Dynamic crushing strength of hexagonal honeycombs

Based on the repeatable collapsing mechanism of cells’ structure under dynamic crushing, an analytical formula of the dynamic crushing strength of regular hexagonal honeycombs is derived in terms of impact velocity and cell walls’ thickness ratio. It is consistent with the equation obtained from the shock wave theory that regards cellular material as continuum, in which the key parameter is approximately measured from the “stress–strain” curve of the cellular material. The effect of unequal thickness of cell walls on the honeycomb's dynamic crushing strength is discussed, and the result shows that the dynamic crushing strength of the hexagonal honeycomb with some double-thickness walls is about 1.3 times of that of the hexagonal honeycomb without double-thickness wall. All of the analytical predictions are compared with the numerical simulation results, showing good agreements.     

The manufacture and characterisation of a novel,low modulus,negative Poisson’s ratio composite

Relatively few negative Poisson’s ratio (auxetic) composites have been manufactured and characterised and none with inherently auxetic phases [Milton G. J. Mech. Phys. Solids 1992;40:1105–37]. This paper presents the use of a novel double-helix yarn that is shown to be auxetic, and an auxetic composite made from this yarn in a woven textile structure. This is the first reported composite to exhibit auxetic behaviour using inherently auxetic yarns. Importantly, both the yarn and the composite are produced using standard manufacturing techniques and are therefore potentially useful in a wide range of engineering applications.     

Dynamic crushing of a one-dimensional chain of type II structures

The development of an analytical model based on a ‘bar–hinge’ idealisation is described. It is used to investigate the structural response of a one-dimensional chain of components each constructed from a pair of slightly bent elastic–plastic struts under axial impact loading. Each component (a typical type II structure) in the chain is modelled as four axially compressible, elastic–plastic, straight bars of infinite bending rigidity connected to each other by elastic–plastic hinges of finite length. A new approach to formulate the constitutive relation between the generalised force and displacement of the bars and hinges is developed. A self-contact algorithm is used for intra-component contact simulation. The ‘bar–hinge’ model is validated using the results from finite-element simulations using ABAQUS. Good agreement is achieved. The analytical model was then used to investigate the crushing features of the chain structure. The effects of the number of the components in the chain and the crookedness angle of the components on the crushing behaviour were studied. It was found that collapse of the components in the chain occurs at the proximal end of the chain first and that the component at the fixed distal end will also collapse at some later time. The single component structure is more ‘inertia sensitive’ than a chain structure of more than two components.     

Interaction of dynamic mode-I cracks with inclined interfaces

In this paper, we report on an experimental study of the deflection/penetration behavior of dynamic mode-I cracks propagating at two different crack velocities (slower and faster) toward inclined weak interfaces of three dissimilar angles (α): 30°, 45° and 60°. A simple wedge-loading specimen configuration as proposed by Xu et al. [Xu LR, Huang YY, Rosakis AJ. Dynamic crack deflection and penetration at interface in homogenous materials: experimental studies and model predictions. J Mech Phys Solids 2003;51:461-86], made of brittle Homalite-100, is used. A modified Hopkinson bar setup is used to achieve well-controlled impact loading conditions. Dynamic photoelasticity in conjunction with high-speed photography is used to capture real-time isochromatics associated with deflected/penetrated cracks.     

A failure criterion for brittle elastic materials under mixed-mode loading

The failure criterion of Leguillon at reentrant corners in brittle elastic materials (Leguillon 2002, Eur J Mech A/Solids 21: 61–72; Leguillon et al. (2003), Eur J Mech A—Solids 22(4): 509–524) validated in (Yosibash et al. 2004, Int J Fract 125(3–4): 307–333) for mode I loading is being extended to mixed mode loading and is being validated by experimental observations. We present an explicit derivation of all quantities involved in the computation of the failure criterion. The failure criterion is validated by predicting the critical load and crack initiation angle of specimens under mixed mode loading and comparison to experimental observations on PMMA (polymer) and Macor (ceramic) V-notched specimens.     

Characterization of deformation instability in modified 9Cr–1Mo steel during thermo-mechanical processing

In this study, various existing instability criteria were employed to delineate the unstable flow regions in modified 9Cr–1Mo steel during hot deformation. Experimental stress–strain data obtained from isothermal hot compression tests, in a wide range of temperatures (1123–1373 K) and strain rates (10⁻³–10 s⁻¹), were employed to develop instability maps. The domains of these instability maps were validated through detailed microstructural study. It has been observed that Hart’s stability criterion, Jonas’s criterion and Semiatin’s criterion under-predicts the instability regions in the studied temperatures and strain rates regime. Gegel’s and Alexander’s criteria as well as Murty’s metallurgical instability criterion, on the other hand, found to over-predict the instability domains. The instability map developed based on Dynamic Materials Model criterion has been found to precisely predict the instability domains. This instability map revealed four major unstable domains. Microscopic examination in these domains revealed that the instability is manifested in the specimens either as localized deformation band primarily along one of the diagonal or inhomogeneous distribution of martensite lath in the prior austenite grains.     

