Dynamic compressive strength properties of aluminium foams. Part I—experimental data and observations

This study of the dynamic compressive strength properties of metal foams is in two parts. Part I presents data from an extensive experimental study of closed-cell Hydro/Cymat aluminium foam, which elucidates a number of key issues and phenomena. Part II focuses on modelling issues.The dynamic compressive response of the foam was investigated using a direct-impact technique for a range of velocities from 10 to . Elastic wave dispersion and attenuation in the pressure bar was corrected using a deconvolution technique.A new method of locating the point of densification in the nominal stress-strain curves of the foam is proposed, which provides a consistent framework for the definition of the plateau stress and the densification strain, both essential parameters of the ‘shock’ model in Part II. Data for the uniaxial, plastic collapse and plateau stresses are presented for two different average cell sizes of approximately 4 and 14 mm. They show that the plastic collapse strength of the foam changes significantly with compression rate. This phenomenon is discussed, and the distinctive roles of microinertia and ‘shock’ formation are described. The effects of compression rates on the initiation, development and distribution of cell crushing are also examined. Tests were carried out to examine the effects of density gradient and specimen gauge length at different rates of compression and the results are discussed. The origin of the conflicting conclusions in the literature on the correlation between nominal strain rate (ratio of the impact velocity V_i to the initial gauge length l_o of the specimen) and the dynamic strength of aluminium alloy foams is identified and explained.     

A ‘stack’ model of rate-independent polycrystals

A novel ‘stack’ model of a rate-independent polycrystal, which extends the ‘ALAMEL’ model of Van Houtte et al. (2005) is proposed. In the ‘stack’ model, stacks of N neighboring ‘ALAMEL’ domains collectively accommodate the imposed macroscopic deformation while deforming such that velocity and traction continuity with their neighbors is maintained. The flow law and consistency conditions are derived and an efficient solution methodology based on the linear programming technique is given. The present model is applied to study plastic deformation of an idealized two-dimensional polycrystal under macroscopically imposed plane-strain tension and simple shear constraints. Qualitative and quantitative variations in the predicted macroscopic and microscopic response with N are presented. The constraint on individual ‘ALAMEL’ domains diminishes with stack size N but saturates for large N. Computational effort associated with the present model is analyzed and found to be well within one order of magnitude greater than that required to solve the classical Taylor model. Furthermore, implementation of the consistency conditions is found to reduce computation time by at least 50%.     

Breakage mechanics—Part I: Theory

Different measures have been suggested for quantifying the amount of fragmentation in randomly compacted crushable aggregates. A most effective and popular measure is to adopt variants of Hardin's [1985. Crushing of soil particles. J. Geotech. Eng. ASCE 111(10), 1177-1192] definition of relative breakage ‘B_r’. In this paper we further develop the concept of breakage to formulate a new continuum mechanics theory for crushable granular materials based on statistical and thermomechanical principles. Analogous to the damage internal variable ‘D’ which is used in continuum damage mechanics (CDM), here the breakage internal variable ‘B’ is adopted. This internal variable represents a particular form of the relative breakage ‘B_r’ and measures the relative distance of the current grain size distribution from the initial and ultimate distributions. Similar to ‘D’, ‘B’ varies from zero to one and describes processes of micro-fractures and the growth of surface area. However, unlike damage that is most suitable to tensioned solid-like materials, the breakage is aimed towards compressed granular matter. While damage effectively represents the opening of micro-cavities and cracks, breakage represents comminution of particles. We term the new theory continuum breakage mechanics (CBM), reflecting the analogy with CDM. A focus is given to developing fundamental concepts and postulates, and identifying the physical meaning of the various variables. In this part of the paper we limit the study to describe an ideal dissipative process that includes breakage without plasticity. Plastic strains are essential, however, in representing aspects that relate to frictional dissipation, and this is covered in Part II of this paper together with model examples.     

