Testing With Bars From Dynamic to Quasi-static

The numerical calculation of the dynamic loading of a structure includes a great number of steps in which various fundamental or engineering problems are involved. Most of them are addressed in the present course at CISM. In this paper, we discuss the testing of materials in order to model their behaviour.

Because of waves induced in the testing device by impulse loadings and short time measurements, the data analysis has to deal with transient effects. Using bars makes easier such an analysis. For this reason, Hopkinson bars are a very commonly used dynamic testing device.

Using the word “dynamic” means that “time” is considered as an active parameter in the evolution process. When dynamically testing a structure (a cylindrical specimen is a common example of such a structure) the effects of time appears in different ways.

There is not static equilibrium in the machine so that measurements at specimen ends cannot be simply deduced from measurements with sensors incorporated in the machine, as it is the case with quasi-static testing. Furthermore, most sensors (like force cells) have a limited high passing band.

Transient effects in the specimen induce waves and the non-homogeneity of mechanical parameters. Consequently, average or global measurements cannot be right away related to local ones.

Stresses cannot be simply related to forces measurements as inertia effects are also involved – the most known effect is the confinement induced by lateral inertia, especially important when testing a big specimen of brittle material.

Short testing times do no allow for isothermal testing – a metallic specimen can have a temperature increase up to 100°C during a SHBP test.

The behaviour of an elementary volume of the material can depend on the rate of change of basic mechanical parameters strain and/or stress. This last effect (strain rate sensitivity) is the (only) one that is expected to be measured, in most cases.

In (dynamic) mechanical testing it is then suitable to consider separately the global measurements made on the specimen (forces applied at a part of the specimen border and displacement measured at another - or the same - part) and the analysis of its mechanical evolution.

This is commonly done in the quasi-static side but is not always, for historical reasons, done in dynamic testing.

The above discussion does not answer the basic question of the boarder between quasi-static and dynamic testing. Theoretically, indeed, waves in solids are still present in quasi-static testing. The common criterion to evaluate this limit is to compare the time

τ

e

needed to reach equilibrium (say < 5% of non homogeneity of stresses and strains) to the measurement duration. Note that

τ

e

mostly depends on the specimen size and on the elastic speed of waves in the material and not on the measurement duration. In the classical SHPB literature, this problem is related to the “impedance matching problem”, misunderstood in many publications, perfectly addressed in 1963 by Davies & Hunter. Based on this criterion, (too) many SHPB tests are considered as quasi-static ones.