Dynamic crushing of aluminum foams: Part I – Experiments

The two-part series of papers presents the results of a study of the crushing behavior of open-cell Al foams under impact. In Part I, direct and stationary impact tests are performed on cylindrical foam specimens at impacts speeds in the range of 20–160 m/s using a gas gun. The stress at one end is recorded using a pressure bar, while the deformation of the entire foam specimen is monitored with high-speed photography. Specimens impacted at velocities of 60 m/s and above developed nearly planar shocks that propagated at well-defined velocities crushing the specimen. The shock speed vs. impact speed, and the strain behind the shock vs. impact speed representations of the Hugoniot were both extracted directly from the high-speed images. The former follows a linear relationship and the latter asymptotically approaches a strain of about 90% at higher velocities. The Hugoniot enables calculation of all problem variables without resorting to an assumed constitutive model. The compaction energy dissipation across the shock is shown to increase with impact velocity and to be significantly greater than the corresponding quasi-static value. Specimens impacted at velocities lower than 40 m/s exhibited response and deformation patterns that are very similar to those observed under quasi-static crushing. Apparently, in this impact speed regime inertia increases the energy absorption capacity very modestly.     

Parameterization and modelling of large off-road tyres for ride analyses: Part 2 – Parameterization and validation of tyre models

Every mathematical model used in a simulation is an idealization and simplification of reality. Vehicle dynamic simulations that go beyond the fundamental investigations require complex multi-body simulation models. The tyre–road interaction presents one of the biggest challenges in creating an accurate vehicle model. Many tyre models have been proposed and developed but proper validation studies are less accessible. These models were mostly developed and validated for passenger car tyres for application on relatively smooth roads. The improvement of ride comfort, safety and structural integrity of large off-road vehicles, over rough terrain, has become more significant in the development process of heavy vehicles. This paper investigates whether existing tyre models can be used to accurately describe the vertical behaviour of large off road tyres while driving over uneven terrain. [1] Presented an extensive set of experimentally determined parameterization and validation data for a large off-road tyre. Both laboratory and field test are performed for various loads, inflation pressures and terrain inputs. The parameterization process of four tyre models or contact models are discussed in detail. The parameterized models are then validated against test results on various hard but rough off-road terrain and the results are discussed.     

Anisotropic response of high-purity α-titanium: Experimental characterization and constitutive modeling

This paper presents a comprehensive experimental and theoretical investigation of the deformation behavior of high-purity, polycrystalline α-titanium under quasi-static conditions at room temperature. The initial material in this study was a cross-rolled plate with a strong basal texture. To quantify the plastic anisotropy and the tension–compression asymmetry of this material, monotonic tensile and compressive tests were conducted, on samples cut along different directions of the plate. A new anisotropic elastic/plastic model was developed to describe the quasi-static macroscopic response of the aggregate. Key in its formulation is the use of an anisotropic yield criterion that captures strength-differential effects and an anisotropic hardening rule that accounts for texture evolution associated to twinning. A very good agreement between FE simulations using the model developed and uniaxial data was obtained.     

Finite deformation pseudo-elasticity of shape memory alloys – Constitutive modelling and finite element implementation

In this paper we suggest a new phenomenological material model for shape memory alloys. In contrast to many earlier concepts of this kind the present approach includes arbitrarily large deformations. The work is motivated by the requirement, also expressed by regulatory agencies, to carry out finite element simulations of NiTi stents. Depending on the quality of the numerical results it is possible to circumvent, at least partially, expensive experimental investigations. Stent structures are usually designed to significantly reduce their diameter during the insertion into a catheter. Thereby large rotations combined with moderate and large strains occur. In this process an agreement of numerical and experimental results is often hard to achieve. One of the reasons for this discrepancy is the use of unrealistic material models which mostly rely on the assumption of small strains. In the present paper we derive a new constitutive model which is no longer limited in this way. Further its efficient implementation into a finite element formulation is shown. One of the key issues in this regard is to fulfil “inelastic” incompressibility in each time increment. Here we suggest a new kind of exponential map where the exponential function is suitably computed by means of the spectral decomposition. A series expansion is completely avoided. Finite element simulations of stent structures show that the new concept is well appropriate to demanding finite element analyses as they occur in practically relevant problems.     

