Brian N. Cox

The mechanics of matrix cracking in brittle-matrix fiber composites

Abstract Matrix fracture in brittle-matrix fiber composites is analyzed for composites that exhibit multiple matrix cracking prior to fiber failure and have purely frictional bonding between the fibers and matrix. The stress for matrix cracking is evaluated using a stress intensity approach, in which the influence of the fibers that bridge the matrix crack is represented by closure tractions at the crack surfaces. Long and short cracks are distinguished. Long cracks approach a steady-state configuration, for which the stress intensity analysis and a previous energy balance analysis are shown to predict identical dependence of matrix cracking stress on material properties. A numerical solution and an approximate analytical solution are obtained for smaller cracks and used to estimate the range of crack sizes over which the steady-state solution applies.

Failure mechanisms of 3D woven composites in tension, compression, and bending

Abstract Observations of failure mechanisms in monotonic loading are reported for graphite/epoxy composites containing three-dimensional (3D) interlock weave reinforcement. The key phenomena are delamination and kink band formation in compression, tow rupture and pullout in tension, and combinations of these in bending. The materials exhibit great potential for damage tolerance and notch insensitivity. This is partly due to the presence of geometrical flaws that are broadly distributed in strength and space; and partly to the coarseness of the reinforcing tows, which leads to extensive debonding and reduced stress intensification around sites of failure. Rules of mixture corrected for the effects of tow irregularity suffice to estimate elastic moduli. Rough estimates of the stress at which the first failure events occur in compression or tension can be made from existing micromechanical models. Ultimate tensile failure might be modeled by regarding failed tows that are being pulled out of the composite as a cohesive zone. The characteristic length estimated for this zone, which is a direct measure of damage tolerance and notch insensitivity, has very large values of order of magnitude 0.1–0.5 m.

Tensile fracture of brittle matrix composites: Influence of fiber strength

Abstract A stress intensity approach is used to analyze tensile failure of brittle matrix composites that contain unidirectionally aligned fibers held in place by friction. In general, failure may initiate either by growth of a crack in the matrix, or by fracture of fibers that bridge the matrix crack. Subsequently, these failure processes may continue either unstably or stably with increasing applied stress. Solutions to the fracture mechanics analysis are obtained numerically in normalized form, with one microstructural variable, the normalized fiber strength. The analysis defines transitions between failure mechanisms and provides strength/crack-size relations for each mechanism. Explicit relations are derived for the matrix cracking stress (noncatastrophic failure mode), the condition for transition to a catastrophic failure mode, and the fracture toughness in a region of catastrophic failure, in terms of microstructural properties of the composite.

A mechanistic approach to the properties of stitched laminates

Abstract New insights are presented into the mechanisms responsible for changes to the in-plane mechanical properties of polymer matrix laminates as a result of through-the-thickness stitching. A critical appraisal of a large amount of published mechanical property data reveals that stitching usually reduces the stiffness, strength and fatigue resistance of a laminate by not more than 10–20%, although in a few cases the properties remain unchanged or increase slightly. This range of changes is observed for loading in compression, tension, bending or shear. Softening and strengthening mechanisms are proposed to account for the changes in properties due to stitching. Areas of research that are needed to further the understanding of the relationships between mechanisms and properties are identified. These include more detailed reporting of the physical properties of the stitched and unstitched laminates (e.g. fiber content, fiber distortions), the stitching conditions (e.g. yarn tension) and more thorough examination of observed mechanisms.

A binary model of textile composites-I: formulation

Abstract This paper presents a finite element model of polymer composites with three-dimensional (3D) reinforcement. The model performs Monte Carlo simulations of failure under monotonic and fatigue loading. The formulation of the model is guided by extensive prior experimental observations of 3D woven composites. Special emphasis is placed on realistic representation of the pattern of reinforcing tows, random irregularity in tow positioning, randomness of the strengths of constituent elements, and the mechanics of stress redistribution around sites of local failure. The constitutive properties of model elements (or their distributions) are based on micromechanical models of observed failure events. Material properties that are appropriately analyzed by the model are contrasted with those amenable to much simpler models. Some illustrative model simulations are presented. Prescriptions for the calibration of the model for design and reliability applications and details of its performance in simulating the elastic and damaged regimes of 3D woven composites will appear in subsequent papers.

