S. Leigh Phoenix

The Chain-of-Bundles Probability Model For the Strength of Fibrous Materials I: Analysis and Conjectures

Analytical results are discussed for the chain-of-bundles probability model for the strength of fibrous materials. Two load sharing rules are considered for failed and nonfailed fibers in a bundle. The first is the equal load sharing rule of classical analysis, and the second is a local load sharing rule which is more realistic for composite materials. A rather detailed discussion of past statistical analysis is given. From a careful study of previous results, several conjectures and key questions about the behavior of the strength are generated. Also, an exact analysis of failure is per formed so that the properties of the strength distribution can be studied. Difficulties of a general analysis are discussed in detail. The sequel will contain a thorough numerical investigation of the model with emphasis on studying the convergence of certain transformed distributions and on answering key questions raised in this study.

A new membrane model for the ballistic impact response and V50 performance of multi-ply fibrous systems

This paper develops an analytical model for the ballistic impact response of fibrous materials of interest in body armor applications. It focuses on an un-tensioned 2D membrane impacted transversely by a blunt-nosed projectile, a problem that has remained unsolved for a half a century. Membrane properties are assumed characteristic of the best current body armor materials (Kevlar®, Spectra®, Zylon®, S2 glass), which have very high stiffness and strength per unit weight, and low strain-to-failure. Successful comparisons will be made with extensive experimental data on such material systems as reported by Cunniff [Decoupled response of textile body armor. Proc. 18th Int. Symp. of Ballistics, San Antonio, Texas, 1999a, pp. 814–821; Vs–Vr relationships in textile system impact. Proc. 18th Int. Symp. of Ballistics, San Antonio, Texas, 1999b; Dimensional parameters for optimization of textile-based body armor systems, Proc. 18th Int. Symp. of Ballistics, San Antonio, Texas, 1999c, pp. 1303–1310]. Our mathematical formulation draws on the seminal work of Rakhmatulin and Dem’yanov [Strength Under High Transient Loads, 1961, pp. 94–152]. Under constant projectile velocity we first develop self-similar solution forms for the tensile ‘implosion’ wave and the curved cone wave that develops in its wake. Through matching boundary conditions at the cone wave front, we obtain an accurate approximate solution for the membrane response including cone wave speed and strain distribution. We then consider projectile deceleration due to membrane reactive forces, and obtain results on cone velocity, displacement and strain concentration versus time. Other results obtained are the membrane ballistic limit, or V50 velocity, and the residual velocity when penetrated above this limit. We then derive an exact functional representation of a V50 ‘master curve’ found empirically by Cunniff [ibid] to reduce data for a wide variety of fabric systems impacted by blunt cylindrical projectiles. This curve is given in terms two dimensionless parameters based only on fiber mechanical properties and the ratio of the fabric areal density to the projectile mass divided by its area of fabric contact. Our functional representation has no fitting parameters beyond one reflecting uncertainty in the effective diameter of the impact zone relative to the projectile diameter, and even then the values are consistent across several experimental systems. The extremely successful comparison of our analytical model to experimental results in the literature raises fundamental questions about many long-held views on fabric system impact behavior and parameters thought to be important.

A Study of Statistical Variability in the Strength of Single Aramid Filaments

Variability in the failure load, tenacity, and linear density of single aramid filaments is studied experimentally. Data indicate that both the failure load and the tenacity of filaments, for a given gauge length and yarn cross section, can be fitted to a two parameter Weibull distribution; however the fit is better for tenacity than for failure load, and the Weibull shape parameter for the former is typically smaller. Within a yam cross section filaments vary significantly in linear density (and diameter), and this variability contributes a component to the variability in failure load, but not to the tenacity. Also, the mean tenacity and the variability in linear density and in failure load may differ greatly from spool to spool. The implications of the variability are discussed in light of the work of Bunsell. The effect of gauge length on the strength distribution of filaments is examined. Weibull statistics are used to separate out this effect. As previously shown for other brittle filaments, the Weibull shape parameter for aramid filaments depends on the gauge length; however values for the shape parameter calculated at a fixed gauge length are substantially lower than those obtained by a procedure based on varying the gauge length. This suggests short range correlations in flaw strengths along a filament.

