Dynamic failure of metallic pyramidal truss core materials – Experiments and modeling

The quasi-static and dynamic compressive behavior of pyramidal truss cores made of 304 stainless steel were investigated using a combination of experimental techniques. Quasi-static tests were performed using a miniature loading stage while a Kolsky bar apparatus was used to investigate intermediate deformation rates. High deformation rates were examined using a light gas gun. Optical imaging of the sample deformation was performed in real time by means of high-speed photography. In this article, we provide a quantification of load-deformation response and associated failure modes across the sample as captured by high-speed photography. A finite element model is formulated and thorough simulations performed to understand the roles of material strain rate hardening and structural microinertia. Deformation modes were identified from acquired images, force-deformation histories and numerical modeling. Comparison between force-deformation histories under quasi-static and Kolsky bar loading reveals a moderate microinertia effect as manifested by a small increase in peak compressive stress. At high deformation rates, gas gun experiments, a totally different deformation mode is manifested with a major increase in peak compressive stress. In this case, the inertia associated to the bending and buckling of truss struts played a significant role. This effect appears to dominate the early truss core response because of two effects: (i) the propagation of a plastic wave along the truss members; (ii) buckling induced lateral motion. These findings are consistent with prior theoretical and computational work carried out by Vaughn et al. (2005) [Vaughn, D. Canning, M., Hutchinson, J.W., 2005. Coupled plastic wave propagation and column buckling. Journal of Applied Mechanics 72 (1), 1–8]. At larger deformations, the material strain rate hardening contribution to the total energy is as pronounced as the contribution arising from microinertia effect.     

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.     

The elastic–plastic behaviour of foam under shock loading

An experimental investigation of the elastic–plastic nature of shock wave propagation in foams was undertaken. The study involved experimental blast wave and shock tube loading of three foams, two polyurethane open-cell foams and a low-density polyethylene closed-cell foam. Evidence of precursor waves was observed in all three foam samples under various compressive wave loadings. Experiments with an impermeable membrane are used to determine if the precursor wave in an open-cell foam is a result of gas filtration or an elastic response of the foam. The differences between quasi-static and shock compression of foams is discussed in terms of their compressive strain histories and the implications for the energy absorption capacity of foam in both loading scenarios. Through a comparison of shock tube and blast wave loading techniques, suggestions are made concerning the accurate measurements of the principal shock Hugoniot in foams.     

Specimen Size and Strain Rate Effects on the Compressive Behavior of Concrete

Three high-performance concrete (HPC) materials with different specimen geometries were characterized using Kolsky compression bar techniques to study the strain rate and specimen size effects on their uniaxial compressive strength. A large-diameter Kolsky bar and recently established annular pulse shaping technique were used to achieve dynamic stress equilibrium and constant strain-rate deformation in the experiments. A complimentary effort was conducted using a 19-mm-diameter Kolsky compression bar to understand the strain rate and specimen size effects on failure strength and dynamic increase factor (DIF) for concrete. It was found that, for all three concrete materials investigated, the failure strength is highly dependent on the specimen geometry, however such a relationship is not apparent for the DIF. The DIF observed in this study shows significantly lower values compared to historical data, which may indicate the importance of well-controlled dynamic testing conditions on the accuracy and validity of experimental results for concrete materials.     

Radial Inertia Effects in Kolsky Bar Testing of Extra-soft Specimens

Impact responses of extra-soft materials, such as ballistic gelatins and biological tissues, are increasingly in demand. The Kolsky bar is a widely used device to characterize high-rate behavior of materials. When a Kolsky bar is used to determine the dynamic compressive response of an extra-soft specimen, a spike-like feature often appears in the initial portion of the measured stress history. It is important to distinguish whether this spike is an experimental artifact or an intrinsic material response. In this research, we examined this phenomenon using experimental, numerical and analytical methods. The results indicate that the spike is the extra stress from specimen radial inertia during the acceleration stage of the axial deformation. Based on this understanding, remedies in both specimen geometry and loading pulse to minimize the artifact are proposed and verified, and thus capture the intrinsic dynamic behavior of the specimen material.     

