Radial Inertia Effects Research Articles

Correction method and verification of radial inertia and friction effects under a unified deformation framework in SHPB experiments on soft materials

During the split Hopkinson pressure bar (SHPB) experiments, significant measurement errors can arise due to severe radial inertia and friction effects. Previous studies have developed various correction methods for these two effects. However, these methods have problems such as over-reliance on the volume invariance assumption of the specimen and inconsistent assumptions on the deformation patterns of the two effects, which limit their universality and effectiveness. Therefore, this paper integrates the radial inertia effect and friction effect in a unified deformation framework through reasonable assumptions, and proposes a method to correct the specimen from a complex stress state to a uniaxial stress state. SHPB numerical simulation experiments demonstrate that this method effectively eliminates the combined effects of radial inertia and friction on measurement results for both elastic and viscoelastic materials, including the size effect associated with these two factors. Additionally, the paper presents a scheme to determine the friction coefficient using the size effect of the specimens when the friction coefficient between the specimen and the bar is unknown. Finally, the method was applied to correct the stresses measured in SHPB experiments on silicone rubber of different diameters. It successfully eliminated discrepancies in the stress-strain relationships between specimens of various sizes and determined a friction coefficient that fell within a reasonable range.

Poisson‘s Ratio Induced Radial Inertia Confinement During Dynamic Compression of Hyperelastic Foams

Hyperelastic foams have excellent impact energy absorption capability and can experience full recovery following impact loading. Consequently, hyperelastic foams are selected for different applications as shock isolators. Obtaining accurate intrinsic dynamic compressive properties of the hyperelastic foams has become a crucial step in shock isolation design and evaluation. Radial inertia is a key issue in dynamic characterization of soft materials. Radial inertia induced stress in the sample is generally caused by axial acceleration and large deformation applied to a soft specimen. In this study, Poisson’s ratio of a typical hyperelastic foam – silicone foam – was experimentally characterized under high strain rate loading and was observed to drastically change across the densification process. A transition in the Poisson’s ratio of the silicone foam specimen during dynamic compression generated radial inertia which consequently resulted in additional axial stress in the silicone foam sample. A new analytical method was developed to address the Poisson’s ratio-induced radial inertia effects for hyperelastic foams during high rate compression.

Pure rate effect on compressive strength of concrete

Dynamic Increase Factor (DIF) has been used to consider the compressive strength enhancement of concrete at the high and intermediate strain rates. However, DIF formulae suggested until now include the inertia effects as well as the rate effect because the DIF formulae has been assumed as a function of only the strain rate and the inertia effects cannot be avoided in tests at the high and intermediate strain rate region. Therefore, applying the DIF to design or analysis of social infrastructures may be dangerous because the resistance by the inertia effects are considered repetitively. In this study, an apparent DIF formula, which includes the inertia effects, was proposed by introducing terms related with the strain acceleration, which represent the axial and radial inertia effects. Then, a nonlinear regression analysis was conducted to determine the coefficients in the apparent DIF formula with results of Split Hopkinson Pressure Bar (SHPB) tests for concrete. Finally, the DIF formula excluding the axial and radial inertia effects was proposed for compressive strength of concrete at the high and intermediate strain rates.

Investigation into Pure Rate Effect on Dynamic Increase Factor for Concrete Compressive Strength

Abstract Dynamic increase factor (DIF) has been used to consider rate effect on compressive strength of concrete in both design and analysis of concrete structures loaded with high rate. Until now, a variety of DIFs have been suggested by various researchers, and these DIFs are adopted in design guidelines and model codes, e.g., ACI 349-13, ACI 370R-14, fib MC2010, and UFC 3-340-02. However, the DIFs includes the axial and radial inertia effects, which cause confining effect and resistance to deformation. Therefore, additional strength enhancement due to the inertia effects is included in the DIFs, so applying the DIFs to design and analysis can lead to nonconservative results. In this study, in consideration of the strain acceleration and geometry of a specimen, a new DIF, which is exclusive of the inertia effects, was suggested based on concrete split Hopkinson pressure bar (SHPB) test results. Furthermore, to validate the new DIF, finite element analyses to which the existing and new DIFs were applied were performed for concrete SHPB tests. The verification results indicated that the DIF proposed in this study improve predictions of the dynamic strength enhancement of concrete compared with the existing DIFs.

The effect of radial inertia on flow localization in ductile rods subjected to dynamic extension

The objective of this work is to investigate the influence of radial inertia on the flow localization in ductile rods subjected to dynamic extension. Using the theory of a straight Cosserat rod which includes normal cross-sectional extension it is possible to obtain an exact solution for nonlinear uniform extension of a rigid-plastic material using a functional form of the yield stress that models the effect of the more general stress field in the necking region of the rod. Linear stability analysis of this exact nonlinear solution yields equations that generalize the formulation reported in Ref. [1] to include radial stretching and inertia. Examples show the quantitative effect of radial inertia on the stabilization of the localization process and on the determination of the expected length of fragments.