Propagation of weak waves in elastic-plastic solids

Wave fronts admitting discontinuities only in the derivatives of the dependent variables are by convention called ‘weak’ waves. For the special case of discontinuous first-order derivatives, the fronts are customarily called ‘acceleration’ waves. If the governing equations are quasi-linear, then the weak waves are necessarily characteristic surfaces. Sometimes, these surfaces are also referred to as ‘singular surfaces’ of order r ? 1, where r stands for the order of the first discontinuous derivatives. This latter approach is adopted in this paper and applied to governing equations which form a set of first-order quasi-linear hyperbolic equations. When these equations are written in terms of singular surface coordinates, simplification occurs upon differencing equations written on the front and rear sides of the surface: a set of algebraic (‘connection’) equations is generated for the discontinuities in the normal derivatives of the dependent variables across the surface. When a similar operation is performed on the governing equations written for second-order derivatives, a set of first-order differential (‘transport’) equations is generated.     

One, no one, and one hundred thousand crack propagation laws: A generalized Barenblatt and Botvina dimensional analysis approach to fatigue crack growth

Barenblatt and Botvina with elegant dimensional analysis arguments have elucidated that Paris’ power-law is a weak form of scaling, so that the Paris’ parameters C and m should not be taken as material constants. On the contrary, they are expected to depend on all the dimensionless parameters of the problem, and are really “constants” only within some specific ranges of all these. In the present paper, the dimensional analysis approach by Barenblatt and Botvina is generalized to explore the functional dependencies of m and C on more dimensionless parameters than the original Barenblatt and Botvina, and experimental results are interpreted for a wider range of materials including both metals and concrete. In particular, we find that the size-scale dependencies of m and C and the resulting correlation between C and m are quite different for metals and for quasi-brittle materials, as it is already suggested from the fact the fatigue crack propagation processes lead to m=2-5 in metals and m=10-50 in quasi-brittle materials. Therefore, according to the concepts of complete and incomplete self-similarities, the experimentally observed breakdowns of the classical Paris’ law are discussed and interpreted within a unified theoretical framework. Finally, we show that most attempts to address the deviations from the Paris’ law or the empirical correlations between the constants can be explained with this approach. We also suggest that “incomplete similarity” corresponds to the difficulties encountered so far by the “damage tolerant” approach which, after nearly 50 years since the introduction of Paris’ law, is still not a reliable calculation of damage, as Paris himself admits in a recent review.     

Interfacial instability in turbulent flow over a liquid film in a channel

We revisit the stability of a deformable interface that separates a fully-developed turbulent gas flow from a thin layer of laminar liquid. Although this problem has received considerable attention previously, a model that requires no fitting parameters and that uses a base-state profile that has been validated against experiments is, as yet, unavailable. Furthermore, the significance of wave-induced perturbations in turbulent stresses remains unclear. To address these outstanding issues, we investigate this problem and introduce a turbulent base-state velocity that requires specification of a flow rate or a pressure drop only; no adjustable parameters are necessary. This base state is validated extensively against available experimental data as well as the results of direct numerical simulations. In addition, the effect of perturbations in the turbulent stress distributions is investigated, and demonstrated to be small for cases wherein the liquid layer is thin. The detailed modelling of the liquid layer also elicits two unstable modes, ‘interfacial’ and ‘internal’, with the former being the more dominant of the two. We show that it is possible for interfacial roughness to reduce the growth rate of the interfacial mode in relation to that of the internal one, promoting the latter, to the status of most dangerous mode. Additionally, we introduce an approximate measure to distinguish between ‘slow’ and ‘fast’ waves, the latter being the case for ‘critical-layer’-induced instabilities; we demonstrate that for the parameter ranges studied, the large majority of the waves are ‘slow’. Finally, comparisons of our linear stability predictions are made with experimental data in terms of critical parameters for onset of wave-formation, wave speeds and wavelengths; these yield agreement within the bounds of experimental error.     

