Delaminations induced by constrained transverse cracking in symmetric composite laminates

Transverse ply cracking and its induced delaminations at the φ/90° interfaces in [. . . /φ_i/φ_m/90_n] _s laminates are theoretically investigated. Three cracked and delaminated model laminates, one five-layer model (FLM) laminate [S^L/φ_m/90_2n/φ_m/S^R] _T and two three-layer model (TLM) laminates I and II, [φ_m/90_2n/φ_m] _T and [S′^L/90_2n/S′^R] _T, are designed to examine constraining mechanisms of the constraining plies of the center 90°-ply group on transverse crack induced delaminations, where S^L, S^R, S′^L and S′^R are sublaminates [. . ./φ_i] _T, [φ_i/. . .] _T, [. . ./φ_i/φ_m] _T and [φ_m/φ_i/. . .] _T, respectively. A sublaminate-wise first-order shear laminate theory is used to analyze stress and strain fields in the three cracked and delaminated laminates loaded in tension. The extension stiffness reduction of the constrained 90°-plies and the strain energy release rate for a local delamination normalized by the square of the laminate strain are calculated as a function of delamination length and transverse crack spacing. The constraining effects of the immediate neighboring plies and the remote plies are identified by conducting comparisons between the three model laminates. It is seen for the examined laminates that the nearest neighboring ply group of the 90°-plies primarily affects the stiffness reduction and also the normalized strain energy release rate, whereas the influences of the remote constraining layers are negligible.     

Influence of temperature on a carbon–fibre epoxy composite subjected to static and fatigue loading under mode-I delamination

In this paper, interlaminar crack initiation and propagation under mode-I with static and fatigue loading of a composite material are experimentally assessed for different test temperatures. The material under study is made of a 3501-6 epoxy matrix reinforced with AS4 unidirectional carbon fibres, with a symmetric laminate configuration [0°]_16/S. In the experimental programme, DCB specimens were tested under static and fatigue loading. Based on the results obtained from static tests, fatigue tests were programmed to analyse the mode-I fatigue behaviour, so the necessary number of cycles was calculated for initiation and propagation of the crack at the different temperatures. G–N curves were determined under fatigue loading, N being the number of cycles at which delamination begins for a given energy release rate. G_ICmax–a, a–N and da/dN–a curves were also determined for different G_cr rates (90%, 85%, 75%, etc.) and different test temperatures: 90 °C, 50 °C, 20 °C, 0 °C, ?30 °C and ?60 °C.     

A unified periodical boundary conditions for representative volume elements of composites and applications

An explicit unified form of boundary conditions for a periodic representative volume element (RVE) is presented which satisfies the periodicity conditions, and is suitable for any combination of multiaxial loads. Starting from a simple 2-D example, we demonstrate that the “homogeneous boundary conditions” are not only over-constrained but they may also violate the boundary traction periodicity conditions. Subsequently, the proposed method is applied to: (a) the simultaneous prediction of nine elastic constants of a unidirectional laminate by applying multiaxial loads to a cubic unit cell model; (b) the prediction of in-plane elastic moduli for [±θ]_n angle-ply laminates. To facilitate the analysis, a meso/micro rhombohedral RVE model has been developed for the [±θ]_n angle-ply laminates. The results obtained are in good agreement with the available theoretical and experimental results.     

A complete mechanical-property characterization of a single composite specimen

For a unidirectional lamina, the in-plane mechanical properties of interest areE ₁₁,E ₂₂, ν₁₂, ν₁₂ andG ₁₂. Only three of the first four listed are independent. The fourth may be calculated from the reciprocal relationshipE ₁₁ ν₂₁ =E ₂₂ ν₁₂, which results from stiffness-tensor symmetry. Measurement of any four of the five properties listed will completely characterize the in-plane mechanical properties. A longitudinal uniaxial tension test was used to evaluate longitudinal modulus and Poisson's ratio. A modified rail-shear apparatus was designed to measure shear modulus in a rail-shear orientation, as well as a totally constrained transverse modulus in a uniaxial tension test with an orientation transverse to the longest dimension of the specimen. A flexure test was also used to estimate transverse moduli, since the achievement of the totally constrained transverse-modulus measurement was impaired by the variability of load distribution and constraint across the breadth (10-in. length) of the specimen. An apparatus was designed that applied a line loading to the specimen, simply supported on two opposite sides, with the two remaining sides free. The feasibility of this technique was documented by comparing measured values of the transverse modulus and out-of-plane displacement for a [0°]₄ specimen to values obtained from the Levy solution for bending of an orthotropic plate.     

