The high strain mechanical response of the wet Kelvin model for open-cell foams

Surface Evolver software was used to create the three-dimensional geometry of a Kelvin open-cell foam, to simulate that of polyurethane flexible foams. Finite Element Analysis (FEA) with 3D elements was used to model large compressive deformation in the [0 0 1] and [1 1 1] directions, using cyclic boundary conditions when necessary, treating the polyurethane as an elastic or elastic–plastic material. The predicted foam Young’s moduli in the [0 0 1] direction are double those of foams with uniform Plateau border cross-section edges, for the same foam density and material properties. For compression in the [1 1 1] direction, the normalized Young’s modulus increases from 0.9 to 1.1 with foam relative density, and the predicted stress–strain relationship can have a plateau, even for a linearly-elastic polymer. As the foam density increases, the predicted effects of material plasticity become larger. For foam of relative density 0.028, edge-to-edge contact is predicted to occur at a 66% strain for [1 1 1] direction compression. The foam is predicted to contract laterally when the [1 1 1] direction compressive strain exceeds 25%.     

Influence of temperature and strain rate on the mechanical behavior of three amorphous polymers: Characterization and modeling of the compressive yield stress

Uniaxial compression stress–strain tests were carried out on three commercial amorphous polymers: polycarbonate (PC), polymethylmethacrylate (PMMA), and polyamideimide (PAI). The experiments were conducted under a wide range of temperatures (−40 °C to 180 °C) and strain rates (0.0001 s⁻¹ up to 5000 s⁻¹). A modified split-Hopkinson pressure bar was used for high strain rate tests. Temperature and strain rate greatly influence the mechanical response of the three polymers. In particular, the yield stress is found to increase with decreasing temperature and with increasing strain rate. The experimental data for the compressive yield stress were modeled for a wide range of strain rates and temperatures according to a new formulation of the cooperative model based on a strain rate/temperature superposition principle. The modeling results of the cooperative model provide evidence on the secondary transition by linking the yield behavior to the energy associated to the β mechanical loss peak. The effect of hydrostatic pressure is also addressed from a modeling perspective.     

Relating inhomogeneous deformation to local texture in zirconium through grain-scale digital image correlation strain mapping experiments

The effect of local texture on inhomogeneous plastic deformation is studied in zirconium subjected to uniaxial compression. Cross-rolled commercially pure Zr 702 plate that had a strong basal (0 0 0 1) texture through the plate thickness, and a non-basal texture in cross-section, was obtained. At a compressive strain rate of 1 s^?1, samples loaded either in the through-thickness or in-plane directions exhibited significant differences in yield strength, hardening response and failure mechanisms. These macroscopic differences are related to microstructural features by combining information from electron backscattered diffraction with real time in situ imaging and subsequent full-field strain measurements obtained using digital image correlation. Experimental results indicate that the through-thickness loaded zirconium samples, which show a strong basal-texture in the loading direction, do not deform homogeneously – implying the lack of a representative volume element. The detailed surface deformation fields provided by digital image correlation allow for a qualitative and quantitative analysis of the relationship between grain orientation and patterns of deformation bands that form as the precursors to development of an adiabatic shear band in the through-thickness loaded sample. For the in-plane loaded samples, inhomogeneities still exist at the microscale, but the collective behavior of several grains leads to a homogeneous response at the macroscale. It is observed that local texture for hcp polycrystals, which are significantly slip restricted, can directly affect both local and global response, even at low to moderate plastic strains.     