Stress concentrations around multiple fiber breaks in an elastic matrix with local yielding or debonding using quadratic influence superposition

Abstract A new computational technique, called the quadratic influence superposition (QIS) technique, is developed to study the stresses around arbitrary arrays of fiber breaks in a unidirectional composite loaded in simple tension, and consisting of elastic fibers in a matrix, which is either elastic-perfectly plastic or which can debond at the interface leaving residual friction. The method involves extending a recently developed break influence superposition (BIS) technique, where to model the behavior of damaged (yielded or debonded) matrix elements, we use special compensating shear stress profiles and develop the corresponding influence functions. The QIS technique appears to be at least an order of magnitude more efficient than other numerical schemes as the computation time is tied mainly to the amount of damage, and it is more accurate than a simpler version of this technique developed earlier. In illustrative examples, the method determines the Mode I fiber and matrix stress distributions around a “center crack” consisting of up to 31 contiguous fiber breaks. Incremental treatment is needed to establish the extent of the inelastic regions and the results, which achieve excellent agreement with exact shear lag analyses, clearly show that QIS calculated these correctly. Results show that the extent of the matrix damage region increases approximately linearly with applied load and nonlinearly with the number of breaks. The stress concentrations and overload profiles along nearby unbroken fibers are altered as compared to the fully elastic case with magnitudes reduced but length scales increased.

Statistics for the strength and size effects of microcomposites with four carbon fibers in epoxy resin

Abstract Experimental results are presented for the strength distribution and length effects of carbon/epoxy microcomposites consisting of four carbon fibers (Hercules AS4) embedded in one of two different epoxies approximately in a square array at a fiber volume fraction of about 0.7. One epoxy (Dow DER 331 with Dow DEH 26 hardener) was stiff in bulk with a strain to failure of less than 10%, and the other (50% by weight of Dow DER 331 blended with 50% Dow DER 732 and Dow DEH 26 hardener) was very flexible in bulk with large apparent ductility and a strain to failure of 40%. Individual fibers tested at two gauge lengths, 1 and 20 cm, yielded a Weibull shape parameter of about 5 at each length, but the strength versus length plot for the scale parameter on log-log coordinates suggested a Weibull shape parameter of about 8.3. Therefore, the ratio a was about 0.6 rather than unity as in previous work in microcomposites. Thus a modified statistical theory for microcomposite strength as a function of length was developed to incorporate this a effect. Experimental verification of both the size effect and a effect are the two major advantages of the present model over previous Weibull fiber composite models. Strength data for microcomposites at two gauge lengths, 1 and 20 cm, for both epoxies were generated and plotted on Weibull coordinates, where both the expected doubling and tripling of the Weibull shape parameter as compared to that of the fiber and the predicted a effect for microcomposite length were observed. The stiff epoxy produced stronger composites at both gauge lengths. On the other hand, the length effect was more severe than predicted in the 20 cm flexible epoxy microcomposite apparently due to a combination of epoxy yielding in shear and fiber/matrix debonding at fiber breaks, and a lower matrix ultimate strength, leading to longer effective load transfer lengths, δ.

Time evolution of overstress profiles near broken fibers in a composite with a viscoelastic matrix

Abstract The shear-lag model is applied to a monolayer, unidirectional, fiber-reinforced composite loaded in tension. The monolayer contains an infinite number of parallel fibers, with an arbitrary number of them broken simultaneously. While the fibers are modeled as linear elastic, a linear viscoelastic constitutive law is assumed for the matrix material. The time evolution of the overstress profiles in the fibers and matrix near breaks is determined and the time dependence of the effective load transfer length is calculated. Exact closed-form solutions as well as approximate evaluations of the above quantities are given for a power-law creep compliance model, suitable for most epoxy thermosetting resins as matrix materials. These results are also extended to the case of sequential breaks in time and the ease of an idealized indentation test.

The Asymptotic Time to Failure of a Mechanical System of Parallel Members

A mechanical system of parallel members differs from the usual parallel element model studied in reliability monographs in that the nonfailed members at any instant must share an applied system load according to some rule. The resulting dependence among members causes difficulty in deriving useful exact results. However under certain reasonable probabilistic assumptions regarding single member failure, system loading and member load sharing, it can be shown that the system failure time is asymptotically normally distributed as the number of members grows large. We give the asymptotic mean and variance as functions of the parameters of loading and single member behavior. We study several cases of practical importance and show that many of the behavioral features of single members carry over to the system. But on a load per member basis, the system is weaker than a single member in many respects. However the variance in system time to failure can be reduced by increasing the number of members.