On the influence of material properties on the wave propagation in Mindlin-type microstructured solids

A mathematical model describing 1D wave propagation in Mindlin-type microstructured solids with nonlinearities in the macro- and microscale is used for studying propagation of solitary waves in such media. The results could be used for the stress analysis as well as for the nondestructive testing of material properties. The model equations are solved numerically under the localized initial conditions and periodic boundary conditions by the pseudospectral method. It is demonstrated how the values of the model parameters influence the wave propagation, the evolution and the interaction of waves under the framework of considered models. For this reason the solutions of the model equations are compared under different parameter combinations against one fixed combination of material parameters which is called ‘the reference case’.     

On the effective stress in unsaturated porous continua with double porosity

Using mixture theory we formulate the balance laws for unsaturated porous media composed of a double-porosity solid matrix infiltrated by liquid and gas. In this context, the term ‘double porosity’ pertains to the microstructural characteristic that allows the pore spaces in a continuum to be classified into two pore subspaces. We use the first law of thermodynamics to identify energy-conjugate variables and derive an expression for the ‘effective’, or constitutive, stress that is energy-conjugate to the rate of deformation of the solid matrix. The effective stress has the form , where σ is the total Cauchy stress tensor, B is the Biot coefficient, and is the mean fluid pressure weighted according to the local degrees of saturation and pore fractions. We identify other emerging energy-conjugate pairs relevant for constitutive modeling of double-porosity unsaturated continua, including the local suction versus degree of saturation pair and the pore volume fraction versus weighted pore pressure difference pair. Finally, we use the second law of thermodynamics to determine conditions for maximum plastic dissipation in the regime of inelastic deformation for the unsaturated two-porosity mixture.     

Bubble migration in a turbulent boundary layer

The wall void peaking distribution observed in an upward turbulent bubbly boundary layer along a flat plate is generated by bubbles that move towards the plate, come into contact with the wall and then slide along it. This transverse ‘migration’ has been studied using flow visualization, high speed video and particle tracking techniques to measure the trajectories of mono-disperse air bubbles at very low void fractions. Investigations have been performed at four Reynolds numbers in the range 280 < Re_θ < 3000, covering both the laminar and turbulent regimes, with mono-disperse bubbles of mean equivalent diameter between 2 mm and 6 mm. Lagrangian statistics calculated from hundreds of trajectories show that the migration only occurs in the turbulent regime and for bubble diameters below some critical value: 3.5 mm < d_eqcrit < 4 mm. Above this size (We > 3), the interface deformation is such that bubbles do not remain at the wall, even when they are released at the surface. Also, bubble migration is shown to be non-systematic, to have a non-deterministic character in the sense that trajectories differ significantly, to increase with Reynolds number and to take place on a short time scale. A series of experiments with isolated bubbles demonstrates that these results are not influenced by bubble–bubble interactions and confirm that two-way coupling in the flow is limited. Flow visualizations show that the migration originates with the capture of bubbles inside the large turbulent structures of the boundary layer (‘bulges’). The bubbles begin to move towards the wall as they cross these structures, and the point at which they reach the wall is strongly correlated with the position of the deep ‘valleys’ which separate the turbulent ‘bulges’. The analysis of the mean Lagrangian trajectories of migrating bubbles confirms these observations. Firstly, the average time of migration calculated from these trajectories coincides with the mean transit time of the bubbles across the structures. Secondly, once the trajectories have been scaled by this transit time and the boundary layer thickness δ, they all have the same form in the region y/δ < 0.4, independent of the Reynolds number.     

