Dear iMechanica community,
I came across a problem in plasticity for which I am unable to find a reference in the literature and kindly ask your advice.
My research topic is the dynamic compression of specimens in a split Hopkinson pressure bar (SHPB). In addition to the common signal recording by strain gages on the bar surfaces, we observe the deformation with high-speed cameras and analyze the pictures with digital image correlation (DIC). For that purpose a paint pattern is sprayed on the specimens prior to testing. Currently I am trying to extract the plastic wave speed from image data.
What is the plastic wave speed in the short cylinder in the axial deformation while being compressed, according to theory ?
It is commonly stated that the speed of waves in bodies is given by sqrt(slope/density), where sqrt is the square root and slope is the slope of the uniaxial stress vs. strain curve.
My questions, for which I'd need a competent advice, are:
1) In the elastic range: slope = Young's modulus E, and this is identical to the speed of so-called elastic "extensional waves" in thin bars. It is also known as the "thin (or: slender) bar velocity" and equal to the speed of the elastic waves in the bars in SHPB testing. Thus, there is a geometrical restriction on the applicability of this equation. The given equation thus seems to be of limited validity, even in the elastic domain.
2) In various textbooks I found another explanation. According to that, three distinct plastic wave speeds are possible in bulk media. One of them is identical to the speed of the elastic shear wave, one is named "fast wave", the other one "slow wave".
I also consulted the reference mentioned:
Plastic Wave Speeds in Isotropically Work-Hardening Materials
T. C. T. Ting
J. Appl. Mech. 44(1), 68-72 (Mar 01, 1977)
abstract:
http://appliedmechanics.asmedigitalcollection.asme.org/Mobile/article.a…
a) Are all 3 waves generated simultaneously or may only one or two of them emerge ? Thus, do I observe a single-mode plastic wave or a superposition of plastic waves ? Not to be forgotten, the elastic wave is still present. Both elastic and plastic waves reverberate within the specimen as they are continuously being reflected and transmitted at the specimen-bar interfaces. Is a single wave speed discernible from image data ? And what dictates which wave(s) is/are generated ?
b) That theory is a 3-dimensional theory. Do the 3 equations from this theory merge into a single one if 2 dimensions are eliminated ?
3) Another concern is prompted by a simplification of the theory: in elasticity the fast and slow wave speeds reduce to sqrt(M/density) and sqrt(G/density); i.e. the wave speeds in infinite media (G = shear modulus; M = P-wave modulus, axial modulus, longitudinal modulus, constrained modulus, defined as the ratio of axial stress to axial strain in a uniaxial strain state). Uniaxial strain can be assumed when lateral dimensions are infinite and no perpendicular strains arise. Put otherwise: M = E only if Poisson's ratio = zero.
But that means: picking the Young's modulus E from a compression test and computing the elastic wave speed when combined with the density is erroneous; i.e. the simple relation sqrt(E/density) as the solution in the elastic domain of the 1-dimensional case is not applicable. One has to compute sqrt(M/density) respectively sqrt(G/density). It is astonishing that the relation sqrt(E/density), omnipresent throughout the literature, is not strict. Here we are at point 1 above again.
4) Hence, what is the appropriate theory for SHPB testing, applicable to the propagation of plastic waves when impacting a short cylinder with an aspect ratio of approximately one (in our case: height = 0.7 cm, diameter = 1.0 cm) ? This body does not qualify as a "continuous medium" (the heading in a textbook where I found this theory), and the one-dimensional theory seems simplistic. The cylinder expands radially and circumferentially when being compressed, thus the strain state is three-dimensional.
I'd be content with a theoretical solution for an isotropic elastic-plastic material with hardening (constant slope in the plastic range of the stress-strain curve). Does that theory preserve a means to relate the wave speed to the slope of the stress-strain curve ? If so, I could construct the stress-strain curve from the wave speed and compare it to the measured dynamic stress-strain curve.
This topic seems to be so elusive. I searched literature databases (Web of Science) and the internet, but didn't come up with clear statement. Both the Web of Science and the WWW list only very few hits when searching for publications citing that publication by Professor Ting as a reference. How come if the results are so fundamental ? Or are the results so unpleasant because wave superposition negates every hope to identify a particular value for the speed of an observed plastic wave ?
I am highly indebted to you for any clarification and highly value your assistance.
Sincerely
Jean Legrand