A rigorous assessment of the benefits of miniaturization in the Kolsky bar system

A rigorous experimental and numerical assessment is made of the benefits and limits of miniaturization in the Kolsky bar system. The primary issues that arise in very high strain rate testing (stress equilibration, inertial effects, wave dispersion, friction, and controllability of deformations) are addressed through experiments coupled with explicit finite element analyses. A miniaturized Kolsky bar system that includes the input bar is developed, together with the use of the laser occlusive radius detector to obtain local measurements of specimen strain during the very high rate deformations. It is demonstrated that this miniaturized Kolsky bar system can be used to provide fully validated results, including the explicit determination of equilibration, over a very wide range of strain rates (1×10³ to 5×10⁴ s⁻¹). The desired high strain rate can be achieved even at low accumulated strains, and the total strain developed can be controlled very effectively. Specific conditions are developed for determining the range of utility of the technique for a given material. The technique is applied to the characterization of 6061-T651 aluminum, and the results are compared with the results obtained using a conventional Kolsky bar.     

An analytical model for shock wave impact on compressible open-cell foam

A simple model is proposed to describe the main features of the complex phenomena that occur when a plane shock wave strikes a open-cell foam block. The complex two-phase interactions can be simplified using the experimental evidence that the foam face may be treated as a contact surface, that the length of the collapsed foam plug grows linearly in time, and that the gas pressure profile in the region between the head of the plug and the undisturbed material may be treated as being pseudo-stationary. These simplifications enable the prominent features of the compression process to be predicted for a variety of foam types. Analytical results for the reflected shock strength, foam front face velocity, transmitted wave velocity and strength, and maximum peak back wall pressures are presented.      

Modification of the kolsky method for studying properties of low‐density materials under high‐velocity cyclic strain

A modification of the Kolsky method with the use of the split Hopkinson bar is proposed, which allows testing lowdensity materials under cyclic loads of an identical sign. Cyclic dynamic testing of specimens is based on the essential difference of acoustic impedances of the material of the specimen tested from the material of pressure bars. The choice of the supportbar length several times greater than the loadingbar length allows registration of strain pulses in several cycles. Results are presented for the proposed modification of the Kolsky method used for tests in compression of foam plastic of two densities under three loading cycles.     

通孔泡沫铝的动态压缩行为

在SHPB装置上对渗流法制备的通孔泡沫铝进行了动态压缩实验,研究了相对密度为0.341~0.419的通孔泡沫铝在10-3~2000 s-1应变率范围内的压缩响应特征和应变率相关性,并用扫描电镜（scanning electron microscope,SEM)分析了泡沫铝的压缩变形特征。实验结果表明,通孔泡沫铝有明显应变率效应,随应变率上升,泡沫铝流动应力提高。SEM观察结果揭示,在动态压缩下,通孔泡沫铝宏观上均匀变形,微观变形机制以泡孔横向伸展坍塌为主。     

Dynamic Tensile Characterization of a 4330-V Steel with Kolsky Bar Techniques

Dynamic tensile experimental techniques of high-strength alloys using a Kolsky tension bar implemented with pulse shaping and advanced analytical and diagnostic techniques have been developed. The issues that include minimizing abnormal stress peak, determining strain in specimen gage section, evaluating uniform deformation, as well as developing pulse shaping for constant strain rate and stress equilibrium have been addressed in this study to ensure valid experimental conditions and obtainment of reliable high-rate tensile stress–strain response of alloys with a Kolsky tension bar. The techniques were applied to characterize the tensile stress–strain response of a 4330-V steel at two high strain rates. Comparing these high-rate results with quasi-static data, the strain rate effect on the tensile stress–strain response of the 4330-V steel was determined. The 4330-V steel exhibits slight work-hardening behavior in tension and the tensile flow stress is significantly sensitive to strain rate.     

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.     

One-dimensional shock wave interaction with rubber and low-porosity foam

One-dimensional interaction between a planar shock wave and a rubber or low-porosity foam is investigated experimentally and numerically. The considered polyurethane foam is of high density (ρ _c=290 kg/m³) and lowporosity (ϕ=0.76), and this corresponds to an intermediate condition between rubber and high-porosity foam. Stress-strain relations for the low-porosity foam are investigated by machine tests, which show larger deformation against compressive force and higher non-linearity in stress-strain curve as compared with rubber. Also the low-porosity foam shows a hysteresis cycle. Experiments on shock wave-foam interactions are conducted by using a shock tube. Experimental time history of the surface stress of the foam at the end of the shock tube does not show shock type stress increase, but continuous excessive stress rise can be seen, and then dumping vibration approaching to gas dynamic pressure of the reflected shock wave is followed, and the highest stress amounts about 3∼4 times of the pressure after the reflected gas dynamic shock wave. Interactive motions of gas and the low-porosity foam are analyzed using the Lagrangean coordinates system. An elastic model for a low-porosity foam is assumed to be a single elastic material with the measured stress-strain relation. Results of numerical simulations are compared with the shock tube experiments, which show essentially same stress variations with experimental results.     