A theory of finite deformation magneto-viscoelasticity

This paper deals with the mathematical modelling of large strain magneto-viscoelastic deformations. Energy dissipation is assumed to occur both due to the mechanical viscoelastic effects as well as the resistance offered by the material to magnetisation. Existence of internal damping mechanisms in the body is considered by decomposing the deformation gradient and the magnetic induction into ‘elastic’ and ‘viscous’ parts. Constitutive laws for material behaviour and evolution equations for the non-equilibrium fields are derived that agree with the laws of thermodynamics. To illustrate the theory the problems of stress relaxation, magnetic field relaxation, time dependent magnetic induction and strain are formulated and solved for a specific form of the constitutive law. The results, that show the effect of several modelling parameters on the deformation and magnetisation process, are illustrated graphically.     

Elastic crystals with a triple point

The peculiar behavior of active crystals is due to the presence of evolving phase mixtures the variety of which depends on the number of coexisting phases and the multiplicity of symmetry-related variants. According to Gibbs’ phase rule, the number of phases in a single-component crystal is maximal at a triple point in the p-T phase diagram. In the vicinity of this special point the number of metastable twinned microstructures will also be the highest—a desired effect for improving performance of smart materials. To illustrate the complexity of the energy landscape in the neighborhood of a triple point, and to produce a workable example for numerical simulations, in this paper we construct a generic Landau strain-energy function for a crystal with the coexisting tetragonal (t), orthorhombic (o), and monoclinic (m) phases. As a guideline, we utilize the experimental observations and crystallographic data on the t-o-m transformations of zirconia (ZrO₂), a major toughening agent for ceramics. After studying the kinematics of the t-o-m phase transformations, we re-evaluate the available experimental data on zirconia polymorphs, and propose a new mechanism for the technologically important t-m transition. In particular, our proposal entails the softening of a different tetragonal modulus from the one previously considered in the literature. We derive the simplest expression for the energy function for a t-o-m crystal with a triple point as the lowest-order polynomial in the relevant strain components, exhibiting the complete set of wells associated with the t-o-m phases and their symmetry-related variants. By adding the potential of a hydrostatic loading, we study the p-T phase diagram and the energy landscape of our crystal in the vicinity of the t-o-m triple point. We show that the simplest assumptions concerning the order-parameter coupling and the temperature dependence of the Landau coefficients produce a phase diagram that is in good qualitative agreement with the experimental diagram of ZrO₂.     

Ductile fracture: Experiments and computations

Numerous criteria have been developed for ductile fracture (DF) prediction in metal plastic deformation. Finding a way to select these DF criteria (DFCs) and identify their applicability and reliability, however, is a non-trivial issue that still needs to be addressed in greater depth. In this study, several criteria under the categories of ‘uncoupled damage criterion’ and the ‘coupled damage criterion’, including the continuum damage mechanics (CDM)-based Lemaitre model and the Gurson-Tvergaard-Needleman (GTN) model, are investigated to determine their reliability in ductile failure prediction. To create diverse stress and strain states and fracture modes, different deformation scenarios are generated using tensile and compression tests of Al-alloy 6061 (T6) with different sample geometries and dimensions. The two categories of criteria are coded into finite element (FE) models based on the unconditional stress integration algorithm in the VUMAT/ABAQUS platform. Through physical experiments, computations and three industrial case studies, the entire correlation panorama of the DFCs, deformation modes and DF mechanisms is established and articulated. The experimental and simulation results show the following. (1) The mixed DF mode exists in every deformation of concern in this study, even in the tensile test of the round bar sample with the smallest notch radius. A decrease of stress triaxiality (η-value) leads to a reduction in the accuracy of DF prediction by the two DFC categories of DFCs, due to the interplay between the principal stress dominant fracture and the shear-stress dominant factor. (2) For deformations with a higher η-value, both categories of DFCs predict the fracture location reasonably well. For those with a lower or even negative η-value, the GTN and CDM-based criteria and some of the uncoupled criteria, including the C&L, Ayada and Oyane models, provide relatively better predictions. Only the Tresca and Freudenthal models can properly predict the shear dominant fracture. The reliability sequence of fracture moment prediction is thus the GTN model, followed by the CDM-based model and the uncoupled models. (3) The applicability of the DFCs depends on the use of suitable damage evolution rules (void nucleation/growth/coalescence and shear band) and consideration of several influential factors, including pressure stress, stress triaxiality, the Lode parameter, and the equivalent plastic strain or shear stress. These parameters determine the deformation mode (shear dominant or maximum principal stress dominant deformation) and, further, the DF mechanism (dimple fracture/shear fracture/mixed fracture).     