Swirl number and nozzle confinement effects in a flat-vane axial swirler

This paper presents the results of a parametric experimental study of free swirling flow at the exit of a flat-vane axial swirler. A total of 16 data sets were acquired by combining four swirler vane angles (22°, 29°, 50.5°, and 58.3°) and four exit nozzles of different diameters (30, 40, 52, and 76 mm). Sophisticated pressure probes consisting of precise microphones and a two-component LDV system were used to investigate the effect of these geometrical parameters on swirling flow regimes characterized by the swirl number. Particular attention was paid to the precessing vortex core (PVC) phenomenon observed at the exit of the swirler nozzle. It has been shown that by varying the vane angle and the diameter of the exit nozzle, it is possible to independently control the swirl number value and the occurrence of a PVC. A distinct correlation has been found between the PVC-induced pressure pulsations detected by acoustic probes and the tangential velocity fluctuations measured by LDV. The use of microphones provides a quick way to measure the frequency response of swirl flow in a wide range of geometries and flow configurations. The PVC effect does not occur at low subcritical values of the integral swirl number (S < 0.5) and in the case of strong swirl flow (S_g = 0.9 and 1.2) in the absence of constriction by the nozzle (D_e/D₀ = 1). The disappearance of the PVC effect for strong swirl flow without constriction is due to the extreme displacement of the flow to the nozzle walls. The absence of a PVC in the flow was inferred not only from measurements of the frequency response of the flow over a wide range of Re numbers, but also from the absence of specific markers in velocity RMS distributions. Measurement results are used to derive an empirical correlation of the integral swirl number and the Strouhal number with a modified geometric swirl number. This allows a generalization of the frequency characteristics of swirling flows with a PVC for flat-vane axial swirlers, which are widely used in engineering.     

Flow structures behind a vibrissa-shaped cylinder at different angles of attack: Complication on vortex-induced vibration

Complementary experimental studies have been conducted with a vibrissa-shaped cylinder at different angles of attack, through vortex-induced vibration (VIV) test in a wind tunnel, along with extensive measurements of wake dynamics in a water channel using time-resolved particle image velocimetry (TR-PIV). The VIV responses of an elastically mounted vibrissa-shaped cylinder are experimentally compared at various angles of attack in the range of θ = 0°–90°. At the reduced velocity of U₀/f₀D_h = 3–10 (f₀ being the system's natural frequency), the cross-flow displacement of the cylinders convincingly demonstrates that the vibrissa-shaped cylinder at a small angle of attack (θ ≤ 30°) is stable, and without appreciable displacement. Beyond θ = 30°, a prominent three-branched VIV response is readily identified, and increasing the angle of attack results in an upward shift of the synchronized region and a considerable intensification of the peak amplitude. Subsequently, TR-PIV measurements are made of the wake flow behind the vibrissa-shaped cylinder, to determine the spatio-temporally varying flow fields in two spanwise planes, i.e., the saddle and the nodal planes. Four systems with different angles of attack are chosen for comparison at Re_D = 1.8 × 10³, i.e., θ = 0°, 30°, 60° and 90°. In the two systems with θ = 0° and 30°, the wake regions feature weak velocity fluctuations in highly limited areas. However, increasing the angles of attack (to θ = 60° and 90°) gives rise to expanded recirculation zones, highly unstable flow reversals immediately behind the cylinder, and strengthened velocity fluctuations in the bulk wake regions. Cross-correlation of the fluctuating longitudinal velocities shows that at θ = 60° and 90° the energetic large-scale vortical structures form earlier, and they exert considerable influence on the near-wake fluid behind the cylinder. Finally, a sophisticated data-driven dynamic mode decomposition (DMD) process is used to extract the dominant unsteady structures in the four systems with different angles of attack. In the system with θ = 0°, two dominant DMD modes at frequencies St= 0.23 and 0.30 are identified in the saddle and the nodal planes, respectively, and those frequencies are St= 0.18 and 0.19 in the system with θ = 30°. The interaction between these dominant events at different frequencies tends to disrupt the formation of a strong vortex-shedding process. Therefore, the hydrodynamic force on the cylinder does not make a concerted contribution to suppressing the VIV behavior along the spanwise direction. In the systems with θ = 60° and 90°, the corresponding DMD modes exhibit much more synchronous, organized characteristics in the saddle and nodal planes, and unsteady events at the same frequencies are detected in both planes, reaching St = 0.14 (for θ = 60°) and 0.12 (for θ = 90°). These effects, along with the intensified vortex-shedding processes in the saddle and nodal planes, exert a concerted hydrodynamic force on the cylinder, causing it to start with an oscillatory state.     