Perforation of sheets by pyramidal weapons such as arrowheads

Perforation experiments were performed through metal sheets employing pyramidal indenters having both regular polygonal bases and also non-regular lozenge-shaped bases. The tools also have a range of ‘point angles’. The evolution of petal radii as perforation proceeded was recorded. In the case of lozenge-based pyramids, fracture occurred only at the sharper edges located at the ends of the longer diagonal, with uncracked stretch zones at the ends of the shorter diagonal. Regular pyramids with a small number of edges produced cracking at every corner but perforations with six-sided indenters resulted in only a limited number of petals and cracks, fewer cracks being produced the ‘flatter’ the point. Experiments with cones also resulted in only limited number of petals and cracks. The results for forces, energy and petal radii of curvature were interpreted using modifications of the Wierzbicki and Thomas [Closed form solution for wedge cutting force through thin sheets. Int J Mech Sci 1993;35:209–29] analysis for the cutting of thin plates by wedges, together with independent determinations of yield strength and fracture toughness of the metal sheets. Experimental results make overall sense in relation to the theory but there is a consistent mismatch that is probably attributable to initial dishing of the sheets before initial perforation is complete. The relevance of the analysis to the performance of arrowheads perforating armour is discussed. What constitutes the best design of weapon against particular armour is investigated.     

Families of asymptotic tensile crack tip stress fields in elastic-perfectly plastic single crystals

In this work, two families of asymptotic near-tip stress fields are constructed in an elastic-ideally plastic FCC single crystal under mode I plane strain conditions. A crack is taken to lie on the (010) plane and its front is aligned along the direction. Finite element analysis is first used to systematically examine the stress distributions corresponding to different constraint levels. The general framework developed by Rice (Mech Mater 6:317–335, 1987) and Drugan (J Mech Phys Solids 49:2155–2176, 2001) is then adopted to generate low triaxiality solutions by introducing an elastic sector near the crack tip. The two families of stress fields are parameterized by the normalized opening stress prevailing in the plastic sector in front of the tip and by the coordinates of a point where elastic unloading commences in stress space. It is found that the angular stress variations obtained from the analytical solutions show good agreement with finite element analysis.     

Ballistic impact studies of a borosilicate glass

The ballistic impact properties of a borosilicate (‘pyrex’) glass was studied using mild steel rods accelerated using a light gas gun. High-speed photography at sub-microsecond framing rates was used along with schlieren optics to investigate the propagation of elastic shock waves and fracture fronts. Flash X-radiography was used to visualise the deformation of rods as they penetrated the comminuted glass normally. The rod was seen initially to dwell on the surface for at least 3 μs creating a Hertzian cone-crack. Later on, between 40 and 60 μs, self-sharpening of the projectile was observed as the ‘wings’ of the heavily deformed front end sheared off. After this event, the front of the rod speeded up. X-rays also showed that the pattern of fissures within the comminuted glass was observed to be very similar shot-to-shot. X-radiography was also used to examine the mechanisms occurring during oblique impact of rods at 45°. In oblique impact, bending of the rod rather than plastic deformation (‘mushrooming’) takes on the role of distributing the load over an area larger than that of the original rod diameter. High-speed photography of the rear surface of a glass block on which a fine grid had been placed confirmed that the comminuted glass moved as larger interlocked blocks. The experiments were modelled using the QinetiQ Eulerian hydrocode GRIM making use of the Goldthorpe fracture model. The model was found to predict well the transition from dwell to penetration.     

Crack tip fields in a viscoplastic solid: monotonic and cyclic loading

The asymptotic stress and strain field near the tip of a plane strain Mode I stationary crack in a viscoplastic material are investigated in this work, using a unified viscoplastic model based on Chaboche (Int J Plast 5(3):247–302, 1989). Asymptotic analysis shows that the near tip stress field is governed by the Hutchinson–Rice–Rosengren (HRR) field (Hutchinson in J Mech Phys Solids 16(1):13–31, 1968; Rice and Rosengren in J Mech Phys Solids 16(1):1–12, 1968) with a time dependent amplitude that depends on the loading history. Finite element analysis is carried out for a single edge crack specimen subjected to a constant applied load and a simple class of cyclic loading history. The focus is on small scale creep where the region of inelasticity is small in comparison with typical specimen dimensions. For the case of constant load, the amplitude of the HRR field is found to vanish at long times and the elastic K field dominates. For the case of cyclic loading, we study the effect of stress ratio on inelastic strain and find that the strain accumulated per cycle decreases with stress ratio.     