Modeling the dynamic response of visco-elastic open-cell foams

This article introduces a mesoscopic formulation for modeling the dynamic response of visco-elastic, open-cell solid foams. The effective material response is obtained by enforcing on a representative 3D unit cell the principle of minimum action for dissipative systems. The resulting model accounts explicitly for the foam topology, the elastic and viscous properties of the cell wall, and the inertial effects arising from non-affine motion within the cells. The microinertial effects become significant in retarding the foam collapse during exceedingly high strain-rate loading. As an application example, a heterogenous case of compressive deformation at high strain rate is simulated utilizing the present model as a constitutive update in a non-linear finite element analysis code. This FEM simulation shows the ability of the model to capture the progressive foam collapse during the dynamic compression as observed in experimental studies. Using the microscopic model, the inertial and viscous strain-rate effects are investigated through the foam density, viscosity, and relative density. Based on the physics incorporated into the local cell model, we provide insight into the physical mechanisms responsible for the experimentally observed strain-rate effects on the behavior of dynamically loaded foam materials.     

On the energy conservation and critical velocities for the propagation of a ＂steady-shock＂ wave in a bar made of cellular material

The propagation of shock waves in a cellular bar is systematically studied in the framework of continuum solids by adopting two idealized material models, viz. the dynamic rigid, perfectly plastic, locking （D-R-PP-L） model and the dynamic rigid, linear hardening plastic, locking （D-R-LHP-L） model, both considering the effects of strain-rate on the material properties. The shock wave speed relevant to these two models is derived. Consider the case of a bar made of one of such material with initial length L 0 and initial velocity v i impinging onto a rigid target. The variations of the stress, strain, particle velocity, specific internal energy across the shock wave and the cease distance of shock wave are all determined analytically. In particular the ＂energy conservation condition＂ and the ＂kinematic existence condition＂ as proposed by Tan et al. （2005） is re-examined, showing that the ＂energy conservation condition＂ and the consequent ＂critical velocity＂, i.e. the shock can only be generated and sustained in R-PP-L bars when the impact velocity is above this critical velocity, is incorrect. Instead, with elastic deformation, strain-hardening and strain-rate sensitivity of the cellular materials being considered, it is appropriate to redefine a first and a second critical impact velocity for the existence and propagation of shock waves in cellular solids. Starting from the basic relations for shock wave propagating in D-R-LHP-L cellular materials, a new method for inversely determining the dynamic stress-strain curve for cellular materials is proposed. By using e.g. a combination of Taylor bar and Hopkinson pressure bar impact experimental technique, the dynamic stress-strain curve of aluminum foam could bedetermined. Finally, it is demonstrated that this new formulation of shock theory in this one-dimensional stress state can be generalized to shocks in a one-dimensional strain state, i.e. for the case of plate impact on cellular materials, by simply making proper replacements of the elastic and plastic constants.     

泡沫铝子弹高速撞击下铝基复合泡沫夹层板的动态响应

应用一级轻气炮驱动泡沫铝弹丸高速撞击加载技术,对实心钢板以及前/后面板为Q235钢板、芯层分别为铝基复合泡沫和普通泡沫铝的夹层板结构,在脉冲载荷作用下的动态力学响应进行实验研究。结果表明:泡沫铝子弹高速撞击靶板可近似模拟爆炸载荷效果;铝基复合泡沫夹层板的变形分为芯层压缩和整体变形两个阶段;与其他靶板相比,铝基复合泡沫夹层板的抗冲击性能最优。基于实验研究,应用LS-DYNA非线性动力有限元软件,对泡沫铝夹层板的动态响应进行数值模拟。结果表明:泡沫铝子弹的长度和初始速度对子弹与夹层板之间的接触作用力影响显著,并且呈线性关系。泡沫芯层强度对等质量及等厚度夹层板的抗冲击性能均有显著影响,夹层板中心挠度对前、后面板的厚度匹配较为敏感,在临界范围内,若背板厚度大于面板厚度,可减小夹层板的最终挠度。夹层板面板宜采用刚度较低、延性好、拉伸破坏应变较大的金属材料。     

