B. N. Cox

Concepts for bridged cracks in fracture and fatigue

Some recent work on length scales in bridged crack problems is reviewed and enlarged upon. Fundamental differences are highlighted between the solutions obtainable from bridged crack models (nonvanishing crack tip stress intensity factor, Ktip) and traditional cohesive zone models (vanishing Ktip); and from models for which the bridging tractions are nonincreasing (softening) and increasing (hardening) functions of the crack opening displacement. The roles of length scales in determining ductility, notch sensitivity, and whether failure is catastrophic or noncatastrophic are discussed. Length scales for monotonic loading are extended to cyclic loading.

In Quest of Virtual Tests for Structural Composites

The difficult challenge of simulating diffuse and complex fracture patterns in tough structural composites is at last beginning to yield to conceptual and computational advances in fracture modeling. Contributing successes include the refinement of cohesive models of fracture and the formulation of hybrid stress-strain and traction-displacement models that combine continuum (spatially averaged) and discrete damage representations in a single calculation. Emerging hierarchical formulations add the potential of tracing the damage mechanisms down through all scales to the atomic. As the models near the fidelity required for their use as virtual experiments, opportunities arise for reducing the number of costly tests needed to certify safety and extending the design space to include material configurations that are too complex to certify by purely empirical methods.

The Macroscopic Elasticity of 3D Woven Composites

The elastic properties of graphite/epoxy composites with three-dimensional interlock weave reinforcement have been measured over length scales somewhat greater than the characteristic length of the weave pattern. Orientation averaging models similar to those developed elsewhere over the last twenty years provide estimates of elastic constants that are in fair agreement with the experimental data. However, in-plane Youngs moduli are consistently too high and properties related to the through-thickness reinforcement show considerable scatter. Most of the discrepancies can be attributed to waviness and other geometrical irregularities in nominally straight tow segments. Much improved agreement with in-plane properties is obtained by measuring and accounting for the out-of-plane waviness of in-plane tows. Other observed distortions of in-plane tows and irregularity in through-thickness tows are very difficult to quantify experimentally. There results a significant and apparently unavoidable uncertainty in predictions of properties that depend strongly on the through-thickness reinforcement. Nevertheless, the utility of orientation averaging models in applications where the in-plane properties are paramount is clearly verified.

On the tensile failure of 3D woven composites

Abstract Tensile tests are reported for some graphite/epoxy composites with three-dimensional woven interlock reinforcement. Composite failure consists of the accumulation of discrete tow rupture events distributed over a band of damage typically 10–20 mm wide. Load—displacement data for gauges spanning the band indicate work of fracture values ranging from 0.4 to 1.1 MJ m −2 . Most of these unusually high vales derives from the ability of the composite to sustain loads near peak load (≈1 GPa) for displacements significantly beyond those at which tows have all failed. The key mechanism is very strong friction or lockup that couples sliding, broken tows to the surrounding composite. Lockup is the product of the geometrical irregularity of nominally straight tows and clamping compressive stresses generated by the through-thickness reinforcement. Lesser contributions to the work of fracture arise from plastic straightening of tows prior to their rupture and the relatively easy but prolonged pull-out of tows following failure of the lockup mechanism.

Simple models for triaxially braided composites

Comparison of measurements and theory shows that the elastic properties of some triaxially braided glass/urethane composites are predicted as well as they can be by a simple laminate model modified to account for measured tow waviness. Any remaining discrepancies are the result of inevitable irregularity in the positioning of tows and are not grounds for more sophisticated modelling. The same modified laminate model provides estimates of the partitioning of in-plane stresses among tows of different orientation. Estimates of these local stresses correlate well with measured compressive and tensile strengths in the primary load-bearing direction. Absolute values for compressive strength are predicted quite accurately by kink band models, using independently measured values for tow misalignment angles and the critical shear stress for resin flow. Some summary design principles are presented based on the model of elasticity and observations and modelling of tensile and compressive strength.