An extension of the Secant Method for the homogenization of the nonlinear behavior of composite materials

We discuss the consequences of a different application of the principle the Modified Secant Method is [C. R. Acad. Sci. Paris Sér. IIb 320 (1995) 563] based on. In fact, we directly compute the second-order averages of the local fields available from the linear elastic homogenization procedure exploited in order to evaluate the effective elastic moduli. This method, which can be seen either as a simplification of the Modified Secant Method or as an extension of the Secant Method [J. Mech. Phys. Solids 26 (1979) 325], may be useful for any composite whose overall elastic constants need to be estimated by modeling the microstructure through Morphologically Representative Patterns [J. Mech. Phys. Solids 44 (1996) 307], which is for instance the case of syntactic foams [Int. J. Solids and Structures 38/40-41 (2001) 7235]. In order to show the accuracy of the proposed method, we apply it to several examples and compare its results with those obtainable by means of other analytical methods available in the literature, with numerical results of Finite Element simulations, and with experimental results. Closed-form solutions are derived for the effective yield stress of porous metals and incompressible composites reinforced with rigid spheres.     

Dugdale model solutions for a penny-shaped crack in three-dimensional transversely isotropic piezoelectric media by boundary-integral equation method

Making use of the Displacement Discontinuity Boundary Integral Equation Method (DDBIEM), the dimension of the plastic zone at the tip of a penny-shaped crack in a three-dimensional elastic medium is determined by the application of the Dugdale model; Furthermore, the solutions for a penny-shaped crack in three-dimensional piezoelectric media are obtained by the use of the Dugdale-like model proposed by Gao et al.[Gao H, Zhang T, Tong P. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 1997;45:491–510], in which the electrical polarization is assumed to reach a saturation limit in a thin annular region in front of a crack while the mechanical stresses have the ordinary singularity.     

A modified effective crack-length formulation in elastic-plastic fracture mechanics

In examining the performance of standard effective crack-length formulations, the authors noted quantitative accuracy up to “high” fractions of limit load under loading conditions for which the elastic T-stress was non-negative, while a pronounced deviation from the corresponding continuum elastic-plastic plane-strain finite-element solutions was seen in shallow-cracked geometries having negative T-stress. This trend can be rationalized by noting that, under modified boundary layer (K_I and T) loading, the maximum plastic zone radius strongly increases as the T-stress decreases from zero (J.R. Rice (1974), J. Mech. Phys. Solids 22, 17–26; S.G. Larsson and A.J. Carlsson (1973), J. Mech. Phys. Solids 21, 263–277; N.P. O'Dowd and C.F. Shih (1991), J. Mech. Phys. Solids 39(8), 989–1015.) Accordingly, we formulate a modified effective crack length to account for the effects of the elastic T-stress.

The new formulation consistently extends the load range for which accurate predictions of compliance, J-integral, and crack-tip constraint are obtained in several plane strain specimen geometries. The magnitude of influence of the T-stress varies with specimen type and relative crack depth. The greatest “improvement” to standard effective crack length approximations occurs in specimens of “moderately” negative T-stress.     

Upgradation of gravity load designed sub-assemblages subjected to seismic type loading

In the present study, a methodology based on strength hierarchy has been proposed for upgradation of original gravity load designed (GLD) reinforced concrete (RC) structures. Exterior beam–column joint of an RC structure has been considered as the target sub-assemblage and the target strength of the deficient sub-assemblage was decided from that of a seismically designed ‘Ductile’ one. Three different types of upgradation schemes were investigated where shear- and flexural-strengthening were provided by Fiber Reinforced Plastics (FRP) and weak joint region of ‘GLD’ sub-assemblage was upgraded by steel plate jacketing. The original (‘GLD’-, ‘NonDuctile’-, ‘Ductile’-) and upgraded-sub-assemblages were investigated under repeated reverse cyclic loading. It was observed that the ‘GLD’ specimen seized to function under reverse loading and subsequent improvements, though not optimal, were observed from ‘NonDuctile’ and ‘Ductile’ specimens. It was further found out that the upgraded specimens showed considerable improvement in strength deterioration, stiffness degradation and energy dissipation. Further, the upgraded specimens with adequate energy dissipation could even be able to shift the plastic hinge from the joint face into the beam which was not observed even in original ‘Ductile’ specimen. The upgraded schemes are simple, practically feasible and efficient as well.     