The influence of a compressive stress on the nonlinear response of ferroelectric crystals

The origin of nonlinearity in a ferroelectric crystal is domain reorientation, and such a process can be affected by the presence of a compressive stress. In this article we examine how a superimposed compression affects the evolution of new domain and how it changes the shape of the hysteresis loop. We start out by considering the thermodynamic driving force for domain reorientation, and then use a dual-phase homogenization theory to calculate the overall response. To uncover the influence of a compressive stress, the theory is used to calculate the hysteresis loop between the electric displacement D and the electric field E of a BaTiO₃ crystal, first without and then with a compression, using a two-consecutive 90° switch model (i.e. 0° → 90° → 180°). It is found that, from the initial 0° position, the compressive stress will increase the thermodynamic driving force and promote an earlier onset of the 90° domain, but its presence will cause a significant delay for the reorientation process to pass through the intermediate 90° state in route to its final 180° configuration. The D vs. E loop then exhibits a more round shape and a lesser steep slope near the coercive field. The delayed passage and more rounded shape are found to be consistent with a recent experimental observation [Burcsu et al., 2004. J. Mech. Phys. Solids 52, 823–846].     

Predictions of first-ply failure load of laminated composite shafts

The failure strengths of laminated composite shafts subjected to static bending load and/or torque are studied using both theoretical and experimental approaches. We investigate optimal angle-ply orientations of laminated composite shafts with the design of symmetric main fiber lay-ups [θ/−θ/...] _S and anti-symmetric reinforcing fiber lay-ups [α/−α/...]_−S for maximum stiffness. Different analytical methods, together with various failure criteria, are used to predict the first-ply failure strengths of the composite shafts. A hand lay-up and pressure bag technique was used to manufacture the composite shafts, which includes the stacking of pieces of prepreg in the desired orientations and the curing process. The first-ply failure loads of laminated composite shafts were determined using acoustic emission. The failure modes of the shafts are studied and experimental results are used to verify the theoretical predictions. The experimental results have been proven to be efficient and effective in the theoretical prediction of first-ply failure loads of laminated composite shafts.     

Finite deformation and stability behaviour of cylindrical membranes subjected to symmetric line loads2

An ‘exact’ analysis of the complete non-linear load-deflection and stability behaviour of cylindrical membranes without end ‘shear walls’ subjected to longitudinal Symmetric line loads is presented. The analysis includes low as well as high profile structures. In order to determine the lateral stability behaviour, infinitesimal lateral displacements are superimposed on the symmetric finite deflection field.Results indicate that such structures may experience vertical and/or lateral instability depending on their initial geometry. Membranes with initial central angles, θ₀ ? 90° are stable, both vertically and laterally, for all load values. For 42.23° < θ₀ < 90° the structure is laterally stable but becomes vertically unstable at a certain ‘limit load’ W?_V while for θ₀< 42.23°, the structure becomes laterally unstable at a load value w?_l<W?_V.The analysis admits contact between vertical segments of the membrane to either side of the line load as well as between the membrane and the horizontal surface next to the supports.     