Numerical and experimental indentation tests considering size effects

A series of nanoindentation experiments with maximum depths varying from 1200 to 2500 nm were conducted to study indentation size effects on copper, aluminium alloy and nickel. As expected, results from classical plasticity simulation deviate significantly from experimental data for indentation at micron and submicron levels. C⁰ continuity finite element analysis incorporating the conventional theory of mechanism-based strain-gradient (CMSG) plasticity has been carried out to simulate spherical and Berkovich indentation tests at micron level where size effect is observed. The results from both numerical and actual spherical and Berkovich indentation tests demonstrate that materials are significantly strengthened for deformation at this level and the proposed approach is able to provide reasonably accurate results.     

Shock enhancement of cellular structures under impact loading: Part I Experiments

This paper aims at showing experimental proof of the existence of a shock front in cellular structures under impact loading, especially at low critical impact velocities around 50 m/s. First, an original testing procedure using a large diameter Nylon Hopkinson bar is introduced. With this large diameter soft Hopkinson bar, tests under two different configurations (pressure bar behind/ahead of the supposed shock front) at the same impact speed are used to obtain the force/time histories behind and ahead of the assumed shock front within the cellular material specimen.Stress jumps (up to 60% of initial stress level) as well as shock front speed are measured for tests at 55 m/s on Alporas foams and nickel hollow sphere agglomerates, whereas no significant shock enhancement is observed for Cymat foams and 5056 aluminium honeycombs. The corresponding rate sensitivity of the studied cellular structures is also measured and it is proven that it is not responsible for the sharp strength enhancement.A photomechanical measurement of the shock front speed is also proposed to obtain a direct experimental proof. The displacement and strain fields during the test are obtained by correlating images shot with a high speed camera. The strain field measurements at different times show that the shock front discontinuity propagates and allows for the measurement of the propagation velocity.All the experimental evidences enable us to confirm the existence of a shock front enhancement even at quite low impact velocities for a number of studied materials.     

Modelling matrix damage and fibre–matrix interfacial decohesion in composite laminates via a multi-fibre multi-layer representative volume element (MRVE)

A three-dimensional multi-fibre multi-layer micromechanical finite element model was developed for the prediction of mechanical behaviour and damage response of composite laminates. Material response and micro-scale damage mechanism of cross-ply, [0/90]_ns, and angle-ply, [±45]_ns, glass-fibre/epoxy laminates were captured using multi-scale modelling via computational micromechanics. The framework of the homogenization theory for periodic media was used for the analysis of the proposed ‘multi-fibre multi-layer representative volume element’ (M²RVE). Each layer in M²RVE was represented by a unit cube with multiple randomly distributed, but longitudinally aligned, fibres of equal diameter and with a volume fraction corresponding to that of each lamina (equal in the present case). Periodic boundary conditions were applied to all the faces of the M²RVE. The non-homogeneous stress–strain fields within the M²RVE were related to the average stresses and strains by using Gauss’ theorem in conjunction with the Hill–Mandal strain energy equivalence principle. The global material response predicted by the M²RVE was found to be in good agreement with experimental results for both laminates. The model was used to study effect of matrix friction angle and cohesive strength of the fibre–matrix interface on the global material response. In addition, the M²RVE was also used to predict initiation and propagation of fibre–matrix interfacial decohesion and propagation at every point in the laminae.     