The influence of cell micro-topology on the in-plane dynamic crushing of honeycombs

The influence of the cell micro-topology on the in-plane dynamic crushing of honeycombs is studied by means of explicit dynamic finite element simulation using ANSYS/LS-DYNA. Firstly, under the assumption that the edge length and thickness are the same, the dynamic properties of the honeycombs filled by cells with different shapes (equilateral triangular or quadratic cells) and micro-arrangements (regular or staggered arrangement) are numerically analyzed. The full-scale in-plane dynamic crushing of the specimen, as well as the micro-structure transformation during the deformation, is discussed. Based on these, the influence of the cell micro-arrangement on the energy absorption ability of the honeycombs is clarified. The results show that owing to the differences in the micro-topology, triangular or quadratic honeycombs display different local deformation properties during the crushing. The variation of the cell arrangement patterns changes the local dynamic evolution characteristic of stress waves. ‘>’ and ‘<’ mode local deformation bands form at the sides of the stagger-arranged honeycombs, which results in lateral compression shrinkage during the crushing. The plateau stresses also increase with the impact velocity by a square law. The empirical equations for honeycombs filled with different cells (equilateral triangular or quadratic cells) and micro-arrangements (regular or staggered arrangement) at high impact velocities are formulated in terms of impact velocity, and the cell geometrical (edge length and thickness) and topology (edge connectivity) parameters.     

Comparison of predictions by mode II or mode III criteria on crack front twisting in three or four point bending experiments

Whatever the external loading, a crack front in a solid tries to reach mode I loading conditions after propagation. In mode I + II, the crack kinks to annihilate mode II, kinking angle being well predicted by the principle of local symmetry (PLS) or by the maximum tangential stress criterion (MTS). In presence of mode III, the problem becomes three-dimensional and the proposed propagation criterion are not yet well proved and established. In particular in three point bending experiments (3PB) with an initially inclined crack, the crack twists around the direction of propagation to finally reach a situation of pure mode I. The aim of the paper is to compare the propagation paths predicted by two different criteria for 3PB fatigue experiments performed on PMMA. The first criterion developed by Schollmann et al. (Int J Fract 117(2):129–141, 2002), is a three-dimensional extension of the MTS criterion and predicts the local angles that annihilates mode II and III at each point of the front. The second one developed by Lazarus et al. (J Mech Phys Solids 49(7):1421–1443, 2001b), predicts an abrupt and then progressive twisting of the front to annihilate mode III. Due to presence of sign changing mode II and almost uniform mode III in the experiments, both criteria give good results. However, since mode III is predominant over mode II in the case under consideration, the global criterion gives better results. Nevertheless, the local type criterion seems to be of greater universality for practical engineering applications.     

Comments on ‘‘Basic properties of Rayleigh surface wave propagation along curved surfaces”, by F. Jin, Z. Wang, K. Kishimoto, International Journal of Engineering Science 43 (2005) 250–261

In this ‘Letter to the Editor’, it is demonstrated that the main results obtained in the work ‘‘Basic properties of Rayleigh surface wave propagation along curved surfaces”, by F. Jin, Z. Wang, K. Kishimoto, International Journal of Engineering Science 43 (2005) 250–261, have been established ten years earlier.     

Fictitious notch rounding concept applied to sharp V-notches: Evaluation of the microstructural support factor for different failure hypotheses. Part I: Basic stress equations

With the aim to perform a comprehensive and accurate evaluation of the microstructural support factor of sharp V-notches (Neuber’s notch rounding concept), in Part I of this contribution, the indispensable theoretical tools, especially the basic stress equations, are reconsidered and amended in respect of accuracy of results. First, the analytical solution derived by Neuber [Neuber H. Kerbspannungslehre. 2nd ed. Berlin: Springer-Verlag; 1958] for sharp rounded V-notches with an arbitrary flank angle under tension loading is considered. The equation of the normal stress has been obtained with the restriction to the notch bisector. Using the Airy stress function suggested by Neuber, this solution is extended to the region outside the notch bisector, and the complete stress field is derived in this manner. A comparison between Neuber’s solution, a more recent solution due to Filippi et al. [Filippi S, Lazzarin P, Tovo R. Developments of some explicit formulas useful to describe elastic stress fields ahead of notches in plates. Int J Solids Struct 2002;39:4543-65] and highly accurate FE results is performed. Filippi’s equations which include Williams’ solution [Williams ML. Stress singularities resulting from various boundary conditions in angular corners on plates in tension. J Appl Mech 1952;19:526-8] for pointed V-notches, are shown to be superior.