Three-dimensional simulations of flow past two circular cylinders in side-by-side arrangements at right and oblique attacks

The flow past two identical circular cylinders in side-by-side arrangements at right and oblique attack angles is numerically investigated by solving the three-dimensional Navier–Stokes equations using the Petrov–Galerkin finite element method. The study is focused on the effect of flow attack angle and gap ratio between the two cylinders on the vortex shedding flow and the hydrodynamic forces of the cylinders. For an oblique flow attack angle, the Reynolds number based on the velocity component perpendicular to the cylinder span is defined as the normal Reynolds number Re_N and that based on the total velocity is defined as the total Reynolds number Re_T. Simulations are conducted for two Reynolds numbers of Re_N=500 and Re_T=500, two flow attack angles of α=0° and 45° and four gap ratios of G/D=0.5, 1, 3 and 5. The biased gap flow for G/D=0.5 and 1 and the flip-flopping bistable gap flow for G/D=1 are observed for both α=0° and 45°. For a constant normal Reynolds number of Re_N=500, the mean drag and lift coefficients at α=0° are very close to those at α=45°. The difference between the root mean square (RMS) lift coefficient at α=0° and that at α=45° is about 20% for large gap ratios of 3 and 5. From small gap ratios of 0.5 and 1, the RMS lift coefficients at α=0° and 45° are similar to each other. The present simulations show that the agreement in the force coefficients between the 0° and 45° flow attack angles for a constant normal Reynolds number is better than that for a constant total Reynolds number. This indicates that the normal Reynolds number should be used in the implementation of the independence principle (i.e., the independence of the force coefficients on the flow attack angle). The effect of Reynolds number on the bistable gap flow is investigated by simulating the flow for Re_N=100–600, α=0° and 45° and G/D=1. Flow for G/D=1 is found to be two-dimensional at Re_N=100 and weak three-dimensional at Re_N=200. While well defined biased flow can be identified for Re_N=300–600, the gap flow for Re_N=100 and 200 changes its biased direction too frequently to allow stable biased flow to develop.     

Effect of laminate construction on residual stresses in graphite/polyimide composites

The influence of ply-stacking sequence and ply orientation on the magnitude of lamination residual stresses in graphite/polyimide angle-ply laminates was investigated. The effect of stacking sequence was investigated with laminates of [0₂/±45]_s, [±45/0₂]_s, [0/+45/0/?45]_s and [+45/0₂/?45]_s layup. The effect of ply orientation was evaluated with additional specimens of [0₂/±15]_s and [0₂/90₂]_s layup. Thermal strains were measured using embedded-gage techniques. Residual strains were determined by comparing thermal strains in the angle-ply laminates with those of a unidirectional laminate. The ply-stacking sequence did not have an influence on the magnitude of residual strains. The highest residual strains occur in the [0₂/90₂]_s laminate and the lowest, approximately one-fourth in magnitude, occur in the [0₂/±15]_s laminate. The maximum residual strains in the [0₂/±45]_s group are slightly lower than those in the [0₂/90₂]_s laminate. Residual stress computations show that, at room temperature, the transverse-to-the-fibers stresses in all plies, except those of the [0₂/±15]_s laminate, exceed the transverse tensile strength of the unidirectional material.     

Out-of-plane deflections of metallic and composite pin-loaded coupons

The projection shadow moiré technique was employed to determine an out-of-plane contouring history of aluminum, [(+45/–45)]_3S , [(0/90₃,0] _S and [(0/90/+45/–45)]_2S fiberglass epoxy pin-loaded specimens experimentally. The contouring interval was limited to 0.0254 mm by experimental concerns. Qualification of the projection shadow moiré optical arrangement was initially accomplished by the out-of-plane contouring of a clamped, centrally loaded, circular aluminum plate. Experimental results indicate significant out-of-plane displacements in the bearing region of the coupon at load levels well below ultimate. Effects of material anisotropy could be seen in the fiber-oriented shape of the displacement contours. Comparisons with three-dimensional finite-element results indicate that experimental out-of-plane contours were significantly larger than their finite-element counterparts in the region above the pin for the [(+45/–45)]_3S , [(0/90)₃, 0] _S laminates. These deviations increased with increasing pin-load level. These variations could be attributed to linear-elastic through-thickness moduli assumptions as well as through-thickness finite-element mesh coarseness.     