Dynamic crushing of sandwich panels with prismatic lattice cores

The dynamic out-of-plane compressive response of stainless steel corrugated and Y-frame sandwich cores have been investigated for impact velocities ranging from quasi-static to 200 ms⁻¹. Laboratory-scale sandwich cores of relative density 2.5% were manufactured and the stresses on the front and rear faces of the dynamically compressed sandwich cores were measured using a direct impact Kolsky bar. Direct observational evidence is provided for micro-inertial stabilisation of both topologies against elastic buckling at impact velocities below 30 ms⁻¹. At higher impact velocities, plastic waves within the core members result in the front face stresses increasing with increasing velocity while the rear face stresses remain approximately constant. While the finite element calculations predict the rear face stresses and dynamic deformation modes to reasonable accuracy, the relatively slow response time of the measurement apparatus results in poor agreement between the measured and predicted front face stresses. The finite element calculations also demonstrate that material strain-rate effects have a negligible effect upon the dynamic compressive response of laboratory-scale and full-scale sandwich cores.     

Integral-based constitutive equation for rubber at high strain rates

High-speed experiments were conducted to characterize the deformation and failure of Styrene Butadiene Rubber at impact rates. Dynamic tensile stress–strain curves of uniaxial strip specimens and force–extension curves of thin sheets were obtained from a Charpy tensile impact apparatus. Results from the uniaxial tension tests indicated that although the rubber became stiffer with increasing strain rates, the stress–strain curves remained virtually the same above 280 s⁻¹. Above this critical strain rate, strength, fracture strain and toughness decreased with increasing strain rates. When strain rates were below 180 s⁻¹, the initial modulus, tensile strength and breaking extension increased as the strain rate increased. Between strain rates of 180 and 280 s⁻¹, the initial modulus and tensile strength increased with increasing strain rates but the extension at break decreased with increasing strain rates. A hyper-viscoelastic constitutive relation of integral form was used to describe the rate-dependent material behavior of the rubber. Two characteristic relaxation times, 5 ms and 0.25 ms, were needed to fit the proposed constitutive equation to the data. The proposed constitutive equation was implemented in ABAQUS Explicit via a user-defined subroutine and used to predict the dynamic response of the rubber sheets in the experiments. Numerical predictions for the transient deformation and failure of the rubber sheet were within 10% of experimental results.     

Mechanics of the rate-dependent elastic–plastic deformation of glassy polymers from low to high strain rates

A combined experimental and analytical investigation has been performed to understand the mechanical behavior of two amorphous polymers—polycarbonate and poly(methyl methacrylate)—at strain rates ranging from 10⁻⁴ to 10⁴ s⁻¹. This range in strain rates was achieved in uniaxial tension and compression tests using a dynamic mechanical analyzer (DMA), a servo-hydraulic testing machine, and an aluminum split-Hopkinson pressure bar. DMA tension tests were used to characterize the viscoelastic behavior of these materials, with focus on the rate-dependent shift of material transition temperatures. Uniaxial compression tests on the servo-hydraulic machine (10⁻⁴ to 1 s⁻¹) and the split-Hopkinson pressure bar (10³ to 10⁴ s⁻¹) were used to characterize the rate-dependent yield and post-yield behavior. Both materials were observed to exhibit increased rate sensitivity of yield under the same strain rate/temperature conditions as the β-transition of the viscoelastic behavior. A physically based constitutive model for large strain deformation of thermoplastics was then extended to encompass high-rate conditions. The model accounts for the contributions of different molecular motions which become operational and important in different frequency regimes. The new features enable the model to not only capture the transition in the yield behavior, but also accurately predict the post-yield, large strain behavior over a wide range of temperatures and strain rates.     