Statistics of fracture for an elastic notched composite lamina containing Weibull fibers— Part I. Features from Monte-Carlo simulation

Abstract Monte-Carlo simulation is used to study the effects of the statistics of fiber strength on the fracture process, the fracture resistance, and the overall strength distribution for an elastic composite lamina with an internal transverse notch of N contiguous, broken fibers (0 ≤ N ≤ 51). To isolate the effects of variability in fiber strength, we assign individual fiber strengths drawn from a Weibull distribution with shape parameter γ ≥ 3 typical of commercial fibers, and we consider a simple case where fiber strength does not vary along the fiber length. The latter forces fibers to fail in the notch plane, eliminating the need to consider staggered breaks, debonding and fiber pullout. So under an increasing tensile load, failure develops through a progression of random fiber fractures governed by an interplay of stress concentrations and variations in fiber strength along the notch plane. Calculation of the fiber stresses for every configuration of surviving and broken fibers that occurs as the load is increased up to catastrophic failure is performed by an efficient, shear-lag based, break influence superposition (BIS) technique. Results show that the mean strength relative to the deterministic value (y = ∞) and mean number of new fiber fractures up to crack instability all increase with N regardless of γ, whereas variability in strength decreases. For smaller γ, we identify mechanisms responsible for flaw intolerance in the short notch regime and for toughness in the long notch regime, and show that variability in fiber strength can manifest as a nonlinear mechanism in an otherwise elastically deforming composite. Indeed as N increases we observe R-curve behavior, which is most pronounced for the smallest γ values where fracture resistance increases markedly and where mean fracture strength scales inversely with the initial notch size slower than the usual power of 1 2 . Compared to simulation results, a weakest-link or first failure model and unique fiber strength model severely underestimate fracture strength, failing to capture the statistical aspects of composite fracture.

Lifetime statistics for single kevlar 49 filaments in creep-rupture

Experimental data are presented for the lifetime of single Kevlar 49 filaments under moderate to high stress levels at standard ambient conditions (21°C, 65% r.h.). Filaments were drawn from two spools, A and B, taken from the same production lot. Previously we found that filaments from spool A were 7% lower in mean strength but much less variable in diameter than filaments from spool B; however, the respective variabilities in failure stress were equivalent. The lifetime data were interpreted in light of a previously developed kinetic model embodying Weibull failure statistics and power law dependence of lifetime on stress level. As predicted, lifetime data at each stress level generally followed a two-parameter Weibull distribution with a shape parameter value near 0.2. Based on absolute stress levels, the filaments drawn from spool B had a Weibull scale parameter for lifetime about ten times greater than those from spool A; however, when the stress-levels were normalized by the respective Weibull scale parameters for short-term strength, these differences disappeared. With respect to power law dependence of lifetime on stress level, three distinct time domains emerged, each marked by a different power law exponent. Similar behaviour was observed earlier for preproduction Kevlar 49/epoxy strands, and the values for the power law exponents for the filaments agree closely with those for the strands.

Time evolution of stress redistribution around multiple fiber breaks in a composite with viscous and viscoelastic matrices

Abstract We develop an efficient computational technique, called viscous break interaction (VBI), to determine the time evolution of fiber and matrix stresses around a large, arbitrary array of fiber breaks in a unidirectional composite with a matrix that creeps. The matrix is assumed to be linearly viscoelastic or viscous in shear following a power-law in time (creep exponent 0 ⩽ α ⩽ 1), and interface debonding or slip is not permitted. Such a law is applicable to polymeric matrices over a wide range of temperatures or to a viscous, glassy interphase in a ceramic composite with elongated microstructure. Specifically, we consider an infinitely large, 2-D composite lamina in the shear-lag framework of Hedgepeth, and the multiple break formulation is built on weighted superposition using influence functions based on the response to an isolated break. We apply the method to problems involving large transverse cracks (i.e., aligned, contiguous breaks), fully bridged cracks, and arrays of interacting, longitudinally staggered breaks. In each case we calculate the time evolution of stress concentrations and displacements of individual fibers. In comparing cracks vs spatially staggered breaks, the results reveal interesting contrasts in the time variation of both peak fiber stress concentrations and break opening displacements. In the latter case, we see behavior consistent with the three stages of creep, and show how the local fiber tensile stresses can rise (and subsequently even fall) at rates depending on various microstructural length scales. The motivation for developing VBI is to provide the computational framework for modeling the statistical features of the lifetime of composites in creep-rupture resulting from an accumulation of many fiber breaks and ultimately localization and collapse.