Strain-rate Dependent Theory Research Articles

A sensing-plate method for measuring force and duration of impact in elastic-plastic impact of bodies

A new method for measuring an impulsive force generated at the impact end in collision and the duration of impact is developed. The measurement method is based on a simple structure, here called the special sensing plate, which produces the same effect as strain gages embedded in a solid. The sensing-plate method is confirmed by calibration tests to be effective for measuring not only the impulsive force generated at the impact end of a body but also the duration of the force. The measurement is almost entirely free from disturbance caused by interference from reflected waves. Plastic impact experiments are also performed for pure aluminum bars with several different lengths by impacting them on a sensing circular plate. Impact specimens are shot out from a high-pressure air gun. The time variation of stress at the impact end of the specimen bars colliding with the sensing plate and the duration of impact are measured at various impact velocities. The strain-rate dependent theory of plastic wave propagation is applied to obtain a theoretical prediction of those results. The theoretical predictions have shown to agree well with the observed results.

Propagation of plastic waves in a long rod with consideration given to temperature rise

The strain and temperature distributions along a long rod subjected to a longitudinal impact at a high strain level are investigated using Malvern's linear strain-rate dependent theory with consideration being given to the temperature rise caused by the plastic work. The results show that a steeply decreasing part of strain appears near the impacted end and the region is followed by an uniform-strain (strain-plateau) one along the rod, and that the formation and growth of the uniform-strain region depends on the strain-hardening sensitivity of the rod. The computed profile of the temperature distribution is similar to that of the strain distribution. However, the growth of a uniform-temperature region (temperature plateau) is much slower that that of the strain plateau.

弾塑性衝突する物体の衝突端応力変動

In the present study, a new method for measuring the time-variation of stress at the impact end in the elastic-plastic collision of a body against another body was developed and the duration of impact was discussed.Firstly, the simple method for measuring the duration of impact, called the sensing cylinder method, was proposed. The sensing cylinder was confirmed by calibration tests to be effective for measuring not only the duration of impact but also the impulsive force generated at the impact end of a body.Secondly, as an application of the measuring method, plastic impact tests were made on pure copper and aluminum specimens with several different lengths. The time-variation of stress at the impact end of the bars colliding with the sensing cylinder and the duration of impact were measured under various impact velocities. The experimental results were compared with the theoretical ones based on the strain-rate dependent theory of plastic wave propagation. Close agreement between the observed and the calculated values was obtained.

Duration of contact at plastic impact of a bar. (2nd report Discussions on a constitutive equation and a method for measuring the duration of impact)

The present paper is concerned with the duration of contact at plastic impact of a finite-length bar on a rigid body. The duration of impact is predicted on the basis of the strain-rate dependent theory of plastic wave propagation. Thus the variations of elastic-plastic stress at the impact end of lead bars are examined by using the constitutive equation obtained from the split Hopkinson bar. Next, a new method for measuring the duration of impact is proposed which is termed the sensing cylinder. This method is effective for measuring not only the duration of impact but also the impulsive force at the impact end. The experimental verifications of the duration of impact are made for lead bars of four different lengths. The time variations of stress at the impact end of the bars colliding with the sensing cylinder under various impact velocities are measured. The theoretical predictions are in good agreement with experimental observations.

Duration of contact at plastic impact of two bars.

The present paper deals with the problem of the duration of impact between two bars. The two bars collide with each other longitudinally, and one is plastically deformed, while the other remains elastic during the collision. In order to estimate the duration of contact between two bars impacting each other, the variations of stress at the impact ends are examined theoretically and experimentally. The theoretical predictions based on the strain-rate dependent theory for plastic wave propagation showed that as the length of a specimen, the impact velocity and the magnitude of the dependence of strain-rate increase, the duration of contact increases regardless of the rising mode of an impact velocity and its rise time. The experimental verification was made by measuring the time variation of elastic-plastic stress at the impact end of a lead specimen in collision with an elastic stress bar made of carbon steel. The theoretical predictions almost agreed with the experimental observations.

Duration of Contact at Plastic Impact of Two Bars

The present paper deals with a problem on the duration of impact between two bars. The two bars collide with each other longitudinally, one being plastically deformed and the other remaining elastic during the collision. In order to estimate the duration of contact between the two bars impacting each other, the variations of stress at the impact end are examined theoretically and experimentally. The theoretical predictions based on the strain-rate dependent theory for plastic wave propagation show that as the length of a specimen, the impact velocity and the magnitude of the dependence of strain-rate increase, the duration of contact increases regardless of the rising mode of the impact velocity and its rise time. The experimental verification was made by measuring the time variation of elastic-plastic stress at the impact end of a lead specimen in collision with an elastic stress bar made of carbon steel. The theoretical predictions almost agreed with the experimental observations.