Strong discontinuity analysis of a class of anisotropic continuum damage constitutive models – Part I: Theoretical considerations

Several modeling techniques aiming at considering cracks as kinematics discontinuities have been proposed for the past years. Within this scope, the embedded finite element method (E-FEM) was introduced a couple of years ago. Among the features of this approach, it has been shown that a kinematic enhancement of the displacement field allows constructing a discrete model (expressed in terms of traction vector–displacement jump) from any continuous model (expressed in terms of stress–strain). This result has been rigorously established if the continuous model is formulated within the framework of either isotropic continuum damage or plasticity theories. The objectives of this study are (i) to extend this result in case where the continuous model belongs to a class of anisotropic continuum damage constitutive models and (ii) to show the main features of a specific traction/separation law derived from the aforementioned class of constitutive models through several numerical case-studies. In this paper, the light is put on the theoretical considerations which allow deriving discrete models in a consistent way.     

A micromechanics-based strain gradient damage model for fracture prediction of brittle materials – Part I: Homogenization methodology and constitutive relations

In this paper, we first describe a homogenization methodology with the aim of establishing strain gradient constitutive relations for heterogeneous materials. The methodology presented in this work includes two main steps. The first one is the construction of the average strain-energy density for a well-chosen RVE by using a homogenization technique. The second one is the transformation of the obtained average strain-energy density to that for the continuum. An important characteristic of this method is its self-consistency with respect to the choice of the RVE: the strain gradient constitutive law built by using the present method is independent of the size and the form of the RVE. In the frame of this homogenization procedure, we have constructed a strain gradient constitutive relation for a two-dimensional elastic material with many microcracks by adopting the self-consistent scheme. It was shown that the effective behavior of cracked solids depends not only on the crack density but also on the average crack size with which the strain gradient is associated. The proposed constitutive relation provides a starting point for the development of an evolution law of damage including strain gradient effect, which will be presented in the second part of this work.     

Strong discontinuity analysis of a class of anisotropic continuum damage constitutive models – Part II: Concrete material application

On the basis of the strong discontinuity analysis, a discrete model expressed in terms of traction vector-displacement jump has been constructed from a continuous model expressed in terms of stress–strain law. In the first part of the paper, this approach has been extended to a class of anisotropic continuum damage constitutive models [1]. In this second part of the paper, the proposed class of discrete anisotropic damage constitutive models is particularized. More precisely, a micromechanical-based anisotropic damage constitutive model is derived. This model accounts in a natural manner for particular crack families orientation. The aims of this paper are (i) to illustrate the capabilities of the proposed discrete enhanced model in reproducing the induced anisotropy appearing in quasi-brittle materials when cracking and (ii) to assess the numerical robustness of the time integration scheme. For this purpose, two numerical examples at the material point level are exposed after a brief presentation of the time integration scheme. The correspondence between the continuous and the discrete model as well as the induced anisotropy features are emphasized.     

Segregation of particulate solids＂ Experiments and DEM simulations

Segregation of granular materials is a complex phenomenon, difficult to measure quantitatively and to predict. Discrete element method （DEM） can be a useful tool to predict segregation effects and to support the industrial design. In this context, a very challenging idea is the characterization of the granular solids to provide the key parameters needed for a successful DEM simulation of segregation processes. Rolling friction, sliding friction and the coefficient of restitution are the critical parameters to be studied. These microscopic simulation parameters are calibrated by comparing the macroscopic behavior of granular matter in standard bulk experiments, which have the advantage of being highly repeatable and reliable. An experimental method is presented to characterize free surface segregation. The effects of different particle properties, particularly, shape and size, on segregation of cohesionless materials were investi- gated. From the experiments, particle size demonstrated a stronger effect on segregation than particle shape. Finally, the corresponding DEM simulations of the segregation experiments were presented. The parameters obtained by calibration were validated by the comparison of the modeled segregation behav- ior with the experimental results. Thus, calibrated DEM simulations are capable of predicting segregation effects.     