Onset of viscous fingering for miscible liquid-liquid displacements in porous media

The displacement of one fluid by another miscible fluid in porous media is an important phenomenon that occurs in petroleum engineering, in groundwater movement, and in the chemical industry. This paper presents a recently developed stability criterion which applies to the most general miscible displacement. Under special conditions, different expressions for the onset of fingering given in the literature can be obtained from the universally applicable criterion. In particular, it is shown that the commonly used equation to predict the stable velocity ignores the effects of dispersion on viscous fingering.Nomenclature C Solvent concentration - Unperturbed solvent concentration - D _L Longitudinal dispersion coefficient [m²/s] - D _T Transverse dispersion coefficient [m²/s] - g Gravitational acceleration [m/s²] - I _sr Instability number - k Permeability [m²] - K Ratio of transverse to longitudinal dispersion coefficient - L Length of the porous medium [m] - L _x Width of the porous medium [m] - L _y Height of the porous medium [m] - M Mobility ratio - V Superficial velocity [m/s] - V _c Critical velocity [m/s] - V _s Velocity at the onset of instability [m/s] - µ Viscosity [Pa/s] - Unperturbed viscosity [Pa/s] - µ ₀,µ _s Viscosities of oil and solvent, respectively [Pa/s] - Density [kg/m³] - ₀, _s Densities of oil and solvent, respectively [kg/m³] - Porosity - Dimensionless length     

TEMPERATURE EFFECT ON LOW-CYCLE FATIGUE BEHAVIOR OF NICKEL-BASED SINGLE CRYSTALLINE SUPERALLOY

The low-cycle fatigue （LCF） behavior of a nickel-based single crystal superalloy with [001] orientation was studied at an intermediate temperature of T0℃ and a higher temperature of To ＋ 250℃ under a constant low strain rate of 10^-3 s^-1 in ambient atmosphere. The superalloy exhibited cyclic tension-compression asymmetry which is dependent on the temperature and applied strain amplitude. Analysis on the fracture surfaces showed that the surface and subsurface casting micropores were the major crack initiation sites. Interior Ta-rich carbides were frequently observed in all specimens. Two distinct types of fracture were suggested by fractogaphy. One type was characterized by Mode-I cracking with a microscopically rough surface at To ＋ 250℃. Whereas the other type at lower temperature T0℃ favored either one or several of the octahedral {111} planes, in contrast to the normal Mode-I growth mode typically observed at low loading frequencies （several Hz）. The failure mechanisms for two cracking modes are shearing of γ＇ precipitates together with the matrix at T0℃ and cracking confined in the matrix and the γ/γ＇interface at To - 250℃.     

Heat transfer by laminar flow of an elastico-viscous fluid in posttreatment analysis of wire coating with linearly varying temperature along the coated wire

The present study focuses on the heat transfer by the laminar flow of an elastico-viscous fluid in posttreatment of wire coating analysis with linearly varying temperature on the surface of coated wire. The surface of wire (uncoated) and the surface of coated wire were subjected to two thermal boundary conditions. The constitutive equation of motion and equation of energy have been solved by using perturbation theory for velocity, pressure distribution along the radial direction and temperature distribution. The theoretical analysis of flow rate, average velocity, shear stress, thickness of coated wire, and force on the total wire were also derived. Moreover, the flow phenomenon has been studied under the influence of elastic number R _e velocity ratio U and the dimensionless number S in the ranges 0?≤?R _e ?≤?20, 0.2?≤?U?≤?1.4 and 0?≤?S?≤?20. We noticed that with the increase in elastic number R _e velocity decreases whereas thickness of the coated wire and force on the total wire increases. Also temperature profile decreases with the increase of non-dimensional parameter S.     