泡沫铝合金动态力学性能实验研究

利用分离式霍布金森压杆(SHPB)实验技术和MTS材料实验机对两组不同孔径、不同密度的开孔泡沫铝合金进行了准静态和动态压缩实验研究。实验结果表明:泡沫铝合金的静态和动态变形过程均具有泡沫材料变形的三个阶段特征。开孔泡沫铝合金的变形是均匀变化过程，并不出现局部的变形带。与相对密度对力学性能的影响相比，孔径大小的影响可以忽略不计。在考察的应变率范围内，屈服应力对应变率并不很敏感。     

A simplified viscoelastic constitutive equation applied to a dynamic finite deformation problem

A single integral type of constitutive equation for finite viscoelastic deformation is proposed. A special case, which is a viscoelastic generalization of the constitutive equation for a neo-Hookean elastic solid, is used to consider the finite deformation problem of shock wave propagation resulting from the sudden application of compressive force at the end of a semi-infinite viscoelastic bar. An approximate method is used to determine the shock front path and shock strength when the viscoelastic dissipative effect is small.     

Dynamic Experiments using Simultaneous Compression and Shear Loading

Material characterization at high strain rates under simultaneous compression and shear loading has been a challenge due to the differing normal and shear wave speeds. An experimental technique utilizing the compression Kolsky bar apparatus was developed to apply dynamic compression and shear loading on a specimen nearly simultaneously. Synchronization between the compression and shear loading was realized by generating the torsion wave near the specimen which minimizes the time difference between the arrival of the compression and torsion waves. This modified Kolsky bar makes it possible to characterize the dynamic response of a material to combined compression and shear impact loading. This method can also be applied to study dynamic friction behavior across an interface under controlled loading conditions. The feasibility of this method is demonstrated in the dynamic characterization of a simulant polymer bonded explosive material.     

On the dynamic mechanical properties of open-cell metal foams – A re-assessment of the ‘simple-shock theory’

Metal foams are increasingly used for energy absorption especially in lightweight structures and to resist blast and impact loads. This requires an understanding of the dynamic response of these materials for modelling purposes. As a supplement to Tan et al., 2005a, Tan et al., 2005b, hereinafter referred to as T–L for brevity, this paper provides experimental data for the dynamic mechanical properties of open-cell Duocel® foams having a three-dimensional (3D) distribution of cells. These confirm significant enhancement of the foam’s compressive strength, accompanied by changes in their deformation pattern in certain loading régimes, particularly what has come to be described as the ‘shock’ régime by Zheng et al. (2012). This paper examines experimentally, in a similar fashion as T–L, how the structural response of the individual cell walls is affected by cell-shape anisotropy at the cell (meso)-scale and how this, in turn, alters the pattern of cell crushing and the dynamic, mechanical properties. The distinctive role of cell microinertia and ‘shock’ formation are discussed in relation to the mechanical properties measured for these 3D cylindrical specimens. For consistency the same procedures described in T–L are used. The features identified are shown to be consistent with those observed in finite-element simulations of two-dimensional (2D) honeycombs as estimated by the one-dimensional (1D) steady-shock theory summarised in T–L. The different deformation patterns that develop in the various loading régimes are categorised according to the compression rate/impact speed. Critical values of impact velocity, corresponding to the transition from one pattern to the other, are quantified and predictive formulae for the compressive uniaxial strengths in the directions of two of the principal axes of the material in each loading régime are derived and discussed. The accuracy of the predictive formula in T–L is shown to critically depend on the ‘densification strain’ of the foam specimens. This parameter and the discussion that follows could assist the formulation and validation of alternative theoretical/computational models on the dynamic deformation of such materials.     

A Kolsky Torsion Bar Technique for Characterization of Dynamic Shear Response of Soft Materials

A novel Kolsky torsion bar technique is developed and successfully utilized to characterize the high strain rate shear response of a rate-independent end-linked polydimethylsiloxane (PDMS) gel rubber with a shear modulus of about10 KPa. The results show that the specimen deforms uniformly under constant strain rate and the measured dynamic shear modulus follows reasonably well the trend determined by dynamic mechanical analysis (DMA) at lower strain rates. For comparison, Kolsky compression bar experiments are also performed on the same gel material with annular disk specimens. The dynamic moduli obtained from compression experiments, however, are an order of magnitude higher than those obtained by the torsional technique, due to the pressure caused by the radial inertia and end constraints.