Effects of radial inertia and end friction in specimen geometry in split Hopkinson pressure bar tests: A computational study

The split Hopkinson pressure bar (SHPB) technique has been used widely for the impact testing of materials in the strain-rate range from 102 to 104s−1. However, some specific problems still remain mainly concerning the effects of radial inertia and end friction in a cylindrical specimen on the accurate determination of dynamic stress–strain curves of materials. In this study, the basic principle of the SHPB technique is revisited based on energy conservation and some modifications are made considering radial momentum conservation. It is pointed out that the radial inertia and end friction effects are coupled to each other in the SHPB specimen. Computational simulations using the commercial finite element (FE) code ABAQUS/Explicit ver. 6.8 are conducted to check the validity of the modifications for ductile pure aluminum specimens. Both rate-independent and rate-dependent models are adopted for the test material. Simulations are performed by varying two different control parameters: a friction coefficient between the specimen and the pressure bars and a slenderness ratio of the specimen (or thickness to diameter ratio).

Comments on the Effect of Radial Inertia in the Kolsky Bar Test for an Incompressible Material

We present equations that show the effect of radial inertia for incompressible samples that are in dynamic force equilibrium during the split Hopkinson pressure bar test or Kolsky bar test. For steel samples the radial inertia effect can be neglected; however, radial inertia can be important for very soft materials.

Dynamic Tensile Testing of Soft Materials

Determination of dynamic tensile response of soft materials has been a challenge because of experimental difficulties. Split Hopkinson tension bar (SHTB) is a commonly used device for the characterization of high-rate tensile behavior of engineering materials. However, when the specimen is soft, it is challenging to design the necessary grips, to measure the weak transmitted signals, and for the specimen to achieve dynamic stress equilibrium. In this work, we modified the SHTB on the loading pulse, the equilibrium-monitoring system, and the specimen geometry. The results obtained using this modified device to characterize a soft rubber indicate that the specimen deforms under dynamic stress equilibrium at a nearly constant strain rate. Axial and radial inertia effects commonly encountered in dynamic characterization of soft materials are also minimized.

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.

Dynamic and quasi-static compressive response of porcine muscle

Research and application activities in impact biomechanics require dynamic response of biological tissues under high-rate loading. However, experimental difficulties have limited the characterization of soft tissues under such loading conditions. In this paper, we identify these technical challenges in dynamic compression experiments using a split Hopkinson pressure bar (SHPB) and present the remedies to overcome them. In order to subject the specimens to valid dynamic testing conditions, in addition to developing new pulse-shaping techniques and incorporating highly sensitive load-measuring transducers, annular thin-disc specimens radically different from regular solid specimens were used to minimize radial inertia effects that may overshadow the intrinsic material properties. By using this modified SHPB, the compressive stress–strain behavior of soft porcine muscle tissue was obtained along and perpendicular to the muscle fiber direction from quasi-static to dynamic strain rates. The results show that the non-linear compressive stress–strain responses in both directions are strongly strain-rate sensitive.

The effect of radial inertia on brittle samples during the split Hopkinson pressure bar test

For a valid split Hopkinson pressure bar (SHPB) or Kolsky compression bar experiment, the sample should be in dynamic stress equilibrium over most of the test duration. In this study, we investigate the effect of radial inertia on elastic samples during a valid SHPB test. We present closed-form equations for the three additional stress components induced by radial inertia for incompressible and compressible, linear elastic samples. These equations should assist in the early experimental designs. As the experiments proceed and more is learned about the sample response, numerical analysis can be used to obtain a more refined account of the sample response and dynamic material strength.

Axial plastic waves in a rod

Abstract Equations are presented to describe nonlinear plastic wave propagation in a rod with the effects of radial inertia and radial shear included. Intrinsic constitutive equations are postulated for plastic deformation, and some of the principal aspects of wave propagation are described. In particular, a scaling argument is used to extract the nonlinear bar speed, which carries the principal pulse of energy, rather than the two characteristic speeds.

Numerical analysis of longitudinal elastic-plastic waves with radial effects

The two-dimensional problem of longitudinal elastic-plastic waves in circular rods taking radial-inertia effects into account is solved based on the finite-element method and an explicit integration algorithm. The elastic-plastic constitutive equations are the yield criterion of von Mises with isotropic work hardening and the Prandtl-Reuss flow rule. Numerical results are shown for the region near the impact end of a semi-infinite rod for two sets of boundary conditions, namely prescribed longitudinal velocity and prescribed longitudinal stress at the bar end. The lateral motion of the struck end is assumed to be unrestrained (zero shear stress). The numerical results show response characteristics which deviate from the one-dimensional solution and which are in a good qualitative agreement with a number of experimental observations reported in the literature. These impact test results have been examined with respect to the predictions in order to separate radial- and strain-rate effects. Several specific calculations for the various test conditions have been performed to obtain quantitative agreement with experimental observations.