Stable and unstable solutions for bridged cracks in various specimens

This paper reviews the formulation of the problem of a bridged crack in an elastic medium as an integral equation, noting explicit forms for specimens of various common shapes. Numerical methods are provided for the convenient and efficient self-consistent solution of the integral equation when the bridging tractions, p, are a function of crack opening displacement, u, rather than an explicit function of position in the crack. Methods are presented for determining physically and computationally unstable crack configurations for various forms of p(u), including functions possessing a peak. Knowledge of both stable and unstable solutions is essential to demarking the transition from noncatastrophic (or ductile) failure to catastrophic (or brittle) failure.

Mixed mode delamination of polymer composite laminates reinforced through the thickness by z-fibers

Abstract The mixed mode delamination behavior of through-thickness reinforced carbon–epoxy laminates was investigated using two different test specimens, a T-stiffener and a mixed-mode bending (MMB) specimen. Small quantities of titanium or carbon z-fibers (short rods) substantially improve delamination resistance in both types of specimen. Reinforcement raises the ultimate strength of the MMB specimen by a factor of three. However, the failure sequence and therefore the ultimate load in the T-stiffeners depend strongly on the test configuration. No change in ultimate load is seen in some cases but up to 40% improvement is observed in others. Improved delamination resistance results from crack bridging by the z-fibers, which reduces the driving force for crack growth. Mode I crack displacement is suppressed more effectively than mode II displacement, resulting in purely mode II cracking in what without z-fibers would be a mixed mode or primarily mode I loading situation. This important consequence of so-called large scale bridging effects confirms recent theoretical results for delamination specimens. The mechanisms of bridging and crack propagation are described here in detail.

Real-time quantitative imaging of failure events in materials under load at temperatures above 1,600 °C

Ceramic matrix composites are the emerging material of choice for structures that will see temperatures above ~1,500 °C in hostile environments, as for example in next-generation gas turbines and hypersonic-flight applications. The safe operation of applications depends on how small cracks forming inside the material are restrained by its microstructure. As with natural tissue such as bone and seashells, the tailored microstructural complexity of ceramic matrix composites imparts them with mechanical toughness, which is essential to avoiding failure. Yet gathering three-dimensional observations of damage evolution in extreme environments has been a challenge. Using synchrotron X-ray computed microtomography, we have fully resolved sequences of microcrack damage as cracks grow under load at temperatures up to 1,750 °C. Our observations are key ingredients for the high-fidelity simulations used to compute failure risks under extreme operating conditions.

Extrinsic factors in the mechanics of bridged cracks

Abstract Calculations for single-edge notch specimens under uniform remote tension are used to demonstrate the profound influence of the extrinsic factors of specimen shape and load distribution on the propagation of bridged cracks when the bridging zone length is comparable to any of the dimensions of the crack or the specimen. The influence of the extrinsic factors is so strong that there is a grave risk of seriously nonconservative predictions of strength and reliability if standard engineering methods are used for materials exhibiting such bridged cracks. However, this difficulty can be resolved by regarding the relationship between the bridging tractions p and the crack opening displacement u as a fundamental material property. Extensive calculations are summarized for one class of relations p(u) that rise to a peak corresponding to ligament failure and then fall gradually to zero at large u. Resistance curves of great variety are displayed. The central role of the “bridging length scale,” the initial crack extension over which the bridging zone matures, is demonstrated. The roles of various features of p(u) in determining the transition from noncatastrophic (or ductile) failure to catastrophic (or brittle) failure are examined.

Concepts for bridged Mode ii delamination cracks

Limiting cases and length scales are detailed for Mode II delamination cracks bridged by through-thickness reinforcement. Analytical results are found for two limits: a steady-state configuration indicative of noncatastrophic failure and a small-scale bridging configuration indicative of catastrophic failure. General large-scale bridging conditions are studied numerically using bending theory for anisotropic plates. The effects of the mechanical properties of the laminate and the reinforcement, notch length, and plate thickness on the transition between the two limiting configurations, notch sensitivity and mechanical behavior are analyzed. All of these effects can be expressed succinctly in terms of a few length scales which are material-structure parameters involving the plate thickness.