ひずみ速度依存性を示す材料における塑性波伝ぱの解析 ひずみプラトーの実験的検証

The present paper deals with plastic wave propagation in a bar subjected to longitudinal impact. It has been shown in authors' previous papers that the existence of a strain plateau adjacent to the impact end of a bar can be predicted by Malvern's strain-rate dependent theory and its appearance requires a certain time which is governed by the strainrate dependence of material and the impact velocity at the impact end. In this paper, the theoretical prediction of the previous papers was verified experimentally. First, the strain distributions in a finite length bar and a semi-infinite bar were predicted and the difference between them was examined. Next, the plastic strain distribution was measured in a cylindrical specimen by subjecting it to a longitudinal compression impact with a stress bar which remains elastic during the test. The duration time of impact was also measured on the stress bar.The results obtained are summarized as follows:(1) The strain plateau in a finite length bar was less uniform and less broad than that in a semi-infinite bar.(2) The experimental results for the lead and copper specimens almost agreed with the theoretical predictions. Consequently, it may be verified that the formation of strain plateau is governed by the factors mentioned above.

The Impact End Stress of a Bar Subjected to Longitudinal Compression Impact

In the present paper the elastic-plastic stress at the impact end of a cylindrical specimen subjected to longitudinal compression impact with a stress bar which remains elastic during the test is investigated experimentally and theoretically. The measured elastic response of the stress bar showed explicitly an elevation of dynamic stress and its relaxation at the impact end of the specimen. The impact end stress of the specimen was analyzed by using the strain-rate dependent theory for plastic wave propagation, taking into account a rise time of impact and impact conditions. An extreme elevation of the dynamic stress was predicted in the case of a step impact, but it went down rapidly with an increase in the rise time of an impact velocity. Taking the stress as an incident stress to the stress bar, the elastic response of the stress bar based on the Love theory for elastic waves almost agreed quantitatively with the experimental results as well as qualitatively.

The impact end stress of a bar subjected to longitudinal compression impact.

In the present paper the elastic-plastic stress of the impact end of a cylindrical specimen subjected to longitudinal compression impact with a stress bar which remains elastic during the test is investigated experimentally and theoretically. The measured elastic response of the stress bar showed explicitly the elevation of dynamic stress and its relaxation at the impact end of the specimen. The impact end stress of specimen was analyzed by using of the strain-rate dependent theory for plastic wave propagation, taking into account a rise time of impact and impact conditions. The extreme elevation of the dynamic stress was predicted in the case of a step impact, but it went down rapidly with the increasing of the rise time of an impact velocity. Taking these stresses as an incident pulse to the stress bar, the elastic response of the stress bar based on the Love theory for elastic waves almost agreed quantitatively with the experimental results as well as qualitatively.

ひずみ速度依存性を示す材料における塑性波伝ぱの解析 衝撃速度変化の影響

The present paper is concerned with a problem of longitudinal plastic waves in a bar subjected to tensile impact. The effect of the variation of impact velocity on the propagation plastic waves and the formation of strain plateau in the strain distribution is discussed experimentally and theoretically. Impact tensile tests of copper specimens were carried out by an impact testing apparatus using a falling weight. The plastic strain distribution of the specimen was measured and the time history of plastic waves at various stations along the specimen was observed, together with the variation of impact velocity during impact. Theoretical calculation was performed by using Malvern's strain-rate dependent theory, taking account of the attenuation of impact velocity.The results obtained are summarized as follows:(1) The experimental results showed that the strain distribution along the specimen decreased with the increase of the distance from the impact end and no strain plateau at neighborhood of the impact end appeared.(2) The theoretical results under the condition of decreasing impact velocity predicted also no strain plateau and almost agreed with the experimental results.(3) The formation of strain plateau may be limited to the case of a constant impact velocity, though it is related to the strain-rate dependent nature of material, the impact velocity and the duration of impact time.