The effect of radial inertia on the longitudinal waves propagating in viscoelastic rods.

The longitudinal waves in a viscoelastic rod of finite thickness are studied, taking into account the effect of radial inertia. The equation of motion used in this work is an application of the Love-Rayleigh theory for an elastic rod. An analytical solution of the transient response is obtained to examine the influence of radial motion on axial strain-time history. In addition, the attenuation and dispersion characteristics in the frequency region are discussed. Some experiments are performed on polymethyl methacrylate (PMMA) specimens with the view to evaluating the adequacy of the equation of motion. It is found that the theoretical predictions provide reasonable estimations of the strain-time history as well as of the attenuation and dispersion properties.

Rain-induced vibration

The purely longitudinal response of an elastic rod structural model of a space vehicle moving through rain at a constant velocity has been calculated. A statistical model and a related deterministic model which describe the rain-induced force on the nose of the space vehicle are formulated. These models are used to predict first- and second-order statistical properties of the excitation and response. Results show that the average force due to the raindrops in a typical weather encounter is small compared to the atmospheric drag. It is further found that the structural response is dominated by frequencies in the 1 MHz regime. A comparison between the responses of the rod theory which allows only longitudinal motion and that which includes the effects of radial inertia and radial shear suggests that the simpler theory provides an upper bound on the axial response of a rod cross section.

Elastic wave propagation in an exponential rod

A theoretical and experimental investigation was undertaken to study wave propagation in an exponential rod composed of aluminum when subjected to impact by steel spheres with diameters from 3·175 to 50·8 mm at velocities up to about 60 m/s. Strain measurements were effected at three gage stations and force inputs were obtained in some cases by sandwiched quartz crystals located at the impact point. Corresponding data were obtained for a 17·8° cone of the same length and material. One-dimensional predictions concerning wave shapes in the rod were obtained employing an analytical representation that neglected the effects of radial inertia and numerical finite difference methods that either neglected or incorporated the effects of this parameter. Good correspondence was found between the predicted and measured strain histories during first pulse passage; subsequent correlation showed increasing divergence both with shorter pulse durations and increasing propagation distance.

The effect of radial inertia on the plastic stress wave speeds in thin-walled tubes under combined stress

The plastic wave speeds in a thin-walled tube of rate insensitive material subject to combined axial and torsional loading may be calculated in closed form. For a given stress state the wave speeds are determined from theories which differ in their formulation only by the inclusion of radial inertia. For axial stress states it is shown that for specific values of the tangent modulus both the fast and slow wave speeds coincide with the elastic shear wave speed and that the inclusion of radial inertia markedly affects these critical values of tangent modulus.

The Effect of Radial Inertia on Combined Plastic Stress Wave Propagation in Thin-Walled Tubes

This investigation is concerned with the analysis of combined longitudinal and torsional inelastic stress wave propagation in thin-walled tubes. The effects of radial inertia are included in the development of the governing hyperbolic differential equations which are solved by the method of characteristics. Numerical results are presented for combined stress wave propagation in statically preloaded tubes of material assumed to be described by rate-independent plasticity. The effects of finite rise time of the imposed dynamic load and of radial inertia are analyzed in detail.

Radial inertia effects on an ideally plastic circular plate under impulsive axial compression

An analysis is presented here of the dynamic problem of a circular rigid-perfectly-plastic plate forged between two rigid plates under a number of input conditions. It is shown that the radial inertia of the deforming plate has a significant effect, and hence the results of this dynamic solution differ for instance, for high axial strain rates from those which would have been obtained under static conditions, for which an exact mathematical solution already exists [1].The approach used is that of finite element analysis. The analysis is applied to some cases of practical interest. Input conditions treated are (a) prescribed axial strain rate, (b) axial strain rate dependent on plate thickness and time, (c) constant forging power and (d) constant compression total force.A previously known static solution [1] is obtained with excellent agreement, as a special case of this analysis, by taking very small axial strain rates.

The strain-rate effect and the incremental plastic wave in copper

An experimental investigation was conducted to determine the degree of sensitivity of commerically pure copper to strain rate and to note the effect of this sensitivity on the velocity of propagation of shearing strain in copper. Thin-walled cylindrical specimens of copper were loaded in torsion to eliminate the effects of radial inertia. All specimens were annealed and then cold worked in torsion to obtain necessary specimen uniformity.