown that the product of d In (– )/d and (- _i) is constant where is the applied stress, _i is the (deformation-induced) internal stress and = d/dt. The quantity d In ( )/d is often presented in the literature as the (experimental) activation volume, and there are many examples in which the above relationship with (- _i) holds true. This is in apparent contradiction to the arguments that lead to the association of the quantity d In (– )/d with the activation volume, since these normally start with the premise that the activation volume is independent of stress. In the modified theory presented here the source of this anomaly is apparent. Similar anomalies arise in the estimation of activation volume from creep or constant strain rate tests and these are also examined from the standpoint of the site model theory. In the derivation presented here full account is taken of the site population distribution and this is the major difference compared to most other analyses. The predicted behaviour is identical to that obtained with the standard linear solid. Consideration is also given to the orientation-dependence of stress-aided activation.     

Internal material damping of polymer matrix composites under off-axis loading

The objective of this paper is to determine theoretically the material damping of short fibre-reinforced polymer matrix composites. The major damping mechanism in such composites is the viscoelastic behaviour of the polymer matrix. The analysis was carried out by developing a finite-element program which is capable of evaluating the stress and strain distribution of short fibre composites under axial loading (see Fig. 1a). Using the concept of balance of force we can express the modulusE _x along the loading direction as a function of the mechanical properties of the fibre and matrix materials, fibre aspect ratio,l/d, loading angle,, and fibre volume fraction,V _f. Then we apply the elastic-viscoelastic correspondence principle to replace all the mechanical properties of the composite, fibre and matrix materials such asE _x,E _f,E _m,G _m, by the corresponding complex moduli such asE _x +iE _x , andE _f +iE _f . After separation of the real and imaginary parts, we can expressE _' ^x/t' andE _x ^t" as functions of the fibre aspect ratio,l/d, loading angle,, stiffness ratio,E _f/E _m, fibre volume fraction,V _f, and damping properties of the fibre and matrix materials such as _f and _m. Numerical results of the composite storage modulus,E _x , loss modulus,E _x , and loss factor (damping), _C, are plotted as functions of parameters such asl/d,,V _f, and are discussed in terms of variations ofl/d,, andE _f/E _m, in detail. It is observed that for a given composite, there exist optimum values ofl/d and at whichE _x and _c are maximized. The results of this paper can be used to optimize the performance of composite structures.Nomenclature A _c,A _f,A _m cross-sectional area of composite, fibre and matrix, respectively - d fibre diameter - E _L longitudinal modulus of composite (along the fibre direction) (see Fig. 1a) - E _T transverse modulus of composite (see Fig. 1a) - E _x modulus of composite along thex-direction (see Fig. 1b) - E _f tensile modulus of fibre - E _m tensile modulus of matrix - G _m shear modulus of matrix - G _LT in-plane shear modulus of composite (see Fig. 1a) - l fibre length - m tip to tip distance between fibres - i (–1)^1/2 - R one-half of centre-to-centre fibre spacing - V _f fibre volume fraction - x distance along fibre from end of fibre - defined in Equation 22 - defined in Equation 3 - ^* defined in Equation 19 - _L extensional (longitudinal strain) of composite - _f, _m extensional (longitudinal strain) of fibre and matrix, respectively - _c, _f, _m extensional loss factor of composite, fibre and matrix respectively - G _m shear loss factor of matrix - angle between fibre and thex-direction - ¯ _c, ¯ _f, ¯ _m average longitudinal stress in composite, fibre and matrix, respectively - longitudinal stress in fibre - shear stress at fibre-matrix interface - defined in Equation 23     

Charged defects-controlled conductivity in Ge-ln-Se glasses

The variation of the d.c. electrical conductivity, , with composition and temperature was investigated for glasses of the Ge-In-Se system. The results indicate a decrease in the activation energy for electrical conductivity, E, and an increase in on introduction of indium into Ge-Se glasses. The changes in E and with composition (selenium content in the glasses) are identical for the Ge_x In₅ Se_95–x and Ge_x In₈Se_92–x families. The results have been traced to the conduction controlled by charged defects in these chalcogenide glasses. The changes in E and have been explained by a shift in the Fermi level, being brought by the introduction of indium.     

Steady state creep of a composite material with short fibres

The shear within a matrix volume is assumed to be an important process during the creep of composite material reinforced with short rigid fibres. The rate of elongation of such a composite with certain fibre distributions can be estimated. The agreement with a few experimental data is reasonably good.List of main symbols V _f volume fraction of fibres in composite - aspect ratio of a fibre - L length of a fibre - h transverse size of a fibre - h interfibre spacing - m, _m , _m constants for creep for a matrix material - n, _f, _f constants of creep for a fibre material _{f m} - v rate of relative motion of two fibres - _* ultimate strength of a fibre - ^* the first critical value of aspect ratio - ^** the second critical value of aspect ratio This work was carried out when the author was a guest worker at the National Physical Laboratory, Teddington, Middlesex, UK.     