On the Dynamic Yield of Metallic Materials under Impact Loading

In the present paper the dynamic yield of metallic materials under impact loading was analyzed by using the strain-rate dependent theory of Malvern for plastic wave propagation along a bar. Impact tests were also made on lead and copper specimens.The results obtained are summarized as follows:(1) In the case of step loading, a large increase of dynamic yield stress was predicted at the section near to the impact end of the bar by the rate dependent theory, but the increase was considerably small when the rise time of impact velocity was taken into consideration.(2) The dynamic yield stress near the impact end increased with increasing impact velocity and rising time but with decreasing the constant K in the constitutive equation. The yield delay time depended mainly on the magnitude of K.(3) The phenomena of the increase in dynamic yield stress and the yield delay at the impact end were clearly observed in the impact experiments on lead and copper specimens. If the rising time of impact velocity is taken into consideration the rate dependent theory can account for those phenomena quantitatively.

On the Existence of a Strain Plateau in the Strain-Rate Dependent Theory of Malvern for Plastic Wave Propagation

On the Existence of a Strain Plateau in the Strain-Rate Dependent Theory of Malvern for Plastic Wave Propagation

ひずみ速度依存性を示す材料における塑性波伝ぱの解析 ひずみプラトーの存在について

The present paper deals with a problem of propagation of longitudinal plastic waves in a semi-infinite bar subjected to a constant impact velocity, according to Malvern's strain-rate dependent theory. Especially, we shall discuss whether or not Malvern's theory can account for the plateau of uniform plastic strain adjacent to the impact end of the bar, which is predicted by Karman's theory neglecting a strain-rate effect and is observed experimentally. Malvern could not explain the strain plateau in his own calculations. This is considered so far to be an only weak point in the strain-rate dependent theory, though it can account for the extensive phenomena of high-rate deformation and plastic wave propagation in a bar.In this study, the following conclusions were obtained.(1) The existence of the strain plateau can be also predicted by Malvern's theory.(2) Its appearance requires a relaxation time which is governed by the strain-rate dependence of material and the impact velocity at the end of a bar.(3) As the strain-rate dependence of material and the impact velocity increase, the relaxation time needed for the appearance of the strain plateau increases.

Plastic waves in a rate sensitive material—I. Waves of uniaxial stress

Summary Experiments have been carried out in which longitudinal plastic waves have been propagated in thin-walled tubes of alpha-titanium. Strain-time profiles recorded in these experiments show evidence of (i) stress levels considerably above quasi-static values at the same strain, (ii) decay of the amplitude of the elastic precursor, and (iii) variation with distance of propagation of the speed at which a given level of strain propagates. These features of the strain-time profiles are interpreted as indicating that a strain-rate dependent theory is necessary to describe the observed wave phenomena. Numerical solutions based on such a theory agree reasonably well with experimental results. For these solutions a plastic strain-rate function is used which agrees with data from quasi-static and split Hopkinson bar tests on the same tubes.

A theoretical and experimental analysis of high-strain-rate tensile tests using Malvern's theory

The theoretical behaviour of a cylindrical specimen of finite length during a high-strain-rate tensile test has been studied with the help of a strain-rate dependent theory. The behaviour is compared with experimental observations of tests on a specimen partially fixed at one extremity and submitted to a constant velocity impact at the other end. The chosen plasticity function is that introduced by Malvern which predicts a linear relationship between the stress and plastic strain rate. The calculations and experiments have been done in two cases: (i) ideal elastic-plastic material and (ii) highly work-hardened material.The main results are as follows: two stages can be distinguished, a `transient' state at the beginning of the test, where the values of state variables are heterogeneous, and a `steady' state where stress and strain rate are uniform in the whole specimen. The strain distribution in the steady state depends markedly upon the `quasi-static' stress-strain relation σ=h(ϵ) and the parameter of the plasticity function. Malvern's theory is a good approximation of the experimental results in the `steady' state, but it cannot predict those of the `transient' state; in particular the theory gives too low values of the peak stress at the extremity of the specimen opposite to the impact section. The use of a more complex plasticity function is briefly investigated.

Velocity of Torsional Waves in Metals Stressed Statically into the Plastic Range

This paper is concerned with the determination of the velocity of propagation of torsional plastic waves in metals stressed statically into the plastic range. A new method was developed in which a tubular test specimen together with a concentric bar of a brittle material was twisted slowly such that when the specimen was stressed beyond its yield the brittle bar broke suddenly and transmitted a plastic torsional stress increment along the specimen. It was found that the velocity of propagation both in copper and mild steel was the same as the elastic shear wave velocity. Although consistent with the strain-rate dependent theory, the result can be explained in terms of the strain-rate independent theory provided the stress-strain curve for the appropiate strain rate is used.