High strain-rate behavior of ice under uniaxial compression

In the present study, a modified split Hopkinson pressure bar (SHPB) is employed to investigate the dynamic response of ice under uniaxial compression in the range of strain rates from 60 to 1400 s^?1 and at initial test temperatures of ?10 and ?30 °C. The compressive strength of ice shows positive strain-rate sensitivity over the range of strain rates employed; a slight influence of ice microstructure is observed, but it is much less than that reported previously for ice deformation under quasi-static loading conditions [Schulson, E.M., IIiescu, D., Frott, A., 2005. Characterization of ice for return-to-flight of the space shuttle. Part 1 – Hard ice. NASA CR-2005-213643-Part 1]. Specimen thickness, within the range studied, was found to have little or no effect on the peak (failure) strength of ice, while lowering the test temperature from ?10 to ?30 °C had a considerable effect, with ice behaving stronger at the lower test temperature. Moreover, unlike in the case of uniaxial quasi-static compression of ice, the effect of specimen end-constraint during the high rate compression was found to be negligible. One important result of these experiments, which may have important implications in modeling ice impacts, involves the post “peak-stress” behavior of the ice in that the ice samples do not catastrophically lose their load carrying capacity even after the attainment of peak stress during dynamic compression. This residual (tail) strength of the damaged/fragmented ice is sizable, and in some cases is larger than the quasi-static compression strength reported for ice. Moreover, this residual strength is observed to be dependent on sample thickness and the strain rate, being higher for thinner samples and at higher strain-rates during dynamic compression.     

The mechanisms governing the activation of dislocation sources in aluminum at different strain rates

This article examines the time to activate Frank–Read sources in response to macroscopic strain rates ranging from 10¹ s⁻¹ to 10¹⁰ s⁻¹ in aluminium under athermal conditions. We develop analytical models of the bowing of a pinned dislocation segment as well as numerical simulations of three dimensional dislocation dynamics. We find that the strain rate has a direct influence on both the activation time and the source strength of Frank–Read sources at strain rates up to 10⁶ s⁻¹, and the source strength increases in almost direct proportion to the strain rate. This contributes to the increase in the yield stress of materials at these strain rates. Above 10⁶ s⁻¹, the speed of the bowing segments reaches values that exceed the domain of validity of the linear viscous drag law, and the drag law is modified to account for inertial effects on the motion of the dislocation. As a result the activation times of Frank–Read sources reach a finite limit at strain rates greater than 10⁸ s⁻¹, suggesting that Frank–Read sources are unable to operate before homogeneous nucleation relaxes elastic stresses at the higher strain rates of shock loading. Elastodynamic calculations are carried out to compare the contributions of Frank–Read sources and homogeneous nucleation of dislocations to plastic relaxation. We find that at strain rates of 5×10⁷ s⁻¹ homogeneous nucleation becomes the dominant generation mechanism.     

Thermo-mechanical response of nylon 101 under uniaxial and multi-axial loadings: Part I,Experimental results over wide ranges of temperatures and strain rates

A comprehensive study of the thermo-mechanical response of a thermoplastic polymer, nylon 101 is presented. Quasi-static and dynamic compression uniaxial and multi-axial experiments (stress states) were performed at a wide range of strain rates (10⁻⁵ to 5000 s⁻¹) and temperatures (−60 to 177 °C or −76 to 350 °F). The material is found to be non-linearly dependent on strain rate and temperature. The change in volume after plastic deformation is investigated and is found to be negligibly small. The relaxation and creep responses at room temperature are found to be dependent on strain rate and the stress–strain level at which these phenomena are initiated. Total deformation is decomposed into visco-elastic and visco-plastic components; these components have been determined at different levels of deformation. Results from non-proportional uniaxial to biaxial compression, and torsion experiments, are also reported for three different strain rates at room temperature. It is shown that nylon 101 has a response dependent on the hydrostatic pressure.     