Delamination and Buckling in 3D Composites

Elementary results are applied to the buckling of stitched laminates and woven composites with three-dimensional (3D) reinforcement that contain delamination cracks. The through-thickness fibers are assumed to provide continuous, linear restoring tractions opposing the deflection of the delaminated layer adjacent to the crack. With the boundary condition that the ends of the delaminated layer are clamped and with deflections permitted in one direction only, there exists a characteristic length a0 for buckling: if the length, 2a, of the delamination crack exceeds 2a0, then, when buckling occurs, it will consist of waves of period 2a, and will usually not span the whole delamination. A simple expression is derived for the minimum density of through thickness reinforcement required to suppress buckling of the delamination layer prior to failure by other mechanisms. It is shown that for typical densities of through-thickness reinforcement in current stitched laminates and 3D woven composites, the length 2a0 will rarely exceed the thickness of the delaminated layer by as much as an order of magnitude. Furthermore, the current generation of these materials are probably considerably overdesigned in the quantity of through-thickness reinforcement they contain for suppressing buckling.

Monte Carlo simulations of the growth of small fatigue cracks

Abstract In this paper are presented Monte Carlo simulations of small, plane embedded or surface-breaking fatigue cracks propagating under mode I cyclic loading through stochastic microstructures. Unlike most prior statistical models of surface-breaking fatigue cracks, which study only the behavior of the visible surface tips of the crack, the simulations calculate the advance of the entire crack front. The rate of growth of any segment of the crack front is determined by empirical or postulated laws that quantify the influence of its immediate microstructural environment. To make feasible the generation of many simulations at reasonable computational expense, approximate, simple algorithms have been derived for estimating the Mode I stress intensity factor around a plane, embedded or surface-breaking crack of any shape. The computational procedures used to carry out the simulations are defined. The potential of the simulations for analysing experimental measurements of fluctuations in crack shape and velocity is explored. The dependence on crack length and the variance of the aspect ratios of small surface-breaking cracks in Al 7075-T6 are accounted for successfully by the effects of microstructure-dependent plasticity-induced closure. Various other statistics of crack shape and velocity that can be predicted by the simulations and compared to experiment are considered. Laws describing the mechanics of crack growth both at and below the surface can be inferred from such comparisons.

Simple, Conservative Criteria for Buckling and Delamination Propagation in the Presence of Stitching

The problem of delamination cracks in laminates that are bridged by stitches is revisited. Analytical solutions for the critical strain for buckling of a bridged, delaminated layer are extended to consider propagation of the delamination crack under in-plane compression in the limit that the crack is long. Numerical estimates for common laminate parameters show that delaminations of interest will not be far from the long crack limit. Therefore, the analytical results should be conservative but not wasteful design guides.

INDUCTIONS FROM MONTE CARLO SIMULATIONS OF SMALL FATIGUE CRACKS

Abstract The relationship between random microstructure and the statistics of small fatigue crack growth is illuminated by Monte Carlo simulations whose formulation has been described in previous work by the author. In this paper, more detailed analysis is presented of fluctuations in crack shape and the observed surface crack velocity, d c d N , using empirical laws of growth proposed elsewhere for Al 2219-T851 and Al 7075-T6. Unusually large values of d c d N are shown to follow retardation or arrest of the surface crack tips while the subsurface crack front has continued to advance. This relation implies that a physically based model of the statistics of d c d N must account for irregular crack shape to predict the distribution of d c d N and remaining lifetime accurately. The value of even rough measurements of surface crack depth in estimating remaining lifetime from nondestructive evaluation of a small crack is demonstrated. It is concluded that the statistics of d c d N require at least two independent random variables for their description. It is also proposed that experimental observations of the degree of irregularity of cracks of various sizes provide the most direct means to date of comparing the mechanics of the growth of surface and subsurface segments of the crack front.

Statistical Mechanics of Early Growth of Fatigue Cracks

This paper presents a mechanistic theory of the statistics of crack initiation and early growth in a polycrystalline alloy subjected to uniform, fully reversed cyclic loads. The physical basis, mathematical form, and numerical solution of the theory are described, and experimental tests are proposed. For brevity and clarity, the simplified version presented here describes surface cracks propagating with one tip permanently blocked.