Convection and diffusion of polymer network strands

A review of several important constitutive equations is herein conducted with an eye towards determining those most suitable for use in modelling polymer melt processing. General principles are invoked for a priori screening of the equations without needing detailed comparison of the model predictions with experimental data. These principles, which are derived from continuum mechanics, thermodynamics and molecular kinetic theory, and dela with convection and diffusion of entangled polymer strands during flow, are: (1) During sudden deformations, the stress is a unique function of the total strain. (2) The second law of thermodynamics holds for all deformations. (3) The constitutive equation can be derived from a plausible molecular model which describes the convection and diffusion. (4) The model parameters can be determined by a reasonable number of rheometric experiments. Based on these principles, it is concluded that separable free energy models are the most promising. These are either BKZ integral models with a kernel factorable into a time-dependent and a strain-dependent part. or sets of Maxwell-type differential equations that employ a generalized convected derivative, and that are linear in stress in the absence of flow.     

A generalized Paris’ law for fatigue crack growth

An extension of the celebrated Paris law for crack propagation is given to take into account some of the deviations from the power-law regime in a simple manner using the Wöhler SN curve of the material, suggesting a more general “unified law”. In particular, using recent proposals by the first author, the stress intensity factor K(a) is replaced with a suitable mean over a material/structural parameter length scale Δa, the “fracture quantum”. In practice, for a Griffith crack, this is seen to correspond to increasing the effective crack length of Δa, similarly to the Dugdale strip-yield models. However, instead of including explicitly information on cyclic plastic yield, short-crack behavior, crack closure, and all other detailed information needed to eventually explain the SN curve of the material, we include directly the SN curve constants as material property. The idea comes as a natural extension of the recent successful proposals by the first author to the static failure and to the infinite life envelopes. Here, we suggest a dependence of this fracture “quantum” on the applied stress range level such that the correct convergence towards the Wöhler-like regime is obtained. Hence, the final law includes both Wöhler's and Paris’ material constants, and can be seen as either a generalized Wöhler's SN curve law in the presence of a crack or a generalized Paris’ law for cracks of any size.     

Shock–Free Breakup of Droplets. Temporal Characteristics

In the present paper, we consider the shock–free breakup of droplets in their encounter with a layer (sheet) of a moving gas in the absence of pressure perturbations when the droplets are affected by a short U–shaped pulse of aerodynamic forces. Under a high pressure of the ambient gas medium p₀ = 20—80 bar, the droplets (ethanol or liquid oxygen) have a chance to break up after stay in a thing (2—5 mm thick) gas layer (jet) moving with a velocity of 1—10 m/sec. A distinctive feature of the process is that the characteristic time of droplet deformation and the period of natural oscillations coincide with the residence time for the droplets in the region of their interaction with the gas stream. Empirical formulas are proposed for determination of the total breakup time and the duration of the droplet disintegration stage in shock–free breakup.     

Process zone in the Single Cantilever Beam under transverse loading - Part II: Experimental

This paper describes an experimental arrangement to evaluate stress/strain fields in the process zone of asymmetric adhesively bonded joints. A transparent polycarbonate flexible beam was bonded to an aluminium alloy rigid block with an epoxy adhesive in a Single Cantilever Beam (SCB) configuration. The flexible adherend was loaded in the direction parallel to the initial crack front at constant rate. To monitor strains induced by bending and shear along the beam, electric strain gauges were attached to the upper surface of the flexible adherend. Thus strain distribution was measured above the bonded surface, which could be used to monitor crack propagation and investigate stress redistribution in the process zone. A Timoshenko beam lying on a Pasternak elastic foundation model was used for the analysis of experimental findings. Subsequently, the Digital Image Correlation technique was used to measure the flexible substrate in-plane displacement field in the vicinity of the crack front and to assess the specimen kinematics. We found that strain gauge instrumentation of the fracture mechanics specimen was a very sensitive technique for experimental analysis of crack propagation under complex loading, offering fine investigation of stress distribution in the cohesive zone.