Full-field characterization of mechanical behavior of polyurethane foams

The foam material of interest in this investigation is a rigid closed-cell polyurethane foam PMDI with a nominal density of 20 pcf (320 kg/m³). Three separate types of compression experiments were conducted on foam specimens. The heterogeneous deformation of foam specimens and strain concentration at the foam–steel interface were obtained using the 3-dimensional digital image correlation (3D-DIC) technique. These experiments demonstrated that the 3D-DIC technique is able to obtain accurate and full-field large deformation of foam specimens, including strain concentrations. The experiments also showed the effects of loading configurations on deformation and strain concentration in foam specimens. These DIC results provided experimental data to validate the previously developed viscoplastic foam model (VFM). In the first experiment, cubic foam specimens were compressed uniaxially up to 60%. The full-field surface displacement and strain distributions obtained using the 3D-DIC technique provided detailed information about the inhomogeneous deformation over the area of interest during compression. In the second experiment, compression tests were conducted for cubic foam specimens with a steel cylinder inclusion, which imitate the deformation of foam components in a package under crush conditions. The strain concentration at the interface between the steel cylinder and the foam specimen was studied in detail. In the third experiment, the foam specimens were loaded by a steel cylinder passing through the center of the specimens rather than from its end surface, which created a loading condition of the foam components similar to a package that has been dropped. To study the effects of confinement, the strain concentration and displacement distribution over the defined sections were compared for cases with and without a confinement fixture.     

Ductility of selected metals under electromagnetic ring test loading conditions

Experimental studies on ductility of selected metals differing mechanical properties under strain rates between 4 × 10³ and 2 × 10⁴ s^?1 are presented in this work. The electromagnetic expanding ring experiment was used as the primary tool for examining the ductility behaviour of metals. Through a use of the Phantom v12 digital high-speed camera and specialised TEMA Automotive software, rings expansion velocities were determined with satisfactory accuracy for all ring tests. In this paper, the experimental observations on cold-rolled copper Cu-ETP, aluminium alloy Al 7075, barrel steel and tungsten heavy alloy are reported. Ductility of studied materials was estimated by measuring changes in cross-sectional areas in the uniform strain portions of the recovered ring fragments. In a similar way the metals ductility was defined at the conventional tensile test condition. Moreover, results of analogue investigation for static and dynamic loading at the temperature of about ?40 °C were described. The experimental observations mainly revealed the different ductility behaviour of metals tested at applied dynamic loadings; Cu-ETP and barrel steel demonstrated an increase in ductility, whereas aluminium alloy Al 7075 and tungsten heavy alloy were characterised by lower ductility in comparison to static loading. These results appear to be partially in contrast with the observations reported recently by some other investigators.     

Dynamic response and energy dissipation characteristics of balsa wood: experiment and analysis

Dynamic response of a cellular sandwich core material, balsa wood, is investigated over its entire density spectrum ranging from 55 to 380 kg/m³. Specimens were compression loaded along the grain direction at a nominal strain rate of 3 × 10³ s⁻¹ using a modified Kolsky (split Hopkinson) bar. The dynamic data are discussed and compared to those of quasi-static experiments reported in a previous study (Mech. Mater. 35 (2003) 523). Results show that while the initial failure stress is very sensitive to the rate of loading, plateau (crushing) stress remains unaffected by the strain rate. As in quasi-static loading, buckling and kink band formation were identified to be two major failure modes in dynamic loading as well. However, the degree of dynamic strength enhancement was observed to be different for these two distinct modes. Kinematics of deformation of the observed failure modes and associated micro-inertial effects are modeled to explain this different behavior. Specific energy dissipation capacity of balsa wood was computed and is found to be comparable with those of fiber-reinforced polymer composites.     

An investigation of a model of the flow pattern transition mechanism in relation to the identification of annular flow of R134a in a vertical tube using various void fraction models and flow regime maps

In the present study, new experimental data are presented for literature on the prediction of film thickness and identification of flow regime during the co-current downward condensation in a vertical smooth copper tube having an inner diameter of 8.1 mm and a length of 500 mm. R134a and water are used as working fluids in the tube side and annular side of a double tube heat exchanger, respectively. Condensation experiments are done at mass fluxes of 300 and 515 kg m^?2 s^?1. The condensing temperatures are between 40 and 50 °C; heat fluxes are between 12.65 and 66.61 kW m^?2. The average experimental heat transfer coefficient of the refrigerant HFC-134a is calculated by applying an energy balance based on the energy transferred from the test section. A mathematical model by Barnea et al. based on the momentum balance of liquid and vapor phases is used to determine the condensation film thickness of R134a. The comparative film thickness values are determined indirectly using relevant measured data together with various void fraction models and correlations reported in the open literature. The effects of heat flux, mass flux, and condensation temperature on the film thickness and condensation heat transfer coefficient are also discussed for the laminar and turbulent flow conditions. There is a good agreement between the film thickness results obtained from the theoretical model and those obtained from six of 35 void fraction models in the high mass flux region of R134a. In spite of their different valid conditions, six well-known flow regime maps from the literature are found to be predictive for the annular flow conditions in the test tube in spite of their different operating conditions.     

