Dear Community,
I'd like to ask for advice on modelling reflection and transmission coefficients when two bars with different cross-sectional areas contact each other and are exposed to pulse loading.
I have modelled this process in Abaqus/Explicit as a two-dimensional simulation. The investigation was conveniently done as a parametric study, defining the lateral dimension as parameter. Both bars are slim (length 2 m, lateral dimensions of the bars are some centimeters) and elastic.
Three material combinations are defined:
a) identical materials: 2 steel bars
b) different materials: first bar is steel, second bar is aluminum (acoustically more dense first)
c) different materials: first bar is aluminum, second bar is steel (acoustically more dense second)
The reflection coefficient R may change sign depending on the material combination and the area ratio whereas the transmission coefficient T is always positive.
Elements:
CAX4R is the only axisymmetric solid element available in Abaqus/Explicit with four nodes
CPE4R is the only plane strain solid element available in Abaqus/Explicit with four nodes
CPS4R is the only plane stress solid element available in Abaqus/Explicit with four nodes
Contact is defined as CONTACT PAIR with the penalty constraint. Surfaces are element-based. Abaqus was run in double precision.
The reflection and transmission coefficients from FEM data are computed as the maximum of the respective signal with respect to the maximum of the incident wave.
Observation:
the FEM data for identical materials (a) and the ones for the reflection coefficient in case of different materials (b and c; in case b though a deviation for larger aspect ratios) match the theoretical result.
A discrepancy arises for
1) the transmission coefficient in case of different materials: the FEM data are too big for the combination b, and too low for for the combination c.
2) the reflection coefficient for the combination b, but only at larger area ratios.
The result is alike for axisymmetric, plain strain and plane stress elements.
The different formulas for computing the area ratio for axisymmetric and rectangular elements is properly accounted for.
Attached are:
an input-file
the psf-file for the parametric study with this input-file
a measured pulse in the first bar, called by the input-file
the superposition of all signals from one of the parametric studies with axisymmetric elements, full vertical range and enlarged
R and T for the axisymmetric case, theory and FEM data
The signals R and T are recorded at different distances from the interface, that's why they are distant along the time axis.
Could someone please advise me on the cause for the discrepancy between theory and FEM ? Is someone aware of
1) an effect not taken into account in the theoretical formulas ?
2) a compilation of measured data ?
3) FEM data published elsewhere ?
Also, how can I suppress this initial non-zero transmitted strain signal in CAX4R-elements ? This phenomenon manifests itself exclusively in CAX4R-elements, and is even more pronounced in the second bar. Both bars are constructed with the same element type in all simulations.
Thank you for generously sharing your expertise,
Frank Richter
Carry out calculations with different values of artificial viscosity.
Dear Valery,
thanks for this advice.
This approach might work, but it might help only in this study. It is a preliminary study to a simulation of a "Split Hopkinson Pressure Bar" (SHPB) where a thin disc to be plastically deformed is sandwiched between the bars.
Even if I succeed in finding a numerical tweak with specific parameters for the bulk viscosity for the above question, it does not imply that the same parametes are correct for the plastic deformation of a specimen, as there is no analytical solution to this problem and a discrepancy to measured data (using strain gages on the bars) might reside anywhere. There is no fixed rule for picking the parameters for bulk viscosity.
I was thinking of a more appropriate definition of contact. The chapters on contact modelling in the manual are hard to understand.
Frank
Dear Frank!
You're right.
The viscosity coefficient is determined in each case separately.
But, the material can also have physical viscosity.
Deformations in 0.15% in the metal can already be plastic.
The effect of contact can be significant for different cross-sectional areas due to friction and transverse deformation.
Best regards,
Valery N. Aptukov
Hello Valery
thank you once more.
I cannot set a particular value for the bulk viscosity for each individual simulation of my experiments in the SHPB.
The simulations referred to here are by definition strictly elastic. Contact is friction-less and the Poisson's ratio of both bars is set to zero.
Meanwhile I also tried simulating this with beam elements, but my simulations did not achieve any transmission into the second bar, responding like a free interface with reflection coefficient = -1. I didn't get beam-to-beam in Abaqus/Explicit to work.
That is too much of a hassle. Have to live with that, it seems.
Strangely, I could not find any discussion of this apparently simple problem in FEM.
Regards
Frank
Elastic waves are difficult to model explicitly.
Try the implicit method.
Hello Valery,
thank you once more for your response here.
I resolved that problem with my reflection and transmission coefficients. It was much easier: the simulations were all correct. But two theoretical formulas exist, for displacement and for stress. I picked the wrong one. Also, I plugged in values for strains instead of stress, neglecting the modification of the formula by different Young's moduli.
I achieve almost perfect harmony between theory and FEM, for all element types, and for both displacement and stress.
Regards
Frank
So everything is in order.
The laws are valid.
I'm happy for you.
Best regards.
Valery
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