The tensile strength and ductility of continuous fibre composites

The plastic instability approach has been applied to the tensile behaviour of a continuous fibre composite. It is shown that the combination of two components with different strengths and degrees of work-hardening produces a new material with a new degree of work-hardening, which may be determined by the present analysis. Expressions for the elongation at rupture and the strength of a composite have been obtained and the results of the calculation are compared with some experimental data.List of symbols V _f volume fraction of fibres in composite - , , true strain of fibre, matrix and composite - s true stress - , , nominal stress on fibre, matrix and composite - _*, _*, _* critical stress of fibre, matrix and composite (ultimate tensile strength) - _*, _* critical strain of separate fibre and matrix - _* critical strain of composite - Q external load - A cross-sectional area - A ₀ initial value of area     

The influence of brittle boundary layers on the strength of fibrous metallic composites

The brittle boundary layers often caused during the production of composites or by their treatment at higher temperatures, may change the mechanical properties. On the steel wire/aluminium system the growth of the intermetallic boundary phase and its influence on the strength of the composite were investigated. Hence followed a maximum strength at small layer thicknesses. By means of fracture investigations new models were developed which allow the calculation of the dependence of strength behaviour on layer thickness.List of symbols E _f Young's modulus of fibre - E _b Young's modulus of boundary layer - _c external load - _f tensile stress in the fibre - _m tensile stress in the matrix - _b tensile stress in the boundary layer - _uc,f,b ultimate strength of the composite, the fibre or the boundary layer, respectively - averaged stress in the fibre - _bf shear stress in the boundary layer-fibre interface - ₀ shear strength of the boundary layer-fibre-interface - _uf ultimate strain of the fibre - fraction of the layer which has grown into the matrix - Weibull parameter - ^–1 characteristic length of stress transfer between fibre and boundary layer - d diameter of the boundary layer - 2l length of the boundary layer segments - r _f fibre radius - u(x) displacement field - v _{f, b,m} volume fraction of fibres, boundary layer or matrix respectively.     

Evaluation of biaxial stress failure surfaces for a glass fabric reinforced polyester resin under static and fatigue loading

In an attempt to evaluate failure theories for a glass fabric reinforced polyester resin over 370 tests have been conducted on thin-walled tubes under combined axial loading and internal pressure, both for static and fatigue loading. For plane stress the results are considered in relation to imaginary failure surfaces in ₁, ₂, ₆ space. A limited measure of agreement between theories and results can be obtained after highly subjective selection of data. Only those theories which involve complex stress properties provide a reasonable fit. The behaviour of tubular specimens is strongly influenced by the presence of joints in the reinforcements.Nomenclature _x , _y nominal hoop and axial (principal) stresses in a thin-walled tube - ₁, ₂, ₆ normal and shear stresses in the direction of the principal material axes - F ₁,F ₂,F ₆ strengths in the principal material directions and the in-plane shear strength - F _1t,F _2t tensile strengths in the principal material directions - F _1c,F _2c compressive strengths in the principal material directions - K ₂ a constant evaluated from a combined stress test - H ₁₂ normal stress interaction component of a strength tensor - off-axis angle - S-N curve conventional stress-log life fatigue curve - R principal stress ratio, _y /gs _x      

Pyramidal yield criteria for epoxides

The tensile, compressive and shear yield strengths of two epoxides were measured under superposed hydrostatic pressure extending to 300 MN m^–2. For both materials, the ratio of the moduli of the tensile, _T, to compressive, _C, yield stress at atmospheric pressure was approximately 34, as has been reported previously for a number of thermoplastics. The ₂= ₃ envelope in stress space was plotted according to these two-parameter ( _C and _T) yield criteria: conical, paraboloidal and pyramidal; the best correlation was with the last. The experimental tensile and compressive data for tests under pressure, however, fit slightly better two straight lines which are consistent with a three-parameter single hexagonal pyramidal yield surface. For plane stress and shear under pressure yield envelopes of these surfaces, the correlation with experimental data is again best for the pyramidal criteria, except for biaxial or triaxial tension when these resins are brittle. The third independent parameter employed in the pyramidal criterion was the equi-biaxial compressive yield stress, determined by tensile experiments under appropriate superposed hydrostatic pressure; alternatively plane strain compressive yield stress, _PC, may be used.     

A modified Weibull treatment for the analysis of strength-test data from non-identical brittle specimens

Powder compacts (e.g., pharmaceutical tablets) manufactured on commerically available machines are not strictly identical but show inevitable variability in their weights, thicknesses and compaction pressures. Consequently, the variability in fracture-stress data obtained from such brittle specimens is greater than that due to the inherent strength variability of the material itself. A modified Weibull analysis has been developed so that a more accurate estimate of the inherent variability of the mechanical strength of the material can be derived from test data obtained from commercially produced compacts; its application is illustrated.Nomenclature D diameter - f() relative frequency of occurrence of specimens with density and volume - F minimization function - i ascending rank number of a fracture stress - m Weibull modulus - N _tot number of specimens in a batch - N() number of specimens with densities in the range to + d and volumes in the range to + d - P _f failure probability - p _u upper punch compaction pressure - t thickness - volume - w weight - W _f fracture load - density - _f fracture stress - ¯ _f mean fracture stress of a batch - ¯ _f() mean fracture stress of specimens with density and volume - ₀ scale parameter or normalizing factor - _u location parameter or threshold stress