Micromechanical analysis of polymer composites reinforced by unidirectional fibres: Part I – Constitutive modelling

Micromechanical analyses of unidirectional continuous-fibre reinforced composite materials were performed to study the mechanisms of deformation and fracture of the constituents, and their influence on the mechanical properties of the composite. Special focus was given to the matrix material behaviour as well as to the interface between constituents. The matrix was modelled using a pressure dependent, elasto-plastic thermodynamically consistent damage model. Cohesive elements were used to model the interface between matrix and fibres. Part I of this paper details the continuum model developed for a typical epoxy matrix. Part II will focus on micromechanical analyses of composite materials and the estimation of its elastic and strength properties.     

Biaxial crushing of honeycombs: —Part 1: Experiments

The in-plane compression and crushing of honeycombs is known to be closely related to the crushing behavior of the broader class of space filling cellular solids. Previously, the authors conducted an extensive study of uniaxial crushing of a polycarbonate honeycomb with circular cells. In this paper the same honeycomb is crushed biaxially. The crushing was performed in a custom testing facility between rigid platens which can be moved independently in two orthogonal directions. The facility allows testing at various biaxiality ratios and volume reductions as high as 95%. The facility was used to conduct several series of biaxial crushing experiments on nearly square honeycomb specimens (18×21 cells) . In each experiment we recorded the true stress–displacement responses in the x- and y-directions as well as full field views of the deformation using a video camera. Biaxial crushing is quite complex and the prevalent mechanisms of collapse depend on the biaxiality ratio (γ) . As is the case in uniaxial crushing, the onset of collapse involves localized instabilities, however, the extent of localized deformation varies with γ. The energy absorption capacity of the material depends on γ. The highest energy is required when the specimen is crushed at the same rates in the two directions.     

The automatic control of boundary-layer transition — Experiments and computation

In recent years there has been an increasing interest in the control of boundary-layer transition through the use of wall suction. In the current work suction is provided through one or more suction panels situated close to the leading edge of a plate. Experiments show that boundary-layer pressure fluctuation measurements can be used to identify the position of transition. Transition can be maintained at a desired location with minimum power consumption by employing an automatic adaptive feedback control loop which regulates the suction flow rates of two independent suction panels. This can be expressed as a constrained optimization problem. To allow the suction flow rates to be updated, a modified least mean squares algorithm is used within the control loop. Experimental measurements show that the control algorithm allows fast and stable convergence towards the optimum suction distribution for a double suction panel configuration. Numerical simulations have also been performed. The two-dimensional boundary layer was calculated allowing the viscous boundary layer to interact with the inviscid outer flow. Following linear stability theory the spatial growth rates are calculated by solving an Orr-Sommerfeld type eigenvalue problem, with the streamwise location of transition predicted via thee ^N -method. Applying the same optimization strategy as in the experiments, good qualitative agreement between computations and experiments was found. The optimization algorithm has been applied to computer models where the relation between suction flow rates and transition location is described by an empirical analytical function. This shows that the controller can in principle be applied to systems with more than two suction panels.Nomenclature b transition location with zero suction - d desired transition location - e(k) error signal - k iteration index - p rms pressure - p _ref reference rms pressure - r sum of the reference pressure - u streamwise velocity - u _e external velocity - inviscid external velocity - A wave amplitude - F( ) cost function - I identity matrix - N maximum amplification factor - P projection matrix - R Reynolds number - Re Reynolds number based on the boundary-layer thickness - R matrix of weights - Tu turbulence level - vector of suction flow rates - v normal velocity - v _wall suction velocity at the surface - x streamwise coordinates - x _m microphone location - x _T(k) measured transition location - y normal coordinate - y(k) sum of the measured pressures - w(k) noise - plate length - r +i _i - free stream velocity - * displacement thickness - gradient vector - Lagrange multiplier - controller gain - disturbance stream function - disturbance amplitude - wave frequency = complex wave number     

Crack tortuousity in the nacreous layer – Topological dependence and biomimetic design guideline

The nacreous layer in seashells is known for two phenomenal aspects: light-weightiness and superior fracture toughness. Of a multitude of toughening mechanisms, the highly meandering nature of the crack path through its staggered architecture has been reported to contribute approximately a third of its overall toughness. In the current article, we are trying to establish the scientific rationale associated with the influence of overlap length on the crack-tip driving force from a local perspective via development of a simplified analytical model. Characteristic overlap lengths computed showed reasonable agreement with the values reported in the nacreous layer and previously published experimental data. Biomimetic design guideline obtained from the current investigation would thereby lead to development of synthetic staggered architecture materials with improved stiffness, load-transfer and toughness.     

