John A. Nairn

The Strain Energy Release Rate of Composite Microcracking: A Variational Approach

A variational analysis approach has been used to determine the two- dimensional thermoelastic stress state in cross-ply laminates of type [0 m /90 n ] s and [90 m / 0 n ] s . The stress analysis was used to calculate the energy release rate due to formation of a new microcrack. The analysis accurately includes the effect of residual thermal stresses. When compared with experiments, the new energy release rate expressions are found to predict typical data using a single value for the critical energy release rate for microcrack ing. This critical energy release rate has a physical interpretation as a microcracking fracture toughness or as an intralaminar fracture toughness. We also used the variational solution to get an analytical expression for the longitudinal expansion coefficient of the microcracked cross-ply laminate as a function of microcrack density. Finally, variational theorems are used to show that this new microcracking fracture analysis is rigorously more accurate than previous attempts at the same problem.

On the Use of Shear-Lag Methods for Analysis of Stress Transfer in Unidirectional Composites

Abstract The ‘shear-lag’ analysis method is frequently used for analysis of stress transfer between the fiber and the matrix in composites. The accuracy of shear-lag methods has not been critically assessed, in part because the assumptions have not been fully understood. This paper starts from the exact equations of elasticity for axisymmetric stress states in transversely isotropic materials and introduces the minimum assumptions required to derive the most commonly used shear-lag equations. These assumptions can now be checked to study the accuracy of shear-lag analysis on any problem. Some sample calculations were done for stress transfer from a matrix into a broken fiber. The shear-lag method did a reasonable job (within 20%) of predicting average axial stress in the fiber and total strain energy in the specimen provided the shear-lag parameter most commonly used in the literature is replaced by a new one derived from the approximate elasticity analysis. The shear-lag method does a much worse job of predicting shear stresses and energy release rates. Furthermore, the shear-lag method does not work for low fiber volume fractions.

2.12 – Matrix Microcracking in Composites

2.13.

The initiation and growth of delaminations induced by matrix microcracks in laminated composites

A recent variational mechanics analysis of microcracking damage in cross-ply laminates of the form [(S)/90n]s, where (S) is any orthotropic sublaminate much stiffer than [90n], has been extended to account for the presence of delaminations emanating from the tips of microcracks in the [902n]T sublaminate. The new two-dimensional stress analysis is used to calculate the total strain energy, effective modulus, and longitudinal thermal expansion coefficient for a laminate having microcracks and delaminations. These results are used to calculate the energy release rate for the initiation and growth of a delamination induced by a matrix microcrack. At low crack densities, [(S)/90n]slaminates are expected to fail by microcracking and to show little or no delamination. At some critical crack density, which is a function of laminate structure and material properties, the energy release rate for delamination exceeds that for microcracking and delamination is predicted to dominate over microcracking. A quasi-three-dimensional model is used to predict the propagation of arbitrarily shaped delamination fronts. All predictions agree with experimental observations.

Matrix solidification and the resulting residual thermal stresses in composites

The disparate thermal expansion properties of the fibres and matrices in high-performance composites lead to an inevitable build up of residual thermal stresses during fabrication. We first discuss the thermal expansion behaviour of thermoplastic and thermoset polymers that may be used as high-performance composite matrices. The three classes of polymers considered are epoxies, amorphous thermoplastics, and semicrystalline thermoplastics. The relevant thermal expansion data for prediction of the magnitude of the residual stresses in composites is the zero (atmospheric)-pressure thermal expansion data; these data are plotted for a range of thermoplastics and a typical epoxy. Using the technique of photoelasticity, we have measured the magnitude of the residual stresses in unidirectional graphite composites with an amorphous thermoplastic matrix (polysulfone) and with an epoxy matrix (BP907). The temperature dependence of the residual stress build up and the resulting magnitude of the residual stresses correlate well with the thermal and physical properties of the matrix resin.

A variational mechanics analysis of the stresses around breaks in embedded fibers

Abstract Embedded single-fiber tests are often used to characterize the fiber/matrix interface, but their interpretation is usually limited by reliance on the qualitative view of the stresses provided by shear-lag analyses. This paper describes a new, three-dimensional, axisymmetric solution for the stresses around breaks in embedded fibers. The new solution is obtained using variational mechanics. It obeys equilibrium and traction boundary conditions exactly, obeys compatibility approximately, includes all components of the stresses, accounts for interacting fiber breaks, and includes residual thermal stresses. We apply the stress analysis to the single-fiber fragmentation test. In some sample calculations, we plot all components of stress at the fiber/matrix interface and give predictions for an “ideal” single-fiber fragmentation test. The stress analysis technique is readily adaptable to new problems such as the single-fiber pull-out test, the microdrop debond test, the description of interfacial fracture or yielding, and the effect of interfacial friction.