Flow boiling of halogenated refrigerants at high saturation temperature in a horizontal smooth tube

Several correlations are available in the open literature for computing the heat transfer coefficient during flow boiling inside plain channels. With respect to halogenated refrigerants, these correlations are usually compared to data taken in a limited range of evaporation temperature and reduced pressure. More recently, the adoption of new refrigerants, such as high pressure HFCs and carbon dioxide, requires to largely extend the pressure range of application of such correlations. Besides, the design of evaporators for some heat pumping applications, where temperatures are set at higher values as compared to usual evaporating temperatures in air-conditioning equipment, requires proper validation of the computing methods.The present paper aims at comparing four well-known predicting models to a new database collected during flow boiling of HCFC (R22) and HFC refrigerants (R134a, R125 and R410A) in a horizontal 8 mm internal diameter tube. This database is characterized by saturation temperature ranging between 25 °C and 45 °C, reduced pressure spanning between 0.19 and 0.53. Mass velocity ranges between 200 and 600 kg m^?2 s^?1 and heat flux between 9 and 53 kW m^?2.Evaporating heat transfer coefficients of halogenated refrigerants at such high temperatures have not been reported in the open literature so far. The discussion of the results will enlighten some similarities with experimental trends presented in the literature for evaporating carbon dioxide.Two models tested here show good prediction capabilities of the present experimental data, but not for all the data sets in the same way. For the purpose of practical use, a simple modification of the correlation by Gungor and Winterton [1] is proposed, showing that this is able to catch the experimental trends of the present database with good agreement.     

Singular,hypersingular and singular free electromagnetic fields at wedge tips in metamaterials

The engineering response of metamaterials has a dramatic impact on the physics, optics and engineering communities, because they offer electromagnetic properties that are difficult or impossible to achieve with conventional materials. In this paper, an asymptotic analysis of the electromagnetic fields at multi-material wedges composed of metamaterials is proposed. This is made possible by removing the assumption of positive electric permittivities and magnetic permeabilities, an hypothesis which usually applies to conventional materials. Exploring the whole range of variability of these electromagnetic properties, it is shown that, in addition to the classical real eigenvalues 0 ? λ < 1 leading to power-law singularities of the type O(r^λ?1) as r → 0, it is also possible to find imaginary eigenvalues leading to hypersingular solutions, as well as nonsingular configurations for a suitable choice of the negative electric permittivities and magnetic permeabilities of the media. Moreover, to fully characterize the asymptotic fields, the analysis is not only restricted to the determination of the lowest real and complex eigenvalues, but is also extended to the evaluation of the higher-order nonsingular ones. The obtained analytical results collected in synthetic diagrams are expected to have impact on the design of micro- and nano-electro-mechanical systems.     

Diagnosing laser-driven,shock-heated foam target with Al absorption spectroscopy on OMEGA EP