Shot peening and peen forming finite element modelling – Towards a quantitative method

Peen forming is commonly used on aluminium alloys in the aerospace industry for wing skin shaping. Numerous analytical, numerical, and experimental studies have been made to better understand the effects of various peening parameters on the final material state and to predict deformed shapes, but conclusions were often limited to trends. The purpose of this study is therefore to develop and verify experimentally quantitative numerical tools for peen forming applications by studying the simple case of peening an Almen-sized AA-2024 aluminium strip in an Almen holder. The first step consisted in improving an existing random dynamic model by determining optimal dimensions. The AA-2024 target mechanical behaviour was characterized experimentally and a combined isotropic-kinematic hardening law was selected to model the material behaviour. The dynamic impact model and material constitutive law provided good prediction of peening-induced stresses in thick AA-2024 for two shot velocities. The sequence-sensitive aspect of the forming process was also investigated and a new shell-based finite element model was proposed. Numerical and experimental results for three shot velocities were compared to evaluate the validity of this numerical simulation method and promising agreement was observed.     

On the directional approach in constitutive modelling: A general thermomechanical framework and exact solutions for Mooney–Rivlin type elasticity in each direction

In order to represent process-induced anisotropies in continuum mechanics or to transfer one-dimensional material models to three spatial dimensions the directional approach is a helpful technique. Since the essential equations are defined in the orientation space it is also denoted as microsphere approach. In the current article, the relation for the directional stress tensor of the second Piola–Kirchhoff type is motivated using the volumetric/isochoric split of the deformation gradient and the Clausius–Duhem inequality. Owing to inherent nonlinearities, numerical discretisation techniques are usually applied to calculate the total stress by averaging the directional stress tensors over the unit sphere. In order to investigate the accuracy of such simulations, the availability of exact solutions in closed form is essential. To this end, the tension/compression behaviour which belongs to a certain direction in the orientation space is modelled by an elasticity relation of the Mooney Rivlin type. The exact solutions are calculated, visualized and discussed for uniaxial tension and compression as well as for equibiaxial tension.     

Ratchetting under tension–torsion loadings: experiments and modelling

This paper is concerned with the mechanical behaviour of 316 austenitic stainless steel under multiaxial loadings and particular attention is paid to ratchetting under tension–torsion non-proportional loadings. First, a series of uniaxial tests and biaxial tests has been carried out in order to calibrate five different cyclic plasticity models based on an isotropic hardening rule and a non-linear kinematic hardening rule. It is shown that this class of models gives quite good agreement between the experimental and numerical results. Second, another series of ratchetting tests has been carried out under tension–torsion loadings in order to test the prediction capacities of the previous models. It is shown that whereas the models have been calibrated with similar loading paths, four of the five selected models give poor predictions.     

On uniqueness and continuous dependence of solutions in viscoelastic mixtures

This note deals with the isothermal linear theory of porous viscoelastic mixtures. Questions of uniqueness and continuous dependence for solutions of various classes of initial boundary value problems in mixtures consisting of two constituents: a porous elastic solid and a porous Kelvin–Voigt material are studied. The Lagrange identity and Logarithmic convexity methods are used to establish uniqueness and continuous dependence results, with no definiteness assumptions upon the internal energy.     

An elastic–plastic constitutive law for the description of uniaxial and multiaxial ratchetting

An elastic–plastic constitutive model is proposed to describe 1-D and 2-D ratchetting. The model is also able to give correct results for 2-D ratchetting when only uniaxial identification is used, while no special threshold or parameter is used for the case of non-proportional loading. The original feature of this model consist in the introduction of a ratchetting stress (material characteristic) along with the maximal stress supported in the history of loading and the plastic strain at the last unloading. In this paper uniaxial and 3-D formulations have been described based on a numerical implementation in the software Code_Aster. Uniaxial and also multiaxial identifications have been used. Simulations have been realized for proportional and non-proportional homogeneous cases, as well as for structures under anisothermal thermomechanical loading. The results of a benchmark on a structure, comparing experiment, simulations by this model and some other phenomenological models, and a polycrystalline model are presented. An analysis of error margin due to the choice of Mises criterion is exposed.