Determination of flow activation energy at viscosity maximum for spherical and wormlike micelles of different lengths and flexibility

Steady shear rheological measurements were carried out on aqueous solutions containing 15 mM cetyltrimethylammonium bromide (CTABr) and a constant value of [MX] and temperature for MX = 2,3-; 2,4-; 2,5-; 2,6-; 3,4-; and 3,5-Cl₂BzNa with Bz^?representing C₆H₃CO_2?. Plots of zero shear viscosity (η ₀) vs. [MX] at 35 °C and 15 mM CTABr show the presence of single maximum and double maxima for MX = 2,3- and 3,5-Cl₂BzNa, respectively. Turbidity data (absorbance at 600 nm vs. [MX]) coupled with η ₀vs. [MX] data at 35 °C reveal indirectly the presence of vesicles along with wormlike micelles (WM) at MX / CTABr > 0. 7 for MX = 3,5-Cl₂BzNa. Temperature dependence of η ₀in the vicinity of the viscosity maximum shows nonlinear and linear Arrhenius behavior, within the temperature range of 20–55 °C, for MX = 2,3-; 2,4-; 2,5-; 3,4-; and 3,5-Cl₂BzNa, respectively. The values of η ₀, $\dot {\gamma }_{\text {cr}} $ (critical shear rate), and flow activation energy correlate with CTABr micellar binding constants of counterions.     

Two-phase geothermal flows with conduction and the connection with Buckley-Leverett theory

Two-phase flows of boiling water and steam in geothermal reservoirs satisfy a pair of conservation equations for mass and energy which can be combined to yield a hyperbolic wave equation for liquid saturation changes. Recent work has established that in the absence of conduction, the geothermal saturation equation is, under certain conditions, asymptotically identical with the Buckley-Leverett equation of oil recovery theory. Here we summarise this work and show that it may be extended to include conduction. In addition we show that the geothermal saturation wave speed is under all conditions formally identical with the Buckley-Leverett wave speed when the latter is written as the saturation derivative of a volumetric flow.Roman Letters C(P, S,q) geothermal saturation wave speed [ms^–1] (14) - c _t (P, S) two-phase compressibility [Pa^–1] (10) - D(P, S) diffusivity [m s^–2] (8) - E(P, S) energy density accumulation [J m^–3] (3) - g gravitational acceleration (positive downwards) [ms^–2] - h _w (P),h _w (P) specific enthalpies [J kg^–1] - J _M (P, S,P) mass flow [kg m^–2 s^–1] (5) - J _E (P, S,P) energy flow [J m^–2s^–1] (5) - k absolute permeability (constant) [m²] - k _w (S),k _s (S) relative permeabilities of liquid and vapour phases - K formation thermal conductivity (constant) [Wm^–1 K^–1] - L lower sheetC<0 in flow plane - m, c gradient and intercept - M(P, S) mass density accumulation [kg m^–3] (3) - O flow plane origin - P(x,t) pressure (primary dependent variable) [Pa] - q volume flow [ms^–1] (6) - S(x, t) liquid saturation (primary dependent variable) - S ^*(x,t) normalised saturation (Appendix) - t time (primary independent variable) [s] - T temperature (degrees Kelvin) [K] - T _sat(P) saturation line temperature [K] - TdT _sat/dP saturation line temperature derivative [K Pa^–1] (4) - T _c ,T _D convective and diffusive time constants [s] - u _w (P),u _s (P),u _r (P) specific internal energies [J kg^–1] - U upper sheetC > 0 in flow plane - U(x,t) shock velocity [m s^–1] - x spatial position (primary independent variable) [m] - X representative length - x, y flow plane coordinates - z depth variable (+z vertically downwards) [m] Greek Letters _P , _S remainder terms [Pa s^–1], [s^–1] - double-valued saturation region in the flow plane - h =h _s –h _w latent heat [J kg^–1] - = _w – _s density difference [kg m^–3] - line envelope - =D _K /D ₀ diffusivity ratio - porosity (constant) - _w (P), _s (P), _t (P, S) dynamic viscosities [Pa s] - v _w (P),v _s (P) kinematic viscosities [m²s^–1] - v ₀ =kh/KT kinematic viscosity constant [m² s^–1] - ₀ =v ₀ dynamic viscosity constant [m² s^–1] - _w (P), _s (P) density [kg m^–3] Suffixes r rock matrix - s steam (vapour) - w water (liquid) - t total - av average - 0 without conduction - K with conduction     