Results on diagnoses of laser-driven, shock-heated foam plasma with time-resolved Al 1s-2p absorption spectroscopy are reported. Experiments were carried out to produce a platform for the study of relativistic electron transport. In cone-guided Fast Ignition (FI), relativistic electrons generated by a high-intensity, short-pulse igniter beam must be transported through a cone tip to an imploded core. Transport of the energetic electrons could be significantly affected by the temperature-dependent resistivity of background plasmas. The experiment was conducted using four UV beams of the OMEGA EP laser at the Laboratory For Laser Energetics. One UV beam (1.2 kJ, 3.5 ns square) was used to launch a shock wave into a foam package target, consisting of 200 mg/cm³ CH foam with aluminum dopant and a solid plastic container surrounding the foam layer. The other three UV beams with the total energy of 3.2 kJ in 2.5 ns pulse duration were tightly focused onto a Sm dot target to produce a point X-ray source in the energy range of 1.4–1.6 keV. The quasi-continuous X ray signal was transmitted through the shock-heated Al-doped, foam layer and recorded with an X-ray streak camera. The measured 1s-2p Al absorption features were analyzed using an atomic physics code FLYCHK. Electron temperature of 40 eV inferred from the spectral analysis is consistent with 2-D DRACO Radiation-hydrodynamic simulations.     

Thermo-mechanical behaviour of TRIP 1000 steel sheets subjected to low velocity perforation by conical projectiles at different temperatures

This paper presents and analyzes the behaviour of TRIP 1000 steel sheets subjected to low velocity perforation by conical projectiles. The relevance of this material resides in the potential transformation of retained austenite to martensite during impact loading. This process leads to an increase in strength and ductility of the material. However, this transformation takes place only under certain loading conditions strongly dependent on the initial temperature and deformation rate. In order to study the material behaviour under impact loading, perforation tests have been performed using a drop weight tower. Experiments were carried out at two different initial temperatures T₀ = 213 K and T₀ = 288 K, and within the range of impact velocities 2.5 m/s ? V₀ ? 4.5 m/s. The experimental setup enabled the measuring of impact velocity, residual velocity, load-time history and failure mode. In addition, dry and lubricated contacts between the striker and the plate have been investigated. Finally, by using X-ray diffraction it has been shown that no martensitic transformation takes place during the perforation process. The causes involving the none-appearance of martensite are examined.     

On the interplay between material flaws and dynamic necking

In this paper we investigate the interplay between material defects and flow localization in elastoplastic bars subjected to dynamic tension. For that task, we have developed a 1D finite difference scheme within a large deformation framework in which the material is modelled using rate-dependent J₂ plasticity. A perturbation of the initial yield stress is introduced in each node of the finite difference mesh to model localized material flaws. Numerical computations are carried out within a wide spectrum of strain rates ranging from 500 s⁻¹ to 2500 s⁻¹. On the one hand, our calculations reveal the effect of the material defects in the necking process. On the other hand, our results show that the necking inception, instead of being a random type process, is the deterministic result of the interplay between the mechanical behaviour of the material and the boundary conditions. This conclusion agrees with the experimental evidence reported by Rittel et al. [1] and Rotbaum et al. [2].     

Some C0-continuous mixed formulations for general dipolar linear gradient elasticity boundary value problems and the associated energy theorems

The goal of this work is a systematic presentation of some classes of mixed weak formulations, for general multi-dimensional dipolar gradient elasticity (fourth order) boundary value problems. The displacement field main variable is accompanied by the double stress tensor and the Cauchy stress tensor (case 1 or μ ? τ ? u formulation), the double stress tensor alone (case 2 or μ ? u formulation), the double stress, the Cauchy stress, the displacement second gradient and the standard strain field (case 3 or μ ? τ ? κ ? ε ? u formulation) and the displacement first gradient, along with the equilibrium stress (case 4 or u ? θ ? γ formulation). In all formulations, the respective essential conditions are built in the structure of the solution spaces. For cases 1, 2 and 4, one-dimensional analogues are presented for the purpose of numerical comparison. Moreover, the standard Galerkin formulation is depicted. It is noted that the standard Galerkin weak form demands C¹-continuous conforming basis functions. On the other hand, up to first order derivatives appear in the bilinear forms of the current mixed formulations. Hence, standard C⁰-continuous conforming basis functions may be employed in the finite element approximations. The main purpose of this work is to provide a reference base for future numerical applications of this type of mixed methods. In all cases, the associated quadratic energy functionals are formed for the purpose of completeness.