Energy release rate analysis for adhesive and laminate double cantilever beam specimens emphasizing the effect of residual stresses

Abstract The mode I energy release rate, including the effect of residual stresses, was evaluated for both adhesive and laminate double cantilever beam specimens. The energy release rate can be partitioned into a mechanical term and a residual-stress term in beam theory. The beam-theory mechanical term is not very accurate, but can be corrected by a slight modification to a previous correction factor. This correction factor accounts for crack tip rotation of the specimen arms. The beam-theory residual-stress term is very accurate for a wide range of specimen geometries; it can be used without correction. The consequence of ignoring residual stresses is that one measures an apparent toughness instead of a true toughness. The error between the apparent toughness and true toughness can be calculated for a given specimen geometry and amount of residual stresses. Such errors can be large and are often larger than the correction required for crack-tip rotation effects. In double cantilever beam specimens used to study laminate delamination, the errors are large when the delaminating arms, considered by themselves, are unsymmetric laminates. Some experimental methods are suggested which can be used to correct for residual stress effects.

Picosecond fluorescence kinetics and energy transfer in chloroplasts and algae

Single-photon timing with picosecond resolution is used to investigate the kinetics of the fluorescence emission of chlorophyll a in chloroplasts from spinach and pea and in the algae Chlorella pyrenoidosa and Chlamydomonas reinhardii. The fluorescence decay is best described by three exponential components in all species. At low light intensity and with open reaction centers of Photosystem II (F0), we find lifetimes of approx. 100, 400 and 1100 ps for the three components. Closing the reaction centers by addition of 3-(3′,4′-dichlorophenyl)-1,1-dimethylurea plus hydroxylamine and by increasing light intensity produces only minor changes in the almost constant fast- and medium-lifetime components; however, there is a dramatic increase in the yield of the slow component, by a factor of about 20, accompanied by only a modest increase in the lifetime to 2200 ps (Fmax). In good agreement with previous fluorescence lifetime measurements, we find an increase in the averaged lifetime of the three components from 0.5 to 2.0 ns, which is proportional to the 4-fold increase in the total fluorescence yield. Our time-resolved results are inconsistent with models which are based on the proportionality between lifetime and yield and which involve a homogeneous origin of fluorescence that is sensitive to the state of the reaction centers. We conclude that the variable part of the fluorescence, which is dominated by the slow phase, reflects the kinetics of charge recombination in the reaction center, as proposed previously (Klimov, V.V., Allakhverdiev, S.I. and Paschenko, V.Z. (1978) Dokl. Akad. Nauk S.S.S.R. 242, 1204–1207). The modest increase in lifetime of the slow phase indicates the presence of some energy transfer between photosynthetic units.

The formation and effect of outer-ply microcracks in cross-ply laminates: A variational approach

Abstract The microcracking process in laminates which have outer-ply 90° plies (e.g. [90 m /0 n ] s ) has some important differences from the microcracking process in laminates which lack outer-ply 90° plies (e.g. [0 n /90 m ] s ). Foremost among the differences is the characteristic damage state. [90 m /0 n ] s laminates form antisymmetric or staggered microcracks while [0 n /90 m ] s laminates form symmetric microcracks. To explain observed differences, this paper presents a variational mechanics analysis of the stresses and the energy release rate in a [[90 m /0 n ] s laminate having staggered microcracks. The new analysis and a previous analysis of [0 m /90 n ] s laminates are used to assess the effect of laminate structure on the mechanical properties and failure properties of cross-ply laminates. The findings are as follows. (1) A given level of microcracking damage causes a greater amount of degradation in mechanical properties in [[90 m /0 n ] s laminates than in the corresponding [0 n /90 m ] s laminates. (2) Although [[90 m /0 n ] s laminates will initiate microcracks at lower loads, the corresponding [0 n /90 m ] s laminates will develop more microcracks after continued loading. (3) A bending effect present in [90 m /0 n ] s laminates but not in [0 n /90 m ] s laminates promotes mode I delamination in [90 m /0 n ] s laminates.

On the use of planar shear-lag methods for stress-transfer analysis of multilayered composites

Shear-lag equations for analysis of stresses in a multilayered composite were derived using a series of approximations to exact two-dimensional elasticity methods. The shear-lag equations derived with the fewest assumptions were termed the optimal, shear-lag analysis for planar problems in composites. A solution method for these equations was outlined based on eigenanalysis of a matrix of shear-lag parameters. The optimal, shear-lag analysis differs from most prior shear-lag methods in the literature. By adding more assumptions, we could reduce the optimal analysis to two common, prior shear-lag methods. These prior methods were labeled as interlayer, shear-lag analysis and parametric, interlayer, shear-lag analysis. Because these two interlayer methods required more assumptions than the optimal method, they are less accurate than that method. Several examples illustrated the types of problems that can be accurately solved by shear-lag analysis and the differences in accuracy between the various shear-lag methods. The results of this paper can be used to guide the derivation of future, improved shear-lag models or to evaluate the quality of prior shear-lag models.