Numerical study of solidification of a drop with a growth angle difference

This study presents the direct numerical results of a drop solidifying on a plate, in which the difference between the growth angles is considered. The drop is two-dimensional with the presence of the left and right triple points, and the method used is a front-tracking technique. The growth angles at the right (ϕ_gr1) and left (ϕ_gr2) triple points are not equal, i.e. Δϕ_gr = ϕ_gr1–ϕ_gr2 ≠ 0°. Unlike the identical growth angles, the growth angle difference results in an asymmetric drop after complete solidification. In the presence of the solid-to-liquid density ratio ρ_sl < 1.0 (i.e. volume expansion), the tip of the solidified drop shifts more to the right as Δϕ_gr increases in the range of 0°–12°. In addition, the angle at the solidified drop top (i.e. tip angle) increases with Δϕ_gr. We also pay attention to the effects of some other parameters (such as the wetting angle ϕ₀, the growth angle ϕ_gr1 and ρ_sl) on the solidification process with the growth angle difference. The results reveal that the growth angle varied in the range of 6°–24° has a minor effect on the movement of the tip to the right while the tip shift increases with an increase in ϕ₀ in the range of 60°–130° or with a decrease in ρ_sl in the range of 0.8–1.1. The tip angle increases with an increase in ρ_sl or with a decrease in ϕ_gr1 or ϕ₀. We also investigate the solidification process under the influence of the Bond number.     

A plastic flow rule at a yield vertex

A yield vertex having three distinct facets at the uniaxial stress point is considered. An associated plastic flow rule is constructed using a new 3 × 3 coupled hardening matrix. This has the property that the incipient shear modulus is less than the elastic value. The approach is novel and distinct from previous work having that conclusion. In particular, all the relevant incipient moduli governing fully active in-plane loading can be fitted, if desired, to those values which J₂-deformation theory would require. To that extent the proposed incremental theory therefore legitimizes the use of the latter moduli, for example in certain ‘paradoxical’ buckling problems. When the moduli of the incremental theory are so chosen, the domain of stress-rate vectors enforcing loading is a calculable pyramid which contracts from the Mises half-space as the stress increases beyond yield. This domain becomes wider, at a given stress, as the initial smooth curvature of the stress-strain curve is imagined to become sharper.     

Biaxial testing of [O2/±45] s graphite/epoxy plates with holes

An experimental investigation was conducted to study the behavior under biaxial-tensile loading of [O₂/±45] _s graphite/epoxy plates with circular holes and to determine the influence of hole diameter on failure. The specimens were 40-cm×40-cm (16-in.×16-in.) graphite/epoxy plates of [O₂/±45] _s layup. Four hole diameters, 2.54 cm (1.00 in.), 1.91 cm (0.75 in.), 1.27 cm (0.50 in.) and 0.64 cm (0.25 in.), were investigated. Deformations and strains were measured using strain gages and birefringent coatings. Biaxial tension in a 2∶1 ratio was applied by means of four whiffle-tree grip linkages and controlled with a servohydraulic system. Stress and strain redistributions occur around the hole at a stress level corresponding to localized failure around the 67.5-deg location and nonlinear strain response at the 0-deg location. Maximum measured strains at failure on the hole boundary are higher (approximately 0.016) than the highest ultimate strain of the unnotched laminate (0.010). Two basic patterns of failure were observed: (a) horizontal cracking initiating at points off the horizontal axis and accompanied by extensive delamination of the subsurface ±45 deg plies, and (b) vertical cracking along vertical tangents to the hole and accompanied by delamination of the outer 0-deg plies. The strength reduction ratios are lower than corresponding values for uniaxial loading by approximately 16 percent, although the stress-concentration factor under biaxial loading is lower.